Parabolized Stability Equations Research Articles

DeHNSSo: The delft harmonic Navier-Stokes solver for nonlinear stability problems with complex geometric features

A nonlinear Harmonic Navier-Stokes (HNS) framework is introduced for simulating instabilities in laminar spanwise-invariant shear layers, featuring sharp and smooth wall surface protuberances. While such cases play a critical role in the process of laminar-to-turbulent transition, classical stability theory analyses such as parabolized or local stability methods fail to provide (accurate) results, due to their underlying assumptions. The generalized incompressible Navier-Stokes (NS) equations are expanded in perturbed form, using a spanwise and temporal Fourier ansatz for flow perturbations. The resulting equations are discretized using spectral collocation in the wall-normal direction and finite-difference methods in the streamwise direction. The equations are then solved using a direct sparse-matrix solver. The nonlinear mode interaction terms are converged iteratively. The solution implementation makes use of a generalized domain transformation to account for geometrical smooth surface features, such as humps. No-slip conditions can be embedded in the interior domain to account for the presence of sharp surface features such as forward- or backward-facing steps. Common difficulties with Navier-Stokes solvers, such as the treatment of the outflow boundary and convergence of nonlinear terms, are considered in detail. The performance of the developed solver is evaluated against several cases of representative boundary layer instability growth, including linear and nonlinear growth of Tollmien-Schlichting waves in a Blasius boundary layer and stationary crossflow instabilities in a swept flat-plate boundary layer. The latter problem is also treated in the presence of a geometrical smooth hump and a sharp forward-facing step at the wall. HNS simulation results, such as perturbation amplitudes, growth rates, and shape functions, are compared to benchmark flow stability analysis methods such as Parabolized Stability Equations (PSE), Adaptive Harmonic Linearized Navier-Stokes (AHLNS), or Direct Numerical Simulations (DNS). Good agreement is observed in all cases. The HNS solver is subjected to a grid convergence study and a simple performance benchmark, namely memory usage and computational cost. The computational cost is found to be considerably lower than high-fidelity DNS at comparable grid resolutions. Program summaryProgram Title: DeHNSSoCPC Library link to program files:https://doi.org/10.17632/9bnms99kk2.1Developer's repository link:https://github.com/SvenWesterbeek/DeHNSSoLicensing provisions: GPLv3Programming language: MatlabSupplementary material: The supplementary material contains the code as well as a user manual.Nature of problem: Fluid flows are subject to laminar-to-turbulent transition following the growth of instabilities. To avoid computationally demanding Direct Numerical Simulations (DNS), perturbation theory is often applied to their analysis. However, classical stability methods based on the Orr-Sommerfeld equation or the Parabolized Stability Equations neglect the influence of streamwise gradients in varying degrees. The validity of these assumptions is difficult to estimate a priori.Solution method: The Delft Harmonic Navier-Stokes Solver (DeHNSSo) solves the harmonic Navier-Stokes equations nonlinearly on domains featuring sharp and smooth spanwise invariant surface features using a generalized grid approach in combination with an embedded boundary method. This allows the user to include the effects of streamwise gradients on flow at a fraction of the cost of DNS.Additional comments including restrictions and unusual features: In DeHNSSo, the equations are solved using direct matrix solvers. As such, memory is treated as a scarce resource. The problem is formulated in a mode-independent manner such that left-hand side matrices need only be computed and stored once despite incorporating nonlinear terms. Additionally, DeHNSSo offers the user the possibility to prescribe inhomogeneous boundary conditions to introduce and solve the receptivity problem or manipulate instabilities. Due to the double Fourier expansion, the solver is restricted to spanwise and temporally periodic problems.

Roughness-Induced Transition in Supersonic Boundary Layers with Wall Heating and Cooling Effects

The laminar–turbulent transition of a supersonic boundary layer induced by an isolated roughness element is investigated using direct numerical simulation, BiGlobal linear stability theory, and the three-dimensional parabolized stability equation. Cylindrical and diamond-shaped roughness elements are investigated in combination with different wall temperatures. Direct numerical simulations show that the cylinder configuration induces an earlier transition than the diamond configuration, with the interaction between the separated shear layer and the counter-rotating vortices causing the transition. BiGlobal analysis and the parabolized stability equation confirm the existence of two unstable instability modes in the wake region, namely, a symmetric mode and an antisymmetric mode, with the former being strongly dominant. The wall cooling and heating effects are studied by changing the wall temperature. Wall heating lifts the inlet boundary layer and weakens the separated shear layer. This, in turn, weakens the wake instability and delays the transition. The antisymmetric model disappears as the wall heating increases. Wall cooling accelerates the transition by enhancing the distortion of the roughness wake, resulting in stronger instabilities. Finally, the Reynolds number based on the momentum defect is used to define the transition criterion, and this is found to be in good agreement with the present simulation results.

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.

Principle of fundamental resonance in hypersonic boundary layers: an asymptotic viewpoint

The fundamental resonance (FR) in the nonlinear phase of the boundary-layer transition to turbulence appears when a dominant planar instability mode reaches a finite amplitude and the low-amplitude oblique travelling modes with the same frequency as the dominant mode, together with the stationary streak modes, undergo the strongest amplification among all the Fourier components. This regime may be the most efficient means to trigger the natural transition in hypersonic boundary layers. In this paper, we aim to reveal the intrinsic mechanism of the FR in the weakly nonlinear framework based on the large-Reynolds-number asymptotic technique. It is found that the FR is, in principle, a triad resonance among a dominant planar fundamental mode, a streak mode and an oblique mode. In the major part of the boundary layer, the nonlinear interaction of the fundamental mode and the streak mode seeds the growth of the oblique mode, whereas the interaction of the oblique mode and the fundamental mode drives the roll components (transverse and lateral velocity) of the streak mode, which leads to a stronger amplification of the streamwise component of the streak mode due to the lift-up mechanism. This asymptotic analysis clearly shows that the dimensionless growth rates of the streak and oblique modes are the same order of magnitude as the dimensionless amplitude of the fundamental mode $(\bar {\epsilon }_{10})$ , and the amplitude of the streak mode is $O(\bar {\epsilon }_{10}^{-1})$ greater than that of the oblique mode. The main-layer solution of the streamwise velocity, spanwise velocity and temperature of both the streak and the oblique modes become singular as the wall is approached, and so a viscous wall layer appears underneath. The wall layer produces an outflux velocity to the main-layer solution, inclusion of which leads to an improved asymptotic theory whose accuracy is confirmed by comparing with the calculations of the nonlinear parabolised stability equations (NPSEs) at moderate Reynolds numbers and the secondary instability analysis (SIA) at sufficiently high Reynolds numbers.

An adjoint-based methodology for calculating manufacturing tolerances for natural laminar flow airfoils susceptible to smooth surface waviness

An adjoint-based method is presented for determining manufacturing tolerances for aerodynamic surfaces with natural laminar flow subjected to wavy excrescences. The growth of convective unstable disturbances is computed by solving Euler, boundary layer, and parabolized stability equations. The gradient of the kinetic energy of disturbances in the boundary layer (E) with respect to surface grid points is calculated by solving adjoints of the governing equations. The accuracy of approximations of ΔE\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Delta E$$\\end{document}, using gradients obtained from adjoint, is investigated for several waviness heights. It is also shown how second-order derivatives increase the accuracy of approximations of ΔE\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Delta E$$\\end{document} when surface deformations are large. Then, for specific flight conditions, using the steepest ascent and the sequential least squares programming methodologies, the waviness profile with minimum L2-\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L2-$$\\end{document}norm that causes a specific increase in the maximum value of N- factor, ΔN\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Delta N$$\\end{document}, is found. Finally, numerical tests are performed using the NLF(2)-0415 airfoil to specify tolerance levels for ΔN\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Delta {N}$$\\end{document} up to 2.0 for different flight conditions. Most simulations are carried out for a Mach number and angle of attack equal to 0.5 and 1.25∘\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$1.25^{\\circ }$$\\end{document}, respectively, and with Reynolds numbers between 9×106\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$9\ imes 10^6$$\\end{document} and 15×106\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$15\ imes 10^6$$\\end{document} and for waviness profiles with different ranges of wavelengths. Finally, some additional studies are presented for different angles of attack and Mach numbers to show their effects on the computed tolerances.Graphic abstract

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.

Influence of free-stream turbulence on the boundary layer stability of a wind turbine airfoil and near wake

Free-stream turbulence (FST) alters the boundary layer of wind turbine blades, changing the hydrodynamic stability and near wake. Large-eddy simulations (LES) of a blade section with a laminar separation bubble for several turbulence intensities (TI) and a Reynolds number of 105 are performed. The effects of boundary-layer streaks generated by FST on Tollmien-Schlichting (TS) and Kelvin-Helmholtz (KH) instabilities are analyzed with a model based on the parabolized stability equations (PSE). Two competing effects on flow stability are identified. The spanwise-averaged mean-flow distortion stabilizes primary TS/KH modes for increasing TI. However, this contribution seems dominant only for TI ≥ 8.6%. For lower TI, the spanwise-oscillating distortion caused by streaks destabilizes the flow, and the growth rates of secondary modal instabilities increase with the streak amplitude. The destabilization occurs mainly at spanwise locations with negative streaks since the inflection point shifts away from the wall, enhancing inviscid instabilities. Inflection points in the spanwise direction formed by the streaks also contribute to the destabilization. The modal structures from PSE and LES agree. Finally, the trailing-edge near-wake coherent structures are more energetic for TI ≥ 8.6% due to the partial stabilization of modal instabilities, delaying the turbulent breakdown.

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.

Hydrodynamic sources of the vortex sound in a two-dimensional shear layer

Development of advanced noise reduction devices requires an in-depth understanding of two fundamental questions: what are the true noise sources and how are the acoustic radiations generated. An accurate separation of the hydrodynamic and acoustic fluctuations helps to reveal the answers, but no consensus exists on its feasibility in the near-field source region of compressible flows. This study proposes a methodology to examine the dynamics of vortex sound generation in a two-dimensional artificially excited subsonic mixing layer. The parabolized stability equation (PSE) is applied to resolve the hydrodynamic fluctuations and the vortex sound theory is used to predict the acoustic pressures. Numerical simulations show that the PSE solutions capture the vortex pairing reasonably accurately and damp the acoustic modes to a negligible level, and that the vortex sound theory recovers the acoustic pressures. Good agreement of both solutions with the direct simulations indicates that a physically reasonable separation of hydrodynamic sources is achieved and can be used to further examine the vortex dynamics and noise source mechanisms.

Neural prediction model for transition onset of a boundary layer in presence of two-dimensional surface defects

Predicting the laminar to turbulent transition is an important aspect of computational fluid dynamics because of its impact on skin friction. Traditional transition prediction methods such as local stability theory or the parabolized stability equation method do not allow for the consideration of strongly non-parallel boundary layer flows, as in the presence of surface defects (bumps, steps, gaps, etc.). A neural network approach, based on an extensive database of two-dimensional incompressible boundary layer stability studies in the presence of gap-like surface defects, is used. These studies consist of linearized Navier–Stokes calculations and provide information on the effect of surface irregularity geometry and aerodynamic conditions on the transition to turbulence. The physical and geometrical parameters characterizing the defect and the flow are then provided to a neural network whose outputs inform about the effect of a given gap on the transition through the ${\rm \Delta} N$ method (where N represents the amplification of the boundary layer instabilities).

Recursive one-way Navier-Stokes equations with PSE-like cost

Spatial marching methods, in which the flow state is spatially evolved in the downstream direction, can be used to produce low-cost models of flows containing a slowly varying direction, such as mixing layers, jets, and boundary layers. The parabolized stability equations (PSE) are popular due to their extremely low cost but can only capture a single instability mode; all other modes are damped or distorted by regularization methods required to stabilize the spatial march, precluding PSE from properly capturing non-modal behavior, acoustics, and interactions between multiple instability mechanisms. The one-way Navier-Stokes (OWNS) equations properly retain all downstream-traveling modes at a cost that is a fraction of that of global methods but still one to two orders of magnitude higher than PSE. In this paper, we introduce a new variant of OWNS whose cost, both in terms of CPU time and memory requirements, approaches that of PSE while still properly capturing the contributions of all downstream-traveling modes. The method is formulated in terms of a projection operator that eliminates upstream-traveling modes. Unlike previous OWNS variants, the action of this operator on a vector can be efficiently approximated using a series of equations that can be solved recursively, i.e., successively one after the next, rather than as a coupled set. In addition to highlighting the improved cost scaling of our method, we derive explicit error expressions and elucidate the relationship with previous OWNS variants. The properties, efficiency, and accuracy of our method are demonstrated for both free-shear and wall-bounded flows.

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.

Stability analysis of roughness-disturbed boundary layer controlled by wall-blowing

Roughness-induced transition control is of considerable importance for high-speed vehicles. In this paper, the instability of a roughness-disturbed boundary layer controlled by spanwise-uniform wall-blowing is investigated through BiGlobal and three-dimensional parabolized stability equation (PSE-3D) analysis. Without wall-blowing, symmetric and antisymmetric unstable modes are observed when using BiGlobal analysis, with PSE-3D analysis suggesting that the symmetric mode is the dominant instability. Both modes are associated with the instability of the entire separated shear layer behind the roughness region rather than the components in certain directions, as both the wall-normal shear and the spanwise component resulting from the bending shear layer contribute to the growth of the disturbance. Upstream wall-blowing delays the roughness-induced transition by modifying the wake instability. The antisymmetric mode is the first to disappear as the blowing intensity increases while the symmetric instability is also suppressed. Upstream wall-blowing also reduces both the strength and bending of the shear layer by affecting the inflow boundary layer. This leads to a decrease in the wall-normal and spanwise contributions to the disturbance energy. Downstream wall-blowing achieves a control effect by decelerating the development of the dominant symmetric mode through the direct interaction between wall-blowing and the wake. Although the reduction in shear strength is not as strong as with upstream wall-blowing, downstream wall-blowing still relaxes the bending of the shear layer, which is related to the production of disturbance energy. In conclusion, two-dimensional wall-blowing can delay the roughness-induced transition by modifying the wake structure and instability.

Transition induced by an egg-crate roughness on a flat plate in supersonic flow

Hot-wire measurements in a Mach 3.5 quiet tunnel were made in the wake of a roughness patch on a flat plate. These measurements were used to determine mode shapes and frequencies of the dominant instabilities leading to boundary-layer transition. The egg-crate roughness pattern is an analytic function described by a sinusoidal equation, similar to an array of discrete elements that are positioned in a spanwise and streamwise grid, but containing both protuberances and dimples. This is an intermediate configuration towards understanding the underlying physics of a pseudorandom distributed roughness, and ultimately, the underlying physics of roughness-induced boundary-layer transition. The roughness pattern had a wavelength of 6.25 mm, with a nominal amplitude of 272 ${\rm \mu}{\rm m}$ (0.49 times the boundary layer thickness at the first row of protuberances). The roughness was positioned near the leading edge of the flat plate and contained 3.5 wavelengths in the streamwise direction and 7.5 wavelengths in the spanwise direction. The dominant instability was centred near 74 kHz at a free stream unit Reynolds number of $12.9\times 10^{6}\,{\rm m}^{-1}$ and resembled an antisymmetric mode downstream of each of the protuberances in the roughness patch. Computations using linear stability analysis based on the plane-marching parabolized stability equations (PSE) showed limited agreement with measurements when comparing the growth of the wake instability. Better agreement with the measurements was observed when considering the modification of first mode waves by the egg-crate roughness patch and the solution of the three-dimensional harmonic linearized Navier–Stokes equations was used as the in-flow to the PSE. The agreement confirms the significance of disturbance growth both upstream of and above a finite length roughness patch and the effect on the growth of instabilities in the wake.

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.

Boundary-Layer Instabilities over a Cone–Cylinder–Flare Model at Mach 6

Computations are performed to investigate the convective and global boundary-layer instabilities over a sharp cone–cylinder–flare model at zero-degree angle of attack. The model geometry and the flow conditions are selected to match the experiments conducted in the Boeing/AFOSR Mach 6 Quiet Tunnel at Purdue University. The geometry consists of a nominally sharp 5 deg half-angle cone, followed by a cylindrical segment, and then a 10 deg flare. Additionally, flare half angles equal to 8 and 12 deg and nosetip radii equal to 0.1, 1, and 5 mm are studied. An axisymmetric separation bubble is generated as a result of the laminar shock–boundary-layer interaction in the cylinder–flare region. The comparison of the laminar flow solution and the schlieren images shows a remarkable agreement between the respective locations of both the boundary-layer edge and the reattachment shock. The linear amplification of first and second Mack mode instabilities that begin to amplify in the cone region are computed with a combination of the parabolized stability equations and the harmonic linearized Navier–Stokes equations. The predicted frequency spectra of the surface pressure fluctuations associated with both planar and oblique instability waves capture the distinct lobes within the disturbance amplification spectra measured with PCB® and Kulite sensors. To our knowledge, this represents the first successful comparison between convective instability analysis and measured surface pressure fluctuations for a hypersonic configuration with a separation bubble. Finally, the global instability analysis shows that the laminar flow becomes supercritical for flare half angles larger than 8 deg.

Stability analysis of streamwise vortices over a blunt inclined cone under a hypersonic flight condition

We studied the stability of leeward streamwise vortices over a hypersonic inclined blunt cone under a flight condition by solving the two-dimensional spatial eigenvalue problem (BiGlobal) and plane-marching parabolized stability equations (PSE3D). The stability analyses were performed based on the laminar flow obtained by direct numerical simulation. Due to the azimuthal pressure gradient and large bluntness, a pair of large-scale inward vortices dominate in the vicinity of the leeward ray. Such vortical structures are different from previous studies, where outward vortices are most prominent. Two types of unstable modes are identified, namely, “inner mode” with low phase velocities and “outer mode” with high phase velocities. The inner modes are unstable in a wide frequency range compared to the outer modes, differing from previous studies where inner modes are deemed to lie in a much lower frequency range compared to the outer modes. Moreover, the inner modes, with supersonic phase velocities, are found to be able to radiate weak acoustics outside the vortices. Mack mode, belonging to the outer-mode instability, is most amplified that it is likely responsible for the breakdown of the vortices. BiGlobal and PSE3D results show good agreement regarding the downstream developments of the Mack mode and inner modes and yet exhibit remarkable discrepancies in tracing the evolution of the outer modes, highlighting the necessity of considering the non-parallel effects when studying the stability of streamwise vortices.