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MAE 5130: VISCOUS FLOWS

Lecture 3: Kinematic Properties

August 24, 2010

Mechanical and Aerospace Engineering Department

Florida Institute of Technology

D. R. Kirk

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CHAPTER 1: CRITICAL READING

1-2 (all)

Know how to derive Eq. (1-3)

1-3 (1-3.11-3.6, 1-3.8, 1-3.121-3.17) Understanding between Lagrangianand Eulerianviewpoints

Detailed understanding of Figure 1-14

Eq. (1-12) use of tan-1vs. sin-1

Familiarity with tensors

1-4 (all) Fluid boundary conditions: physical and mathematical understanding

Comments

Note error in Figure 1-14

Problem 1-8 should read, Using Eq. (1-3) for inviscid flow past a cylinder

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KINEMATIC PROPERTIES: TWO VIEWS OF MOTION

1. Lagrangian Description

Follow individual particle trajectories

Choice in solid mechanics Control mass analyses

Mass, momentum, and energy usually formulated for particles or systems of fixed

identity (ex., F=d/dt(mV) is Lagrandian in nature)

2. Eulerian Description Study field as a function of position and time; not follow any specific particle paths

Usually choice in fluid mechanics

Control volume analyses

Eulerian velocity vector field:

Knowing scalars u, v, w as f(x,y,z,t) is a solution

ktzyxwjtzyxvitzyxutzyxVtrV ,,,,,,,,,,,,,

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KINEMATIC PROPERTIES

Let Q represent any property of thefluid (r, T, p, etc.)

Total differential change in Q

Spatial increments

Time derivative of Q of a particularelemental particle

Substantial derivative, particle

derivative or material derivative

Particle acceleration vector

9 spatial derivatives

3 local (temporal) derivates

VVt

V

Dt

VD

QVt

Q

Dt

DQ

z

Qw

y

Qv

x

Qu

t

Q

dt

dQ

wdtdz

vdtdy

udtdx

dtt

Qdz

z

Qdy

y

Qdx

x

QdQ

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4 TYPES OF MOTION

In fluid mechanics we are interested in general motion, deformation, and rate of

deformation of particles

Fluid element can undergo 4 types of motion or deformation:1. Translation

2. Rotation

3. Shear strain

4. Extensional strain or dilatation

We will show that all kinematic properties of fluid flow

Acceleration

Translation

Angular velocity

Rate of dilatation

Shear strain

are directly related to fluid velocity vector V= (u, v, w)

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1. TRANSLATION

dx

dy

A

B C

D

y

x

+

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1. TRANSLATION

dx

dy

A

B C

D

A

B C

D

udt

vdt

y

x

+

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2. ROTATION

dx

dy

A

B C

D

y

x

+

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2. ROTATION

Angular rotation of element about z-axis is defined as the average

counterclockwise rotationof the two sides BC and BA

Or the rotation of the diagonal DB to BD

dx

dy

A

B C

DA

B

C

D

y

x

+

da

db

ba ddd

z

2

1

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2. ROTATION

dydty

u

y

u

x

v

dt

d

dtx

v

dx

dxdtx

v

d

dty

u

dy

dydty

u

d

ddd

z

z

2

1

tan

tan

2

1

1

1

a

b

ba

A

B

C

D

da

db

y

x

+

dxdtx

v

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3. SHEAR STRAIN

dx

dy

A

B C

D

y

x

+

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3. SHEAR STRAIN

dx

dy

A

B C

D

y

x

+

db

da

Defined as the average decrease of the angle between two lines which are

initially perpendicular in the unstrained state(AB and BC)

dt

d

dt

d

dd

xy

ba

ba

2

1

2

1Shear-strain increment

Shear-strain rate

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COMMENTS: STRAIN VS. STRAIN RATE

Strain is non-dimensional

Example: Change in length DL divided by initial length, L: DL/L

In solid mechanics this is often given the symbol , non-dimensional Recall Hookes Law: s = E

Modulus of elasticity

In fluid mechanics, we are interested in rates

Example: Change in length DL divided by initial length, L, per unit time:

DL/Lt gives units of [1/s]

In fluid mechanics we will use the symbol for strain rate, [1/s]

Strain rates will be written as velocity derivates

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4. EXTENSIONAL STRAIN (DILATATION)

dx

dy

A

B C

D

y

x+

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4. EXTENSIONAL STRAIN (DILATATION)

dx

dy

A

B C

D

A

B C

D

Extensional strain in x-direction is defined as the fractional increase in length of the

horizontal side of the element

y

x

+

dtx

u

dx

dxdxdtx

udx

dtxx

dxdtx

udx

Extensional strain in x-direction

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FIGURE 1-14: DISTORTION OF A MOVING FLUID ELEMENT

dxdtxv

Note:Mistakeintext

bookFigure1-14

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COMMENTS ON ANGULAR ROTATION

Recall: angular rotation of element about z-axis is

defined as average counterclockwise rotation of

two sides BC and BA BC has rotated CCW da

BA has rotated CW (-db)

Overall CCW rotation since da> db

daand dbboth related to velocity derivates

through calculus limits

Rates of angular rotation (angular velocity)

3 components of angular velocity vector d

dt

Very closely related to vorticity

Recall: the vorticity, w, is equal to twice the local

angular velocity, d/dt (see example in Lecture 2)

dt

d

x

w

z

ud

z

v

y

wd

y

u

x

vd

dt

y

u

dydtyvdx

dydty

u

d

dtx

v

dxdtx

udx

dxdtx

v

d

ddd

y

x

z

dt

dt

z

2

2

1

2

1

2

1

tanlim

tanlim

2

1

1

0

1

0

w

b

a

ba

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COMMENTS ON SHEAR STRAIN

Recall: defined as the average decrease of

the angle between two lines which are

initially perpendicular in the unstrainedstate (AB and BC)

Shear-strain rates

Shear-strain rates are symmetric

jiij

zx

yz

xy

dx

dw

dz

du

dz

dv

dy

dwdy

du

dx

dv

dt

d

dt

d

dd

ba

ba

2

1

2

12

1

2

1

2

1

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COMMENTS ON EXTENSIONAL STRAIN RATES

Recall: the extensional strain in the x-

direction is defined as the fractional increase

in length of the horizontal side of the element

Extensional strains

z

w

y

v

x

u

dtxu

dx

dxdxdtx

udx

dt

zz

yy

xx

xx

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STRAIN RATE TENSOR

Taken together, shear and extensional

strain rates constitute a symmetric 2nd

order tensor

Tensor components vary with change of

axes x, y, z

Follows transformation laws ofsymmetric tensors

For all symmetric tensors there exists

one and only one set of axes for which

the off-diagonal terms (the shear-strainrates) vanish

These are called the principal axes

3

2

1

0000

00

zzzyzx

yzyyyx

xzxyxx

ij

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USEFUL SHORT-HAND NOTATION

Short-hand notation

i and j are any two coordinate directions

Vector can be split into two parts

Symmetric

Antisymmetric

Each velocity derivative can be resolved

into a strain rate ()plus an angularvelocity (d/dt)

dt

d

x

u

uuuuu

x

uu

ij

ij

j

i

ijjiijjiji

j

iji

,,,,,

,

2

1

2

1

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DEVELOPMENT OF N/S EQUATIONS: ACCELERATION

surface

body

externalsurfacebody

externalsurfacebody

fg

Dt

VD

gf

fff

Dt

VD

Dt

VDa

ffffV

Fa

Fam

rr

r

r

r

Momentum equation, Newton

Concerned with:

Body forces

Gravity

Electromagnetic potential

Surface forces

Friction (shear, drag)

Pressure External forces

Eulerian description of acceleration

Substitution in to momentum

Recall that body forces apply to entire mass of fluid

element

Now ready to develop detailed expressions for surface

forces (and how they related to strain, which are

related to velocity derivatives)

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SUMMARY

All kinematic properties of fluid flow

Acceleration: DV/Dt

Translation: udt, vdt, wdt Angular velocity: d/dt

dx/dt, dy/dt, dz/dt

Also related to vorticity

Shear-strain rate: xy

=yx

, xz

=zx

, yz

=zy

Rate of dilatation: xx, yy, zz

are directly related to the fluid velocity vector V= (u, v, w)

Translation and angular velocity do not distort the fluid element

Strains (shear and dilation) distort the fluid element and cause viscous stresses