Evolution of disturbances in three-dimensional boundary layers

In the earlier paper (Jeyasingham and Balakumar 1999), the stability characteristics of a supersonic boundary layer over a sharp cone with 5” half-angle at 2’ angle of attack were investigated. The results showed that the eigenmodes are clustered into isolated confined regions and there exist multiple eigenvalues for a given wavenumber or a given frequency. In this paper, evolution of disturbances induced by harmonic sources are investigated by using generalized Fourier transform method. The disturbances originated from the harmonic source in a short distance away from it evolve into sinusoidal waves in the axial direction. The disturbances grow exponentially with different growth rates along the constant azimuthal direction, The disturbances form a confined isolated region in the cross sectional plane and appear as inclined waves in the streamwise and azimuthal plane with the inclination of about 8’ to the inviscid stream lines for the parameters considered in this work. By comparing with the 2-D eigenvalue computations it appears that the disturbances downstream of the source consist of the most amplified eigenmodes in the cross sectional plane. Introduction Our objectives in this work are to investigate what type of disturbances or modes are generated from external disturbances in three-dimensional boundary layers and how do these disturbances evolve downstream. The disturbances are introduced from localized harmonic sources. Balakumar and Malik (1992) investigated the disturbance field induced due to harmonic point source in two-dimensional supersonic flatplate boundary layers using the Fourier, Laplace and Brigg’s method. It was observed that the contribution from the continuous spectra is large closer to the source and decays slowly in the downstream direction. The results also showed that the contribution to the solution from the stable discrete modes is much larger near the source than that from the most unstable mode. Since these stable modes decay in the downstream direction, eventually the unstable mode will become dominant in the region away from the source. In general three-dimensional boundary layers, the mean boundary layer profiles vary in all three directions: streamwise, spanwise or azimuthal and normal directions. Since the meanflow is not uniform in the azimuthal direction, the Fourier transform in the streamwise direction yields a Partial Differential Equations (PDEs) in the normal and azimuthal direction for the disturbance. In the earlier paper (Jeyasingham and Balakumar 1999) we investigated the stability characteristics of the boundary layer over a sharp cone with 5” half-angle at 2’ angle of attack by solving this PDEs as a two-dimensional eigenvalue problem. It was found that due to the variation of the meanflow quantities in the azimuthal direction, the eigenmodes are confined into isolated regions and the amplifications are smaller than that is obtained from the one-dimensional parallel theory. In this work we investigate the evolution of disturbances which are generated from harmonic sources in the threedimensional supersonic boundary layer over a sharp cone with 5” half-angle at 2’ angle of attack at a free stream Mach number of 3.5. It is difficult to determine all the discrete and the continuous spectrum of the PDEs, and we will concentrate on the disturbances in the region away from the source. Formulation We consider three-dimensional compressible boundary layer flow over a sharp cone at an angle of attack (figure 1.). A harmonic source of disturbance is embedded in the wall at streamwise location 11 = ~10. A body oriented coordinate system is used as shown in figure I, with xl taken along the streamwise direction, xz along the azimuthal direction and x3 normal to the surface. Let Q(I~.x~,I~) = {Iri.Uz.LTs!T.p.q} represent the meanflow quantities, velocities, temperature, density and pressure and the corresponding disturbance quantities are represented by Q(x~,Q,z~,/) = {u,,u~,u~.8,r.p}. We are interested in the response of the boundary layer to three dimensional cxcitation of the source starting at time t=O. In this paper, we will investigate the generation and evolution of disturbances in a parallel flow, i.e., the mean flow is given by I.Y,(~i.z.~), L!2(rz, ~a), 0, T(xz. 23). 4~2. ~a), P(xz. 13) (figure 2.). This permits us to use the Fourier transform method to solve the governing equations. The linearized Navier-Stokes equations can be written as (c)2000 American Institute of Aeronautics & Astronautics or published with permission of author(s) and/or author(s)’ sponsoring organization. a2q @q a29 0% 41-t A27 t .A-+ A,*8% 8X; ax; (3X{ dXlBX2 + A13 ax, ax,3 a29 89 + ‘123ax2ar3 t RlG 89 89 -IR?G + RaJ-& t cq = 0. (1) Here 9 = {ut? u?,us. p,B}’ is the disturbance variables. The boundary conditions at the wall are u1 = 112 = 0 = 0, U3(xl.x2,0,t) = sin(wol)~(x1 510) f(x2) H(1), (2) where tin is the frequency of the oscillations of the source, 6 is Dirac’s delta function and H(t) is the unit step function, which indicates that the motion starts from rest at t=O and f(x2) determines the type of disturbances in the azimuthal direction. We require that the disturbances decay far from the plate. The physical disturbance field may be related to the spectral space by using the generalized double Fourier transform defined as q(krr2rx3,~) = dt J J dXl Q(Xl, x2,53> t) dk=+): -cc -cc (3) where k is the complex wavenumber in the xt direction and (J is the complex frequency. Here, the tilde represents the transformed variable. Thus Eq. (1) can be transformed using Eq. (3) into the following system of equations for 4. ,j @i+, a2q+, a2cj 3 ax; "ax; 21-+B2g+B2~+& =O. I axzaxa (4) Transforming the boundary conditions yields