伴随方法在高超声速边界层转捩中的运用

47th AIAA Fluid Dynamics Conference : Stabilization of hypersonic boundary layers by linear and nonlinear optimal perturbations Page 1 Stabilization of Hypersonic Boundary Layers …

… -dimensional disturbance in a hypersonic boundary layer. The problem … It is shown that the hypersonic boundary layer is highly … We assume that solutions of the adjoint problem (46) are …

… stability equations is used in a variational approach to extend the previous body of … in boundary-layer flows. This paper investigates the optimal growth characteristics in the hypersonic …

… stabiliy of hypersonic boundary layers, several modes could lie close by and local methods fail … at the end of each iteration cycle k the eigenfunction and its adjoint are normalized so that …

In this study, the disturbance energy budget is analyzed on the derived disturbance energy norm in hypersonic and high-enthalpy boundary layers with thermal-chemical nonequilibrium (TCNE) effects. The disturbance growth rate is decomposed to quantitatively evaluate the contribution from various classified terms. Hypersonic flat-plate flows are investigated with various free-stream Mach numbers, free-stream temperatures, and wall temperatures. The linear and nonlinear evolutions of disturbances are predicted using linear stability theory and parabolized stability equations. The results show that in the first-mode region, the disturbance growth rates are determined by the production term (destabilizing) and the viscous term (stabilizing), while the former nearly offset the latter. In the second-mode region, the viscous term decreases to the minimum, resulting in the dominance of the second mode. The disturbance of the TCNE source term has a stabilizing effect on the second mode, but at most it reduces the growth rate by 6% in a Mach 10 adiabatic case with the highest free-stream temperature of 900 K. The production term is mainly responsible for the second-mode growth rate difference between the TCNE flow and calorically perfect gas flow. TCNE changes the disturbance characteristics mainly through the mean flow modification. In the oblique-mode breakdown case, the intensive energy transfer between the selected modes and their harmonic waves is found to occur where they interact strongly with the mean flow.

… Additionally, the sensitivity changes signs within the boundary layer. Because the … change in sign most likely relates to the overlap between the momentum and mass boundary layers. In …

HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés. Numerical Study of Hypersonic Boundary-Layer Transition Delay through Second-Mode Absorption Romain Fiévet, Hughes Deniau, Jean-Philippe Brazier, Estelle Piot

… boundary-layer transition at supersonic and hypersonic speeds is investigated. Computations for sharp cones, using the eN method … mode is responsible for transition at adiabatic wall …

… transition of high-speed boundary layers. Previous DNS studies of supersonic and hypersonic boundary layer transition … flow over flat-plate boundary layers without shock waves. For …

Abstract During the process of aerodynamic shape design of supersonic and hypersonic space planes, laminar flow design and boundary layer transition prediction play important roles in aero-thermal numerical simulations and aero-thermal protection design. Therefore, in this study, a computational fluid dynamics compatible transition closure model for high speed laminar-to-turbulent transitional flows is formulated with consideration of the analysis results from stability theory. The proposed model contains two transport equations to describe the transition mechanism using local variables. Specifically, the eddy viscosity of laminar fluctuations and intermittency factor are chosen to be the characteristic parameters and modeled by transport equations. Accounting for the dominant instability modes at supersonic/hypersonic conditions, the first- and second- modes are modeled using local variables through the analysis of laminar self-similar boundary layers. Then, the present transition model is applied with compressibility corrected k $k$-ω $\omega$ shear stress transport turbulence model. Thus, as the main significance of the current work, the present model is enabled to capture the overshoot phenomena as well as predict the transition onset position. Finally, comparisons between the predictions using the present model and the wind tunnel experimental results of several well-documented flow cases are provided to validate the proposed transition turbulence model.

… The transition is detected from pitot pressure measurements in the boundary layer. Transition onset occurs very near the nose at Mach 4 and is delayed to the middle part of the …

… ago that moderate bluntness on cones delayed hypersonic boundarylayer transition.Above a certain critical level, bluntness leads to early transition, especially on highly cooled bodies …

The compressible-boundary-layer stability can be considerably influenced by base flow distortion. The distortion may originate from perturbations of flow parameters, such as the Mach number. In this paper, sensitivities of the boundary layer stability to certain flow parameters are derived analytically by utilizing the homotopy analysis (with codes shared), in conjunction with a direct-adjoint stability theory. The sensitivities can be categorized according to the routes the distortion evolves. Route I is that parameters distort the base flow (Sensitivity A), which, in turn, affect the eigenvalue of the linear stability equation (Sensitivity B). Route II gives rise to the effects of flow parameters onto eigenvalues caused by direct perturbation of the linear operators (Sensitivity C). Results indicate that Sensitivity A is characterized by the only peak found on the sensitivity profile that corresponds to the maximum gradient of base flow; for Sensitivity B, production terms, e.g., the mean-shear terms, are found to be significant, while for Sensitivity C, which is rarely discussed in existing literature, the pressure gradient terms in the momentum equations are dominant in affecting the stability via route II. Furthermore, route II can be more significant than route I. Having examined the variation of the mean shear gradient, d(ρ¯du¯/dy)/dy, near the critical layer yc, it is proven that the sensitivity of the eigenvalue to the velocity or temperature distortion is negative at yc under certain assumptions, particularly for the temperature-relevant sensitivity that has hardly been discussed before.

… Parallel discrete direct and adjoint sensitivity analysis capabilities are developed for arbitrary Mach flows on mixed‐element unstructured grids. The discrete direct and adjoint methods …

… layer interaction makes the flow topology even more sensitive to transitional mechanisms, since … of adjoint modes (modes of the adjoint linearized operator): the structural sensitivity is …

… accurate gradients (that incorporate the sensitivities of the transition criterion) have been … [25], the direct (or flow-sensitivity) method, and the discrete-adjoint gradient method, which is …

… a new method for predicting flow transition and explores its … and the adjoint sensitivity problem, as shown in Figure 7(b). … damped, we also obtain one discrete mode that is unstable. This …

… and remains almost constant inside the boundary layer before it … on the receptivity process in hypersonic flows over several … As previously mentioned, modal analysis using adjoint …

Direct numerical simulation of receptivity in a boundary layer over a sharp wedge of half-angle 5:3 degrees was carried out with two-dimensional perturbations introduced into the ∞ow by periodic-in-time blowing-suction through a slot. The free stream Mach number was equal to 8. The perturbation ∞ow fleld downstream from the slot was decomposed into normal modes with the help of the biorthogonal eigenfunction system. Filtered-out amplitudes of two discrete normal modes and of the fast acoustic modes are compared with the linear receptivity problem solution. The examples ilustrate how the multimode decomposition technique may serve as a tool for gaining insight into computational results. I. Introduction The progress being made in computational ∞uid dynamics provides an opportunity for reliable simulation of such complex phenomena as laminar-turbulent transition. The dynamics of ∞ow transition depends on the instability of small perturbations excited by external sources. Computational results provide complete information about the ∞ow fleld, which would be impossible to measure in real experiments. However, validation of the results might be a challenging problem. Sometimes, numerical simulations of small perturbations in boundary layers are accompanied by comparisons with results obtained within the scope of the linear stability theory. In principle, this is possible in the case of a ∞ow possessing an unstable mode. Far downstream from the actuator, the perturbations might be dominated by the unstable mode, and one may compare the computational results for the velocity and temperature perturbation proflles and their growth rates with the linear stability theory. This analysis does not work when the amplitude of the unstable mode is comparable to that of other modes, or when one needs to evaluate the amplitude of a decaying mode. Recently, a method of normal mode decomposition was developed for two- and three- dimensional perturbations in compressible and incompressible boundary layers. 1{3 The method is based on the expansion of solutions of linearized Navier{Stokes equations for perturbations of prescribed frequency into the normal modes of discrete and continuous spectra. The instability modes belong to the discrete spectrum, whereas the continuous spectrum is associated with vorticity, entropy, and acoustic modes. Because the problem of perturbations within the scope of the linearized Navier{Stokes equations is not self-adjoint, the eigenfunctions representing the normal modes are not orthogonal. Therefore, the eigenfunctions of the adjoint problem are involved in the computation of the normal modes’ weights. Originally, the method based on the expansion into the normal modes was used for analysis of discrete modes (Tollmien{Schlichting{like modes) only. After clariflcation of uncertainties associated with the continuous spectra in Ref. 1, the method was also applied to the analysis of roughness-induced perturbations. 4{6 In order to flnd the amplitude of a normal mode, one needs proflles of the velocity, temperature, and pressure perturbations, together with some of their streamwise derivatives given at only one station downstream from the disturbance source. Because computational results can provide all the necessary information about the perturbation fleld, the application of the multimode decomposition is straightforward. However, the flrst

Boundary layer transition can be initiated differently given the nature of the external disturbances. Receptivity, which refers to the interaction between external disturbances and the boundary layer, introduces an initial disturbance amplitude into the flow for all scenarios. Different transition prediction tools consider receptivity to different extents. The popular $e^N$ method primarily focuses on the growth rate, also called the $N$ factor, of disturbances and neglects the initial amplitude. To address this issue, Mack (1977) proposed the amplitude method to incorporate receptivity, nonlinear effects, and broadband characteristics of disturbances. In amplitude method, an accurate evaluation of the receptivity coefficient at the branch I neutral stability location is significant in obtaining the initial amplitude of the disturbance wave. Current evaluation of the receptivity coefficient are the experimental fitting method used by Marineau (2017) and the back-tracking numerical method implemented by He and Zhong (2021). Although both approaches have the capability of obtaining the receptivity coefficient, the evaluation is not directly at the branch I neutral stability location for their case studies. Furthermore, the DNS results by He indicated a need for a multimode analysis to obtain the true initial amplitude at the branch I neutral stability location. To facilitate this analysis, the bi-orthogonal eigenfunction decomposition, proposed by Tumin (2007), can be applied to decompose the DNS flow field into normal modes including discrete and continuous modes to obtain the modal amplitude for receptivity evaluation. By employing the high order finite difference method by Zou and Zhong (2023), the bi-orthogonal eigenfunction system of the hypersonic boundary layer over a blunt cone is obtained, allowing for the decomposition of the receptivity flowfield. From the preliminary results of the decomposition, an overall trend of the discrete mode S amplitude and growth rate agrees with previous $e^N$ and LST results, showing a reduction in multimode effects. The receptivity coefficient for a band of frequencies are also computed for demonstration. Moreover, in addition to the discrete modes F and S, a discrete entropy layer mode has been identified near the synchronization region. Decomposition results confirm that the discrete mode S being the dominant mode in downstream of the second mode unstable region while the discrete entropy layer mode and the discrete mode F both contribute to the flow near the synchronization region.

Temperatures within the boundary layers of high-enthalpy hypersonic flows can soar to thousands or even tens of thousands of degrees, leading to significant real gas phenomena. Although there has been significant research on real gas effects on hypersonic boundary layer stability, their impact on the boundary layer’s receptive stage is still poorly understood. Most aerodynamic boundary layers in flight vehicles are three-dimensional. Because of complex geometry and significant crossflow effects, the crossflow mode in three-dimensional boundary layers is crucial in hypersonic vehicle design. In this study, a linear stability analysis (LST) accounting for chemical nonequilibrium effects (CNE) and its adjoint form (ALST) is developed to investigate the real gas effects on the stability and receptivity of stationary crossflow modes. The results indicate that real gas effects significantly influence the receptivity of stationary crossflow modes. Specifically, chemical nonequilibrium effects destabilize the crossflow modes but reduce the receptivity coefficients of the stationary crossflow modes. The Mach number effect was also investigated. It was found that increasing the Mach number stabilizes the stationary crossflow modes, but the receptivity coefficients increase. As the Mach number progressively rises, these effects alternately dominate, leading to a non-monotonic shift in the transition position.

… of wave packets in hypersonic flow over a compression corner. … Contours of constant amplification rate, αi, are shown in the … Taking into account the adjoint equation and the boundary …

… for compressible boundary layers. Receptivity efficiency functions predicted by the adjoint PSE … The adjoint PSE solution does not contain any continuous spectrum or other eigenmode …

… (§V), the continuous spectra provide correct representation of the Mach waves in supersonic flows. … of modes belonging to continuous spectra, whereas the adjoint method in the parallel-…

… We consider supersonic flow over an axisymmetric 7 deg halfangle cone at zero angle of … boundary layer remain at a constant value equal to the wave number in the outer part of the …

The receptivity of high-speed compressible boundary layers to kinetic fluctuations is considered within the framework of fluctuating hydrodynamics. The formulation is based on the idea that KF-indu...

The generation of the first-mode instability through scattering of an acoustic wave by localised surface roughness, suction or heating is studied with a time-harmonic compressible adjoint linearised Navier–Stokes (AHLNS) approach for subsonic flow conditions. High Strouhal number analytical solutions to the compressible Stokes layer problem are deduced and shown to be in better agreement with numerical solutions compared to previous works. The adjoint methodology of Hill in the context of acoustic receptivity is extended to the compressible flow regime and an alternative formulation to predict sensitivity to the angle of incidence of an acoustic wave is proposed. Good agreement of the acoustic AHLNS receptivity model is found with published direct numerical simulations and the simpler finite Reynolds number approach. Parametric investigations of the influence of the acoustic wave angle on receptivity amplitudes reveal that the linearised unsteady boundary layer equations are a valid model of the acoustic signature for a large range of acoustic wave obliqueness values, failing only where the wave is highly oblique and travels upstream. An extensive parametric study of the influence of frequency, spanwise wavenumber, local Reynolds number and free-stream Mach number over the efficiency function for the different types of wall perturbation mechanisms is undertaken.

The receptivity of boundary-layer flow to a three-dimensional hump (an array of humps) at a finite Reynolds number is solved with the help of an expansion of the linearized solution of Navier-Stokes equations into the biorthogonal eigenfunction system. There are two counterrotating vortices behind the roughness element that bring the high-speed fluid down into the wake region. Depending on the geometry, two relatively high-speed streaks could be observed in the wake downstream from the hump. The results for the flow-field structure are in qualitative agreement with available computational data. The quantitative discrepancy is attributed to the nonlinear character of the receptivity mechanism at the parameters considered in the computational studies.

… Eigenfunctions of the direct and adjoint problems were obtained as a linear combination of … The wall is adiabatic so that the flow temperature is constant across the boundary layer. …

As a high-fidelity approach to transition prediction, the coupled Reynolds-averaged Navier–Stokes (RANS) and linear stability theory (LST)-based [Formula: see text] method is widely used in engineering applications and is the preferred method for laminar flow optimization. However, the further development of gradient-based laminar flow wing optimization schemes is hindered by a lack of efficient and accurate derivative computation methods for LST-based eigenvalue problems with a large number of design variables. To address this deficiency and to compute the derivatives in the LST-based solution solver, we apply the adjoint method and analytical reverse algorithm differentiation (RAD), which scale well with the number of inputs. The core of this paper is the computation of the standard eigenvalue and eigenvector derivatives for the LST problem, which involves a complex matrix. We develop an adjoint method to compute these derivatives, and we couple this method with RAD to reduce computational costs. In addition, we incorporate the LST-based partial derivatives into the laminar–turbulent transition prediction framework for the computation of total derivatives. We verify our proposed method with reference to finite difference (FD) results for an infinite swept wing. Both the intermediate derivatives from the transition module and total derivatives agree with the FD reference results to at least three digits, demonstrating the accuracy of our proposed approach. The fully adjoint and the coupled adjoint–RAD methods both have considerable advantages in terms of computational efficiency compared with iterative RAD and FD methods. The LST-based transition method and the proposed method for efficient and accurate derivative computations have prospects for wide application to laminar flow optimization in aerodynamic design.

… different transition models that have proven to greatly enhance the boundary layer prediction … optimization within our in-house UNstructured PArallel Compressible Design Optimization …

… An adjoint-based optimization technique was used to optimize the input disturbance at the … by Luchini to the case of compressible boundary layers. They also investigated the effect of …

… The laminar-turbulence transition in boundary layer flows is … instability theory for both compressible and incompressible flows, … reveal the efficiency of adjoint based tools in obtaining …

… of momentum in a compressible boundary layer in framework of … , we compare the adjoint based gradients to those obtained … approach for the prediction of boundary-layer receptivity in …

… models by comparing the predicted viscous stress and heat … , including shock waves and boundary layers. Alternatively, … the single-component, compressible Navier–Stokes equations…

We investigate flow instability created by an oblique shock wave impinging on a Mach 5.92 laminar boundary layer at a transitional Reynolds number. The adverse pressure gradient of the oblique shock causes the boundary layer to separate from the wall, resulting in the formation of a recirculation bubble. For sufficiently large oblique shock angles, the recirculation bubble is unstable to three-dimensional perturbations and the flow bifurcates from its original laminar state. We utilize Direct Numerical Simulation (DNS) and Global Stability Analysis (GSA) to show that this first occurs at a critical shock angle of $\theta = 12.9^o$. At bifurcation, the least stable global mode is non-oscillatory, and it takes place at a spanwise wavenumber $\beta=0.25$, in good agreement with DNS results. Examination of the critical global mode reveals that it originates from an interaction between small spanwise corrugations at the base of the incident shock, streamwise vortices inside the recirculation bubble, and spanwise modulation of the bubble strength. The global mode drives the formation of long streamwise streaks downstream of the bubble. While the streaks may be amplified by either the lift-up effect or by G\"ortler instability, we show that centrifugal instability plays no role in the upstream self-sustaining mechanism of the global mode. We employ an adjoint solver to corroborate our physical interpretation by showing that the critical global mode is most sensitive to base flow modifications that are entirely contained inside the recirculation bubble.

… transitions. Hence, in this subsection, we conduct a comparison between the global stability … LST in predicting the stability characteristics of 3D boundary layers, ie, the windward side …

… Precisely the extensibility of the novel PSE-3D algorithm developed in the framework of the present thesis to study nonlinear flow instability permits transition prediction in flows of …

To realize the drag reduction benefit of boundary-layer transition control strategies, it is crucial to integrate transition prediction into the vehicle design through an optimization process. The integration of transition prediction based on linear stability analysis into adjoint-based design optimization requires coupling an adjoint-enabled computational fluid dynamics (CFD) solver with an adjoint-enabled linear stability code. In particular, the boundary-layer transition location is often predicted using the [Formula: see text]-factor method based on linear stability theory (LST). Thus, the sensitivity of the linear-stability eigenvalues constitutes an essential building block for optimizing the laminar flow performance. The present paper describes an implementation of LST eigenvalue sensitivity analysis that can be easily coupled with a CFD solver. Specifically, we describe a discrete adjoint formulation for the transition location prediction based on the [Formula: see text]-factor method. The verification of this formulation is carried out by comparing the adjoint-based sensitivity of the local growth rate of a given instability mode with respect to the disturbance frequency and the adjoint-based sensitivity of the transition location with respect to the spanwise wavenumber with those sensitivities computed using a finite-difference approximation. Finally, the adjoint LST formulation is applied to flat-plate boundary-layer flows at transonic, supersonic, and hypersonic conditions to determine the behavior and sensitivities of the transition location with respect to a range of disturbance spanwise wavenumbers.

… It covers both the problem of how a laminar boundary layer transitions to turbulence in the supersonic and hypersonic regime and the problem of how compressibility of a fluid affects …

Abstract The efficacy of steady large-amplitude blowing/suction on instability and transition control for a hypersonic flat plate boundary layer with Mach number 5.86 is investigated systematically. The influence of the blowing/suction flux and amplitude on instability is examined through direct numerical simulation and resolvent analysis. When a relatively small flux is used, the two-dimensional instability critical frequency that distinguishes the promotion/suppression mode effect closely aligns with the synchronisation frequency. For the oblique wave, as the spanwise wavenumber increases, the suppression effects would become weaker and the mode suppression bandwidth diminishes/increases in general in the blowing/suction control. Increasing the blowing/suction flux can effectively broaden the frequency bandwidth of disturbance suppression. The influence of amplitude on disturbance suppression is weak in a scenario of constant flux. To gain a deeper insight into disturbance suppression mechanism, momentum potential theory (MPT) and kinetic energy budget analysis are further employed in analysing disturbance evolution with and without control. When the disturbance is suppressed, the blowing induces the transport of certain acoustic components along the compression wave out of the boundary layer, whereas the suction does not. The velocity fluctuations are derived from the momentum fluctuations of the MPT. Compared with the momentum fluctuations, the evolutions indicated by each component's velocity fluctuations greatly facilitate the investigations of the acoustic nature of the second mode. The rapid variation of disturbance amplitude near the blowing is caused by the oscillations of the acoustic component and phase speed differences between vortical and thermal components. Kinetic energy budget analysis is performed to address the non-parallel effect of the boundary layer introduced by blowing/suction, which tends to suppress disturbances near the blowing. Moreover, viscous effects leading to energy dissipation are identified to be stronger in regions where the boundary layer is rapidly thickening. Finally, it is demonstrated that a flat plate boundary layer transition triggered by a random disturbance can be delayed by a blowing/suction combination control. The resolvent analysis further demonstrates that disturbances with frequencies that dominate the early transition stage are dampened in the controlled base flow.

… A shift-invert transformation in the Arnoldi algorithm is … axis are sought, a simple transformation is used in order to convert the … not affected by this transformation. Specifically, defining D 1 …

The objective of this article is to review some developments in the use of adjoint equations in hydrodynamic stability theory. Adjoint-based sensitivity analysis finds both analytical and numerical applications much beyond those originally imagined. It can be used to identify optimal perturbations, pinpoint the most receptive path to break down, select the most destabilizing base-flow defect in a nominally stable configuration, and map the structural sensitivity of an oscillator. We focus on two flow cases more closely: the noise-amplifying instability of a boundary layer and the global mode occurring in the wake of a cylinder. For both cases, the clever interpretation and use of direct and adjoint modes provide key insight into the process of the transition to turbulence.

Laminar–turbulent transition in the boundary layer at supersonic speeds can be initiated by small solid particles present in the free stream. Particulates interacting with the boundary-layer flow generate unstable wavepackets related to Tollmien–Schlichting (TS) waves. The latter grow downstream and ultimately break down to turbulent spots. This scenario of TS-dominated transition is modelled using the Mack amplitude method. A theoretical model describing the receptivity mechanism is developed to predict the initial spectrum of TS waves. With these initial conditions the downstream growth of TS instability is calculated using the linear stability theory. The transition onset is associated with the point where the disturbance amplitude reaches a threshold value. As an example, calculations are carried out for a 14° half-angle sharp wedge flying in the standard atmosphere at altitude 20 km, Mach number 4 and zero angle of attack. It is shown that spherical particles of radius from $10$ to $20~\unicode[.5,0][STIXGeneral,Times]{x03BC} \mathrm{m} $ and density ${\geqslant }1~\mathrm{g} ~{\mathrm{cm} }^{- 3} $ can cause transition onset corresponding to the amplification factor $N= 9{\unicode{x2013}} 10$, which is in the empirical range of flight data. This indicates that atmospheric particulates may be a major source of TS-dominated transition on aerodynamically smooth surfaces at supersonic speeds. The receptivity model provides a foundation for further treatments of different cases associated with transition in dusty environments. It can also be used for predictions of particle-induced transition at subsonic and hypersonic speeds.

The dependence on initial conditions of the three-dimensional algebraic spatial instability of the Blasius boundary layer is examined by a recently developed method of receptivity analysis based on the upstream integration of adjoint equations. This method allows us to determine optimal perturbations, i.e. initial perturbations that maximize the energy growth, even in the wavenumber range where the problem is not amenable to a mode analysis, and thus to complement a previous paper in which the small-wavenumber regime was described.

… and/or body-generated, interact with the mean-flow field and constitute the initial values for the boundary-layer instabilities. This receptivity mechanism provides the initial conditions for …

Understanding the receptivity of hypersonic flows to free-stream disturbances is crucial for predicting laminar to turbulent boundary layer transition. Input-output analysis as a receptivity tool considers which free-stream disturbances lead to the largest response from the boundary layer using the global linear dynamics. Two technical challenges are addressed. First, we extend recent work by Kamal et al. (Kamal, 2023) and restrict the allowable forcing to physically realizable inputs via a free-stream boundary modification to the classic input-output formulation. Second, we develop a hierarchical input-output (H-IO) analysis which allows us to solve the three-dimensional problem at a fraction of the computational cost otherwise associated with directly inverting the fully three-dimensional resolvent operator. Next, we consider Mach 5.8 flows over a sharp cone and two blunt cones with 3.6 mm and 7.2 mm spherically blunt tips. H-IO correctly predicts that the sharp cone boundary layer is most receptive to slow acoustic waves at an optimal incidence angle of 10 degrees, validating the method. We then investigate the effect of free-stream disturbances on the blunt cone boundary layer, and identify two distinct vorticity-dominated receptivity mechanisms for the oblique first mode instability at 10 kHz and an entropy layer instability at 40 and 70 kHz. Our results reveal these receptivity processes to be highly three-dimensional in nature, involving both the nose-tip and excitation along narrow bands at certain azimuthal angles along the oblique shock downstream. We interpret these processes in terms of critical angles from linear shock/perturbation interaction theory. Finally, we show how these novel receptivity processes vary with frequency and nose tip bluntness, and demonstrate how this methodology might be applied to transition prediction from first principles.

… the linear receptivity of a swept-wing boundary layer to … adjoint approach, receptivity coefficients for a Falkner-Skan-Cooke similarity solution and a transonic swept-wing boundary layer …

… While studies have been done on the receptivity to effects … boundary layer is receptive to disturbances behind the shock … ^{\dagger }\) is the adjoint transfer function. We leverage this fact …

… interacting harmonics (or Fourier modes) of the governing compressible Navier-Stokes equations. This enables the study of boundary layer receptivity at … q we obtain the adjoint system, …

… linear receptivity analysis exploiting the properties of adjoint … and point-source receptivity monotonically decrease with … region of high receptivity (determined by the adjoint mode profile) …

… boundary-layer flow forced by small free-stream disturbances [2]. Hereafter we consider acoustic receptivity … acoustic waves effectively excite the boundary-layer mode F near the plate …

Understanding transition mechanisms in high speed boundary layers is important for predictive design and control in aeronautics. Shock boundary layer interaction, in which the inviscid pressure rise causes the incoming boundary layer to separate and reattach downstream, has a destabilizing effect on the flow. The presence of a recirculation bubble and highly concave curvature of the streamlines near reattachment can support large growth of perturbations. In this paper, we investigate the receptivity properties of an asymptotically stable shock boundary layer interaction on a slender double wedge. The optimal frequency response of the two-dimensional steady state flow subjected to external perturbations is computed. It is found that the flow is highly receptive to low-frequency streamwise vorticity perturbations of a specific spanwise wavelength in the incoming boundary layer. This results in growth of streaks post-reattachment. The most amplified spanwise wavelength scales with approximately twice the boundary layer thickness at reattachment. By excluding the role of bubble dynamics in the input-output analysis, we find that recirculation bubble plays an important role in the perturbation growth and the spanwise wavelength selection. The present work demonstrates the efficacy of input-output analysis for investigating the stability and sensitivity of compressible boundary layer flows.

… the second class of natural receptivity mechanisms, which involve abrupt adjustments of an already developed boundary layer. The first solution of a receptivity problem in this class was …

We investigate the nonmodal physical mechanisms responsible for transient growth in a hypersonic laminar boundary layer and in the interaction of this boundary layer with an incident oblique shock wave. The optimal disturbances and growth curves are computed using an adjoint looping approach. We validate this iterative approach by applying it to several parallel boundary layers that have been studied before in great detail. For these parallel flows, the lift-up effect is generally the dominant transient growth mechanism. However, for a Mach 5.92 spatially developing boundary layer, the inviscid Orr mechanism and convective instabilities are responsible for the large transient response. Furthermore, the optimal initial condition corresponding to the oblique shock wave/boundary layer interaction could be related to both the inviscid Orr mechanism and the lift-up effect. Tilted streamwise streaks that oppose the mean shear are present in the upstream boundary layer, and centrifugal instability near the apex of the separation bubble creates small vortices that grow into elongated streamwise structures with time. Due to the strong spatial non-normality of the oblique shock wave/boundary layer interaction, one cannot obtain an accurate lower bound of the transient growth using the direct or adjoint information separately. Therefore, the nonmodal technique, which takes advantage of both the direct and adjoint operators, is an elegant solution to this problem.

… This formulation allows spatially-targeted control of boundary layer instabilities. To determine the optimal control location and magnitude, we solve an adjoint sensitivity problem. The …

Hypersonic boundary layer transition is critical to the design of all hypersonic vehicles due to its effect on the heat transfer into the vehicle surface and potential drag enhancement or reduction during reentry. Boundary layer transition and boundary layer stability analysis under hypersonic conditions has been studied for decades, yet there is ample room for improved accuracy and further investigations into the relevant phenomena. In this work, we present a recent implementation of chemical equilibrium, finite-rate chemistry, and thermochemical nonequilibrium capabilities into LASTRAC, an existing well-established boundary-layer stability analysis code. Verification against existing numerical results in the literature are presented. LASTRAC was previously able to address calorically perfect flows. By using solutions of the Parabolized Stability Equations (PSE) with chemical and thermal nonequilibrium, we are able to investigate the effects of chemical and thermal nonequilibrium on a variety of phenomena including stationary crossflow instability on a swept wing and 2 mode instabilities over a wedge.

This study is dedicated to delaying the transition induced by broad-spectrum disturbances in a Mach 5.86 hypersonic flat-plate boundary layer through local steady blowing–suction control. The optimal blowing–suction parameters, including the position and amplitude of a single blowing suction as well as the positions of the dual blowing–suction with fixed amplitudes, are determined through direct numerical simulation employing Bayesian optimization. It is revealed that the effective single blowing–suction is located at a certain distance downstream of the synchronization point of the fast/slow mode corresponding to the dominant second mode. The effective dual blowing–suction positions are distributed around the position of the single blowing–suction. Within the parameter range considered in the study of the dual blowing–suction, the transition delay effect tends to improve with an increase in the sum of control amplitudes. The analysis of the transition mechanism found that the baseline transition case involves the presence of broad-spectrum second modes, oblique second modes, and first-mode oblique waves, resulting in the coexistence of multiple canonical transition mechanisms. Analysis of the control mechanism reveals that the transition delay primarily depends on the significant attenuation of the second mode and the maintenance of [Formula: see text]-shaped vortices induced by the first mode along the streamwise direction. A detailed investigation of the linear and nonlinear mechanisms behind the transition delay is conducted using resolvent analysis and wavelet bispectrum analysis. Results show that the optimized blowing–suction can suppress both the broad-spectrum second and first modes. Furthermore, the wavelet bispectrum analysis indicates that the suppressed nonlinear interactions between the second mode and the first mode oblique wave result in the inhibition of the evolution of [Formula: see text]-shaped vortices to the hairpin vortices. Finally, the robustness of the control strategy is explored. As the broad-spectrum disturbances present stronger three-dimensional effects, the transition delay effect weakens but eventually converges to half of the optimal delay effect.

We investigate by direct numerical simulation the active control of laminar-turbulent transition in a hypersonic flat-plate boundary layer at a freestream Mach number of 5.86. The control mechanism is a synthetic jet. Based upon the linear stability theory of Mack, in hypersonic flow the important path to transition involves a high-frequency, second-mode fundamental resonance. Through systematic investigation, we reveal that the forcing the boundary layer with a synthetic jet at appropriate combinations of amplitude and frequency suppresses the second mode and delays transition. To gain physical insights into the major control mechanism, we employ the momentum potential theory (MPT) to analyse the flows with and without control. Essentially, the underlying control mechanism relies on an intriguing effect of the synthetic jet via generating the outwards radiated wave structures}, which are identified to split the upstream acoustic and vortical components. The splitting treatment presents the second-mode energy to drop sharply after the flow passes through the synthetic jet slot. The MPT source-term analysis reveals that the significantly suppressed near-wall source terms are responsible for suppressing the second mode downstream. Compared with the vortical and thermal source terms, the acoustic source term is found to be suppressed most. The kinetic budget analysis further reveals that the splitting treatment is related to the non-parallel effect and the nonlinear interaction.

Hypersonic boundary layer transition is a hot yet challenging problem restricting the development and breakthrough of hypersonic aerodynamics. In recent years, despite great progress made by wind tunnel experiment, transition mechanism and transition prediction, only partial knowledge has been gained so far. In this paper, firstly, the specific scenarios of hypersonic boundary layer transition control are clarified. Secondly, the experimental research progress and mechanism of passive control and active control methods under different hypersonic transition control demands are summarized, with their advantages and disadvantages being analyzed separately. Plasma actuation is easy to produce controllable broadband aerodynamic actuation, which has potential in the field of boundary layer transition control. Hence, the following part of the paper focuses on plasma flow control. The feasibility of plasma actuation to control the hypersonic boundary layer transition is demonstrated and the research ideas are presented. Finally, hypersonic boundary layer transition control methods are summarized and the direction of future research is prospected.

Accurate prediction of linear amplification of disturbances in hypersonic boundary layers is computationally challenging. While direct numerical simulations (DNS) and global analysis can be used to compute optimal (worst-case) disturbances and forced responses, their large computational expense render these tools less practical for large design parameter spaces. At the same time, parabolized stability equations can be unreliable for problems involving multimodal and non-modal interactions. To bridge this gap, we apply an approximate fast marching technique, the One-Way Navier-Stokes (OWNS) Equations, in iterative fashion to solve for optimal disturbances. OWNS approximates a rigorous parabolization of the equations of motion by removing disturbances with upstream group velocity using a higher-order recursive filter. UsingOWNS,we aim to characterize disturbances of flat-plate hypersonic boundary layers over a range of Mach numbers, and find optimal disturbances under different cost functions that define corresponding receptivity problems. The calculation of optimal disturbances reveals multi-modal transition scenarios depending on the spatial support, frequency, and physical nature of the external disturbances.

… [11] employed an adjoint-based optimization framework to design roughness-element shapes that delay second-mode–induced boundary-layer transition in a hypersonic flow over an …

Perturbations of flow control parameters may yield a significant alteration in the boundary layer stability. Based on the previously established parameter-associated sensitivity, the present work derives the optimal minor parameter perturbation analytically under the constraint of base flow energy variation. Specifically, the steady blowing-suction factor and the generalized Hartree parameter are examined at Mach number 4.5 to stabilize the mode S. Good agreement between the linear stability theory calculation, sensitivity theory and Lagrangian approach is achieved for the optimal parametric state. The optimal state occurs if the contribution of the base velocity distortion has the greatest advantage over the temperature counterpart. Contributions of various physical sources to the growth rate behave similarly and collapse onto one correlation if normalized by the maximum, particularly for the major four: advection, mean shear, base temperature gradient and pressure gradient. When the parameter perturbation further becomes finite, although the favorable pressure gradient and wall suction stabilize the broadband mode S, an unusual opposite tendency may occur for single-frequency disturbances. In this unusual parametric range, positive contributions of both the major and minor physical sources to the growth rate are promoted. The contributive increase of major and minor sources are attributed to the enhancement of mean shear and viscous effect, respectively. Whether the parametric influence is stabilization or destabilization is intrinsically determined by the sensitivities, and the intermediate process is analyzed. Finally, given the modification to the critical Reynolds number, the input control parameter perturbation is inversely obtained and verified.

… Hypersonic boundary layers exhibit diverse transition pathways, influenced by various flow conditions and environments. Nonmodal mechanisms, such as the lift-up effect, are …

The classical four-equation γ-Re transition model has presented excellent accuracy in low-speed boundary layer transition prediction. However, once the incoming flow reaches hypersonic speed, the original model is no longer applicable due to the compressibility problem and the appearance of multiple instability modes. Recently, there has been widespread interest in data-driven modeling for quantifying uncertainty or improving model prediction accuracy. In this paper, a data-driven framework based on field inversion and machine learning is performed to extend the prediction capability of the original γ-Re transition model for the hypersonic boundary layer transition.First, the iterative regularized ensemble Kalman filter method is applied to obtain the spatial distribution of the perturbation correction term β for the switching function Fonset1, and the effectiveness of this method is initially verified in the hypersonic flat plate case. Then, the random forest algorithm is adopted to construct a mapping from the average flow features to β. The generalizability of the well-trained learning model is fully validated in the blunt cone cases with different unit Reynolds numbers, free-stream flow temperature, and bluntness. The simulation results indicate that the performance of the original γ-Re transition model in the hypersonic boundary layer transition prediction is significantly improved, and the boundary layer transition onset location and the length of transition zone can be correctly obtained. In addition, the machine learning model investigates the importance of the input features and confirms that the effective length scale plays a significant role in the numerical simulation of the hypersonic boundary layer transition.

A high-order spectral difference flow solver is used to perform direct numerical simulations (DNS) of a hypersonic laminar boundary layer on an ultrasonically absorptive coating (UAC), in order to ...

… Examples of transition often classified as bypass transition include the subcritical transition … , D., “Sensitivity analysis using adjoint parabolized stability equations for compressible flows,” …

… The most important area of application, however, is the use of the PSE approach for transition analysis in aerodynamic design. Together with the adjoint linear problem, PSE methods …

… in order to obtain the delay in transition in a channel flow. … , as the initial stages of the transition of laminar shear flows is a … , derived using the adjoint equations. Such method could in …

… of the input model in this study has revealed, and transition. … This energy growth is linked to the known bypass-transition … -self-adjoint nature of the three-dimensional stability problem …

The inclusion of transition-to-turbulence effects in computational fluid dynamics simulations is essential to accurately predict drag reduction from the use of laminar flow technologies. The parabo...

We consider the interaction of free-stream disturbances with the leading edge of a body and its effect on the transition point. We present a method which combines an asymptotic receptivity approach, and a numerical method which marches through the Orr–Sommerfeld region. The asymptotic receptivity analysis produces a three-deck eigensolution which in its far downstream limiting form produces an upstream boundary condition for our numerical parabolized stability equation (PSE). We discuss the advantages of this method compared to existing numerical and asymptotic analysis and present results which justify this method for the case of a semi-infinite flat plate, where asymptotic results exist in the Orr–Sommerfeld region. We also discuss the limitations of the PSE and comment on the validity of the upstream boundary conditions. Good agreement is found between the present results and the numerical results of Haddad & Corke (1998).

… This paper presents the Adjoint Parabolized Stability Equations (APSE) which are used to … against solutions of the Adjoint Navier-Stokes (ANS) equations which demonstrates that APSE …

… -to-turbulent transition of boundary layers based on parabolized stability equations (PSE) and the 𝑵-factor envelope. Our goal is to couple the PSE-based transition model with the …

We identify and quantify a seemingly overlook mechanism for energy transfer between adjacent frequency disturbances in the Nonlinear Parabolized Stability Equations method. Physically, this energy transfer results from the finite-bandwidth nature of actual disturbance spectrums versus the common numerical assumption of a discrete spectrum representation. Both quiet wind tunnel and flight conditions are considered and it is found that, for Mack’s second-mode instability, the mechanism is most significant in the 0.1–1% disturbance amplitude range (based on normalized pressure) and is responsible for a 15–30% increase in predicted disturbance amplitude.

… a parabolized stability equation transition analysis code … for stability analysis of three-dimensional boundary layers over swept wings. In the first step, the parabolized stability equation …

… a the linearized Navier–Stokes equations and b the parabolized stability equations. The LNS … using adjoint parabolized stability equations for compressible flows. Flow Turbul. Combust. …