Compressible unsteady Görtler vortices subject to free-stream vortical disturbances

Abstract

The perturbations triggered by free-stream vortical disturbances in compressible boundary layers developing over concave walls are studied numerically and through asymptotic methods. We employ an asymptotic framework based on the limit of high Görtler number, the scaled parameter defining the centrifugal effects; we use an eigenvalue formulation where the free-stream forcing is neglected; and we solve the receptivity problem by integrating the compressible boundary-region equations complemented by appropriate initial and boundary conditions that synthesize the influence of the free-stream vortical flow. Near the leading edge, the boundary-layer perturbations develop as thermal Klebanoff modes and, when centrifugal effects become influential, these modes turn into thermal Görtler vortices, i.e. streamwise rolls characterized by intense velocity and temperature perturbations. The high-Görtler-number asymptotic analysis reveals the condition for which the Görtler vortices start to grow. The Mach number is destabilizing when the spanwise diffusion is negligible and stabilizing when the boundary-layer thickness is comparable with the spanwise wavelength of the vortices. When the Görtler number is large, the theoretical analysis also shows that the vortices move towards the wall as the Mach number increases. These results are confirmed by the receptivity analysis, which additionally clarifies that the temperature perturbations respond to this reversed behaviour further downstream than the velocity perturbations. A matched-asymptotic composite profile, found by combining the inviscid core solution and the near-wall viscous solution, agrees well with the receptivity profile sufficiently downstream and at high Görtler number. The Görtler vortices tend to move towards the boundary-layer core when the flow is more stable, i.e. as the frequency or the Mach number increase, or when the curvature decreases. As a consequence, a region of unperturbed flow is generated near the wall. We also find that the streamwise length scale of the boundary-layer perturbations is always smaller than the free-stream streamwise wavelength. During the initial development of the vortices, only the receptivity calculations are accurate. At streamwise locations where the free-stream disturbances have fully decayed, the growth rate and wavelength are computed with sufficient accuracy by the eigenvalue analysis, although the correct amplitude and evolution of the Görtler vortices can only be determined by the receptivity calculations. It is further proved that the eigenvalue predictions of the growth rate and wavenumber worsen as the Mach number increases as these quantities show a dependence on the wall-normal direction. We conclude by qualitatively comparing our results with the direct numerical simulations available in the literature.

Neutral stability curves of compressible Görtler flow generated by low-frequency free-stream vortical disturbances

“…In the limits of large Reynolds number R≫1, small perturbations ǫ≪R −1 , and low frequency k x ≪1, the linearized unsteady boundary-region (LUBR) equations for the perturbation flow are recovered by inserting the decomposed q(x, t) into the full compressible Navier-Stokes and continuity equations, using (2.2), and collecting O(ǫ) terms. The LUBR equations are coupled with the initial and boundary conditions derived through asymptotic matching to synthesize the effect of the oncoming free-stream vortical disturbances on the boundary layer (Leib et al 1999;Viaro & Ricco 2019). A second-order implicit finite-difference scheme is employed to solve the LUBR equations (Ricco & Wu 2007), which are parabolic along the streamwise direction.…”

“…The boundary-layer velocity, pressure and temperature are decomposed into their mean and perturbation as . As in Viaro & Ricco (2019), the mean flow is the compressible Blasius boundary layer without an externally imposed pressure gradient. The Dorodnitsyn–Howarth transformation is used to scale the mean-flow equations in similarity form, for which the independent similarity variable is , where .…”

“…The LUBR equations are coupled with the initial and boundary conditions derived through asymptotic matching to synthesize the effect of the oncoming free-stream vortical disturbances on the boundary layer (Leib et al. 1999; Viaro & Ricco 2019). A second-order implicit finite-difference scheme is employed to solve the LUBR equations (Ricco & Wu 2007), which are parabolic along the streamwise direction.…”

“…We herein use the receptivity theory of Viaro & Ricco (2019) for compressible flows over streamwise-concave surfaces to analyze the effect of compressibility on the neutral curves computed by Viaro & Ricco (2018). We adopt the same notation, mathematical framework, and numerical approach of Viaro & Ricco (2019), so the reader is referred to that publication for further details.…”

“…We herein use the receptivity theory of Viaro & Ricco (2019) for compressible flows over streamwise-concave surfaces to analyse the effect of compressibility on the neutral curves computed by Viaro & Ricco (2018). We adopt the same notation, mathematical framework and numerical approach as Viaro & Ricco (2019), so the reader is referred to that publication for further details. We investigate the effect of the Görtler number, the frequency and the Mach number, and in particular the impact of the oblique Tollmien–Schlichting ( ) waves, studied by Ricco & Wu (2007) in the flat-plate case, on the neutral curves.…”