2. Turbulent Flow
• Most flows encountered in engineering practice are turbulent
• it is important to understand how turbulence affects wall shear stress
• turbulent flow is a complex mechanism dominated by fluctuations,
• theory of turbulent flow remains largely undeveloped.
• we must rely on experiments and the empirical or semi-empirical correlations developed for various
situations.
• Turbulent flow is characterized by random and rapid fluctuations of swirling regions of fluid, called
eddies,
• In laminar flow, fluid particles flow in an orderly manner along pathlines, and momentum and energy
are transferred across streamlines by molecular diffusion.
• In turbulent flow, the swirling eddies transport mass, momentum, and energy to other regions of flow
much more rapidly than molecular diffusion, greatly enhancing mass, momentum, and heat transfer.
3. The intense mixing in turbulent flow
brings fluid particles at different
momentums into close contact and
thus, enhances momentum transfer
As a result,
turbulent flow is associated with much higher values of friction, heat transfer,
and mass transfer coefficients
4. Fluctuations of the velocity
component u with time at a specified
location in turbulent flow.
velocity, temperature, pressure,
and even density (in compressible flow)
time average of fluctuating components is zero,
Independent of
time
chaotic fluctuations
Hot wire anemometer probes with single, double, and
triple sensors to measure
5. Total shear stress
friction between layers in the flow direction
friction between the fluctuating fluid particles and the fluid body
The velocity profile and the variation
of shear stress with radial distance for
turbulent flow in a pipe.
6. Turbulent shear stress
• The mass flow rate of the fluid particles rising
through dA is
• and its net effect on the layer above dA is a reduction
in its average flow velocity because of momentum
transfer to the fluid particles with lower average flow
velocity.
• This momentum transfer causes the horizontal
velocity of the fluid particles to increase by u`
• and thus its momentum in the horizontal direction to
increase at a rate of (ρv`/dA)u`,
• force in a given direction is equal to the rate of
change of momentum in that direction,
Consider turbulent flow in a horizontal pipe
Fluid particle moving upward through
a differential area dA as a result of the
velocity fluctuation v`
Reynolds Stress
7. • momentum transport by eddies in turbulent flows is analogous to the
molecular momentum diffusion(random motion of molecules in a gas—
colliding with each other after traveling a certain distance and exchanging
momentum in the process)
• Simpler turbulence model- Know as Boussinesq
• where µt is the eddy viscosity or turbulent viscosity, which accounts for
momentum transport by turbulent eddies.
8. • the total shear stress can be expressed conveniently as
• eddy viscosity must be modeled as a function of the average flow variables;
we call this eddy viscosity closure
• Prandle mixing length lm, which is related to the average size of the eddies
that are primarily responsible for mixing,
µt kinematic eddy viscosity or kinematic turbulent viscosity
The velocity gradients at the wall, and thus
the wall shear stress, are much larger for
turbulent flow than they are for laminar
flow, even though the turbulent boundary
layer is thicker than the laminar one for the
same value of free-stream velocity.
9. • The velocity profile in this Viscous sublayer is very nearly linear,
and the flow is streamlined.
• the buffer layer, in which turbulent effects are becoming significant,
but the flow is still dominated by viscous effects.
• Above the buffer layer is the overlap (or transition) layer, also
called the inertial sublayer, in which the turbulent effects are much
more significant, but still not dominant
• outer (or turbulent) layer in the remaining part of the flow in which
turbulent effects dominate over molecular diffusion (viscous) effects
10. Viscous sublayer
• The thickness of the viscous sublayer is very small (typically, much less than 1 percent of the pipe
diameter)
• The wall dampens any eddy motion, and thus the flow in this layer is essentially laminar and the
shear stress consists of laminar shear stress which is proportional to the fluid viscosity
• the velocity profile in this layer to be very nearly linear,
• and the wall shear stress can be expressed as
• Dimension of velocity and it is known as friction velocity.
• the velocity profile in the viscous sublayer can be expressed in dimensionless form as
This equation is known as the law of the wall,
11. Comparison of the law of the wall and
the logarithmic-law velocity profiles
with experimental data for fully
developed turbulent flow in a pipe
• In Normalized low of the Wall U+ = y+
12. Overlap layer
• the experimental data for velocity are observed to line up on a straight line when plotted against the
logarithm of distance from the wall.
к=0.40 and B= 5.0
13. Outer Turbulent Layer
• Boundary condition to solve for B is maximum velocity umax in a pipe occurs at the centerline where r =0
• setting y =R - r =R and U= Umax
The deviation of velocity from the centerline value umax - u is called the
velocity defect is called the velocity defect law.
This relation shows that the normalized velocity profile in the core region of turbulent flow in a pipe depends
on the distance from the centerline and is independent of the viscosity of the fluid. This is not surprising since
the eddy motion is dominant in this region, and the effect of fluid viscosity is negligible.
14. power-law velocity profile
• Numerous other empirical velocity profiles exist for turbulent pipe flow. Among those, the simplest
and the best known is the power-law velocity profile expressed as
• power-law profile cannot be used to calculate wall shear stress since it gives a velocity gradient of
infinity there, and it fails to give zero slope at the centerline
The value of n increases with increasing
Reynolds number
15. Turbulent Boundary layer over flat plate
Illustration of the unsteadiness
of a turbulent boundary layer;
the thin, wavy black lines are
instantaneous profiles, and the
thick blue line is a long time-
averaged profile.
One common empirical approximation for the
time averaged velocity profile of a turbulent
flat plate boundary layer is the one seventh-
power law,
16. Skin Friction Coefficient
Comparison of laminar and turbulent
flat plate boundary layer profiles,
nondimensionalized by boundary
layer thickness.
17. Summary of expressions for laminar and turbulent boundary layers on a smooth
flat plate aligned parallel to a uniform stream*
Obtained from one-seventh
power law
Obtained from one-seventh powe
law with empirical data for turbule
flow through smooth pipes
19. Solution: To compare laminar versus turbulent boundary layer profiles, local skin
friction coefficient, and boundary layer thickness at the end of a flat plate.
Assumption
1 The plate is smooth, and the free stream is calm and uniform.
2 The flow is steady in the mean.
3 The plate is infinitesimally thin and is aligned parallel to the free stream.
Properties The kinematic viscosity of air at 20°C is ν= 1.516 ×10-2 m2/s.
20. For Pipe flow
if Re < 2000 the flow is laminar
if 2000 < Re < 2900 the flow is transient
if Re > 2900 the flow is turbulent
For a flat Plate flow,
if Re < 5 × 105 the flow is laminar
if 5 × 105 < Re < 107 the flow is transient
if Re > 107 the flow is turbulent
22. Compressible Flow
Outcome
• Understanding the concepts of stagnation state, speed of sound, and Mach number for
compressible flows
• The relationships bet for isentropic flows of ideal gasesween the static and stagnation fluid
properties
• The effects of area changes for one dimensional isentropic subsonic and supersonic flows
23. Example
• Oscillating Piston- How the pressure waves to initiate the fluid
motion. This pressure wave compress the fluid and causes the density
variation
• Couette flow problem- How heat is generating in the bearing- How
the shear profile generate thermal energy by viscous dissipation. The
temperature gradient leads to density variation.
• High speed airplanes and rockets because there is a layer of shear
flow next to the surface.
24. Internal Energy and Enthalphy
Energy stored in it by virtue of its molecular motion
U=mCvT
In compressible flow internal energy is combined with the flow energy (pv) is
know as enthalpy (h)- It is convenient
So h= U+pv= U+p/ρ
For perfect gas
h = U+mRT
=mCvT+mRT
=mT(Cv+R)
=mCpT
R= Cp-CV
Cp specific heat at constant pressure
Cv= Specific heat at constant volume
25. Stagnation Properties
• it is convenient to combine the enthalpy and the kinetic energy of the fluid into a single term called
stagnation (or total) enthalpy h0,
• Whenever the kinetic and potential energies of the fluid are negligible, as is often the case, the
enthalpy represents the total energy of a fluid.
• For high-speed flows, such as those encountered in jet engines, the potential energy of the fluid is
still negligible, but the kinetic energy is not.
• the ordinary enthalpy h is referred to as the static enthalpy
Aircraft and jet engines involve high
speeds, and thus the kinetic energy
term should always be considered
when analyzing them
26. Steady flow of a fluid through an
adiabatic duct.
energy balance relation (Ein = Eout)
in the absence of any heat and work interactions and any changes in potential energy, the stagnation enthalpy of
a fluid remains constant during a steady-flow process.
If the fluid were brought to a complete stop, then the velocity at state 2 would be zero
Thus the stagnation enthalpy represents the enthalpy of a fluid when it is brought to rest adiabatically
The temperature of an ideal gas
flowing at a velocity V rises by V2/2cp
when it is brought to a complete stop.
27. During a stagnation process, the kinetic energy of a fluid is converted to enthalpy (internal energy+flow energy),
which results in an increase in the fluid temperature and pressure
The properties of a fluid at the stagnation state are called stagnation properties
stagnation temperature
stagnation pressure,
stagnation density
isentropic stagnation state
reversible as well as adiabatic (i.e., isentropic).
actual stagnation pressure is lower than the isentropic
stagnation pressure since entropy increases during the
actual stagnation process as a result of fluid friction
Here T0 is called the stagnation (or total) temperature
dynamic temperature
energy balance for a single-stream, steady-flow device can be expressed as
28. ONE DIMENSIONAL
COMPRESSIBLE FLOW
CONTINUITY EQUATION
Mass is conserved
Over a short time, interval Δt
This equation is called the continuity equation for
steady one-dimensional flow
29. Momentum Equation
Conservation of momentum requires that the time rate of change of momentum in a given direction is
equal to the sum of the forces acting in that direction. This is known as Newton’s second law of motion
and in the model used here the forces concerned are gravitational (body) forces and the surface
forces.
around the curved surface the pressure may
be taken to be the mean value
neglecting quantities of small order such as (dp/ds)dsdA
and cancelling,
(1)
(2)
The stream tube and element for the momentum equation
The forces on the element
30. Newton’s second law of motion (force = mass x acceleration) applied
After simplification
Integrating
This result is known as Bernoulli’s equation for compressible flow
(3)
(4)
(5)
(6)
(7)
(8)
31. ENERGY EQUATION:
Conservation of energy implies that changes in energy, heat transferred, and work done by a system
in steady operation are in balance.
Control volume for the energy equation
@1 unit mass of fluid entering the system through section
will possess internal energy cvT1, kinetic energy v1
2 /2 and
potential energy gz1, i.e.
Likewise @2 on exit from the system across section 2 unit
mass will possess energy
Flow Energy: Now to enter the system, unit mass possesses a volume 1/ρ1 which must push against the
pressure p1 and utilize energy to the value of p1 x 1/ρ1 pressure x (specific) volume.
32. Now to enter the system, unit mass possesses a volume 1/ρ1 which must push against the pressure p1 and
utilize energy to the value of p1 x 1/ρ1 pressure x (specific) volume.
At exit p2 /ρ2 is utilized in a similar manner.
In the meantime, the system accepts, or rejects, heat q per unit mass.
So, In steady flow energy equation, the energy entering plus the heat transfer must equal the energy leaving.
For an adiabatic horizontal flow - no heat transfer
33. Mach Number
• Mach No= √(Inertia Force/ elastic force) =√ (ρAc2/KA)=
√ (ρAc2/ρa2A)= √ (c2/a2)
• Speed of sound
• M= c/a = fluid velocity / local velocity of sound
34. Variation of flow velocity with flow Area
couplings among the velocity, density, and flow areas for isentropic duct flow are rather complex
To develop relationship between static-to-stagnation property ratios with the Mach number for pressure,
temperature, and density.
seeking relationships among the pressure,
temperature, density, velocity, flow area, and Mach number for one-dimensional
isentropic flow.
Consider the mass balance for a steady-flow process
Take log and differentiate the mass flow rate,
(1)
35. (2)
Energy Balance for Isentropic flow with no work interaction
Substitute (2) in (1)
This relation is also the differential form of Bernoulli’s equation when changes in potential energy are
negligible, which is a form of the conservation of momentum principle for steady-flow control volumes
the speed of sound in a fluid is c
isentropic flow in ducts
it describes the variation of pressure
with flow area.
36. Type of Flow Mach Number
Subsonic
flow
For subsonic flow (Ma
<1),
(1 - Ma2) is positive
Sonic Flow For supersonic flow
(Ma=1)
(1 - Ma2) is zero
Supersonic
flow
For sonic flow (Ma>1), (1 - Ma2) is
Negative
38. Problem
An aircraft is flying at a cruising speed of 250 m/s at an altitude of 5000 m
where the atmospheric pressure is 54.05 kPa and the ambient air
temperature is 255.7 K. The ambient air is first decelerated in a diffuser
before it enters the compressor as shown in Fig. Assuming both the
diffuser and the compressor to be isentropic, determine (a) the stagnation
pressure at the compressor inlet and (b) the required compressor work
per unit mass if the stagnation pressure ratio of the compressor is 8.
39. SOLUTION High-speed air enters the diffuser and the compressor of an aircraft. The stagnation
pressure of the air and the compressor work input are to be determined.
Assumptions 1 Both the diffuser and the compressor are isentropic. 2 Air is an ideal gas with
constant specific heats at room temperature.
Given Data : cp = 1.005 kJ/kg°K and k =1.4
Formulae
That is, the temperature of air would increase by 31.1°C and the pressure by 26.72 kPa as air is decelerated from
250 m/s to zero velocity. These increases in the temperature and pressure of air are due to the conversion of the
kinetic energy into enthalpy.
40. Problem
Air enters a diffuser shown in Figure with a velocity of 200 m/s.
Determine (a) the speed of sound and (b) the Mach number at the
diffuser inlet when the air temperature is 30°C.
SOLUTION Air enters a diffuser with a high velocity. The speed of sound
and the Mach number are to be determined at the diffuser inlet.
Assumption Air at specified conditions behaves as an ideal gas.
Properties The gas constant of air is R = 0.287 kJ/kg · K, and its specific
heat ratio at 30°C is 1.4. (from Gas Table)
Discussion The flow at the diffuser inlet is subsonic since Ma <1.
41. Problem
Carbon dioxide flows steadily through a varying cross-sectional area duct such as a
nozzle shown in Figure at a mass flow rate of 3 kg/s. The carbon dioxide enters the
duct at a pressure of 1400 kPa and 200°C with a low velocity, and it expands in the
nozzle to a pressure of 200 kPa. The duct is designed so that the flow can be
approximated as isentropic. Determine the density, velocity, flow area, and Mach
number at each location along the duct that corresponds to a pressure drop of 200
kPa.
SOLUTION Carbon dioxide enters a varying cross-sectional
area duct at specified conditions. The flow properties are to be
determined along the duct.
Assumptions 1 Carbon dioxide is an ideal gas with constant
specific heats at room temperature. 2 Flow through the duct is
steady, one-dimensional, and isentropic.
Properties For simplicity we use cp = 0.846 kJ/kg · K and k =
1.289
throughout the calculations, which are the constant-pressure
specific heat and specific heat ratio values of carbon dioxide
at room temperatures. The gas constant of carbon dioxide is
R = 0.1889 kJ/kg · K.
43. One Dimensional
Isentropic flow
During fluid flow through many devices such as nozzles,
diffusers, and turbine blade passages, flow quantities
vary primarily in the flow direction only, and the flow can
be approximated as one-dimensional isentropic flow
with good accuracy.
45. Critical Temperature and Pressure in Gas
compressible flow to let the superscript asterisk (*) represent the
critical values. Setting Ma = 1
46. Problem
Carbon dioxide flows steadily through a varying cross-sectional
area duct such as a nozzle shown in Figure at a mass flow rate
of 3 kg/s. The carbon dioxide enters the duct at a pressure of
1400 kPa and 200°C with a low velocity, and it expands in the
nozzle to a pressure of 200 kPa. The duct is designed so that the
flow can be approximated as isentropic. Calculate the critical
pressure and temperature of carbon dioxide for the given flow
conditions
Editor's Notes
onsider the steady flow of fluid through a duct (that is, the inlet and outlet flows do not vary with time). The inflow and outflow are one-dimensional, so that the velocity V and density r are constant over the area A. Now we apply the principle of mass conservation. Since there is no flow through the side walls of the duct, what mass comes in over area A1 goes out of area A2, (the flow is steady so that there is no mass accumulation). Over a short time interval Δt,
