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Transcript of T.saad - Introduction to Turbulent Flow (March2004)
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TURBULENT FLOW
A BEGINNERS APPROACH
Tony Saad
March 2004
http://tsaad.utsi.edu - [email protected]
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CONTENTS
Introduction
Random processes
The energy cascade mechanism
The Kolmogorov hypotheses
The closure problem Some Turbulence Models
Conclusion
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INTRODUCTION
Most flows encountered in engineeringpractice are Turbulent
Turbulent Flows are characterized by afluctuating velocity field (in both positionand time). We say that the velocity field isRandom.
Turbulence highly enhances the rates ofmixing of momentum, heat etc
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INTRODUCTION
The motivations to study turbulent flows aresummarized as follows: The vast majority of flows is turbulent
The transport and mixing of matter, momentum, andheat in turbulent flows is of great practical importance
Turbulence enhances the rates of the aboveprocesses
The basic approaches to investigating turbulentflows are shown in the following chart
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INTRODUCTION
Experiment Empirical Correlations
High Cost $$$$
Numerical MethodsAveraging of Equations.
Turbulence Modeling
Filtering of Equations.Large Eddy Simulation
Full Resolution of the FlowDirect Numerical Simulation
Time ConsumingResource Consuming
MO
D
E
L
I
NG
A
P
P
RO
A
C
H
E
S
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RANDOM PROCESSES
Consider a typical pipe flow where theinstantaneous velocity is to be measured as a
function of time. (see dye moviesee velocity
profile movie)
http://dye.mov/http://velocity_profile.mov/http://velocity_profile.mov/http://velocity_profile.mov/http://velocity_profile.mov/http://dye.mov/ -
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RANDOM PROCESSES
It is clear that the velocity is fluctuating with time
We say that the velocity field is RANDOM
What does that mean? Why is it so?
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RANDOM PROCESSES
Consider the pipe flow experiment that is to berepeated several times (say 395804735 times)
under the same set of conditions: On the same planet
In the same country
In the same lab
Using the same apparatus
Supervised by the same person
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RANDOM PROCESSES
We are interested in measuring the streamwisevelocity component at a given pointA(x1,y1,z1) &
at a given time during the experiment: U(A,t1)
Velocity Measurment
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RANDOM PROCESSES
Consider the event denoted by E such that:
{ }sm
tAUE 10),( 1
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RANDOM PROCESSES
The word Random does not hold anysophisticated significance as it is usually
assigned The event E is random means only that it
may or may not occur
And this answers our first question; i.e. themeaning of a random variable
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RANDOM PROCESSES
We can now check the results of our experiment. After 40 repetitionsof the experiment the measured velocity was recorded as shownbelow:
Magnitude of U
0
5
10
15
20
0 5 10 15 20 25 30 35 40
Figure 1.1: Magnitude of U as a func tion of Exper iment Number
U
(m/s)
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RANDOM PROCESSES
Okay. Why is the velocity field random in aturbulent flow?
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RANDOM PROCESSES
Obviously, if there were no determinism (at least in theunderlying structure of the universe), why would we doscience in the first place? As science aims @ predictingthe behavior of physical phenomena
To answer the question, we have to resort to the theoryof dynamical systems. After all, the governing equationsof fluid mechanics represent a set of dynamicalequations.
Dynamical systems are systems that are extremelysensitive to minute changes in the surroundingenvironment, i.e. initial & boundary condit ions
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RANDOM PROCESSES
An illustrative example is the Lorenz dynamic system.
Consider the following set of ODEs
( )x y x
y x y xz
z z xy
= =
= +
Assume we fix =10 and =8 3 and vary
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RANDOM PROCESSES The solution is easy to get numerically and is as
follows for various values of
0 10 20 30 40 50 6030
0
30
6060
30
x t( )
T0 t
5=
t
0 10 20 30 40 50 6030
0
30
60
60
30
x1 t( )
T0 t
5=
t
0 10 20 30 40 50 6030
0
30
6060
30
x t( ) x1 t( )
T0 t
5=
Difference between the two
solutions
= 5
First set ofinitial conditions
Second set ofinitial conditions
Difference between thetwo solutions
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RANDOM PROCESSES Nothing much happens for some values of
0 10 20 30 40 50 6030
0
30
6060
30
x t( )
T0 t
20=
t
0 10 20 30 40 50 6030
0
30
60
60
30
x1 t( )
T0 t
20=
t
0 10 20 30 40 50 6030
0
30
6060
30
x t( ) x1 t( )
T0 t
20=
= 20
First set ofinitial conditions
Second set ofinitial conditions
Difference between thetwo solutions
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RANDOM PROCESSES What is so special about = 24.2
0 10 20 30 40 50 6030
0
30
6060
30
x t( )
T0 t
24.2=
t
0 10 20 30 40 50 60
30
0
30
6060
30
x1 t( )
T0 t
24.2=
t
0 10 20 30 40 50 6030
0
30
6060
30
x t( ) x1 t( )
T0 t
24.2=
= 24.2
First set ofinitial conditions
Second set ofinitial conditions
Difference between thetwo solutions
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RANDOM PROCESSES = 24.2 denotes the onset of sensitivity!!!
0 10 20 30 40 50 6030
0
30
6060
30
x t( )
T0 t
24.3=
t
0 10 20 30 40 50 6030
0
30
60
60
30
x1 t( )
T0 t
24.3=
t
0 10 20 30 40 50 6030
0
30
6060
30
x t( ) x1 t( )
T0 t
24.3=
See animation
= 24.3First set of
initial conditions
Second set ofinitial conditions
Difference between thetwo solutions
http://lorentz%20system.avi/http://lorentz%20system.avi/ -
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RANDOM PROCESSES As we can see, the figures show the sensitivity
of the system. In fact, for a critical value of thecoefficients (fix , ) =24.3, the system
becomes highly sensitive to small perturbations. This coefficient corresponds to the Reynolds
number in fluid flow. Beyond a critical value, theflow becomes extremely sensitive to any minute
disturbance whether from the boundary or fromthe internal structure of the flow (an eddy)
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RANDOM PROCESSES Conclusion:
A theory that predicts exact values of the velocityfield is definitely prone to be wrong. Such a theory
should aim at predicting the probability of occurrenceof certain events
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THE ENERGY CASCADE One of the first attacks to theoretically tackle the
problem of turbulence was done by Richardsonin 1922
His hypothesis is called the Energy CascadeMechanism & is now one of the fundamentalphysical models underlying any approach tosolving turbulent flow problems
Turbulence can be considered to be composedof eddies of different sizes
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THE ENERGY CASCADE An eddy is considered to be a turbulent motion
localized within a region of size l
These sizes range from the Flow length scale L
to the smallest eddies.
Each eddy of length size has a characteristic
velocity u() and timescale () = /u()
The largest eddies have length scalescomparable to L
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THE ENERGY CASCADE
Movie
http://eddies.mov/http://eddies.mov/ -
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THE ENERGY CASCADE Each eddy has a Reynolds number
For large eddies, Re is large, i.e. viscous effectsare negligible.
The idea is that the large eddies are unstableand break up transferring energy to the smallereddies.
The smaller eddies undergo the same processand so on
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THE ENERGY CASCADE
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THE ENERGY CASCADE This energy cascade continues until the
Reynolds number is sufficiently small thatenergy is dissipated by viscous effects: the eddy
motion is stable, and molecular viscosity isresponsible for dissipation.
Big whorls have litt le whorls,
which feed on their velocity;
and l ittle whorls have lesser whorls,and so on to viscosity
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THE KOLMOGOROV HYPOTHESES
What is the size of the smallest eddies???
The KOLMOGOROV Hypotheses address thisfundamental question along with many others
There are three propositions that Kolmogorovmade:
The hypothesis of local isotropy
First similarity
Second similarity
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THE KOLMOGOROV HYPOTHESES
We first introduce some useful scales
Denote by L0 the length scale of an eddycomparable in size to the flow geometry
Denote by the demarcation scalebetween the largest (L > LEI) eddies and the smallest
eddies (L < LEI)
061 LLEI
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THE KOLMOGOROV HYPOTHESES
Kolmogorovs hypothesis of local isotropy:
At sufficiently high Reynolds number, the small-
scale motions (L < L0) are statist ically isotropic.
As the energy passes down the cascade, allinformation about the directional properties ofthe large eddies (determined by the flowgeometry) is also lost. As a consequence, the
small enough eddies have a somehow universalcharacter, independent of the flow geometry
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THE KOLMOGOROV HYPOTHESES
Kolmogorovs first similarity hypothesis:
In every turbulent flow at sufficiently high
Reynolds number, the statistics of the small-
scale motions (L < LEI
) have a universal form that
is uniquely determined by and .
Just as the directional properties are lost in theenergy cascade, all information about the
boundaries of the flow is also lost, therefore, theimportant processes responsible for dissipationare determined uniquely by
and
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THE KOLMOGOROV HYPOTHESES
Kolmogorovs second similarity hypothesis:
In every turbulent flow at sufficiently high
Reynolds number, there exists a range of scales
whose properties are uniquely determined by
The three hypothesis are shown in the sketch tofollow
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THE KOLMOGOROV HYPOTHESES
DI EI 0
Universal Equilibrium Range Energy Containing Range
Inertial SubrangeDissipation Range
Governed byGoverned by &
The smallest scales
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Mass Conservation:
Momentum Conservation:
Source
sgx
p
x
u
xx
uuuuii
ij
i
jj
iji ++
=
+
DiffusionConvectionAccum.
0=
+
i
i
x
u
THE CLOSURE PROBLEM
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THE CLOSURE PROBLEM
The infinitude of scales in a turbulent flowrenders the flow equations impossible to tacklemathematically in that the velocity field
represents a random variable A clever proposition, addressed by Reynolds
reduces the numbers of scales from infinity to 1or 2
After all, for all practical purposes, we are onlyinterested in the mean flow and in the mean fluidproperties
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THE CLOSURE PROBLEM Problems: Elliptic & Non-Linear Equations; Pressure-Velocity
Coupling & Temperature-Velocity Coupling; Turbulence: Unsteady,
3-D, Chaotic, Diffusive, Dissipative, Intermittent, ...; Infinite Numberof Scales (Grid
Re9/4), Etc.
Solution: Reduce the number of scales Reynolds Decomposition.
' +=
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Turbulence: RANS Result: Reynolds Averaged Navier-Stokes (RANS) Equations.
Closure Problem: Turbulent Stresses ( ) are unknowns
Turbulence Models.
0x
U
i
i =
+
ui
i
ji
j
i
jj
iji Sx
Puu
x
U
xx
UUU+
=
+
''
jiuu ''
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THE CLOSURE PROBLEM
One can suggest to write transport equations forthe reynolds stresses, however, one would endup with triple correlationsand so forth
The idea is that we have to stop somewhereand model the correlation using physicalarguments
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THE CLOSURE PROBLEM
( )
( ) ( )
( )( )
( )
( )ij
j
k
k
i
i
k
k
j
k
j
k
i
ij
i
j
j
i
ij
i
k
j
j
k
i
k
ji
k
ijkjikkji
k
ijijji
ij
k
i
kj
k
j
ki
k
jikji
x
u'
x
u'
x
u'
x
u'
x
u'
x
u'
x
u'p
x
u'p
D
x
u'u
x
u'u
x
u'u'
x
upupuuux
Gtugtug
Px
Uuu
x
Uuu
x
u'u'U
u'u'
+
+
+
+
+
+
++
+
+
=
+
2
''
'''''''
''
''''
ExampleOf transport
EquationFor the
TurbulentStress
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THE CLOSURE PROBLEM
Physical mechanisms governing the change of turbulent stresses &fluxes.
Convection
Viscous
Turbulent
Pressure
Diffusion
Net BalanceGeneration
Dissipation
Distribution
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THE CLOSURE PROBLEM Unknown Correlations:
-) Turbulent & Pressure Diffusion (Dij, Dit)
-) Pressure-Strain Correlations (ij, it)
-) Dissipation Rates (ij, it)
-) Possible Sources (Sit)
Modeling is required in order to close the governing equations.
Turbulence Models:
-) Algebraic (zero-equation) models: Mixing length, etc.
-) One-equation models: k-model, t-model, etc.
-) Two-equation models: k-, k-kl, k-2, low-Reynolds k-, etc.
-) Algebraic stress models: ASM, etc.
-) Reynolds stress models: RSM, etc.
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THE CLOSURE PROBLEM
Reynolds Stress Models
Algebraic Stress Models
Two-Equation Models
One-Equation Models
Zero-Equation Models
Constant/'' =kuu ji
=P
L/k 2/3=
( )2dydUk L=
FirstOrderModels
SecondOrderMod
els
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First-Order Models Governing equations of turbulent stresses and fluxes are very
complex Analogy between laminar and turbulent flows.
Laminar Flow:
Turbulent Flow:
Generalized Boussinesq Hypothesis
j
j
iji
j
j
i
ij x
u
3
2
x
u
x
u
+
=
k32
xU
xUuu ij
i
j
j
itji
tij
+==
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First-Order Models Zero Equation Models
As the name implies, these models do notmake use of any additional equation. In such
models we contend with a simple algebraicrelation for the turbulent viscosity
Zero equation models are based on the mixinglength theory which is the length over whichthere is high interaction between vortices
Note that the turbulent viscosity is not aproperty of the fluid! It is a flow property
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Zero-Equation Models Mixing Length Theory: Dimensional analysis
lm is determined experimentally. For boundary layers :
lm = y for y
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First-Order Models One Equation Models
As the name implies, we develop one equationfor a given variable. The variable of choice is
the turbulent kinetic energyWe write a transport equation for turbulent
kinetic energy and use algebraic modeling forthe new variables introduced
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One-Equation Models Turbulent Kinetic Energy:
Dimensional analysis
u : velocity scale - proportional to k1/2
l : length scale mixing length lm
Other choices: (Bradshaw et al.), etc.
Direct calculation of t using a transport equation for the eddyviscosity (Nee & Kovasznay).
lu t
( )222 wvu
21k ++=
mt kC l=
21
uvu
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One-Equation Models Turbulent Kinetic Energy:
Unknown Correlations that require modeling:-) Turbulent & Pressure Diffusion;-) Dissipation Rates.
( )
( )
( )kj
i
j
i
j
jj
2
i
j
kkii
j
iji
j
j
x
u
x
u
Dx
kpuuu
2
1
x
GPtugx
Uuu
x
kUk
k
+
+
=
+
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Conclusion Turbulence is a very important phenomenon Turbulence modeling is a highly active area of
research
The future of CFD has a high dependence on
turbulence models LES seems to be the prevailing method for
the next two decades
DNS, with the ever increasing computational
power and parallel computing technologies,remains a dream for all researchers. Will itever be feasible on a personal computer?