t.saad - introduction to turbulent flow (march2004)

8/10/2019 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





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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





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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


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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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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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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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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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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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?

t.saad - introduction to turbulent flow (march2004)

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