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Aerodynamics part i

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Solo Hermelin
Solo Hermelin

Aerodynamics Part I of 3 describes aerodynamics of wings and bodies in subsonic flight. For comments please contact me at solo.hermelin@gmail.com. For more presentations on different subjects visit my website at http://www.solohermelin.com.

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1 AERODYNAMICS Part I SOLO HERMELIN http://www.solohermelin.com
2 Table of Content AERODYNAMICS Earth Atmosphere Mathematical Notations SOLO Basic Laws in Fluid Dynamics Conservation of Mass (C.M.) Conservation of Linear Momentum (C.L.M.) Conservation of Moment-of-Momentum (C.M.M.) The First Law of Thermodynamics The Second Law of Thermodynamics and Entropy Production Constitutive Relations for Gases Newtonian Fluid Definitions – Navier–Stokes Equations State Equation Thermally Perfect Gas and Calorically Perfect Gas Boundary Conditions Dimensionless Equations Boundary Layer and Reynolds Number
3 Table of Content (continue – 1) AERODYNAMICS SOLO Circulation Biot-Savart Formula Helmholtz Vortex Theorems 2-D Inviscid Incompressible Flow Stream Function ψ, Velocity Potential φ in 2-D Incompressible Irrotational Flow Aerodynamic Forces and Moments Blasius Theorem Kutta Condition Kutta-Joukovsky Theorem Joukovsky Airfoils Theodorsen Airfoil Design Method Profile Theory by the Method of Singularities Airfoil Design Flow Description Streamlines, Streaklines, and Pathlines
4 Table of Content (continue – 2) AERODYNAMICS SOLO Lifting-Line Theory Subsonic Incompressible Flow (ρ∞ = const.) about Wings of Finite Span (AR < ∞) 3D Lifting-Surface Theory through Vortex Lattice Method (VLM) Incompressible Potential Flow Using Panel Methods Wing Configurations Wing Parameters References
5 Table of Content (continue – 3) AERODYNAMICS SOLO Linearized Flow Equations Cylindrical Coordinates Small Perturbation Flow Applications: Nonsteady One-Dimensional Flow Applications: Two Dimensional Flow Shock & Expansion Waves Shock Wave Definition Normal Shock Wave Oblique Shock Wave Prandtl-Meyer Expansion Waves Movement of Shocks with Increasing Mach Number Drag Variation with Mach Number Swept Wings Drag Variation Variation of Aerodynamic Efficiency with Mach Number AERO
6 Table of Content (continue – 4) AERODYNAMICS SOLO Analytic Theory and CFD Transonic Area Rule Aircraft Flight Control AERO
7 Wright Brothers First Flight AERODYNAMICS SOLO
SOLO Atmosphere Continuum Flow Low-density and Free-molecular Flow Viscous Flow Inviscid Flow Incompressible Flow Compressible Flow Subsonic Flow Transonic Flow Supersonic Flow Hypersonic Flow AERODYNAMICS AERODYNAMICS
9 Percent composition of dry atmosphere, by volume ppmv: parts per million by volume Gas Volume Nitrogen (N2) 78.084% Oxygen (O2) 20.946% Argon (Ar) 0.9340% Carbon dioxide (CO2) 365 ppmv Neon (Ne) 18.18 ppmv Helium (He) 5.24 ppmv Methane (CH4) 1.745 ppmv Krypton (Kr) 1.14 ppmv Hydrogen (H2) 0.55 ppmv Not included in above dry atmosphere: Water vapor (highly variable) typically 1% Gas Volume nitrous oxide 0.5 ppmv xenon 0.09 ppmv ozone 0.0 to 0.07 ppmv (0.0 to 0.02 ppmv in winter) nitrogen dioxide 0.02 ppmv iodine 0.01 ppmv carbon monoxide trace ammonia trace •The mean molecular mass of air is 28.97 g/mol. Minor components of air not listed above include: Composition of Earth's atmosphere. The lower pie represents the trace gases which together compose 0.039% of the atmosphere. Values normalized for illustration. The numbers are from a variety of years (mainly 1987, with CO2 and methane from 2009) and do not represent any single source Earth AtmosphereSOLO
10 Earth AtmosphereSOLO
Earth AtmosphereSOLO
The Earth Atmosphere might be described as a Thermodynamic Medium in a Gravitational Field and in Hydrostatic Equilibrium set by Solar Radiation. Since Solar Radiation and Atmospheric Reradiation varies diurnally and annually and with latitude and longitude, the Standard Atmosphere is only an approximation. SOLO 12 The purpose of the Standard Atmosphere has been defined by the World Metheorological Organization (WMO). The accepted standards are the COESA (Committee on Extension to the Standard Atmosphere) US Standard Atmosphere 1962, updated by US Standard Atmosphere 1976. Earth Atmosphere
The basic variables representing the thermodynamics state of the gas are the Density, ρ, Temperature, T and Pressure, p. SOLO 13 The Density, ρ, is defined as the mass, m, per unit volume, v, and has units of kg/m3 . v m v ∆ ∆ = →∆ 0 limρ The Temperature, T, with units in degrees Kelvin ( ͦ K). Is a measure of the average kinetic energy of gas particles. The Pressure, p, exerted by a gas on a solid surface is defined as the rate of change of normal momentum of the gas particles striking per unit area. It has units of N/m2 . Other pressure units are millibar (mbar), Pascal (Pa), millimeter of mercury height (mHg) S f p n S ∆ ∆ = →∆ 0 lim kPamNbar 100/101 25 == ( ) mmHginHgkPamkNmbar 00.7609213.29/325.10125.1013 2 === The Atmospheric Pressure at Sea Level is: Earth Atmosphere
14 Physical Foundations of Atmospheric Model The Atmospheric Model contains the values of Density, Temperature and Pressure as function of Altitude. Atmospheric Equilibrium (Barometric) Equation In figure we see an atmospheric element under equilibrium under pressure and gravitational forces ( )[ ] 0=⋅+−+⋅⋅⋅− APdPPHdAg gρ or ( ) gg HdHgPd ⋅⋅=− ρ In addition, we assume the atmosphere to be a thermodynamic fluid. At altitude bellow 100 km we assume the Equation of an Ideal Gas where V – is the volume of the gas N – is the number of moles in the volume V m – the mass of gas in the volume V R* - Universal gas constant TRNVP ⋅⋅=⋅ * V m M m N == ρ& MTRP /* ⋅⋅= ρ Earth AtmosphereSOLO
( ) mmHginHgkPamkNmbar 00.7609213.29/325.10125.1013 2 === Earth AtmosphereSOLO
We must make a distinction between: - Kinetic Temperature (T): measures the molecular kinetic energy and for all practical purposes is identical to thermometer measurements at low altitudes. - Molecular Temperature (TM): assumes (not true) that the Molecular Weight at any altitude (M) remains constant and is given by sea-level value (M0) SOLO 16 T M M TM ⋅= 0 To simplify the computation let introduce: - Geopotential Altitude H - Geometric Altitude Hg Newton Gravitational Law implies: ( ) 2 0         + ⋅= gE E g HR R gHg The Barometric Equation is ( ) gg HdHgPd ⋅⋅=− ρ The Geopotential Equation is defined as HdgPd ⋅⋅=− 0ρ This means that g gE E g Hd HR R Hd g g Hd ⋅         + =⋅= 2 0 Integrating we obtain g gE E H HR R H ⋅         + = Earth Atmosphere
17 Atmospheric Constants Definition Symbol Value Units Sea-level pressure P0 1.013250 x 105 N/m2 Sea-level temperature T0 288.15 ͦ K Sea-level density ρ0 1.225 kg/m3 Avogadro’s Number Na 6.0220978 x 1023 /kg-mole Universal Gas Constant R* 8.31432 x 103 J/kg-mole -ͦ K Gas constant (air) Ra=R*/M0 287.0 J/kg--ͦ K Adiabatic polytropic constant γ 1.405 Sea-level molecular weight M0 28.96643 Sea-level gravity acceleration g0 9.80665 m/s2 Radius of Earth (Equator) Re 6.3781 x 106 m Thermal Constant β 1.458 x 10-6 Kg/(m-s-ͦ K1/2) Sutherland’s Constant S 110.4 ͦ K Collision diameter σ 3.65 x 10-10 m Earth AtmosphereSOLO
18 Physical Foundations of Atmospheric Model Atmospheric Equilibrium Equation HdgPd ⋅⋅=− 0ρ At altitude bellow 100 km we assume t6he Equation of an Ideal Gas TRMTRP a MRR a aa ⋅⋅=⋅⋅= = ρρ / * * / Hd TR g P Pd a ⋅=− 0 Combining those two equations we obtain Assume that T = T (H), i.e. function of Geopotential Altitude only. The Standard Model defines the variation of T with altitude based on experimental data. The 1976 Standard Model for altitudes between 0.0 to 86.0 km is divided in 7 layers. In each layer dT/d H = Lapse-rate is constant. Earth AtmosphereSOLO
19 Layer Index Geopotential Altitude Z, km Geometric Altitude Z; km Molecular Temperature T, ͦ K 0 0.0 0.0 288.150 1 11.0 11.0102 216.650 2 20.0 20.0631 216.650 3 32.0 32.1619 228.650 4 47.0 47.3501 270.650 5 51.0 51.4125 270.650 6 71.0 71.8020 214.650 7 84.8420 86.0 186.946 1976 Standard Atmosphere : Seven-Layer Atmosphere Lapse Rate Lh; ͦ K/km -6.3 0.0 +1.0 +2.8 0.0 -2.8 -2.0 Earth AtmosphereSOLO
20 Physical Foundations of Atmospheric Model • Troposphere (0.0 km to 11.0 km). We have ρ (6.7 km)/ρ (0) = 1/e=0.3679, meaning that 63% of the atmosphere lies below an altitude of 6.7 km. ( ) Hd HLTR g Hd TR g P Pd aa ⋅ ⋅+ =⋅=− 0 00 kmKLHLTT /3.60  −=⋅+= Integrating this equation we obtain ( )∫∫ ⋅ ⋅+ =− H a P P Hd HLTR g P PdS S 0 0 0 1 0 ( ) 0 00 lnln 0 T HLT RL g P P aS S ⋅+ ⋅ ⋅ −= Hence aRL g SS H T L PP ⋅ −       ⋅+⋅= 0 0 0 1 and           −         ⋅= ⋅ 1 0 0 0 g RL S S a P P L T H Earth AtmosphereSOLO
21 Physical Foundations of Atmospheric Model Hd TR g P Pd Ta ⋅=− * 0 Integrating this equation we obtain ( )T TaS S HH TR g P P T −⋅ ⋅ −= * 0 ln Hence ( )T Ta T HH TR g SS ePP −⋅ ⋅ − ⋅= * 0 and S STTa T P P g TR HH ln 0 * ⋅ ⋅ += ∫∫ =− H HTa P P T S TS Hd TR g P Pd * 0 • Stratosphere Region (HT=11.0 km to 20.0 km). Temperature T = 216.65 ͦ K = TT* is constant (isothermal layer), PST=22632 Pa Earth AtmosphereSOLO
22 Physical Foundations of Atmospheric Model ( )[ ] Hd HHLTR g Hd TR g P Pd SSTaa ⋅ −⋅+⋅ =⋅=− * 00 ( ) ( ) PaPHPkmKLHHLTT SSSSSST 5474.9,/0.1 * ===−⋅−=  Integrating this equation we obtain ( )[ ]∫∫ ⋅ −⋅+ =− H H SSTa P P S S SS Hd HHLTR g P Pd * 0 1 ( )[ ] * * 0 lnln T ST aSSS S T HHLT RL g P P −⋅+ ⋅ ⋅ = Hence ( ) aRL g S T S SSS HH T L PP ⋅ −         −⋅+⋅= 0 * 1 and           −      ⋅+= ⋅ 1 0 * g RL SS S S T S aS P P L T HH Stratosphere Region (HS=20.0 km to 32.0 km). Earth AtmosphereSOLO
23 1962 Standard Atmosphere from 86 km to 700 km Layer Index Geometric Altitude km Molecular Yemperature , K Kinetic Temperature K Molecular Weight Lapse Rate K/km 7 86.0 186.946 186.946 28.9644 +1.6481 8 100.0 210.65 210.02 28.88 +5.0 9 110.0 260.65 257.00 28.56 +10.0 10 120.0 360.65 349.49 28.08 +20.0 11 150.0 960.65 892.79 26.92 +15.0 12 160.0 1110.65 1022.20 26.66 +10.0 13 170.0 1210.65 1103.40 26.49 +7.0 14 190.0 1350.65 1205.40 25.85 +5.0 15 230.0 1550.65 132230 24.70 +4.0 16 300.0 1830.65 1432.10 22.65 +3.3 17 400.0 2160.65 1487.40 19.94 +2.6 18 500.0 2420.65 1506.10 16.84 +1.7 19 600.0 2590.65 1506.10 16.84 +1.1 20 700.0 2700.65 1507.60 16.70 Earth AtmosphereSOLO
24 1976 Standard Atmosphere from 86 km to 1000 km Geometric Altitude Range: from 86.0 km to 91.0 km (index 7 – 8) 78 /0.0 TT kmK Zd Td = =  Geometric Altitude Range: from 91.0 km to 110.0 km (index 8 – 9) 2/12 8 2 8 2/12 8 1 1 −               − −      − ⋅−=               − −⋅+= a ZZ a ZZ a A Zd Td a ZZ ATT C kma KA KTC 9429.19 3232.76 1902.263 −= −= =   Geometric Altitude Range: from 110.0 km to 120.0 km (index 9 – 10) ( ) kmK Zd Td ZZLTT Z /0.12 99  += −⋅+= Geometric Altitude Range: from 120.0 km to 1000.0 km (index 10 – 11) ( ) ( ) ( ) ( )       + + ⋅−=       + + ⋅−⋅= ⋅−⋅−−= ∞ ∞∞ ZR ZR ZZ kmK ZR ZR TT Zd Td TTTT E E E E 10 10 10 10 10 / exp ξ λ ξλ  KT kmR km E  1000 10356766.6 /01875.0 3 = ×= = ∞ λ Earth AtmosphereSOLO
25 Sea Level Values Pressure p0 = 101,325 N/m2 Density ρ0 = 1.225 kg/m3 Temperature = 288.15 ͦ K (15 ͦ C) Acceleration of gravity g0 = 9.807 m/sec2 Speed of Sound a0 = 340.294 m/sec Earth AtmosphereSOLO
26 Earth AtmosphereSOLO
27 Winds Winds represents the relative motion of the Atmosphere Earth Atmosphere Although in the standard atmosphere the air is motionless with respect to the Earth, it is known that the air mass through which an airplane flies is constantly in a state of motion with respect to the surface of the Earth. Its motion is variable both in time and space and is exceedingly complex. The motion may be divided into two classes: (1) large- scale motions and (2) small-scale motions. Large- scale motions of the atmosphere (or winds) affect the navigation and the performance of an aircraft. SOLO Return to Table of Content
28 FLUID DYNAMICS 1. MATHEMATICAL NOTATIONS VECTOR NOTATION CARTESIAN TENSOR NOTATION 1.1 VECTOR 1.2 SCALAR PRODUCT 1.3 VECTOR PRODUCT u kk = 1 2 3, ,     u u e u e u e= + +1 1 2 2 3 3   u v u v u v u v⋅ = + +1 1 2 2 3 3 u v kk k = 1 2 3, ,   u v u u u u u u v v v × = − − −                     0 0 0 3 2 3 1 2 1 1 2 3      =    − + ± =−= ji permutjiodd permutjieven CevittaLevi vu ij jiij 0 ., ., 1 ε ε SOLO
29 FLUID DYNAMICS 1. MATHEMATICAL NOTATIONS (CONTINUE) VECTOR NOTATION CARTESIAN TENSOR NOTATION 1.5 ROTOR OF A VECTOR 1.4 DIVERGENCE OF A VECTOR 1.6 GRADIENT OF A SCALAR ∇⋅ = + +  u u x u x u x ∂ ∂ ∂ ∂ ∂ ∂ 1 1 2 2 3 3 i i x u ∂ ∂ ∇× = −       + −       + −           u u x u x e u x u x e u x u x e ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ 3 2 2 3 1 1 3 3 1 2 1 2 2 1 3     u u u u u×∇× =∇       − ⋅∇ 2 2 ∂ ∂ ∂ ∂ u x u x i k k i − i k j k i i x u u x u u ∂ ∂ ∂ ∂ − ∇ = + + =              φ ∂ φ ∂ ∂ φ ∂ ∂ φ ∂ ∂ φ ∂ ∂ φ ∂ ∂ φ ∂ x e x e x e x x x 1 1 2 2 31 3 1 2 3    ∂ φ ∂ xk SOLO
30 FLUID DYNAMICS 1.MATHEMATICAL NOTATIONS (CONTINUE) VECTOR NOTATION CARTESIAN TENSOR NOTATION 1.7GRADIENT OF A VECTOR ∇ = ∇ + ∇ + ∇     u u e u e u e1 1 2 2 3 3 ∇ =                    u u x u x u x u x u x u x u x u x u x ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ 1 1 1 2 1 3 2 1 2 2 2 3 3 1 3 2 3 3 ∇ = + + + + + + + + +                      u u x u x u x u x u x u x u x u x u x u x u x u x u x u x u x u x u x u x D ik 1 2 1 1 1 1 1 2 2 1 1 3 3 1 2 1 1 2 2 2 2 2 2 3 3 1 3 1 1 3 3 2 2 3 3 3 3 3 ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂  +    ik x u x u x u x u x u x u x u x u x u x u x u x u Ω                     −− −− −− + 0 0 0 2 1 3 2 2 3 3 1 1 3 1 3 3 2 2 1 1 2 1 3 3 1 1 2 2 1 ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ u x i k ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ u x u x u x u x u x i k i k k i i k k i = +       + −       1 2 1 2 D u x u x ik i k k i = +       ∆ 1 2 ∂ ∂ ∂ ∂ Ω ∆ ik i k k i u x u x = −       1 2 ∂ ∂ ∂ ∂ SOLO
31 FLUID DYNAMICS 1. MATHEMATICAL NOTATIONS (CONTINUE) VECTOR NOTATION CARTESIAN TENSOR NOTATION 1.8 GAUSS’ THEOREMS ds A V ∇⋅  A analytic in V ↓ = =    A C C const vectorη . ( ) ∫∫ ∫∫∫∇= S V dvsdGAUSS ηη  2 ∇η analytic in V ∫∫ ∫∫∫= S k k V dv s ds ∂ η∂ η SOLO Johann Carl Friederich Gauss 1777-1855 ( ) ∫∫ ∫∫∫ ⋅∇=⋅ S V dvAsdAGAUSS  1 ∫∫ ∫∫∫= S k k kk V dv x A dsA ∂ ∂
32 FLUID DYNAMICS 1.MATHEMATICAL NOTATIONS (CONTINUE) VECTOR NOTATION CARTESIAN TENSOR NOTATION 1.8GAUSS’ THEOREMS (CONTINUE) ( ) ( ) ( )∫∫ ∫∫∫ ⋅∇=⋅ S V dvAsdAGAUSS  ηη3 ( )= ⋅∇ + ∇⋅∫∫∫   A A dvη η η∇⋅∇ ,A  analytic inV ( )η ∂ η ∂ A ds A x dv V k k k kS = ∫∫∫∫∫ ∫∫∫       += V k k k k x A x A ∂ ∂ η ∂ η∂ ↓ = + +     B e e eη η η1 1 2 2 3 3 ( ) ( ) ( )[ ]∫∫ ∫∫∫ ⋅∇+∇⋅=⋅ S V dvABBAsdABGAUSS  4 B A ds A B x B A x dv V i k k k i k i k kS = +      ∫∫∫∫∫ ∂ ∂ ∂ ∂ ∇ ×  A analytic inV( ) ∫∫ ∫∫∫ ×∇=× S V dvAAsdGAUSS  5 ( )ds A ds A A x A x dv V i j j i j i i jS − = −      ∫∫∫∫∫ ∂ ∂ ∂ ∂ SOLO
33 FLUID DYNAMICS 1.MATHEMATICAL NOTATIONS (CONTINUE) VECTOR NOTATION CARTESIAN TENSOR NOTATION 1.9STOCKES’ THEOREM     A d r A d s C S ⋅ = ∇ × ⋅∫ ∫∫ ∇ ×  A analytic on S A d r A x A x d si i C j i i j k S ∫ ∫∫= −       ∂ ∂ ∂ ∂ Gauss’ and Stokes’ Theorems are generalizations of the Fundamental Theorem Of CALCULUS ( )A b A a d A x d x d x a b ( ) ( )− = ∫ George Stokes 1819-1903 SOLO
SOLO Variational Principles of Hydrodynamics Joseph-Louis Lagrange 1736-1813 Leonhard Euler 1707-1783 FIXED IN SPACE (CONSTANT VOLUME) EULER LAGRANGE MOVING WITH THE FLUID (CONSTANT MASS) 1e 3 e 2 e u The phenomena considered in Hydrodynamics are macroscopic and the atomic or molecular nature of the fluid is neglected. The fluid is regarded as a continuous medium. Any small volume element is always supposed to be so large that it still contains a large number of molecules. There are two representations normally employed in the study of Hydrodynamics: - Euler representation: The fluid passes through a Constant Volume Fixed in Space - Lagrange representation: The fluid Mass is kept constant during its motion in Space. Hydrodynamic Field
SOLO Variational Principles of Hydrodynamics Material Derivatives (M.D.) Vector Notation Cartesian Tensor Notation 1 e 2e 3 e r  u  b  rd ( ) Frddt t F trFd    ∇⋅+= ∂ ∂ , ( ) d dt F r t F t dr dt F      , = + ⋅∇ ∂ ∂ ( ) d dt F r t F t b F b      , = + ⋅∇ ∂ ∂ rdanyfor  ( )d F r t F t dt d r F x i k i k i k , = + ∂ ∂ ∂ ∂ ( ) d dt F r t F t d r dt F x i k i k i k , = + ∂ ∂ ∂ ∂ ( ) d dt F r t F t b F xb i k i k i k , = + ∂ ∂ ∂ ∂ vectoranybbtd rd  = ( ) Fu t F F tD D trF td d u    ∇⋅+=≡ ∂ ∂ , ( ) k i k i ki u x F u t F F tD D trF td d ∂ ∂ ∂ ∂ +=≡, velocityfluiduu td rd If   = uu u t u uu t u u tD D      ×∇×−      ∇+= ∇⋅+= 2 2 ∂ ∂ ∂ ∂       ⋅−⋅−       += += k i k i j j j i i k i k i i x u u x u u u xt u x u u t u u tD D ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ 2 2 1 Acceleration Of The Fluid 1 e 2 e 3 e r  u  duu +  dr Material Derivatives = = Derivative Along A Fluid Path (Streamline)tD D Hydrodynamic Field
36 FLUID DYNAMICS 1. MATHEMATICAL NOTATIONS (CONTINUE) VECTOR NOTATION CARTESIAN TENSOR NOTATION 1.10 MATERIAL DERIVATIVES (CONTINUE) d u u t dt dr u     = + ⋅∇ ∂ ∂ du u t dt dx u x i i k i k = + ⋅ ∂ ∂ ∂ ∂ rdrdDtd t u xd xd xd x u x u x u x u x u x u x u x u x u t u t u t u ud ud ud ikik   Ω++=                                                   =           ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ 3 2 1 3 3 2 3 1 3 3 2 2 2 1 2 3 1 2 1 1 1 3 2 1 3 2 1  d u u t d t u x u x d x u x u x d x i i Translation i k k i Dilation k i k k i Rotation k = + + +       + −       ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ 1 2 1 2      ( ) ( ) ( ) ( )[ ] Dilationrduu rdurdu urdrdurdu rdurdurdD T u u ik ⇒⋅∇+∇= ⋅∇+⋅∇= ∇⋅−⋅∇+⋅∇= ××∇−⋅∇=     2 1 2 1 2 1 2 1 2 1 2 1 ( )Ωik dr u dr Rotation    = ∇ × × ⇒ 1 2 SOLO
37 REYNOLDS’ TRANSPORT THEOREM -any system of coordinatesOxyz - any continuous and differentiable functions in ( ) ( )trtr OO ,,, ,,  ηχ ( )tandrO,  ( )trO ,,  ρ - flow density at point and time t Or,  SOLO - mass flow through the element .mdsdVS   =⋅− ,ρ sd  - any control volume, changing shape, bounded by a closed surface S(t)v (t) - flow velocity, relative to O, at point and time t( )trV OOflow ,,,  Or,  - position and velocity, relative to O, of an element of surface, part of the control surface S(t). OSOS Vr ,, ,  - area of the opening i, in the control surface S(t).iopenS - gradient operator in O frame.O,∇ - flow relative to the opening i, in the control surface S(t).OSiOflowSi VVV ,,,  −= - differential of any vector , in O frame. O td d ζ  ζ  FLUID DYNAMICS
38 Start with LEIBNIZ THEOREM from CALCULUS: ( ) ( )    ChangeBoundariesthetodueChange tb ta tb ta td tad ttaf td tbd ttbfdx t txf dxtxf td d LEIBNITZ       −+= ∫∫ )),(()),(( ),( ),(:: )( )( )( )( ∂ ∂ and generalized it for a 3 dimensional vector space on a volume v(t) bounded by the surface S(t). Using LEIBNIZ THEOREM followed by GAUSS THEOREM (GAUSS 4): ( ) ( ) ( ) ( ) ∫∫∫∫∫       ⋅∇+∇⋅+=⋅+ → = tv OSOOOSGAUSS Opotolative dsofMovement thetodueChage tS OS tv O LEIBNITZ O tv vdVV t GAUSS sdVvd t vd td d ,,,,)4( intRe )( ,      χχ ∂ χ∂ χ ∂ χ∂ χ This is REYNOLDS’ TRANSPORT THEOREM OSBORNE REYNOLDS 1842-1912 SOLO GOTTFRIED WILHELM von LEIBNIZ 1646-1716 REYNOLDS’ TRANSPORT THEOREM FLUID DYNAMICS 1. MATHEMATICAL NOTATIONS (CONTINUE)
39 FLUID DYNAMICS 1. MATHEMATICAL NOTATIONS (CONTINUE) 1.11 REYNOLDS’ TRANSPORT THEOREM (CONTINUE) VECTOR NOTATION CARTESIAN TENSOR NOTATION ( ) ∫∫∫ ∫∫∫∫∫∫∫∫         ⋅∇+∇⋅+= ⋅+= )( ,,,,)4( , )()()( tv OSOOOS O GAUSS OS tStv O LEIBNITZ O tv vdVV t GAUSS sdVvd t vd td d     χχ ∂ χ∂ χ ∂ χ∂ χ ∫∫∫ ∫∫∫∫∫∫∫∫         ++= += )( , ,)4( , )()()( tv k kOS i k i kOS i GAUSS kkOS tS i tv i LEIBNITZ tv i vd x V x V t GAUSS sdVvd t vd td d ∂ ∂ χ ∂ χ∂ ∂ χ∂ χ ∂ χ∂ χ SOLO
40 FLUID DYNAMICS 1. MATHEMATICAL NOTATIONS (CONTINUE) 1.11 REYNOLDS’ TRANSPORT THEOREM (CONTINUE) VECTOR NOTATION CARTESIAN TENSOR NOTATION O OOS td Rd uV   == ,, CASE 1 (CONTROL VOLUME vF ATTACHED TO THE FLUID) kkOS uV =, ( ) ∫∫∫ ∫∫∫∫∫∫∫∫         ⋅∇+∇⋅+= ⋅+= )( ,,,)4( , )()()( tv OOO O GAUSS O tStv OO tv F FFF vduu t GAUSS sduvd t vd td d      χχ ∂ χ∂ χ ∂ χ∂ χ ∫∫∫ ∫∫∫∫∫∫∫∫         ++= += )( )4( )()()( tv k k I k I k I GAUSS kK tS I tv I tv I F FFF vd x u x u t GAUSS sduvd t vd td d ∂ ∂ χ ∂ χ∂ ∂ χ∂ χ ∂ χ∂ χ SOLO
41 FLUID DYNAMICS 1. MATHEMATICAL NOTATIONS (CONTINUE) 1.11 REYNOLDS’ TRANSPORT THEOREM (CONTINUE) VECTOR NOTATION CARTESIAN TENSOR NOTATION 1&, == χkkOS uV1&, == χuV OS  CASE 2 (CONTROL VOLUME vF ATTACHED TO THE FLUID AND )1=χ ∫∫∫∫∫∫∫∫ ⋅∇=⋅== )( ,, )( , )( )( tv OO tS O tv F FFF vdusduvd td d td tvd  ∫∫∫∫∫∫∫∫ === )()()( )( tv k k k tS k tv F FFF dv x u dsudv td d td tvd ∂ ∂               =⋅∇ → td tvd tv u F F tv OO F )( )( 1 lim0)( ,,                = → td tvd tvx u F F tv k k F )( )( 1 lim0)(∂ ∂ EULER 1755 SOLO
42 FLUID DYNAMICS 1. MATHEMATICAL NOTATIONS (CONTINUE) 1.11 REYNOLDS’ TRANSPORT THEOREM (CONTINUE) VECTOR NOTATION CARTESIAN TENSOR NOTATION CASE 3 (CONTROL VOLUME vF ATTACHED TO THE FLUID AND ) ρχ == &, kkOS uVρχ == &, uV OS  ρχ = or, since this is true for any attached volume vF(t) ( )∫∫∫ ∫∫∫∫∫ ∫∫∫       ⋅∇+= ⋅+=== )( ,, )( , )( )( )( 0 tv OO tS O tv tv F FF F vdu t sduvd t vd td d td tmd   ρ ∂ ρ∂ ρ ∂ ρ∂ ρ ( )∫∫∫ ∫∫∫∫∫ ∫∫∫       += +=== )( )()( )( )( 0 tv k k tS kk tv tv F FF F vdu xt sduvd t dv td d td tmd ρ ∂ ∂ ∂ ρ∂ ρ ∂ ρ∂ ρ Because the Control Volume vF is attached to the fluid and they are not sources or sinks, the mass is constant. ( ) OOOOOO uu t u t ,,,,,,0  ⋅∇+∇⋅+=⋅∇+= ρρ ∂ ρ∂ ρ ∂ ρ∂ ( ) k k k k k x u x u t u xt ∂ ∂ ρ ∂ ρ∂ ∂ ρ∂ ρ ∂ ∂ ∂ ρ∂ ++=+=  0 SOLO
43 FLUID DYNAMICS 1. MATHEMATICAL NOTATIONS (CONTINUE) 1.11 REYNOLDS’ TRANSPORT THEOREM (CONTINUE) VECTOR NOTATION CARTESIAN TENSOR NOTATION CASE 4 (CONTROL VOLUME WITH FIXED SHAPE C.V. )0,  =OS V Define ∫∫∫∫∫∫ = .... VC OO VC vd t vd td d ∂ χ∂ χ   ∫∫∫∫∫∫ = .... VC i VC i vd t vd td d ∂ χ∂ χ ( ) ( ) ( )      χ ρ ηr t r t r t, , ,≡ ( ) ( ) ( )χ ρ ηi k k i kx t x t x t, , ,≡ ( )∫∫ ∫∫∫∫∫∫ ⋅+         += )( , )()( tS OS tv OO tv sdV vd tt vd td d     ηρ ∂ ρ∂ η ∂ η∂ ρηρ k tS kOSi tv i i tv i sdV vd tt vd td d FF ∫∫ ∫∫∫∫∫∫ +       += )( , )()( ηρ ∂ ρ∂ η ∂ η∂ ρηρ We have but ( ) ( )OOOO u t u t ,,,, 0  ρη ∂ ρ∂ ηρ ∂ ρ∂ ⋅∇−=⇒=⋅∇+ ( ) ( )k k iik k u xt u xt ρ ∂ ∂ η ∂ ρ∂ ηρ ∂ ∂ ∂ ρ∂ −=⇒=+ 0 CASE 5 ( ) ( ) ( )     χ ρ ηr t r t r t, , ,≡ SOLO
44 FLUID DYNAMICS 1. MATHEMATICAL NOTATIONS (CONTINUE) 1.11 REYNOLDS’ TRANSPORT THEOREM (CONTINUE) VECTOR NOTATION CARTESIAN TENSOR NOTATION We have ( ) ( ) ( ) ( )[ ] ( ) ( )[ ]∫∫∫∫∫= ∫∫ ∫∫∫ ∫∫ ∫∫∫∫∫∫ ⋅−+ ⋅+         ⋅∇+∇⋅−         ∇⋅+= ⋅+         ⋅∇−= + + )( ,, )( 4 . )( , )( ,,,,,, )( , )( ,, )( tS OOS tv O MDG DerMat GAUSS tS OS tv OOOOOO O tS OS tv OO OO tv sduVvd tD D sdV vduuu t sdV vdu t vd td d          ρηρ η ρη ρηηρη ∂ η∂ ρ ρη ρηρ ∂ η∂ ρη ( ) ( ) ( ) ( ) ( ) ( )[ ]∫∫∫∫∫= ∫∫ ∫∫∫ ∫∫ ∫∫∫∫∫∫ −+ +               +−      += +       −= + + )( , )( 4 . )( , )( )( , )()( tS kkkOSi tv i MDG DerMat GAUSS tS kkOSi tv k k i k i k k i k i tS kkOSi tv k k i i tv i sduVvd tD D sdV vd x u x u x u t sdV vd x u t vd td d ρηρ η ρη ∂ ρ∂ η ∂ η∂ ρ ∂ η∂ ∂ η∂ ρ ρη ∂ ρ∂ ηρ ∂ η∂ ρη CASE 5 ( ) ( ) ( )     χ ρ ηr t r t r t, , ,≡ SOLO
45 FLUID DYNAMICS 1. MATHEMATICAL NOTATIONS (CONTINUE) 1.11 REYNOLDS’ TRANSPORT THEOREM (CONTINUE) VECTOR NOTATION CARTESIAN TENSOR NOTATION REYNOLDS 1 ( )[ ]       ⋅−+= ∫∫∫∫∫ ∫∫∫ )( ,, )( )( tS OOS tv O O tv sduVvd tD D vd td d    ρηρ η ρη ( )[ ]       −+= ∫∫∫∫∫ ∫∫∫ )( , )( )( tS kkkOSi tv i tv i sduVvd tD D dv td d ρηρ η ρη REYNOLDS 2 ( )[ ]        = ⋅−+ ∫∫∫ ∫∫∫∫∫ )( )( ,, )( tv O tS OSO O tv vd tD D sdVuvd td d ρ η ρηρη   ( )[ ]        = −+ ∫∫∫ ∫∫∫∫∫ )( )( , )( tv i tS kkOSki tv i vd tD D sdVuvd td d ρ η ρηρη CASE 5 ( ) ( ) ( )      χ ρ ηr t r t r t, , ,≡ SOLO
46 FLUID DYNAMICS 1. MATHEMATICAL NOTATIONS (CONTINUE) 1.11 REYNOLDS’ TRANSPORT THEOREM (CONTINUE) VECTOR NOTATION CARTESIAN TENSOR NOTATION REYNOLDS 3 CASE 1 (CONTROL VOLUME ATTACHED TO THE FLUID vF(t) ) kkOS uV =, ∫∫∫∫∫∫ = )()( tv OO tv FF vd tD D vd td d ρ η ρη   ∫∫∫∫∫∫ = )()( tv i tv i FF vd tD D vd td d ρ η ρη SOLO O OOS td Rd uV   == ,, ( ) ( ) ( )     χ ρ ηr t r t r t, , ,≡ CASE 4 (CONTROL VOLUME WITH FIXED SHAPE C.V. )0,  =OS V REYNOLDS 4 ( )       ⋅+= ∫∫∫∫∫ ∫∫∫ .. , .. .. SC O O VC VC O sduvd td d vd tD D   ρηρη ρ η ( )       += ∫∫∫∫∫ ∫∫∫ .... .. SC kki VC i VC i sduvd td d vd tD D ρηρη ρ η Return to Table of Content
47 FLUID DYNAMICS 2. BASIC LAWS IN FLUID DYNAMICS THE FLUID DYNAMICS IS DESCRIBED BY THE FOLLOWING FIVE LAWS: SOLO (1) CONSERVATION OF MASS (C.M.) (2) CONSERVATION OF LINEAR MOMENTUM (C.L.M.) (3) CONSERVATION OF MOMENT OF MOMENTUM (C.M.M.) (4) THE FIRST LAW OF THERMODYNAMICS (5) THE SECOND LAW OF THERMODYNAMICS Return to Table of Content
48 FLUID DYNAMICS 2. BASIC LAWS IN FLUID DYNAMICS (CONTINUE) (2.1) CONSERVATION OF MASS (C.M.) Control Volume attached to the fluid (containing a constant mass m) bounded by the Control Surface SF (t). ( )tvF ( )tr,  ρ ( )3 /mkgFlow density SOLO Because vF(t) is attached to the fluid and there are no sources or sinks in this volume, the Conservation of Mass requires that: d m t d t ( ) = 0 ( ) ( )trVtru OfluidO ,, ,,  = Flow Velocity relative to a predefined Coordinate System O (Inertial or Not-Inertial) ( )sm/
49 FLUID DYNAMICS 2. BASIC LAWS IN FLUID DYNAMICS (CONTINUE) (2.1) CONSERVATION OF MASS (CONTINUE - 1) VECTOR NOTATION CARTESIAN TENSOR NOTATION d m t d t ( ) = 0 ( )∫∫∫= ∫∫∫∫∫ ∫∫∫       ⋅∇+ ⋅+=== )( ,, 1 )( , )( )( )( 0 tv OO GAUSS tS O tv tv REYNOLDS F FF F vdu t sduvd t dv td d td tmd   ρ ∂ ρ∂ ρ ∂ ρ∂ ρ ( )∫∫∫= ∫∫∫∫∫ ∫∫∫       + +=== )( 1 )()( )( )( 0 tv k k GAUSS tS kk tv tv REYNOLDS F FF F vdu xt sduvd t dv td d td tmd ρ ∂ ∂ ∂ ρ∂ ρ ∂ ρ∂ ρ The Control Volume mass rate is zero as long as vF(t) is attached to the fluid and therefore contains the same amount of mass. 0),,,( )( =∫∫∫ tvF vdtzyx td d ρ is true in any Coordinate System (O) and so is: ( ) ( ) ( ) ( )( ) 0,,,,,, ,,, ,,, )( ,, )( =      ⋅∇+= ∫∫∫∫∫∫ tv OO tv FF vdtzyxutzyx t tzyx vdtzyx td d  ρ ∂ ρ∂ ρ SOLO
50 FLUID DYNAMICS 2. BASIC LAWS IN FLUID DYNAMICS (CONTINUE) (2.1) CONSERVATION OF MASS (CONTINUE - 2) VECTOR NOTATION CARTESIAN TENSOR NOTATION For any Control Volume v (t) (not necessarily attached to the fluid) The following is true for any Coordinate System (for points that are not sources or sinks – mathematically equivalent to analytic) ( )OO u t ,,,  ρ ∂ ρ∂ ⋅∇ ( ) OOOOOO uu t u t ,,,,,,0  ⋅∇+∇⋅+=⋅∇+= ρρ ∂ ρ∂ ρ ∂ ρ∂ ( ) k k k kO k x u x u t u xt ∂ ∂ ρ ∂ ρ∂ ∂ ρ∂ ρ ∂ ∂ ∂ ρ∂ ++=+= ,0  ( ) ( ) 0 )( ,, 4 ).( , ).()( ≠=      ⋅∇+ ⋅+ ∫∫∫= ∫∫∫∫∫=∫∫∫ mvdV t sdVvd t vd td d tv OSO GAUSS tS OS tv LEIBNITZ tv    ρ ∂ ρ∂ ρ ∂ ρ∂ ρ ( ) 0 )( , 4 ).( , ).()( ≠=      + + ∫∫∫= ∫∫∫∫∫=∫∫∫ mvdV xt sdVvd t vd td d tv kOS k GAUSS tS kkOS tv LEIBNITZ tv ρ ∂ ∂ ∂ ρ∂ ρ ∂ ρ∂ ρ The integral above is not zero because the mass in v (t) is not constant. SOLO
51 FLUID DYNAMICS 2. BASIC LAWS IN FLUID DYNAMICS (CONTINUE) (2.1) CONSERVATION OF MASS (CONTINUE - 3) Material Derivative of vdmd ρ= Let use EULER’s 1755 expression ( ) ( ) ( ) ( )vd tD D vdtd tvd tv u F F tv OO F 11 lim 0 ,, =            =⋅∇ →  and the (C.M.): to develop the following: ( ) 0,, =⋅∇+ OO u t  ρ ∂ ρ∂ ( ) ( ) ( ) 0,,,,,, ,,,, =      ⋅∇+ ∂ ∂ =      ⋅∇+∇⋅+ ∂ ∂ = ⋅∇+      ∇⋅+ ∂ ∂ =+== vdu t vduu t uvdvdu t vd tD D vd tD D vd tD D tD mD OOOOOO OOOO   ρ ρ ρρ ρ ρρ ρ ρ ρ ρ SOLO
52 FLUID DYNAMICS ∑+= openings i iopenW SCSCSC .... 2. BASIC LAWS IN FLUID DYNAMICS (CONTINUE) (2.1) CONSERVATION OF MASS (CONTINUE – 4) SOLO Control Volume with fixed shape C.V. and boundary C.S. in O Coordinates( ) 0,  =OS V There are no sources or sinks in the volume C.V. The change in the mass of the system is due only to the flow through the surface openings C.Sopen i (i=1,2,…). The surface C.S. can be divided in: • C.Sw the impermeable wall through which the fluid can not escape .         =−= 0 0 ,,,  OSOs VuV • C.Sopen i the openings (i=1,2,…) through which the fluid enters or exits .( )0>m ( )0<m  ∑∑ ∫∫∫∫∫∫∫∫∫ =⋅−⋅−=⋅−== openings i i openings i m SC O SC O SC O VC msdusdusduvd td d td md i iopenw     . , . 0 , .. , .. ρρρρ Therefore where is the flow rate entering through the opening Sopen i.∫∫ ⋅−= iopenSC Oi sdum . ,   ρ Return to Table of Content
53 FLUID DYNAMICS 2. BASIC LAWS IN FLUID DYNAMICS (CONTINUE) (2.2) CONSERVATION OF LINEAR MOMENTUM (C.L.M.) -Fluid density at he point and time t( )tr,  ρ  r ( )3 / mKg -Fluid inertial velocity at the point and time t ( )tru I ,,   r ( )sec/m -Surface Stress ( )2 / mNT  -Pressure (force per unit surface) of the surrounding on the control surface ( )2 / mN p -Stress tensor (force per unit surface) of the surrounding on the control surface ( )2 / mN σ~ -Body forces acceleration -(gravitation, electromagnetic,..) G  ( )2 sec/m nnpnT ˆ~ˆˆ~ ⋅+−=⋅= τσ  Consider a volume vF(t) attached to the fluid, bounded by the closed surface SF(t). SOLO -unit vector normal to the surface S(t) and pointing outside the volume v (t)nˆ vF (t) m SF (t) O x y z r u,O np ˆ− nˆ~⋅τ nˆ~⋅σ dSnˆ -Shear stress tensor (force per unit surface) of the surrounding on the control surface ( )2 / mN τ~
54 FLUID DYNAMICS 2. BASIC LAWS IN FLUID DYNAMICS (CONTINUE) (2.2) CONSERVATION OF LINEAR MOMENTUM (CONTINUE - 1) Derivation From Integral Form The LINEAR MOMENTUM of the Constant Mass in vF(t) is given by: ∫∫∫= )( , tv I F vduP  ρ The External Forces acting on the mass are Body and Surface Forces: ( )     ForcesSurface tS ForcesBody tv external FF sdTvdGF ∫∫∫∫∫∑ += )( ρ According to NEWTON’s Second Law, for a constant mass in vF(t), we have: I external td Pd F   =∑ SOLO
55 FLUID DYNAMICS 2. BASIC LAWS IN FLUID DYNAMICS (CONTINUE) (2.2) CONSERVATION OF LINEAR MOMENTUM (CONTINUE - 2) VECTOR NOTATION CARTESIAN TENSOR NOTATION ( ) I IMomentumLinear tv I REYNOLDS tv I I ForcesSurface tS ForcesBody tv external P td d vdu td d vd tD uD sdvdGF FF FF         === ⋅+= ∫∫∫∫∫∫ ∫∫∫∫∫∑ )( , 3 )( , )()( ~ ρρ σρ i tv i REYNOLDS tv i tS kik tv iiex P td d vdu dt d vd tD uD dsvdGF FF FF === += ∫∫∫∫∫∫ ∫∫∫∫∫∑ )( 3 )( )()( _ ρρ σρ C.L.M.-1      T ds n ds ds ds n ds = ⋅ ⋅ = =~ ~σ σ T ds n ds dsi ik k ds n ds ik k k k = = =σ σ C.L.M.-2 ( )∫∫∫ ∫∫∫∫∫∫∫∫ ⋅∇+= ⋅+= )( , )()()( , ~ ~ tv I tStvtv I I F FFF vdG sdvdGvd tD uD σρ σρρ   ∫∫∫ ∫∫∫∫∫∫∫∫       += += )( )()()( tv i ik i tS kik tv i tv i F FFF vd x G sdvdGvd tD uD ∂ σ∂ ρ σρρ SOLO Derivation From Integral Form (Continue)
56 FLUID DYNAMICS 2. BASIC LAWS IN FLUID DYNAMICS (CONTINUE) (2.2) CONSERVATION OF LINEAR MOMENTUM (CONTINUE - 3) VECTOR NOTATION CARTESIAN TENSOR NOTATION C.L.M.-2 Since this is true for all volumes vF (t) attached to the fluid we can drop the volume integral. [ ] [ ] [ ]τσ τρσρ ∂ ∂ ρ ∂ ∂ ρρ ~~ ~~ 2 1 ,,, , 2 , , .).( +−= ⋅∇+∇−=⋅∇+=         ×∇×−      ∇+=         ∇⋅+= Ip pGG uuu t u uu t u tD uD III II I I I DM I      ikikik i ik i i i ik i k i k i j jjj i i k i k i DM i p xx p G x G x u u x u uuu xt u x u u t u tD uD τδσ ∂ τ∂ ∂ ∂ ρ ∂ σ∂ ρ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ρ ∂ ∂ ∂ ∂ ρρ +−= +−=+=             ⋅−⋅−      +=       ⋅+= 2 1 .).( SOLO Derivation From Integral Form (Continue) ( )∫∫∫ ∫∫∫∫∫∫∫∫ ⋅∇+= ⋅+= )( , )()()( , ~ ~ tv I tStvtv I I F FFF vdG sdvdGvd tD uD σρ σρρ   ∫∫∫ ∫∫∫∫∫∫∫∫       += += )( )()()( tv i ik i tS kik tv i tv i F FFF vd x G sdvdGvd tD uD ∂ σ∂ ρ σρρ
57 FLUID DYNAMICS 2. BASIC LAWS IN FLUID DYNAMICS (CONTINUE) (2.2) CONSERVATION OF LINEAR MOMENTUM (CONTINUE - 4) Derivation From a Cartesian Differential Volume VECTOR NOTATION CARTESIAN TENSOR NOTATION σ ∂ σ ∂ xx xx x dx+ 1 2 σ ∂ σ ∂ xx xx x dx− 1 2 τ ∂ τ ∂ yx yx y dy+ 1 2 τ ∂ τ ∂ yz yz y dy− 1 2 τ ∂τ ∂ zx zx z dz+ 1 2 τ ∂τ ∂ zx zx z dz− 1 2 τ ∂ τ ∂ xy xy x dx+ 1 2 τ ∂ τ ∂ xy xy x dx− 1 2 σ ∂ σ ∂ yy yy y dy+ 1 2 σ ∂ σ ∂ yy yy y dy− 1 2 τ ∂τ ∂ zy zy z dz+ 1 2 τ ∂τ ∂ zy zy z dz− 1 2 τ ∂ τ ∂ xz xz x dx− 1 2 τ ∂ τ ∂ yz yz y dy+ 1 2 τ ∂ τ ∂ yx yx y dy− 1 2 σ ∂ σ ∂ zz zz z dz+ 1 2 σ ∂ σ ∂ zz zz z dz− 1 2 z y x dy dx dz O τ ∂ τ ∂ xz xz x dx+ 1 2 ∂σ ∂ ∂τ ∂ ∂τ ∂ ρ ρ ∂τ ∂ ∂σ ∂ ∂τ ∂ ρ ρ ∂τ ∂ ∂τ ∂ ∂σ ∂ ρ ρ xx yx zx xB x xy yy zy yB y xz yz zz zB z x y z G a x y z G a x y z G a + + + = + + + = + + + = CAUCHY’s First Law of Motion I tD uD a aG    = =+⋅∇ ρρσ~ tD uD a aG x i i ii i ij = =+ ρρ ∂ σ∂ SOLO AUGUSTIN LOUIS CAUCHY )1789-1857(
58 FLUID DYNAMICS 2. BASIC LAWS IN FLUID DYNAMICS (CONTINUE) (2.2) CONSERVATION OF LINEAR MOMENTUM (CONTINUE-5) Derivation For Any Control Volume v (t) (the velocity of an element of surface is )d s  IS V ,  V(t) b ds V*(t) I T d s= ⋅~σ G m u Use REYNOLDS’ Transport Theorem (REYNOLDS 2) with and O = I, and then the Conservation of Linear Momentum (C.L.M.) I u,  =η VECTOR NOTATION CARTESIAN TENSOR NOTATION ( )[ ] ( ) ( ) ( ) ∑∫∫∫∫∫ ∫∫∫∫∫∫ ∫∫∫∫∫ =⋅+= ⋅∇+== ⋅−+ iexternal tStv tv I MLC tv I REYNOLDS tS ISII I tv I FsdvdG vdGvd tD uD sdVuuvdu td d FF FF FF    )()( )( , ... )( 2 )( ,,, )( , ~ ~ σρ σρρ ρρ ( ) ( ) ( ) ∑∫∫∫∫∫ ∫∫∫∫∫∫ ∫∫∫∫∫ =+=         +== −+ iexternal tS kik tv i tv k ik i MLC tv i REYNOLDS tS kkISki tv i FsdvdG vd x Gvd tD uD sdVuuvdu td d )()( )( ... )( 2 )( , )( σρ ∂ σ∂ ρρ ρρ SOLO Return to Table of Content
59 ( ) ( ) PdRRvdVRRHd OOO  ×−=×−= ρ, 2. BASIC LAWS IN FLUID DYNAMICS (2.3) CONSERVATION OF MOMENT-OF-MOMENTUM (C.M.M.) SOLO The Absolute Angular Momentum, of the differential mass and Inertial Velocity ,relative to a reference point O is defined as vdmd ρ= V  The Absolute Angular Momentum of the mass enclosed by C.V. is defined as ( ) ( )∫∫∫∫∫∫ ×−=×−= .... , VC O VC OOCV PdRRvdVRRH  ρ Let differentiate the Absolute Angular Momentum and use Reynolds’ Transport Theorem ( ) ( ) ( ) ( )∫∫∫∫∫∫∫∫ ⋅−×−+ ×− =×−= .. , .... , SC md SO VC I O REYNOLDS I VC O I OCV sdVVRRvd tD VRRD vdVRR td d td Hd       ρρρ We have ( ) ( ) ( ) ( ) ( ) VV tD VD RRVVV tD VD RR V tD RD tD RD tD VD RR tD VRRD O I OO I O I O II O I O          ×−×−=×−+×−= ×         −+×−= ×− FLUID DYNAMICS
60 ( ) ( ) ( ) int, : fdRRfdRRvd tD VD RRMd OextO I OO    ×−+×−=×−= ρ ( ) ( ) ( ) ( )∫∫∫∫∫∫∫∫∫∫∫ ⋅−×−+×−×−=×−= .. , ...... , SC md SO P VC O VC I O REYNOLDS I VC O I OCV sdVVRRvdVVvd tD VD RRvdVRR td d td Hd CV         ρρρρ 2. BASIC LAWS IN FLUID DYNAMICS (2.3) CONSERVATION OF MOMENT-OF-MOMENTUM (C.M.M.) SOLO The Moment, of the differential mass dm = ρdv, relative to a reference point O is defined as Therefore Let integrate this equation over the control volume C.V. ( ) ( ) ( )        0 .. int .... , ∫∫∫∫∫∫∫∫∫∑ ×−+×−=×−= VC O VC extO VC I OOCV fdRRfdRRvd tD VD RRM ρ Using the differential of Angular Momentum equation we obtain ( ) ( ) ( )∫∫∫∫∫∑∫∫∫ ⋅−×−+×−=×−= .. , .. , .. , SC md SO P VC OOCV I VC O I OCV sdVVRRvdVVMvdVRR td d td Hd CV       ρρρ ( ) ( ) ( ) ( ) ( ) ∑∑∫∫∫∫∫∫∫∫∑ +×−++−×−+×−=×−= =⋅ k k j jOj SC sdTsd O VC O VC extOOtCV MFRRsdtfnpRRvdgRRfdRRM      ...... , 11 σ ρ Also ( )∑ ×− j jOj FRR  - Moment, relative to O, of discrete forces exerting by the surrounding at point jR  - Discrete Moments exerting by the surrounding.∑ k k M  FLUID DYNAMICS
61 ( ) ( ) ( ) ∑∑∫∫∫∫∫∫∫∫ +×+×+×=×+⋅−×−× k k j jO tv O tv extO P VC O SC md SO I VC O MFrfdrfdrvdVVsdVVrvdVr td d CV         , 0 int,, .... ,, .. , ρρρ 2. BASIC LAWS IN FLUID DYNAMICS (2.3) CONSERVATION OF MOMENT-OF-MOMENTUM (C.M.M.) SOLO Let find the equation of moment around the turbomachine axis. We shall use polar coordinates , where z is the turbomachine axis. zr ,,θ zzrrrO ˆˆ, +=  zVVrVV zr ˆˆˆ ++= θθ  zFFrFF zr ˆˆˆ ++= θθ  ( ) zVrVrVzrVz VVV zr zr Vr zrz zr O ˆˆ0 ˆˆˆ , θ θ θ +−+−==×  ( )  ( ) ∑∑∫∫∫∫ ++=×+⋅−− k kz j j tv extCVO SC S VC MFrdfrPVsdVVrvdVr td d θθθθ ρρ   0 .. , .. The moment of momentum equation around the turbomachine z axis. Example FLUID DYNAMICS
62 2. BASIC LAWS IN FLUID DYNAMICS (2.3) CONSERVATION OF MOMENT-OF-MOMENTUM (C.M.M.) SOLO ( ) ( ) ( ) ( ) ( ) ( )          systemoutsidefromexertedTorque M l lz j j tv ext AVVrAVVr SC S statesteady VC zSnSn MFrdfrsdVVrvdVr td d ∑∑∫∫∫∫ ++=⋅−− +−−→ θθ ρρ θθ θθ ρρ 22,21111,122 .. , 0 .. We obtain ( ) ( )[ ] zflow MQVrVr =− 111122 ρθθ or ( ) ( ) ( ) ( ) ( ) ( )[ ] ( ) zSnSnSn MAVVrVrAVVrAVVr =−=− 11,1112211,11122,222 ρρρ θθθθ Euler Turbine Equation ρ1 - mean fluid density one inlet (1) of area A1. where ρ2 - mean fluid density one outlet (2) of area A2. (Vθ )1, r1 - mean fluid tangential velocity and radius one inlet (1) of area A1. (Vθ )2, r2 - mean fluid tangential velocity and radius one outlet (2) of area A2. (V,Sn )1 - mean fluid normal velocity (relative to A1) one inlet (1) of area A1. (V,Sn )2 - mean fluid normal velocity (relative to A2) one outlet (2) of area A2. - mean flow rate one outlet (1) of area A1.( ) 11,1 : AVQ Snflow = FLUID DYNAMICS Return to Table of Content
63 FLUID DYNAMICS 2. BASIC LAWS IN FLUID DYNAMICS (CONTINUE) (2.4) CONSERVATION OF ENERGY (C.E.) – THE FIRST LAW OF THERMODYNAMICS (DIFFERENTIAL FORM) -Fluid mean velocity [m/sec[( )  u r t, -Body Forces Acceleration -(gravitation, electromagnetic,..) G  -Surface Stress [N/m2 [T  nnpnT ˆ~ˆˆ~ ⋅+−=⋅= τσ  m V(t) G q T n= ⋅~σ d E d t ∂ ∂ Q t uu d s n ds= -Internal Energy of Fluid molecules (vibration, rotation, translation per mass [W/kg[ e - Rate of Heat transferred to the Control Volume (chemical, external sources of heat) [W/m3 [ ∂ ∂ Q t - Rate of Work change done on fluid by the surrounding (rotating shaft, others) positive for a compressor, negative for a turbine) [W[td Ed SOLO Consider a volume vF(t) attached to the fluid, bounded by the closed surface SF(t). -Rate of Conduction and Radiation of Heat from the Control Surface (per unit surface) [W/m3 [ q
64 FLUID DYNAMICS 2. BASIC LAWS IN FLUID DYNAMICS (CONTINUE) (2.4) CONSERVATION OF ENERGY (C.E.) – THE FIRST LAW OF THERMODYNAMICS (CONTINUE - 1) - The Internal Energy of the molecules of the fluid plus the Kinetic Energy of the mass moving relative to an Inertial System (I) The FIRST LAW OF THERMODYNAMICS CHANGE OF INTERNAL ENERGY + KINETIC ENERGY = CHANGE DUE TO HEAT + WORK + ENERGY SUPPLIED BY SUROUNDING SOLO The energy of the constant mass m in the volume vF(t) attached to the fluid, bounded by the closed surface SF(t) is This energy will change due to - The Work done by the surrounding - Absorption of Heat - Other forms of energy supplied to the mass (electromagnetic, chemical,…)
65 FLUID DYNAMICS 2. BASIC LAWS IN FLUID DYNAMICS (CONTINUE) (2.4) CONSERVATION OF ENERGY (C.E.) – THE FIRST LAW OF THERMODYNAMICS (CONTINUE - 2) VECTOR NOTATION CARTESIAN TENSOR NOTATION C.E.-1               systementering td Qd tSv systemontnmenenvirobydone td Wd shaft tSv v REYNOLDS KineticInternal tv FF FF FF sdqvd t Q td Wd ForcesSurface sdTu ForcesBody vdGu vdue tD D vdue td d ∫∫∫∫∫ ∫∫∫∫∫ ∫∫∫∫∫∫ ⋅−+ +⋅+⋅=       +=      + + )( )( 2 )3( )( 2 2 1 2 1 ∂ ∂ ρ ρρ             systementering td Qd tS kk tv systemontnemnoenvirbydone td Wd shaft tS kk tv kk tv REYNOLDS KineticInternal tv FF FF FF dsqvd t Q td Wd ForcesSurface sdTu ForcesBody vdGu vdue tD D vdue td d ∫∫∫∫∫ ∫∫∫∫∫ ∫∫∫∫∫∫ −+ ++=       +=      + + )()( )()( )( 2 )3( )( 2 2 1 2 1 ∂ ∂ ρ ρρ SOLO
66 FLUID DYNAMICS 2. BASIC LAWS IN FLUID DYNAMICS (CONTINUE) (2.4) CONSERVATION OF ENERGY (C.E.) – THE FIRST LAW OF THERMODYNAMICS (CONTINUE - 3) VECTOR NOTATION CARTESIAN TENSOR NOTATIONC.E.-2 ( ) ( ) ∫∫∫∫∫∫ ∫∫∫∫∫∫∫∫∫ ∫∫∫∫∫ ∫∫∫∫∫∫∫ ∫∫∫ ⋅∇−+ ⋅⋅∇+⋅∇−⋅= ⋅−+ ⋅⋅+⋅−⋅= +       + )()( )()()( )1( )()( )()()( )( 2 ~ ~ 2 1 tvtv tvtvtv GAUSS td Qd tStv td Wd tStStv tv FF FFF FF FFF F vdqvd t Q vduvdupvdGu sdqvd t Q sdusdupvdGu KineticInternal vdue tD D              ∂ ∂ τρ ∂ ∂ τρ ρ ( ) ( ) ∫∫∫∫∫∫ ∫∫∫∫∫∫∫∫∫= ∫∫∫∫∫ ∫∫∫∫∫∫∫ ∫∫∫ −+ +− −+ +−=       + + )()( )()()( )1( )()( )()()( )( 2 2 1 tV s s tV tV k k iki tV k k k tV kk GAUSS td Qd tS kk tV td Wd tS kiki tS kk tV kk KineticInternal tV vd x q vd t Q ds x u ds x up vdGu dsqvd t Q dsudsupvdGu vdue tD D ∂ ∂ ∂ ∂ ∂ τ∂ ∂ ∂ ρ ∂ ∂ τρ ρ                T n pn n ds n ds= ⋅ = − + ⋅ =~ ~ &σ τ0= td Wd shaft assume and use SOLO
67 FLUID DYNAMICS 2. BASIC LAWS IN FLUID DYNAMICS (CONTINUE) (2.4) CONSERVATION OF ENERGY (C.E.) – THE FIRST LAW OF THERMODYNAMICS (CONTINUE-4) VECTOR NOTATION CARTESIAN TENSOR NOTATIONC.E.-3 Since the last equation is valid for each vF(t) we can drop the integral and obtain: ( ) ( ) q t Q uGuupue tD D   ⋅∇−+ ⋅+⋅⋅∇+⋅−∇=      + ∂ ∂ ρτρ ~ 2 1 2 ( ) ( ) k k kk k iik k k x q t Q uG x u x up ue tD D ∂ ∂ ∂ ∂ ρ ∂ τ∂ ∂ ∂ ρ −+ ++−=      + 2 2 1 Multiply (C.L.M.-2) by  u τρρ ~⋅∇⋅+∇⋅−⋅=⋅ upuuG tD uD u   ( ) k ik i k kkk i i x u x p uuGu tD D tD uD u ∂ τ∂ ∂ ∂ ρρρ +−== 2 Subtract this equation from (C.E.-3) C.E.-4 ( )[ ]ρ τ τ ∂ ∂ D e D t p u u u Q t q = − ∇⋅ + ∇⋅ ⋅ − ⋅∇⋅ + −∇⋅        ~ ~ Φ ρ ∂ ∂ τ ∂ ∂ ∂ ∂ ∂ ∂ D e D t p u x u u x Q t q x k k ik i k k k =− + + − Φ   ( )Φ ≡ ∇⋅ ⋅ − ⋅∇ ⋅ >~ ~τ τ   u u 0 Φ ≡ >τ ∂ ∂ ik i k u x 0 (Proof of inequality given later) SOLO
68 FLUID DYNAMICS 2. BASIC LAWS IN FLUID DYNAMICS (CONTINUE) (2.4) CONSERVATION OF ENERGY (C.E.) – THE FIRST LAW OF THERMODYNAMICS (CONTINUE - 5) VECTOR NOTATION CARTESIAN TENSOR NOTATION Enthalpy Use this result and (C.E.-4) C.E.-5 ρ p eh +=: ( ) tD pD up tD hD u p tD pD tD hD tD Dp tD pD tD hD tD pD tD hD tD eD −⋅∇−=⋅∇−+−= +−=       −=  ρρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρρρ 2 tD pD x u p tD hD x up tD pD tD hD tD pDp tD hD tD pD tD pD tD hD tD eD k k k k −−=        −+−= +−=       −= ∂ ∂ ρ ∂ ∂ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρρρ 2 Φ++⋅∇−= t Q q tD pD tD hD ∂ ∂ ρ  Φ++−= t Q x q tD pD tD hD k k ∂ ∂ ∂ ∂ ρ SOLO ( )Φ ≡ ∇⋅ ⋅ − ⋅∇ ⋅ >~ ~τ τ   u u 0 Φ ≡ >τ ∂ ∂ ik i k u x 0
69 FLUID DYNAMICS 2. BASIC LAWS IN FLUID DYNAMICS (CONTINUE) (2.4) CONSERVATION OF ENERGY (C.E.) – THE FIRST LAW OF THERMODYNAMICS (CONTINUE - 6) VECTOR NOTATION CARTESIAN TENSOR NOTATION Total Enthalpy Use this result and (C.E.-3) C.E.-6 22 2 1 2 1 : u p euhH ++=+= ρ ( ) t p up tD HD tD pD up tD HD p tD D tD HD ue tD D ∂ ∂ ρρ ρ ρρρ −⋅∇−=−⋅∇−=       −=      +  2 2 1 ( ) t p up xtD HD tD pD x u p tD HD p tD D tD HD ue tD D kk k ∂ ∂ ∂ ∂ ρ ∂ ∂ ρ ρ ρρρ −−=−−=       −=      +  2 2 1 ( ) q t Q uGu t p tD HD  ⋅∇−+⋅+⋅⋅∇+= ∂ ∂ ρτ ∂ ∂ ρ ~ ( ) k k kk k iik x q t Q uG x u t p tD HD ∂ ∂ ∂ ∂ ρ ∂ τ∂ ∂ ∂ ρ −+++= SOLO Return to Table of Content
70 FLUID DYNAMICS 2. BASIC LAWS IN FLUID DYNAMICS (CONTINUE) SOLO THERMODYNAMIC PROCESSES 1. ADIABATIC PROCESSES 2. REVERSIBLE PROCESSES 3. ISENTROPIC PROCESSES No Heat is added or taken away from the System No dissipative phenomena (viscosity, thermal, conductivity, mass diffusion, friction, etc) Both adiabatic and reversible (2.5) THE SECOND LAW OF THERMODYNAMICS AND ENTROPY PRODUCTION
71 FLUID DYNAMICS 2. BASIC LAWS IN FLUID DYNAMICS (CONTINUE) (2.5) THE SECOND LAW OF THERMODYNAMICS AND ENTROPY PRODUCTION 2nd LAW OF THERMODYNAMICS Using GAUSS’ THEOREM 0 )()( ≥+ ∫∫∫∫∫ tStv FF Ad T q vds td d  ρ 00 )( )1( )()( ≥            ⋅∇+⇒≥+ ∫∫∫∫∫∫∫∫ tv GAUSS tStv FFF vd T q tD sD Ad T q vd tD sD  ρρ - Change in Entropy per unit volumed s - Local TemperatureT [ ]K - Fluid Densityρ [ ]3 / mKg d e q w T ds pdv= + = −δ δ d s d e T p T dv= + SOLO For a Reversible Process - Rate of Conduction and Radiation of Heat from the System per unit surface q  [ ]2 / mW
72 FLUID DYNAMICS 2. BASIC LAWS IN FLUID DYNAMICS (CONTINUE) (2.5) THE SECOND LAW OF THERMODYNAMICS (CONTINUE - 1) d e q w T ds pdv= + = −δ δ d s d e T p T dv= + u T p tD eD T u T p tD eD T tD D T p tD eD TtD D T p tD eD TtD vD T p tD eD TtD sD u tD D MC v   ⋅∇+=      ⋅∇+=       −+=      +=+= ⋅∇−= = ρ ρ ρ ρρ ρ ρ ρρ ρ ρρρρ ρ ρ ρ ρ 2 .).( 2 1 1 11 The Energy Equation (C.E.-4) is ( ) k i ik x u oruu t Q qup tD eD ∂ ∂ τττ ∂ ∂ ρ =Φ⋅∇⋅−⋅⋅∇=ΦΦ++⋅∇−⋅∇−= ~~  Tt Q TT q up tD eD TtD sD Φ ++ ⋅∇ −=      ⋅∇+= ∂ ∂ ρ 11   or Φ++⋅−∇= t Q q tD sD T ∂ ∂ ρ  SOLO
73 FLUID DYNAMICS 2. BASIC LAWS IN FLUID DYNAMICS (CONTINUE) (2.5) THE SECOND LAW OF THERMODYNAMICS (CONTINUE - 2) Define ρ ∂ ∂ T D s Dt q Q t = −∇ ⋅ + +  Φ Θ ≡ + ∇ ⋅       ≥ρ Ds Dt q T  0 Entropy Production Rate per unit volume Therefore ( ) Θ Φ Θ= − ∇ ⋅ + + + ∇ ⋅       ≥∫∫∫   q T T Q t T q T dv V t 1 0 ∂ ∂ & SOLO or    0 1 ≥Φ++⋅∇⋅−=Θ nDissipatio System toadded Heat System from Radiation Heat t Q Tq T T ∂ ∂
74 FLUID DYNAMICS 2. BASIC LAWS IN FLUID DYNAMICS (CONTINUE) (2.5) THE SECOND LAW OF THERMODYNAMICS (CONTINUE - 3)     q q q q conduction rate per unit surface q radiation rate per unit surfacec r c r = +     q K T K FOURIER s Conduction Lawc = − ∇ > 0 ' ( )− ∇ ⋅ + ∇ ⋅       = − ∇ ⋅ + ∇ ⋅ + ⋅∇       = ⋅∇       = − ∇ + ⋅∇       = − ∇ ⋅ − ∇       + ⋅∇       = ∇      + ⋅∇                q T q T q T T q q T q T K T q T K T T T q T K T T q T r r r 1 1 1 1 1 1 1 2 2 Θ Φ Φ= ∇      + + + ⋅∇       > > >      K T T T T Q t q T K T r 2 1 1 0 0 0 ∂ ∂  Θ Φ ≡ + ∇⋅       = ∇      + + + ⋅∇       ≥ρ ∂ ∂ D s D t q T K T T T T Q t q Tr   2 1 1 0 SOLO JEAN FOURIER 1768-1830
75 FLUID DYNAMICS 2. BASIC LAWS IN FLUID DYNAMICS (CONTINUE) (2.5) THE SECOND LAW OF THERMODYNAMICS (CONTINUE - 4) SOLO Gibbs Function Helmholtz Function sThG ⋅−=: sTeH ⋅−=: Josiah Willard Gibbs (1839-1903) Hermann Ludwig Ferdinand von Helmholtz (1821 – 1894) Using the Relations vdpsdTed ⋅−⋅= ( ) pdvsdTvpdedhd ⋅+⋅=⋅+=vpe p eh ⋅+=+= ρ : pdvTdssdTTdshdGd ⋅+⋅−=⋅−⋅−= vdpTdsTdssdTedHd ⋅−⋅−=⋅−⋅−= dv T p T ed sd +=
76 FLUID DYNAMICS 2. BASIC LAWS IN FLUID DYNAMICS (CONTINUE) (2.5) THE SECOND LAW OF THERMODYNAMICS (CONTINUE - 5) SOLO Maxwell’s Relations vdpsdTed ⋅−⋅= pdvsdThd ⋅+⋅= pdvTdsGd ⋅+⋅−= vdpTdsHd ⋅−⋅−= Ts pv v F p v e s h T s e       ∂ ∂ =−=      ∂ ∂       ∂ ∂ ==      ∂ ∂ vp Ts T F s T G p G v p h       ∂ ∂ =−=      ∂ ∂       ∂ ∂ ==      ∂ ∂ ps vs s v p T s p v T       ∂ ∂ =      ∂ ∂       ∂ ∂ −=      ∂ ∂ vT pT T p v s T v p s       ∂ ∂ =      ∂ ∂       ∂ ∂ −=      ∂ ∂ James Clerk Maxwell (1831-1879) Return to Table of Content
77 FLUID DYNAMICS 2. BASIC LAWS IN FLUID DYNAMICS (CONTINUE) (2.6) CONSTITUTIVE RELATIONS FOR GASES (2.6.1) NEWTONIAN FLUID DEFINITION – NAVIER–STOKES EQUATIONS [ ] τσ ~~ +−= Ip Stress NEWTONIAN FLUID: The Shear Stress on A Surface Parallel To the Flow = Distance Rate of Change of Velocity SOLO CARTESIAN TENSOR NOTATION ikikik p τδσ +−= VECTOR NOTATION - Stress tensor (force per unit surface) of the surrounding on the control surface ( )2 / mN σ~ -Shear stress tensor (force per unit surface) of the surrounding on the control surface ( )2 / mN τ~
78 FLUID DYNAMICS 2. BASIC LAWS IN FLUID DYNAMICS (CONTINUE) (2.6) CONSTITUTIVE RELATIONS (2.6.1) NEWTONIAN FLUID DEFINITION – NAVIER–STOKES EQUATIONS M. NAVIER 1822 INCOMPRESSIBLE FLUIDS (MOLECULAR MODEL) G.G. STOKES 1845 COMPRESSIBLE FLUIDS (MACROSCOPIC MODEL) VECTOR NOTATION CARTESIAN TENSOR NOTATION [ ] [ ] ( )[ ] [ ]IuuuIpIp T  ∇+∇+∇+−=+−= λµτσ ~~ ik k k i k k i ikikikik x u x u x u pp δ ∂ ∂ λ ∂ ∂ ∂ ∂ µδτδσ +      ++−=+−= ( )[ ] [ ]( ) ( ) ( ) µλλµλµτ 3 2 32~0 −=⇒∇+∇=∇+∇+∇== utrutrIutruutrtr T  ( ) µλ ∂ ∂ λµδ ∂ ∂ λ ∂ ∂ µτ 3 2 0322 −=⇒=+=+= i i ik k k i i ii x u x u x u SOLO STOKES ASSUMPTION µλ 3 2 −=0~ =τtrace μ, λ - Lamé parameters from Elasticity
79 FLUID DYNAMICS 2. BASIC LAWS IN FLUID DYNAMICS (CONTINUE) (2.6) CONSTITUTIVE RELATIONS (2.6.1) NAVIER–STOKES EQUATIONS (CONTINUE) (2.6.1.2) VECTORIAL DERIVATION I x y z T n= ⋅~σ d s n ds= r dru u +du( )unrdtd t u urdtd t u ud      ∇⋅+=∇⋅+= 1 ∂ ∂ ∂ ∂ ( ) ( ) ( ) rdnurdnuuntd t u ud RotationnTranslatio         1 2 1 1 2 1 1 ××∇+      ××∇−∇⋅+= ∂ ∂ OR DEFINITION OF NEWTONIAN FLUID, NAVIER-STOKES EQUATION ( ) ( ) nnunuunnpT nTranslatio      1~11 2 1 121 ⋅=⋅∇+      ××∇−∇⋅+−≡ σλµ CONSERVATION OF LINEAR MOMENTUM EQUATIONS SOLO
80 FLUID DYNAMICS 2. BASIC LAWS IN FLUID DYNAMICS (CONTINUE) (2.6) CONSTITUTIVE RELATIONS (2.6.1) NAVIER–STOKES EQUATIONS (CONTINUE) (2.6.1.2)VECTORIAL DERIVATION (CONTINUE) I x y z T n= ⋅~σ d s n ds= r dru u + du CONSERVATION OF LINEAR MOMENTUM EQUATIONS ( ) ( ) ( ) ( ) ( ) ( )  ( ) ( ) ( ) ( ) ( ) ( ) ∫∫∫ ∫∫∫∫∫∫∫∫∫∫∫ ∫∫∫∫∫∫∫∫∫∫∫∫∫∫∫           ⋅∇∇+×∇×∇+∇⋅∇+∇−= =⋅∇+×∇×+∇⋅+−=       ⋅∇+      ××∇−∇⋅+⋅−=+= )( )()()()()( )()()()()()( 251 2 2 2 11 2 1 121 tV GAUSS tStStStStV tStStVtStVtV vd GAUSS u GAUSS u GAUSS u GAUSS pG usdusdusdsdpvdG sdnunuunsdnpvdGdsTvdGvd tD uD         λµµρ λµµρ λµρρρ BUT ( ) ( ) ( )∇× ∇× ≡ ∇ ∇⋅ − ∇⋅ ∇2 2 2µ µ µ    u u u ( ) ( ) ( ) ( )∇⋅ ∇ + ∇× ∇× = ∇ ∇⋅ − ∇× ∇×2 2µ µ µ µ     u u u u THEN SOLO
81 FLUID DYNAMICS 2. BASIC LAWS IN FLUID DYNAMICS (CONTINUE) I x y z T n= ⋅~σ d s n ds= r dru u + du THEREFORE ( ) ( ) ( ){ }∫∫∫∫∫∫ ⋅∇∇+×∇×∇−⋅∇∇+∇−= )()( 2 tVtV vduuupGvd tD uD  λµµρρ OR ( ) ( )[ ]uupG tD uD  ⋅∇+∇+×∇×∇−∇−= µλµρρ 2 SOLO (2.6) CONSTITUTIVE RELATIONS (2.6.1) NAVIER–STOKES EQUATIONS (CONTINUE) (2.6.1.2)VECTORIAL DERIVATION (CONTINUE)
82 FLUID DYNAMICS 2. BASIC LAWS IN FLUID DYNAMICS (CONTINUE) VECTOR NOTATION CARTESIAN TENSOR NOTATION CONSERVATION OF LINEAR MOMENTUM ( ) ( )[ ]∇ ⋅ = − ∇ − ∇ × ∇ × + ∇ + ∇ ⋅~σ µ µ λp u u   2 ( )       ++            ++−= k k ii k k i iii ik x u xx u x u xx p x ∂ ∂ λµ ∂ ∂ ∂ ∂ ∂ ∂ µ ∂ ∂ ∂ ∂ ∂ σ∂ 2 ( ) ( )[ ] ρ ρ σ ρ µ µ λ Du Dt G G p u u      = + ∇ ⋅ = − ∇ − ∇ × ∇ × + ∇ + ∇ ⋅ ~ 2 ( )       ++            ++−= += k k ii k k i ii i i ik i i x u xx u x u xx p G x G tD uD ∂ ∂ λµ ∂ ∂ ∂ ∂ ∂ ∂ µ ∂ ∂ ∂ ∂ ρ ∂ σ∂ ρρ 2 USING STOKES ASSUMPTION tr ~τ λ µ= ⇒ = −0 2 3 ( )     ⋅∇∇+×∇×∇−∇−= ⋅∇+= uupG G tD uD   µµρ σρρ 3 4 ~       +            ++−= += k k ki k k i ii i i ik i i x u i xx u x u xx p G x G tD uD ∂ ∂ µ ∂ ∂ ∂ ∂ ∂ ∂ µ ∂ ∂ ∂ ∂ ρ ∂ σ∂ ρρ 3 4 SOLO (2.6) CONSTITUTIVE RELATIONS (2.6.1) NAVIER–STOKES EQUATIONS (CONTINUE)
83 FLUID DYNAMICS 2. BASIC LAWS IN FLUID DYNAMICS (CONTINUE) VECTOR NOTATION CARTESIAN TENSOR NOTATION Euler Equations are obtained by assuming Inviscid Flow 0 3 2 0~ =−=⇒= µλτ pG tD uD ∇−=  ρρ i i i x p G tD uD ∂ ∂ ρρ −= SOLO (2.6) CONSTITUTIVE RELATIONS (2.6.2) EULER EQUATIONS Leonhard Euler (1707-1783) pGuu t u ∇−=      ∇⋅+ ∂ ∂   ρρ i i k i k i x p G x u u t u ∂ ∂ ρρ −=      ∂ ∂ + ∂ ∂ or or
84 FLUID DYNAMICS 2. BASIC LAWS IN FLUID DYNAMICS (CONTINUE) (2.6.1.3) COMPUTATION BUT Φ Φ = = +       = +             = = τ ∂ ∂ τ ∂ ∂ τ ∂ ∂ τ ∂ ∂ ∂ ∂ τ τ τ ik i k ik i k ki k i ik i k k i ik ik u x u x u x u x u x D ik ki1 2 1 2 τ µ λ δik ik kk ikD D= +2 HENCE ( )Φ = = +τ µ λ δik ik ik kk ik ikD D D D2 OR ( )[ ] ( )[ ] ( )[ ] ( ) Φ = + + + + + + + + + + + + + + + + + ⇒ = 2 2 2 2 11 11 22 33 11 22 11 22 33 22 33 11 22 33 33 12 2 21 2 13 2 31 2 23 2 32 2 µ λ µ λ µ λ µ D D D D D D D D D D D D D D D D D D D D D D Dij ji ( ) ( )Φ = + + + + + + + +2 2 2 211 2 22 2 33 2 12 2 13 2 23 2 11 22 33 2 µ λD D D D D D D D DOR SOLO (2.6) CONSTITUTIVE RELATIONS (2.6.1) NAVIER–STOKES EQUATIONS (CONTINUE)
85 FLUID DYNAMICS 2. BASIC LAWS IN FLUID DYNAMICS (CONTINUE) (2.6.1.3) COMPUTATION (CONTINUE) USING STOKES ASSUMPTION: tr ~τ λ µ= ⇒ = −0 2 3 Φ ( ) ( )Φ = + + + + + + + +2 2 2 211 2 22 2 33 2 12 2 13 2 23 2 11 22 33 2 µ λD D D D D D D D D ( ) ( ) ( ) ( ) ( )  ( ) Φ = + + − + + + + + + + + − + + + + 2 3 4 3 4 3 4 2 3 11 22 33 2 11 22 11 33 22 33 11 2 22 2 33 2 2 12 2 13 2 23 2 11 22 33 2 11 2 22 2 33 2 µ µ µ µ µ λ µ D D D D D D D D D D D D D D D D D D D D D    OR ( ) ( ) ( )[ ] ( )Φ = − + − + − + + + > 2 3 4 011 22 2 11 33 2 22 33 2 12 2 13 2 23 2µ µD D D D D D D D D SOLO (2.6) CONSTITUTIVE RELATIONS (2.6.1) NAVIER–STOKES EQUATIONS (CONTINUE)
86 FLUID DYNAMICS 2. BASIC LAWS IN FLUID DYNAMICS (CONTINUE) (2.6.1.4) ENTROPY AND VORTICITY From (C.L.M.) or ( ) ( )[ ]Du Dt u t u u u G p u u        = + ∇       − × ∇ × = − ∇ − ∇ × ∇ × + ∇ + ∇ ⋅ ∂ ∂ ρ ρ µ ρ λ µ 2 2 1 1 1 2 GIBBS EQUATION: T d s d h d p = − ρ       ∀      +⋅∇−      +⋅∇=      +⋅∇ →→→→ tld pd td t p ldp hd td t h ldh sd td t s ldsT & 1        ∂ ∂ ρ∂ ∂ ∂ ∂ Since this is true for all d l t → & T s h p T s t h t p t ∇ = ∇ − ∇ = − ρ ∂ ∂ ∂ ∂ ρ ∂ ∂ & 1 SOLO (2.6) CONSTITUTIVE RELATIONS (2.6.1) NAVIER–STOKES EQUATIONS (CONTINUE) Josiah Willard Gibbs (1903 – 1839)
87 FLUID DYNAMICS 2. BASIC LAWS IN FLUID DYNAMICS (CONTINUE) (2.6.1.4) ENTROPY AND VORTICITY from (C.L.M.) or GIBBS EQUATION: T d s d h d p = − ρ       ∀      +⋅∇−      +⋅∇=      +⋅∇ →→→→ tld pd td t p ldp hd td t h ldh sd td t s ldsT & 1        ∂ ∂ ρ∂ ∂ ∂ ∂ Since this is true for all d l t → & T s h p T s t h t p t ∇ = ∇ − ∇ = − ρ ∂ ∂ ∂ ∂ ρ ∂ ∂ & 1 SOLO hsTG p Guuu t u II III II I ,, ,,, , 2 , ~~ 2 1 ∇−∇+ ⋅∇ += ⋅∇ + ∇ −=         ×∇×−      ∇+ ρ τ ρ τ ρ∂ ∂   ρ p hsT dlpdp dlhdh dlsds ∇ −∇=∇→               ⋅∇= ⋅∇= ⋅∇=
88 Luigi Crocco 1909-1986 FLUID DYNAMICS 2. BASIC LAWS IN FLUID DYNAMICS (CONTINUE) (2.6.1.4) ENTROPY AND VORTICITY (CONTINUE) Define Let take the CURL of this equation Vorticityu  ×∇≡Ω If , then from (C.L.M.) we get:  G = −∇Ψ CRROCO’s EQUATION (1937)  ( ) ( )       ⋅∇×∇+      Ψ++∇×∇−∇×∇=×Ω×∇+×∇ Ω τ ρ∂ ∂ ~1 0 2 2      u hsTuu t SOLO ρ τ ∂ ∂ ~ 2 1 ,2 ,, ⋅∇ +      Ψ++∇−∇=×Ω+ I II I uhsTu t u  hsTGuuu t u II I II I ,, , , 2 , ~ 2 1 ∇−∇+ ⋅∇ +=         ×∇×−      ∇+ ρ τ ∂ ∂   From
89 FLUID DYNAMICS 2. BASIC LAWS IN FLUID DYNAMICS (CONTINUE) (2.6.1.4) ENTROPY AND VORTICITY (CONTINUE) ( ) ( ) ( ) ( ) ( )∇ × × = ⋅∇ − ∇ ⋅ + ∇ ⋅ − ⋅∇ ← ∇ ⋅ = ∇ ⋅∇ × =               Ω Ω Ω Ω Ω Ωu u u u u u 0 0 ( )∇ × ∇ = ∇ × ∇T s T s τ ρ τ ρ τ ρ ~ 0 1~1~1 ⋅∇×∇+⋅∇×      ∇=       ⋅∇×∇  Therefore ( ) ( ) ( ) τ ρ∂ ∂ ~1 ⋅∇×      ∇−∇×∇=∇⋅Ω−Ω⋅∇+Ω∇⋅+ Ω sTuuu t   SOLO ( ) ( ) τ ρ ~1 ⋅∇×      ∇−∇×∇+⋅∇Ω−∇⋅Ω= Ω sTuu tD D   or
90 FLUID DYNAMICS 2. BASIC LAWS IN FLUID DYNAMICS (CONTINUE) (2.6) CONSTITUTIVE RELATIONS (2.6.1) NAVIER–STOKES EQUATIONS (CONTINUE) (2.6.1.4) ENTROPY AND VORTICITY (CONTINUE) ( ) ( ) τ ρ ~1 ⋅∇×      ∇−∇×∇+⋅∇Ω−∇⋅Ω= Ω sTuu tD D   FLUID WITHOUT VORTICITY WILL REMAIN FOREVER WITHOUT VORTICITY IN ABSENSE OF ENTROPY GRADIENTS OR VISCOUS FORCES - FOR AN INVISCID FLUID ( )λ µ τ= = → =0 0~ ~ ( ) ( ) sTuu tD D INVISCID ∇×∇+⋅∇Ω−∇⋅Ω= Ω =   0 ~~τ - FOR AN HOMENTROPIC FLUID INITIALLY AT REST s const everywhere i e s s t . ; . . &∇ = =      0 0 ∂ ∂( )( )   Ω 0 0= ( ) D Dt s     Ω Ω= = = ∇ =0 0 0 0 0~ ~ , ,τ SOLO Return to Table of Content
91 FLUID DYNAMICS 2. BASIC LAWS IN FLUID DYNAMICS (CONTINUE) (2.6) CONSTITUTIVE RELATIONS (2.6.2) STATE EQUATION p - PRESSURE (FORCE / SURFACE) V - VOLUME OF GAS M - MASS OF GAS R - 8314 - 286.9 T - GAS TEMPERATURE - GAS DENSITY [ ]m3 [ ]kg [ ]J kg mol Ko / ( )⋅ [ ]J kg Ko / ( )⋅R [ ]kgmol /−η [ ]o K [ ]kg m/ 3 ρ [ ]2 / mN IDEAL GAS TRMVp η= TMVp R= DEFINE: ρ ρ = = = ∆ ∆M V v V M & 1 pv T= R p T= ρ R OR SOLO
92 FLUID DYNAMICS 2. BASIC LAWS IN FLUID DYNAMICS (CONTINUE) IDEAL GAS TMVp R= SOLO (2.6) CONSTITUTIVE RELATIONS (2.6.2) STATE EQUATION Return to Table of Content
93 FLUID DYNAMICS 2. BASIC LAWS IN FLUID DYNAMICS (CONTINUE) (2.6) CONSTITUTIVE RELATIONS (2.6.3) THERMALLY PERFECT GAS AND CALORICALLLY PERFECT GAS A THERMALLY PERFECT GAS IS DEFINED AS A GAS FOR WHICH THE INTERNAL ENERGY e IS A FUNCTION ONLY OF THE TEMPERATURE T. ( ) ( )h e T p e T RT h T= + = + =/ ( )ρ THERMALLY PERFECT GAS DEFINE C C v V V p p p p p e T q T h T de pdv v d p d T de pdv d T dq d T = = = = = =                   + +      +            ∆ ∆ ∂ ∂ ∂ ∂ ∂ ∂ A CALORICALLY PERFECT GAS IS DEFINED AS A GAS FOR WHICH Cv IS CONSTANT CALORICALLY PERFECT GASe C Tv= SOLO
94 FLUID DYNAMICS 2. BASIC LAWS IN FLUID DYNAMICS (CONTINUE) (2.6) CONSTITUTIVE RELATIONS (2.6.3) CALORICALLLY PERFECT GAS (CONTINUE) A CALORICALLY PERFECT GAS IS DEFINED AS A GAS FOR WHICH Cv IS CONSTANT CALORICALLY PERFECT GASe C Tv= FOR A CALORICALLY PERFECT GAS ( )h C T RT C R T C T C C Rv v p p v= + = + = → = + γ γ γ γ = ⇒ = − ⇒ = − = + = −∆ C C C R C Rp v C C R p R C C v p v p v 1 1 γ air = 14. SOLO
95 FLUID DYNAMICS 2. BASIC LAWS IN FLUID DYNAMICS (CONTINUE) (2.6) CONSTITUTIVE RELATIONS (2.6.3) CALORICALLLY PERFECT GAS (CONTINUE) (2.6.3.1) ENTROPY CALCULATIONS FOR A CALORICALLLY PERFECT GAS pv T= R p T= ρ R IDEAL GAS ( ) ds de pdv T de pdv vdp vdp T dh vdp T = + = + + − = −∆ ds C dT T R dv v s s C T T R v v C T T Rv v v= + → − = + = −2 1 2 1 2 1 2 1 2 1 ln ln ln ln ρ ρ 1 2 1 2 12 lnln p p R T T Css p dp R T dT Cds pp −=−→−= s s C p p R C p p Cv v p2 1 2 1 1 2 2 1 2 1 2 1 − = ⋅       − = −ln ln ln ln ρ ρ ρ ρ ρ ρ ENTROPY SOLO
96 FLUID DYNAMICS 2. BASIC LAWS IN FLUID DYNAMICS (CONTINUE) (2.6) CONSTITUTIVE RELATIONS (2.6.3) CALORICALLLY PERFECT GAS (CONTINUE) (2.6.3.1) ENTROPY CALCULATIONS FOR A CALORICALLLY PERFECT GAS p p T T e T T e p p T T C R s s R s s R isentropic s s p 2 1 2 1 2 1 1 2 1 2 1 12 1 2 1 2 1 =       =       =       − − − − − = − ⇒ γ γ γ γ ρ ρ ρ ρ γ γ γ 2 1 2 1 2 1 1 1 2 1 2 1 1 12 1 2 1 2 1 =       =       =       − − − − − = − ⇒ T T e T T e T T C R s s R s s R isentropic s s v p p e e p p C C s s R s s R isentropic s s p v 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 =       =       =       − − − − = ⇒ ρ ρ ρ ρ ρ ρ γ γ T T h h p p e p p e T T h h p p s s C s s C isentropic s s v p2 1 2 1 2 1 2 1 2 1 1 2 1 1 2 1 2 1 2 1 1 2 1 12 1 2 1 2 1 = = ⋅ =       =       = =       =       − − − − − − = − − ⇒ ρ ρ ρ ρ ρ ρ γ γ γ γ γ γ ISENTROPIC CHAIN SOLO Return to Table of Content
97 FLUID DYNAMICS BASIC LAWS IN FLUID DYNAMICS (CONTINUE) BOUNDARY CONDITIONS SOLO Return to Table of Content
98 SOLO Dimensionless Equations Dimensionless Variables are: 0/~ ρρρ = 0/ ~ Uuu = gGG / ~ = ( )2 00/~ Upp ρ= 0/~ lUtt = 2 0/ ~ UCTT p=( )2 00/~ Uρττ = 2 0/ ~ UHH = 2 0/ ~ Uhh = 2 0/~ Uee = ( )2 00/~ Uqq ρ= ( )2 / ~ UQQ = ∇=∇ 0 ~ l Field Equations (C.M.): ( ) 00 0 0 U l u t ρ ρ ∂ ρ∂ =⋅∇+  ( ) 2 00 0 ~ 3 4 U l uupGuu t u ρ µµρ ∂ ∂ ρ τ      ⋅∇       ⋅∇∇+×∇×∇−∇−=      ∇⋅+(C.L.M.): ( ) ( ) 3 00 0~ U l Tk t Q uGu t p Hu t H q ρ∂ ∂ ρτ ∂ ∂ ρ    ∇⋅∇−+⋅+⋅⋅∇+=      ∇⋅+ ∂ ∂ (C.E.): ( ) ( ) ( ) 0 / / 00 0 00 0 =      ⋅∇+ U u l lUt  ρ ρ ∂ ρρ∂ ( ) ( ) ( ) ( ) ( )       ⋅∇∇      +      ×∇×∇      − ∇−=       ∇⋅+ 0 00 000 0 0 0 0 0 000 0 2 00 02 0 0 00 0 000 0 0 3 4 / / U u ll UlU u ll Ul U p l g G U lg U u l U u lUt Uu   ρ µ µ µ ρ µ ρρ ρ ∂ ∂ ρ ρ ( ) ( ) ( ) ( ) ( ) ( )         ∇⋅∇               −+⋅+        ⋅⋅∇+        ∂ ∂ =        2 0 0 0 0 0 0 000 0 2 00000 2 0 0 0 2 00 02 0000 2 00000 / ~ // U CT l k k l C k UlU Q lUtU u g G U gl U u U l U p lUtU H lUtD D p pµρ µ ∂ ∂ ρ ρ ρ τ ρρρ ρ  0/~ ρρρ = 0/ ~ Uuu = gGG / ~ = ( )2 00/~ Upp ρ= 0/~ lUtt = 2 0/ ~ UCTT p=( )2 00/~ Uρττ = 2 0/ ~ UHH = 2 0/ ~ Uhh = 2 0/~ Uee = ( )2 00/~ Uqq ρ= ( )2 / ~ UQQ = ∇=∇ 0 ~ l 0/~ µµµ = 0/ ~ kkk = Reference Quantities: ρ0(density), U0(velocity), l0 (length), g (gravity), μ0 (viscosity), k0 (Fourier Constant), λ0 (mean free path) 0/ ~ λλλ =
99 SOLO Dimensionless Equations Dimentionless Field Equations (C.M.): ( ) 0 ~~~~ =⋅∇+ u t  ρ ∂ ρ∂ ( ) ( )u R u R pG F uu t u eer ~~~~1 3 4~~~~1~~~~1~~~ ~ ~ ~ 2   ⋅∇∇+×∇×∇−∇−=        ∇⋅+ µµρ ∂ ∂ ρ(C.L.M.): ( ) ( )Tk PRt Q uG F u t p Hu t H rer ∇⋅∇−+⋅+⋅⋅∇+=        ∇⋅+ ∂ ∂ 11 ~ ~ ~~~1~~~ ~ ~~~~ ~ ~ ~ 2 ∂ ∂ ρτ ∂ ∂ ρ  (C.E.): Reynolds: 0 000 µ ρ lU Re = Prandtl: 0 0 k C P p r µ = Froude: 0 0 gl U Fr = 0/~ ρρρ = 0/ ~ Uuu = gGG / ~ = ( )2 00/~ Upp ρ= 0/~ lUtt = 2 0/ ~ UCTT p=( )2 00/~ Uρττ = 2 0/ ~ UHH = 2 0/ ~ Uhh = 2 0/~ Uee = ( )2 00/~ Uqq ρ= ( )2 / ~ UQQ = ∇=∇ 0 ~ l 0/~ ρρρ = 0/ ~ Uuu = gGG / ~ = ( )2 00/~ Upp ρ= 0/~ lUtt = 2 0/ ~ UCTT p=( )2 00/~ Uρττ = 2 0/ ~ UHH = 2 0/ ~ Uhh = 2 0/~ Uee = ( )2 00/~ Uqq ρ= ( )2 / ~ UQQ = ∇=∇ 0 ~ l 0/~ µµµ = 0/ ~ kkk = Dimensionless Variables are: Reference Quantities: ρ0(density), U0(velocity), l0 (length), g (gravity), μ0 (viscosity), k0 (Fourier Constant), λ0 (mean free path) 0/ ~ λλλ = Knudsen l Kn 0 0 : λ =
100 SOLO Dimensionless Equations Constitutive Relations TRp ρ= 2 2 1 uTCH p += Tkq ∇−=  TCh p=      − == 2 00 2 00 2 00 1 U TC U TC C R U p pp p ρ ρ γ γ ρ ρ ρ       =      2 0 2 0 U TC U h p 2 0 2 0 2 0 2 1       +      =      U u U TC U H p ( )       ∇               −= 2 0 0 00 0 000 0 3 00 U TC l k k C k UlU q p p µρ µ ρ  ( ) [ ]3 3 2~ Iuuu T  ⋅∇−∇+∇= µµτ [ ]3 0 0 0000 0 0 0 0 0 0000 0 00 3 2~ I U u l UlU u l U u l UlU T  ⋅∇      −      ∇+∇      = µ µ ρ µ µ µ ρ µ ρ τ 0/~ ρρρ = 0/ ~ Uuu = gGG / ~ = ( )2 00/~ Upp ρ= 0/~ lUtt = 2 0/ ~ UCTT p=( )2 00/~ Uρττ = 2 0/ ~ UHH = 2 0/ ~ Uhh = 2 0/~ Uee = ( )2 00/~ Uqq ρ= ( )2 / ~ UQQ = ∇=∇ 0 ~ l 0/~ ρρρ = 0/ ~ Uuu = gGG / ~ = ( )2 00/~ Upp ρ= 0/~ lUtt = 2 0/ ~ UCTT p=( )2 00/~ Uρττ = 2 0/ ~ UHH = 2 0/ ~ Uhh = 2 0/~ Uee = ( )2 00/~ Uqq ρ= ( )2 / ~ UQQ = ∇=∇ 0 ~ l 0/~ µµµ = 0/ ~ kkk = Dimensionless Variables are: Reference Quantities: ρ0(density), U0(velocity), l0 (length), g (gravity), μ0 (viscosity), k0 (Fourier Constant), λ0 (mean free path) 0/ ~ λλλ =
101 SOLO Dimensionless Equations Dimensionless Constitutive Relations 2~ 2 1~~ uTH += Tp ~~1~ ρ γ γ − = Ideal Gas ( ) [ ]3 ~~~ 3 2~~~~~~~ Iu R uu R e T e  ⋅∇−∇+∇= µµ τ Navier-Stokes Th ~~ = Calorically Perfect Gas Tk PR q re ~~~11~ ∇−=  Fourier Law Reynolds: 0 000 µ ρ lU Re = Prandtl: 0 0 k C P p r µ = 0/~ ρρρ = 0/ ~ Uuu = gGG / ~ = ( )2 00/~ Upp ρ= 0/~ lUtt = 2 0/ ~ UCTT p=( )2 00/~ Uρττ = 2 0/ ~ UHH = 2 0/ ~ Uhh = 2 0/~ Uee = ( )2 00/~ Uqq ρ= ( )2 / ~ UQQ = ∇=∇ 0 ~ l 0/~ ρρρ = 0/ ~ Uuu = gGG / ~ = ( )2 00/~ Upp ρ= 0/~ lUtt = 2 0/ ~ UCTT p=( )2 00/~ Uρττ = 2 0/ ~ UHH = 2 0/ ~ Uhh = 2 0/~ Uee = ( )2 00/~ Uqq ρ= ( )2 / ~ UQQ = ∇=∇ 0 ~ l 0/~ µµµ = 0/ ~ kkk = Dimensionless Variables are: Reference Quantities: ρ0(density), U0(velocity), l0 (length), g (gravity), μ0 (viscosity), k0 (Fourier Constant), λ0 (mean free path) 0/ ~ λλλ = Return to Table of Content
102 SOLO Mach Number Mach number (M or Ma) / is a dimensionless quantity representing the ratio of speed of an object moving through a fluid and the local speed of sound. • M is the Mach number, • U0 is the velocity of the source relative to the medium, and • a0 is the speed of sound Mach: 0 0 a U M = The Mach number is named after Austrian physicist and philosopher Ernst Mach, a designation proposed by aeronautical engineer Jakob Ackeret. Ernst Mach (1838–1916) Jakob Ackeret (1898–1981) m Tk Mo TR a Bγγ ==0 • R is the Universal gas constant, (in SI, 8.314 47215 J K−1 mol−1 ), [M1 L2 T−2 θ−1 'mol'−1 ] • γ is the rate of specific heat constants Cp/Cv and is dimensionless γair = 1.4. • T is the thermodynamic temperature [θ1 ] • Mo is the molar mass, [M1 'mol'−1 ] • m is the molecular mass, [M1 ] AERODYNAMICS
103 SOLO Mach Number – Flow Regimes Regime Mach mph km/h m/s General plane characteristics Subsonic <0.8 <610 <980 <270 Most often propeller-driven and commercial turbofan aircraft with high aspect-ratio (slender) wings, and rounded features like the nose and leading edges. Transonic 0.8-1.2 610- 915 980-1,470 270-410 Transonic aircraft nearly always have swept wings, delaying drag- divergence, and often feature design adhering to the principles of the Whitcomb Area rule. Supersonic 1.2–5.0 915- 3,840 1,470– 6,150 410–1,710 Aircraft designed to fly at supersonic speeds show large differences in their aerodynamic design because of the radical differences in the behaviour of flows above Mach 1. Sharp edges, thin aerofoil- sections, and all-moving tailplane/canards are common. Modern combat aircraft must compromise in order to maintain low-speed handling; "true" supersonic designs include the F-104 Starfighter, SR-71 Blackbird and BAC/Aérospatiale Concorde. Hypersonic 5.0–10.0 3,840– 7,680 6,150– 12,300 1,710– 3,415 Cooled nickel-titanium skin; highly integrated (due to domination of interference effects: non-linear behaviour means that superposition of results for separate components is invalid), small wings, such as those on the X-51A Waverider High- hypersonic 10.0–25.0 7,680– 16,250 12,300– 30,740 3,415– 8,465 Thermal control becomes a dominant design consideration. Structure must either be designed to operate hot, or be protected by special silicate tiles or similar. Chemically reacting flow can also cause corrosion of the vehicle's skin, with free-atomic oxygen featuring in very high-speed flows. Hypersonic designs are often forced into blunt configurations because of the aerodynamic heating rising with a reduced radius of curvature. Re-entry speeds >25.0 >16,25 0 >30,740 >8,465 Ablative heat shield; small or no wings; blunt shape
104 SOLO Different Regimes of Flow Mach Number – Flow Regimes AERODYNAMICS
105 where ρ = air density V = true speed l = characteristic length μ = absolute (dynamic) viscosity υ = kinematic viscosity Reynolds: υµ ρ ρ µ υ lVlV Re = == Osborne Reynolds (1842 –1912) It was observed by Reynolds in 1884 that a Fluid Flow changes from Laminar to Turbulent at approximately the same value of the dimensionless ratio (ρ V l/ μ) where l is the Characteristic Length for the object in the Flow. This ratio is called the Reynolds number, and is the governing parameter for Viscous Flow. Reynolds Number and Boundary Layer SOLO 1884AERODYNAMICS
106 Boundary Layer SOLO 1904AERODYNAMICS Ludwig Prandtl (1875 – 1953) In 1904 at the Third Mathematical Congress, held at Heidelberg, Germany, Ludwig Prandtl (29 years old) introduced the concept of Boundary Layer. He theorized that the fluid friction was the cause of the fluid adjacent to surface to stick to surface – no slip condition, zero local velocity, at the surface – and the frictional effects were experienced only in the boundary layer a thin region near the surface. Outside the boundary layer the flow may be considered as inviscid (frictionless) flow. In the Boundary Layer on can calculate the •Boundary Layer width •Dynamic friction coefficient μ •Friction Drag Coefficient CDf
107 The flow within the Boundary Layer can be of two types: •The first one is Laminar Flow, consists of layers of flow sliding one over other in a regular fashion without mixing. •The second one is called Turbulent Flow and consists of particles of flow that moves in a random and irregular fashion with no clear individual path, In specifying the velocity profile within a Boundary Layer, one must look at the mean velocity distribution measured over a long period of time. There is usually a transition region between these two types of Boundary-Layer Flow SOLO AERODYNAMICS
108 Normalized Velocity profiles within a Boundary-Layer, comparison between Laminar and Turbulent Flow. SOLO Boundary-Layer AERODYNAMICS
109 Flow Characteristics around a Cylindrical Body as a Function of Reynolds Number (Viscosity) AERODYNAMICS SOLO
110 Relative Drag Force as a Function of Reynolds Number (Viscosity) AERODYNAMICS Drag CD0 due to Flow Separation SOLO
111 Relative Drag Force as a Function of Reynolds Number (Viscosity) AERODYNAMICS Drag due to Viscosity: 1.Skin Friction 2.Flow Separation (Drop in pressure behind body) ∫∫ ∫∫         ⋅+⋅ − −=         ⋅+⋅−= ∧∧ ∞ ∧∧ W W S S fpD ds w t V f w n V pp S ds w tC w nC S C xx xx 11 11 ˆ 2/ ˆ 2/ 1 ˆˆ 1 22 ρρ SOLO
112 Parasite Drag Pressure Differential, Viscous Shear Stress, and Separation AERODYNAMICS Relative Drag Force as a Function of Reynolds Number (Viscosity)       DragFrictionSkin ET EL ll ET EL uu DragPressure ET EL ll ET EL uu sdfsdf sdpsdpD ∫∫ ∫∫ ++ +−= .. .. .. .. .. .. .. .. coscos sinsin θθ θθ SOLO
113 AERODYNAMICS Relative Drag Force as a Function of Reynolds Number (Viscosity) • Blunt Body: Most of Drag is Pressure Drag. • Streamlined Body: Most of Drag is Skin Friction Drag. SOLO
114 AERODYNAMICS Relative Drag Force as a Function of Reynolds Number (Viscosity) SOLO
115 AERODYNAMICS Relative Drag Force as a Function of Reynolds Number (Viscosity) SOLO
116 AERODYNAMICS Relative Drag Force as a Function of Reynolds Number (Viscosity) SOLO Variation of total skin-friction coefficient with Reynolds number for a smooth, flat plate.[From Dommasch, et al. (1967).]
117 Typical Effect of Reynolds Number on Parasitic Drag Flow may stay attached farther at high Re, reducing the drag AERODYNAMICSSOLO Return to Table of Content
118 FluidsSOLO Knudsen number (Kn) is a dimensionless number defined as the ratio of the molecular mean free path length to a representative physical length scale. This length scale could be, for example, the radius of the body in a fluid. The number is named after Danish physicist Martin Knudsen. Knudsen l Kn 0 0 : λ = Martin Knudsen (1871–1949). For a Boltzmann gas, the mean free path may be readily calculated as: • kB is the Boltzmann constant (1.3806504(24) × 10−23 J/K in SI units), [M1 L2 T−2 θ−1 ] p TkB 20 2 σπ λ = • T is the thermodynamic temperature [θ1 ] λ0 = mean free path [L1 ] Knudsen Number l0 = representative physical length scale [L1 ]. • σ is the particle hard shell diameter, [L1 ] • p is the total pressure, [M1 L−1 T−2 ]. See “Kinetic Theory of Gases” Presentation For particle dynamics in the atmosphere and assuming standard atmosphere pressure i.e. 25 °C and 1 atm, we have λ0 ≈ 8x10-8 m.
119 FluidsSOLO Martin Knudsen (1871–1949). Knudsen Number (continue – 1) Relationship to Mach and Reynolds numbers Dynamic viscosity, Average molecule speed (from Maxwell–Boltzmann distribution), thus the mean free path, where • kB is the Boltzmann constant (1.3806504(24) × 10−23 J/K in SI units), [M1 L2 T−2 θ−1 ] • T is the thermodynamic temperature [θ1 ] • ĉ is the average molecular speed from the Maxwell–Boltzmann distribution, [L1 T−1 ] • μ is the dynamic viscosity, [M1 L−1 T−1 ] • m is the molecular mass, [M1 ] • ρ is the density, [M1 L−3 ]. 0 2 1 λρµ c= m Tk c B π 8 = Tk m B2 0 π ρ µ λ =
120 FluidsSOLO Martin Knudsen (1871–1949). Knudsen Number (continue – 2) Relationship to Mach and Reynolds numbers (continue – 1) The dimensionless Reynolds number can be written: Dividing the Mach number by the Reynolds number, and by multiplying by yields the Knudsen number. The Mach, Reynolds and Knudsen numbers are therefore related by: Reynolds:Re 0 000 µ ρ lU = Tk m lmTklallU aUM BB γρ µ γρ µ ρ µ µρ 00 0 00 0 000 0 0000 00 // / Re ==== Kn Tk m lTk m l BB == 22 00 0 00 0 π ρ µπγ γρ µ 2Re πγM Kn =
121 FluidsSOLO Knudsen Number (continue – 3) Relationship to Mach and Reynolds numbers (continue –2) According to the Knudsen Number the Gas Flow can be divided in three regions: 1.Free Molecular Flow (Kn >> 1): M/Re > 3 molecule-interface interaction negligible between incident and reflected particles 2.Transition (from molecular to continuum flow) regime: 3 > M/Re and M/(Re)1/2 > 0.01 (Re >> 1). Both intermolecular and molecule-surface collision are important. 3.Continuum Flow (Kn << 1): 0.01 > M/(Re)1/2 . Dominated by intermolecular collisions. 2Re πγM Kn =
FluidsSOLO Knudsen Number (continue – 4) Inviscid Limit Free Molecular LimitKnudsen Number Boltzman Equation Collisionless Boltzman Equation Discrete Particle model Euler Equation Navier-Stokes Equation Continuum model Conservation Equation do not form a closed set Validity of conventional mathematical models as a function of local Knudsen Number Return to Table of Content
123 AERODYNAMICS Fluid flow is characterized by a velocity vector field in three-dimensional space, within the framework of continuum mechanics. Streamlines, Streaklines and Pathlines are field lines resulting from this vector field description of the flow. They differ only when the flow changes with time: that is, when the flow is not steady. • Streamlines are a family of curves that are instantaneously tangent to the velocity vector of the flow. These show the direction a fluid element will travel in at any point in time. • Streaklines are the locus of points of all the fluid particles that have passed continuously through a particular spatial point in the past. Dye steadily injected into the fluid at a fixed point extends along a streakline • Pathlines are the trajectories that individual fluid particles follow. These can be thought of as a "recording" of the path a fluid element in the flow takes over a certain period. The direction the path takes will be determined by the streamlines of the fluid at each moment in time. • Timelines are the lines formed by a set of fluid particles that were marked at a previous instant in time, creating a line or a curve that is displaced in time as the particles move. The red particle moves in a flowing fluid; its pathline is traced in red; the tip of the trail of blue ink released from the origin follows the particle, but unlike the static pathline (which records the earlier motion of the dot), ink released after the red dot departs continues to move up with the flow. (This is a streakline.) The dashed lines represent contours of the velocity field (streamlines), showing the motion of the whole field at the same time. (See high resolution version. Flow Description SOLO
124 3-D Flow Flow Description SOLO Steady Motion: If at various points of the flow field quantities (velocity, density, pressure) associated with the fluid flow remain unchanged with time, the motion is said to be steady. ( ) ( ) ( )zyxppzyxzyxuu ,,,,,,,, === ρρ  Unsteady Motion: If at various points of the flow field quantities (velocity, density, pressure) associated with the fluid flow change with time, the motion is said to be unsteady. ( ) ( ) ( )tzyxpptzyxtzyxuu ,,,,,,,,,,, === ρρ  Path Line: The curve described in space by a moving fluid element is known as its trajectory or path line. tt tt ∆+ t tt ∆+ tt ∆+ 2 t tt ∆+ tt ∆+ 2 Path Line (steady flow) t tt ∆+ t tt ∆+ 2 tt ∆+ t Path Line (unsteady flow) tt ∆+ 2 tt ∆+ t
125 3-D Flow Flow Description SOLO Path Line: The curve described in space by a moving fluid element is known as its trajectory or path line. t tt ∆+ tt ∆+ 2 Streamlines: The family of curves such that each curve is tangent at each point to the velocity direction at that point are called streamlines. Consider the coordinate of a point P and the direction of the streamline passing through this point. If is the velocity vector of the flow passing through P at a time t, then and parallel, or: r  rd u  u  rd 0=×urd  ( ) ( ) ( ) 0 1 1 1111 =             − − − = zdyudxv ydxwdzu xdzvdyw wvu dzdydx zyx w zd v yd u xd == Cartesian t u  r  rd
126 3-D Flow Flow Description SOLO Path Line: The curve described in space by a moving fluid element is known as its trajectory or path line. Streamlines: The family of curves such that each curve is tangent at each point to the velocity direction at that point are called streamlines. ( ) ( ) ( )tzyxw zd tzyxv yd tzyxu xd ,,,,,,,,, == t u  r  rd Those are two independent differential equations for a streamline. Given a point the streamline is defined from those equations.( )0000 ,,, tzyxr  ( ) ( ) ( ) ( ) ( ) ( )tzyxw zd tzyxv yd tzyxv yd tzyxu xd ,,,,,, 2 ,,,,,, 1 = = ( ) ( ) ( ) ( ) ( ) ( ) 0,,,,,,,,, 0,,,,,,,,, 222 111 =++ =++ zdtzyxcydtzyxbxdtzyxa zdtzyxcydtzyxbxdtzyxa ( ) ( ) ( ) ( )21 21 22 11 •+• •+• βα βα 0 22 11 ≠ βα βα Pfaffian Differential Equations For a given a point the solution of those equations is of the form:( )0000 ,,, tzyxr  ( ) ( ) 2,,, 1,,, 02 01 consttzyx consttzyx = = ψ ψ u  ( )0 tr  rd 0t ( ) 11 cr =  ψ ( ) 22 cr =  ψ Streamline Those are two surfaces, the intersection of which is the streamline.
127 3-D Flow Flow Description SOLO Path Line: The curve described in space by a moving fluid element is known as its trajectory or path line. Streamlines: The family of curves such that each curve is tangent at each point to the velocity direction at that point are called streamlines. ( ) ( ) ( )tzyxw zd tzyxv yd tzyxu xd ,,,,,,,,, == t u  r  rd For a given a point the solution of those equations is of the form:( )0000 ,,, tzyxr  ( ) ( ) 2,,, 1,,, 02 01 consttzyx consttzyx = = ψ ψ u  ( )0 tr  rd 0 t ( ) 11 cr =  ψ ( ) 22 cr =  ψ Streamline Those are two surfaces, the intersection of which is the streamline. The streamline is perpendicular to the gradients (normals) of those two surfaces. ( ) ( ) ( )0201 ,, trtrVr  ψψµ ∇×∇= where μ is a factor that must satisfy the following constraint. ( )( ) ( ) ( ) 0,, 0201 =∇×∇⋅∇=⋅∇ trtrVr  ψψµ Return to Table of Content
128 AERODYNAMICS Streamlines, Streaklines, and Pathlines Mathematical description Streamlines If the components of the velocity are written and those of the streamline as we deduce which shows that the curves are parallel to the velocity vector Pathlines Streaklines where, is the velocity of a particle P at location and time t . The parameter , parametrizes the streakline and 0 ≤ τP ≤ t0 , where t0 is a time of interest . The suffix P indicates that we are following the motion of a fluid particle. Note that at point the curve is parallel to the flow velocity vector where the velocity vector is evaluated at the position of the particle at that time t . SOLO
129 ∞V Airfoil Pressure Field variation with α AERODYNAMICS Airfoil Velocity Field variation with αAirfoil Streamline variation with αAirfoil Streakline with α Streamlines, Streaklines, and Pathlines SOLO
130 AERODYNAMICS Streamlines, Streaklines, and Pathlines SOLO
131 AERODYNAMICSSOLO
132 AERODYNAMICS SOLO
133 AERODYNAMICS Streamlines, Streaklines, and Pathlines SOLO Return to Table of Content
134 3-D Inviscid Incompressible Flow Circulation SOLO Circulation Definition: tV ∆  ( ) tVV ∆∆+  S∆ Sn ∆1 V  ×∇ t r  ∆ ttr ∆+∆  t C tt C ∆+ ∫ ⋅=Γ C rdV  : Material Derivative of the Circulation ( )∫∫∫ ⋅+⋅=         ⋅= Γ CCC rd tD D Vrd tD VD rdV tD D tD D    From the Figure we can see that: ( ) tVrtVVr ttt ∆+∆=∆∆++∆ ∆+  ( ) Vdrd tD D V t rr t ttt  =→∆= ∆ ∆−∆ →∆ ∆+ 0 ( ) 0 2 2 =      =⋅=⋅ ∫∫∫ CCC V dVdVrd tD D V  Therefore: ∫ ⋅= Γ C rd tD VD tD D  integral of an exact differential on a closed curve. C – a closed curve
135 3-D Inviscid Incompressible FlowSOLO tV ∆  ( ) tVV ∆∆+  S∆ Sn ∆1 V  ×∇ t r  ∆ ttr ∆+∆  t C tt C ∆+ S ∫ ⋅=Γ tC rdV  : Material Derivative of the Circulation (second derivation) Subtract those equations: tVrdSn t ∆×=∆  1 ( )∫∆+ ⋅∆+=Γ∆+Γ ttC rdVV  : ( ) ( )∫∫∫∫ ∆⋅×∇=⋅∆+−⋅=Γ∆− ∆+ S TheoremsStoke CC SnVrdVVrdV ttt 1 '  S is the surface bounded by the curves Ct and C t+Δ t ( ) ( ) ( ) tVVrdtVrdVSnV S t S t S ∆         ×∇×⋅=∆×⋅×∇=∆⋅×∇=Γ∆− ∫∫∫∫∫∫  1 td d ttd rd t V ttD D rdd Γ + ∂ Γ∂ =Γ∇⋅+ ∂ Γ∂ =Γ∇⋅+ ∂ Γ∂ = Γ Γ∇⋅=Γ Computation of: ∫ ⋅ ∂ ∂ = ∂ Γ∂ tC rd t V t  Computation of: td d Γ
136 3-D Inviscid Incompressible FlowSOLO tV ∆  ( ) tVV ∆∆+  S∆ Sn ∆1 V  ×∇ t r  ∆ tt r ∆+ ∆  t C tt C ∆+ Material Derivative of the Circulation (second derivation) ( ) tVVrd S t ∆         ×∇×⋅=Γ∆− ∫∫  When Δ t → 0 the surface S shrinks to the curve C=Ct and the surface integral transforms to a curvilinear integral: ( ) ( ) ( )∫∫∫∫∫ ∇⋅⋅+      −=∇⋅⋅+      ∇⋅−=×∇×⋅−= Γ C t CC t C t C t VVrd V dVVrd V rdVVrd td d    0 22 22 Computation of: (continue) td d Γ Finally we obtain: ( ) ∫∫∫ ⋅=∇⋅⋅+⋅ ∂ ∂ = Γ + ∂ Γ∂ = Γ tt CC t C rd tD VD VVrdrd t V td d ttD D   
137 3-D Inviscid Incompressible FlowSOLO tV ∆  ( ) tVV ∆∆+  S∆ Sn ∆1 V  ×∇ t r  ∆ tt r ∆+ ∆  t C tt C ∆+ Material Derivative of the Circulation We obtained: ∫ ⋅= Γ tC rd tD VD tD D  Use C.L.M.: hsT p VV t V tD VD II I G II II ,, , ,, ~ ∇−∇+ ⋅∇ +Ψ∇=         ∇⋅+= τ ∂ ∂     ( ) ( )  0 , ,, , , ~~ ∫∫∫∫ −Ψ+⋅      ⋅∇ +∇=⋅∇−Ψ∇+⋅      ⋅∇ +∇= Γ tttt CC I I C I C I I I hddrd p sTrdhrd p sT tD D ττ to obtain: ∫ ⋅      ⋅∇ +∇= Γ tC I I I rd p sT tD D τ~ , , or: Kelvin’s Theorem William Thomson Lord Kelvin (1824-1907) In an inviscid , isentropic flow d s = 0 with conservative body forces the circulation Γ around a closed fluid line remains constant with respect to time. 0 ~~ =τ Ψ∇=G Return to Table of Content 1869
138 3-D Inviscid Incompressible FlowSOLO Circulation Definition: ∫ ⋅=Γ C rdV  : C – a closed curve Biot-Savart Formula 1820 Jean-Baptiste Biot 1774 - 1862 VorticityV  ×∇≡Ω ∫ − Ω = Space dV sr A    π4 1 ( )lddSn sr Ad     ⋅ − Ω = π4 1 The contribution of a length dl of the Vortex Filament to isA  ∫∫∫∫∫ ⋅Ω=⋅×∇=⋅=Γ SS Stokes C SdnSdnVrdV  : If the Flow is Incompressible 0=⋅∇ u  so we can write , where is the Vector Potential. We are free to choose so we choose it to satisfy . AV  ×∇= A  A  0=⋅∇ A  We obtain the Poisson Equation that defines the Vector Potential A  Ω−=∇  A2 Poisson Equation Solution( ) ∫ − Ω = Space dv sr rA    π4 1 Félix Savart 1791 - 1841 Biot-Savart Formula
139 3-D Inviscid Incompressible FlowSOLO Circulation Definition: ∫ ⋅=Γ C rdV  : C – a closed curve Biot-Savart Formula (continue - 1) 1820 Jean-Baptiste Biot 1774 - 1862 VorticityV  ×∇≡Ω ( )lddSn sr Ad     ⋅ − Ω = π4 1 We found ∫∫∫∫∫ ⋅Ω=⋅×∇=⋅=Γ SS Stokes C SdnSdnVrdV  : also we have dlld Ω Ω =   ( ) ( ) ∫∫∫∫∫ × − ∇⋅Ω=⋅ − Ω ×∇=×∇= Γ Ω Ω = ld sr dSnlddSn sr AdrV r S dlld v rr          1 4 1 4 1 ππ ( ) ( ) ∫ − −×Γ = 3 4 sr srld rV    π Biot-Savart Formula Félix Savart 1791 - 1841 Biot-Savart Formula
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Aerodynamics part i

  • 2. 2 Table of Content AERODYNAMICS Earth Atmosphere Mathematical Notations SOLO Basic Laws in Fluid Dynamics Conservation of Mass (C.M.) Conservation of Linear Momentum (C.L.M.) Conservation of Moment-of-Momentum (C.M.M.) The First Law of Thermodynamics The Second Law of Thermodynamics and Entropy Production Constitutive Relations for Gases Newtonian Fluid Definitions – Navier–Stokes Equations State Equation Thermally Perfect Gas and Calorically Perfect Gas Boundary Conditions Dimensionless Equations Boundary Layer and Reynolds Number
  • 3. 3 Table of Content (continue – 1) AERODYNAMICS SOLO Circulation Biot-Savart Formula Helmholtz Vortex Theorems 2-D Inviscid Incompressible Flow Stream Function ψ, Velocity Potential φ in 2-D Incompressible Irrotational Flow Aerodynamic Forces and Moments Blasius Theorem Kutta Condition Kutta-Joukovsky Theorem Joukovsky Airfoils Theodorsen Airfoil Design Method Profile Theory by the Method of Singularities Airfoil Design Flow Description Streamlines, Streaklines, and Pathlines
  • 4. 4 Table of Content (continue – 2) AERODYNAMICS SOLO Lifting-Line Theory Subsonic Incompressible Flow (ρ∞ = const.) about Wings of Finite Span (AR < ∞) 3D Lifting-Surface Theory through Vortex Lattice Method (VLM) Incompressible Potential Flow Using Panel Methods Wing Configurations Wing Parameters References
  • 5. 5 Table of Content (continue – 3) AERODYNAMICS SOLO Linearized Flow Equations Cylindrical Coordinates Small Perturbation Flow Applications: Nonsteady One-Dimensional Flow Applications: Two Dimensional Flow Shock & Expansion Waves Shock Wave Definition Normal Shock Wave Oblique Shock Wave Prandtl-Meyer Expansion Waves Movement of Shocks with Increasing Mach Number Drag Variation with Mach Number Swept Wings Drag Variation Variation of Aerodynamic Efficiency with Mach Number AERO
  • 6. 6 Table of Content (continue – 4) AERODYNAMICS SOLO Analytic Theory and CFD Transonic Area Rule Aircraft Flight Control AERO
  • 7. 7 Wright Brothers First Flight AERODYNAMICS SOLO
  • 8. SOLO Atmosphere Continuum Flow Low-density and Free-molecular Flow Viscous Flow Inviscid Flow Incompressible Flow Compressible Flow Subsonic Flow Transonic Flow Supersonic Flow Hypersonic Flow AERODYNAMICS AERODYNAMICS
  • 9. 9 Percent composition of dry atmosphere, by volume ppmv: parts per million by volume Gas Volume Nitrogen (N2) 78.084% Oxygen (O2) 20.946% Argon (Ar) 0.9340% Carbon dioxide (CO2) 365 ppmv Neon (Ne) 18.18 ppmv Helium (He) 5.24 ppmv Methane (CH4) 1.745 ppmv Krypton (Kr) 1.14 ppmv Hydrogen (H2) 0.55 ppmv Not included in above dry atmosphere: Water vapor (highly variable) typically 1% Gas Volume nitrous oxide 0.5 ppmv xenon 0.09 ppmv ozone 0.0 to 0.07 ppmv (0.0 to 0.02 ppmv in winter) nitrogen dioxide 0.02 ppmv iodine 0.01 ppmv carbon monoxide trace ammonia trace •The mean molecular mass of air is 28.97 g/mol. Minor components of air not listed above include: Composition of Earth's atmosphere. The lower pie represents the trace gases which together compose 0.039% of the atmosphere. Values normalized for illustration. The numbers are from a variety of years (mainly 1987, with CO2 and methane from 2009) and do not represent any single source Earth AtmosphereSOLO
  • 12. The Earth Atmosphere might be described as a Thermodynamic Medium in a Gravitational Field and in Hydrostatic Equilibrium set by Solar Radiation. Since Solar Radiation and Atmospheric Reradiation varies diurnally and annually and with latitude and longitude, the Standard Atmosphere is only an approximation. SOLO 12 The purpose of the Standard Atmosphere has been defined by the World Metheorological Organization (WMO). The accepted standards are the COESA (Committee on Extension to the Standard Atmosphere) US Standard Atmosphere 1962, updated by US Standard Atmosphere 1976. Earth Atmosphere
  • 13. The basic variables representing the thermodynamics state of the gas are the Density, ρ, Temperature, T and Pressure, p. SOLO 13 The Density, ρ, is defined as the mass, m, per unit volume, v, and has units of kg/m3 . v m v ∆ ∆ = →∆ 0 limρ The Temperature, T, with units in degrees Kelvin ( ͦ K). Is a measure of the average kinetic energy of gas particles. The Pressure, p, exerted by a gas on a solid surface is defined as the rate of change of normal momentum of the gas particles striking per unit area. It has units of N/m2 . Other pressure units are millibar (mbar), Pascal (Pa), millimeter of mercury height (mHg) S f p n S ∆ ∆ = →∆ 0 lim kPamNbar 100/101 25 == ( ) mmHginHgkPamkNmbar 00.7609213.29/325.10125.1013 2 === The Atmospheric Pressure at Sea Level is: Earth Atmosphere
  • 14. 14 Physical Foundations of Atmospheric Model The Atmospheric Model contains the values of Density, Temperature and Pressure as function of Altitude. Atmospheric Equilibrium (Barometric) Equation In figure we see an atmospheric element under equilibrium under pressure and gravitational forces ( )[ ] 0=⋅+−+⋅⋅⋅− APdPPHdAg gρ or ( ) gg HdHgPd ⋅⋅=− ρ In addition, we assume the atmosphere to be a thermodynamic fluid. At altitude bellow 100 km we assume the Equation of an Ideal Gas where V – is the volume of the gas N – is the number of moles in the volume V m – the mass of gas in the volume V R* - Universal gas constant TRNVP ⋅⋅=⋅ * V m M m N == ρ& MTRP /* ⋅⋅= ρ Earth AtmosphereSOLO
  • 15. ( ) mmHginHgkPamkNmbar 00.7609213.29/325.10125.1013 2 === Earth AtmosphereSOLO
  • 16. We must make a distinction between: - Kinetic Temperature (T): measures the molecular kinetic energy and for all practical purposes is identical to thermometer measurements at low altitudes. - Molecular Temperature (TM): assumes (not true) that the Molecular Weight at any altitude (M) remains constant and is given by sea-level value (M0) SOLO 16 T M M TM ⋅= 0 To simplify the computation let introduce: - Geopotential Altitude H - Geometric Altitude Hg Newton Gravitational Law implies: ( ) 2 0         + ⋅= gE E g HR R gHg The Barometric Equation is ( ) gg HdHgPd ⋅⋅=− ρ The Geopotential Equation is defined as HdgPd ⋅⋅=− 0ρ This means that g gE E g Hd HR R Hd g g Hd ⋅         + =⋅= 2 0 Integrating we obtain g gE E H HR R H ⋅         + = Earth Atmosphere
  • 17. 17 Atmospheric Constants Definition Symbol Value Units Sea-level pressure P0 1.013250 x 105 N/m2 Sea-level temperature T0 288.15 ͦ K Sea-level density ρ0 1.225 kg/m3 Avogadro’s Number Na 6.0220978 x 1023 /kg-mole Universal Gas Constant R* 8.31432 x 103 J/kg-mole -ͦ K Gas constant (air) Ra=R*/M0 287.0 J/kg--ͦ K Adiabatic polytropic constant γ 1.405 Sea-level molecular weight M0 28.96643 Sea-level gravity acceleration g0 9.80665 m/s2 Radius of Earth (Equator) Re 6.3781 x 106 m Thermal Constant β 1.458 x 10-6 Kg/(m-s-ͦ K1/2) Sutherland’s Constant S 110.4 ͦ K Collision diameter σ 3.65 x 10-10 m Earth AtmosphereSOLO
  • 18. 18 Physical Foundations of Atmospheric Model Atmospheric Equilibrium Equation HdgPd ⋅⋅=− 0ρ At altitude bellow 100 km we assume t6he Equation of an Ideal Gas TRMTRP a MRR a aa ⋅⋅=⋅⋅= = ρρ / * * / Hd TR g P Pd a ⋅=− 0 Combining those two equations we obtain Assume that T = T (H), i.e. function of Geopotential Altitude only. The Standard Model defines the variation of T with altitude based on experimental data. The 1976 Standard Model for altitudes between 0.0 to 86.0 km is divided in 7 layers. In each layer dT/d H = Lapse-rate is constant. Earth AtmosphereSOLO
  • 19. 19 Layer Index Geopotential Altitude Z, km Geometric Altitude Z; km Molecular Temperature T, ͦ K 0 0.0 0.0 288.150 1 11.0 11.0102 216.650 2 20.0 20.0631 216.650 3 32.0 32.1619 228.650 4 47.0 47.3501 270.650 5 51.0 51.4125 270.650 6 71.0 71.8020 214.650 7 84.8420 86.0 186.946 1976 Standard Atmosphere : Seven-Layer Atmosphere Lapse Rate Lh; ͦ K/km -6.3 0.0 +1.0 +2.8 0.0 -2.8 -2.0 Earth AtmosphereSOLO
  • 20. 20 Physical Foundations of Atmospheric Model • Troposphere (0.0 km to 11.0 km). We have ρ (6.7 km)/ρ (0) = 1/e=0.3679, meaning that 63% of the atmosphere lies below an altitude of 6.7 km. ( ) Hd HLTR g Hd TR g P Pd aa ⋅ ⋅+ =⋅=− 0 00 kmKLHLTT /3.60  −=⋅+= Integrating this equation we obtain ( )∫∫ ⋅ ⋅+ =− H a P P Hd HLTR g P PdS S 0 0 0 1 0 ( ) 0 00 lnln 0 T HLT RL g P P aS S ⋅+ ⋅ ⋅ −= Hence aRL g SS H T L PP ⋅ −       ⋅+⋅= 0 0 0 1 and           −         ⋅= ⋅ 1 0 0 0 g RL S S a P P L T H Earth AtmosphereSOLO
  • 21. 21 Physical Foundations of Atmospheric Model Hd TR g P Pd Ta ⋅=− * 0 Integrating this equation we obtain ( )T TaS S HH TR g P P T −⋅ ⋅ −= * 0 ln Hence ( )T Ta T HH TR g SS ePP −⋅ ⋅ − ⋅= * 0 and S STTa T P P g TR HH ln 0 * ⋅ ⋅ += ∫∫ =− H HTa P P T S TS Hd TR g P Pd * 0 • Stratosphere Region (HT=11.0 km to 20.0 km). Temperature T = 216.65 ͦ K = TT* is constant (isothermal layer), PST=22632 Pa Earth AtmosphereSOLO
  • 22. 22 Physical Foundations of Atmospheric Model ( )[ ] Hd HHLTR g Hd TR g P Pd SSTaa ⋅ −⋅+⋅ =⋅=− * 00 ( ) ( ) PaPHPkmKLHHLTT SSSSSST 5474.9,/0.1 * ===−⋅−=  Integrating this equation we obtain ( )[ ]∫∫ ⋅ −⋅+ =− H H SSTa P P S S SS Hd HHLTR g P Pd * 0 1 ( )[ ] * * 0 lnln T ST aSSS S T HHLT RL g P P −⋅+ ⋅ ⋅ = Hence ( ) aRL g S T S SSS HH T L PP ⋅ −         −⋅+⋅= 0 * 1 and           −      ⋅+= ⋅ 1 0 * g RL SS S S T S aS P P L T HH Stratosphere Region (HS=20.0 km to 32.0 km). Earth AtmosphereSOLO
  • 23. 23 1962 Standard Atmosphere from 86 km to 700 km Layer Index Geometric Altitude km Molecular Yemperature , K Kinetic Temperature K Molecular Weight Lapse Rate K/km 7 86.0 186.946 186.946 28.9644 +1.6481 8 100.0 210.65 210.02 28.88 +5.0 9 110.0 260.65 257.00 28.56 +10.0 10 120.0 360.65 349.49 28.08 +20.0 11 150.0 960.65 892.79 26.92 +15.0 12 160.0 1110.65 1022.20 26.66 +10.0 13 170.0 1210.65 1103.40 26.49 +7.0 14 190.0 1350.65 1205.40 25.85 +5.0 15 230.0 1550.65 132230 24.70 +4.0 16 300.0 1830.65 1432.10 22.65 +3.3 17 400.0 2160.65 1487.40 19.94 +2.6 18 500.0 2420.65 1506.10 16.84 +1.7 19 600.0 2590.65 1506.10 16.84 +1.1 20 700.0 2700.65 1507.60 16.70 Earth AtmosphereSOLO
  • 24. 24 1976 Standard Atmosphere from 86 km to 1000 km Geometric Altitude Range: from 86.0 km to 91.0 km (index 7 – 8) 78 /0.0 TT kmK Zd Td = =  Geometric Altitude Range: from 91.0 km to 110.0 km (index 8 – 9) 2/12 8 2 8 2/12 8 1 1 −               − −      − ⋅−=               − −⋅+= a ZZ a ZZ a A Zd Td a ZZ ATT C kma KA KTC 9429.19 3232.76 1902.263 −= −= =   Geometric Altitude Range: from 110.0 km to 120.0 km (index 9 – 10) ( ) kmK Zd Td ZZLTT Z /0.12 99  += −⋅+= Geometric Altitude Range: from 120.0 km to 1000.0 km (index 10 – 11) ( ) ( ) ( ) ( )       + + ⋅−=       + + ⋅−⋅= ⋅−⋅−−= ∞ ∞∞ ZR ZR ZZ kmK ZR ZR TT Zd Td TTTT E E E E 10 10 10 10 10 / exp ξ λ ξλ  KT kmR km E  1000 10356766.6 /01875.0 3 = ×= = ∞ λ Earth AtmosphereSOLO
  • 25. 25 Sea Level Values Pressure p0 = 101,325 N/m2 Density ρ0 = 1.225 kg/m3 Temperature = 288.15 ͦ K (15 ͦ C) Acceleration of gravity g0 = 9.807 m/sec2 Speed of Sound a0 = 340.294 m/sec Earth AtmosphereSOLO
  • 27. 27 Winds Winds represents the relative motion of the Atmosphere Earth Atmosphere Although in the standard atmosphere the air is motionless with respect to the Earth, it is known that the air mass through which an airplane flies is constantly in a state of motion with respect to the surface of the Earth. Its motion is variable both in time and space and is exceedingly complex. The motion may be divided into two classes: (1) large- scale motions and (2) small-scale motions. Large- scale motions of the atmosphere (or winds) affect the navigation and the performance of an aircraft. SOLO Return to Table of Content
  • 28. 28 FLUID DYNAMICS 1. MATHEMATICAL NOTATIONS VECTOR NOTATION CARTESIAN TENSOR NOTATION 1.1 VECTOR 1.2 SCALAR PRODUCT 1.3 VECTOR PRODUCT u kk = 1 2 3, ,     u u e u e u e= + +1 1 2 2 3 3   u v u v u v u v⋅ = + +1 1 2 2 3 3 u v kk k = 1 2 3, ,   u v u u u u u u v v v × = − − −                     0 0 0 3 2 3 1 2 1 1 2 3      =    − + ± =−= ji permutjiodd permutjieven CevittaLevi vu ij jiij 0 ., ., 1 ε ε SOLO
  • 29. 29 FLUID DYNAMICS 1. MATHEMATICAL NOTATIONS (CONTINUE) VECTOR NOTATION CARTESIAN TENSOR NOTATION 1.5 ROTOR OF A VECTOR 1.4 DIVERGENCE OF A VECTOR 1.6 GRADIENT OF A SCALAR ∇⋅ = + +  u u x u x u x ∂ ∂ ∂ ∂ ∂ ∂ 1 1 2 2 3 3 i i x u ∂ ∂ ∇× = −       + −       + −           u u x u x e u x u x e u x u x e ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ 3 2 2 3 1 1 3 3 1 2 1 2 2 1 3     u u u u u×∇× =∇       − ⋅∇ 2 2 ∂ ∂ ∂ ∂ u x u x i k k i − i k j k i i x u u x u u ∂ ∂ ∂ ∂ − ∇ = + + =              φ ∂ φ ∂ ∂ φ ∂ ∂ φ ∂ ∂ φ ∂ ∂ φ ∂ ∂ φ ∂ x e x e x e x x x 1 1 2 2 31 3 1 2 3    ∂ φ ∂ xk SOLO
  • 30. 30 FLUID DYNAMICS 1.MATHEMATICAL NOTATIONS (CONTINUE) VECTOR NOTATION CARTESIAN TENSOR NOTATION 1.7GRADIENT OF A VECTOR ∇ = ∇ + ∇ + ∇     u u e u e u e1 1 2 2 3 3 ∇ =                    u u x u x u x u x u x u x u x u x u x ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ 1 1 1 2 1 3 2 1 2 2 2 3 3 1 3 2 3 3 ∇ = + + + + + + + + +                      u u x u x u x u x u x u x u x u x u x u x u x u x u x u x u x u x u x u x D ik 1 2 1 1 1 1 1 2 2 1 1 3 3 1 2 1 1 2 2 2 2 2 2 3 3 1 3 1 1 3 3 2 2 3 3 3 3 3 ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂  +    ik x u x u x u x u x u x u x u x u x u x u x u x u Ω                     −− −− −− + 0 0 0 2 1 3 2 2 3 3 1 1 3 1 3 3 2 2 1 1 2 1 3 3 1 1 2 2 1 ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ u x i k ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ u x u x u x u x u x i k i k k i i k k i = +       + −       1 2 1 2 D u x u x ik i k k i = +       ∆ 1 2 ∂ ∂ ∂ ∂ Ω ∆ ik i k k i u x u x = −       1 2 ∂ ∂ ∂ ∂ SOLO
  • 31. 31 FLUID DYNAMICS 1. MATHEMATICAL NOTATIONS (CONTINUE) VECTOR NOTATION CARTESIAN TENSOR NOTATION 1.8 GAUSS’ THEOREMS ds A V ∇⋅  A analytic in V ↓ = =    A C C const vectorη . ( ) ∫∫ ∫∫∫∇= S V dvsdGAUSS ηη  2 ∇η analytic in V ∫∫ ∫∫∫= S k k V dv s ds ∂ η∂ η SOLO Johann Carl Friederich Gauss 1777-1855 ( ) ∫∫ ∫∫∫ ⋅∇=⋅ S V dvAsdAGAUSS  1 ∫∫ ∫∫∫= S k k kk V dv x A dsA ∂ ∂
  • 32. 32 FLUID DYNAMICS 1.MATHEMATICAL NOTATIONS (CONTINUE) VECTOR NOTATION CARTESIAN TENSOR NOTATION 1.8GAUSS’ THEOREMS (CONTINUE) ( ) ( ) ( )∫∫ ∫∫∫ ⋅∇=⋅ S V dvAsdAGAUSS  ηη3 ( )= ⋅∇ + ∇⋅∫∫∫   A A dvη η η∇⋅∇ ,A  analytic inV ( )η ∂ η ∂ A ds A x dv V k k k kS = ∫∫∫∫∫ ∫∫∫       += V k k k k x A x A ∂ ∂ η ∂ η∂ ↓ = + +     B e e eη η η1 1 2 2 3 3 ( ) ( ) ( )[ ]∫∫ ∫∫∫ ⋅∇+∇⋅=⋅ S V dvABBAsdABGAUSS  4 B A ds A B x B A x dv V i k k k i k i k kS = +      ∫∫∫∫∫ ∂ ∂ ∂ ∂ ∇ ×  A analytic inV( ) ∫∫ ∫∫∫ ×∇=× S V dvAAsdGAUSS  5 ( )ds A ds A A x A x dv V i j j i j i i jS − = −      ∫∫∫∫∫ ∂ ∂ ∂ ∂ SOLO
  • 33. 33 FLUID DYNAMICS 1.MATHEMATICAL NOTATIONS (CONTINUE) VECTOR NOTATION CARTESIAN TENSOR NOTATION 1.9STOCKES’ THEOREM     A d r A d s C S ⋅ = ∇ × ⋅∫ ∫∫ ∇ ×  A analytic on S A d r A x A x d si i C j i i j k S ∫ ∫∫= −       ∂ ∂ ∂ ∂ Gauss’ and Stokes’ Theorems are generalizations of the Fundamental Theorem Of CALCULUS ( )A b A a d A x d x d x a b ( ) ( )− = ∫ George Stokes 1819-1903 SOLO
  • 34. SOLO Variational Principles of Hydrodynamics Joseph-Louis Lagrange 1736-1813 Leonhard Euler 1707-1783 FIXED IN SPACE (CONSTANT VOLUME) EULER LAGRANGE MOVING WITH THE FLUID (CONSTANT MASS) 1e 3 e 2 e u The phenomena considered in Hydrodynamics are macroscopic and the atomic or molecular nature of the fluid is neglected. The fluid is regarded as a continuous medium. Any small volume element is always supposed to be so large that it still contains a large number of molecules. There are two representations normally employed in the study of Hydrodynamics: - Euler representation: The fluid passes through a Constant Volume Fixed in Space - Lagrange representation: The fluid Mass is kept constant during its motion in Space. Hydrodynamic Field
  • 35. SOLO Variational Principles of Hydrodynamics Material Derivatives (M.D.) Vector Notation Cartesian Tensor Notation 1 e 2e 3 e r  u  b  rd ( ) Frddt t F trFd    ∇⋅+= ∂ ∂ , ( ) d dt F r t F t dr dt F      , = + ⋅∇ ∂ ∂ ( ) d dt F r t F t b F b      , = + ⋅∇ ∂ ∂ rdanyfor  ( )d F r t F t dt d r F x i k i k i k , = + ∂ ∂ ∂ ∂ ( ) d dt F r t F t d r dt F x i k i k i k , = + ∂ ∂ ∂ ∂ ( ) d dt F r t F t b F xb i k i k i k , = + ∂ ∂ ∂ ∂ vectoranybbtd rd  = ( ) Fu t F F tD D trF td d u    ∇⋅+=≡ ∂ ∂ , ( ) k i k i ki u x F u t F F tD D trF td d ∂ ∂ ∂ ∂ +=≡, velocityfluiduu td rd If   = uu u t u uu t u u tD D      ×∇×−      ∇+= ∇⋅+= 2 2 ∂ ∂ ∂ ∂       ⋅−⋅−       += += k i k i j j j i i k i k i i x u u x u u u xt u x u u t u u tD D ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ 2 2 1 Acceleration Of The Fluid 1 e 2 e 3 e r  u  duu +  dr Material Derivatives = = Derivative Along A Fluid Path (Streamline)tD D Hydrodynamic Field
  • 36. 36 FLUID DYNAMICS 1. MATHEMATICAL NOTATIONS (CONTINUE) VECTOR NOTATION CARTESIAN TENSOR NOTATION 1.10 MATERIAL DERIVATIVES (CONTINUE) d u u t dt dr u     = + ⋅∇ ∂ ∂ du u t dt dx u x i i k i k = + ⋅ ∂ ∂ ∂ ∂ rdrdDtd t u xd xd xd x u x u x u x u x u x u x u x u x u t u t u t u ud ud ud ikik   Ω++=                                                   =           ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ 3 2 1 3 3 2 3 1 3 3 2 2 2 1 2 3 1 2 1 1 1 3 2 1 3 2 1  d u u t d t u x u x d x u x u x d x i i Translation i k k i Dilation k i k k i Rotation k = + + +       + −       ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ 1 2 1 2      ( ) ( ) ( ) ( )[ ] Dilationrduu rdurdu urdrdurdu rdurdurdD T u u ik ⇒⋅∇+∇= ⋅∇+⋅∇= ∇⋅−⋅∇+⋅∇= ××∇−⋅∇=     2 1 2 1 2 1 2 1 2 1 2 1 ( )Ωik dr u dr Rotation    = ∇ × × ⇒ 1 2 SOLO
  • 37. 37 REYNOLDS’ TRANSPORT THEOREM -any system of coordinatesOxyz - any continuous and differentiable functions in ( ) ( )trtr OO ,,, ,,  ηχ ( )tandrO,  ( )trO ,,  ρ - flow density at point and time t Or,  SOLO - mass flow through the element .mdsdVS   =⋅− ,ρ sd  - any control volume, changing shape, bounded by a closed surface S(t)v (t) - flow velocity, relative to O, at point and time t( )trV OOflow ,,,  Or,  - position and velocity, relative to O, of an element of surface, part of the control surface S(t). OSOS Vr ,, ,  - area of the opening i, in the control surface S(t).iopenS - gradient operator in O frame.O,∇ - flow relative to the opening i, in the control surface S(t).OSiOflowSi VVV ,,,  −= - differential of any vector , in O frame. O td d ζ  ζ  FLUID DYNAMICS
  • 38. 38 Start with LEIBNIZ THEOREM from CALCULUS: ( ) ( )    ChangeBoundariesthetodueChange tb ta tb ta td tad ttaf td tbd ttbfdx t txf dxtxf td d LEIBNITZ       −+= ∫∫ )),(()),(( ),( ),(:: )( )( )( )( ∂ ∂ and generalized it for a 3 dimensional vector space on a volume v(t) bounded by the surface S(t). Using LEIBNIZ THEOREM followed by GAUSS THEOREM (GAUSS 4): ( ) ( ) ( ) ( ) ∫∫∫∫∫       ⋅∇+∇⋅+=⋅+ → = tv OSOOOSGAUSS Opotolative dsofMovement thetodueChage tS OS tv O LEIBNITZ O tv vdVV t GAUSS sdVvd t vd td d ,,,,)4( intRe )( ,      χχ ∂ χ∂ χ ∂ χ∂ χ This is REYNOLDS’ TRANSPORT THEOREM OSBORNE REYNOLDS 1842-1912 SOLO GOTTFRIED WILHELM von LEIBNIZ 1646-1716 REYNOLDS’ TRANSPORT THEOREM FLUID DYNAMICS 1. MATHEMATICAL NOTATIONS (CONTINUE)
  • 39. 39 FLUID DYNAMICS 1. MATHEMATICAL NOTATIONS (CONTINUE) 1.11 REYNOLDS’ TRANSPORT THEOREM (CONTINUE) VECTOR NOTATION CARTESIAN TENSOR NOTATION ( ) ∫∫∫ ∫∫∫∫∫∫∫∫         ⋅∇+∇⋅+= ⋅+= )( ,,,,)4( , )()()( tv OSOOOS O GAUSS OS tStv O LEIBNITZ O tv vdVV t GAUSS sdVvd t vd td d     χχ ∂ χ∂ χ ∂ χ∂ χ ∫∫∫ ∫∫∫∫∫∫∫∫         ++= += )( , ,)4( , )()()( tv k kOS i k i kOS i GAUSS kkOS tS i tv i LEIBNITZ tv i vd x V x V t GAUSS sdVvd t vd td d ∂ ∂ χ ∂ χ∂ ∂ χ∂ χ ∂ χ∂ χ SOLO
  • 40. 40 FLUID DYNAMICS 1. MATHEMATICAL NOTATIONS (CONTINUE) 1.11 REYNOLDS’ TRANSPORT THEOREM (CONTINUE) VECTOR NOTATION CARTESIAN TENSOR NOTATION O OOS td Rd uV   == ,, CASE 1 (CONTROL VOLUME vF ATTACHED TO THE FLUID) kkOS uV =, ( ) ∫∫∫ ∫∫∫∫∫∫∫∫         ⋅∇+∇⋅+= ⋅+= )( ,,,)4( , )()()( tv OOO O GAUSS O tStv OO tv F FFF vduu t GAUSS sduvd t vd td d      χχ ∂ χ∂ χ ∂ χ∂ χ ∫∫∫ ∫∫∫∫∫∫∫∫         ++= += )( )4( )()()( tv k k I k I k I GAUSS kK tS I tv I tv I F FFF vd x u x u t GAUSS sduvd t vd td d ∂ ∂ χ ∂ χ∂ ∂ χ∂ χ ∂ χ∂ χ SOLO
  • 41. 41 FLUID DYNAMICS 1. MATHEMATICAL NOTATIONS (CONTINUE) 1.11 REYNOLDS’ TRANSPORT THEOREM (CONTINUE) VECTOR NOTATION CARTESIAN TENSOR NOTATION 1&, == χkkOS uV1&, == χuV OS  CASE 2 (CONTROL VOLUME vF ATTACHED TO THE FLUID AND )1=χ ∫∫∫∫∫∫∫∫ ⋅∇=⋅== )( ,, )( , )( )( tv OO tS O tv F FFF vdusduvd td d td tvd  ∫∫∫∫∫∫∫∫ === )()()( )( tv k k k tS k tv F FFF dv x u dsudv td d td tvd ∂ ∂               =⋅∇ → td tvd tv u F F tv OO F )( )( 1 lim0)( ,,                = → td tvd tvx u F F tv k k F )( )( 1 lim0)(∂ ∂ EULER 1755 SOLO
  • 42. 42 FLUID DYNAMICS 1. MATHEMATICAL NOTATIONS (CONTINUE) 1.11 REYNOLDS’ TRANSPORT THEOREM (CONTINUE) VECTOR NOTATION CARTESIAN TENSOR NOTATION CASE 3 (CONTROL VOLUME vF ATTACHED TO THE FLUID AND ) ρχ == &, kkOS uVρχ == &, uV OS  ρχ = or, since this is true for any attached volume vF(t) ( )∫∫∫ ∫∫∫∫∫ ∫∫∫       ⋅∇+= ⋅+=== )( ,, )( , )( )( )( 0 tv OO tS O tv tv F FF F vdu t sduvd t vd td d td tmd   ρ ∂ ρ∂ ρ ∂ ρ∂ ρ ( )∫∫∫ ∫∫∫∫∫ ∫∫∫       += +=== )( )()( )( )( 0 tv k k tS kk tv tv F FF F vdu xt sduvd t dv td d td tmd ρ ∂ ∂ ∂ ρ∂ ρ ∂ ρ∂ ρ Because the Control Volume vF is attached to the fluid and they are not sources or sinks, the mass is constant. ( ) OOOOOO uu t u t ,,,,,,0  ⋅∇+∇⋅+=⋅∇+= ρρ ∂ ρ∂ ρ ∂ ρ∂ ( ) k k k k k x u x u t u xt ∂ ∂ ρ ∂ ρ∂ ∂ ρ∂ ρ ∂ ∂ ∂ ρ∂ ++=+=  0 SOLO
  • 43. 43 FLUID DYNAMICS 1. MATHEMATICAL NOTATIONS (CONTINUE) 1.11 REYNOLDS’ TRANSPORT THEOREM (CONTINUE) VECTOR NOTATION CARTESIAN TENSOR NOTATION CASE 4 (CONTROL VOLUME WITH FIXED SHAPE C.V. )0,  =OS V Define ∫∫∫∫∫∫ = .... VC OO VC vd t vd td d ∂ χ∂ χ   ∫∫∫∫∫∫ = .... VC i VC i vd t vd td d ∂ χ∂ χ ( ) ( ) ( )      χ ρ ηr t r t r t, , ,≡ ( ) ( ) ( )χ ρ ηi k k i kx t x t x t, , ,≡ ( )∫∫ ∫∫∫∫∫∫ ⋅+         += )( , )()( tS OS tv OO tv sdV vd tt vd td d     ηρ ∂ ρ∂ η ∂ η∂ ρηρ k tS kOSi tv i i tv i sdV vd tt vd td d FF ∫∫ ∫∫∫∫∫∫ +       += )( , )()( ηρ ∂ ρ∂ η ∂ η∂ ρηρ We have but ( ) ( )OOOO u t u t ,,,, 0  ρη ∂ ρ∂ ηρ ∂ ρ∂ ⋅∇−=⇒=⋅∇+ ( ) ( )k k iik k u xt u xt ρ ∂ ∂ η ∂ ρ∂ ηρ ∂ ∂ ∂ ρ∂ −=⇒=+ 0 CASE 5 ( ) ( ) ( )     χ ρ ηr t r t r t, , ,≡ SOLO
  • 44. 44 FLUID DYNAMICS 1. MATHEMATICAL NOTATIONS (CONTINUE) 1.11 REYNOLDS’ TRANSPORT THEOREM (CONTINUE) VECTOR NOTATION CARTESIAN TENSOR NOTATION We have ( ) ( ) ( ) ( )[ ] ( ) ( )[ ]∫∫∫∫∫= ∫∫ ∫∫∫ ∫∫ ∫∫∫∫∫∫ ⋅−+ ⋅+         ⋅∇+∇⋅−         ∇⋅+= ⋅+         ⋅∇−= + + )( ,, )( 4 . )( , )( ,,,,,, )( , )( ,, )( tS OOS tv O MDG DerMat GAUSS tS OS tv OOOOOO O tS OS tv OO OO tv sduVvd tD D sdV vduuu t sdV vdu t vd td d          ρηρ η ρη ρηηρη ∂ η∂ ρ ρη ρηρ ∂ η∂ ρη ( ) ( ) ( ) ( ) ( ) ( )[ ]∫∫∫∫∫= ∫∫ ∫∫∫ ∫∫ ∫∫∫∫∫∫ −+ +               +−      += +       −= + + )( , )( 4 . )( , )( )( , )()( tS kkkOSi tv i MDG DerMat GAUSS tS kkOSi tv k k i k i k k i k i tS kkOSi tv k k i i tv i sduVvd tD D sdV vd x u x u x u t sdV vd x u t vd td d ρηρ η ρη ∂ ρ∂ η ∂ η∂ ρ ∂ η∂ ∂ η∂ ρ ρη ∂ ρ∂ ηρ ∂ η∂ ρη CASE 5 ( ) ( ) ( )     χ ρ ηr t r t r t, , ,≡ SOLO
  • 45. 45 FLUID DYNAMICS 1. MATHEMATICAL NOTATIONS (CONTINUE) 1.11 REYNOLDS’ TRANSPORT THEOREM (CONTINUE) VECTOR NOTATION CARTESIAN TENSOR NOTATION REYNOLDS 1 ( )[ ]       ⋅−+= ∫∫∫∫∫ ∫∫∫ )( ,, )( )( tS OOS tv O O tv sduVvd tD D vd td d    ρηρ η ρη ( )[ ]       −+= ∫∫∫∫∫ ∫∫∫ )( , )( )( tS kkkOSi tv i tv i sduVvd tD D dv td d ρηρ η ρη REYNOLDS 2 ( )[ ]        = ⋅−+ ∫∫∫ ∫∫∫∫∫ )( )( ,, )( tv O tS OSO O tv vd tD D sdVuvd td d ρ η ρηρη   ( )[ ]        = −+ ∫∫∫ ∫∫∫∫∫ )( )( , )( tv i tS kkOSki tv i vd tD D sdVuvd td d ρ η ρηρη CASE 5 ( ) ( ) ( )      χ ρ ηr t r t r t, , ,≡ SOLO
  • 46. 46 FLUID DYNAMICS 1. MATHEMATICAL NOTATIONS (CONTINUE) 1.11 REYNOLDS’ TRANSPORT THEOREM (CONTINUE) VECTOR NOTATION CARTESIAN TENSOR NOTATION REYNOLDS 3 CASE 1 (CONTROL VOLUME ATTACHED TO THE FLUID vF(t) ) kkOS uV =, ∫∫∫∫∫∫ = )()( tv OO tv FF vd tD D vd td d ρ η ρη   ∫∫∫∫∫∫ = )()( tv i tv i FF vd tD D vd td d ρ η ρη SOLO O OOS td Rd uV   == ,, ( ) ( ) ( )     χ ρ ηr t r t r t, , ,≡ CASE 4 (CONTROL VOLUME WITH FIXED SHAPE C.V. )0,  =OS V REYNOLDS 4 ( )       ⋅+= ∫∫∫∫∫ ∫∫∫ .. , .. .. SC O O VC VC O sduvd td d vd tD D   ρηρη ρ η ( )       += ∫∫∫∫∫ ∫∫∫ .... .. SC kki VC i VC i sduvd td d vd tD D ρηρη ρ η Return to Table of Content
  • 47. 47 FLUID DYNAMICS 2. BASIC LAWS IN FLUID DYNAMICS THE FLUID DYNAMICS IS DESCRIBED BY THE FOLLOWING FIVE LAWS: SOLO (1) CONSERVATION OF MASS (C.M.) (2) CONSERVATION OF LINEAR MOMENTUM (C.L.M.) (3) CONSERVATION OF MOMENT OF MOMENTUM (C.M.M.) (4) THE FIRST LAW OF THERMODYNAMICS (5) THE SECOND LAW OF THERMODYNAMICS Return to Table of Content
  • 48. 48 FLUID DYNAMICS 2. BASIC LAWS IN FLUID DYNAMICS (CONTINUE) (2.1) CONSERVATION OF MASS (C.M.) Control Volume attached to the fluid (containing a constant mass m) bounded by the Control Surface SF (t). ( )tvF ( )tr,  ρ ( )3 /mkgFlow density SOLO Because vF(t) is attached to the fluid and there are no sources or sinks in this volume, the Conservation of Mass requires that: d m t d t ( ) = 0 ( ) ( )trVtru OfluidO ,, ,,  = Flow Velocity relative to a predefined Coordinate System O (Inertial or Not-Inertial) ( )sm/
  • 49. 49 FLUID DYNAMICS 2. BASIC LAWS IN FLUID DYNAMICS (CONTINUE) (2.1) CONSERVATION OF MASS (CONTINUE - 1) VECTOR NOTATION CARTESIAN TENSOR NOTATION d m t d t ( ) = 0 ( )∫∫∫= ∫∫∫∫∫ ∫∫∫       ⋅∇+ ⋅+=== )( ,, 1 )( , )( )( )( 0 tv OO GAUSS tS O tv tv REYNOLDS F FF F vdu t sduvd t dv td d td tmd   ρ ∂ ρ∂ ρ ∂ ρ∂ ρ ( )∫∫∫= ∫∫∫∫∫ ∫∫∫       + +=== )( 1 )()( )( )( 0 tv k k GAUSS tS kk tv tv REYNOLDS F FF F vdu xt sduvd t dv td d td tmd ρ ∂ ∂ ∂ ρ∂ ρ ∂ ρ∂ ρ The Control Volume mass rate is zero as long as vF(t) is attached to the fluid and therefore contains the same amount of mass. 0),,,( )( =∫∫∫ tvF vdtzyx td d ρ is true in any Coordinate System (O) and so is: ( ) ( ) ( ) ( )( ) 0,,,,,, ,,, ,,, )( ,, )( =      ⋅∇+= ∫∫∫∫∫∫ tv OO tv FF vdtzyxutzyx t tzyx vdtzyx td d  ρ ∂ ρ∂ ρ SOLO
  • 50. 50 FLUID DYNAMICS 2. BASIC LAWS IN FLUID DYNAMICS (CONTINUE) (2.1) CONSERVATION OF MASS (CONTINUE - 2) VECTOR NOTATION CARTESIAN TENSOR NOTATION For any Control Volume v (t) (not necessarily attached to the fluid) The following is true for any Coordinate System (for points that are not sources or sinks – mathematically equivalent to analytic) ( )OO u t ,,,  ρ ∂ ρ∂ ⋅∇ ( ) OOOOOO uu t u t ,,,,,,0  ⋅∇+∇⋅+=⋅∇+= ρρ ∂ ρ∂ ρ ∂ ρ∂ ( ) k k k kO k x u x u t u xt ∂ ∂ ρ ∂ ρ∂ ∂ ρ∂ ρ ∂ ∂ ∂ ρ∂ ++=+= ,0  ( ) ( ) 0 )( ,, 4 ).( , ).()( ≠=      ⋅∇+ ⋅+ ∫∫∫= ∫∫∫∫∫=∫∫∫ mvdV t sdVvd t vd td d tv OSO GAUSS tS OS tv LEIBNITZ tv    ρ ∂ ρ∂ ρ ∂ ρ∂ ρ ( ) 0 )( , 4 ).( , ).()( ≠=      + + ∫∫∫= ∫∫∫∫∫=∫∫∫ mvdV xt sdVvd t vd td d tv kOS k GAUSS tS kkOS tv LEIBNITZ tv ρ ∂ ∂ ∂ ρ∂ ρ ∂ ρ∂ ρ The integral above is not zero because the mass in v (t) is not constant. SOLO
  • 51. 51 FLUID DYNAMICS 2. BASIC LAWS IN FLUID DYNAMICS (CONTINUE) (2.1) CONSERVATION OF MASS (CONTINUE - 3) Material Derivative of vdmd ρ= Let use EULER’s 1755 expression ( ) ( ) ( ) ( )vd tD D vdtd tvd tv u F F tv OO F 11 lim 0 ,, =            =⋅∇ →  and the (C.M.): to develop the following: ( ) 0,, =⋅∇+ OO u t  ρ ∂ ρ∂ ( ) ( ) ( ) 0,,,,,, ,,,, =      ⋅∇+ ∂ ∂ =      ⋅∇+∇⋅+ ∂ ∂ = ⋅∇+      ∇⋅+ ∂ ∂ =+== vdu t vduu t uvdvdu t vd tD D vd tD D vd tD D tD mD OOOOOO OOOO   ρ ρ ρρ ρ ρρ ρ ρ ρ ρ SOLO
  • 52. 52 FLUID DYNAMICS ∑+= openings i iopenW SCSCSC .... 2. BASIC LAWS IN FLUID DYNAMICS (CONTINUE) (2.1) CONSERVATION OF MASS (CONTINUE – 4) SOLO Control Volume with fixed shape C.V. and boundary C.S. in O Coordinates( ) 0,  =OS V There are no sources or sinks in the volume C.V. The change in the mass of the system is due only to the flow through the surface openings C.Sopen i (i=1,2,…). The surface C.S. can be divided in: • C.Sw the impermeable wall through which the fluid can not escape .         =−= 0 0 ,,,  OSOs VuV • C.Sopen i the openings (i=1,2,…) through which the fluid enters or exits .( )0>m ( )0<m  ∑∑ ∫∫∫∫∫∫∫∫∫ =⋅−⋅−=⋅−== openings i i openings i m SC O SC O SC O VC msdusdusduvd td d td md i iopenw     . , . 0 , .. , .. ρρρρ Therefore where is the flow rate entering through the opening Sopen i.∫∫ ⋅−= iopenSC Oi sdum . ,   ρ Return to Table of Content
  • 53. 53 FLUID DYNAMICS 2. BASIC LAWS IN FLUID DYNAMICS (CONTINUE) (2.2) CONSERVATION OF LINEAR MOMENTUM (C.L.M.) -Fluid density at he point and time t( )tr,  ρ  r ( )3 / mKg -Fluid inertial velocity at the point and time t ( )tru I ,,   r ( )sec/m -Surface Stress ( )2 / mNT  -Pressure (force per unit surface) of the surrounding on the control surface ( )2 / mN p -Stress tensor (force per unit surface) of the surrounding on the control surface ( )2 / mN σ~ -Body forces acceleration -(gravitation, electromagnetic,..) G  ( )2 sec/m nnpnT ˆ~ˆˆ~ ⋅+−=⋅= τσ  Consider a volume vF(t) attached to the fluid, bounded by the closed surface SF(t). SOLO -unit vector normal to the surface S(t) and pointing outside the volume v (t)nˆ vF (t) m SF (t) O x y z r u,O np ˆ− nˆ~⋅τ nˆ~⋅σ dSnˆ -Shear stress tensor (force per unit surface) of the surrounding on the control surface ( )2 / mN τ~
  • 54. 54 FLUID DYNAMICS 2. BASIC LAWS IN FLUID DYNAMICS (CONTINUE) (2.2) CONSERVATION OF LINEAR MOMENTUM (CONTINUE - 1) Derivation From Integral Form The LINEAR MOMENTUM of the Constant Mass in vF(t) is given by: ∫∫∫= )( , tv I F vduP  ρ The External Forces acting on the mass are Body and Surface Forces: ( )     ForcesSurface tS ForcesBody tv external FF sdTvdGF ∫∫∫∫∫∑ += )( ρ According to NEWTON’s Second Law, for a constant mass in vF(t), we have: I external td Pd F   =∑ SOLO
  • 55. 55 FLUID DYNAMICS 2. BASIC LAWS IN FLUID DYNAMICS (CONTINUE) (2.2) CONSERVATION OF LINEAR MOMENTUM (CONTINUE - 2) VECTOR NOTATION CARTESIAN TENSOR NOTATION ( ) I IMomentumLinear tv I REYNOLDS tv I I ForcesSurface tS ForcesBody tv external P td d vdu td d vd tD uD sdvdGF FF FF         === ⋅+= ∫∫∫∫∫∫ ∫∫∫∫∫∑ )( , 3 )( , )()( ~ ρρ σρ i tv i REYNOLDS tv i tS kik tv iiex P td d vdu dt d vd tD uD dsvdGF FF FF === += ∫∫∫∫∫∫ ∫∫∫∫∫∑ )( 3 )( )()( _ ρρ σρ C.L.M.-1      T ds n ds ds ds n ds = ⋅ ⋅ = =~ ~σ σ T ds n ds dsi ik k ds n ds ik k k k = = =σ σ C.L.M.-2 ( )∫∫∫ ∫∫∫∫∫∫∫∫ ⋅∇+= ⋅+= )( , )()()( , ~ ~ tv I tStvtv I I F FFF vdG sdvdGvd tD uD σρ σρρ   ∫∫∫ ∫∫∫∫∫∫∫∫       += += )( )()()( tv i ik i tS kik tv i tv i F FFF vd x G sdvdGvd tD uD ∂ σ∂ ρ σρρ SOLO Derivation From Integral Form (Continue)
  • 56. 56 FLUID DYNAMICS 2. BASIC LAWS IN FLUID DYNAMICS (CONTINUE) (2.2) CONSERVATION OF LINEAR MOMENTUM (CONTINUE - 3) VECTOR NOTATION CARTESIAN TENSOR NOTATION C.L.M.-2 Since this is true for all volumes vF (t) attached to the fluid we can drop the volume integral. [ ] [ ] [ ]τσ τρσρ ∂ ∂ ρ ∂ ∂ ρρ ~~ ~~ 2 1 ,,, , 2 , , .).( +−= ⋅∇+∇−=⋅∇+=         ×∇×−      ∇+=         ∇⋅+= Ip pGG uuu t u uu t u tD uD III II I I I DM I      ikikik i ik i i i ik i k i k i j jjj i i k i k i DM i p xx p G x G x u u x u uuu xt u x u u t u tD uD τδσ ∂ τ∂ ∂ ∂ ρ ∂ σ∂ ρ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ρ ∂ ∂ ∂ ∂ ρρ +−= +−=+=             ⋅−⋅−      +=       ⋅+= 2 1 .).( SOLO Derivation From Integral Form (Continue) ( )∫∫∫ ∫∫∫∫∫∫∫∫ ⋅∇+= ⋅+= )( , )()()( , ~ ~ tv I tStvtv I I F FFF vdG sdvdGvd tD uD σρ σρρ   ∫∫∫ ∫∫∫∫∫∫∫∫       += += )( )()()( tv i ik i tS kik tv i tv i F FFF vd x G sdvdGvd tD uD ∂ σ∂ ρ σρρ
  • 57. 57 FLUID DYNAMICS 2. BASIC LAWS IN FLUID DYNAMICS (CONTINUE) (2.2) CONSERVATION OF LINEAR MOMENTUM (CONTINUE - 4) Derivation From a Cartesian Differential Volume VECTOR NOTATION CARTESIAN TENSOR NOTATION σ ∂ σ ∂ xx xx x dx+ 1 2 σ ∂ σ ∂ xx xx x dx− 1 2 τ ∂ τ ∂ yx yx y dy+ 1 2 τ ∂ τ ∂ yz yz y dy− 1 2 τ ∂τ ∂ zx zx z dz+ 1 2 τ ∂τ ∂ zx zx z dz− 1 2 τ ∂ τ ∂ xy xy x dx+ 1 2 τ ∂ τ ∂ xy xy x dx− 1 2 σ ∂ σ ∂ yy yy y dy+ 1 2 σ ∂ σ ∂ yy yy y dy− 1 2 τ ∂τ ∂ zy zy z dz+ 1 2 τ ∂τ ∂ zy zy z dz− 1 2 τ ∂ τ ∂ xz xz x dx− 1 2 τ ∂ τ ∂ yz yz y dy+ 1 2 τ ∂ τ ∂ yx yx y dy− 1 2 σ ∂ σ ∂ zz zz z dz+ 1 2 σ ∂ σ ∂ zz zz z dz− 1 2 z y x dy dx dz O τ ∂ τ ∂ xz xz x dx+ 1 2 ∂σ ∂ ∂τ ∂ ∂τ ∂ ρ ρ ∂τ ∂ ∂σ ∂ ∂τ ∂ ρ ρ ∂τ ∂ ∂τ ∂ ∂σ ∂ ρ ρ xx yx zx xB x xy yy zy yB y xz yz zz zB z x y z G a x y z G a x y z G a + + + = + + + = + + + = CAUCHY’s First Law of Motion I tD uD a aG    = =+⋅∇ ρρσ~ tD uD a aG x i i ii i ij = =+ ρρ ∂ σ∂ SOLO AUGUSTIN LOUIS CAUCHY )1789-1857(
  • 58. 58 FLUID DYNAMICS 2. BASIC LAWS IN FLUID DYNAMICS (CONTINUE) (2.2) CONSERVATION OF LINEAR MOMENTUM (CONTINUE-5) Derivation For Any Control Volume v (t) (the velocity of an element of surface is )d s  IS V ,  V(t) b ds V*(t) I T d s= ⋅~σ G m u Use REYNOLDS’ Transport Theorem (REYNOLDS 2) with and O = I, and then the Conservation of Linear Momentum (C.L.M.) I u,  =η VECTOR NOTATION CARTESIAN TENSOR NOTATION ( )[ ] ( ) ( ) ( ) ∑∫∫∫∫∫ ∫∫∫∫∫∫ ∫∫∫∫∫ =⋅+= ⋅∇+== ⋅−+ iexternal tStv tv I MLC tv I REYNOLDS tS ISII I tv I FsdvdG vdGvd tD uD sdVuuvdu td d FF FF FF    )()( )( , ... )( 2 )( ,,, )( , ~ ~ σρ σρρ ρρ ( ) ( ) ( ) ∑∫∫∫∫∫ ∫∫∫∫∫∫ ∫∫∫∫∫ =+=         +== −+ iexternal tS kik tv i tv k ik i MLC tv i REYNOLDS tS kkISki tv i FsdvdG vd x Gvd tD uD sdVuuvdu td d )()( )( ... )( 2 )( , )( σρ ∂ σ∂ ρρ ρρ SOLO Return to Table of Content
  • 59. 59 ( ) ( ) PdRRvdVRRHd OOO  ×−=×−= ρ, 2. BASIC LAWS IN FLUID DYNAMICS (2.3) CONSERVATION OF MOMENT-OF-MOMENTUM (C.M.M.) SOLO The Absolute Angular Momentum, of the differential mass and Inertial Velocity ,relative to a reference point O is defined as vdmd ρ= V  The Absolute Angular Momentum of the mass enclosed by C.V. is defined as ( ) ( )∫∫∫∫∫∫ ×−=×−= .... , VC O VC OOCV PdRRvdVRRH  ρ Let differentiate the Absolute Angular Momentum and use Reynolds’ Transport Theorem ( ) ( ) ( ) ( )∫∫∫∫∫∫∫∫ ⋅−×−+ ×− =×−= .. , .... , SC md SO VC I O REYNOLDS I VC O I OCV sdVVRRvd tD VRRD vdVRR td d td Hd       ρρρ We have ( ) ( ) ( ) ( ) ( ) VV tD VD RRVVV tD VD RR V tD RD tD RD tD VD RR tD VRRD O I OO I O I O II O I O          ×−×−=×−+×−= ×         −+×−= ×− FLUID DYNAMICS
  • 60. 60 ( ) ( ) ( ) int, : fdRRfdRRvd tD VD RRMd OextO I OO    ×−+×−=×−= ρ ( ) ( ) ( ) ( )∫∫∫∫∫∫∫∫∫∫∫ ⋅−×−+×−×−=×−= .. , ...... , SC md SO P VC O VC I O REYNOLDS I VC O I OCV sdVVRRvdVVvd tD VD RRvdVRR td d td Hd CV         ρρρρ 2. BASIC LAWS IN FLUID DYNAMICS (2.3) CONSERVATION OF MOMENT-OF-MOMENTUM (C.M.M.) SOLO The Moment, of the differential mass dm = ρdv, relative to a reference point O is defined as Therefore Let integrate this equation over the control volume C.V. ( ) ( ) ( )        0 .. int .... , ∫∫∫∫∫∫∫∫∫∑ ×−+×−=×−= VC O VC extO VC I OOCV fdRRfdRRvd tD VD RRM ρ Using the differential of Angular Momentum equation we obtain ( ) ( ) ( )∫∫∫∫∫∑∫∫∫ ⋅−×−+×−=×−= .. , .. , .. , SC md SO P VC OOCV I VC O I OCV sdVVRRvdVVMvdVRR td d td Hd CV       ρρρ ( ) ( ) ( ) ( ) ( ) ∑∑∫∫∫∫∫∫∫∫∑ +×−++−×−+×−=×−= =⋅ k k j jOj SC sdTsd O VC O VC extOOtCV MFRRsdtfnpRRvdgRRfdRRM      ...... , 11 σ ρ Also ( )∑ ×− j jOj FRR  - Moment, relative to O, of discrete forces exerting by the surrounding at point jR  - Discrete Moments exerting by the surrounding.∑ k k M  FLUID DYNAMICS
  • 61. 61 ( ) ( ) ( ) ∑∑∫∫∫∫∫∫∫∫ +×+×+×=×+⋅−×−× k k j jO tv O tv extO P VC O SC md SO I VC O MFrfdrfdrvdVVsdVVrvdVr td d CV         , 0 int,, .... ,, .. , ρρρ 2. BASIC LAWS IN FLUID DYNAMICS (2.3) CONSERVATION OF MOMENT-OF-MOMENTUM (C.M.M.) SOLO Let find the equation of moment around the turbomachine axis. We shall use polar coordinates , where z is the turbomachine axis. zr ,,θ zzrrrO ˆˆ, +=  zVVrVV zr ˆˆˆ ++= θθ  zFFrFF zr ˆˆˆ ++= θθ  ( ) zVrVrVzrVz VVV zr zr Vr zrz zr O ˆˆ0 ˆˆˆ , θ θ θ +−+−==×  ( )  ( ) ∑∑∫∫∫∫ ++=×+⋅−− k kz j j tv extCVO SC S VC MFrdfrPVsdVVrvdVr td d θθθθ ρρ   0 .. , .. The moment of momentum equation around the turbomachine z axis. Example FLUID DYNAMICS
  • 62. 62 2. BASIC LAWS IN FLUID DYNAMICS (2.3) CONSERVATION OF MOMENT-OF-MOMENTUM (C.M.M.) SOLO ( ) ( ) ( ) ( ) ( ) ( )          systemoutsidefromexertedTorque M l lz j j tv ext AVVrAVVr SC S statesteady VC zSnSn MFrdfrsdVVrvdVr td d ∑∑∫∫∫∫ ++=⋅−− +−−→ θθ ρρ θθ θθ ρρ 22,21111,122 .. , 0 .. We obtain ( ) ( )[ ] zflow MQVrVr =− 111122 ρθθ or ( ) ( ) ( ) ( ) ( ) ( )[ ] ( ) zSnSnSn MAVVrVrAVVrAVVr =−=− 11,1112211,11122,222 ρρρ θθθθ Euler Turbine Equation ρ1 - mean fluid density one inlet (1) of area A1. where ρ2 - mean fluid density one outlet (2) of area A2. (Vθ )1, r1 - mean fluid tangential velocity and radius one inlet (1) of area A1. (Vθ )2, r2 - mean fluid tangential velocity and radius one outlet (2) of area A2. (V,Sn )1 - mean fluid normal velocity (relative to A1) one inlet (1) of area A1. (V,Sn )2 - mean fluid normal velocity (relative to A2) one outlet (2) of area A2. - mean flow rate one outlet (1) of area A1.( ) 11,1 : AVQ Snflow = FLUID DYNAMICS Return to Table of Content
  • 63. 63 FLUID DYNAMICS 2. BASIC LAWS IN FLUID DYNAMICS (CONTINUE) (2.4) CONSERVATION OF ENERGY (C.E.) – THE FIRST LAW OF THERMODYNAMICS (DIFFERENTIAL FORM) -Fluid mean velocity [m/sec[( )  u r t, -Body Forces Acceleration -(gravitation, electromagnetic,..) G  -Surface Stress [N/m2 [T  nnpnT ˆ~ˆˆ~ ⋅+−=⋅= τσ  m V(t) G q T n= ⋅~σ d E d t ∂ ∂ Q t uu d s n ds= -Internal Energy of Fluid molecules (vibration, rotation, translation per mass [W/kg[ e - Rate of Heat transferred to the Control Volume (chemical, external sources of heat) [W/m3 [ ∂ ∂ Q t - Rate of Work change done on fluid by the surrounding (rotating shaft, others) positive for a compressor, negative for a turbine) [W[td Ed SOLO Consider a volume vF(t) attached to the fluid, bounded by the closed surface SF(t). -Rate of Conduction and Radiation of Heat from the Control Surface (per unit surface) [W/m3 [ q
  • 64. 64 FLUID DYNAMICS 2. BASIC LAWS IN FLUID DYNAMICS (CONTINUE) (2.4) CONSERVATION OF ENERGY (C.E.) – THE FIRST LAW OF THERMODYNAMICS (CONTINUE - 1) - The Internal Energy of the molecules of the fluid plus the Kinetic Energy of the mass moving relative to an Inertial System (I) The FIRST LAW OF THERMODYNAMICS CHANGE OF INTERNAL ENERGY + KINETIC ENERGY = CHANGE DUE TO HEAT + WORK + ENERGY SUPPLIED BY SUROUNDING SOLO The energy of the constant mass m in the volume vF(t) attached to the fluid, bounded by the closed surface SF(t) is This energy will change due to - The Work done by the surrounding - Absorption of Heat - Other forms of energy supplied to the mass (electromagnetic, chemical,…)
  • 65. 65 FLUID DYNAMICS 2. BASIC LAWS IN FLUID DYNAMICS (CONTINUE) (2.4) CONSERVATION OF ENERGY (C.E.) – THE FIRST LAW OF THERMODYNAMICS (CONTINUE - 2) VECTOR NOTATION CARTESIAN TENSOR NOTATION C.E.-1               systementering td Qd tSv systemontnmenenvirobydone td Wd shaft tSv v REYNOLDS KineticInternal tv FF FF FF sdqvd t Q td Wd ForcesSurface sdTu ForcesBody vdGu vdue tD D vdue td d ∫∫∫∫∫ ∫∫∫∫∫ ∫∫∫∫∫∫ ⋅−+ +⋅+⋅=       +=      + + )( )( 2 )3( )( 2 2 1 2 1 ∂ ∂ ρ ρρ             systementering td Qd tS kk tv systemontnemnoenvirbydone td Wd shaft tS kk tv kk tv REYNOLDS KineticInternal tv FF FF FF dsqvd t Q td Wd ForcesSurface sdTu ForcesBody vdGu vdue tD D vdue td d ∫∫∫∫∫ ∫∫∫∫∫ ∫∫∫∫∫∫ −+ ++=       +=      + + )()( )()( )( 2 )3( )( 2 2 1 2 1 ∂ ∂ ρ ρρ SOLO
  • 66. 66 FLUID DYNAMICS 2. BASIC LAWS IN FLUID DYNAMICS (CONTINUE) (2.4) CONSERVATION OF ENERGY (C.E.) – THE FIRST LAW OF THERMODYNAMICS (CONTINUE - 3) VECTOR NOTATION CARTESIAN TENSOR NOTATIONC.E.-2 ( ) ( ) ∫∫∫∫∫∫ ∫∫∫∫∫∫∫∫∫ ∫∫∫∫∫ ∫∫∫∫∫∫∫ ∫∫∫ ⋅∇−+ ⋅⋅∇+⋅∇−⋅= ⋅−+ ⋅⋅+⋅−⋅= +       + )()( )()()( )1( )()( )()()( )( 2 ~ ~ 2 1 tvtv tvtvtv GAUSS td Qd tStv td Wd tStStv tv FF FFF FF FFF F vdqvd t Q vduvdupvdGu sdqvd t Q sdusdupvdGu KineticInternal vdue tD D              ∂ ∂ τρ ∂ ∂ τρ ρ ( ) ( ) ∫∫∫∫∫∫ ∫∫∫∫∫∫∫∫∫= ∫∫∫∫∫ ∫∫∫∫∫∫∫ ∫∫∫ −+ +− −+ +−=       + + )()( )()()( )1( )()( )()()( )( 2 2 1 tV s s tV tV k k iki tV k k k tV kk GAUSS td Qd tS kk tV td Wd tS kiki tS kk tV kk KineticInternal tV vd x q vd t Q ds x u ds x up vdGu dsqvd t Q dsudsupvdGu vdue tD D ∂ ∂ ∂ ∂ ∂ τ∂ ∂ ∂ ρ ∂ ∂ τρ ρ                T n pn n ds n ds= ⋅ = − + ⋅ =~ ~ &σ τ0= td Wd shaft assume and use SOLO
  • 67. 67 FLUID DYNAMICS 2. BASIC LAWS IN FLUID DYNAMICS (CONTINUE) (2.4) CONSERVATION OF ENERGY (C.E.) – THE FIRST LAW OF THERMODYNAMICS (CONTINUE-4) VECTOR NOTATION CARTESIAN TENSOR NOTATIONC.E.-3 Since the last equation is valid for each vF(t) we can drop the integral and obtain: ( ) ( ) q t Q uGuupue tD D   ⋅∇−+ ⋅+⋅⋅∇+⋅−∇=      + ∂ ∂ ρτρ ~ 2 1 2 ( ) ( ) k k kk k iik k k x q t Q uG x u x up ue tD D ∂ ∂ ∂ ∂ ρ ∂ τ∂ ∂ ∂ ρ −+ ++−=      + 2 2 1 Multiply (C.L.M.-2) by  u τρρ ~⋅∇⋅+∇⋅−⋅=⋅ upuuG tD uD u   ( ) k ik i k kkk i i x u x p uuGu tD D tD uD u ∂ τ∂ ∂ ∂ ρρρ +−== 2 Subtract this equation from (C.E.-3) C.E.-4 ( )[ ]ρ τ τ ∂ ∂ D e D t p u u u Q t q = − ∇⋅ + ∇⋅ ⋅ − ⋅∇⋅ + −∇⋅        ~ ~ Φ ρ ∂ ∂ τ ∂ ∂ ∂ ∂ ∂ ∂ D e D t p u x u u x Q t q x k k ik i k k k =− + + − Φ   ( )Φ ≡ ∇⋅ ⋅ − ⋅∇ ⋅ >~ ~τ τ   u u 0 Φ ≡ >τ ∂ ∂ ik i k u x 0 (Proof of inequality given later) SOLO
  • 68. 68 FLUID DYNAMICS 2. BASIC LAWS IN FLUID DYNAMICS (CONTINUE) (2.4) CONSERVATION OF ENERGY (C.E.) – THE FIRST LAW OF THERMODYNAMICS (CONTINUE - 5) VECTOR NOTATION CARTESIAN TENSOR NOTATION Enthalpy Use this result and (C.E.-4) C.E.-5 ρ p eh +=: ( ) tD pD up tD hD u p tD pD tD hD tD Dp tD pD tD hD tD pD tD hD tD eD −⋅∇−=⋅∇−+−= +−=       −=  ρρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρρρ 2 tD pD x u p tD hD x up tD pD tD hD tD pDp tD hD tD pD tD pD tD hD tD eD k k k k −−=        −+−= +−=       −= ∂ ∂ ρ ∂ ∂ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρρρ 2 Φ++⋅∇−= t Q q tD pD tD hD ∂ ∂ ρ  Φ++−= t Q x q tD pD tD hD k k ∂ ∂ ∂ ∂ ρ SOLO ( )Φ ≡ ∇⋅ ⋅ − ⋅∇ ⋅ >~ ~τ τ   u u 0 Φ ≡ >τ ∂ ∂ ik i k u x 0
  • 69. 69 FLUID DYNAMICS 2. BASIC LAWS IN FLUID DYNAMICS (CONTINUE) (2.4) CONSERVATION OF ENERGY (C.E.) – THE FIRST LAW OF THERMODYNAMICS (CONTINUE - 6) VECTOR NOTATION CARTESIAN TENSOR NOTATION Total Enthalpy Use this result and (C.E.-3) C.E.-6 22 2 1 2 1 : u p euhH ++=+= ρ ( ) t p up tD HD tD pD up tD HD p tD D tD HD ue tD D ∂ ∂ ρρ ρ ρρρ −⋅∇−=−⋅∇−=       −=      +  2 2 1 ( ) t p up xtD HD tD pD x u p tD HD p tD D tD HD ue tD D kk k ∂ ∂ ∂ ∂ ρ ∂ ∂ ρ ρ ρρρ −−=−−=       −=      +  2 2 1 ( ) q t Q uGu t p tD HD  ⋅∇−+⋅+⋅⋅∇+= ∂ ∂ ρτ ∂ ∂ ρ ~ ( ) k k kk k iik x q t Q uG x u t p tD HD ∂ ∂ ∂ ∂ ρ ∂ τ∂ ∂ ∂ ρ −+++= SOLO Return to Table of Content
  • 70. 70 FLUID DYNAMICS 2. BASIC LAWS IN FLUID DYNAMICS (CONTINUE) SOLO THERMODYNAMIC PROCESSES 1. ADIABATIC PROCESSES 2. REVERSIBLE PROCESSES 3. ISENTROPIC PROCESSES No Heat is added or taken away from the System No dissipative phenomena (viscosity, thermal, conductivity, mass diffusion, friction, etc) Both adiabatic and reversible (2.5) THE SECOND LAW OF THERMODYNAMICS AND ENTROPY PRODUCTION
  • 71. 71 FLUID DYNAMICS 2. BASIC LAWS IN FLUID DYNAMICS (CONTINUE) (2.5) THE SECOND LAW OF THERMODYNAMICS AND ENTROPY PRODUCTION 2nd LAW OF THERMODYNAMICS Using GAUSS’ THEOREM 0 )()( ≥+ ∫∫∫∫∫ tStv FF Ad T q vds td d  ρ 00 )( )1( )()( ≥            ⋅∇+⇒≥+ ∫∫∫∫∫∫∫∫ tv GAUSS tStv FFF vd T q tD sD Ad T q vd tD sD  ρρ - Change in Entropy per unit volumed s - Local TemperatureT [ ]K - Fluid Densityρ [ ]3 / mKg d e q w T ds pdv= + = −δ δ d s d e T p T dv= + SOLO For a Reversible Process - Rate of Conduction and Radiation of Heat from the System per unit surface q  [ ]2 / mW
  • 72. 72 FLUID DYNAMICS 2. BASIC LAWS IN FLUID DYNAMICS (CONTINUE) (2.5) THE SECOND LAW OF THERMODYNAMICS (CONTINUE - 1) d e q w T ds pdv= + = −δ δ d s d e T p T dv= + u T p tD eD T u T p tD eD T tD D T p tD eD TtD D T p tD eD TtD vD T p tD eD TtD sD u tD D MC v   ⋅∇+=      ⋅∇+=       −+=      +=+= ⋅∇−= = ρ ρ ρ ρρ ρ ρ ρρ ρ ρρρρ ρ ρ ρ ρ 2 .).( 2 1 1 11 The Energy Equation (C.E.-4) is ( ) k i ik x u oruu t Q qup tD eD ∂ ∂ τττ ∂ ∂ ρ =Φ⋅∇⋅−⋅⋅∇=ΦΦ++⋅∇−⋅∇−= ~~  Tt Q TT q up tD eD TtD sD Φ ++ ⋅∇ −=      ⋅∇+= ∂ ∂ ρ 11   or Φ++⋅−∇= t Q q tD sD T ∂ ∂ ρ  SOLO
  • 73. 73 FLUID DYNAMICS 2. BASIC LAWS IN FLUID DYNAMICS (CONTINUE) (2.5) THE SECOND LAW OF THERMODYNAMICS (CONTINUE - 2) Define ρ ∂ ∂ T D s Dt q Q t = −∇ ⋅ + +  Φ Θ ≡ + ∇ ⋅       ≥ρ Ds Dt q T  0 Entropy Production Rate per unit volume Therefore ( ) Θ Φ Θ= − ∇ ⋅ + + + ∇ ⋅       ≥∫∫∫   q T T Q t T q T dv V t 1 0 ∂ ∂ & SOLO or    0 1 ≥Φ++⋅∇⋅−=Θ nDissipatio System toadded Heat System from Radiation Heat t Q Tq T T ∂ ∂
  • 74. 74 FLUID DYNAMICS 2. BASIC LAWS IN FLUID DYNAMICS (CONTINUE) (2.5) THE SECOND LAW OF THERMODYNAMICS (CONTINUE - 3)     q q q q conduction rate per unit surface q radiation rate per unit surfacec r c r = +     q K T K FOURIER s Conduction Lawc = − ∇ > 0 ' ( )− ∇ ⋅ + ∇ ⋅       = − ∇ ⋅ + ∇ ⋅ + ⋅∇       = ⋅∇       = − ∇ + ⋅∇       = − ∇ ⋅ − ∇       + ⋅∇       = ∇      + ⋅∇                q T q T q T T q q T q T K T q T K T T T q T K T T q T r r r 1 1 1 1 1 1 1 2 2 Θ Φ Φ= ∇      + + + ⋅∇       > > >      K T T T T Q t q T K T r 2 1 1 0 0 0 ∂ ∂  Θ Φ ≡ + ∇⋅       = ∇      + + + ⋅∇       ≥ρ ∂ ∂ D s D t q T K T T T T Q t q Tr   2 1 1 0 SOLO JEAN FOURIER 1768-1830
  • 75. 75 FLUID DYNAMICS 2. BASIC LAWS IN FLUID DYNAMICS (CONTINUE) (2.5) THE SECOND LAW OF THERMODYNAMICS (CONTINUE - 4) SOLO Gibbs Function Helmholtz Function sThG ⋅−=: sTeH ⋅−=: Josiah Willard Gibbs (1839-1903) Hermann Ludwig Ferdinand von Helmholtz (1821 – 1894) Using the Relations vdpsdTed ⋅−⋅= ( ) pdvsdTvpdedhd ⋅+⋅=⋅+=vpe p eh ⋅+=+= ρ : pdvTdssdTTdshdGd ⋅+⋅−=⋅−⋅−= vdpTdsTdssdTedHd ⋅−⋅−=⋅−⋅−= dv T p T ed sd +=
  • 76. 76 FLUID DYNAMICS 2. BASIC LAWS IN FLUID DYNAMICS (CONTINUE) (2.5) THE SECOND LAW OF THERMODYNAMICS (CONTINUE - 5) SOLO Maxwell’s Relations vdpsdTed ⋅−⋅= pdvsdThd ⋅+⋅= pdvTdsGd ⋅+⋅−= vdpTdsHd ⋅−⋅−= Ts pv v F p v e s h T s e       ∂ ∂ =−=      ∂ ∂       ∂ ∂ ==      ∂ ∂ vp Ts T F s T G p G v p h       ∂ ∂ =−=      ∂ ∂       ∂ ∂ ==      ∂ ∂ ps vs s v p T s p v T       ∂ ∂ =      ∂ ∂       ∂ ∂ −=      ∂ ∂ vT pT T p v s T v p s       ∂ ∂ =      ∂ ∂       ∂ ∂ −=      ∂ ∂ James Clerk Maxwell (1831-1879) Return to Table of Content
  • 77. 77 FLUID DYNAMICS 2. BASIC LAWS IN FLUID DYNAMICS (CONTINUE) (2.6) CONSTITUTIVE RELATIONS FOR GASES (2.6.1) NEWTONIAN FLUID DEFINITION – NAVIER–STOKES EQUATIONS [ ] τσ ~~ +−= Ip Stress NEWTONIAN FLUID: The Shear Stress on A Surface Parallel To the Flow = Distance Rate of Change of Velocity SOLO CARTESIAN TENSOR NOTATION ikikik p τδσ +−= VECTOR NOTATION - Stress tensor (force per unit surface) of the surrounding on the control surface ( )2 / mN σ~ -Shear stress tensor (force per unit surface) of the surrounding on the control surface ( )2 / mN τ~
  • 78. 78 FLUID DYNAMICS 2. BASIC LAWS IN FLUID DYNAMICS (CONTINUE) (2.6) CONSTITUTIVE RELATIONS (2.6.1) NEWTONIAN FLUID DEFINITION – NAVIER–STOKES EQUATIONS M. NAVIER 1822 INCOMPRESSIBLE FLUIDS (MOLECULAR MODEL) G.G. STOKES 1845 COMPRESSIBLE FLUIDS (MACROSCOPIC MODEL) VECTOR NOTATION CARTESIAN TENSOR NOTATION [ ] [ ] ( )[ ] [ ]IuuuIpIp T  ∇+∇+∇+−=+−= λµτσ ~~ ik k k i k k i ikikikik x u x u x u pp δ ∂ ∂ λ ∂ ∂ ∂ ∂ µδτδσ +      ++−=+−= ( )[ ] [ ]( ) ( ) ( ) µλλµλµτ 3 2 32~0 −=⇒∇+∇=∇+∇+∇== utrutrIutruutrtr T  ( ) µλ ∂ ∂ λµδ ∂ ∂ λ ∂ ∂ µτ 3 2 0322 −=⇒=+=+= i i ik k k i i ii x u x u x u SOLO STOKES ASSUMPTION µλ 3 2 −=0~ =τtrace μ, λ - Lamé parameters from Elasticity
  • 79. 79 FLUID DYNAMICS 2. BASIC LAWS IN FLUID DYNAMICS (CONTINUE) (2.6) CONSTITUTIVE RELATIONS (2.6.1) NAVIER–STOKES EQUATIONS (CONTINUE) (2.6.1.2) VECTORIAL DERIVATION I x y z T n= ⋅~σ d s n ds= r dru u +du( )unrdtd t u urdtd t u ud      ∇⋅+=∇⋅+= 1 ∂ ∂ ∂ ∂ ( ) ( ) ( ) rdnurdnuuntd t u ud RotationnTranslatio         1 2 1 1 2 1 1 ××∇+      ××∇−∇⋅+= ∂ ∂ OR DEFINITION OF NEWTONIAN FLUID, NAVIER-STOKES EQUATION ( ) ( ) nnunuunnpT nTranslatio      1~11 2 1 121 ⋅=⋅∇+      ××∇−∇⋅+−≡ σλµ CONSERVATION OF LINEAR MOMENTUM EQUATIONS SOLO
  • 80. 80 FLUID DYNAMICS 2. BASIC LAWS IN FLUID DYNAMICS (CONTINUE) (2.6) CONSTITUTIVE RELATIONS (2.6.1) NAVIER–STOKES EQUATIONS (CONTINUE) (2.6.1.2)VECTORIAL DERIVATION (CONTINUE) I x y z T n= ⋅~σ d s n ds= r dru u + du CONSERVATION OF LINEAR MOMENTUM EQUATIONS ( ) ( ) ( ) ( ) ( ) ( )  ( ) ( ) ( ) ( ) ( ) ( ) ∫∫∫ ∫∫∫∫∫∫∫∫∫∫∫ ∫∫∫∫∫∫∫∫∫∫∫∫∫∫∫           ⋅∇∇+×∇×∇+∇⋅∇+∇−= =⋅∇+×∇×+∇⋅+−=       ⋅∇+      ××∇−∇⋅+⋅−=+= )( )()()()()( )()()()()()( 251 2 2 2 11 2 1 121 tV GAUSS tStStStStV tStStVtStVtV vd GAUSS u GAUSS u GAUSS u GAUSS pG usdusdusdsdpvdG sdnunuunsdnpvdGdsTvdGvd tD uD         λµµρ λµµρ λµρρρ BUT ( ) ( ) ( )∇× ∇× ≡ ∇ ∇⋅ − ∇⋅ ∇2 2 2µ µ µ    u u u ( ) ( ) ( ) ( )∇⋅ ∇ + ∇× ∇× = ∇ ∇⋅ − ∇× ∇×2 2µ µ µ µ     u u u u THEN SOLO
  • 81. 81 FLUID DYNAMICS 2. BASIC LAWS IN FLUID DYNAMICS (CONTINUE) I x y z T n= ⋅~σ d s n ds= r dru u + du THEREFORE ( ) ( ) ( ){ }∫∫∫∫∫∫ ⋅∇∇+×∇×∇−⋅∇∇+∇−= )()( 2 tVtV vduuupGvd tD uD  λµµρρ OR ( ) ( )[ ]uupG tD uD  ⋅∇+∇+×∇×∇−∇−= µλµρρ 2 SOLO (2.6) CONSTITUTIVE RELATIONS (2.6.1) NAVIER–STOKES EQUATIONS (CONTINUE) (2.6.1.2)VECTORIAL DERIVATION (CONTINUE)
  • 82. 82 FLUID DYNAMICS 2. BASIC LAWS IN FLUID DYNAMICS (CONTINUE) VECTOR NOTATION CARTESIAN TENSOR NOTATION CONSERVATION OF LINEAR MOMENTUM ( ) ( )[ ]∇ ⋅ = − ∇ − ∇ × ∇ × + ∇ + ∇ ⋅~σ µ µ λp u u   2 ( )       ++            ++−= k k ii k k i iii ik x u xx u x u xx p x ∂ ∂ λµ ∂ ∂ ∂ ∂ ∂ ∂ µ ∂ ∂ ∂ ∂ ∂ σ∂ 2 ( ) ( )[ ] ρ ρ σ ρ µ µ λ Du Dt G G p u u      = + ∇ ⋅ = − ∇ − ∇ × ∇ × + ∇ + ∇ ⋅ ~ 2 ( )       ++            ++−= += k k ii k k i ii i i ik i i x u xx u x u xx p G x G tD uD ∂ ∂ λµ ∂ ∂ ∂ ∂ ∂ ∂ µ ∂ ∂ ∂ ∂ ρ ∂ σ∂ ρρ 2 USING STOKES ASSUMPTION tr ~τ λ µ= ⇒ = −0 2 3 ( )     ⋅∇∇+×∇×∇−∇−= ⋅∇+= uupG G tD uD   µµρ σρρ 3 4 ~       +            ++−= += k k ki k k i ii i i ik i i x u i xx u x u xx p G x G tD uD ∂ ∂ µ ∂ ∂ ∂ ∂ ∂ ∂ µ ∂ ∂ ∂ ∂ ρ ∂ σ∂ ρρ 3 4 SOLO (2.6) CONSTITUTIVE RELATIONS (2.6.1) NAVIER–STOKES EQUATIONS (CONTINUE)
  • 83. 83 FLUID DYNAMICS 2. BASIC LAWS IN FLUID DYNAMICS (CONTINUE) VECTOR NOTATION CARTESIAN TENSOR NOTATION Euler Equations are obtained by assuming Inviscid Flow 0 3 2 0~ =−=⇒= µλτ pG tD uD ∇−=  ρρ i i i x p G tD uD ∂ ∂ ρρ −= SOLO (2.6) CONSTITUTIVE RELATIONS (2.6.2) EULER EQUATIONS Leonhard Euler (1707-1783) pGuu t u ∇−=      ∇⋅+ ∂ ∂   ρρ i i k i k i x p G x u u t u ∂ ∂ ρρ −=      ∂ ∂ + ∂ ∂ or or
  • 84. 84 FLUID DYNAMICS 2. BASIC LAWS IN FLUID DYNAMICS (CONTINUE) (2.6.1.3) COMPUTATION BUT Φ Φ = = +       = +             = = τ ∂ ∂ τ ∂ ∂ τ ∂ ∂ τ ∂ ∂ ∂ ∂ τ τ τ ik i k ik i k ki k i ik i k k i ik ik u x u x u x u x u x D ik ki1 2 1 2 τ µ λ δik ik kk ikD D= +2 HENCE ( )Φ = = +τ µ λ δik ik ik kk ik ikD D D D2 OR ( )[ ] ( )[ ] ( )[ ] ( ) Φ = + + + + + + + + + + + + + + + + + ⇒ = 2 2 2 2 11 11 22 33 11 22 11 22 33 22 33 11 22 33 33 12 2 21 2 13 2 31 2 23 2 32 2 µ λ µ λ µ λ µ D D D D D D D D D D D D D D D D D D D D D D Dij ji ( ) ( )Φ = + + + + + + + +2 2 2 211 2 22 2 33 2 12 2 13 2 23 2 11 22 33 2 µ λD D D D D D D D DOR SOLO (2.6) CONSTITUTIVE RELATIONS (2.6.1) NAVIER–STOKES EQUATIONS (CONTINUE)
  • 85. 85 FLUID DYNAMICS 2. BASIC LAWS IN FLUID DYNAMICS (CONTINUE) (2.6.1.3) COMPUTATION (CONTINUE) USING STOKES ASSUMPTION: tr ~τ λ µ= ⇒ = −0 2 3 Φ ( ) ( )Φ = + + + + + + + +2 2 2 211 2 22 2 33 2 12 2 13 2 23 2 11 22 33 2 µ λD D D D D D D D D ( ) ( ) ( ) ( ) ( )  ( ) Φ = + + − + + + + + + + + − + + + + 2 3 4 3 4 3 4 2 3 11 22 33 2 11 22 11 33 22 33 11 2 22 2 33 2 2 12 2 13 2 23 2 11 22 33 2 11 2 22 2 33 2 µ µ µ µ µ λ µ D D D D D D D D D D D D D D D D D D D D D    OR ( ) ( ) ( )[ ] ( )Φ = − + − + − + + + > 2 3 4 011 22 2 11 33 2 22 33 2 12 2 13 2 23 2µ µD D D D D D D D D SOLO (2.6) CONSTITUTIVE RELATIONS (2.6.1) NAVIER–STOKES EQUATIONS (CONTINUE)
  • 86. 86 FLUID DYNAMICS 2. BASIC LAWS IN FLUID DYNAMICS (CONTINUE) (2.6.1.4) ENTROPY AND VORTICITY From (C.L.M.) or ( ) ( )[ ]Du Dt u t u u u G p u u        = + ∇       − × ∇ × = − ∇ − ∇ × ∇ × + ∇ + ∇ ⋅ ∂ ∂ ρ ρ µ ρ λ µ 2 2 1 1 1 2 GIBBS EQUATION: T d s d h d p = − ρ       ∀      +⋅∇−      +⋅∇=      +⋅∇ →→→→ tld pd td t p ldp hd td t h ldh sd td t s ldsT & 1        ∂ ∂ ρ∂ ∂ ∂ ∂ Since this is true for all d l t → & T s h p T s t h t p t ∇ = ∇ − ∇ = − ρ ∂ ∂ ∂ ∂ ρ ∂ ∂ & 1 SOLO (2.6) CONSTITUTIVE RELATIONS (2.6.1) NAVIER–STOKES EQUATIONS (CONTINUE) Josiah Willard Gibbs (1903 – 1839)
  • 87. 87 FLUID DYNAMICS 2. BASIC LAWS IN FLUID DYNAMICS (CONTINUE) (2.6.1.4) ENTROPY AND VORTICITY from (C.L.M.) or GIBBS EQUATION: T d s d h d p = − ρ       ∀      +⋅∇−      +⋅∇=      +⋅∇ →→→→ tld pd td t p ldp hd td t h ldh sd td t s ldsT & 1        ∂ ∂ ρ∂ ∂ ∂ ∂ Since this is true for all d l t → & T s h p T s t h t p t ∇ = ∇ − ∇ = − ρ ∂ ∂ ∂ ∂ ρ ∂ ∂ & 1 SOLO hsTG p Guuu t u II III II I ,, ,,, , 2 , ~~ 2 1 ∇−∇+ ⋅∇ += ⋅∇ + ∇ −=         ×∇×−      ∇+ ρ τ ρ τ ρ∂ ∂   ρ p hsT dlpdp dlhdh dlsds ∇ −∇=∇→               ⋅∇= ⋅∇= ⋅∇=
  • 88. 88 Luigi Crocco 1909-1986 FLUID DYNAMICS 2. BASIC LAWS IN FLUID DYNAMICS (CONTINUE) (2.6.1.4) ENTROPY AND VORTICITY (CONTINUE) Define Let take the CURL of this equation Vorticityu  ×∇≡Ω If , then from (C.L.M.) we get:  G = −∇Ψ CRROCO’s EQUATION (1937)  ( ) ( )       ⋅∇×∇+      Ψ++∇×∇−∇×∇=×Ω×∇+×∇ Ω τ ρ∂ ∂ ~1 0 2 2      u hsTuu t SOLO ρ τ ∂ ∂ ~ 2 1 ,2 ,, ⋅∇ +      Ψ++∇−∇=×Ω+ I II I uhsTu t u  hsTGuuu t u II I II I ,, , , 2 , ~ 2 1 ∇−∇+ ⋅∇ +=         ×∇×−      ∇+ ρ τ ∂ ∂   From
  • 89. 89 FLUID DYNAMICS 2. BASIC LAWS IN FLUID DYNAMICS (CONTINUE) (2.6.1.4) ENTROPY AND VORTICITY (CONTINUE) ( ) ( ) ( ) ( ) ( )∇ × × = ⋅∇ − ∇ ⋅ + ∇ ⋅ − ⋅∇ ← ∇ ⋅ = ∇ ⋅∇ × =               Ω Ω Ω Ω Ω Ωu u u u u u 0 0 ( )∇ × ∇ = ∇ × ∇T s T s τ ρ τ ρ τ ρ ~ 0 1~1~1 ⋅∇×∇+⋅∇×      ∇=       ⋅∇×∇  Therefore ( ) ( ) ( ) τ ρ∂ ∂ ~1 ⋅∇×      ∇−∇×∇=∇⋅Ω−Ω⋅∇+Ω∇⋅+ Ω sTuuu t   SOLO ( ) ( ) τ ρ ~1 ⋅∇×      ∇−∇×∇+⋅∇Ω−∇⋅Ω= Ω sTuu tD D   or
  • 90. 90 FLUID DYNAMICS 2. BASIC LAWS IN FLUID DYNAMICS (CONTINUE) (2.6) CONSTITUTIVE RELATIONS (2.6.1) NAVIER–STOKES EQUATIONS (CONTINUE) (2.6.1.4) ENTROPY AND VORTICITY (CONTINUE) ( ) ( ) τ ρ ~1 ⋅∇×      ∇−∇×∇+⋅∇Ω−∇⋅Ω= Ω sTuu tD D   FLUID WITHOUT VORTICITY WILL REMAIN FOREVER WITHOUT VORTICITY IN ABSENSE OF ENTROPY GRADIENTS OR VISCOUS FORCES - FOR AN INVISCID FLUID ( )λ µ τ= = → =0 0~ ~ ( ) ( ) sTuu tD D INVISCID ∇×∇+⋅∇Ω−∇⋅Ω= Ω =   0 ~~τ - FOR AN HOMENTROPIC FLUID INITIALLY AT REST s const everywhere i e s s t . ; . . &∇ = =      0 0 ∂ ∂( )( )   Ω 0 0= ( ) D Dt s     Ω Ω= = = ∇ =0 0 0 0 0~ ~ , ,τ SOLO Return to Table of Content
  • 91. 91 FLUID DYNAMICS 2. BASIC LAWS IN FLUID DYNAMICS (CONTINUE) (2.6) CONSTITUTIVE RELATIONS (2.6.2) STATE EQUATION p - PRESSURE (FORCE / SURFACE) V - VOLUME OF GAS M - MASS OF GAS R - 8314 - 286.9 T - GAS TEMPERATURE - GAS DENSITY [ ]m3 [ ]kg [ ]J kg mol Ko / ( )⋅ [ ]J kg Ko / ( )⋅R [ ]kgmol /−η [ ]o K [ ]kg m/ 3 ρ [ ]2 / mN IDEAL GAS TRMVp η= TMVp R= DEFINE: ρ ρ = = = ∆ ∆M V v V M & 1 pv T= R p T= ρ R OR SOLO
  • 92. 92 FLUID DYNAMICS 2. BASIC LAWS IN FLUID DYNAMICS (CONTINUE) IDEAL GAS TMVp R= SOLO (2.6) CONSTITUTIVE RELATIONS (2.6.2) STATE EQUATION Return to Table of Content
  • 93. 93 FLUID DYNAMICS 2. BASIC LAWS IN FLUID DYNAMICS (CONTINUE) (2.6) CONSTITUTIVE RELATIONS (2.6.3) THERMALLY PERFECT GAS AND CALORICALLLY PERFECT GAS A THERMALLY PERFECT GAS IS DEFINED AS A GAS FOR WHICH THE INTERNAL ENERGY e IS A FUNCTION ONLY OF THE TEMPERATURE T. ( ) ( )h e T p e T RT h T= + = + =/ ( )ρ THERMALLY PERFECT GAS DEFINE C C v V V p p p p p e T q T h T de pdv v d p d T de pdv d T dq d T = = = = = =                   + +      +            ∆ ∆ ∂ ∂ ∂ ∂ ∂ ∂ A CALORICALLY PERFECT GAS IS DEFINED AS A GAS FOR WHICH Cv IS CONSTANT CALORICALLY PERFECT GASe C Tv= SOLO
  • 94. 94 FLUID DYNAMICS 2. BASIC LAWS IN FLUID DYNAMICS (CONTINUE) (2.6) CONSTITUTIVE RELATIONS (2.6.3) CALORICALLLY PERFECT GAS (CONTINUE) A CALORICALLY PERFECT GAS IS DEFINED AS A GAS FOR WHICH Cv IS CONSTANT CALORICALLY PERFECT GASe C Tv= FOR A CALORICALLY PERFECT GAS ( )h C T RT C R T C T C C Rv v p p v= + = + = → = + γ γ γ γ = ⇒ = − ⇒ = − = + = −∆ C C C R C Rp v C C R p R C C v p v p v 1 1 γ air = 14. SOLO
  • 95. 95 FLUID DYNAMICS 2. BASIC LAWS IN FLUID DYNAMICS (CONTINUE) (2.6) CONSTITUTIVE RELATIONS (2.6.3) CALORICALLLY PERFECT GAS (CONTINUE) (2.6.3.1) ENTROPY CALCULATIONS FOR A CALORICALLLY PERFECT GAS pv T= R p T= ρ R IDEAL GAS ( ) ds de pdv T de pdv vdp vdp T dh vdp T = + = + + − = −∆ ds C dT T R dv v s s C T T R v v C T T Rv v v= + → − = + = −2 1 2 1 2 1 2 1 2 1 ln ln ln ln ρ ρ 1 2 1 2 12 lnln p p R T T Css p dp R T dT Cds pp −=−→−= s s C p p R C p p Cv v p2 1 2 1 1 2 2 1 2 1 2 1 − = ⋅       − = −ln ln ln ln ρ ρ ρ ρ ρ ρ ENTROPY SOLO
  • 96. 96 FLUID DYNAMICS 2. BASIC LAWS IN FLUID DYNAMICS (CONTINUE) (2.6) CONSTITUTIVE RELATIONS (2.6.3) CALORICALLLY PERFECT GAS (CONTINUE) (2.6.3.1) ENTROPY CALCULATIONS FOR A CALORICALLLY PERFECT GAS p p T T e T T e p p T T C R s s R s s R isentropic s s p 2 1 2 1 2 1 1 2 1 2 1 12 1 2 1 2 1 =       =       =       − − − − − = − ⇒ γ γ γ γ ρ ρ ρ ρ γ γ γ 2 1 2 1 2 1 1 1 2 1 2 1 1 12 1 2 1 2 1 =       =       =       − − − − − = − ⇒ T T e T T e T T C R s s R s s R isentropic s s v p p e e p p C C s s R s s R isentropic s s p v 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 =       =       =       − − − − = ⇒ ρ ρ ρ ρ ρ ρ γ γ T T h h p p e p p e T T h h p p s s C s s C isentropic s s v p2 1 2 1 2 1 2 1 2 1 1 2 1 1 2 1 2 1 2 1 1 2 1 12 1 2 1 2 1 = = ⋅ =       =       = =       =       − − − − − − = − − ⇒ ρ ρ ρ ρ ρ ρ γ γ γ γ γ γ ISENTROPIC CHAIN SOLO Return to Table of Content
  • 97. 97 FLUID DYNAMICS BASIC LAWS IN FLUID DYNAMICS (CONTINUE) BOUNDARY CONDITIONS SOLO Return to Table of Content
  • 98. 98 SOLO Dimensionless Equations Dimensionless Variables are: 0/~ ρρρ = 0/ ~ Uuu = gGG / ~ = ( )2 00/~ Upp ρ= 0/~ lUtt = 2 0/ ~ UCTT p=( )2 00/~ Uρττ = 2 0/ ~ UHH = 2 0/ ~ Uhh = 2 0/~ Uee = ( )2 00/~ Uqq ρ= ( )2 / ~ UQQ = ∇=∇ 0 ~ l Field Equations (C.M.): ( ) 00 0 0 U l u t ρ ρ ∂ ρ∂ =⋅∇+  ( ) 2 00 0 ~ 3 4 U l uupGuu t u ρ µµρ ∂ ∂ ρ τ      ⋅∇       ⋅∇∇+×∇×∇−∇−=      ∇⋅+(C.L.M.): ( ) ( ) 3 00 0~ U l Tk t Q uGu t p Hu t H q ρ∂ ∂ ρτ ∂ ∂ ρ    ∇⋅∇−+⋅+⋅⋅∇+=      ∇⋅+ ∂ ∂ (C.E.): ( ) ( ) ( ) 0 / / 00 0 00 0 =      ⋅∇+ U u l lUt  ρ ρ ∂ ρρ∂ ( ) ( ) ( ) ( ) ( )       ⋅∇∇      +      ×∇×∇      − ∇−=       ∇⋅+ 0 00 000 0 0 0 0 0 000 0 2 00 02 0 0 00 0 000 0 0 3 4 / / U u ll UlU u ll Ul U p l g G U lg U u l U u lUt Uu   ρ µ µ µ ρ µ ρρ ρ ∂ ∂ ρ ρ ( ) ( ) ( ) ( ) ( ) ( )         ∇⋅∇               −+⋅+        ⋅⋅∇+        ∂ ∂ =        2 0 0 0 0 0 0 000 0 2 00000 2 0 0 0 2 00 02 0000 2 00000 / ~ // U CT l k k l C k UlU Q lUtU u g G U gl U u U l U p lUtU H lUtD D p pµρ µ ∂ ∂ ρ ρ ρ τ ρρρ ρ  0/~ ρρρ = 0/ ~ Uuu = gGG / ~ = ( )2 00/~ Upp ρ= 0/~ lUtt = 2 0/ ~ UCTT p=( )2 00/~ Uρττ = 2 0/ ~ UHH = 2 0/ ~ Uhh = 2 0/~ Uee = ( )2 00/~ Uqq ρ= ( )2 / ~ UQQ = ∇=∇ 0 ~ l 0/~ µµµ = 0/ ~ kkk = Reference Quantities: ρ0(density), U0(velocity), l0 (length), g (gravity), μ0 (viscosity), k0 (Fourier Constant), λ0 (mean free path) 0/ ~ λλλ =
  • 99. 99 SOLO Dimensionless Equations Dimentionless Field Equations (C.M.): ( ) 0 ~~~~ =⋅∇+ u t  ρ ∂ ρ∂ ( ) ( )u R u R pG F uu t u eer ~~~~1 3 4~~~~1~~~~1~~~ ~ ~ ~ 2   ⋅∇∇+×∇×∇−∇−=        ∇⋅+ µµρ ∂ ∂ ρ(C.L.M.): ( ) ( )Tk PRt Q uG F u t p Hu t H rer ∇⋅∇−+⋅+⋅⋅∇+=        ∇⋅+ ∂ ∂ 11 ~ ~ ~~~1~~~ ~ ~~~~ ~ ~ ~ 2 ∂ ∂ ρτ ∂ ∂ ρ  (C.E.): Reynolds: 0 000 µ ρ lU Re = Prandtl: 0 0 k C P p r µ = Froude: 0 0 gl U Fr = 0/~ ρρρ = 0/ ~ Uuu = gGG / ~ = ( )2 00/~ Upp ρ= 0/~ lUtt = 2 0/ ~ UCTT p=( )2 00/~ Uρττ = 2 0/ ~ UHH = 2 0/ ~ Uhh = 2 0/~ Uee = ( )2 00/~ Uqq ρ= ( )2 / ~ UQQ = ∇=∇ 0 ~ l 0/~ ρρρ = 0/ ~ Uuu = gGG / ~ = ( )2 00/~ Upp ρ= 0/~ lUtt = 2 0/ ~ UCTT p=( )2 00/~ Uρττ = 2 0/ ~ UHH = 2 0/ ~ Uhh = 2 0/~ Uee = ( )2 00/~ Uqq ρ= ( )2 / ~ UQQ = ∇=∇ 0 ~ l 0/~ µµµ = 0/ ~ kkk = Dimensionless Variables are: Reference Quantities: ρ0(density), U0(velocity), l0 (length), g (gravity), μ0 (viscosity), k0 (Fourier Constant), λ0 (mean free path) 0/ ~ λλλ = Knudsen l Kn 0 0 : λ =
  • 100. 100 SOLO Dimensionless Equations Constitutive Relations TRp ρ= 2 2 1 uTCH p += Tkq ∇−=  TCh p=      − == 2 00 2 00 2 00 1 U TC U TC C R U p pp p ρ ρ γ γ ρ ρ ρ       =      2 0 2 0 U TC U h p 2 0 2 0 2 0 2 1       +      =      U u U TC U H p ( )       ∇               −= 2 0 0 00 0 000 0 3 00 U TC l k k C k UlU q p p µρ µ ρ  ( ) [ ]3 3 2~ Iuuu T  ⋅∇−∇+∇= µµτ [ ]3 0 0 0000 0 0 0 0 0 0000 0 00 3 2~ I U u l UlU u l U u l UlU T  ⋅∇      −      ∇+∇      = µ µ ρ µ µ µ ρ µ ρ τ 0/~ ρρρ = 0/ ~ Uuu = gGG / ~ = ( )2 00/~ Upp ρ= 0/~ lUtt = 2 0/ ~ UCTT p=( )2 00/~ Uρττ = 2 0/ ~ UHH = 2 0/ ~ Uhh = 2 0/~ Uee = ( )2 00/~ Uqq ρ= ( )2 / ~ UQQ = ∇=∇ 0 ~ l 0/~ ρρρ = 0/ ~ Uuu = gGG / ~ = ( )2 00/~ Upp ρ= 0/~ lUtt = 2 0/ ~ UCTT p=( )2 00/~ Uρττ = 2 0/ ~ UHH = 2 0/ ~ Uhh = 2 0/~ Uee = ( )2 00/~ Uqq ρ= ( )2 / ~ UQQ = ∇=∇ 0 ~ l 0/~ µµµ = 0/ ~ kkk = Dimensionless Variables are: Reference Quantities: ρ0(density), U0(velocity), l0 (length), g (gravity), μ0 (viscosity), k0 (Fourier Constant), λ0 (mean free path) 0/ ~ λλλ =
  • 101. 101 SOLO Dimensionless Equations Dimensionless Constitutive Relations 2~ 2 1~~ uTH += Tp ~~1~ ρ γ γ − = Ideal Gas ( ) [ ]3 ~~~ 3 2~~~~~~~ Iu R uu R e T e  ⋅∇−∇+∇= µµ τ Navier-Stokes Th ~~ = Calorically Perfect Gas Tk PR q re ~~~11~ ∇−=  Fourier Law Reynolds: 0 000 µ ρ lU Re = Prandtl: 0 0 k C P p r µ = 0/~ ρρρ = 0/ ~ Uuu = gGG / ~ = ( )2 00/~ Upp ρ= 0/~ lUtt = 2 0/ ~ UCTT p=( )2 00/~ Uρττ = 2 0/ ~ UHH = 2 0/ ~ Uhh = 2 0/~ Uee = ( )2 00/~ Uqq ρ= ( )2 / ~ UQQ = ∇=∇ 0 ~ l 0/~ ρρρ = 0/ ~ Uuu = gGG / ~ = ( )2 00/~ Upp ρ= 0/~ lUtt = 2 0/ ~ UCTT p=( )2 00/~ Uρττ = 2 0/ ~ UHH = 2 0/ ~ Uhh = 2 0/~ Uee = ( )2 00/~ Uqq ρ= ( )2 / ~ UQQ = ∇=∇ 0 ~ l 0/~ µµµ = 0/ ~ kkk = Dimensionless Variables are: Reference Quantities: ρ0(density), U0(velocity), l0 (length), g (gravity), μ0 (viscosity), k0 (Fourier Constant), λ0 (mean free path) 0/ ~ λλλ = Return to Table of Content
  • 102. 102 SOLO Mach Number Mach number (M or Ma) / is a dimensionless quantity representing the ratio of speed of an object moving through a fluid and the local speed of sound. • M is the Mach number, • U0 is the velocity of the source relative to the medium, and • a0 is the speed of sound Mach: 0 0 a U M = The Mach number is named after Austrian physicist and philosopher Ernst Mach, a designation proposed by aeronautical engineer Jakob Ackeret. Ernst Mach (1838–1916) Jakob Ackeret (1898–1981) m Tk Mo TR a Bγγ ==0 • R is the Universal gas constant, (in SI, 8.314 47215 J K−1 mol−1 ), [M1 L2 T−2 θ−1 'mol'−1 ] • γ is the rate of specific heat constants Cp/Cv and is dimensionless γair = 1.4. • T is the thermodynamic temperature [θ1 ] • Mo is the molar mass, [M1 'mol'−1 ] • m is the molecular mass, [M1 ] AERODYNAMICS
  • 103. 103 SOLO Mach Number – Flow Regimes Regime Mach mph km/h m/s General plane characteristics Subsonic <0.8 <610 <980 <270 Most often propeller-driven and commercial turbofan aircraft with high aspect-ratio (slender) wings, and rounded features like the nose and leading edges. Transonic 0.8-1.2 610- 915 980-1,470 270-410 Transonic aircraft nearly always have swept wings, delaying drag- divergence, and often feature design adhering to the principles of the Whitcomb Area rule. Supersonic 1.2–5.0 915- 3,840 1,470– 6,150 410–1,710 Aircraft designed to fly at supersonic speeds show large differences in their aerodynamic design because of the radical differences in the behaviour of flows above Mach 1. Sharp edges, thin aerofoil- sections, and all-moving tailplane/canards are common. Modern combat aircraft must compromise in order to maintain low-speed handling; "true" supersonic designs include the F-104 Starfighter, SR-71 Blackbird and BAC/Aérospatiale Concorde. Hypersonic 5.0–10.0 3,840– 7,680 6,150– 12,300 1,710– 3,415 Cooled nickel-titanium skin; highly integrated (due to domination of interference effects: non-linear behaviour means that superposition of results for separate components is invalid), small wings, such as those on the X-51A Waverider High- hypersonic 10.0–25.0 7,680– 16,250 12,300– 30,740 3,415– 8,465 Thermal control becomes a dominant design consideration. Structure must either be designed to operate hot, or be protected by special silicate tiles or similar. Chemically reacting flow can also cause corrosion of the vehicle's skin, with free-atomic oxygen featuring in very high-speed flows. Hypersonic designs are often forced into blunt configurations because of the aerodynamic heating rising with a reduced radius of curvature. Re-entry speeds >25.0 >16,25 0 >30,740 >8,465 Ablative heat shield; small or no wings; blunt shape
  • 104. 104 SOLO Different Regimes of Flow Mach Number – Flow Regimes AERODYNAMICS
  • 105. 105 where ρ = air density V = true speed l = characteristic length μ = absolute (dynamic) viscosity υ = kinematic viscosity Reynolds: υµ ρ ρ µ υ lVlV Re = == Osborne Reynolds (1842 –1912) It was observed by Reynolds in 1884 that a Fluid Flow changes from Laminar to Turbulent at approximately the same value of the dimensionless ratio (ρ V l/ μ) where l is the Characteristic Length for the object in the Flow. This ratio is called the Reynolds number, and is the governing parameter for Viscous Flow. Reynolds Number and Boundary Layer SOLO 1884AERODYNAMICS
  • 106. 106 Boundary Layer SOLO 1904AERODYNAMICS Ludwig Prandtl (1875 – 1953) In 1904 at the Third Mathematical Congress, held at Heidelberg, Germany, Ludwig Prandtl (29 years old) introduced the concept of Boundary Layer. He theorized that the fluid friction was the cause of the fluid adjacent to surface to stick to surface – no slip condition, zero local velocity, at the surface – and the frictional effects were experienced only in the boundary layer a thin region near the surface. Outside the boundary layer the flow may be considered as inviscid (frictionless) flow. In the Boundary Layer on can calculate the •Boundary Layer width •Dynamic friction coefficient μ •Friction Drag Coefficient CDf
  • 107. 107 The flow within the Boundary Layer can be of two types: •The first one is Laminar Flow, consists of layers of flow sliding one over other in a regular fashion without mixing. •The second one is called Turbulent Flow and consists of particles of flow that moves in a random and irregular fashion with no clear individual path, In specifying the velocity profile within a Boundary Layer, one must look at the mean velocity distribution measured over a long period of time. There is usually a transition region between these two types of Boundary-Layer Flow SOLO AERODYNAMICS
  • 108. 108 Normalized Velocity profiles within a Boundary-Layer, comparison between Laminar and Turbulent Flow. SOLO Boundary-Layer AERODYNAMICS
  • 109. 109 Flow Characteristics around a Cylindrical Body as a Function of Reynolds Number (Viscosity) AERODYNAMICS SOLO
  • 110. 110 Relative Drag Force as a Function of Reynolds Number (Viscosity) AERODYNAMICS Drag CD0 due to Flow Separation SOLO
  • 111. 111 Relative Drag Force as a Function of Reynolds Number (Viscosity) AERODYNAMICS Drag due to Viscosity: 1.Skin Friction 2.Flow Separation (Drop in pressure behind body) ∫∫ ∫∫         ⋅+⋅ − −=         ⋅+⋅−= ∧∧ ∞ ∧∧ W W S S fpD ds w t V f w n V pp S ds w tC w nC S C xx xx 11 11 ˆ 2/ ˆ 2/ 1 ˆˆ 1 22 ρρ SOLO
  • 112. 112 Parasite Drag Pressure Differential, Viscous Shear Stress, and Separation AERODYNAMICS Relative Drag Force as a Function of Reynolds Number (Viscosity)       DragFrictionSkin ET EL ll ET EL uu DragPressure ET EL ll ET EL uu sdfsdf sdpsdpD ∫∫ ∫∫ ++ +−= .. .. .. .. .. .. .. .. coscos sinsin θθ θθ SOLO
  • 113. 113 AERODYNAMICS Relative Drag Force as a Function of Reynolds Number (Viscosity) • Blunt Body: Most of Drag is Pressure Drag. • Streamlined Body: Most of Drag is Skin Friction Drag. SOLO
  • 114. 114 AERODYNAMICS Relative Drag Force as a Function of Reynolds Number (Viscosity) SOLO
  • 115. 115 AERODYNAMICS Relative Drag Force as a Function of Reynolds Number (Viscosity) SOLO
  • 116. 116 AERODYNAMICS Relative Drag Force as a Function of Reynolds Number (Viscosity) SOLO Variation of total skin-friction coefficient with Reynolds number for a smooth, flat plate.[From Dommasch, et al. (1967).]
  • 117. 117 Typical Effect of Reynolds Number on Parasitic Drag Flow may stay attached farther at high Re, reducing the drag AERODYNAMICSSOLO Return to Table of Content
  • 118. 118 FluidsSOLO Knudsen number (Kn) is a dimensionless number defined as the ratio of the molecular mean free path length to a representative physical length scale. This length scale could be, for example, the radius of the body in a fluid. The number is named after Danish physicist Martin Knudsen. Knudsen l Kn 0 0 : λ = Martin Knudsen (1871–1949). For a Boltzmann gas, the mean free path may be readily calculated as: • kB is the Boltzmann constant (1.3806504(24) × 10−23 J/K in SI units), [M1 L2 T−2 θ−1 ] p TkB 20 2 σπ λ = • T is the thermodynamic temperature [θ1 ] λ0 = mean free path [L1 ] Knudsen Number l0 = representative physical length scale [L1 ]. • σ is the particle hard shell diameter, [L1 ] • p is the total pressure, [M1 L−1 T−2 ]. See “Kinetic Theory of Gases” Presentation For particle dynamics in the atmosphere and assuming standard atmosphere pressure i.e. 25 °C and 1 atm, we have λ0 ≈ 8x10-8 m.
  • 119. 119 FluidsSOLO Martin Knudsen (1871–1949). Knudsen Number (continue – 1) Relationship to Mach and Reynolds numbers Dynamic viscosity, Average molecule speed (from Maxwell–Boltzmann distribution), thus the mean free path, where • kB is the Boltzmann constant (1.3806504(24) × 10−23 J/K in SI units), [M1 L2 T−2 θ−1 ] • T is the thermodynamic temperature [θ1 ] • ĉ is the average molecular speed from the Maxwell–Boltzmann distribution, [L1 T−1 ] • μ is the dynamic viscosity, [M1 L−1 T−1 ] • m is the molecular mass, [M1 ] • ρ is the density, [M1 L−3 ]. 0 2 1 λρµ c= m Tk c B π 8 = Tk m B2 0 π ρ µ λ =
  • 120. 120 FluidsSOLO Martin Knudsen (1871–1949). Knudsen Number (continue – 2) Relationship to Mach and Reynolds numbers (continue – 1) The dimensionless Reynolds number can be written: Dividing the Mach number by the Reynolds number, and by multiplying by yields the Knudsen number. The Mach, Reynolds and Knudsen numbers are therefore related by: Reynolds:Re 0 000 µ ρ lU = Tk m lmTklallU aUM BB γρ µ γρ µ ρ µ µρ 00 0 00 0 000 0 0000 00 // / Re ==== Kn Tk m lTk m l BB == 22 00 0 00 0 π ρ µπγ γρ µ 2Re πγM Kn =
  • 121. 121 FluidsSOLO Knudsen Number (continue – 3) Relationship to Mach and Reynolds numbers (continue –2) According to the Knudsen Number the Gas Flow can be divided in three regions: 1.Free Molecular Flow (Kn >> 1): M/Re > 3 molecule-interface interaction negligible between incident and reflected particles 2.Transition (from molecular to continuum flow) regime: 3 > M/Re and M/(Re)1/2 > 0.01 (Re >> 1). Both intermolecular and molecule-surface collision are important. 3.Continuum Flow (Kn << 1): 0.01 > M/(Re)1/2 . Dominated by intermolecular collisions. 2Re πγM Kn =
  • 122. FluidsSOLO Knudsen Number (continue – 4) Inviscid Limit Free Molecular LimitKnudsen Number Boltzman Equation Collisionless Boltzman Equation Discrete Particle model Euler Equation Navier-Stokes Equation Continuum model Conservation Equation do not form a closed set Validity of conventional mathematical models as a function of local Knudsen Number Return to Table of Content
  • 123. 123 AERODYNAMICS Fluid flow is characterized by a velocity vector field in three-dimensional space, within the framework of continuum mechanics. Streamlines, Streaklines and Pathlines are field lines resulting from this vector field description of the flow. They differ only when the flow changes with time: that is, when the flow is not steady. • Streamlines are a family of curves that are instantaneously tangent to the velocity vector of the flow. These show the direction a fluid element will travel in at any point in time. • Streaklines are the locus of points of all the fluid particles that have passed continuously through a particular spatial point in the past. Dye steadily injected into the fluid at a fixed point extends along a streakline • Pathlines are the trajectories that individual fluid particles follow. These can be thought of as a "recording" of the path a fluid element in the flow takes over a certain period. The direction the path takes will be determined by the streamlines of the fluid at each moment in time. • Timelines are the lines formed by a set of fluid particles that were marked at a previous instant in time, creating a line or a curve that is displaced in time as the particles move. The red particle moves in a flowing fluid; its pathline is traced in red; the tip of the trail of blue ink released from the origin follows the particle, but unlike the static pathline (which records the earlier motion of the dot), ink released after the red dot departs continues to move up with the flow. (This is a streakline.) The dashed lines represent contours of the velocity field (streamlines), showing the motion of the whole field at the same time. (See high resolution version. Flow Description SOLO
  • 124. 124 3-D Flow Flow Description SOLO Steady Motion: If at various points of the flow field quantities (velocity, density, pressure) associated with the fluid flow remain unchanged with time, the motion is said to be steady. ( ) ( ) ( )zyxppzyxzyxuu ,,,,,,,, === ρρ  Unsteady Motion: If at various points of the flow field quantities (velocity, density, pressure) associated with the fluid flow change with time, the motion is said to be unsteady. ( ) ( ) ( )tzyxpptzyxtzyxuu ,,,,,,,,,,, === ρρ  Path Line: The curve described in space by a moving fluid element is known as its trajectory or path line. tt tt ∆+ t tt ∆+ tt ∆+ 2 t tt ∆+ tt ∆+ 2 Path Line (steady flow) t tt ∆+ t tt ∆+ 2 tt ∆+ t Path Line (unsteady flow) tt ∆+ 2 tt ∆+ t
  • 125. 125 3-D Flow Flow Description SOLO Path Line: The curve described in space by a moving fluid element is known as its trajectory or path line. t tt ∆+ tt ∆+ 2 Streamlines: The family of curves such that each curve is tangent at each point to the velocity direction at that point are called streamlines. Consider the coordinate of a point P and the direction of the streamline passing through this point. If is the velocity vector of the flow passing through P at a time t, then and parallel, or: r  rd u  u  rd 0=×urd  ( ) ( ) ( ) 0 1 1 1111 =             − − − = zdyudxv ydxwdzu xdzvdyw wvu dzdydx zyx w zd v yd u xd == Cartesian t u  r  rd
  • 126. 126 3-D Flow Flow Description SOLO Path Line: The curve described in space by a moving fluid element is known as its trajectory or path line. Streamlines: The family of curves such that each curve is tangent at each point to the velocity direction at that point are called streamlines. ( ) ( ) ( )tzyxw zd tzyxv yd tzyxu xd ,,,,,,,,, == t u  r  rd Those are two independent differential equations for a streamline. Given a point the streamline is defined from those equations.( )0000 ,,, tzyxr  ( ) ( ) ( ) ( ) ( ) ( )tzyxw zd tzyxv yd tzyxv yd tzyxu xd ,,,,,, 2 ,,,,,, 1 = = ( ) ( ) ( ) ( ) ( ) ( ) 0,,,,,,,,, 0,,,,,,,,, 222 111 =++ =++ zdtzyxcydtzyxbxdtzyxa zdtzyxcydtzyxbxdtzyxa ( ) ( ) ( ) ( )21 21 22 11 •+• •+• βα βα 0 22 11 ≠ βα βα Pfaffian Differential Equations For a given a point the solution of those equations is of the form:( )0000 ,,, tzyxr  ( ) ( ) 2,,, 1,,, 02 01 consttzyx consttzyx = = ψ ψ u  ( )0 tr  rd 0t ( ) 11 cr =  ψ ( ) 22 cr =  ψ Streamline Those are two surfaces, the intersection of which is the streamline.
  • 127. 127 3-D Flow Flow Description SOLO Path Line: The curve described in space by a moving fluid element is known as its trajectory or path line. Streamlines: The family of curves such that each curve is tangent at each point to the velocity direction at that point are called streamlines. ( ) ( ) ( )tzyxw zd tzyxv yd tzyxu xd ,,,,,,,,, == t u  r  rd For a given a point the solution of those equations is of the form:( )0000 ,,, tzyxr  ( ) ( ) 2,,, 1,,, 02 01 consttzyx consttzyx = = ψ ψ u  ( )0 tr  rd 0 t ( ) 11 cr =  ψ ( ) 22 cr =  ψ Streamline Those are two surfaces, the intersection of which is the streamline. The streamline is perpendicular to the gradients (normals) of those two surfaces. ( ) ( ) ( )0201 ,, trtrVr  ψψµ ∇×∇= where μ is a factor that must satisfy the following constraint. ( )( ) ( ) ( ) 0,, 0201 =∇×∇⋅∇=⋅∇ trtrVr  ψψµ Return to Table of Content
  • 128. 128 AERODYNAMICS Streamlines, Streaklines, and Pathlines Mathematical description Streamlines If the components of the velocity are written and those of the streamline as we deduce which shows that the curves are parallel to the velocity vector Pathlines Streaklines where, is the velocity of a particle P at location and time t . The parameter , parametrizes the streakline and 0 ≤ τP ≤ t0 , where t0 is a time of interest . The suffix P indicates that we are following the motion of a fluid particle. Note that at point the curve is parallel to the flow velocity vector where the velocity vector is evaluated at the position of the particle at that time t . SOLO
  • 129. 129 ∞V Airfoil Pressure Field variation with α AERODYNAMICS Airfoil Velocity Field variation with αAirfoil Streamline variation with αAirfoil Streakline with α Streamlines, Streaklines, and Pathlines SOLO
  • 133. 133 AERODYNAMICS Streamlines, Streaklines, and Pathlines SOLO Return to Table of Content
  • 134. 134 3-D Inviscid Incompressible Flow Circulation SOLO Circulation Definition: tV ∆  ( ) tVV ∆∆+  S∆ Sn ∆1 V  ×∇ t r  ∆ ttr ∆+∆  t C tt C ∆+ ∫ ⋅=Γ C rdV  : Material Derivative of the Circulation ( )∫∫∫ ⋅+⋅=         ⋅= Γ CCC rd tD D Vrd tD VD rdV tD D tD D    From the Figure we can see that: ( ) tVrtVVr ttt ∆+∆=∆∆++∆ ∆+  ( ) Vdrd tD D V t rr t ttt  =→∆= ∆ ∆−∆ →∆ ∆+ 0 ( ) 0 2 2 =      =⋅=⋅ ∫∫∫ CCC V dVdVrd tD D V  Therefore: ∫ ⋅= Γ C rd tD VD tD D  integral of an exact differential on a closed curve. C – a closed curve
  • 135. 135 3-D Inviscid Incompressible FlowSOLO tV ∆  ( ) tVV ∆∆+  S∆ Sn ∆1 V  ×∇ t r  ∆ ttr ∆+∆  t C tt C ∆+ S ∫ ⋅=Γ tC rdV  : Material Derivative of the Circulation (second derivation) Subtract those equations: tVrdSn t ∆×=∆  1 ( )∫∆+ ⋅∆+=Γ∆+Γ ttC rdVV  : ( ) ( )∫∫∫∫ ∆⋅×∇=⋅∆+−⋅=Γ∆− ∆+ S TheoremsStoke CC SnVrdVVrdV ttt 1 '  S is the surface bounded by the curves Ct and C t+Δ t ( ) ( ) ( ) tVVrdtVrdVSnV S t S t S ∆         ×∇×⋅=∆×⋅×∇=∆⋅×∇=Γ∆− ∫∫∫∫∫∫  1 td d ttd rd t V ttD D rdd Γ + ∂ Γ∂ =Γ∇⋅+ ∂ Γ∂ =Γ∇⋅+ ∂ Γ∂ = Γ Γ∇⋅=Γ Computation of: ∫ ⋅ ∂ ∂ = ∂ Γ∂ tC rd t V t  Computation of: td d Γ
  • 136. 136 3-D Inviscid Incompressible FlowSOLO tV ∆  ( ) tVV ∆∆+  S∆ Sn ∆1 V  ×∇ t r  ∆ tt r ∆+ ∆  t C tt C ∆+ Material Derivative of the Circulation (second derivation) ( ) tVVrd S t ∆         ×∇×⋅=Γ∆− ∫∫  When Δ t → 0 the surface S shrinks to the curve C=Ct and the surface integral transforms to a curvilinear integral: ( ) ( ) ( )∫∫∫∫∫ ∇⋅⋅+      −=∇⋅⋅+      ∇⋅−=×∇×⋅−= Γ C t CC t C t C t VVrd V dVVrd V rdVVrd td d    0 22 22 Computation of: (continue) td d Γ Finally we obtain: ( ) ∫∫∫ ⋅=∇⋅⋅+⋅ ∂ ∂ = Γ + ∂ Γ∂ = Γ tt CC t C rd tD VD VVrdrd t V td d ttD D   
  • 137. 137 3-D Inviscid Incompressible FlowSOLO tV ∆  ( ) tVV ∆∆+  S∆ Sn ∆1 V  ×∇ t r  ∆ tt r ∆+ ∆  t C tt C ∆+ Material Derivative of the Circulation We obtained: ∫ ⋅= Γ tC rd tD VD tD D  Use C.L.M.: hsT p VV t V tD VD II I G II II ,, , ,, ~ ∇−∇+ ⋅∇ +Ψ∇=         ∇⋅+= τ ∂ ∂     ( ) ( )  0 , ,, , , ~~ ∫∫∫∫ −Ψ+⋅      ⋅∇ +∇=⋅∇−Ψ∇+⋅      ⋅∇ +∇= Γ tttt CC I I C I C I I I hddrd p sTrdhrd p sT tD D ττ to obtain: ∫ ⋅      ⋅∇ +∇= Γ tC I I I rd p sT tD D τ~ , , or: Kelvin’s Theorem William Thomson Lord Kelvin (1824-1907) In an inviscid , isentropic flow d s = 0 with conservative body forces the circulation Γ around a closed fluid line remains constant with respect to time. 0 ~~ =τ Ψ∇=G Return to Table of Content 1869
  • 138. 138 3-D Inviscid Incompressible FlowSOLO Circulation Definition: ∫ ⋅=Γ C rdV  : C – a closed curve Biot-Savart Formula 1820 Jean-Baptiste Biot 1774 - 1862 VorticityV  ×∇≡Ω ∫ − Ω = Space dV sr A    π4 1 ( )lddSn sr Ad     ⋅ − Ω = π4 1 The contribution of a length dl of the Vortex Filament to isA  ∫∫∫∫∫ ⋅Ω=⋅×∇=⋅=Γ SS Stokes C SdnSdnVrdV  : If the Flow is Incompressible 0=⋅∇ u  so we can write , where is the Vector Potential. We are free to choose so we choose it to satisfy . AV  ×∇= A  A  0=⋅∇ A  We obtain the Poisson Equation that defines the Vector Potential A  Ω−=∇  A2 Poisson Equation Solution( ) ∫ − Ω = Space dv sr rA    π4 1 Félix Savart 1791 - 1841 Biot-Savart Formula
  • 139. 139 3-D Inviscid Incompressible FlowSOLO Circulation Definition: ∫ ⋅=Γ C rdV  : C – a closed curve Biot-Savart Formula (continue - 1) 1820 Jean-Baptiste Biot 1774 - 1862 VorticityV  ×∇≡Ω ( )lddSn sr Ad     ⋅ − Ω = π4 1 We found ∫∫∫∫∫ ⋅Ω=⋅×∇=⋅=Γ SS Stokes C SdnSdnVrdV  : also we have dlld Ω Ω =   ( ) ( ) ∫∫∫∫∫ × − ∇⋅Ω=⋅ − Ω ×∇=×∇= Γ Ω Ω = ld sr dSnlddSn sr AdrV r S dlld v rr          1 4 1 4 1 ππ ( ) ( ) ∫ − −×Γ = 3 4 sr srld rV    π Biot-Savart Formula Félix Savart 1791 - 1841 Biot-Savart Formula

Editor's Notes

  1. J.D. Anderson, Jr, “Fundamentals of Aerodynamics”, McGraw-Hill, 3th Ed., 1984, 1991, 2001
  2. http://en.wikipedia.org/wiki/Earth%27s_atmosphere
  3. “Introduction to the Aerodynamics of Flight”, NASA History Office, SP-367, Talay, 1975
  4. http://en.wikipedia.org/wiki/File:Atmosphere_composition_diagram.jpg
  5. Frank J, Regan, Satya M. Anandakrishnan, “Dynamics of Atmospheric Re-Entry”, AIAA Education Series, 1993
  6. Frank J, Regan, Satya M. Anandakrishnan, “Dynamics of Atmospheric Re-Entry”, AIAA Education Series, 1993
  7. Frank J, Regan, Satya M. Anandakrishnan, “Dynamics of Atmospheric Re-Entry”, AIAA Education Series, 1993
  8. R.P.G. Collinson, “Introduction to Avionics”, Chapman &amp; Hall, Inc., 1996, 1997, 1998 Frank J, Regan, Satya M. Anandakrishnan, “Dynamics of Atmospheric Re-Entry”, AIAA Education Series, 1993
  9. Frank J, Regan, Satya M. Anandakrishnan, “Dynamics of Atmospheric Re-Entry”, AIAA Education Series, 1993
  10. R.P.G. Collinson, “Introduction to Avionics”, Chapman &amp; Hall, Inc., 1996, 1997, 1998 Frank J, Regan, Satya M. Anandakrishnan, “Dynamics of Atmospheric Re-Entry”, AIAA Education Series, 1993
  11. R.P.G. Collinson, “Introduction to Avionics”, Chapman &amp; Hall, Inc., 1996, 1997, 1998 Frank J, Regan, Satya M. Anandakrishnan, “Dynamics of Atmospheric Re-Entry”, AIAA Education Series, 1993
  12. R.P.G. Collinson, “Introduction to Avionics”, Chapman &amp; Hall, Inc., 1996, 1997, 1998 Frank J, Regan, Satya M. Anandakrishnan, “Dynamics of Atmospheric Re-Entry”, AIAA Education Series, 1993
  13. Frank J, Regan, Satya M. Anandakrishnan, “Dynamics of Atmospheric Re-Entry”, AIAA Education Series, 1993
  14. Frank J, Regan, Satya M. Anandakrishnan, “Dynamics of Atmospheric Re-Entry”, AIAA Education Series, 1993
  15. “Introduction to the Aerodynamics of Flight”, NASA History Office, SP-367, Talay, 1975
  16. “Introduction to the Aerodynamics of Flight”, NASA History Office, SP-367, Talay, 1975
  17. W. Yourgrau, S. Mandelstam, “Variational Principles in Dynamics and Quantum Theory”, Dover, 1968, § 13, “Variational Principles in Hydrodynamics”, pp. 142-161 N.N. Moiseyev, V.V. Rumyantsev, “Dynamic Stability of Bodies Containing Fluids”, Springer-Verlag, 1968
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