6 HW 6

In this class,

• Questions must be answered in order asked.
• Solutions must be neat.
• You must use the given symbols.
• You must show all reasoning.
• Copying is never allowed, even when working together.

1. In the Euler equations given in your notes, ignore gravity. Now use the energy equation to convert the ${\rm D}\rho/{\rm D}t$ derivative in the continuity equation by a ${\rm D}p/{\rm D}t$ one. That achieves that continuity and $x$-momentum involve derivatives with respect to only $p$ and $\vec v$. Write out these equations for the special case of one-dimensional, unsteady, inviscid, compressible flow (nonlinear acoustics in a pipe). Now add $\rho a^2$ times your continuity equation to $\pm a$ times your momentum equation and show that the result is of the form:
   
   $$\left[ \frac{\partial p}{\partial t} + (u \pm a) \frac{\pa... ...al t} + (u \pm a) \frac{\partial u}{\partial x} \right] = 0$$
   
   
   Argue from your knowledge of calculus that the terms in the square brackets are derivatives ${\rm d}p/{\rm d}t$ and ${\rm d}p/{\rm d}t$, not along particle paths, but along the paths of sound waves moving left and right. The above two equations (one for each sign) are called the compatibility equations.

2. Continuing the previous question, for normal acoustics the variations in $u$ and $p$ are small enough that the $u$ in $u\pm a$ can be ignored, and that the coefficients $a$ and $\rho$ can be assumed to be constants. Write the equations under these conditions. Now you know from calculus (shifting a function to the right) that a pressure wave moving to the right without changing shape is given by $p=f_1(x-at)$ (where function $f_1$ describes the shape of the pressure wave). Similarly a wave in the velocity field moving in the same direction takes the form $u=f_2(x-at)$. Show that to satisfy both compatibility equations, the shapes of the pressure and velocity waves must be related as $f_1'=\rho a f_2'$. (The energy equation then gives the perturbation in density as the final of the three unknowns in a one-dimensional compressible flow.)

3. Write the nondimensionalized incompressible Navier-Stokes equations out fully in terms of the scaled Cartesian velocity components $(u^*,v^*,w^*)$ and the scaled pressure $p^*$. Include the continuity equation too!

4. A fan makes a lot of aerodynamic noise. Assume that the power $P$ of the emitted acoustic noise and the volumetric flow rate $Q$ produced by the fan depend on the fan diameter $D$, its frequency $\Omega$, the air density $\rho$, and the air speed of sound $a$. Use the Buckingham $\Pi$ theorem to find simplified expressions for the acoustic power and the volumetric flow rate. As selected parameters, use the two parameters that are beyond your control as a fan designer, and the fan diameter.

5. Assuming that the volumetric flow rate is also a given, what can you say about the likelyhood of improving the noise generated by the fan by messing around with fan diameter and frequency? Hints: consider the acoustic power per unit flow rate. And think about the physical interpretation of the relevant nondimensional parameter(s).

6. Using the formulae in the scanned class notes, and corresponding notations, find the stress tensor for Stokes flow around a sphere, in spherical coordinates.

7. Find the drag force on the sphere by finding the stresses on the surface of the sphere, taking $z$-components of them, and then integrating over the surface.