Linear Receptivity Research Articles

Linear and nonlinear receptivity mechanisms in boundary layers subject to free-stream turbulence

Large-eddy simulations of a flat-plate boundary layer, without a leading edge, subject to multiple levels of incoming free-stream turbulence are considered in the present work. Within an input–output model, where nonlinear terms of the incompressible Navier–Stokes equations are treated as an external forcing, we manage to separate inputs related to perturbations coming through the intake of the numerical domain, whose evolution represents a linear mechanism, and the volumetric nonlinear forcing due to triadic interactions. With these, we perform the full reconstruction of the statistics of the flow, as measured in the simulations, to quantify pairs of wavenumbers and frequencies more affected by either linear or nonlinear receptivity mechanisms. Inside the boundary layer, different wavenumbers at near-zero frequency reveal streaky structures. Those that are amplified predominantly via linear interactions with the incoming vorticity occur upstream and display transient growth, while those generated by the nonlinear forcing are the most energetic and appear in more downstream positions. The latter feature vortices growing proportionally to the laminar boundary layer thickness, along with a velocity profile that agrees with the optimal amplification obtained by linear transient growth theory. The numerical approach presented is general and could potentially be extended to any simulation for which receptivity to incoming perturbations needs to be assessed.

Numerical study of the hydrodynamic stability of a wind-turbine airfoil with a laminar separation bubble under free-stream turbulence

The interaction of several instabilities and the influence of free-stream turbulence on laminar-turbulent transition on a 20% thick wind-turbine blade section with a laminar separation bubble (LSB) are investigated with wall-resolved large-eddy simulations (LES). Turbulence intensities (TI) of 0%, 2.2%, 4.5%, 8.6%, and 15.6% at chord Reynolds number 105 are considered. Linear receptivity occurs for the most energetic disturbances; high-frequency perturbations are excited via non-linear mechanisms for TI≥8.6%. Unstable Tollmien–Schlichting (TS) waves appear in the inflectional flow region for TI≤4.5%, shifting to inviscid Kelvin–Helmholtz (KH) modes upon separation and forming spanwise rolls. Sub-harmonic secondary instability occurs for TI=0%, with rolls intertwining before transition. Streaks spanwise modulate the rolls and increase their growth rates with TI for TI≤4.5%, reducing separation and shifting transition upstream. The TI=4.5% case presents the highest perturbations, leading to the smallest LSB and most upstream transition. Earlier inception of TS/KH modes occurs on low-speed streaks, inducing premature transition. However, for TI=8.6%, the effect of the streaks is to stabilize the attached mean flow and front part of the LSB. This occurs due to the near-wall momentum deficit alleviation, leading to the transition delay and larger LSB than TI=4.5%. This also suppresses separation and completely stabilizes TS/KH modes for TI=15.6%. Linear stability theory predicts well the modal evolution for TI≤8.6%. Optimal perturbation analysis accurately computes the streak development upstream of the inflectional flow region but indicates higher amplification than LES downstream due to the capture of low-frequency, oblique modal instabilities from the LSB. Only low-amplitude [O(1%)] streaks displayed exponential growth in the LES since non-linearity precludes the appearance of these modes.

On boundary-layer receptivity to entropy waves

In this paper, we consider the generation of the Tollmien–Schlichting waves in the boundary layer on the surface of a wing exposed to entropy waves. It is well known that the free-stream turbulence is composed of two perturbation modes: the vorticity waves and the entropy waves. The receptivity of the boundary layer to the vorticity waves has been studied extensively by various authors. The entropy waves have not attracted such attention. We show that, in high speed subsonic flows, the entropy waves are as important for the receptivity as the vorticity waves. Methodologically, our work relies on the asymptotic analysis of the Navier–Stokes equations at large values of the Reynolds number, which results in the formulation of a suitably modified triple-deck theory. The entropy waves produce oscillations of the gas temperature and density, but the velocity and the pressure remain unperturbed to the leading order. This precludes the entropy waves from penetrating the boundary layer, as happens, for example, with the acoustic waves. Our analysis reveals that the entropy waves decay rapidly in the transition layer that forms near the outer edge of the boundary layer. We find that an entropy wave alone cannot generate the Tollmien–Schlichting waves. However, when the boundary layer encounters a wall roughness, the flow near the roughness appears to be perturbed not only inside the boundary layer but also in the inviscid region outside the boundary layer. The latter comes into the interaction with the density perturbations in the entropy wave. As a result, a localised ‘forcing’ is created that produces the Tollmien–Schlichting waves. In this paper we present the results of a linear and nonlinear receptivity analysis. We find that the nonlinearity enhances the receptivity significantly, especially when a local separation region forms on the roughness.

Effects of Nose Bluntness on Hypersonic Boundary-Layer Receptivity and Stability

High-speed laminar-to-turbulent transition over blunt bodies is relevant to a variety of aerodynamic applications. Experiments have observed that beyond a critical value of the nose radius the initially downstream movement of transition location is reversed. Linear stability and receptivity analyses have been unsuccessful at predicting this reversal. The current Paper uses a random forcing approach in conjunction with the Navier–Stokes equations to understand this phenomenon using blunted flat plates of different nose radii at . Specifically, steady validated base flows are perturbed to generate multiple scales associated with both the boundary layer as well as the entropy layer. The dominant frequencies and corresponding growth rates in the frequency spectrum are identified with spectral decomposition techniques. With increasing nose bluntness, a reversal in growth rate of maximum amplified frequency is observed beyond a critical value. At low bluntness, the invariance of the scaled frequency parameter indicates that the dominant instability is the Mack mode, as confirmed with through dynamic mode decomposition. For nose radii higher than the critical value, the mode associated with the peak frequency is different, with more prominent support in the entropy layer, indicating the presence of non-Mack modes.

On the linear receptivity of trailing vortices

Abstract

Effects of forced frequency oscillations and free stream turbulence on the separation-induced transition in pressure gradient dominated flows

The individual and cumulative effects of forced frequency oscillations and free stream turbulence (FST), on the separation-induced transition caused by an adverse pressure gradient on a flat plate geometry, are numerically investigated by solving the Navier–Stokes equation. The flat plate is subjected to a streamwise pressure gradient via a contoured wall, representative of a low pressure turbine blade. The results are validated against existing numerical data, and insights into flow physics of the pre-transitional and turbulent regimes are gained using instantaneous, time-averaged, and phase-averaged flow fields, turbulent kinetic energy budgets, and a disturbance enstrophy-based nonlinear analysis based on direct numerical simulations. Oscillations are imposed at two values of reduced frequency (kosc). Higher reduced frequency oscillations are more effective in triggering early transition and reducing the separated region. This suppression of the separated region is more pronounced with FST. A reduction in the size of the separation bubble is noted along with enhanced near-wall mixing resulting in a shift in the inflection point of the velocity profile toward the wall, with FST and oscillations. The budget of the turbulent kinetic energy shows the dominance of the production term for the case with FST and forced oscillations. The disturbance enstrophy-based analysis provides insight into budget terms of the enstrophy equation, specifically on the role of vortex stretching. Finally, spatiotemporal linear receptivity of the case with oscillations near the separation bubble is reported. This provides a quantitative description of the instability triggered by the pressure gradient and oscillations.

Effects of forced frequency oscillations and unsteady wakes on the separation-induced transition in pressure gradient dominated flows

The individual and cumulative effects of forced frequency oscillations and unsteady wakes, on the separation-induced transition caused by an adverse pressure gradient on a flat plate geometry, are investigated. The flat plate is subjected to a streamwise pressure gradient via an inviscid contour wall, which is representative of the pressure gradient on the suction surface of a low pressure turbine. Periodic Gaussian wakes are imposed at the inflow to mimic the effect of upstream stator wakes. The transitional and turbulent regimes of the flow are investigated using instantaneous and time-averaged flow fields, turbulent kinetic energy (TKE) budgets and a disturbance enstrophy-based nonlinear receptivity analysis based on direct numerical simulations. The impression of the wake in the boundary layer convects 50%–70% slower than the wake. The case with oscillations and wakes is most effective in triggering early transition and reducing the region of separation, specifically the secondary separation bubble is completely suppressed. Enhanced near-wall mixing is observed with a shift in the inflection point of the velocity profile toward the wall. The budget of the TKE shows the dominance of the production term for all cases considered. The disturbance enstrophy transport equation shows the role of the vortex stretching, enstrophy diffusion and dissipation for flows with wakes. Finally, a linear spatio-temporal receptivity analysis is performed for velocity profiles in the attached and separated boundary layer. This provides a description of the effects of pressure gradient and unsteady wakes in the transition to turbulence.

Tracking disturbances in transitional and turbulent flows: Coherent structures

Tracking transitional and turbulent flows requires methods other than the classical techniques, which capture coherent structures via locating pressure minima, after the disturbance field has evolved to late-transitional and turbulent flow stages. Keeping it in mind, transition to turbulence of zero pressure gradient flow is studied, following two routes of excitation, by solving the three-dimensional Navier-Stokes equation in derived variable formulation, with vorticity as one of the dependent variables. For such flows, disturbance structures should be traced from the receptivity to the coherent structure stage for the fully developed turbulent flow. The coherent structures in turbulent flows are identified by the Q- and λ2-criteria, based on the occurrence of pressure minima at the vortex cores. In the proposed study here, the zero pressure gradient boundary layer is excited (i) at the wall with a monochromatic source and (ii) causing transition to turbulence, by a convecting vortex in the free stream. The main aim here is to trace the incipient disturbances from the onset to the turbulent state in terms of physical quantities, such as the disturbance mechanical energy introduced by Sengupta et al. [“Vortex-induced instability of an incompressible wall-bounded shear layer,” J. Fluid Mech. 493, 277–286 (2003)] and disturbance enstrophy transport equation, as proposed by Sengupta et al. [“An enstrophy-based linear and nonlinear receptivity theory,” Phys. Fluids 30(5), 054106 (2018)]. Such methods are capable of tracing disturbance structures from the onset to the evolved stage. We compare these methods with Q- and λ2-criteria to trace disturbance evolution.

A time-spectral approximate Jacobian based linearized compressible Navier-Stokes solver for high-speed boundary-layer receptivity and stability

A numerical method for conducting linear receptivity and stability investigations of high-speed wall-bounded flows based on the linearized compressible Navier-Stokes equations is presented. The current approach is directly applicable for stability investigations of arbitrarily complex geometries. The left-hand-side operator for the linear system of equations is obtained by computing numerical right-hand-side Jacobians while the right-hand-side is build from exact flux Jacobians. Utilizing the numerical right-hand-side Jacobian approach avoids lengthy, error prone derivation of the stability equations in the context of generalized curvilinear coordinates. The governing equations are solved using either time-stepping or time-spectral discretizations. Three different time-spectral approaches, i.e., direct inversion, unfactored and factored schemes, are presented and their numerical characteristics for the solution of the linearized Navier-Stokes equations for linear receptivity and stability analysis for large-scale transition problems are explored. Linear receptivity and stability calculation results are provided for different solver options. Performance comparison of the three schemes are presented for a wide range of test cases: An incompressible cross-flow for a swept flat plate boundary layer, a supersonic shock-boundary-layer interaction, hypersonic boundary layers on a flat plate and a flared cone, and, finally, for the receptivity of a hypersonic boundary layer for a right sharp cone.

Effect of frequency and wavenumber on the three-dimensional routes of transition by wall excitation

Three-dimensional (3D) routes of transition affected by the frequency of monochromatic wall excitation started impulsively are studied by performing direct numerical simulation. The computed results for the two frequencies of excitation reported here resemble the experimental setup of Klebanoff et al. [“The three-dimensional nature of boundary-layer instability,” J. Fluid Mech. 12(1), 1–34 (1962)], where a two-dimensional boundary layer is excited using 3D disturbances. Such a monochromatic wall excitation creates three-component disturbance field: a near-field followed by the Tollmien-Schlichting wave-packet and a spatiotemporal wave-front (STWF), which is responsible for eventual transition. It is noted that the case of moderate frequency of excitation shows a complete noninteracting nature of the near-field solution and the STWF. We report another route of transition computationally for a lower frequency of excitation. This case shows an interacting nature of the near-field solution and the STWF. While both frequency of excitations can cause transition for moderate spanwise wavelength (λz) disturbances, dependence of transition on λz is reported here for the first time. It is noted that doubling the spanwise wavenumber leads to the disappearance of STWF and no transition of the flow. We use the recently developed disturbance enstrophy transport equation in Sengupta et al. [“An enstrophy based linear and nonlinear receptivity theory,” Phys. Fluids 30, 054106 (2018)] for a better quantitative method to trace the evolution of disturbance field of the imposed 3D disturbances.

An enstrophy-based linear and nonlinear receptivity theory

In the present research, a new theory of instability based on enstrophy is presented for incompressible flows. Explaining instability through enstrophy is counter-intuitive, as it has been usually associated with dissipation for the Navier-Stokes equation (NSE). This developed theory is valid for both linear and nonlinear stages of disturbance growth. A previously developed nonlinear theory of incompressible flow instability based on total mechanical energy described in the work of Sengupta et al. [“Vortex-induced instability of an incompressible wall-bounded shear layer,” J. Fluid Mech. 493, 277–286 (2003)] is used to compare with the present enstrophy based theory. The developed equations for disturbance enstrophy and disturbance mechanical energy are derived from NSE without any simplifying assumptions, as compared to other classical linear/nonlinear theories. The theory is tested for bypass transition caused by free stream convecting vortex over a zero pressure gradient boundary layer. We explain the creation of smaller scales in the flow by a cascade of enstrophy, which creates rotationality, in general inhomogeneous flows. Linear and nonlinear versions of the theory help explain the vortex-induced instability problem under consideration.

Systematic study of distributed excitation of unsteady Görtler modes by freestream vortices

The paper is devoted to experimental investigation of an efficient mechanism of distributed excitation of nonstationary Görtler vortices in a boundary layer on concave wall due to scattering of three-dimensional, streamwise oriented freestream vortices on natural two-dimensional base-flow nonuniformity associated with the boundary layer growth. The investigations are performed in a broad range of disturbance frequencies and spanwise wavenumbers. The measurements have shown that the distributed receptivity mechanism is able to modify significantly the disturbance growth rates. This mechanism is found to be able to compete successfully with the linear instability mechanism. It is found that at relatively low frequencies and not too large freestream disturbance amplitudes, the Görtler modes’ development is dominated basically by the corresponding linear instability mechanism. Meanwhile, this mechanism becomes weaker with growth of frequency and, simultaneously, the distributed receptivity mechanism gets stronger. This leads to enhancement of role of the latter in amplification of Görtler instability modes, and at high frequencies the role of the distributed receptivity mechanism becomes very important and often even decisive one. Based on deep processing of the experimental data, quantitative values of the distributed linear receptivity coefficients are obtained for the case of excitation of unsteady Görtler modes and estimated for the case of steady Görtler vortices by means of extrapolation to the zero frequency. A complementary experiment devoted to investigation of the problem of linear unsteady Görtler instability is performed for the particular conditions of the distributed receptivity experiments. All main stability characteristics are obtained and compared with calculated ones. In general, the experimental information presented in this paper can be used for verification of various instability and vortex receptivitytheories.

Linear and nonlinear receptivity of the boundary layer in transonic flows

In this paper we analyse the process of the generation of Tollmien–Schlichting waves in a laminar boundary layer on an aircraft wing in the transonic flow regime. We assume that the boundary layer is exposed to a weak acoustic noise. As it penetrates the boundary layer, the Stokes layer forms on the wing surface. We further assume that the boundary layer encounters a local roughness on the wing surface in the form of a gap, step or hump. The interaction of the unsteady perturbations in the Stokes layer with steady perturbations produced by the wall roughness is shown to lead to the formation of the Tollmien–Schlichting wave behind the roughness. The ability of the flow in the boundary layer to convert ‘external perturbations’ into instability modes is termed the receptivity of the boundary layer. In this paper we first develop the linear receptivity theory. Assuming the Reynolds number to be large, we use the transonic version of the viscous–inviscid interaction theory that is known to describe the stability of the boundary layer on the lower branch of the neutral curve. The linear receptivity theory holds when the acoustic noise level is weak, and the roughness height is small. In this case we were able to deduce an analytic formula for the amplitude of the generated Tollmien–Schlichting wave. In the second part of the paper we lift the restriction on the roughness height, which allows us to study the flows with local separation regions. A new ‘direct’ numerical method has been developed for this purpose. We performed the calculations for different values of the Kármán–Guderley parameter, and found that the flow separation leads to a significant enhancement of the receptivity process.

A numerical evaluation of the asymptotic theory of receptivity for subsonic compressible boundary layers

The capabilities of the triple-deck theory of receptivity for subsonic compressible boundary layers have been thoroughly investigated through comparisons with numerical simulations of the compressible Navier–Stokes equations. The analysis focused on the two Tollmien–Schlichting wave linear receptivity problems arising due to the interaction between a low-amplitude acoustic wave and a small isolated roughness element, and the low-amplitude time-periodic vibrations of a ribbon placed on the wall of a flat plate. A parametric study was carried out to look at the effects of roughness element and vibrating ribbon longitudinal dimensions, Reynolds number, Mach number and Tollmien–Schlichting wave frequency. The flat plate is considered isothermal, with a temperature equal to the laminar adiabatic-wall temperature. Numerical simulations of the full and the linearised compressible Navier–Stokes equations have been carried out using high-order finite differences to obtain, respectively, the steady basic flows and the unsteady disturbance fields for the different flow configurations analysed. The results show that the asymptotic theory and the Navier–Stokes simulations are in good agreement. The initial Tollmien–Schlichting wave amplitudes and, in particular, the trends indicated by the theory across the whole parameter space are in excellent agreement with the numerical results. An important finding of the present study is that the behaviour of the theoretical solutions obtained for$\mathit{Re}\rightarrow \infty$holds at finite Reynolds numbers and the only conditions needed for the theoretical predictions to be accurate are that the receptivity process be linear and the free-stream Mach number be subsonic.

Effects of base flow modifications on noise amplifications: flow past a backward-facing step

Amplifications of flow past a backward-facing step with respect to optimal inflow and initial perturbations are investigated at Reynolds number 500. Two mechanisms of receptivity to inflow noise are identified: the bubble-induced inflectional point instability and the misalignment effect downstream of the secondary bubble. Further development of the misalignment results in decay of perturbations from $x=28$ onwards (the step is located at $x=0$), as has been observed in previous non-normality studies (Blackburn et al., J. Fluid Mech., vol. 603, 2008, pp. 271–304), and eventually limits the receptivity. The receptivity is found to be maximized at an inflow perturbation frequency of ${\it\omega}=0.50$ and a spanwise wavenumber of ${\it\beta}=0$, where the inflow noise takes full advantage of both mechanisms and is amplified over two orders of magnitude in terms of the velocity magnitude. In direct numerical simulations (DNS) of the flow perturbed by optimal or random inflow noise, vortex shedding, flapping of bubbles, three-dimensionality and turbulence are observed in succession as the magnitude of the inflow noise increases. Similar features of linear and nonlinear receptivity are observed at higher Reynolds numbers. The Strouhal number of the bubble flapping is 0.08, at which the receptivity to inflow noise reaches a maximum. This Strouhal number is close to reported values extracted from DNS or large eddy simulations (LES) at larger Reynolds numbers (Le et al., J. Fluid Mech., vol. 330, 1997, pp. 349–374; Kaiktsis et al., J. Fluid Mech., vol. 321, 1996, pp. 157–187; Métais, New Trends in Turbulence, 2001, Springer; Wee et al., Phys. Fluids, vol. 16, 2004, pp. 3361–3373). Methods to further clarify the mechanisms of receptivity and to suppress the noise amplifications by modifying the base flow using a linearly optimal body force are proposed. It is observed that the mechanisms of optimal noise amplification are fully revealed by the distribution of the base flow modification, which weakens the bubble instabilities and misalignment effects and subsequently reduces the receptivity significantly. Comparing the base flow modifications with respect to amplifications of inflow and initial perturbations, it is found that the maximum receptivity to initial perturbations is highly correlated with the receptivity to inflow noise at the optimal frequency ${\it\omega}=0.50$, and the correlation reduces as the inflow frequency deviates from this optimal value.

Nonlinear effects on the receptivity of cross-flow in the swept Hiemenz flow

AbstractNonlinear effects on the receptivity of cross-flow in the swept Hiemenz boundary layer are investigated. Numerical simulations are generated using a vorticity form of the Navier–Stokes equations. Steady perturbations are established using surface suction and blowing distributed along the spanwise direction as either a periodic strip or a band of small holes. The method of excitation, the size and the location of the prescribed forcing are shown to have a significant influence on the receptivity of the boundary layer. Blowing holes are found to excite perturbations with considerably larger magnitudes than those generated using a periodic suction and blowing strip. A semi-logarithmic relationship is derived that relates the initial amplitude of the linear-only disturbances with the location at which the absolute magnitude of the chordwise primary Fourier harmonic attains a stationary point or a size of approximately one-tenth of the free-stream spanwise velocity. Furthermore, the size of the physical chordwise velocity perturbation about this position can be estimated directly from the linear-only solutions. This would suggest that, for sufficiently small initial amplitudes, the onset of some nonlinear flow development properties can be predicted directly from a linear receptivity analysis.

Compressibility Effects on Dielectric Barrier Discharge Actuation and Boundary-Layer Receptivity

Dielectric barrier discharge actuators are promising candidates for supersonic boundary-layer transition control because of their ability to concentrate energy transfer in receptive regions in the depth of the boundary layer. A detailed understanding of the plasma–flow coupling mechanisms and optimum scaling in compressible flows is necessary to develop efficient actuation schemes. In this work, the dielectric barrier discharge momentum transfer to an unstable laminar boundary layer over a flat plate under different Reynolds and freestream Mach numbers is studied. A new coupled plasma–compressible flow solver using a block-structured adaptive mesh refinement algorithm is presented, which was developed to model the force field generated by the actuator. The spectral components of the force are used as source terms for the study of the influence of flow conditions and length scales on linear receptivity.

Role of Time Integration in Computing Transitional Flows Caused by Wall Excitation

Numerical investigation of receptivity and flow transition in spatio-temporal framework have shown the central role of spatio-temporal wave-front (STWF) created by wall excitation for transition of a two-dimensional (2D) zero pressure gradient boundary layer (ZPGBL) in Sengupta and Bhaumik (Phys Rev Lett 107:154501, 2011). Although the STWF is created by linear mechanism, it is the later nonlinear stage of evolution revealed by the solution of Navier---Stokes equation (NSE), which causes formation of turbulent spots merging together to create fully developed turbulent flow. Thus, computing STWF for ZPGBL from NSE is of prime importance, which has been reported by the present authors following earlier theoretical investigation. Similar computational efforts using NSE by other researchers do not report finding the STWF. In the present investigation we identify the main reason for other researchers to miss STWF, as due to taking a very short computational domain. Secondly, we show that even one takes a long enough domain and detect STWF, use of traditional low accuracy method will not produce the correct dynamics as reported by Sengupta and Bhaumik (2011). The role of time integration plays a very strong role in the dynamics of transitional flows. We have shown here that implicit methods are more error prone, as compared to explicit time integration methods during flow transition. For the present problem, it is noted that the classical Crank---Nicolson method is unstable for 2D NSE. Same error-prone nature will also be noted for hybrid implicit---explicit time integration methods (known as the IMEX methods). One of the main feature of present analysis is to highlight the accuracy of computations by compact schemes used by the present investigators over a significantly longer domain and over unlimited time, as opposed to those reported earlier in the literature for the wall excitation problem. A consequence of taking long streamwise domain enables one to detect special properties of STWF and its nonlinear growth. The main focus of the present research is to highlight the importance of STWF, which is a new class of spatio-temporal solution obtained from the linear receptivity by solving Orr---Sommerfeld equation and nonlinear analysis of NSE.

Generation of nonstationary Görtler vortices by localized surface nonuniformities. Receptivity coefficients

The mechanism of production of nonstationary Gortler vortices in a boundary layer on concave wall by surface nonuniformities (vibrations and roughness) has been experimentally examined. The nonuniformities were produced by a specially developed disturbance source. They were controlled, localized along the streamwise coordinate, and periodic over the span of the experimental model. Tests in a low-turbulence wind tunnel have proved that the disturbance source is an efficient means of experimental study of the receptivity and stability problem for boundary layers dominated by Gortler instability. The operation of the disturbance source leads to the production of small-amplitude nonstationary Gortler vortices (tenth or hundredth fractions of a per cent of the free-stream velocity) with predefined characteristics (frequency and spanwise wavelength). In our experiments, we quantitatively examined the problem of linear receptivity of boundary layer to surface nonuniformities in a broad range of frequencies for the most dangerous spanwise scales of Gortler vortices. The values of the amplitudes and phases of the receptivity coefficients were determined. The amplitudes proved to be much smaller in magnitude in comparison with the excitation of modes of hydrodynamic instabilities of other types (Tollmien-Schlichting waves and cross-flow-instability modes). It was found that, with increasing the frequency, the amplitudes of the receptivity coefficients showed a distinct growth while for high frequencies those amplitudes also exhibited a growth with the spanwise scale of perturbations, although for stationary surface roughness no effect due to this scale was observed. It was found that the dependences on frequency of the efficiency of the mechanisms of stability and receptivity showed opposing behaviors, were in competition, and could partially compensate each other, promoting, thus, the production of boundary-layer Gortler vortices in a broad range of frequencies.

Nonlinear receptivity to oblique vortical modes in flow past an elliptic leading edge

Nonlinear boundary-layer receptivity to pairs of unsteady oblique freestream vortical modes is studied in direct numerical simulation of flow over a flat plate with an elliptic leading edge. The freestream is perturbed by three types of oblique Fourier modes, differing in the magnitude of the three vorticity components. The vortical modes excite steady boundary-layer streaks. The associated receptivity mechanism, described in detail, is quadratic in the forcing amplitude. Elliptic leading edges with two different aspect ratios are considered. We find that – and explain why – the streak amplitudes in nonlinear receptivity are largely unaffected by the leading-edge bluntness for the types of external disturbances studied. As linear receptivity is the predominant mechanism at low forcing frequencies, the nonlinear mechanism comes into play when high-frequency vortices are present in the freestream. Nonlinear receptivity is therefore expected to contribute to the excitation of boundary-layer streaks by freestream turbulence.