3rd Lecture : Momentum Transfer

MechanismAdvanced Theory of Ship Flow Control

Dept. of Naval Architecture and Ocean Engineering

Lecturer : Inwon Lee

1

Reynolds Stress• What is Reynolds stress ?

– Correlated velocity fluctuations

– Correlation coefficient

– Cauchy-Schwartz inequality

• -1 or 1 : Perfect correlation• 0 : no correlation

2

jiuu

2j

2i

jiij

uu

uuc

1c1 ij

– Pure shear flow• Homogeneous in x1, x3 direction

• Only nonzero velocity component : • Only nonzero stress component in laminar flow• Instantaneous velocity components :• If v1 & v2 is correlated, x1 momentum mv1 of a molecule is

transported in the x2 direction.• Momentum transport per molecule• N molecules per unit volume, transport of x1 momentum in

the x2 direction per unit time and unit area(average for N molecules; not for time)

• Minus sign should be attached to give positive shear stress for positive strain rate

• Baseline mechanism for momentum transfer : molecular collision

Momentum Transfer Mechanism (I)3

0xx 31

21 xU

2

112 dx

dU

21 v,v

• Kinetic Theory of Gas : molecular transport in laminar flow– Model to understand turbulent momentum transport

21vmv~

21vvNm~

2112 vv

• Mean free path : average length between collision• The momentum transported to the molecule at x2=0

by the molecule at x2=- :

• Taylor series expansion

• Higher order terms neglected if

• A local length scale is defined

assumed (in room temp. = 710-8 m)

Momentum Transfer Mechanism (II)4

11 U0UmM

• Kinetic Theory of Gas : molecular transport in laminar flow– Molecular collisions : moving molecule

22

122

2

111 x

U2x

UmU0UmM

22

12

2

1

xU

21

xU

221

221

xUxUL

21L

• Momentum transfer is then approximated by , this is part of v1 correlated with v2

• Number of molecule per unit volume N, rms molecular velocity a (=speed of sound)• Shear stress = momentum transfer per unit area and time :

2

1

xUmM

a32,

xU

xUa

32

xUNmaNaM

2

1

2

1

2

112

• Example) Air at room temp. : = 710-8 m, a = 340m/s = 1.510-5 m2/s

• Knudsen number

• Molecular time scale , Mean flow time scale

• Thermal motion of the molecules should not be disturbed very much of the flow. Molecules collide many thousands of times before the flow has been changed appreciably.

• The dynamical equilibrium of the thermal motion of the molecules in shear flow is the equilibrium state at rest.

Momentum Transfer Mechanism (III)5

1ReMa

23

UaU

23

a23Kn

• Kinetic Theory of Gas : molecular transport in laminar flow– Molecular collisions

sec10a

~ 10 sec10

xU1~ 4

21

• Turbulent fluctuations u1 & u2 are usually correlated negatively.

• The energy of the eddies has to be maintained by the shear flow because they are continuously losing energy to smaller eddies and dissipation.

• Most effective eddy to absorb energy from the mean shear : vortices with axis being aligned with shear flow.

• Vortex stretching by mean shear thinner vortex tube stronger vorticity due to the angular momentum conservation

• 3-D Vortical motion enhances correlation between u1 & u2.Cf) 2-D vortical motion can’t maintain correlation under mean shear

Momentum Transfer Mechanism (IV)6

• Estimation of Reynolds Stress– Vortex stretching

• For a moving particle starting from x2 = 0, t = 0 to pass x2 at time t (Let’s assume particle does not lose momentum during travel to x2).

• Momentum deficit per unit volume

Momentum Transfer Mechanism (V)7

• Estimation of Reynolds Stress : cont’d– Mixing length model : moving particle

2

12

121121

121

xUx

0,0ut,xu0,0Ut,xU0,0u~t,xu~M

negligible

• Fluid volume transported per unit area and unit time

• Then, the average momentum flux/(unit area·unit time) shear stress

• Conservation of momentum assumption : lead to continuous increase of as x2 increase not realistic, we have to assume u2 and x2 become essentially uncorrelated at values of x2 comparable to some transverse length scale l .

• This gives and the length scale l is called “Mixing Length”.

dtdxu~ 2

2

222

12

2

12212 u~x

xUu~

xUxu~M

22u~x

2

222122 uuuc~u~x

• Shear stress becomes

• Let us introduce a turbulent (eddy) viscosity ,

• Problem : u2 and l is flow properties, not fluid properties again function of position and time.• In reality, turbulence consists of fluctuating motion in a broad spectrum of length scales. One

may argue that large eddies contribute more to the momentum transfer than small ones.

– Length scale problem• Momentum deficit approximation is based on the

assumption the local length scale is much larger than the transverse length scale

i.e., (Recall, molecular collision model in kinetic theory).

• But this is not true for turbulent flows, because• Strictly speaking, this “gradient transport model” should not be used to estimate turbulent

momentum transport.

Momentum Transfer Mechanism (VI)8

• Estimation of Reynolds Stress : cont’d– Mixing length model : moving particle

2

12122

2

112 x

Uucu~xxU

T 21T2

121

2

1T12 uc

xUuc

xU

212121 xUx~0,0u~t,xu~M

221

221

xUxUL

21

xUxUL 2

212

21

~L

• Dispersion rate

• Let us investigate how to define l how x2 increase as the fluid particle travel away from x2=0.

• For a stationary process (time origin is meaningless),

• Introducing Correlation coefficient

• Lets define a Lagrangian (average related to moving particles) integral time scale

• Dispersion rate becomes

Momentum Transfer Mechanism (VII)9

• Estimation of Reynolds Stress : cont’d– Physical meaning of mixing length : integral scale

212222 ucu~xx

dtd

21

t

022

t

022

t

02

t

02

t

02

t

02

22

t

022

tdtu~tu~tdtu~tu~tdtu~tdtu~dtdtdtu~tdtu~

dtd

21x

dtd

21

tdtu~tx

t

022

22 dtu~tu~x

dtd

21ttLet

t

0

22

222

2

22 dcu~xdtd

21

u~tu~tu~c

0

dc

22

22 u~x

dtd

21

• If we define Lagrangian integral length scale

• Time scale T is hard to determine, but the length scale lL is comparable to the transverse Eulerian integral scale l (overbar indicates time mean)

– Turbulent stress vs. laminar stress

• Again, we use

Momentum Transfer Mechanism (VIII)10

• Estimation of Reynolds Stress : cont’d– Physical meaning of mixing length : integral scale

0

222222 dx0u~xu~u~

L222

222L uu~x

dtd

21u

1RecucxUx

Uuc 12

1T

21

12

2

12112

• Turbulent time scale , Mean flow time scale

• Time scales are almost comparable Turbulence could not become equilibrium which is independent of the mean flow. The turbulence is continually trying to adjust to its environment, without ever succeeding.

• Cf) for laminar flow, Molecular time scale , Mean flow time scale

• Exceptional case : slowly developing in the downstream direction (boundary layer)

in this case, turbulence may be in approximate equilibrium with respect to its environment.

– Reference• Tennekes, H. and Lumley, J. L., 1972, “A First Course in Turbulence,” The MIT Press, pp.27~52

Momentum Transfer Mechanism (IX)11

• Estimation of Reynolds Stress : cont’d– Comparison of time scales

12

2

1

1

1

1

2 xu~xU

xU~

xu

sec10a

~ 10

sec10~ 4

sec10u

~ 4

2

sec10

xU1~ 4

21

Chapter 4: Fluid KinematicsNO23834 : Fluid Mechanics 12

MV(to)

MV(t)

pathlinesetc. )t,r(VV

)t,r(pp )t,r(rr

Pathline

o

o

o

4-1 Lagrangian DescriptionParticle approach : tracks the position and velocityBased upon Newton's laws of motion. Difficult to use for practical flow analysis.

Fluids are composed of billions of molecules.Interaction between molecules hard to describe/model.

However, useful for specialized applicationsSprays, particles, bubble dynamics, rarefied gases.Coupled Eulerian-Lagrangian methods.

Chapter 4: Fluid KinematicsNO23834 : Fluid Mechanics 13

CV(t)etc. )t,r(VV

)t,r(pp )t,r(

Field approach : a control volume is defined. We define field variables which are functions of space and time.

Velocity field,Well suited for formulation of initial boundary-value problems (PDE's).

4-1 Eulerian Description

Chapter 4: Fluid KinematicsNO23834 : Fluid Mechanics 14

4-1 Eulerian vs. Lagrangian DescriptionLagrangian Approach

man-to-man, tag each individual particledifficult to track many particles at a time

Eulerian Approachzone defense, observe fluid particles in a CVeasy to setup experiment or computational domain

Lagrangian

Eulerian