Mae 5130 viscous flows
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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. 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)

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MAE 5130: VISCOUS FLOWS

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Mae 5130 viscous flows

MAE 5130: VISCOUS FLOWS

Lecture 3: Kinematic Properties

August 24, 2010

Mechanical and Aerospace Engineering Department

Florida Institute of Technology

D. R. Kirk


Chapter 1 critical reading

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…’


Kinematic properties two views of motion

KINEMATIC PROPERTIES: TWO ‘VIEWS’ OF MOTION

  • 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)

  • 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


Kinematic properties

KINEMATIC PROPERTIES

  • Let Q represent any property of the fluid (r, 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


4 types of motion

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:

    • Translation

    • Rotation

    • Shear strain

    • 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)


1 translation

y

+

x

1. TRANSLATION

D

A

dy

B

C

dx


1 translation1

y

+

x

1. TRANSLATION

D’

A’

D

A

dy

B’

C’

vdt

B

C

dx

udt


2 rotation

y

+

x

2. ROTATION

D

A

dy

B

C

dx


2 rotation1

y

+

x

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’

D

A

A’

db

D’

dy

B’

da

B

C

dx

C’


2 rotation2

y

+

x

2. ROTATION

A’

db

D’

B’

da

C’


3 shear strain

y

+

x

3. SHEAR STRAIN

D

A

dy

B

C

dx


3 shear strain1

y

+

x

3. SHEAR STRAIN

  • Defined as the average decrease of the angle between two lines which are initially perpendicular in the unstrained state (AB and BC)

D

A

db

dy

da

B

C

dx

Shear-strain increment

Shear-strain rate


Comments strain vs strain rate

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 e, non-dimensional

    • Recall Hooke’s Law: s = Ee

      • 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 e for strain rate, [1/s]

    • Strain rates will be written as velocity derivates


4 extensional strain dilatation

y

+

x

4. EXTENSIONAL STRAIN (DILATATION)

D

A

dy

B

C

dx


4 extensional strain dilatation1

y

+

x

4. EXTENSIONAL STRAIN (DILATATION)

  • Extensional strain in x-direction is defined as the fractional increase in length of the horizontal side of the element

A’

D’

D

A

dy

B’

C’

B

C

dx

Extensional strain in x-direction


Figure 1 14 distortion of a moving fluid element

FIGURE 1-14: DISTORTION OF A MOVING FLUID ELEMENT

Note: Mistake in text book Figure 1-14


Comments on angular rotation

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

  • da and db both related to velocity derivates through calculus limits

  • Rates of angular rotation (angular velocity)

  • 3 components of angular velocity vector dW/dt

  • Very closely related to vorticity

  • Recall: the vorticity, w, is equal to twice the local angular velocity, dW/dt (see example in Lecture 2)


Comments on shear strain

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


Comments on extensional strain rates

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


Strain rate tensor

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


Useful short hand notation

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 (e) plus an angular velocity (dW/dt)


Development of n s equations acceleration

DEVELOPMENT OF N/S EQUATIONS: ACCELERATION

  • 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)


Summary

SUMMARY

  • All kinematic properties of fluid flow

    • Acceleration: DV/Dt

    • Translation: udt, vdt, wdt

    • Angular velocity: dW/dt

      • dWx/dt, dWy/dt, dWz/dt

      • Also related to vorticity

    • Shear-strain rate: exy=eyx, exz=ezx, eyz=ezy

    • Rate of dilatation: exx, eyy, ezz

      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


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