14 



oscillation, is added to the basic laminar flow which is known to be well rep- 

 resented by the Navier-Stokes equations. 



For two-dimensional flows of the boundary- layer type a periodic dis- 

 turbance wave is specified by 



. , . ia{x- ct) , 



0(x,y,t) = 0(y)e [1] 



where is the disturbing stream function, 



<j> is the amplitude of the disturbance flow (<f> is complex), 



a is the wave number -t-, I being the wave length, 



C = C +iC. where C is the wave velocity and C, is the amplification factor, 

 r i r "l 



x is a coordinate measured from the leading edge of the body in the dl- 



direction of flow, 



y is a coordinate measured perpendicular to the flow from the boundary 

 of the body, and 



t is the time . 



The following linearized disturbance differential equation in <t> is 

 then derived from the Navier-Stokes equations: 



[u(y) - c][0" - a 2 <6] - U> =^(* m * - 2c*V + oV) [2] 



where u(y) is the dimensionless velocity distribution of the main flow, and 

 R is a boundary layer Reynolds number defined by ^y- , 6* being the displace- 

 ment thiekness and v the kinematic viscosity of the fluid. All velocities 

 have been referred to a velocity U and all lengths to a boundary -layer dis- 

 placement thickness 6* so that Equation [2] is dimensionless. Primes Indicate 

 differentiation with respect to y/d*. 



The problem becomes one of finding a relationship of the form 



C 1 = g(o,R) 



from which the curve of neutral stability C. = may be drawn in the a,R plane 

 In this way the stable areas (C. < 0) may be graphically separated from the 

 values of a and R which will give instability, (C i > 0). 



The results of such a calculation made by Lin for the Blasius flat- 

 plate flow are given in Figure 10. Here it is seen that below the lower crit- 

 ical Reynolds number ^- = 420 all disturbances are damped. The theory has 

 been successfully checked by experiment; 3 the agreement of theory and experi- 

 ment is indicated in Figure 10. 



Similar calculations made by Schlichtlng and Ulrich for boundary- 

 layer flows with various pressure gradients are of great help in interpreting 

 the flow about a ship model. They computed the stability regions for flows of 



