Parabolized Stability Equations Research Articles

Instability wave-streak interactions in a supersonic boundary layer.

Optimal initial conditions for transient growth in a two-dimensional boundary layer flow correspond to stationary, counter-rotating vortices that subsequently develop into streamwise elongated streaks, which are characterized by an alternating pattern of low and high streamwise velocity. For incompressible flows, previous studies have shown that boundary layer modulation due to streaks below a threshold amplitude level can stabilize the Tollmien-Schlichting instability waves, resulting in a delay in the onset of laminar-turbulent transition. In the supersonic regime, the linearly, most-amplified waves become three-dimensional, corresponding to oblique, first-mode waves. This change in the character of dominant instabilities leads to an important change in the transition process, which is now dominated by oblique breakdown via nonlinear interactions between pairs of first-mode waves that propagate at equal but opposite angles with respect to the free stream. Because the oblique breakdown process is characterized by a rapid amplification of stationary streamwise streaks, artificial excitation of such streaks may be expected to promote transition in a supersonic boundary layer. Indeed, suppression of those streaks has been shown to delay the onset of transition in prior literature. Consistent with those findings, the present study shows that optimally growing stationary streaks indeed destabilize the first-mode waves, but only when the spanwise wavelength of the instability waves is equal to or smaller than twice the streak spacing. Transition in a benign disturbance environment typically involves first-mode waves with significantly longer spanwise wavelengths, and hence, these waves are stabilized by the optimal growth streaks. Thus, as long as the amplification factors for the destabilized, short wavelength instability waves remain below the threshold level for transition, a significant net stabilization is achieved, yielding a transition delay that is comparable to the length of the laminar region in the uncontrolled case.

High-frequency wavepackets in turbulent jets

Wavepackets obtained as solutions of the flow equations linearised around the mean flow have been shown in recent work to yield good agreement, in terms of amplitude and phase, with those educed from turbulent jets. Compelling agreement has been demonstrated, for the axisymmetric and first helical mode, up to Strouhal numbers close to unity. We here extend the range of validity of wavepacket models to Strouhal number $St=4.0$ and azimuthal wavenumber $m=4$ by comparing solutions of the parabolised stability equations with a well-validated large-eddy simulation of a Mach 0.9 turbulent jet. The results show that the near-nozzle dynamics can be correctly described by the homogeneous linear model, the initial growth rates being accurately predicted for the entire range of frequencies and azimuthal wavenumbers considered. Similarly to the lower-frequency wavepackets reported prior to this work, the high-frequency linear waves deviate from the data downstream of their stabilisation locations, which move progressively upstream as the frequency increases.

Algebraic disturbance growth by interaction of Orr and lift-up mechanisms

Algebraic disturbance growth in spatially developing boundary-layer flows is investigated using an optimization approach. The methodology builds on the framework of the parabolized stability equations and avoids some of the limitations associated with adjoint-based schemes. In the Blasius boundary layer, non-parallel effects are shown to significantly enhance the energy gain due to algebraic growth mechanisms. In contrast to parallel flow, the most energetic perturbations have finite frequency and are generated by the simultaneous activity of the Orr and lift-up mechanisms. The highest amplification occurs in a limited region of the parameter space that is characterized by a linear relation between the wavenumber and frequency of the disturbances. The frequency of the most highly amplified perturbations decreases with Reynolds number. Adverse streamwise pressure gradient further enhances the amplification of disturbances while preserving the linear trend between the wavenumber and frequency of the most energetic perturbations.

Control of stationary crossflow modes in swept Hiemenz flows with dielectric barrier discharge plasma actuators

Sensitivity analyses and non-linear parabolized stability equations are solved to provide a computational assessment of the potential use of a Dielectric Barrier Discharge (DBD) plasma actuator for a prolonging laminar region in swept Hiemenz flow. The derivative of the kinetic energy with respect to the body force is deduced, and its components in different directions are defined as sensitivity functions. The results of sensitivity analyses and non-linear parabolized stability equations both indicate that the introduction of a body force as the plasma actuator at the bottom of a crossflow vortex can mitigate instability to delay flow transition. In addition, the actuator is more effective when placed more upstream until the neutral point. In fact, if the actuator is sufficiently close to the neutral point, it is likely to act as a strong disturbance over-riding the natural disturbance and dominating transition. Different operating voltages of the DBD actuators are tested, resulting in an optimal practice for transition delay. The results demonstrate that plasma actuators offer great potential for transition control.

Interaction of instability waves and a three-dimensional roughness element in a boundary layer

The influence of a single roughness element on the evolution of two-dimensional (2-D) Tollmien–Schlichting (TS) waves is investigated experimentally. Experiments are carried out in a region of zero pressure gradient of an airfoil section. Downstream from the disturbance source, TS waves interact with a cylindrical roughness element with a slowly oscillating height. The oscillation frequency of the roughness was approximately 1500 times lower than the wave frequency and approximately 250 times slower than the characteristic time of flow passing the region of transition development. Therefore, the roughness behaved as a quasi-steady disturbance. The set-up enabled us to perform hot-wire measurements phase locked to the waves and to the roughness movement. Experimental results show a scattering of the 2-D waves into oblique ones and a relatively weak distortion of the mean flow for roughness heights as large as 0.2 times the boundary layer displacement thickness ($\unicode[STIX]{x1D6FF}^{\ast }$). Transfer functions for TS wave scattering at the roughness are obtained. Results show an unexpected coincidence in shape with acoustic receptivity functions found in Würz et al. (J. Fluid Mech., vol. 478, 2003, pp. 135–163) for the problem of excitation of TS waves by scattering of acoustic waves at surface roughness. In the present work, the ratio between the incoming 2-D wave amplitude to the amplitude of the scattered oblique waves scaled linearly with the roughness height only for very shallow roughness. For roughness elements higher than $0.08\unicode[STIX]{x1D6FF}^{\ast }$ and below $0.2\unicode[STIX]{x1D6FF}^{\ast }$, the wave scattering exhibited a quadratic variation with respect to the roughness height. In addition, this feature did not vary significantly with respect to TS wave frequency. An analysis of the weakly nonlinear interactions triggered by the roughness element is also carried out, assisted by numerical solution of nonlinear parabolized stability equations, performed for a two-dimensional Blasius boundary layer. A comparison between experiments and simulations reveals that the weakly nonlinear interactions observed are not substantially affected by mean flow distortions that could be produced in the wake of the small and medium sized roughness elements ($h<0.2\unicode[STIX]{x1D6FF}^{\ast }$). From a practical perspective, results suggest that scattering coefficients might be employed to include the effect of isolated and medium sized roughness elements in transition prediction tools developed for smooth surfaces.

Unsteady flow dynamics reconstruction from mean flow and point sensors: an experimental study

This article presents a reconstruction of the unsteady behaviour of a round jet at a Reynolds number equal to 3300, from the sole knowledge of the time-averaged flow field and one pointwise unsteady measurement. The reconstruction approach is an application of the work of Beneddine et al. (J. Fluid Mech., vol. 798, 2016, pp. 485–504) and relies on the computation of the dominant resolvent modes of the flow, using a parabolised stability equations analysis. To validate the procedure, the unsteady velocity field of the jet has been characterised by time-resolved particle image velocimetry (TR-PIV), yielding an experimental reference. We first show that the dominant resolvent modes are proportional to the experimental Fourier modes, as predicted by Beneddine et al. (J. Fluid Mech., vol. 798, 2016, pp. 485–504). From these results, it is then possible to fully reconstruct the unsteady velocity and pressure fluctuation fields, yielding a flow field that displays good agreement with the experimental reference. Finally, it is found that the robustness of the reconstruction mainly depends on the location of the pointwise unsteady measurement, which should be within energetic regions of the flow, and this robustness as well as the quality of the reconstruction can be greatly improved by considering a few pointwise measurements instead of a single one. The effects of other experimental parameters on the reconstruction, such as the size of the interrogation window used for the TR-PIV processing and the accuracy of the positioning of the sensors, are also investigated in this paper.

Selective enhancement of oblique waves caused by finite amplitude second mode in supersonic boundary layer

Nonlinear interactions of the two-dimensional (2D) second mode with oblique modes are studied numerically in a Mach 6.0 flat-plate boundary layer, focusing on its selective enhancement effect on amplification of different oblique waves. Evolution of oblique modes with various frequencies and spanwise wavenumbers in the presence of 2D second mode is simulated successively, using a modified parabolized stability equation (PSE) method, which is able to simulate interaction of two modes with different frequencies efficiently. Numerical results show that oblique modes in a broad band of frequencies and spanwise wavenumbers can be enhanced by the finite amplitude 2D second mode instability wave. The enhancement effect is accomplished by interaction of the 2D second mode, the oblique mode, and a forced mode with difference frequency. Two types of oblique modes are found to be more amplified, i.e., oblique modes with frequency close to that of the 2D second mode and low-frequency first mode oblique waves. Each of them may correspond to one type of transition routes found in transition experiments. The spanwise wavenumber of the oblique wave preferred by the nonlinear interaction is also determined by numerical simulations.

Real-time modelling of wavepackets in turbulent jets

Three methods are considered for estimating the downstream evolution of wavepackets in turbulent jets based on upstream measurements. The parabolised stability equations are used to compute a transfer function between axially and radially separated points in the flow, and the performance of this theoretical model is compared with that of two empirical approaches, direct transfer function calculation and autoregressive moving-average exogenous system identification, both of which require unsteady experimental data. The three approaches, which perform equally well, prove suitable for estimation of the downstream evolution of wavepackets using pressure data measured in the near-nozzle region. Over distances of the order of a couple of jet diameters, correlations of up to 80 % are observed between estimation and measurement. The performance deteriorates as axial separation between input and output is increased. While the two empirical approaches are limited in terms of both the number of input–output pairs and the number of flow variables that can be reasonably considered, the parabolised stability equations-based approach has no such limitation and can be used to perform full-field estimates comprising all of the dependent variables; in this it constitutes a potentially formidable means by which to perform single-input–multiple-output estimation. It has the further advantage of not requiring unsteady data for its construction, the only necessary ingredients being the mean flow and the linearised equations of motion.

Blunt-body paradox and transient growth on a hypersonic spherical forebody

The laminar-turbulent transition in hypersonic flow on a blunt hemisphere is studied with Navier-Stokes and parabolized stability equations. Maximum transient kinetic energy growth is found close to the sonic point, which could explain the transition observed in wind tunnel and flight experiments.

Applications of EPSE method for predicting crossflow instability in swept-wing boundary layers

The nth-order expansion of the parabolized stability equation (EPSEn) is obtained from the Taylor expansion of the linear parabolized stability equation (LPSE) in the streamwise direction. The EPSE together with the homogeneous boundary conditions forms a local eigenvalue problem, in which the streamwise variations of the mean flow and the disturbance shape function are considered. The first-order EPSE (EPSE1) and the second-order EPSE (EPSE2) are used to study the crossflow instability in the swept NLF(2)-0415 wing boundary layer. The non-parallelism degree of the boundary layer is strong. Compared with the growth rates predicted by the linear stability theory (LST), the results given by the EPSE1 and EPSE2 agree well with those given by the LPSE. In particular, the results given by the EPSE2 are almost the same as those given by the LPSE. The prediction of the EPSE1 is more accurate than the prediction of the LST, and is more efficient than the predictions of the EPSE2 and LPSE. Therefore, the EPSE1 is an efficient e N prediction tool for the crossflow instability in swept-wing boundary-layer flows.

Discrete- and finite-bandwidth-frequency distributions in nonlinear stability applications

A new “wave packet” formulation of the parabolized stability equations method is presented. This method accounts for the influence of finite-bandwidth-frequency distributions on nonlinear stability calculations. The methodology is motivated by convolution integrals and is found to appropriately represent nonlinear energy transfer between primary modes and harmonics, in particular nonlinear feedback, via a “nonlinear coupling coefficient.” It is found that traditional discrete mode formulations overestimate nonlinear feedback by approximately 70%. This results in smaller maximum disturbance amplitudes than those observed experimentally. The new formulation corrects this overestimation, accounts for the generation of side lobes responsible for spectral broadening, and results in disturbance representation more consistent with the experiment than traditional formulations. A Mach 6 flared-cone example is presented.

Amplitude method of prediction of laminar-turbulent transition on a swept-wing

The amplitudemethod of prediction of laminar-turbulent transition on a swept-wing initiated by the simultaneous action of free-stream turbulence and surface roughness is developed. Generation of growing intrinsic perturbations in the boundary layer is described by the mechanism of distributed generation, i.e., an external perturbation generates an instability mode with the same wavelength and frequency. Generation occurs in a small neighborhood of the neutral point at which the parameters of an external perturbation and the instability modes coincide. The development of proper perturbations in the boundary layer is described by the nonlinear method of parabolized stability equations (PSE). The criterion of laminar-turbulent transition is the combined amplitude. i.e., transition begins at a point at which the sum of amplitudes of steady and traveling modes reaches a critical value.

Secondary instability analysis of crossflow on a hypersonic yawed straight circular cone

The purpose of this paper is to provide secondary instability analysis of stationary crossflow vortices on a hypersonic yawed straight circular cone with a $7^{\circ }$ half-angle at $6^{\circ }$ angle of attack, free-stream Mach number 6 and unit Reynolds number $10.09\times 10^{6}~\text{m}^{-1}$. At an angle of attack, a three-dimensional boundary layer is developed between the windward and leeward symmetry planes. Under the action of azimuthal pressure gradients, the flow near the surface is deflected more than the flow near the edge of the boundary layer. This results in an inflectional velocity profile that can sustain the growth of crossflow vortices. The stationary crossflow instability is computed by means of the nonlinear parabolized stability equations, including a methodology to predict the stationary-crossflow marching path and variation of the spanwise number of waves in the marching direction solely from the basic state. Secondary instability analysis is performed using spatial BiGlobal equations based on two-dimensional partial differential equations. The secondary instabilities are calculated at different axial locations along two crossflow vortex trajectories selected to complement experiments conducted in the Mach 6 Quiet Tunnel at Texas A&M University and in the Boeing/AFOSR Mach 6 Quiet Tunnel at Purdue University. The secondary instability analysis captures various instability modes. Similar to observations in the low-speed regime for an infinite swept wing, secondary shear-layer instabilities are amplified as a consequence of the three-dimensional shear layer formed by crossflow vortices. Also, low-frequency travelling crossflow and high-frequency second modes coexist with the shear-layer instabilities. These results are shown to be in good agreement with the two sets of hypersonic yawed cone experiments (one with natural surface roughness and one with artificial discrete roughness) and compare well with experimental measurements of an incompressible swept wing.

Sensitivity of wavepackets in jets to nonlinear effects: the role of the critical layer

Linear instability waves, or wavepackets, are key building blocks for the jet-noise problem. It has been shown in previous work that linear models correctly predict the evolution of axisymmetric wavepackets up to the end of the potential core of subsonic turbulent jets. Beyond this station, linear models fail, and nonlinearity is the likely missing piece. The essential underlying nonlinear mechanisms are unknown, and it remains unclear how these should be incorporated in a reduced-order model. The nonlinear interactions are considered in this work as an ‘external’ harmonic forcing added to the standard linear model. This modelling framework is explored using a locally parallel resolvent analysis to determine optimal forcing and associated responses, and a global approach based on 4D-Var data assimilation aimed at finding the optimal forcing of the parabolised stability equations that would minimise errors in the predictions of wavepackets. In all of the problems considered, the critical layer is found to be relevant: it is the position where sensitivity of wavepackets to nonlinearity is greatest. It is seen that disturbances are forced around the critical layer, and tilted by shear as they are advected, in a manner suggestive of an Orr-like mechanism. The ensemble of results suggests that critical-layer effects play a central role in the dynamics of wavepackets in subsonic turbulent jets, and that inclusion of such effects may remedy the shortcomings of linear reduced-order models.

Modal analysis of jet near-field pressure fluctuation

Purpose Unstable Kelvin-Helmholtz waves are suspected to be responsible for a large part of low-frequency noise radiation in high-speed jets. The purpose of this paper is to check the coherence of numerical and experimental data concerning this phenomenon, in the particular case of a cold subsonic jet. Design/methodology/approach In the present work, a cross-investigation of the near pressure field is performed on three different data sets: large-eddy simulation (LES) computations, parabolised stability equations (PSE) semi-modal computations and microphone measurements, in order to determine the local amplitudes of unstable waves. Findings The large coherent structures are found in both LES and experimental results and they are also in good agreement with direct semi-modal computations carried out with the PSE approach. Originality/value This work confirms that the unstable wave packets can be extracted from both LES and experimental results, provided that an appropriate modal decomposition is performed.

Nonlinear interaction of instability waves and vortex-pairing noise in axisymmetric subsonic jets

A direct simulation with selected inflow forcing is performed for an accurate description of the jet flow field and far-field noise. The effects of the Mach number and heating on the acoustic field are studied in detail. The beam patterns and acoustic intensities are both varied as the change of the Mach number and temperature. The decomposition of the source terms of the Lilley–Goldstein (L–G) equation shows that the momentum and thermodynamic components lead to distinctly different beam patterns. Significant cancellation is found between the momentum and thermodynamic components at low polar angles for the isothermal jet and large polar angles for the hot jet. The cancellation leads to the minimum values of the far-field sound. Based on linear parabolized stability equation solutions, the nonlinear interaction model for sound prediction is built in combination with the L–G equation. The dominant beam patterns and their original locations predicted by the nonlinear model are in good agreement with the direct simulation results, and the predictions of sound pressure level (SPL) by the nonlinear model are relatively reasonable.

Three-dimensional instability analysis of boundary layers perturbed by streamwise vortices

A parametric study is presented for the incompressible, zero-pressure-gradient flat-plate boundary layer perturbed by streamwise vortices. The vortices are placed near the leading edge and model the vortices induced by miniature vortex generators (MVGs), which consist in a spanwise-periodic array of small winglet pairs. The introduction of MVGs has been experimentally proved to be a successful passive flow control strategy for delaying laminar-turbulent transition caused by Tollmien–Schlichting (TS) waves. The counter-rotating vortex pairs induce non-modal, transient growth that leads to a streaky boundary layer flow. The initial intensity of the vortices and their wall-normal distances to the plate wall are varied with the aim of finding the most effective location for streak generation and the effect on the instability characteristics of the perturbed flow. The study includes the solution of the three-dimensional, stationary, streaky boundary layer flows by using the boundary region equations, and the three-dimensional instability analysis of the resulting basic flows by using the plane-marching parabolized stability equations. Depending on the initial circulation and positioning of the vortices, planar TS waves are stabilized by the presence of the streaks, resulting in a reduction in the region of instability and shrink of the neutral stability curve. For a fixed maximum streak amplitude below the threshold for secondary instability (SI), the most effective wall-normal distance for the formation of the streaks is found to also offer the most stabilization of TS waves. By setting a maximum streak amplitude above the threshold for SI, sinuous shear layer modes become unstable, as well as another instability mode that is amplified in a narrow region near the vortex inlet position.

Linear spatial instability analysis in 3D boundary layers using plane-marching 3D-LPSE

It is widely accepted that a robust and efficient method to compute the linear spatial amplified rate ought to be developed in three-dimensional (3D) boundary layers to predict the transition with the e N method, especially when the boundary layer varies significantly in the spanwise direction. The 3D-linear parabolized stability equation (3D-LPSE) approach, a 3D extension of the two-dimensional LPSE (2D-LPSE), is developed with a plane-marching procedure for investigating the instability of a 3D boundary layer with a significant spanwise variation. The method is suitable for a full Mach number region, and is validated by computing the unstable modes in 2D and 3D boundary layers, in both global and local instability problems. The predictions are in better agreement with the ones of the direct numerical simulation (DNS) rather than a 2D-eigenvalue problem (EVP) procedure. These results suggest that the plane-marching 3D-LPSE approach is a robust, efficient, and accurate choice for the local and global instability analysis in 2D and 3D boundary layers for all free-stream Mach numbers.

Improved algorithm for solving nonlinear parabolized stability equations**Project supported by the National Natural Science Foundation of China (Grant Nos. 11332007 and 11402167).

Due to its high computational efficiency and ability to consider nonparallel and nonlinear effects, nonlinear parabolized stability equations (NPSE) approach has been widely used to study the stability and transition mechanisms. However, it often diverges in hypersonic boundary layers when the amplitude of disturbance reaches a certain level. In this study, an improved algorithm for solving NPSE is developed. In this algorithm, the mean flow distortion is included into the linear operator instead of into the nonlinear forcing terms in NPSE. An under-relaxation factor for computing the nonlinear terms is introduced during the iteration process to guarantee the robustness of the algorithm. Two case studies, the nonlinear development of stationary crossflow vortices and the fundamental resonance of the second mode disturbance in hypersonic boundary layers, are presented to validate the proposed algorithm for NPSE. Results from direct numerical simulation (DNS) are regarded as the baseline for comparison. Good agreement can be found between the proposed algorithm and DNS, which indicates the great potential of the proposed method on studying the crossflow and streamwise instability in hypersonic boundary layers.

Parabolized Stability Equations Code with Automatic Inflow for Swept Wing Transition Analysis

This paper aims at developing a parabolized stability equation transition analysis code coupled to the industrial in-house accurate Reynolds-averaged Navier–Stokes code developed at Israel Aerospace Industries for stability analysis of three-dimensional boundary layers over swept wings. In the first step, the parabolized stability equation derivation in a body-fitted coordinate system is exposed, as well as its classical initialization, using the solution procedure of the global Chebyshev eigenvalue problem for the linear stability equations. This initialization procedure demands large user interactions, and it prevents the parabolized stability equation method to be of practical use in an industrial environment. Therefore, one exposes in the second step a new way to automatically generate the inflow solution of the linear stability equations needed by the parabolized stability equations. This new robust approach paves the way for innovative automatic industrial use of the parabolized stability equation technology without the need to involve the user. This approach supplies the inflow conditions required for the parabolized stability equation marching procedure without the need to solve the standalone linear stability global eigenvalue problem. The solution of the linear stability equations at initialization is now obtained by iteratively converging the local linear stability solver starting from the new inflow empirical approximation obtained in two special coordinate systems for crossflow and Tollmien–Schlichting, respectively. These formulas represent some invariant functions of the stability characteristics of the exact Navier–Stokes laminar boundary-layer profiles instead of self-similar boundary-layer profiles like the Falkner–Skan–Cooke. Standard academic validation test cases and industrial results of the parabolized stability equation/Reynolds-average Navier-Stokes codes on a generic unmanned aerial vehicle swept wing are exposed.