Title:

A numerical study of the stability of the swept attachment line boundary layer

A number of numerical schemes are employed in order to gam insight into the stability of the infinite swept attachment line boundary layer. The basic flow is taken to be of the Hiemenz class, i.e. a twodimensional stagnation line flow with an added crossflow giving rise to a constant thickness boundary layer along the attachment line. For the perturbation flow quantities the assumption is made that the chordwise velocity components is linearly dependent 011 the chordwise coordinate x in 2D, while the spanwise and normal velocity components are taken to be independent of x. In all numerical schemes, a secondorderaccurate finitedifferencing scheme is tlsf'd in the normal to the wall direction, a pseudospectral approach is emplo)wl in the other directions; temporally, an implicit CrankNicolson scheme is used and the resulting system of equations is solved using an initialvalueproblem approach. Extensive use of the efficient Fast Fourier Transform (FFT) algorithm has been made in order to transform the solution betw( > en real and Fouriertransform spaces; the FFT was chosen because of the substantial savings in computingcost which result from its implementation. The linear twodimensional results (i.e. results obtained for the perturbation flow quantities when small amplitude twodimensional perturbation waves are introduced into the basic flow) of previous investigations were accurately reproduced using this initialvalueprohlem approach, thus departing from normalmode analyses which has invariably been the tool for earlier work. A timeperiodic scheme is also employed, producing identical results to those of the timemarching approach in the case of the timeperiodic forcings. The twodimensional work was extended to study nonlinear effects in the perturbation (Le. nonsmall perturbation of the basic flow), although the cost of this study generally proved prohibitively high. However, some (very expensive) nonlinear results obtained suggest that the grid used for the linear calculations (which resulted in very large code sizes) has to be further refined in the spanwise direction in order to account for the growth of the nonlinear terms. This result is, as expected, Reynolds number dependent and should be taken into consideration when an investigation of the subcritical regime is undertaken. The smalltime behaviour of the flow, i.e. the flow development shortly after smallamplitude perturbations have been introduced into the basic flow, has been studied analytically using the matchf'd asymptot.ics expansions method. The boundary layer is taken to consist of two flow regions, an 'inner' and an 'outer' one. The parameter upon which perturbation qnantities are expanded is in both cases the normal coordinate, taken to be small in the inner region and of order one in the outer region. The stability of the flow is studied at stat.ions off the attachmentline, when threedimensional perturbations are introduced into the Hiemenz flow; in this case assumptions analogous to those of parallelflow are used, namely that there exist two chordwise lengthscales, a slow one corresponding to the slow boundary layer acceleration and a fast one upon which perturbation quantities are dependent. An important result obtained is that, at a station off the attachment line, the critical for the onset of instability Reynolds numher is an order of magnitude smaller than the corresponding twodimensional critical Reynolds number at the attachment line. Difficulties were experienced when trying to determine a lower branch of the neutral loop in spanwise wavenumber  Reynolds number space; a lower branch was not found for wavenumbers as low as the accuracy of the numerical method employed, a result suggesting that a lower branch of the neutral curve may not exist.
