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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

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 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 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