introduction to turbulent flow
TRANSCRIPT
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Preliminary Remarks
• This section is a very brief introduction to turbulent flows.
• Emphasis is on how to modify the governing equations (Navier-Stokes, boundary layer, integral methods) and model turbulent flows.– Skin friction drag, heat transfer, boundary layer
thickness, etc
• A more details treatment is found in the follow on course AE 6012.
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Jets at two different Reynolds numbers
Source: Tennekes & Lumley. Page 22.
• L.F. Richardson (“Weather Prediction by Numerical Process.” Cambridge University Press, 1922):
Big whirls have little whirlsWhich feed on their velocity;And little whirls have lesser whirls,And so on to viscosityin the molecular sense.
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Features of Turbulent Flow
• Irregular fluctuations in species concentrations, temperature, velocity.
– Hopelessly complex, defies formal mathematical treatment
• Turbulent mixing is very important in many applications.
– 100, 1000, or even 106 (gazillion) times more powerful than molecular mixing.
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Historyhttp://www.engr.uky.edu/~acfd/lctr-notes634.pdf
• da Vinci (circa 1500).• Boussinesq (1877), defined an “eddy
viscosity” that has the same dimensions as kinematic viscosity, and related turbulent stresses to strain rate times the eddy viscosity.
– Form similar to Stokes relations
• Reynolds pipe flow studies of transition to turbulence (1894)
• Poincare studied chaos nonlinear dynamical systems and chaos (1899)
• Prandtl developed a first eddy viscosity model for flow over flat plate (1925)
• G. I. Taylor used statistical mathematics tools (1935)
• Kolmogorov came with an entirely independent way of looking at turbulent flow (1941)
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http://www.engr.uky.edu/~acfd/lctr-notes634.pdf
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Splitting a field into a mean flow and fluctuations
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Decomposition of Flow Properties
• (u,v,w,p)=(U,V,W,P)+(u′,v′,w′,p′)
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Contribution of Velocity fluctuations toTransport of Momentum
• Consider a control volume.• Consider a face of area DA,
with an outwardly facing normal along the negative x-direction as shown.
• The u-velocity component is U+u’
• Mass flow rate crossing this face is r(U+u’)DA.
• Momentum rate crossing this face is r(U+u’)2DA
• If we directly compute U and u’ both, it is called a Direct Numerical Simulation (DNS).
Face of areaDA
U+u’
Mean flow and fluctuation
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Time averaged Momentum Rate
• Because DNS is expensive, and mathematical/analytical prediction of u’ is difficult, Reynolds proposed averaging the flux over a time period, T.
• It must be large enough so that fluctuations u’ cancel out, but the mean flow unsteadiness, in any, which occurs over a much longer period, is preserved.
Face of areaDA
U+u’
Mean flow and fluctuation
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Time averaged Momentum RateLow Speed (Incompressible) Flow
TTT
dtuT
dtuT
UUdtuU
T0
2
0
2
0
2 2'
1 rrrr
The second term on the right side vanishes, since the fluctuations u’ average tozero over a sufficiently long time interval T.
The third term persists, and must be computed (DNS) or modeled.
We use the <> to indicate time averaged value of this quantity.Most books use a bar on top of a quantity to indicate time averaging has been done.
22
0
2'
1uUdtuU
T
T
rrr
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Reynolds Averaging
• Reynolds was the first one to propose such averaging of the momentum (and energy flux) terms.
• This process is therefore called Reynolds averaging.
• The fluctuating quantities r<u’2>, r<u’v’> etc are called Reynolds stress components, and are usually brought to the right hand side of the momentum (and energy equation).
• In incompressible flows, the fluctuations in density are small, and have been neglected.
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Viscous Stresses vs Molecular Stresses
• On the right side of momentum and energy equations, we now have derivatives of Reynolds stresses.
• Because an averaging has taken place, we can no longer solve for these quantities in a deterministic fashion and get u’.
• We need to rely on modeling these terms.
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Reynolds Averaged Navier-Stokes Equations (RANS)
• U-Momentum
w'v22 u
zu
yu
xx
pUW
zUV
yU
xU
txzxyxx rrrrrrr
Molecular stress
Reynolds Stress
We can similarly write the V-, W- and energy transport equations.
Energy transport equation will have derivatives of terms like rCpUT on the left side,and derivatives of terms like rCp<-u’T’> on the right side.
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Turbulent Flow over a flat Plate• We turn our attention to flow over a flat plate.
– Simplest external flow
– First modeled by Prandtl
– Lot of experimental data exists
– This flow is used to develop turbulence closures (also called turbulence models) for modeling the Reynolds stress terms discussed earlier.
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Boussinesq Hypothesis (1877)• Newton and Stokes suggested that viscous stresses be
written as a product of molecular viscosity (mrn) times the strain rate.
– e.g. xy= m(u/y+ v/x)
• Boussinesq suggested that the Reynolds stresses be written as a product of density times eddy viscosity nT
(which has dimensions of n) times the strain rate.
– e.g. <-u’v’>= nT(u/y+ v/x)
• This reduces turbulence modeling to computing a physically meaningful value of eddy viscosity at every point in the flow, hiding the details of the complex flow physics.
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Flow over the Flat Plate• Prandtl was the first one to develop a suitable eddy
viscosity model in the “near wall” region of turbulent flow over a flat plate (1925).
• Theodore von Karman established an empirical constant in Prandtl’s model
• Van Driest further improved the model for region very close to the wall.
• Cebeci and Smith (1960s) developed a suitable eddy viscosity in the outer layers of turbulent flow.
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Different Regions• Turbulent boundary layer over a surface, including
flat plates, has regions where different physical phenomena dominate.
– Viscous sub-layer, very close to the wall where laminar viscous effects dominate
– Buffer region where laminar and turbulent transport both play a roll
– Inertial sub-layer
– Outer “wake” or velocity “defect” layer
• An appropriate mathematical/empirical definition of the velocity profile must be developed for each of these regions.
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Different Regions of a turbulent Flow over a Surface
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Viscous Sublayer
• We start with the region very close to the wall.
• In this region, convection effects are very small due to the very low mean flow velocity.
• Mixing due to turbulent eddies is not dominant, due to the damping or dissipation produced by viscosity.
• The u-momentum equation simplifies considerably.
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Viscous Sublayer
2 2' ' v 'xyxxP
U UV u ux y x x y x y
r r r r
These terms are
small near the wall,
since U and V, time
averaged mean
velocities are small
Flat plate
No pressure gradient
Prandtl’s
Boundary
layer
Hypothesis
says this
term is
small
Viscous effects dissipate
eddies, no significant
Turbulent transport
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Viscous Sub-layer• With these simplifications, the u-momentum
equation may be solved for the mean flow velocity profile U(x,y) very close to the wall:
y
y
U
x
V
y
U
y
W all
W all
W all
W allxy
xy
m
m
m
y)U(x,
wall.at the 0apply U Integrate,
.hypothesislayer boundary with term,secondNeglect
:Integrate
0
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Viscous Sub-layer
• From the previous slide, we notice that U(x,y) linearly varies with y very close to the wall.
• Continuing,
yu
yu
u
U
yu
yu
yxU
u
yyxU
wall
wall
n
nm
r
r
m
22
2
),(
Use
),(
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Different Regions of a turbulent Flow over a Surface
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Buffer Layer
2 2' ' v 'xyxxP
U UV u ux y x x y x y
r r r r
These terms are
small near the wall,
since U and V, time
averaged mean
velocities are small
Flat plate
No pressure gradient
Prandtl’s
Boundary
layer
Hypothesis
says this
term is
small
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Buffer Layer• In the buffer layer, u-momentum simplifies.
• It states that the sum of laminar stress and the Reynolds stress is approximately a constant.
• The sum must equal wall shear stress to ensure that the viscous sub-layer and buffer layer blend smoothly with each other.
wallvuy
U
vuyy
U
y
rm
rm
''
0''
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Inertial Layer• We skip the buffer layer (which serves as a buffer
region between the viscous sub-layer and inertial layer) for now.
• In the inertial layer, the convection effects are still small, and are neglected.
• Laminar viscous stresses progressively become less important and go to zero.
• The Reynolds stress remains constant in this region, and equals wall shear stress to ensure continuity of properties with the buffer region.
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Inertial Layer
2 2' ' v 'xyxxP
U UV u ux y x x y x y
r r r r
These terms are
small near the wall,
since U and V, time
averaged mean
velocities are small
Flat plate
No pressure gradient
Prandtl’s
Boundary
layer
Hypothesis
says this
term is
small
Molecular
Mixing effects
are
small
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Inertial Sub-layer
• Momentum equation simplifies to:
sity.eddy visco theis where
,hypothesis Boussinesq Invoking
v-
:properties of continuity enforce Integrate,
0v
T
T
n
rn
r
r
wall
wall
y
U
u
uy
We need to develop a model for eddy viscosity to proceed further.
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Prandtl’s Mixing Length Model
• Prandtl developed the very first eddy viscosity model near the wall in 1925, which is still in use 89 years later!
• He used dimensional analysis to get started.
• The eddy viscosity nT must have the same dimension as as the kinematic viscosity n(m2/sec).
– Product of a suitable velocity times a suitable length.
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Prandtl’s Mixing Length Model: Length Scale
• For a suitable length l, he assumed that the size of the largest eddies (which bring about much of the transport of momentum) must be proportional to the distance y of the point from the wall.
• That is, l equals ky, where k is a constant to be empirically determined.– Von Karman determined this constant
using measurements done by Prandtl and his students.
Eddies
of size ky
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Prandtl’s Velocity Scale
• From dimensional arguments, Prandtl hypothesized that the velocity scale is l|U/y|.
• It must be a positive number.
• Thus, Prandtl proposed the following nT=l2|U/y|
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Velocity Profile in the Inertial Layer
• Once we have a suitable eddy viscosity model in the inertial layer, we can integrate the u-momentum equation derived for this layer a few slides ago.
wall
T
wallT
y
U
y
Uy
y
Uy
y
U
y
U
rk
kn
rn
22
22. Using
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Velocity Profile in the Inertial Layer
• Continuing from the previous slide, recognizing thaat the velocity profile for a flat plate has a positive slope and the |..| is not necessary,
Byk
u
Byu
ku
Cyu
uy
Uy
u
y
Uy
wall
wall
)log(1
log1U(y)
logU(y)
:Integrate
Using 2
2
22
n
k
k
r
rk
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Different Regions of a turbulent Flow over a Surface
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Von Karman Constant k
• Theodore von Karman plotted u+ vs y+ on a semi-log plot, and determined the constant kas 0.4.
• The constant B may also be found from such plots as 5.5
100030 5.5log1
yyuk
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Empirical Relationships for Flat Plate
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Important Definitions
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Numerical Example(Anderson, Fundamentals of Aerodynamics)
• Piper Cherokee Wing
– 9.75 m span, 1.6 m chord
– 141 miles/hour
– Sea level
• Compute skin friction drag
– For laminar flow
– For turbulent flow
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Laminar Drag Solution
• V∞=141 miles per hour=63.04 m/sec
• Look up atmospheric density, viscosity, compute Reynolds number based on chord as 6.93 Million
• Compute Cd, equals 1.328/(Re)1/2 as, 0.000504
• Compute drag force on both sides as 2 * density *1/2* V∞
2 * Cd * chord * span
– Laminar drag force = 38.4 N
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Turbulent Drag Solution
• Integrate Cf with respect to x/c to get Cd on one side.
• Result: Cd=.0721/Re1/5
• Use this Cd instead of Cd for laminar flow to get drag.
• Result: ~240 N
– 6.25 times more than laminar flow
5/1Re
371.0
xx
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How thick is the boundary layer?
• For laminar flow over the wing in the previous example,
– Boundary layer thickness = 5 c/(Re)1/2=3.04 mm
• For turbulent flow,
– = 0.371 c/ (Re)1/5=25.4 mm
• Turbulent boundary layer is more than 8 times thicker!
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Effects of Transition• If transition occurs at a Reynolds number based on x of
500,000 compute the effect on drag for the wing considered earlier.
• Rex= 500000 corresponds to x=0.1154 m
• Compute laminar flow drag coefficient from x=0 to 0.1154m.– 1.328/(500000)1/2 = 0.001878, Drag force=5.16 N
• Compute turbulent drag coefficient for the chord neglecting transition.– From previous slide, Cd, turbulent=.00536 Drag force= 240 N
• Compute turbulent drag contribution from x=0 to x=0.1154 m as Cd=.0721/Re1/5
– For Re=500000, Cd= 0.00536, Drag force over the first 0.1154m of chord is 14.73 N
– Net drag = 240 -14.73+5.16= 230.43N