introduction to turbulent flow
TRANSCRIPT
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.
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.
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.
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)
http://www.engr.uky.edu/~acfd/lctr-notes634.pdf
Splitting a field into a mean flow and fluctuations
Decomposition of Flow Properties
• (u,v,w,p)=(U,V,W,P)+(u′,v′,w′,p′)
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
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
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
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.
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.
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.
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.
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.
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.
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.
Different Regions of a turbulent Flow over a Surface
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.
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
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
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
),(
Different Regions of a turbulent Flow over a Surface
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
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''
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.
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
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.
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.
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
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|
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
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
Different Regions of a turbulent Flow over a Surface
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
Empirical Relationships for Flat Plate
Important Definitions
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
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
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
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!
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