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EML 5709 Homework 2 Spring 1997

1. For the velocity field

   

   sketch a bundle of streamlines at a typical time t=0.5 both in 3D and their projections onto the x,y- and z,x-planes. Solution.

2. For the same flow, sketch the projection of a typical path line onto the x,y-plane. Also sketch the projection of the pathline on the z,x-plane and in 3D. Eliminate any unnecessary parameter from the solution. Solution.
3. For the same flow, find and draw the streakline generated by a smoke generator at the point (1,0,0) both at time t=0 and at . Eliminate any unnecessary parameter from the solution. Find the difference between the shapes of the pathlines and the streakline. Solution.
4. The following flow field models the formation of a tornado:

   

   where and are constants. Since R and t can be normalized, we can assume without loss of generality that and . Find and draw a couple of streamlines for a time t>0. Solution.

5. For the same flow field, find and draw a typical particle path. Have the particle paths and streamlines the same shape? Solution.
6. For the same flow field, find as a function of R for a streakline. Sketch the streakline for a time and for a time . Solution.
7. (Essentially question 2.3 in the 2nd edition of the book). Given the velocity field , find the circulation around the rectangle , by integrating around its perimeter. Solution.
8. (Essentially question 2.3 in the 2nd edition of the book). Also find the circulation by finding the vorticity and integrating . Check Stokes' theorem. Solution.
9. (Essentially question 2.5 in the 2nd edition of the book). For the two-dimensional velocity field

   

   compute the circulation around the square . . Solution.

10. (Essentially question 2.5 in the 2nd edition of the book). For the same flow, compute the vorticity and integrate . Check Stokes' theorem. Solution.
11. Repeat the previous two questions but use polar coordinates. Check that you get the same results as before. How much is in polar coordinates? So how much is the circulation around any contour around the origin? How much is the circulation around any contour not around the origin? Solution.
12. (Expanded question 2.6 in the 2nd edition of the book). For the three flows

    (a)

    , with );

    (b)

    , with );

    (c)

    , with );

    compute the circulation around an arbitrary circle around the origin. Also compute the volumetric flow rate coming out of a cylinder of unit length around the z-axis. Solution.
13. For the three flows of the previous question, compute the vorticity and see whether Stokes' theorem applies. If not, explain. Solution.
14. For the three flows of the previous two questions, compute the divergence of the velocity and see whether the divergence theorem applies to the flow out of the cylinder. If not, explain. Solution.
15. Consider viscous flow through a pipe of arbitrary cross-section and length. Show that the product of cross sectional area times average axial component of vorticity, , is constant along the pipe, the same way that the volume flow u A is constant for an incompressible flow. What is the value of the constant? Solution.
16. Discuss where you would find vorticity in the flow about a car. Must the vortex lines always meet the surface of the car tangentially? If not, give cases in which they do not. What numerical advantage could there be in describing the flow field about the car by means of its vorticity, instead of its velocity? Solution.

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