MAE 5130: VISCOUS FLOWS

1 / 23

# MAE 5130: VISCOUS FLOWS - PowerPoint PPT Presentation

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)

I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.

## PowerPoint Slideshow about 'MAE 5130: VISCOUS FLOWS' - tobias-stewart

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.

- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript

### 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
• 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
• Lagrangian Description
• 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
• 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
• 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

1. TRANSLATION

D

A

dy

B

C

dx

y

+

x

1. TRANSLATION

D’

A’

D

A

dy

B’

C’

vdt

B

C

dx

udt

y

+

x

2. ROTATION

D

A

dy

B

C

dx

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’

y

+

x

2. ROTATION

A’

db

D’

B’

da

C’

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

• 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

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

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
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
• 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
• 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
• 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