Turbulent Flow

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Chemical & Petroleum Engineering Department

Sharif University of Technology

TURBULENT FLOW

Presented by:

Prof. D.Rashtchian

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Turbulent transport of momentum Turbulence of random velocity fluctuations- Use statistical methods

Turbulent velocity iu~

ˆ ~iii uUu

Component Velocity

gFluctuatin Mean

*************************

Fig.1 Interpret Ui as a time averaged velocity defined by:

T

iiiT

T

iTi udtuUT

dtuT

U00

~ˆ1lim~1lim

T

iiiTi

T

iTi UUdtUT

UdtuT

u00

01limˆ1limˆ

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i.e. the mean value (or time average) of the fluctuating quantity is zero. Assume that Ui the mean flow is steady (∂Ui/∂t = 0) Note: Time averaging commutes w.r.t. differentiation.

ijj

iT

ij

T

j

i

j

i uxx

Udtu

Txdt

xu

Txu ~~1~1~

00

The time average of the fluctuation iu is zero, but the average of the square of the

fluctuation is not zero and the quantity i

i

Uu 2ˆ

is used as a convenient measure of the

turbulent fluctuation-known as the "intensity of turbulence" and ranges from 0.01 to 0.1 for most turbulent flows.

2ˆiu r.m.s. velocity.

Mean K.E./unit volume = nsfluctuatioflowmean

ˆ21ˆ

21 22

iiii uUuUKE

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Equations for the mean flow Consider the momentum and continuity equations. These apply to the instantaneous velocity in a turbulent field.

jj

i

ij

ij xx

uxp

xuu

~~1~

~2

0~

i

i

xu

)1(

The equations must apply on average

ˆ~iii uUu

Continuity

0ˆ~~1lim

0

i

iii

ii

iT

i

i

T xU

uUxx

udt

xu

T (2)

The mean value satisfies continuity. It is the mean value of velocity that we measure and require in applications.

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Momentum: The equations of motion for the mean flow Ui are obtained by taking the time average of all terms in the resulting equation. Consider each term:

(2.1) )ˆˆ(ˆˆ

ˆˆˆˆ

ˆˆ~

~)~~(~~ (i)

ijjj

ijjiji

j

ijjiijijj

iijjjj

jiij

jj

ij

uuxx

UUuuUUx

uUuUuuUUx

uUuUxx

uuuu

xxuu

(2.2) 1)ˆ(1~1 (ii)i

iii x

PpPxx

p

(2.3) ˆ~

(iii)222

jj

iii

jjjj

i

xxUuU

xxxxu

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Hence

(3) )ˆˆ(1 2

ijjjj

i

ij

ij uu

xxxU

xP

xU

U

Equation for mean flow has an additional term.(Drop the ^ ijij uuuu ˆˆ )

Term j

ij

xuu

is analogous to the convective term j

ij x

UU ;

It represents the mean transport of fluctuating momentum by turbulent velocity fluctuations. If iu and ju uncorrelated i.e. 0ijuu - no turbulent momentum transfer but

experience shows that 0ijuu - momentum transfer is a key feature of turbulent motion.

Term )ˆˆ( ijj

uux thus exchanges momentum between the turbulence and the mean

flow (equation 2.1)even though the mean momentumof the turbulent velocity fluctuations is zero ( 0~ iu ).

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Because of the decomposition iii uUu ˆ~ , turbulent motion can be perceived as something which produces stresses in the mean flow. For this reason, equation (3) may be rearrange so that all stress can be put together.

)(ˆˆ jij

iji

j

j

iji

jj

ij T

xuu

xU

xU

Pxx

UU

- mean stress tensor.( ˆ~ T )

i

j

j

ijiijjijiji x

UxU

uuPT ; ˆˆ

(shear) (normal) The contribution of the turbulent motion to the mean stress tensor is ij

Tji uu called the

Reynolds stress tensor. Define, Tjiji ji

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Turbulent shearing stresses Time averaging of the equations of motion leads to the Reynolds stress tensor, ijuu ˆˆ .

iu and ju are the velocity fluctuations in the ji directions at one point and jiuu is a measure of the "correlation" between the fluctuations. Correlated variables jijijjiiji uuUUuUuUuu ˆˆ~~ If 0jiuu , iu and ju are said to be correlated i.e. dependent.

If 0jiuu , uncorrelated i.e. iu and ju are independent.

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Fig2(a)

10

12

21

Ruu

Fig2(b)

10

12

21

Ruu

Fig2(c)

10

12

21

Ruu

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A measure of the degree of correlation between 1u and 2u is obtained from: 21

22

21

21

ˆ.ˆ

ˆˆ

uu

uu

21

0

22

021

2112 ˆ1limˆ :

ˆˆ1lim

T

iTi

T

iTdtu

Tuudt

uuuu

TR

21

2112 uu

uuR

N.B. abbaba )(210)( 222

Hence

T

Tdt

uu

uu

TR

0 22

22

21

21

12 1ˆˆ

ˆˆ1lim

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Pure shear flow Consider a turbulent shear flow with U1(x2) the only non-zero velocity component.

Ω12 is the only component of the mean stress tensor, 122

112 ˆˆ uu

xU

Ω12 – stress in 1 direction on face, normal in 2 direction and must result from molecular transport of momentum in the x2 direction, and turbulent transport.

Assume 02

1

xU .

A fluid particle with positive 2u is being carried by turbulence in positive x2 direction. It is coming from a region where the mean velocity is smaller i.e. is likely to be moving downstream more slowly than its new environment. Thus 1u is negative. Similarly negative 2u associated with positive 1u .

************************ Fig3

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{Momentum/unit volume of flow at A in 1-direction} = )ˆ(~111 uUu

The x1-momentum is transported in the x2-direction if u1 and u2 are correlated. {Flux of x1-momentum in x2-direction} = 211 ˆ)ˆ( uuU {Average flux of x1-momentun in x2-direction} = 21 ˆˆ uu

1u and 2u are negatively correlated: 122112 uuTT

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Turbulent Channel Flow The Navier-Stokes equations in rectangular coordinates are

)(1 2

jijjj

i

ij

ij uu

xxxUv

xP

xUU

For parallel, fully developed, 2 D flow

0...0;0

0

31

32

SHL

xU

xU

UU

ii

0)(1

jiuux

; 0)( 33

uux i

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Hence the equations can be written in the simplified form,

)(10 2

2

uvyy

UvxP

(1)

)(10 2vyy

P

(2)

At the walls 2v = 0, P = P0(x) . Hence form (2)

20 vPP

(3)

dxdP

xP

xP 00

(4)

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Hence (1) can be integrated from y=0 to y with 00

y

uv

uvyUv

yUv

dxdPy

y

0

0 )(0

At y=h, uv=0, 0/ yU (zero velocity gradient, no correlation)

0

0 )(

y

w

yU

dxdPh

Defining a friction velocity u*

2 uw

Substituting in (5)

)1(2* h

yuyUvuv

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Equation (8) may be written in dimensionless form in 2 ways.

(I) )1()/()/( *

*2* h

yhyduUd

huv

uuv

R* =u* h/v. As R* becomes large, (R* is of course a Reynolds number), the viscous stress is suppressed. Such a limit will not applied because viscous forces must always dominate near solid boundaries.

(II) *

*

*

*2*

.1)/()/(

huv

vyu

vyuduUd

uuv

In this case as R* becomes large the change in total stress becomes small. Defining appropriate dimensionless variables

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;*

vyu

y

uUu ; h

y

Then

11*2

* ddu

Ruuv

(11)

*2*

1Ry

dydu

uuv

(12)

Law of wall For large R* (from 12)

12*

dydu

uuv

(13)

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The solution of this equation must be of the form,

)(2*

ygu

vu; )( yfu (law of the wall) (14)

For sufficiently small y+, turbulent stress negligible.

1

dydu

; with u+(0)=0 (15)

u+ = y+ Core region For large R* (from 11)

12uuv

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This equation gives no information, about U itself. However h and u* are the only feasible length and velocity scales, we can write

ddF

hu

dydU * Where F( ) is some function of . (17)

Integration from the center where U=U0

)(*

0 Fu

UU

(18)

From equation (14),

)(*

yfuU

;

dy

ydfv

udydU )(2

* (19)

Matching (17) & (19),

dydfu

ddF

hu

2

. ; Kdy

dfyddF 1

(20)

.ln1)( constK

F .ln1)( constyK

yf

Hence

.ln1

*

0 constKu

UU

.ln1

*

constyKu

U

Discussion

To simplify (12) to (14) requires 1*

Ry (a)

To simplify (11) to (16) requires

1*

1ddu

R (b)

Matching only possible if y 0

In practice it is found that

1.0100

y

are sufficient

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Now 1.0 ; 1.0*

Ry

(cf.(a))

And 100y 100* R

/100* R

1.0

1000* R Experimentally

5.2

ddu

*5.2

*1

Rddu

R

Hence )1(*5.2

*1

Rd

duR ( cf . (b))

Also from (20) Kdydf

y 1

.)ln(1)( constyK

yf (21)

Experimentally

0.1ln5.2*

0

u

UU

0.5ln5.2*

yuU

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Application 1. For Engineering Purposes these equations have been used for 1.0 , i.e. to describe the core region, and also for 0 . Note as

0 , */ uUu 2. Sometimes the Universal Velocity profile is used. Equn. (15) u+=y+ for y+ 5 Equn. (21) u+=2.5lny++5.0for y+ 30. Limits determined experimentally. A curve fit for 5<y+<30 Is u+=5.0 ln y+ - 3.05

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Example of use of turbulent velocity profiles. momentum transfer

Friction factor 2

2*

2

2

21 U

u

Uf

Using the velocity defect law for flow in a tube

*

0

0 *

02

2.1u

UUrdy

uUU

h

hy

y

1

02 }1ln5.2{2

dhhr

Now y-hr ; hddyhy / ; )1( hr

Hence

1

0*

0 }1ln5.2){1(2)(

du

UU

Also from experimental results

0.1ln5.2**

0 uU

uU

0.50.1ln5.2ln5.2 y 0.6*ln5.2 R

22

Re2

**2 fuf

vh

vhuR

6]22

Reln[5.2*0

fuU

10

222

*

]24

5ln2

525ln5[6]22

Reln[5.2 f

uU

53.022

Relog07.4110

f

f

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Mass transfer: Turbulent Taylor Analysis. Proc. Royal. Soc. (1954), A223, P446, for Axial Dispersion in turbulent pipe flow. Consider diffusion equation in rectangular coordinates for simplicity.

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Turbulence in pipe flows Scope of Turbulence Most flows in nature: rivers, the atmosphere Engineer: pipe flow, packed and plate column Pipe Flow Laminar sublayer - viscous forces dominate, very thin Transition region - region of damped turbulence because of nearby wall,

eddy size y. Turbulent core - region of fully developed turbulence, eddies of size d,

velocity nearly constant.

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Turbulent Velocities

- local downstream velocity fluctuates due to turbulent eddies .decompose

velocityeddy

velocitylocalmean

velocityusInstantane

uuut ˆ

- definition of u (mean velocity)

T

tdtuT

u0

1

- clearly the average of the eddy velocity is zero

0ˆ1

ˆ11)ˆ(1

0

0 00

dtuT

dtuT

udtT

dtuuT

u

T

T TT

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- the magnitude of turbulent velocities is characterized by the RMS -

2

1

0

2ˆ1

T

dtuT

u

(RMS fluctuating or eddy velocity.) - the turbulence intensity is defined by,

turbulent intensity = uu

(typically up to 0.1) i.e. the average eddy velocity

may be 1/10 of the mean velocity.

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Properties of turbulent flows (with particular reference to pipe flows)

1) Irregularity local velocities fluctuate in random manner. But all turbulent flows are irregular. E.g. smoke plume. 2) 3D Nature pipe flows are normally considered as 1 dimension in that downstream velocity depends only on radius. However in turbulent flows normal velocity components, though zero on average, have fluctuating components, (V and W ). These give rise to turbulent stresses (remember the mail bag example) and are important in turbulent energy processes. This 3D nature adds the mathematical difficulty.

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3) Turbulence is a property of the flow not of the fluid writing Newton’s law for a flow involving turbulent stresses.

dydu

T )(

; [divided by ]

Where is the kinematics viscosity .

TL 2

T eddy viscosity In laminar sub layer T Transition region ~T Turbulent core T Thus T varies with environment and is a flow property.

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4) Mixing in turbulent flows-diffusivity Rewrite Newton’s law in the form, explicit in shear stress.

dy

udT

)(

Dimensions :

L

LUM

TL

TLUM 32

2 .

i.e. (momentum flux) = (diffusivity) * (gradient of momentum / volume) - this fundamental relation shows how transport (here of momentum) is related to the driving force (momentum gradient).the coefficient, , is the momentum diffusivity. It shows how large a flux is produced by a given gradient. Exactly analogous laws apply for heat transfer (Fourier's Law) and mass transfer (Flick's law).

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- Now T in turbulent flows. Turbulent is a very effective mixer of momentum which accounts for the almost constant velocity of the core region will usually be of almost constant temperature and composition. But T in laminar sublayer.

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In laminar flow it is the molecular motion which transports momentum. (Remember mail bag example). Hence lower rates of transport for a given driving force. Alternatively if we consider heat transfer from the wall to bulk, heat conduction across the laminar sub layer dominates the process (Heat transfer comes later). - Pictorially

Eddy gives rise to normal velocity V . This transports x directional momentum in the y direction gives rise to a momentum flux, or shear, .

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5) Dissipative Nature of Turbulence. - Turbulence comprises eddies of all sizes. - The largest eddies are as big as the flow field. They extract energy from the flow but are not efficient at dissipating energy. In the absence of an energy source, however, turbulence dies away . - There is an energy cascade from the large eddies, through eddies of progressively smaller size until a lower limit is reached. This lower limit is controlled by viscous dissipation of energy and Kinematics viscousity and the rate of energy supply are the important quantities. Based on dimensional analysis this lower limit of eddy size is given by:

43

43

Re

dud

Where size of small eddies; kinematics viscosity

d = size of largest eddies; u' = RMS turbulent velocity

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dvsi 6

ddd

vsii 12

468 3

2

ddd

vsiii 24

16664 3

2

vs

volumesurfaceratendissipatio

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High Reynolds number phenomenon

- Express Newton's Law of viscosity in dimensionless form.

dydu

dyd

uud

dydu

dud

u Re1

Re1

22

Reynolds's number arises in dimensionless form of Newton's Law. - Similarity: compare two flows in similar geometries(same shape but different size)i.e. flows exhibiting geometrical similarity. Suppose Reynolds numbers of each flow are the same though d,u, and of each flow may be individually different. Then as a consequence of the above equation each flow will have the same dimensionless distribution of stress and velocity gradient as a function of position,

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Stress and velocity gradient as a function of position, provided each has the same Reynolds number; i.e. u+ = f(y+) kinetic similarity τ+ = g(y+) dynamic similarity. The consequence is that friction factor (dimensionless wall shear stress) can be considered a unique function of Re. Consider a cylindrical element of diameter d and length of δx

Viscous forces xddrdu .

Interia forces dtduxde

4

2

Redtdrd

cesViscousForcesInteriaFor

High Re-interia forces dominate → Turbulent flow Low Re-viscous forces dominate → Laminar flow

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Summery Notes on Turbulence

Most flows are turbulent both in nature and engineering. A turbulent pipe flow can be divided into three regions:

a) Laminar sublayer - no eddies. b) Transition region – damped eddies (size y) c) Turbulent core – undamped eddies (size d)

Turbulent velocities: uuut ˆ {Instantaneous = local mean + fluctuant}

T

tdtuT

u0

1 {T is a time long enough to include many eddies}

21

0

2 ]ˆ1[ T

dtuT

u {RMS velocity characterizes turbulence}

wvu {Turbulence is homogenous} 1.0/ uu {Turbulence intensity}

Newton's Law in turbulent flows. It is tempting to write

dydu

t )(/

TurbulenceforityVisEddyMolecularforityVisKinematic

t coscos

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ν is a fluid property and constant. νt is a flow property and depends on environment (eddy size)

Rewrite above equation as

dy

udvv t)()( dimensions 22 ][

][][][

TL

LM

(momentum flux)=(momentum diffusivity)(gradient of mom/vol) Large ν implies rapid mixing. Diffusivity has dim. [L]2/[T] νT >> ν : Turbulent flows are rapidly mixed due to eddies.

Energy in turbulent flows: turbulent dissipates considerable energy. Large eddies take energy from mean floe, but are not efficient in

dispersing energy. Small eddies do dissipate energy efficiently. There is a transfer of energy to the small eddies, which appears as heat due to frictional effects.

Smallest eddy size, η, si given by ( dimensional analysis)

43

)(

vdu

d {η is also a good estimate of laminar sub-layer thickness}

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Rynolds Number arises in Newton's Law in dimensionless form

dydu

dyduud

dudydu

uu Re1

)/()/(

222

It may be interpreted as the ratio (interia forces / viscous forces).

Large Re implies dominance of interia forces which promote turbulence. Small Re will dominance of friction (viscous) forces gives laminar flows.

Similarity (Consider different flows of same Reynolds Number) If we have geometric similarity (e.g. two different pipe flows) then we will have kinematic similarity (same du+/dy+) and dynamic similarity (same τ+).

Result f = τw

+ = f(Re)

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Turbulent Flow

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