Consider the subsonic compressible flow over the wavy wall treated in Example 2.1. Derive the equation for the velocity potential for this flow as a function of x and y.

EXAMPLE 2.1

The subsonic compressible flow over a cosine-shaped (wavy) wall is illustrated in Figure 2.17. The wavelength and amplitude of the wall are l and h, respectively, as shown in Figure 2.17. The streamlines exhibit the same qualitative shape as the wall, but with diminishing amplitude as distance above the wall increases. Finally, as y → ∞, the

Figure 2.17 Subsonic compressible flow over a wavy wall; the streamline pattern.

streamline becomes straight. Along this straight streamline, the freestream velocity and Mach numbers are V_∞ and M_∞, respectively. The velocity field in cartesian coordinates is given by

Consider the particular flow that exists for the case where ℓ = 1.0 m, h = 0.01 m, V_∞ = 240 m/s, and M_∞ = 0.7. Also, consider a fluid element of fixed mass moving along a streamline in the flow field. The fluid element passes through the point (x/ℓ, y/ℓ) = (1/4, 1). At this point, calculate the time rate of change of the volume of the fluid element, per unit volume.

V. M Streamline at o