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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 (1-3.1 – 1-3.6, 1-3.8, 1-3.12 – 1-3.17)

– Understanding between Lagrangian and Eulerian viewpoints

– Detailed understanding of Figure 1-14

– Eq. (1-12) use of tan-1 vs. 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 ˆ,,,ˆ,,,ˆ,,,,,,,

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KINEMATIC PROPERTIES

• Let Q represent any property of the fluid (, T, p, etc.)

• Total differential change in Q

• Spatial increments

• Time derivative of Q of a particular elemental 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 rotation of the two sides BC and BA

– Or the rotation of the diagonal DB to B’D’

dx

dy

A

B C

D A’

B’

C’

D’

y

x

+

d

d

ddd z 2

1

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2. ROTATION

dydty

u

y

u

x

v

dt

d

dtx

v

dx

dxdtxv

d

dty

u

dy

dydtyu

d

ddd

z

z

2

1

tan

tan

2

1

1

1

A’

B’

C’

D’

d

d

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

+

d

d

• 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

2

12

1Shear-strain increment

Shear-strain rate

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COMMENTS: STRAIN VS. STRAIN RATE

• Strain is non-dimensional

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

– In solid mechanics this is often given the symbol , non-dimensional

– Recall Hooke’s Law: = E• Modulus of elasticity

• In fluid mechanics, we are interested in rates

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

dxdxdtxu

dxdtxx

dxdtx

udx

Extensional strain in x-direction

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FIGURE 1-14: DISTORTION OF A MOVING FLUID ELEMENT

dxdtx

v

Not

e: M

ista

ke in

text

boo

k F

igur

e 1-

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 d• BA has rotated CW (-d)

• Overall CCW rotation since d > d• d and d both related to velocity derivates

through calculus limits

• Rates of angular rotation (angular velocity)

• 3 components of angular velocity vector ddt

• Very closely related to vorticity

• Recall: the vorticity, , 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

dty

u

dydtyv

dx

dydtyu

d

dtx

v

dxdtxu

dx

dxdtxv

d

ddd

y

x

z

dt

dt

z

2

2

1

2

1

2

1

tanlim

tanlim

2

1

1

0

1

0

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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 unstrained state (AB and BC)

• Shear-strain rates

• Shear-strain rates are symmetric

jiij

zx

yz

xy

dx

dw

dz

du

dz

dv

dy

dw

dy

du

dx

dv

dt

d

dt

d

dd

2

1

2

1

2

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

vx

u

dtx

u

dx

dxdxdtxu

dxdt

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 of symmetric tensors

• For all symmetric tensors there exists one and only one set of axes for which the off-diagonal terms (the shear-strain rates) vanish

– These are called the principal axes

3

2

1

00

00

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 angular velocity (d/dt)

dt

d

x

u

uuuuu

x

uu

ijij

j

i

ijjiijjiji

j

iji

,,,,,

,

2

1

2

1

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DEVELOPMENT OF N/S EQUATIONS: ACCELERATION

surface

body

externalsurfacebody

externalsurfacebody

fgDt

VD

gf

fffDt

VD

Dt

VDa

ffffV

Fa

Fam

• 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