Finite Volume Analysis:   Application of Conservation of Linear Momentum   

 

Example 1  (continued)      The figure (Step 1 in strategy) below shows the control volume, control surface, uniform pressures at sections 1 and 2, uniform velocity profile at section 1, linearly distributed, axisymmetric velocity profile at section 2,  drag force, FD, exerted by the screen on the air, the outward unit normals, n1 and n2,  the diameter, d, of the circular duct,

and the positive X-axis.

 

                                   

 

Next

Step 2

Write the equation for conservation of mass and simplify for known values.  Check for steady flow.  Use to find the velocity profile at section 2.

 

                           d/dt ∫ ρ dV  +  ∫ ρ V ● n  dS   =   0

                                 cv            cs

 

Since the flow is steady (established flow)    d/dt ∫ ρ dV  = 0.

                                                                              cv

 

The flow crosses the control surface at sections 1 and 2.  So the second term becomes

 

                         ∫ ρ V1 ● n1  dS1   +    ∫ ρ V2 ● n2  dS2   =  0

                       cs1                           cs2

 

Now the uniform velocity profile at section 1 is    V1 i   and the unit outward normal  n1 = - i

 

                              So  V1 ● n1  =  - V1        Also assume that  ρ = constant

                                                                                                                                                                      

So          ∫ ρ V1 ● n1  dS1   =  - ρ V1 (π d2/4)

           cs1

 

Click here to continue with this example.

 

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