26 



instability of oscillatory flows. We will be con- 

 cerned mainly with moderate and high frequency os- 

 cillations comparable to the oscillation frequencies 

 of unstable Tollmien-Schlichting waves. Thus, a 

 direct comparison of our results with the low- 

 frequency experimental results cited by Loerke, 

 Morkovin, and Fejer (1975) will not be possible. 



The method used here for analyzing the instability 

 of the oscillatory boundary layer is a combination 

 numerical and perturbation method [Yakubovich and 

 Starzhinskii (1975)]. In this method, the changes 

 in the amplification rates of the free disturbances 

 of the underlying steady boundary layer are computed 

 as perturbation series in the amplitude parameter, 

 Ui/Uq, for any positive value of the frequency, co. 

 Certain resonant and combination frequencies are of 

 particular interest. The numerical method used here 

 to evaluate the perturbation series allows the ef- 

 ficient and easy generation of many terms of the 

 series. 



The plan of this paper is as follows: In Sec- 

 tion 2, we formulate the basic flow whose instability 

 is to be examined along with the associated theory 

 instability problem. Section 3 outlines the solu- 

 tion method. Section 4 discussed the numerical 

 results. Some concluding remarks concerning the 

 instability of womewhat more complex oscillatory 

 boundary layers are contained in Section 5. 



dimensional linear instability theory for steady 

 boundary layers. In particular, the quasi-parallel 

 temporal instability theory as outlined by Rosen- 

 head (1963) is followed. The restriction to two- 

 dimensional disturbances can be justified based on 

 an extended version of Squires' theorem [see von 

 Kerczek and Davis (1974)]. The perturbation ve- 

 locities (u,v) are determined from the stream func- 



tion, ij;(x,ri,t) 



¥'(ri,t)e-' 



if iax 

 Re ^ = - Re -— e 



8i|; 



(3a, b) 



Re T-^ = ReiaVe 

 dx 



The disturbance equation for the perturbation ve- 

 locities is then given by 



0* 



where £ = 3^/311^ - a^, i 



(4) 



-1, Re (a) denotes the 

 real part of a, and a is the wave number of the 

 sinusoidally varying disturbance in the x-direction. 

 The boundary conditions are 



3^ 

 3n 



0, at n 



(5a) 



INSTABILITY THEORY 



and 



The basic flow field whose instability is to be in- 

 vestigated is the oscillatory boundary layer formed 

 on a flat plate in a unidirectional stream with 

 speed Uq + Uj cos ojt perpendicular to the leading 

 edge of the plate and parallel to its plane. Let 

 the cartesian coordinate frame (x,y,z) be placed 

 with its origin in the leading edge of the plate, 

 the X-axis pointing downstream parallel to the plate, 

 the y-axis perpendicular to the plane of the plate 

 and the z-axis pointing in the spanwise direction. 

 For values of the parameter, (woj/Uq) >> 1, the 

 ratio , gi = 6/63, of the boundary layer thickness, 

 6 = /xv/Uq , to th e oscillatory Stokes layer thick- 

 ness, 6g = /2v/u), is large and the oscillatory 

 boundary layer resulting for small values of A = 

 Uj/Uq can be approximated well [see Ackerberg and 

 Phillips (1972)] by the sum of the Blasius profile 

 U3(y) [see Rosenhead (1963), p. 225] and the Stokes 

 layer profile U2(y, t) [Rosenhead (1963), p. 381]. 



Let us scale the x- and y-coordinates by the 

 local value of the displacement thickness 



(5* = 1.7208 /xv/U 



(1) 



Then the transverse coordinate , 



r 1 = 



is defined by 

 n = y/6* and x' = x/6*. The time scale is 6*/Uq 

 so that dimensionless time is t' = tUg/6* and hen 

 forth the primes will be dropped. Then the basic 

 oscillatory boundary layer profile is given ap- 

 proximately by 



U{ri,t) = fB(n) + ARe 



.],.- 



(l+i)Bn, ifit 

 e ) e 



6*/6 



S' 



(2) 



n = 



where fB(n) is the Blasius profile, 

 (d5*/Uo = 26^/R{^, and R^^, = Uq6*/v. 



We shall consider the instability of the basic 

 flow (2) in a similar manner to the standard two- 



3<^ 



-> as n ->■ 



{5b) 



By analogy with Floquet theory for ordinary dif- 

 ferential, equations with periodic coefficients 

 [Coddington and Levinson (1958)], we seek solutions 

 of (4) and (5) in the form 



g(n,t) e 



Xt 



(6) 



where g(n,t) is a periodic function of t with period 

 2Tr/fl. This is a reasonable choice of solution be- 

 cause we are mainly interested in the oscillation 

 induced changes of the principal disturbance mode 

 of the Blasius flow. The principal disturbance mode 

 of Blasius flow has multiplicity one. 



We shall adopt in this study an absolute defini- 

 tion of instability which requires that some measure 

 of the disturbance amplitude becomes infinite as 

 t -s- ■». If the amplitude remains bounded as t ->■ °°, 

 then the flow is defined to be stable to infinites- 

 imal disturbances. However, we must keep in mind 

 that the local instantaneous amplitude may be im- 

 portant in this linear theory because a disturbance 

 may be transiently so large (but bounded) that the 

 linear instability theory is no longer valid. Fur- 

 thermore, the instantaneous magnitude as a multiple 

 of the initial magnitude of the disturbance is an 

 important .quantity for assessing the likelihood of 

 transition from laminar to turbulent flow. Thus we 

 shall consider in detail the gross amplification 

 rate G of a disturbance which we define by 



1 T 



e„ dt 



T 



(7) 



where erp is the total energy of the disturbance de- 

 fined by 



