Ghosh - 550 Page 3 11/29/00

Worked Out Examples

(Continuity Equation)

Example 1. (Evaluation of y(x,y)):

Determine the family of y functions that will yield the velocity field .

Solution

1. Statement of the Problem

a) Given

· Velocity field: Þ &

b) Find

· Family of y functions from the given velocity field.

2. System Diagram

It is not necessary for this problem.

3. Assumptions

· Steady state condition

· Incompressible fluid flow

· 2 - D problem

4. Governing Equations

Stream function (incompressible fluid flow version) definition: &

5. Detailed Solution

With the definition of stream function and the given velocity components:

… 

… ‚

 Þ Þ Þ ,

where f(x) is any function of x including constants.

Using this y obtained, take that will be … ƒ

Comparing ‚ with ƒ, , that represents f(x) = constant.

Finally,

6. Critical Assessment

The stream function, y, exists; therefore, the velocity field satisfies the continuity equation, , and also it can be said that it is a valid velocity field.

Example 2. (Use of y and its properties):

In a parallel one-dimensional flow in the positive x direction, the velocity varies linearly from zero at y = 0 to 100 ft/s at y = 4 ft. Determine an expression for the stream function, y. Also determine the y coordinate above which the volume flow rate is half the total between y = 0 and y = 4 ft.

1. Statement of the Problem

a) Given

· 1 - D flow parallel to the positive x direction.

· Velocity varies linearly, u(y = 0 ft) = 0 ft/s & u(y = 4 ft) = 100 ft/s.

b) Find

· An expression for the stream function, y.

· y coordinate above which the volume flow rate is half the total between y = 0 and y = 4 ft.

2. System Diagram

3. Assumptions

· Steady state condition

· Incompressible fluid flow

· 2 - D problem

4. Governing Equations

Stream function (incompressible fluid flow version) definition: &

5. Detailed Solution

Find the velocity from the given information, u(y = 0 ft) = 0 ft/s & u(y = 4 ft) = 100 ft/s.

Since the velocity varies linearly, .

There is no flow going on in the y direction. Þ v = 0 ft/s.

Using the definition of stream function,

… 

… ‚

 becomes: Þ Þ

where f(x) is any function of x including constants.

Using this y, take that will be . … ƒ

Comparing ƒ with ‚, . Þ f(x) = constant.

Finally,

The volume flow rate, Q, across AB in the diagram can be evaluated as follows:

For a unit depth (dimension perpendicular to the xy plane), the flow rate across AB is

Along AB, x = constant, and . Therefore,

The total volume flow rate is

The half of the total volume flow rate is then,

\

6. Critical Assessment

The stream function, y, exists; therefore, the velocity field satisfies the continuity equation, . Also it can be said that it is a valid velocity field for 2-Dimensional incompressible flows. This problem also demonstrates how volumetric flow rate may be computed alternatively (without using ) by the use of a y property)