312 Solutions Manual x Fluid Mechanics, Fifth Edition 1.2 1.0 0.8 0.6 0.4 0.2 0.0 -0.2 -0.2 O -0.1 0.0 0.1 0.2 0.3 0.4 The streamlines are logarithmic spirals moving out from the origin. [They have axisymmetry about O.] This simple distribution is often used to simulate a swirling flow such as a tornado. 4.17 A reasonable approximation for the two-dimensional incompressible laminar boundary layer on the flat surface in Fig. P4.17 is § 2 y y2 · u U¨ 2¸ ©G G ¹ for y d G Fig. P4.17 where G | Cx1/2 , C const (a) Assuming a no-slip condition at the wall, find an expression for the velocity component v(x, y) for y d G. (b) Then find the maximum value of v at the station x 1 m, for the particular case of airflow, when U 3 m/s and G 1.1 cm. Solution: The two-dimensional incompressible continuity equation yields wv wy wu wx or: v § 2 y dG 2 y 2 dG · U ¨ 2 , or: v © G dx G 3 dx ¸¹ 2U dG § y2 y3 · dG , where 2 3¸ ¨ dx © 2G dx 3G ¹ 2U y dG § y y 2 · dy_x dx ³0 ¨© G 2 G 3 ¸¹ C G 2 x 2x (b) We see that v increases monotonically with y, thus vmax occurs at y Ans. (a) G: const 313 Chapter 4 x Differential Relations for a Fluid Particle vmax v_y UG 6x G (3 m/s )(0.011 m ) 6(1 m ) m s 0.0055 Ans. (b) This estimate is within 4% of the exact vmax computed from boundary layer theory. 4.18 A piston compresses gas in a cylinder by moving at constant speed V, as in Fig. P4.18. Let the gas density and length at t 0 be Uo and Lo, respectively. Let the gas velocity vary linearly from u V at the piston face to u 0 at x L. If the gas density varies only with time, find an expression for U(t). V = constant p(t) x x = L(t) x=0 Fig. P4.18 Solution: The one-dimensional unsteady continuity equation reduces to wU w (Uu) wt wx Enter dU wu U , where u dt wx wu wx V L x· § V ¨1 ¸ , L © L¹ and separate variables: The solution is ln(U/Uo ) U ³ Uo ln(1 Vt/L o ), or: U L o Vt, U dU U t V³ o U(t) only dt L o Vt § Lo · © Lo Vt ¸¹ Uo ¨ Ans. 4.19 An incompressible flow field has the cylindrical velocity components XT Cr, Xz K(R2 – r2), Xr 0, where C and K are constants and r d R, z d L. Does this flow satisfy continuity? What might it represent physically? Solution: We check the incompressible continuity relation in cylindrical coordinates: 1w 1 w vT w vz (rv r ) r wr r wT wz 0 0 0 0 satisfied identically Ans. This flow also satisfies (cylindrical) momentum and could represent laminar flow inside a tube of radius R whose outer wall (r R) is rotating at uniform angular velocity. 4.20 A two-dimensional incompressible velocity field has u K(1 – e–ay), for x d L and 0 d y d f. What is the most general form of v(x, y) for which continuity is satisfied and v vo at y 0? What are the proper dimensions for constants K and a?