The disturbance flow field in a hypersonic boundary layer excited by particle impingement was investigated with a focus on the first stage of the laminar-to-turbulent transition process, namely the receptivity process. A previously validated direct numerical simulation approach adopting disturbance flow tracking is used to simulate the particle-induced transition process. Particle impingement generates a highly complex disturbance flow field that can be characterised by a wide range of frequencies and wavenumbers. After providing some insight about the spectral characteristics of the disturbance flow field in the frequency and wavenumber domains, biorthogonal decomposition is employed to reveal the composition of the disturbance flow field consisting of different continuous and discrete eigenmodes that are triggered through particle impingement. The disturbance flow characteristics for different frequency and wavenumber pairs are discussed where large contributions in the disturbance flow spectrum are observed in the vicinity of the impingement location. A significant amount of the disturbance energy is diverted into the free stream leading to large coefficients of projection for the slow and fast acoustic branches while contributions to the entropy and vorticity branches are negligible. In addition to the continuous acoustic spectra, the first-, second- and other higher-order Mack modes are activated and provide large contributions to the disturbance flow field inside the boundary layer. Finally, it is demonstrated that the disturbance flow field in the vicinity of the impingement location can be reconstructed with a maximum relative error of $2.3\,\%$ by employing a theoretical biorthogonal eigenfunction system expansion and by considering contributions from fast and slow acoustic waves and at most four discrete modes only.

Optimal perturbations in compressible, non-parallel boundary layers are considered here. The flows past a flat plate and past a sphere are analysed. The governing equations are derived from the linearized Navier-Stokes equations by employing a scaling that relies on the presence of streamwise vortices, which are well-known for being responsible for the 'lift-up' effect. Consequently, the energy norm of the inlet perturbation encompasses the wall-normal and spanwise velocity components only. The effect of different choices of the energy norm at the outlet is studied, testing full (all velocity components and temperature) and partial (streamwise velocity and temperature only) norms. Optimal perturbations are computed via an iterative algorithm completely derived in the discrete framework. The latter simplifies the derivation of the adjoint equations and the coupling conditions at the inlet and outlet.Results for the flat plate show that when the Reynolds number is of the order of 10(3), a significant difference in the energy growth is found between the cases of full and partial energy norms at the outlet. The effect of the wall temperature is in agreement with previous parallel-flow results, with cooling being a destabilizing factor for both flat plate and sphere. Flow divergence, which characterizes the boundary layer past the sphere, has significant effects on the transient growth phenomenon. In particular, an increase of the sphere radius leads to a larger transient growth, with stronger effects in the vicinity of the stagnation point. In the range of interesting values of the Reynolds number that are typical of wind tunnel tests and flight conditions for a sphere, no significant role is played by the wall-normal and streamwise velocity components at the outlet.

An initial-value problem is formulated for a three-dimensional wave packet in a compressible boundary layer flow. The problem is solved using a Laplace transform with respect to time and Fourier transforms with respect to the streamwise and spanwise coordinates. The solution can be presented as a sum of modes consisting of continuous and discrete spectra of temporal stability theory. Two discrete modes, known as mode S and mode F, are of interest in high-speed flows since they may be involved in a laminar-turbulent transition scenario. The continuous and discrete spectrum are analyzed numerically for a hypersonic flow with Mach number M=5.6. The following features are revealed: (1) The synchronism of mode S with acoustic waves at a streamwise wave number α→0 is primarily two-dimensional; (2) at high angles of disturbance propagation, mode F is no longer synchronized with entropy and vorticity waves; (3) at high angles of disturbance propagation, the synchronism between mode S and mode F is not accompanied by a mode S instability, and at even higher angles of disturbance propagation, mode S and mode F are not synchronized.

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.

Two-dimensional spatially growing perturbations in a two-dimensional compressible boundary layer are considered within the scope of linearized Navier-Stokes equations. Because a spatially growing solution can be expanded into a biorthogonal eigenfunction system, the latter can be utilized for decomposition of flow fields derived from computational studies when pressure, temperature, and all velocity components, together with their derivatives, are available. The method can be used also in a case where partial data are available when a priori information leads to consideration of a finite number of modes. Nomenclature (, , ) x y t A = vector-function of 9 components j A = j-th component of vector A D = d dy 2/ 3 e = ratio of the bulk viscosity to the dynamic viscosity , ij ij

Linear and nonlinear stability analyses of the entropy layer over a blunted plate are discussed. Results of the linear stability theory are compared with the direct numerical solution of the Euler equations when a disturbance of prescribed frequency is imposed on the mean flow. A solver of the Euler equations based on the space-time conservation element/solution element method predicts linear and nonlinear dynamics of unstable disturbances with high accuracy. The nonlinear effect demonstrates a trend to saturation of the entropy-layer disturbances.

An analysis of the non-modal growth of perturbations in a boundary layer in the presence of a streamwise pressure gradient is presented. The analysis is based on PSE equations for an incompressible fluid. Examples with Falkner- Skan profiles indicate that a favorable pressure gradient decreases the non-modal growth while an unfavorable pressure gradient leads to an increase of the amplification. It is suggested that the transient growth mechanism be utilized to choose optimal parameters of tripping elements on a low-pressure turbine (LPT) airfoil. As an example, a boundary-layer flow with a streamwise pressure gradient corresponding to the pressure distribution over a LPT airfoil is considered. It is shown that there is an optimal spacing of the tripping elements and that the transient growth effect depends on the starting point. The amplification is found to be small at the LPT s very low Reynolds numbers, but there is a possibility to enhance the transient energy growth by means of wall cooling.

An initial-value problem is analyzed for a two-dimensional wave packet induced by a local two-dimensional disturbance in a hypersonic boundary layer. The problem is solved using Fourier transform with respect to the streamwise coordinate and Laplace transform with respect to time. The temporal continuous spectrum is revisited, and the uncertainty associated with the overlapping of continuous-spectrum branches is resolved. It is shown that the discrete spectrum's dispersion relationship is nonanalytic because of the synchronization of the first mode with the vorticity/entropy waves of the continuous spectrum. However, the inverse Laplace transform is regular at the synchronism point. Characteristics of the wave packet generated by an initial temperature spot are numerically calculated. It is shown that the hypersonic boundary layer is highly receptive to vorticity/entropy disturbances in the synchronism region. The feasibility of experimental verification of this receptivity mechanism is discussed.

The spatial stability of a natural convection flow on upward-facing, heated, inclined plates is revisited. The eigenvalue problem is solved numerically employing two methods. the collocation method with Chebyshev polynomials and the fourth-order Runge-Kutta method. Two modes, traveling waves and stationary longitudinal vortices, are considered. Previous theoretical models indicated that nonparallel effects of the mean flow are significant for the vortex instability mode, but most of them ignored the fact that the eigenfunctions are dependent on the streamwise coordinate as well. In the present work the method of multiple scales is applied to take the nonparallel flow effects into consideration. The results demonstrate the stabilizing character of the nonparallel flow effects. The vortex instability mode is also considered within the scope of partial differential equations. The results demonstrate dependence of the neutral point on the initial conditions but, farther downstream, the results collapse onto one curve. The marching method is compared with the quasi-parallel normal mode analysis and with theoretical results including correction to nonparallel flow effects. The marching method provides better agreement of theoretical and experimental growth rates.

The problem of transient growth in compressible boundary layers is considered within the scope of partial differential equations including the nonparallel flow effects. As follows from previous investigations, the optimal disturbances correspond to steady counter-rotating streamwise vortices. The corresponding scaling of the perturbations leads to the governing equations for a Gortler-type of instability, with the Gortler number equal to zero. The iteration procedure employs back and forth marching solutions of the adjoint and original systems of equations

Three-dimensional spatially growing perturbations in a two-dimensional incompressible boundary layer are considered within the scope of linearized Navier–Stokes equations. The Cauchy problem is solved under the assumption of a finite growth rate of the disturbances. It is shown that the solution can be presented as an expansion into a biorthogonal eigenfunction system. The result can be utilized for decomposition of flow fields derived from computational studies when pressure and all velocity components, together with their derivatives, are available. The method can be used also in a case where partial data are available when a priori information leads to consideration of a finite number of modes. In the case of a continuous spectrum, the problem of decomposition based on partial information is ill-posed, but the method might be applied under additional assumptions about the perturbations.

The theory of optimal, spatially growing disturbances is applied to a compressible turbulent mixing layer. It is suggested that the weak, steady disturbances generated by tiny vortex generators be considered as in the triple-decomposition method. The linearized equations for the disturbances are closed via the eddy viscosity model with a turbulent Prandtl number equal to one. The optimization procedure is formulated in terms of Mack's energy norm. As an example, a parallel flow with a hyperbolic tangent velocity profile and a temperature profile following from the Crocco-Busemann relation is analyzed

A theoretical model of harmonic perturbations in far turbulent wakes is considered. The proposed model is based on the triple decomposition method. It is assumed that the instantaneous velocities and pressures consist of three distinctive components: the mean (time-averaged), the coherent (phase-averaged), and the random (turbulent) motion. The interaction between incoherent turbulent fluctuations and large-scale coherent disturbances is incorporated by means of a Newtonian eddy viscosity model. For high-amplitude perturbations, the nonlinear feedback to the mean flow is taken into account by means of the coherent Reynolds stresses. The equations for the mean flow are coupled with the linearized equations for the disturbances, taking into account the mean flow nonparallel effects. The model resolves uncertainties noted in previous theories and provides a correct comparison with available experimental data. The effect of the harmonic perturbations on the turbulent wake growth at high amplitudes is investigated as well.

A theoretical model of harmonic perturbations in a turbulent mixing layer is proposed. The model is based on the triple decomposition method. It is assumed that the instantaneous velocities and pressure consist of three distinctive components: the mean (time average), the coherent (phase average), and the random (turbulent) motion. The interaction between incoherent turbulent fluctuations and large-scale coherent disturbances is incorporated by the Newtonian eddy viscosity model. A slight divergence of the flow is also taken into account, and the results are compared with experimental data. For a high amplitude of the perturbations, the nonlinear feedback to the mean flow is taken into account by means of the coherent Reynolds stresses. The results reveal the possibility of a negative spreading rate of the mixing layer. A simultaneous consideration of the mean flow divergence and nonlinear self-interaction results in Landau-like amplitude equations. It is observed that the nonlinear term in the amplitude equation is not significant at the levels of amplitude considered. The velocity disturbance profiles of the second harmonic are also presented and, at low-level amplitude, they are in good agreement with experiments.

Steady three-dimensional disturbances are considered within the scope of the linearized boundary layer equations. The disturbances are assumed to be periodic in the spanwise direction, and their spanwise scale is assumed to be much larger than the boundary layer thickness. The model is an extension of Luchini’s theory [J. Fluid Mech. 327, 101 (1996)] to the Falkner–Skan boundary layer.

A spatial theory is proposed for the linear transient growth of disturbances in a parallel boundary layer. Following from the consideration of a signaling problem, the spatial development of disturbances downstream of a source may be presented as a sum of decaying eigenmodes and Tollmien–Schlichting (TS) like instability modes. Therefore, the problem of optimal disturbances may be considered as an initial value problem on the subset of the decaying eigenmodes and a TS wave, and a standard optimization procedure may be applied for evaluation of the optimal transient growth. The results indicate that the most significant transient growth is associated with stationary streamwise vortices. Numerical examples illustrate that favorable pressure gradient decreases the overall amplification. Effects of compressibility and the wall cooling are investigated as well.

A spatial theory of linear transient growth for disturbances in a circular pipe is presented. Following from the consideration of a signaling problem, the spatial development of disturbances downstream of a source may be presented as a sum of decaying eigenmodes. Therefore, the problem of optimal disturbances in the pipe flow may be considered as an initial value problem on the subset of the downstream decaying eigenmodes, and a standard optimization procedure may be applied for evaluation of the optimal transient growth. Examples are presented for spatial transient growth of axisymmetric and nonaxisymmetric disturbances. It is shown that stationary disturbances may achieve more significant transient growth than nonstationary ones. The maximum of the transient growth exists at azimuthal index m=1 for stationary perturbations, whereas nonstationary perturbations may achieve their maxima at higher azimuthal indices.