3D global optimal forcing and response of the supersonic boundary layer
Introduction
Depending on their dynamics, open flows can be divided into oscillator and noise selective amplifiers [2]. Whereas the first ones have an intrinsic dynamics related to the physical parameters of the baseflow, the second ones only amplify perturbations in specific ranges of frequencies, which grow in space and advected downstream. In terms of stability analysis, these considerations lead to distinguish absolute from convective instabilities. Local stability analysis [3] have extensively been employed to study the dynamics of various open flows (boundary layer [4], wakes [5], jet flows [6], etc.). This approach allows, in particular, to discriminate absolute and convective instabilities by computing the growth rate of zero group velocity waves [7]. The assumption of a (weakly) parallel baseflow is however required in order to expand perturbations into Fourier-Laplace modes along the streamwise direction.
Focusing on the convectively unstable compressible boundary layer, first stability computations were based on a local approach [8], [9], [10]. Along with theoretical developments [11], these seminal studies established the main features of compressible instabilities, especially noting their inviscid nature caused by the existence of a generalised inflection point and the prevailing growth of 3D perturbations (Squire theorem [12] does not hold for compressible flows [9]). Later, local stability analysis allowed to suggest the existence of an additional unstable mode [13] (generally referred to as second mode, or Mack mode) in the case of sufficiently high supersonic Mach numbers (), soon confirmed by experimental work [14], [15]. Afterwards, more sophisticated local stability analysis taking into account the weak non-parallel effects produced more accurate results [16], [17]. Following the work of Farrell [18] for incompressible flows, several local analysis then focused on computing non-modal growth for compressible boundary layer. Optimal growth in a temporal formulation was first proposed by Hanifi et al. [19] who were able to observe the non-modal growth of compressible streaks. A spatial version of this analysis was suggested by Tumin and Reshotko [20], afterwards improved by considering non-parallel effects [21], [22] and 3D baseflows [23]. These approaches were coupled with a PSE method [24], resulting in a more general framework to study non-modal growth in weakly non-parallel flows [25], [26]. However, these approaches can not be considered as universal as it does not allow to study fully non-parallel flows.
With the increase of computational resources, global stability analysis (in the sense of Theofilis [27]) became affordable. In this framework, the streamwise direction is solved as an eigen-direction which authorise to consider fully non-parallel baseflows. It offers a relevant tool to study globally unstable flows such as the bifurcations occurring in cavity flows [28] and shock wave/boundary layer interactions [29] or the onset of the transonic buffet on an airfoil [30]). Global stability analysis is however not suited to describe the dynamics of convectively unstable flows, which are globally stable. Instead, characterising the response of these flows subject to an external forcing constitutes a more relevant analysis as it is directly related to their noise amplifier nature [31]. In practice, this approach is related to the resolvent operator and an optimisation framework is employed to compute the optimal forcing and response for different frequencies. Such an analysis was first implemented for an incompressible boundary layer by performing a projection of the response onto a restricted number of global modes [32], [33]. Another strategy was afterwards developed by Monokrousos et al. [34] using a time-stepping technique associated with an adjoint-based optimisation method. More recently, Sipp and Marquet [35] suggested to solve a singular value problem associated with the global resolvent operator and showed that, additionally, the left and right singular vectors constituted an orthonormal basis onto which the forcing and response fields could be expanded. Besides, it should be pointed out that these optimal response and forcing approaches are non-modal in nature. Indeed, the optimal response resulting from an optimisation problem can be seen as a superposition of global modes: both modal resonance and non-modal pseudo-resonance are thus taken into account [36]. These non-modal effects are a consequence of two types of non-normalities, associated with the non-normal nature of the linearised Navier-Stokes equations [37], [38]. On the one hand, the convective-type non-normality (the term in the linearised momentum equation), ubiquitous in convectively unstable flows, stems from the advection of perturbations by the baseflow. It was furthermore observed to cause a spatial separation of the forcing and response fields, respectively upstream and downstream [35]. On the other hand, the component-type or lift-up non-normality (the term in the linearised momentum equation) is caused by the transport of baseflow momentum by the perturbations. It was shown to produce component-wise transfer of energy between the forcing and response fields as in the case of the lift-up mechanism [39] or the Orr mechanism [40].
In compressible flows, a global approach taking into account non-modal effects was first implemented for jet flows as an optimal growth problem where an optimal initial condition was looked for [41], [42]. Global optimal forcing based on resolvent computation was then developed and applied to the receptivity of a turbulent shock wave/boundary layer interaction [43]. However, to our knowledge, no work dealing with non-modal growth of 3D global perturbations in compressible flows has been published to date. Given that 3D convective instabilities are especially prevailing in this regime, an efficient numerical framework appears to be missing to tackle this problem.
In this paper, we propose a numerical method to study 3D global linear perturbations developing in convectively unstable, fully non-parallel, compressible 2D baseflows. This approach is based on the computation of the optimal gain and the associated optimal forcing and response, which is achieved by solving a singular value problem associated with the global resolvent operator [35]. The explicit numerical computation of the Jacobian matrix - the first step of the numerical method - uses the discrete framework presented by Mettot et al. [1] which is here extended to 3D perturbation. This point constitutes the main original point of the present work and the mathematical derivation will be fully detailed. An application to the 3D receptivity of the supersonic boundary layer at is presented in order to demonstrate the potential of the method.
The paper is organised as follows. Governing equations and the theoretical approach involved in optimal gain computations are introduced in section 2. The numerical framework is developed in section 3, especially emphasising the computation of the 3D Jacobian matrix (section 3.3). Validation of the numerical framework is given in section 4. Finally, a detailed study of the 3D receptivity of the supersonic boundary layer is presented in section 5.
Section snippets
Governing equations
The flow is governed by the compressible Navier-Stokes equations. Variables are made non-dimensional according to
In the following, the ∼ symbol will be dropped in order to lighten notations. The ∞ symbol refers to far-field quantities. Conservative variables are used, where ρ, and E respectively are the fluid density, the velocity vector and the total energy. T, p, η and λ respectively stand for
Compressible Navier-Stokes solver
The baseflow is computed by means of a finite volume CFD solver as a steady solution of the nonlinear equations (3a), (3b), (3c). Spatial discretisation of convective fluxes is performed using AUSM+ scheme [46] associated with a fifth-order MUSCL extrapolation [47]. Viscous fluxes at cell interfaces are obtained by a second-order centered finite difference scheme. The unsteady equations are marched in time until a steady state is reached. An implicit dual time stepping method with local time
Validation of the present method
In this section, solvers presented in section 3 are validated against data from existing studies. Solutions from the CFD solver are compared to the self-similar solution of the compressible boundary layer. Optimal gain computations are first validated against 3D global results for a Blasius boundary layer. Afterwards, a validation against 3D non-global results for a supersonic boundary layer is performed since no results for 3D global optimal perturbations are known for compressible flows (as
Baseflow
The baseflow used for optimal gain computations (section 5.2) is presented in this section. A boundary layer developing over an adiabatic flat plate is considered at , at which local stability analysis show that Mack mode reaches its maximum growth rate [59]. Physical and numerical parameters are reported in Table 4 where the Reynolds number is computed according to different reference length scales. Local stability studies usually take Blasius length as reference, which is
Conclusion
A numerical method allowing to compute the Jacobian matrix associated with 3D global perturbations developing over a 2D baseflow has been proposed. This method is an extension of the discretised-then-linearised procedure introduced by Mettot et al. [1] for 2D global perturbations which is particularly suited for compressible flows. Because a Fourier expansion is performed in the transverse direction, modifications of the 2D method have been required: the Jacobian matrix has been separated into
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgements
This work was supported by the French government (three-year doctoral contract).
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