On the fully-elliptic representation of convective instabilities in streamwise shear flows

Laminar-turbulent transition is ubiquitous in viscous fluid flows and is critical in many engineering applications. While it can be beneficial to enhance mixing, increase heat transfer or avoid flow separation, it also increases the skin friction that, for instance, drastically increases the fuel consumption of aircraft. Understanding the laminar-turbulent transition is thus key to devising control strategies that can promote or delay the onset of turbulence. In many cases, the laminar-turbulent transition originates from external disturbances that enter the flow field and are amplified by instability mechanisms. Among all types of instability, linear convective instability mechanisms are usually the most dominant in flows that evolve from a laminar to a turbulent regime in space. These mechanisms amplify and propagate the disturbances in a preferential direction, away from the source. Once the amplified disturbances have reached a threshold amplitude, nonlinear interactions rapidly break them down into large-scale turbulent motions. By using linear stability equations, the instability mechanisms supported in a flow field can be studied before the nonlinear breakdown. However, existing numerical methods are not suitable for describing convective instability mechanisms in real flow applications. Traditional one-dimensional equations can only tackle flows that slowly develop towards the propagation direction of the disturbances and, if more dimensions are included in the equations, the results become sensitive to the numerical domain boundaries. The description of convective instabilities in complex flows is thus always compromised by insufficient result accuracy. The objective of this work is to propose a methodology that enables a fully-elliptic representation of convective instability mechanisms in two-dimensional flows. The proposed approach solves the stability equations formulated as an eigenvalue problem in a moving frame of reference to obtain eigenfunctions that decay towards the truncation boundaries, i.e., remain localized within the computational domain. Hence, the corresponding solutions are independent of the numerical domain length and boundary conditions. The eigenfunctions are then introduced into the stationary-frame flow field and integrated in time to obtain the finite-time dynamics of the instability mechanisms. After decomposing the resulting time-dependent wave packets into their individual frequencies, the traditional N-factor and neutral curves are reconstructed. The approach is validated by considering the incompressible flat-plate boundary layer. Whereas the traditional one-dimensional stability equations are acknowledged for delivering valid results for this slowly developing flow, two-dimensional stability analyses in the stationary frame of reference are notoriously tainted by the sensitivity of the results to the domain length. By using the proposed moving-frame approach, this sensitivity issue is eliminated and the resulting N-factor and neutral curves are shown to be in excellent agreement with one-dimensional methods. The moving-frame approach is then used to study the linear stability of a laminar shock-wave/boundary-layer interaction. Because this flow case supports strong in-plane gradients, it provides the opportunity to demonstrate the effectiveness of the moving-frame methodology for capturing convective instability mechanisms in highly two-dimensional flows. The dominant spanwise wavenumber and frequency yielding the largest amplification of perturbations in the present shock-induced recirculation bubble are identified. The convective instability mechanisms are then characterized by decomposing the growth rates of both localized eigenfunctions and time-dependent wave packets into their physical energy-production processes. Finally, a remarkably good agreement is found between the time-dependent wave packets and frame-speed-dependent eigensolutions.