Linear Parabolized Stability Equations Research Articles

Crossflow Instability on a Swept Fin–Cone with Variable Nose Bluntness in Mach 6 Flow

Experiments and computational analysis have been conducted to study instabilities in the boundary layer over a swept fin–cone geometry in hypersonic flow. Experiments were carried out at the United States Air Force Academy’s Mach 6 Ludwieg Tube. Infrared thermography was employed to assess the surface temperature distribution, particularly the thermal striations on the fin that indicated a crossflow-dominated transition. Increasing the unit Reynolds number increased heating and moved the crossflow-transition front closer to the fin leading edge. Stability analysis based on the linear parabolized stability equations identified unstable stationary crossflow modes across the range of tested unit Reynolds numbers, and the most amplified disturbances were found to have wavelengths of 2–4 mm. The experimentally observed thermal striations were analyzed and found to have initial wavelengths of 2–5 mm but eventually formed higher-amplitude structures with a wavelength of approximately 10 mm. Discrete roughness elements of different diameters and spacings were implemented near the fin leading edge. Boundary-layer transition onset was found to be delayed with a subsequent lowering of the fin surface heating, with discrete roughness elements having a wavelength of 13.33 mm.

Influence of Nose Bluntness on Crossflow Instability

Much work has been focused on the initiation of crossflow instability due to roughness-induced mode excitation. Here we emphasize another aspect of the problem; how changes in nose bluntness (possibly due to ablation or other effects) may also alter the dynamics of crossflow instability via entropy layer modification. The effect of nose bluntness on crossflow instability for a Mach 6 yawed cone is considered at two different angles of attack. Linear parabolized stability equations investigation is performed on crossflow vortices along vortex lines calculated by the inflection point method. It is found that small changes in nose bluntness significantly alter the trajectory of crossflow vortices, the strength of the crossflow instability, and the most unstable wavenumber of the crossflow vortices. These results may be of particular importance to experimentalists studying the effect of ablation on crossflow instability.

The influence of nonlinearities on jet noise modeling based on parabolized stability equation

With the aid of a large eddy simulation (LES) model of a turbulent jet, we study the modeling of jet noise based on wavepackets by considering a certain degree of nonlinearity. Linear parabolized stability equations (PSEs) are utilized to solve the spatial evolution of wavepackets with the base flow obtained from the LES. The spectral proper orthogonal decomposition (SPOD) is performed to extract the most energetic coherent structures. Since the mean flow includes partial nonlinearity, improved agreement of hydrodynamic pressure fields between SPOD-filtered results and linear-PSE solutions is obtained in the near field. Deviations only occur when the coherent structures decay. Although linear-PSE solutions represent the near-field hydrodynamics reasonably, the far-field noise propagated from this linear model shows a large deviation from the LES results. Then, a small external harmonic forcing is added to the right-hand side of the PSE to mimic the effects of nonlinearity due to incoherent fluctuations on the late evolution of near-field wavepackets, and an adjoint approach is further utilized to search for optimal forcing distribution. Optimized forcing is mainly located near the critical layer; enhances the energy of wavepackets; and raises the sound radiation efficiency, but to a limited extent. Meanwhile, the coherence-matched PSE wavepacket is proposed to incorporate the coherence decay of wavepackets calculated based on LES. An improved agreement in far-field sound pressure levels for low-frequency components is achieved. In short, these findings all prove the vital role of nonlinearity in jet noise modeling, and the current modeling approaches have made some progress. However, a more physics-based and generalized nonlinear model is still required.

Wall Cooling Effect on High-Enthalpy Supersonic Modes over a Cone

High-enthalpy hypersonic boundary-layer transition plays an important role in many entry/descent vehicles. The aerothermodynamic performance of these vehicles strongly depends on the transition location on the surface. However, detailed transition flow physics in these chemically reacting boundary layers are poorly understood and transition estimates during the design phase rely heavily on empirically derived transition criteria such as . One of the most intriguing characteristics in hypersonic boundary layers is the presence of unstable supersonic modes, first identified in the 1990s. Due to a recent surge in hypersonic applications, there has been renewed interest in studying the flow physics of supersonic modes using either theory or direct numerical simulations. This paper investigates the rise of supersonic modes in a high-enthalpy hypersonic flow over a 5 deg half-angle cone at various wall temperatures using both quasi-parallel linear stability theory and linear parabolized stability equations (PSEs). It was found that supersonic modes exist in all wall temperature conditions, including the adiabatic wall case. A cooler wall temperature causes the second Mack mode to become an unstable supersonic mode naturally downstream of the upper-branch neutral location when the wall is sufficiently cooled. In terms of the integrated growth, the second mode is still the dominant mode. Nonetheless, supersonic modes can cause additional series of relatively weaker growth than that of the second mode beyond the peak amplitude location. According to the present linear PSE results, the formation of unstable supersonic modes is mainly associated with the synchronization of phase speed between the instability and acoustic waves in a nonparallel boundary layer, and not due to nonlinear modal interaction as suggested in the literature.

Parabolized Stability Analysis of Hypersonic Thermal–Chemical Nonequilibrium Boundary-Layer Flows

In this work, the stability of Mach 20 flows past a 6 deg wedge in thermal–chemical nonequilibrium (TCNE) is studied by means of linear parabolized stability equations (PSEs) in combination with advanced thermodynamic and transport models. First, verifications are performed against existing numerical results in the literature. Good agreement is achieved in two benchmark cases. The present PSE results match the direct numerical simulation data from the literature fairly well, especially for the growth-rate oscillation behavior of the supersonic mode disturbance. Then, we investigate the TCNE effects on the supersonic mode instabilities over the 6 deg wedge. It is found that the flow stability is affected mainly by the TCNE effect on mean flows rather than that on disturbances. Subsequently, more PSE calculations are carried out with a variation of wall temperatures and disturbance frequencies. With the wall temperature beyond 30% of the adiabatic one, the unstable supersonic mode is found to return to the subsonic one. Downstream of this turning point, the disturbance growth-rate oscillation dramatically decays. With the increase of the disturbance frequency, a similar trend to wall heating is observed in the growth-rate and phase velocity curves.

Amplitude Scaling of Wave Packets in Turbulent Jets

This paper studies the amplitude of large-scale coherent wave-packet structures in jets, modeled by the parabolized stability equations (PSEs). Linear PSEs can retrieve the shape of the wave packets, but linearity leads to solutions with a free amplitude, which has traditionally been obtained in an ad hoc manner using limited data. We systematically determine the free amplitude as a function of frequency and azimuthal wave number by comparing the fluctuation fields retrieved from PSEs with coherent structures educed from large-eddy simulation data using spectral proper orthogonal decomposition. The wave-packet amplitude is shown to decay exponentially with the Strouhal number for axisymmetric and helical modes at both Mach numbers considered in the study: 0.4 and 0.9. Analytical fit functions are proposed, and the scaled wave packets provide reasonable reconstructions of pressure and velocity spectra on the jet centerline and lip line over a range of streamwise positions.

On the receptivity of surface plasma actuation in high-speed boundary layers

Significant amounts of work have been conducted in the area of plasma flow control, while the receptivity of plasma actuation in high-speed boundary layers has not had much attention over the last two decades. In the present study, the receptivity of a Mach 4.5 flat-plate boundary layer to plasma heating actuation produced by pulsed-DC surface dielectric barrier discharge (SDBD) has been studied by direct numerical simulation (DNS) and stability analysis. With the help of multimode decomposition technology, the amplitude of normal modes can be obtained. The results show that both fast and slow modes can be excited by plasma actuation, and the receptivity maximum is observed near the lower neutral branch. Because the pulsed-DC SDBD actuation is typical periodic pulse signals, when the total power remains constant, the Fourier components with multiples of actuation frequency have the same energy, regardless of the waveform, period, and width of the actuation signal. Such characteristics benefit the robustness of the pulsed-DC SDBD actuator. A theoretical prediction method by combining the receptivity model and linear parabolized stability equations is considered in the present study, and good agreement with the DNS results is achieved.

Secondary instability of stationary Görtler vortices originating from first/second Mack mode

This work investigates the origination of the secondary instability in Görtler vortices using the linear stability theory, BiGlobal analysis, three-dimensional linear parabolized stability equations (3DLPSEs), and direct numerical simulation (DNS). The flow over a concave wall suffering from the Görtler instability and first/second Mack mode instability is selected. Furthermore, this work simulates the evolution of infinitesimal Mack mode disturbance in a flow perturbed by finite-amplitude Görtler vortices by using DNS and 3DLPSE methods. The 3DLPSE approach accurately predicts the process of Mack mode disturbance evolving into the secondary instability of Görtler vortices, and a perfect agreement with results by DNS is obtained. The results indicate that the secondary instability of stationary Görtler vortices can originate from the first/second Mack mode. The evolutions of first/second Mack mode with different spanwise wavenumbers are performed based on 3DLPSE and compared against the BiGlobal method. The results show that the shape functions and growth rates of disturbances always tend to the results of dominant modes obtained by the BiGlobal method. Because the dominant mode might shift from one to another, the overall evolution cannot be predicted only by the BiGlobal method based on a fixed mode. According to our computations, it is deduced that the Mack modes with the same frequency and symmetric characteristics would finally develop into the secondary instability with similar shapes.

Harmonic linearized Navier-Stokes equation on describing the effect of surface roughness on hypersonic boundary-layer transition

Laminar-turbulent transition is crucially influenced by wall roughness. This paper develops a numerical approach based on the harmonic linearized Navier-Stokes (HLNS) equations to accommodate the scattering effect of the rapidly distorted mean flow induced by a two-dimensional hump or indentation at the wall on the oncoming instability modes (including the Mack first and second modes) in a hypersonic boundary layer. Due to the ellipticity of the scattering system when the roughness width is comparable with the instability wavelength, the traditional linear stability theory and the linear parabolized stability equation do not apply, and therefore, the HLNS approach has advantages in both accuracy and efficiency. The impact of a roughness is characterized by a transmission coefficient, which is the ratio of the asymptotic amplitude downstream of the roughness to that upstream. At a Mach number of 5.92, the dependence of the transmission coefficient on the frequency and the oblique angle of the oncoming mode and the size and location of the hump/indentation is studied systematically. It is confirmed that the synchronization frequency appears as a critical frequency, above and below which the oncoming instability modes are suppressed and enhanced by the roughness, respectively, which provides fundamental basis to the laminar-flow control in hypersonic boundary layers.

Modeling mode interactions in boundary layer flows via the parabolized Floquet equations

In this paper, we develop a model based on successive linearization to study interactions between different modes in boundary layer flows. Our method consists of two steps. First, we augment the Blasius boundary layer profile with a disturbance field resulting from the linear Parabolized Stability Equations (PSE) to obtain the modified base flow; and, second, we draw on Floquet decomposition to capture the effect of mode interactions on the spatial evolution of flow fluctuations via a sequence of linear progressions. The resulting Parabolized Floquet Equations (PFE) can be conveniently advanced downstream to examine the interaction between different modes in slowly varying shear flows. We apply our framework to two canonical settings of transition in boundary layers; the H-type transition scenario that is initiated by exponential instabilities, and streamwise elongated laminar streaks that are triggered by the lift-up mechanism. We demonstrate that the PFE capture the growth of various harmonics and provide excellent agreement with the results obtained in direct numerical simulations and in experiments.

Sensitivity of high-speed boundary-layer stability to base-flow distortion

The linear stability of high-speed boundary layers can be altered by distortions to the base velocity and temperature profiles. An analytic expression for the sensitivity is derived for parallel and spatially developing boundary layers, the latter using linear parabolized stability equations and their adjoint. Both the slow mode, S, and the fast mode, F, are investigated at Mach number 4.5. The mode S is more sensitive with respect to distortion in base velocity than in base temperature. The sensitivity is largest within the boundary layer away from the wall. Near the critical layer, where the phase speed of the mode equals the base streamwise velocity, the sensitivity to the base streamwise velocity is negative. For the mode F, there is a discontinuous jump in the sensitivity when the phase speed is below unity, and a critical layer is established. The sensitivity of the two modes increases with the Reynolds number, but there is a sudden drop and a jump in the sensitivities of the modes S and F, respectively, near the synchronization point where the phase speeds of the two modes are equal. Furthermore, the maximum uncertainty bounds are obtained for the distorted base state that maximizes the destabilization or stabilization of the modes by solving the Lagrangian optimization problem for the sensitivity. The sensitivity of the flow stability to surface heating is then studied, and changes in growth rate and the$N$-factor are evaluated. The formulation provides a clear physical interpretation of these changes, and establishes uncertainty bounds for stability predictions for a given level of uncertainty in wall temperature.

Parabolized Stability Equation Analysis of Crossflow Instability on HIFiRE-5b Flight Test

Basic state calculations were performed for the Hypersonic International Flight Research Experiment 5 geometry for both flight and wind-tunnel conditions using the US3D flow solver. Linear parabolized stability equation calculations were performed for the crossflow instabilities, and linear stability theory calculations were performed for the second-mode instabilities using the LASTRAC3d (the three-dimensional Langley stability and transition analysis code) stability suite. Conditions were chosen to represent the flight conditions at a time when a three-lobed transitional front was observed experimentally. Preliminary N-factor thresholds of for the second mode and for stationary crossflow provided a lower bound for the transitional N factor for the observed transitional front during flight conditions over much of the acreage of the vehicle. Applying the same criterion to conditions of recent wind-tunnel tests also reasonably reproduced observed transitional behavior, indicating that the relative physics were accurately captured.

Investigation of the second-mode instability at Mach 14 using calibrated schlieren

Second-mode wave growth within the hypersonic boundary layer of a slender cone is investigated experimentally using high-speed schlieren visualizations. Experiments were performed in AEDC Tunnel 9 over a range of unit Reynolds number conditions at a Mach number of approximately 14. A thin lens with a known density profile placed within the field of view enables calibration of the schlieren set-up, and the relatively high camera frame rates employed allow for the reconstruction of time-resolved pixel intensities at discrete streamwise locations. The calibration in conjunction with the reconstructed signals enables integrated spatial amplification rates ($N$ factors) to be calculated for each unit Reynolds number condition and compared to $N$ factors computed from both pressure transducer measurements and linear parabolized stability equation (PSE) solutions. Good agreement is observed between $N$ factors computed from the schlieren measurements and those computed from the PSE solutions for the most-amplified second-mode frequencies. The streamwise development of $N$ factors calculated from the schlieren measurements compares favourably to that calculated from the pressure measurements with slight variations in the $N$ factor magnitudes calculated for harmonic frequencies. Finally, a bispectral analysis is carried out to identify nonlinear phase-coupled quadratic interactions present within the boundary layer. Multiple interactions are identified and revealed to be associated with the growth of disturbances at higher harmonic frequencies.

A combined criteria-based method for hypersonic three-dimensional boundary layer transition prediction

In this paper, a new method for predicting hypersonic three-dimensional (3D) boundary layer transition is developed. It is based on the Reθ/Me criterion for streamwise instability and on the Recf,new criterion for crossflow instability. An intermittency function is also formulated and applied to combine laminar and turbulent flows. Additionally, a computational grid pretreating method, compatible with modern computational fluid dynamics (CFD) techniques based on parallel execution is adopted in order to obtain the boundary layer parameters. Four criteria are compared to define the boundary layer edge. A HIFiRE-5 elliptic cone at different Reynolds numbers is adopted to validate the performance of the criteria-based transition model and the effectiveness of the four criteria for boundary layer edge definition. The results show that the boundary layer edge of complex hypersonic 3D flows could be obtained properly with the computational grid pretreating method and the combination of h0/h0,∞=0.995 and (du/dH)/(du/dH)w=0.1 criteria. Moreover, the computed Reθ/Me and Recf,new distributions in the region between the leading edge and the centerline are similar to the N-factor for streamwise and crossflow instabilities from linear parabolized stability equation (PSE) methods. The shape and trend of the transition onsets predicted by the criteria-based transition model between the centerline and leading edge of HIFiRE-5 agree well with the experiment. However, as for the transition on the centerline, which is dominated by the inflection point in streamwise velocity profiles, using the criteria-based transition model related to boundary layer thickness would predict the transition onset prematurely.

Experimental study of turbulent-jet wave packets and their acoustic efficiency

This paper details the statistical and time-resolved analysis of the relationship between the near-field pressure fluctuations of unforced, subsonic free jets (0.4 ≤ M ≤ 0.6) and their far-field sound emissions. Near-field and far-field microphone measurements were taken on a conical array close to the jets and an azimuthal ring at 20∘ to the jet axis, respectively. Recent velocity and pressure measurements indicate the presence of linear wave packets in the near field by closely matching predictions from the linear homogenous parabolized stability equations, but the agreement breaks down both beyond the end of the potential core and when considering higher order statistical moments, such as the two-point coherence. Proper orthogonal decomposition (POD), interpreted in terms of inhomogeneous linear models using the resolvent framework allows us to understand these discrepancies. A new technique is developed for projecting time-domain pressure measurements onto a statistically obtained POD basis, yielding the time-resolved activity of each POD mode and its correlation with the far field. A single POD mode, interpreted as an optimal high-gain structure that arises due to turbulent forcing, captures the salient near-field–far-field correlation signature; further, the signatures of the next two modes, understood as suboptimally forced structures, suggest that these POD modes represent higher order, acoustically important near-field behavior. An existing Green's-function-based technique is used to make far-field predictions, and results are interpreted in terms of POD/resolvent modes, indicating the acoustic importance of this higher order behavior. The technique is extended to provide time-domain far-field predictions.

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.

Toward robust prediction of crossflow-wave instability in hypersonic boundary layers

Abstract Traveling and stationary crossflow-wave instabilities in a laminar Mach 6 boundary layer are investigated on a 38.1% scale model of the Fifth Hypersonic International Flight Research Experiment (HIFiRE-5) elliptic cone at zero angle of attack and yaw. The Langley Stability and Transition Analysis Code (LASTRAC) was used to analyze the crossflow dominated boundary layer in the mid-span region near the downstream end of the model. Disturbance growth rates, wave angles, and phase speeds are computed with LASTRAC using quasi-parallel Linear Stability Theory (LST), Linear Parabolized StabilityEquations (LPSE), and two-plane or surface marching LPSE (2pLPSE). The predicted wave angles and phase speeds are validated using experimental data, and are found to be in better agreement than previous computations. Further numerical analysis is conducted using the Spatial BiGlobal technique (SBG), which simultaneously accounts for wall-normal and spanwise gradients in the mean boundary layer at a particular axial station. For the first time in the literature, a comparison is made between crossflow wave growth rates computed using LST, LPSE, and 2pLPSE and those computed using SBG, accounting for curvature and geometric divergence of the elliptic cone. The agreement between LST, LPSE, and SBG is fair at best, but excellent agreement is realized between 2pLPSE and SBG. This result constitutes a co-verification of the LASTRAC and SBG stability codes, and provides evidence that 2pLPSE accurately models the physics of the traveling-crossflow instability.

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.

Parabolized stability analysis of jets from serrated nozzles

We study the viscous spatial linear stability characteristics of the time-averaged flow in turbulent subsonic jets issuing from serrated (chevroned) nozzles, and compare them to analogous round jet results. Linear parabolized stability equations (PSE) are used in the calculations to account for the non-parallel base flow. By exploiting the symmetries of the mean flow due to the regular arrangement of serrations, we obtain a series of coupled two-dimensional PSE problems from the original three-dimensional problem. This reduces the solution cost and manifests the symmetries of the stability modes. In the parallel-flow linear stability theory (LST) calculations that are performed near the nozzle to initiate the PSE, we find that the serrated nozzle reduces the growth rates of the most unstable eigenmodes of the jet, but their phase speeds are approximately similar. We obtain encouraging validation of our linear PSE instability wave results vis-à-vis near-field hydrodynamic pressure data acquired on a phased microphone array in experiments, after filtering the latter with proper orthogonal decomposition (POD) to extract the energetically dominant coherent part. Additionally, a large-eddy simulation database of the same serrated jet is investigated, and its POD-filtered pressure field is found to compare favourably with the corresponding PSE solution within the jet plume. We conclude that the coherent hydrodynamic pressure fluctuations of jets from both round and serrated nozzles are reasonably consistent with the linear instability modes of the turbulent mean flow.

亚声速热射流声源模型的应用

The noise generation mechanisms associated with instability waves in the heated subsonic transitional jet are studied, which are compared with its cold counterpart. The spatial evolution of instability waves is obtained by solving linear parabolized stability equations (LPSE) based on the time-averaged flow field of the large eddy simulation (LES). Then, the linear and nonlinear models for jet noise are built based on the LPSE solutions, coupled with the acoustic analogy. The LPSE results show that heating increases the spatial-growth rate and leads to earlier saturation. For high-frequency components, the sound pressure levels (SPL) are raised by heating as shown in the linear model. In general, compared with that for the cold jet, the gap of SPL between the linear model and the LES is reduced for the heated jet, which indicates that the linear mechanism plays a more important role in the hot jet. For a cold subsonic jet, previous studies have shown that the nonlinear model is able to raise acoustic efficiency. Here, it is found that the gap of SPL between the nonlinear model and the LES could be further decreased in the hot jet, and the thermodynamic sound source terms play a bigger role.