Answer Happy

EXAMPLE 2.1 The subsonic compressible flow over a cosine-shaped (wavy) wall is illustrated in Fig. ure 2.17. The wavelength and amplitude of the wall are / 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 → o, the Streamline at 2h 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 number are V and Mo, respectively. The velocity field in cartesian coordinates is given by 2πχ uV h 27 ве 2. By/4 (2.35) =v=[1+*+ ( E cos 278 ) e<31851] v = -Vh24 (sin 24,8) and 2.17 2лх e-2By/l (2.36) where B = V1 - M 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/2) = (1,1). At this point, calculate the time rate of change of the volume of the fluid element, per unit volume. Solution From Section 2.3.4, we know that the time rate of change of the volume of a moving fluid element of fixed mass, per unit volume, is given by the divergence of the velocity V.V. In cartesian coordinates, from Equation (2.19), we have au av V.V= (2.37) ду From Equation (2.35), + ax 2 2л 2.TX -Van () * (sin 2,5) , e-2ßy/l au ax and from Equation (2.36), (2.38) 2 av = :+V2ch + () B sin In 27.0) e-2.By/l (2.39) ay
Substituting Equation (2.38) and (2.39) into (2.37), we have 277 2лх *. - (-) (**) * (sin 2–5). (B Vach e-2xByle (2.40) Evaluating Equation (2.40) at the point x/l = 1 and y/l = 1, = 2 V.V=B -(8-) \n(?). Vah e-278 (2.41) Equation (2.41) gives the time rate of change of the volume of the fluid element, per unit volume, as it passes through the point (x/l, y/2) = (1, 1). Note that it is a finite (nonzero) value; the volume of the fluid element is changing as it moves along the streamline. This CHAPTER 2 Aerodynamics: Some Fundamental Principles and Equations 125 is consistent with the definition of a compressible flow, where the density is a variable and hence the volume of a fixed mass must also be variable. Note from Equation (2.40) that V.V = 0 only along vertical lines denoted by x/l = 0, 7, 1, 1},..., where the sin(27x/l) goes to zero. This is a peculiarity associated with the cyclical nature of the flow field over the cosine-shaped wall. For the particular flow considered here, where l = 1.0 m, h = 0.01 m, V = 240 m/s, and M2 = 0.7, where B = 1 - M = (1 - (0.75 = 0.714 =
and hence the volume of a fixed mass must also be variable. Note from Equation (2.40) that V.V = 0 only along vertical lines denoted by x/l = 0, 1, 1, 15, ... , where the sin(27x/l) goes to zero. This is a peculiarity associated with the cyclical nature of the flow field over the cosine-shaped wall. For the particular flow considered here, where l = 1.0 m, h = 0.01 m, V = 240 m/s, and M2 = 0.7, where = B = V1 - M. = V1- (0.7)2 = 0.714 Equation (2.41) yields 2.31 v.v = (0,714 (0714-0.714) (240) (0.01) () , -2(0.714) -0.7327 s-1 The physical significance of this result is that, as the fluid element is passing through the point (, 1) in the flow, it is experiencing a 73 percent rate of decrease of volume per second (the negative quantity denotes a decrease in volume). That is, the density of the fluid element is increasing. Hence, the point (5, 1) is in a compression region of the flow, where the fluid element will experience an increase in density. Expansion regions are defined by values ofx/l which yield negative values of the sine function in Equation (2.40), which in turn yields a positive value for 7. V. This gives an increase in volume of the fluid element, hence a decrease in density. Clearly, as the fluid element continues its path through this flow field, it experiences cyclical increases and decreases in density, as well as the other flow field properties.