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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

- 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…’

- 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

- 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

- 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)

y

+

x

D

A

dy

B

C

dx

y

+

x

D’

A’

D

A

dy

B’

C’

vdt

B

C

dx

udt

y

+

x

D

A

dy

B

C

dx

y

+

x

- 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’

y

+

x

A’

db

D’

B’

da

C’

y

+

x

D

A

dy

B

C

dx

y

+

x

- 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

- 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

y

+

x

D

A

dy

B

C

dx

y

+

x

- 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

Note: Mistake in text book Figure 1-14

- 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)

- 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

- 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

- 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

- 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)

- Momentum equation, Newton
- Concerned with:
- Body forces
- Gravity
- Electromagnetic potential

- Surface forces
- Friction (shear, drag)
- Pressure

- External forces

- Body 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)

- 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