BASICS OF COMPUTATIONAL FLUID DYNAMICS ANALYSIS MEEN 5330 Presented By Chaitanya Vudutha Parimal Nilangekar Ravindranath GouniSatish Kumar Boppana Albert Koether Pages-28
Overview Introduction History of CFD Basic concepts CFD Process Derivation of Navier-Stokes Duhem Equation Example Problem Applications Conclusion References
Introduction What is CFD? Prediction fluid flow with the complications of simultaneous flow of heat, mass transfer, phase change, chemical reaction, etc using computers History of CFD Since 1940s analytical solution to most fluid dynamics problems was available for idealized solutions. Methods for solution of ODEs or PDEs were conceived only on paper due to absence of personal computer. Daimler Chrysler was the first company to use CFD in Automotive sector. Speedo was the first swimwear company to use CFD. There are number of companies and softwares in CFD field in the world. Some softwares by American companies are FLUENT, TIDAL, C-MOLD, GASP, FLOTRAN, SPLASH, Tetrex, ViGPLOT, VGRID, etc.
−6 volumes no smaller than say (1*10 m3) Molecular Particles of FluidBasic fluid motion can be described as some combination of1) Translation: [ motion of the center of mass ] 2) Dilatation: [ volume change ]3) Rotation: [About one, two or 3 axes ].4) Shear Strain
Compressible and Incompressible flow A fluid flow is said to be compressible when the pressure variation in the flow field is large enough to cause substantial changes in the density of fluid. dqi 1 ~ µ = f i − p,i + qi , jj dt ρ ρViscous and Inviscid Flow In a viscous flow the fluid friction has significant effects on the solution where the viscous forces are more significant than inertial forces ∂ ∂ ( ρ u ) + ( ρ v) = 0 ∂x ∂y
Steady and Unsteady Flow Whether a problem is steady or unsteady depends on the frame of reference Laminar and Turbulent FlowNewtonian Fluids and Non-Newtonian Fluids In Newtonian Fluids such as water, ethanol, benzene and air, the plot of shear stress versus shear rate at a given temperature is a straight line
Initial or Boundary Conditions Initial condition involves knowing the state of pressure (p) and initial velocity (u) at all points in the flow. Boundary conditions such as walls, inlets and outlets largely specify what the solution will be.
Discretization Methods Finite volume • Where Q - vector of conserved variables method • F - vector of fluxes∂ • V - cell volume∂t ∫∫∫ Qdv + ∫∫ FdA = 0 • A –Cell surface area Finite Element method Ri=Equation residual at an element vertex Q- Conservation equation expressed on element basis Ri = ∫∫∫Wi Qdv e Wi= Weight Factor
Finite difference method ∂Q ∂F ∂G ∂H + + + =0 ∂t ∂x ∂y ∂z Q – Vector of conserved variables F,G,H – Fluxes in the x ,y, z directionsBoundary element method The boundary occupied by the fluid is divided into surface mesh
CFD PROCESS Geometry of problem is defined . Volume occupied by fluid is divided into discrete cells.
CFD PROCESS cont..Physical modeling is defined.Boundary conditions are definedwhich involves specifying of fluidbehavior and properties at theboundaries.Equations are solved iterativelyas steady state or transient state.Analysis and visualization ofresulting solution. post processing
DERIVATION OF NAVIER-STOKES-DUHEM EQUATION The Navier-Stokes equations are the fundamental partial differentials equations that describe the flow of incompressible fluids. Two of the alternative forms of equations of motion, using the Eulerian description, were given as Equation (1) and Equation (2) respectively: ∂( ρqi ) + ( ρq i q j ) , j = ρf i + σ ji , j (1) ∂t dq i ∂q i 1 = + q j qi , j = f i + σ ji , j . (2) dt ∂t ρ
DERIVATION (Cont’d) If we assume that the fluid is isotropic , homogeneous , and Newtonian, then : ~ ~ σ ij = −( p − λ ∈kk )δ ij + 2 µ ∈ij . (3) Substituting Equ(3) into Equ(2), and utilizing the Eulerian relationship for linear stress tensor we get : dqi 1 ~ ~ µ +λ ~ µ = f i − p ,i + q j , ji + qi , jj , (4) dt ρ ρ ρ ( for compressible fluids )
DERIVATION(Cont’d)For incompressible fluid flow the Navier-Stokes-Duhem equation is: dqi 1 ~ µ = f i − p,i + qi , jj dt ρ ρIf the fluid medium is a monatomic ideal gas, then : ~ 2 ~ λ =− µ 3
DERIVATION (Cont’d)Navier stokes equation for compressible flow ofmonatomic ideal gas is : dqi 1 ~ 1µ ~ µ = f i − p ,i + q j , ji + qi , jj , dt ρ 3ρ ρ
EXAMPLE PROBLEMNeglecting the gravity field, describe the steady two-dimensional flow of an isotropic , homogeneous,Newtonian fluid due to a constant pressure gradientbetween two infinite, flat, parallel, plates. State thenecessary assumptions. Assume that the fluid has auniform density.
SOLUTION (Cont’d)The Navier – stokes equations for incompressible flow is: dqi 1 ~ µ + q j qi , j = f i − p,i + qi , jj dt ρ ρSince the flow is steady and the body forces areneglected, the Navier-stokes equation becomes: 1 ~ µ q j qi , j = − p,i + qi , jj ρ ρ
SOLUTION (Cont’d)The no slip boundary conditions for viscous flow are: qi = 0 at y2 = ±aUsing the boundary conditions ( q2= 0 at y2=+/- a )Thus, the first Navier-stokes equations becomes d 2 q1 dp µ 2 = dy 2 dy1
SOLUTION (Cont’d) Integrating twice, we obtain q1 = 1 dp 2 2 µ dy1 ( y2 − a 2 ) The results, assumptions and boundary conditions of this problem in terms of, mathematical symbols are as follows: ρ = Constant ∂ ) ( =0 ∂( ) fi = 0 =0 ∂ t ∂y 3 q1 = 1 dp 2 2µ dy1 ( y2 − a 2 )
HOMEWORK PROBLEM Using the Navier-Stokes equations investigate the flow (q i) between two stationary, infinite, parallel plates a distance h apart. Assuming that you have laminar flow of a constant-density, Newtonian fluid and the pressure gradient is constant (partial derivative of P with respect to 1).
Types of Errors and ProblemsTypes of Errors: Modeling Error. Discretization Error. Convergence Error.Reasons due to which Errors occur: Stability. Consistency. Conservedness and Boundedness.
Applications of CFD1. Industrial Applications: CFD is used in wide variety of disciplines and industries, including aerospace, automotive, power generation, chemical manufacturing, polymer processing, petroleum exploration, pulp and paper operation, medical research, meteorology, and astrophysics.Example: Analysis of AirplaneCFD allows one to simulate the reactorwithout making any assumptions about themacroscopic flow pattern and thus todesign the vessel properly the first time.
Application (Contd..)2. Two Dimensional Transfer Chute Analyses Using a Continuum Method: Fluent is used in chute designing tasks like predicting flow shape, stream velocity, wear index and location of flow recirculation zones.3. Bio-Medical Engineering: The following figure shows pressure contours and a cutaway view that reveals velocity vectors in a blood pump that assumes the role of heart in open-heart surgery. Pressure Contours in Blood Pump
Application (Contd..)4. Blast Interaction with a Generic Ship HullThe figure shows theinteraction of an explosionwith a generic ship hull.The structure was modeledwith quadrilateral shellelements and the fluid as amixture of high explosivesand air. The structuralelements were assumed tofail once the average strain Results in a cut plane for the interaction of anin an element exceeded 60 explosion with a generic ship hull: (a) Surfacepercent at 20msec (b) Pressure at 20msec (c) Surface at 50msec and (d) Pressure at 50msec
Application (Contd..) 5. Automotive Applications: Streamlines in a vehicle without (left) and with rear center and B-pillar ventilation (right)In above figure, influence of the rear center and B-pillar ventilation on therear passenger comfort is assessed. The streamlines marking the rearcenter and B-pillar ventilation jets are colored in red. With the rear centerand B-pillar ventilation, the rear passengers are passed by more cool air. Inthe system without rear center and B-pillar ventilation, the upper part of thebody, in particular chest and belly is too warm.
Conclusion Nearer the conditions of the experiment to those which concern the user, more closely the predictions agree with those data, the greater is the reliance which can be prudently placed on the predictions. CFD iterative Methods like Jacobi and Gauss-Seidel Method are used because the cost of direct methods is too high and discretization error is larger than the accuracy of the computer arithmetic. Many software’s offer the possibility of solving fully nonlinear coupled equations in a production environment. In the future we can have a multidisciplinary, database linked framework accessed from anywhere on demand simulations with unprecedented detail and realism carried out in fast succession so that designers and engineers anywhere in the world can discuss and analyze new ideas and first principles driven virtual reality
References1. Hoffmann, Klaus A, and Chiang, Steve.T “Computational fluid dynamics for engineer’s” vol. I and vol. II2. Rajesh Bhaskaran, Lance Collins “Introduction to CFD Basics”3. http://www.cham.co.uk/website/new/cfdintro.htm accessed on 11/10/06.4. Adapted from notes by: Tao Xing and Fred Stern, The University of Iowa.5. http://www.cfd-online.com/Wiki/Historical_perspective accessed on 11/12/06.6. Frederick and Chang,T.S.,”Continuum Mechanics”7. http://navier-stokes-equations.search.ipupdate.com/8. http://en.wikipedia.org/wiki/Computational_fluid_dynamics#Discretization_ method s, ”Discretization Methods”9. McIlvenna P and Mossad R “Two Dimensional Transfer Chute Analysis Using a Continuum Method”, Third International Conference on CFD in the Minerals and Process Industries, Dec 2003.10. Subramanian R.S. “Non-Newtonian Flows”.11. Lohner R., Cebral J., Yand C., “Large Scale Fluid Structure Interaction Simulations, IEEE June 2004”.12. http://www.cd- adapco.com/press_room/dynamics/23/behr.html,“Predicting Passenger Comfort13. http://www.adl.gatech.edu/classes/lowspdaero/lospd2/lospd2.html, “Types of Fluid Motion”