1 MECH 221 FLUID MECHANICS (Fall 06/07) Chapter 4: FLUID KINETMATICS Instructor: Professor C. T. HSU

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MECH 221 FLUID MECHANICS(Fall 06/07)

Chapter 4: FLUID KINETMATICS

Instructor: Professor C. T. HSU

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MECH 221 – Chapter 4

4. FLUID KINETMATICS Fluid kinematics concerns the motion of fluid

element. As the fluid flows, a fluid particle (element) can translate, rotate, and deform linearly and angularly

Translation Rotation

Linear deformation Circulation

Dilation Viscous stress

Angular deformation

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4.1. Translation The translation considers mainly the velocity and

acceleration along the trajectory of fluid element in linear motion

z

y

x

0

r

d

r

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4.1. Translation For the fluid element moving along the

trajectory r(t), the velocity is simply given by v =dr/dt = (u,v,w). As the description is basically Lagrangian, the acceleration a is given by

which, for steady flows, reduces to

zw

yv

xu

tDt

D

vvvvv

a

zw

yv

xu

Dt

D

vvvv

a

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4.2. Linear Deformation (Strain) Deformation: change of shape of fluid element

For easily understanding, we illustrate here in two-dimensions. The results then can be easily extended to 3-dimensions. Consider the rectangular fluid element at the initial time instant given in the following picture

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4.2. Linear Deformation (Strain) The initial distance between points A and B is ∆x

and between A and C is ∆y. After a short time of ∆t, the distances then become ∆x+∆Lx and ∆y+∆Ly due to different velocities at B and C from A

A u( x) ; B u( x)+uxx

A v(y) ; C v( y)+vy y

Lx=(u B–uA)t=u

xxt

L x

t=uxx

Ly=(vC–vA)t=vyyt

L y

t=uyy

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4.2. Linear Deformation (Strain) The linear strain rate in x and y directions are

then given by

Similarly, for 3-D flows we have in the z-direction,

xu

tx

L

tx

xx

lim

0

yv

ty

L

ty

yy

lim

0

zw

tz

L

tz

zz

lim

0

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4.3. Dilation Volumetric expansion & contraction

The fluid dilation is defined as the change of volume per unit volume. We are more interested in the rate of dilation that determines the compressibility of fluids. For 2-D flows,

yx)y

Ly)(x

LxV1yxV ( ;

t

yx)tyyv

y)(txxu

x(

yx

1

t

V

V

1

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4.3. Dilation Then, the rate of dilatation becomes,

It is easy to generalize this dilation rate for 3-D flows and to reach

For incompressible flow, the rate of dilation is zero,

v

y

v

x

u

txyAxyA

tV

V

lim0Δt

lim0Δt

z

w

y

v

x

u

v

0 v

for 2-D flows

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4.4. Angular Deformation (Strain) Now consider the deformation between A and B

caused by the change in velocity v, and the deformation between A and C by change in u

A v ( x ) , B v ( x )+ v

x x ; =

v

x x t

txv

xη

αα ΔΔΔ

)tan(ΔΔ

A u ( y ) , C u ( y )+ u

y

y ; = u

y

y t

tyu

yΔ

ΔΔ

)tan(ΔΔ

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4.4. Angular Deformation (Strain) For , the counter clockwise rotation of AB is

equal to clockwise rotation of AC; therefore, the fluid element is in pure angular strain without net rotation and the angular strain is equal to either or . However, if ≠ , the strain then is equal to

. The rate of angular strain is then given by

2/)(

)(21

2)(1

0lim

yu

xv

ttyxxy

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4.4. Angular Deformation (Strain) Similarly, we can extend to other planes y-z and z-

x to obtain:

)(21

zv

yw

zyyz

)(21

xw

zu

xzzx

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4.5. Rotation If then the fluid element is under rigid

body rotation on the x-y plane. No angular strain is experienced, i.e.,

0 yxxy with yu

xv

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4.5. Rotation When ≠ , the rotation of fluid element in x-y

plane is the average rotation of the two mutually perpendicular lines AB and AC; therefore,

where a counter clockwise rotation is chosen as positive and the rotation axis is in the z direction

y

u

x

v

2

1

2t

1Ω

ΔΔlim

0Δtz

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4.5. Rotation Rotation is a vector quantity for fluid elements in

3-D motion. A fluid particle moving in a general 3-D flow field may rotate about all three coordinate axes, thus:

kjiΩ zΩyΩxΩ and so,

yu

xv

Ω

xw

zu

Ω

zv

yw

Ω

z

y

x

21

,21

,21

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4.5. Rotation The vorticity of a flow field is defined as

wvuzyx

kji

ω v

kji

yu

xv

xw

zu

zv

yw

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4.5. Rotation Therefore,

The flow vorticity is twice the rotation

In 2-D flow, ∂/∂z=0 and w=0 (or const.), so there is only one component of vorticity,

Irrotational flow is defined as having

ω

Ωω 2 v

0 v

Ω

zΩ

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4.5. Rotation A fluid particle moving, without rotation, in a flow

field cannot develop a rotation under the action of a body force or normal surface force. If fluid is initially without rotation, the development of rotation requires the action of shear stresses. The presence of viscous forces implies the flow is rotational

The condition of irrotationality can be a valid assumption only when the viscous forces are negligible. (as example, for flow at very high Reynolds number, Re, but not near a solid boundary)

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4.6. Circulation Consider the flow field as shown below

The circulation, , is defined as the line integral of the tangential velocity about a closed curve fixed in the flow,

Γ

V

ds

sdΓ v

where ds is the tangential vector along the integration loop. i.e. with being the unit tangential vector

Tdcd is Ti

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4.6. Circulation Where is the line-element vector tangent to the

closed loop C of the integral. It is possible to decompose the integral loop C into the sum of small sub-loops, i.e.,

Without loss of generality, each sub-loop can be a rectangular grid as illustrated below.

C

s)d(dΓΓ v

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4.6. Circulation Therefore,

As a result, we have

yvxyyu

uyxxv

vxud ΔΔΔΔΔΔ

Ayxyxyu

xv

dΓ zz ΔΔΔΔΔ

dAdAdΓAA z ks )( vv

where A is the area enclosed the contour

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4.6. Circulation

Stokes' theorem in 2-D:

The circulation around a closed contour (loop) is the sum of the vorticity (flux) passing through the loop

This is an expression to illustrate the Green’s Theorem. In fact, the surface A can be a curved surface

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4.6. Circulation Then for each sub-loop on the surface, we

have locally

where is the vorticity normal to the surface enclosed by the small increment loop C

nA

d

0A

Csv

lim

n

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4.7. Viscous Stresses The strain rate tensor S is a symmetric tensor

that measures the rate of linear and angular deformations of fluid element. The strain rate tensor is expressed as:

where the superscript “T” represents the transpose

T)( vv S

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4.7. Viscous Stresses In term of a Cartesian coordinate system,

they are expressed as:

z/wz/vz/u

y/wy/vy/u

x/wx/vx/u

v

zwywzvxwzu

ywzvyvxvyu

xwzuxvyuxu

/2)//()//(

)//(/2)//(

)//()//(/2

S

and

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4.7. Viscous Stresses Following the Stokes’ hypothesis, the viscous

stress tensor is linearly related to the rate of dilation and the strain rate tensor by

where I represents the unit tensor, i.e.,

SIσ )(3

2v v)(

100

010

001

I

vσ

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4.7. Viscous Stresses The proportional constants of the above linear

relation are the volume viscosity and shear viscosity of the fluid respectively. It is seen that the fluid viscosity leads to additional normal stresses, as well as shear stresses. Note that is a symmetric tensor, i.e.,

Total stress is given by

vσ

xzzxzyyzyxxy ττ,ττ,ττ

ISIσσσ ))(( v 3

2pvp

σ

and

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1 MECH 221 FLUID MECHANICS (Fall 06/07) Chapter 4: FLUID KINETMATICS Instructor: Professor C. T. HSU