Three-Dimensional Effects in 

 Boundary Layer Stability 



Leslie M. Mack 



California Institute of Technology 



Pasadena, California 



SUMMARY 



Most work in linearized boundary-layer stability 

 theory has been carried out either on the basis 

 of two-dimensional mean flow and plane wave dis- 

 turbances with the wavenumber in the flow direction, 

 or, for a more general case, by a transformation 

 of the equations to two-dimensional form. This 

 procedure can obscure important physical aspects 

 of wave propagation in two space dimensions . In 

 this paper the stability equations are retained 

 in three-dimensional form throughout. A method 

 for treating spatially amplifying disturbances with 

 a complex group velocity is adopted and applied 

 first to oblique waves in a two-dimensional bound- 

 ary layer, and then to the two-parameter yawed 

 Falkner-Skan boundary layers . One parameter is 

 the spanwise to chordwise velocity. For boundary 

 layers with small crossflow, the maximum amplifi- 

 cation rate with respect to frequency is calculated 

 as a function of flow angle for waves whose normal 

 is aligned with the flow. Next, the minimum crit- 

 ical Reynolds number of zero-frequency crossflow 

 instability is obtained for both large and small 

 pressure gradients , and finally the instability 

 properties of two particular boundary layers with 

 crossflow instability are determined for all un- 

 stable frequencies. 



1 . INTRODUCTION 



Most work in linearized boundary- layer stability 

 theory has been restricted to two-dimensional mean 

 flows, and, for these flows, even further restricted 

 to plane-wave disturbances with the wave normal in 

 the flow direction.* The latter restriction is 

 normally justified by reference to the theorem of 



*Such a wave is called two-dimensional because 

 it has only two disturbance velocity components. 

 All other plane waves have three velocity components 

 in any coordinate system, and are called three 

 dimensional. 



Squire (1933) , which states that in a two-dimensional 

 incompressible boundary layer, the minimum critical 

 Reynolds number is given by a two-dimensional wave. 

 Even though the most unstable wave at a given 

 Reynolds number is two dimensional in accordance 

 with the theorem, the most unstable wave of a 

 particular frequency can well be three dimensional. 

 Furthermore , the iinstable three-dimensional waves 

 can have phase orientation angles (the angle between 

 the local freestream direction and the wave normal) 

 up to almost 80°. Any method for the estimation 

 of transition that is based on stability theory 

 must take this large range of unstable three di- 

 mensional waves into account. For a supersonic 

 two-dimensional boundary layer, even the most un- 

 stable plane wave at a given Reynolds number is 

 three dimensional. The two-dimensional waves be- 

 come of little importance as the Mach number in- 

 creases above one until the hypersonic regime is 

 reached, where a two-dimensional second-mode wave 

 is the most unstable. 



When we turn to three-dimensional boundary layers , 

 there are no two-dimensional waves , but the trans- 

 formation of Stuart [Gregory et al. (1955)] reduces 

 the three-dimensional temporal stability problem 

 to a series of two-dimensional problems. That is, 

 the temporal amplification rate can be obtained by 

 solving a two-dimensional problem for the boundary- 

 layer profile in the direction of the wave normal. 

 This approach was carried through numerically by 

 Brown (1961) for the rotating disk and a limited 

 number of swept-wing boundary layers . When the 

 same approach is applied to the spatial theory, 

 it leads to complex velocity profiles and loses 

 much of its utility except as a computational device. 



Instead of trying to make a two-dimensional 

 world out of a three-dimensional world, it might 

 as well be accepted that boundary- layer instability 

 is inherently three dimensional , even with two- 

 dimensional mean flow, and to formulate the insta- 

 bility problem directly as three dimensional [Mack, 

 (1977); this paper will be referred to as M77]. A 

 transformation of the dependent variables reduces 

 the order of the incompressible eigenvalue problem 



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