Nonlinear Parabolized Stability Equations Research Articles

Reused LU factorization as a preconditioner for efficient solution of the parabolized stability equations

The parabolized stability equations (PSE) are a widely used, efficient method to calculate the evolution of streamwise traveling instabilities in spatially developing, weakly nonparallel flows. Although the PSE are very economical, computational time can be significant due to the repeated solution of linear systems of equations in each marching step. The linear systems of equations are typically solved by LU factorization because of the moderate size and the sparsity of the matrices. In this paper, instead of repeated calculation of the LU factorization, a single LU factorization is calculated in each marching step and used as a preconditioner for an iterative solver. Numerical experiments are conducted on the plane-marching (3D) incompressible PSE, nonlinear PSE (NLPSE) with explicit discretization of the nonlinear terms, and NLPSE with implicit treatment of the nonlinearities. It is shown that the time spent on the solution of the linear systems of equations was reduced by a factor of 2.5, 4.5–7.5, and 20–60 for the three cases, respectively. The only requirement of the proposed numerical techniques is that the solution varies slowly in the marching direction; therefore it is expected to be applicable to similar boundary layer stability problems or slowly varying parabolic partial differential equations. This is supported by the fact that the numerical method was applied to the Boundary Region Equations (BRE) and the line-marching (2D) PSE, the latter being a much smaller problem than its plane-marching counterpart. In these cases, a factor of 50% and 15–30% decrease in the net time of the linear equation solution is reported, respectively.

Interactions between oblique second mode and oblique waves in a high-speed boundary layer

Interactions between oblique second mode and oblique waves at a high-speed boundary at Mach 4.5 are studied using linear stability theory, nonlinear parabolized stability equations (NPSE) and direct numerical simulation (DNS). Parametric analysis based on the NPSE suggests that the oblique second mode can amplify both the oblique first and second modes, with the former experiencing a higher amplification. Our analysis reveals that the mean-flow distortion and difference mode contribute to this enhancement, with the latter exerting a key influence through the parametric resonance process. Kinetic energy transfer analysis demonstrates that the oblique waves gain energy from the mean flow, rather than from the oblique second mode. Furthermore, we find that the mechanism underlying the interaction between a pair of second oblique waves and a single oblique wave is similar to that between an oblique second mode and an oblique wave, as the steady modes generated by the pair of oblique second modes have a limited impact on the oblique wave. Finally, DNS confirms the validity of two transition paths proposed in this study based on the NPSE results. The first path suggests that a pair of low-amplitude second oblique waves alone are insufficient to cause oblique breakdown, but the introduction of a pair of low-amplitude damping first oblique modes could lead to boundary layer breakdown. The second path involves the formation of a domino-like effect through the combination of different types of oblique waves with the appropriate parameters. These two nonlinear paths can lead to a fully developed turbulent boundary layer.

Multi-fidelity approach for transitional boundary layer

A multi-fidelity computational approach is proposed for transitional boundary layer flow in order to obtain both high fidelity and high efficiency in flow simulation. Because we can categorize the transitional flow into three regions, i.e., laminar region with instabilities, transition region, and turbulent region, three models are judiciously selected. A boundary layer stability method, here nonlinear parabolized stability equations (NPSE), is chosen for the laminar region incorporating nonlinear interactions between instabilities. Large-eddy simulation (LES) is applied for the transition region including the beginning of the turbulent flow. Reynolds-averaged Navier–Stokes (RANS) simulation is used for the downstream turbulent flow. Since the NPSE-coupled LES method has been validated for transitional boundary layers including both incompressible (Kim et al., 2018, 2019, 2020, 2021) and compressible (Lim et al., 2021) flows, appropriate treatments for the interface between LES and RANS are mainly investigated in this study. The current multi-fidelity approach is tested for a transitional boundary layer on a flat plate. Computational fidelity and cost are compared with previous high-fidelity computations. We successfully demonstrate that the proposed multi-fidelity approach provides both high fidelity and high efficiency in transitional flow computations.

Effect of porous coatings on the nonlinear evolution of Mack modes in hypersonic boundary layers

The influence of a porous wall on the nonlinear evolution of Mack modes in a hypersonic boundary layer is studied by solving the nonlinear parabolized stability equations. The fundamental resonance of the second mode is particularly considered. It is found that the porous effect leads to (1) a much stronger mean-flow distortion in an indirect way, (2) a greater suppression of the saturated fundamental mode, and (3) slower amplification rates of the secondary instability modes, which eventually delays the transition onset. Detailed explanations of the three mechanisms are provided.

Revisit of the oblique-breakdown regime in supersonic boundary layers and mechanism of the extra amplification of streak modes

In the low-Mach-number supersonic boundary layers, the oblique-breakdown regime may be the most efficient route to trigger the laminar–turbulent transition, because the most unstable Mack first mode always appears as an oblique wave. In this paper, we revisit this issue by use of the nonlinear parabolized stability equation (NPSE) approach and particularly focus on the extra amplification of the streak mode generated by the direct interaction of the introduced oblique modes. This mechanism is then well explained based on the weakly nonlinear analysis, and its predictions on the evolution of the streak mode and the two-dimensional traveling-wave mode are quantitatively confirmed by the NPSE calculations. Additionally, the important role of the streak mode, leading to the secondary instability, on triggering the transition onset is identified.

Cross-flow vortices and their secondary instabilities in hypersonic and high-enthalpy boundary layers

Compared to the streamwise instability, the cross-flow instability in high-enthalpy flows has received relatively less attention, but the latter is of vital importance in the flow transition for practical configurations. This work aims to investigate the cross-flow primary and secondary instabilities in hypersonic and high-enthalpy boundary layers, considering thermochemical non-equilibrium (TCNE) effects. The numerical tools adopted include a high-order shock-fitting solver, nonlinear parabolized stability equations and secondary instability theory (SIT). The flow over a swept parabola is calculated at a free-stream Mach number of 16. It is found that TCNE has a destabilizing effect on the cross-flow mode with a non-catalytic wall. Two important non-dimensional parameters are summarized to explain this effect. One is the ratio between the wall and boundary-layer edge temperatures, and the other is the cross-flow Mach number. Due to nonlinear effects, the stationary cross-flow vortices evolve and exhibit the classic rollover structures as in lower-speed flows. Two different disturbance energy norms are used in the energy budget analysis to classify the secondary cross-flow instability modes. The results from SIT highlight the importance of type-IV modes in TCNE flows at the downwash region of the vortex. The type-IV modes arise with the combined contribution from the wall-normal (on top and trough of the vortex) and spanwise (in the downwash region) production terms. The type-I mode is dominant in the calorically perfect gas case with an adiabatic wall, whereas the type-IV mode has the largest growth rate in the TCNE cases irrespective of wall temperature variation.

On Energy Redistribution for the Nonlinear Parabolized Stability Equations Method

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

Control of first-mode oblique breakdown in a supersonic boundary layer using micro-groove coating strips

This paper describes a method for controlling the first-mode oblique breakdown in a Mach 2.0 supersonic boundary layer using micro-groove coating strips. The analysis is performed using the nonlinear parabolized stability equations. Micro-groove coating strips with various combinations of the groove half-width, porosity coefficient, and strip width are placed at different streamwise locations to control the transition. The results show that the micro-groove coating strip significantly suppresses the first-mode oblique breakdown. This effect is enhanced as the groove half-width, porosity coefficient, and strip width increase. When the micro-groove coating strip is located in the linear development region of the oblique breakdown, the suppression effect comes from the stabilization of the fundamental oblique mode by the micro-groove coating. When the strip is employed in the early nonlinear or late nonlinear development regions, a second stabilization mechanism, in addition to the stabilization effect of the micro-groove coating strip, is produced by the high-spanwise-wavenumber steady modes. Compared with the stabilization effect of the micro-groove coating strip, the second stabilization effect is more effective. The combination of these two stabilization mechanisms pushes the breakdown farther downstream.

Krypton: Nonlinear Parabolized Stability Equation Solver for Transonic Flow in Curvilinear Coordinates

Krypton: Nonlinear Parabolized Stability Equation Solver for Transonic Flow in Curvilinear Coordinates

Oblique-mode breakdown in hypersonic and high-enthalpy boundary layers over a blunt cone

The nonlinear analyses of the hypersonic and high-enthalpy boundary-layer transition had received little attention compared with the widely-studied linear instabilities. In this work, the oblique-mode breakdown, as one of the most available transition mechanisms, is studied using the nonlinear parabolized stability equations (NPSE) with consideration of the thermal-chemical non-equilibrium effects. The flow over a blunt cone is computed at a free-stream Mach-number of 15. The rope-like structures and the spontaneous radiation of sound waves are observed in the schlieren-like picture. It is also illustrated that the disturbances of the species mass and vibrational temperature near the wall are mainly generated by the product term of the wall-normal velocity disturbance and the mean-flow gradient. In comparison to the CPG flow, the TCNE effects destabilize the second mode and push upstream the N factor envelope. The higher growth rate of the oblique wave leads to stronger growth of the streamwise vortices and harmonic waves.

Investigation of Görtler vortices in high-speed boundary layers via an efficient numerical solution to the non-linear boundary region equations

Streamwise vortices and the associated streaks evolve in boundary layers over flat or concave surfaces due to disturbances initiated upstream or triggered by the wall surface. Following the transient growth phase, the fully-developed vortex structures become susceptible to inviscid secondary instabilities resulting in early transition to turbulence via `bursting' processes. In high-speed boundary layers, more complications arise due to compressibility and thermal effects, which become more significant for higher Mach numbers. In this paper, we study G\"{o}rtler vortices developing in high-speed boundary layers using the boundary region equations (BRE) formalism, which we solve using an efficient numerical algorithm. Streaks are excited using a small transpiration velocity at the wall. Our BRE-based algorithm is found to be superior to direct numerical simulation (DNS) and ad-hoc nonlinear parabolized stability equation (PSE) models. BRE solutions are less computationally costly than a full DNS and have a more rigorous theoretical foundation than PSE-based models. For example, the full development of a G\"{o}rtler vortex system in high-speed boundary layers can be predicted in a matter of minutes using a single processor via the BRE approach. This substantial reduction in calculation time is one of the major achievements of this work. We show, among other things, that it allows investigation into feedback control in reasonable total computational times. We investigate the development of the G\"{o}rtler vortex system via the BRE solution with feedback control parametrically at various freestream Mach numbers $M_\infty$ and spanwise separations $\lambda$ of the inflow disturbances.

Evolution of instability modes in Mach 10 frozen and chemically reacting boundary layers by means of non-linear parabolized stability equations

The physical mechanisms and laminar-to-turbulent transition scenarios characterizing hypersonic flows are not yet completely identified and well described as in low-supersonic and subsonic regimes. Particularly, there is a lack of knowledge about the role chemistry plays and the effects that high temperature has on the transition non-linear stages. In this study, transition in a Mach 10 adiabatic flat-plate boundary layer is investigated by means of non-linear parabolized stability equations, considering both frozen and chemical non-equilibrium (CNE) flow assumptions. A tendency to switch from a H- to a K-type transition scenario at the increment of the primary-mode initial amplitude is found, similar to what occurs in subsonic and low-supersonic regimes. Secondary-waves' initial amplitudes do not affect directly the occurring breakdown type, but they significantly influence the evolution of the higher harmonics, in particular the generation of streamwise steady vortices. Chemical reactions play an indirect role in transition through the determination of the laminar base flow and the linear stability characteristics of the primary instability. Excitation of secondary modes is weaker in CNE than in frozen conditions, but qualitatively similar. Contrary to previous studies, assuming frozen perturbations in a chemically reacting base flow is found to have a significant effect on the higher-harmonics amplitudes, in particular on the relative evolution of the secondary modes.

Nonlinear interaction between two second-mode disturbances with the same frequency in a hypersonic boundary layer over a swept blunt plate

Zero- or low-frequency crossflow instabilities play a dominant role in the hypersonic boundary layer over a swept blunt plate. However, high-frequency second-mode disturbances dominate and lead to transition due to the wall cooling effect. The nonlinear interaction of high-frequency disturbances generates zero- or low-frequency disturbances. This paper investigates the relationship between the generated disturbances and the crossflow instability disturbances through nonlinear parabolized stability equations (NPSE) and direct numerical simulations. It is found that a special zero-frequency disturbance is generated by two second-mode disturbances with the same frequency but different spanwise wavenumbers, defined as a forced solution in this paper. This differs from the so-called eigenvalue solution obtained by linear stability theory, with the growth rate of the former being much larger than that of the latter. Additionally, the profile of the forced solution is similar to that of the second-mode disturbances, but not the crossflow-mode disturbances. The evolution of the forced solution is determined by the initial amplitude, spanwise wavenumber, and frequency of the second-mode disturbances, and is independent of the initial amplitude of crossflow-mode disturbances. If the second-mode disturbances begin to attenuate, the forced solution cannot maintain itself and transitions to the eigenvalue solution. The amplitude of the forced solution can be predicted using magnitude analysis through the amplitude of the second-mode disturbances, and the theoretical predictions are consistent with those of the NPSE when the second-mode disturbances are unstable. The results demonstrate that the forced solution is a mechanism for the second-mode disturbances to promote transition.

Secondary instability of the hypersonic high-enthalpy boundary layers with thermal–chemical nonequilibrium effects

Compared to the widely studied linear instabilities, the nonlinear analyses of the high-enthalpy and hypersonic boundary-layer transition had received much less attention. In this work, secondary instabilities of high-enthalpy boundary layers are studied to highlight the impacts of thermal–chemical nonequilibrium (TCNE). The numerical tools adopted here are the nonlinear parabolized stability equation (NPSE) and the Floquet analysis (or the secondary instability theory, SIT). The flow over a blunt cone is computed at a free-stream Mach number of 20. The fundamental resonance is found to dominate over the subharmonic resonance. The development of secondary waves from SIT agrees well with that from NPSE, suggesting satisfactory cross-validation of both solvers. It is also illustrated that the disturbances of the species mass and vibrational temperature are mainly generated by the product term of the wall-normal velocity disturbance and the mean-flow gradient. The typical rope-like structures are observed near the boundary-layer edge. The effects of TCNE on the secondary instability are to cause larger maximum secondary growth rate and the corresponding azimuthal wavenumber.

Pressure gradient effects on the secondary instability of Mack mode disturbances in hypersonic boundary layers

In hypersonic boundary layers, Mack mode disturbances play an important role in laminar–turbulence transition. Understanding the secondary instability mechanism of Mack mode disturbances will provide physical insight into turbulence generation for the researchers, which is also meaningful for transition control. According to the previous studies over straight cones and flared cones, it seems that a pressure gradient may affect the primary instability and secondary instability of Mack mode disturbances obviously. In this paper, we are trying to make it clear that what the pressure gradient effect on the secondary instability of Mack modes is and what the influence rule is. Four hypersonic flat plate cases with various pressure gradients at Mach 6 are analyzed through linear stability theory, non-linear parabolized stability equations, and spatial secondary instability theory methods. We found that the essence of the pressure gradient influence on the secondary instability mechanism is by affecting the primary amplitude of Mack modes, rather than other routes or factors. An adverse pressure gradient can enlarge both the primary instability and secondary instability growth rates and advance the transition. Moreover, an adverse pressure gradient will form a larger primary amplitude of the Mack mode, leading to a fundamental resonance dominated secondary instability. In contrast, the favorable pressure gradient will suppress the primary amplitude so that the subharmonic resonance may dominate the secondary instability. Therefore, it is very meaningful and valuable for transition prediction and turbulence generation to conduct the present study of pressure gradient effects on the secondary instability of Mack mode disturbances.

Improved nonlinear parabolized stability equations approach for hypersonic boundary layers* *Project supported by the National Natural Science Foundation of China (Grant Nos. 12072232 and 11672351) and the National Key Project of China (Grant No. GJXM92579).

The nonlinear parabolized stability equations (NPSEs) approach is widely used to study the evolution of disturbances in hypersonic boundary layers owing to its high computational efficiency. However, divergence of the NPSEs will occur when disturbances imposed at the inlet no longer play a leading role or when the nonlinear effect becomes very strong. Two major improvements are proposed here to deal with the divergence of the NPSEs. First, all disturbances are divided into two types: dominant waves and non-dominant waves. Disturbances imposed at the inlet or playing a leading role are defined as dominant waves, with all others being defined as non-dominant waves. Second, the streamwise wavenumbers of the non-dominant waves are obtained using the phase-locked method, while those of the dominant waves are obtained using an iterative method. Two reference wavenumbers are introduced in the phase-locked method, and methods for calculating them for different numbers of dominant waves are discussed. Direct numerical simulation (DNS) is performed to verify and validate the predictions of the improved NPSEs in a hypersonic boundary layer on an isothermal swept blunt plate. The results from the improved NPSEs approach are in good agreement with those of DNS, whereas the traditional NPSEs approach is subject to divergence, indicating that the improved NPSEs approach exhibits greater robustness.

Preflight Boundary-Layer Stability Analysis of BOLT Geometry

Boundary-layer stability and laminar-to-turbulent transition are examined on the Boundary Layer Transition flight-test vehicle using predicted flow conditions at ascent and descent flight times. The parabolized stability equations technique is used predominantly for this analysis because it has proven to be efficient and accurate in modeling the transition process on other three-dimensional geometries. The various potential transition mechanisms are identified in the region outboard of the centerline to the leading edge, and their linear amplification is discussed. These mechanisms include second-mode and crossflow instabilities. Stationary crossflow is predicted to be the dominant mechanism for breakdown away from the centerline and leading edge, and nonlinear analysis is performed. The nonlinear parabolized stability equations are used for this analysis, and comparisons with a direct numerical simulation are performed. Partially through this verification, a physics-based technique to model the heating rates of nonlinearly developing stationary crossflow is assessed.

Mechanism for frustum transition over blunt cones at hypersonic speeds

Numerical and experimental studies have demonstrated laminar–turbulent transition in hypersonic boundary layers over sharp cones via the modal growth of planar Mack-mode instabilities. However, due to the strong reduction in Mack-mode growth at higher nose bluntness values, the mechanisms underlying the observed onset of transition over the cone frustum are currently unknown. Linear non-modal growth analysis has shown that both planar and oblique travelling disturbances that peak within the entropy layer experience appreciable energy amplification for moderate to large nose bluntness. However, due to their weak signature within the boundary-layer region, the route to transition onset via non-modal growth of travelling disturbances remains unclear. Nonlinear parabolized stability equations (NPSE) and direct numerical simulations (DNS) are used to identify a potential mechanism for transition over a 7-degree blunt cone that was tested in the AFRL Mach-6 high-Reynolds-number facility. Specifically, computations are conducted to study the nonlinear development of a pair of oblique, unsteady non-modal disturbances in the regime of moderately blunt nose tips. Excellent agreement was demonstrated between the NPSE and DNS predictions. Results reveal that, even though the linear non-modal disturbances are primarily concentrated outside the boundary layer, their nonlinear interaction can generate stationary streaks that penetrate and amplify within the boundary layer, eventually inducing the onset of transition via the breakdown of these streaks. The results indicate that a pair of oblique, controlled non-modal disturbances can produce transition at the location measured in the experiment when their initial amplitude is chosen to be approximately 0.15 % of the free-stream velocity.

Structure evolution at early stage of boundary-layer transition: simulation and experiment

The beginning of laminar–turbulent transition is usually associated with a wave-like disturbance, but its evolution and role in precipitating the development of other flow structures are not well understood from a structure-based view. Nonlinear parabolized stability equations (NPSE) were solved numerically to simulate the transition of K-regime, N-regime and O-regime. However, only the K-regime transition was examined experimentally using both hydrogen bubble visualization and time-resolved tomographic particle image velocimetry (tomo-PIV). Based on the ‘NPSE visualization’ and ‘tomographic visualization’, at least four common characteristics of the generic transition process were identified: (i) inflectional regions representing high-shear layers (HSL) that develop in vertical velocity profiles, accompanied by ejection–sweep behaviours; (ii) low-speed streak (LSS) patterns, manifested in horizontal timelines, that seem to consist of several three-dimensional (3-D) waves; (iii) a warped wave front (WWF) pattern, displaying multiple folding processes, which develops adjacent to the LSS in the near-wall region, prior to the appearance of 𝛬-vortices; (iv) a coherent 3-D wave front, similar to a soliton, in the upper boundary layer, accompanied by regions of depression along the flanks of the wave. It was determined that the amplification and lift-up of a 3-D wave causes the development of the HSL, WWF and multiple folding behaviour of material surfaces, that all contribute to the development of a 𝛬-vortex. The amplified 3-D wave is hypothesized as a soliton-like coherent structure. Based on our results, a path to transition is proposed, which hypothesizes the function of the WWF in boundary-layer transition.

Numerical simulation of K-, N- and O-regime boundary-layer transition at early nonlinear stage

Boundary layer transition is usually associated with the amplification and breakdown of three-dimensional waves (3D waves). However, the details of the development of 3D waves in different transition regimes are not well resolved. The present study attempts to examine the transition process at early nonlinear stages of K-, N-, and O-regimes by the solution of nonlinear parabolized stability equations (NPSE). The spatial and temporal variations of streamwise velocity are compared at the one-spike stage. Timelines and material surfaces are presented using Lagrangian tracking method. The development of 3D waves are observed to contribute to the upward lift and downward sweep of flow behaviors, causing inflectional regions in the boundary layer. The evolution of material surfaces reveals that the nonlinear 3D wave manifests as a warped wave front in the near-wall region and as a soliton-like structure at the upper region of the boundary layer. The 3D wave structure is hypothesized as a soliton-like coherent structure. Markedly similar flow behaviors are observed between the regimes, which indicates a similar underlying physical mechanism.