14 12/08 M

1. Streamlines around a cylinder or sphere for very low Reynolds number, very viscous Stokes flow look superficially the same as those for high Reynolds number ideal inviscid flows: both are symmetric front/rear. But do they really look the same? Find out. Unfortunately, Stokes flow around a cylinder is tricky. For truly low Reynolds numbers, the flow in the vicinity of the cylinder becomes negligible, so the streamlines become infinitely widely spaced. But Stokes flow around a sphere is more reasonable. So, use the Matlab program streamlines.m to plot the streamlines for potential flow around a cylinder and for both potential and Stokes flow around a sphere. You will need to put in the appropriate streamfunctions at the mesh points; potential flow around a sphere is in section 19.8, and Stokes flow in 21.8. You will need also need to set what contour values of the streamfunction to plot; the program creates equally spaced $y$-values for the streamlines to plot far upstream. However, you will need to translate the upstream $y$-values of the streamlines to their values of the streamfunction.

   Note both differences in streamline spacing near the surface and streamline curvature away from the sphere.

2. This question is about how the vorticity is different for low Reynolds number flow around a sphere versus high Reynolds number flow around it. For the low Reynolds number flow, the velocity field was given in homework question 6.1, and a more extensive discussion is in section 21.8.

   For high Reynolds number flow, assume that the sphere has just been impulsively started from rest, and has moved only a small distance compared to the fluid at infinity. In that case, the velocity inside the boundary layer can be taken to be, (in boundary layer variables),
   

   $$u(x,y) = u_e {\rm erf}\left(y/\sqrt{4\nu t}\right)$$
   
   
   where $u_e$ is the velocity immediately above the boundary layer, which can be deduced from section 19.8. The error function erf is in the Math Handbook, see the Index, and can be differentiated readily. The vorticity is $-\partial{u}/\partial{y}$ (explain why). Take $\sqrt{4\nu t}$ as a relative small fraction of the radius, but not so small you can no longer see the individual vorticity contours.

   Now use the Matlab program vorticity.m to plot lines of constant vorticity for both flows.

   Do keep in mind that for later times, the differences between Stokes flow and high Reynolds number flow are far more dramatic still, as the high Reynolds number flow will have a huge wake enclosed by strong vorticity layers at one side.

3. Summarize in your own words why there is an energy cascade in turbulent flows, what are Kolmogorov scales and why they are there, what is the inertial range and why it exists. For this and the subsequent questions, use the revised notes on turbulence [pdf]

4. For the turbulent mixing layer where the fluid at one side is at rest, give and plot the theoretical half shear layer thickness versus $x$. Use data in the book for any constants. Do the flows shown in the pictures seem consistent with the obtained relationship?

5. For the plane and axisymmetric turbulent jets, give and plot the half jet thickness versus $x$. Get any constant from the book. Also plot the maximum jet velocity versus $x$ for the jets. The book says somewhere that the axisymmetric (round) jet has a faster centerline (maximum) velocity decay as the two-dimensional jet. How did they get that crazy idea?

6. For the plane and axisymmetric turbulent wakes, plot the wake thickness versus x. How do they compare? Also plot the maximum velocity defects versus $x$ and compare those. What is the problem with the axisymmetric turbulent wake for large enough $x$?