Relative Velocity, Velocity Gradient, and Deformation Shear Deformation

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# Relative Velocity, Velocity Gradient, and Deformation Shear Deformation - PowerPoint PPT Presentation

3.2 Introduction to Motion and Velocity Field: Deformation, Boundary Conditions for Velocity Field, and Boundary Layer. Relative Velocity, Velocity Gradient, and Deformation Shear Deformation Boundary Conditions for Velocity Field No-Penetration No-Slip Conditions

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Presentation Transcript
3.2 Introduction to Motion and Velocity Field: Deformation, Boundary Conditions for Velocity Field, and Boundary Layer
• Relative Velocity, Velocity Gradient, and Deformation
• Shear Deformation
• Boundary Conditions for Velocity Field
• No-Penetration
• No-Slip Conditions
• No-Slip Condition and Boundary Layer
Very Brief Summary of Important Points and Equations
• Deformation

In order to have deformation,

• relative velocity between neighboring point , or

is required.

• Boundary condition for velocity field at a solid wall
• Boundary Layer: Thin shear layer near a solid wall where [because of no-slip condition (and viscous effect)] shear stress is relatively large and viscous/frictional effect is important.

B

B

t

A

B

t + dt

t

t + dt

A

B

A

A

Relative velocity in the transverse direction:

Relative velocity in the axial direction:

B

B

t

A

B

t

t + dt

t + dt

A

B

A

A

If , no deformation.

If , no deformation.

If , shear deformation (/rotation –component of).

If , deformation/stretching.

Relative Velocity, Velocity Gradient, and Deformation (Rate)
• In other words, in order to have deformation,
• relative velocity between neighboring point , or

is required.

d l

d u = uB – uA= relative velocity of B with respect to A

B (t)

B (t + d t)

Fluid element at time t + d t

d y

d a

Fluid element at time t

A

Shear deformation rate =

Dimension:

Shear deformation rate =

Shear Deformation Rate / Shear Strain Rate

y

Smaller du/dy at B – lower shear deformation rate

Velocity profile at x = xo

B

Larger du/dy at A– higher shear deformation rate

A

x

u

x = xo

Velocity Profile (and Shear Deformation Rate)
• u = u (y): The plot of the variation of the velocity u with the transverse coordinate y is referred to as the velocity profile.

From The Japan Society of Mechanical Engineers, 1988, Visualized flow: Fluid motion in basic and engineering situations revealed by flow visualization, Pergamon Press.

Flow Image

w

• Consider the relative velocity of fluid with respect to a solid wall:
• Let = fluid velocity wrt earth at point w on the wall
• = solid wall velocity wrt earth at point w on the wall
• Hence, the relative velocity of fluid wrt the solid wall at w is given by
• Decomposing the relative velocity into the normal (to the local surface) and tangential components:

Real: No-slip / No-penetration

Ideal: Free-slip / No-penetration

w

• Hence, (for impermeable wall)
• for real fluid/flow
• for ideal fluid/flow
• For an impermeable wall, fluid cannot penetrate into solid, hence the normal component of the relative velocity vanishes.
• This is called no-penetration condition.
• This is not the case for, e.g., porous wall.

For ideal fluid/flow:

free-slip

(No constraint)

For real fluid/flow, it is observedno-slip

Freestream:

Low du/dy  Lowtyx

approximately inviscid region.

Freestream velocity

Viscous/frictional stress

Flow

Boundary Layer:

high du/dy hightyx

highly viscous region.

Solid wall

Zero fluid velocity at the wall (no slip)

No-Slip Condition, Viscosity, and Boundary Layer

Laminar boundary layer

(From Van Dyke, M., 1982,

An Album of Fluid Motion, Parabolic Press.)

• Due to no-slip condition at the solid wall, fluid velocity decreases from the value in the freestream to zero value at the wall as the wall is approached.
• This creates a thin layer of highdu/dy, and thus a highly viscous region. This thin region is called boundary layer.

Freestream velocity

Solid wall

No-slip

Decreasing velocity as the wall is approached.

Outside flow is ~ inviscid

Boundary layer is highly viscous

y

y

Still air

B

B

Flow

U

x

A

A

Plate is moving at the speedU.

Flow between two stationary parallel plates

(Channel flow)

Example: Qualitatively sketch velocity profile from the knowledge of BC
• Use the knowledge of the boundary condition for a velocity field to qualitatively sketch the velocity profile along the traverse AB
• Assume that the only dominant velocity isu (Vx)