Parabolized Stability 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.

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.

Effect of cone rotation on the nonlinear evolution of Mack modes in supersonic boundary layers

In this paper, we present a systematic study of the nonlinear evolution of the travelling Mack modes in a Mach 3 supersonic boundary layer over a rotating cone with a $7^{\circ }$ half-apex angle using the nonlinear parabolic stability equation (NPSE). To quantify the effect of cone rotation, six cases with different rotation rates are considered, and from the same streamwise position, a pair of oblique Mack modes with the same frequency but opposite circumferential wavenumbers are introduced as the initial perturbations for NPSE calculations. As the angular rotation rate $\varOmega$ increases such that $\bar \varOmega$ (defined as the ratio of the rotation speed of the cone to the streamwise velocity at the boundary-layer edge) varies from 0 to $O(1)$ , three distinguished nonlinear regimes appear, namely the oblique-mode breakdown, the generalised fundamental resonance and the centrifugal-instability-induced transition. For each regime, the mechanisms for the amplifications of the streak mode and the harmonic travelling waves are explained in detail, and the dominant role of the streak mode in triggering the breakdown of the laminar flow is particularly highlighted. Additionally, from the linear stability theory, the dominant travelling mode undergoes the greatest amplification for a moderate $\varOmega$ , which, according to the $e^N$ transition-prediction method, indicates premature transition to turbulence. However, this is in contrast to the NPSE results, in which a delay of the transition onset is observed for a moderate $\varOmega$ . Such a disagreement is attributed to the different nonlinear regimes appearing for different rotation rates. Therefore, the traditional transition-prediction method based on the linear instability should be carefully employed if multiple nonlinear regimes may appear.

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.

Direct numerical simulation of interaction between a stationary crossflow instability and forward-facing steps

The interaction between forward-facing steps of several heights and a pre-existing critical stationary crossflow instability of a swept-wing boundary layer is analysed. Direct numerical simulations (DNS) are performed of the incompressible three-dimensional laminar base flow and the stationary distorted flow that arise from the interaction between an imposed primary stationary crossflow perturbation and the steps. These DNS are complemented with solutions of the linear and the nonlinear parabolised stability equations, used towards identifying the influence of linearity and non-parallelism near the step. A fully stationary solution of the Navier–Stokes equations is enforced numerically, in order to isolate the mechanisms pertaining to the interaction of the stationary disturbance with the step. Results provide insight into the salient modifications of the base laminar boundary layer due to the step, and the response of the incoming crossflow instability to these changes. The fundamental spanwise Fourier mode of the disturbance field gradually lifts up as it approaches the step and passes over it. The flow environment around the step is characterised by a sudden spanwise modulation of the base-flow streamlines. Additional stationary perturbation structures are induced at the step, which manifest in the form of spanwise-aligned velocity streaks near the wall. Shortly downstream of the step, the fundamental component of the crossflow perturbation maintains a rather constant amplification for the smallest steps studied. For the largest step, however, the fundamental crossflow perturbation is stabilised significantly shortly downstream of the largest step. This surprising result is ascribed to a modulation of the kinetic energy transfer between the base flow and the fundamental perturbation field, which is brought forward as a new step interaction mechanism. Possible non-modal growth effects at the step are discussed. Furthermore, the results from DNS indicate significant amplification of the high-order harmonic crossflow components downstream of the step.

Receptivity of crossflow instability to discrete roughness amplitude and location

The effect of discrete roughness elements on the development and breakdown of stationary crossflow instability on a swept wing is explored. Receptivity to various element heights and chordwise locations is explored using a combination of experimental and theoretical tools. Forcing configurations, determined based on linear stability predictions, are manufactured and applied on the wing in a low turbulence facility. Measurements are performed using infrared thermography, quantifying the transition front location, and planar particle image velocimetry, providing a reconstruction of stationary crossflow instabilities and their associated growth. Measurements are corroborated with simulations based on nonlinear parabolised stability equations. Results confirm the efficacy of discrete roughness elements in introducing and conditioning stationary crossflow instabilities. Primary instability amplitudes and resulting laminar-turbulent transition location are found to strongly depend on both roughness amplitude and chordwise location. The Reynolds number based on element height is found to satisfactorily approximate the initial forcing amplitude, revealing the importance of local velocity effects in non-zero-pressure gradient flows. Direct estimation of initial perturbation amplitudes from nonlinear simulations suggests the existence of pertinent flow mechanisms in the element vicinity, active in conditioning the onset of modal instabilities. Dedicated velocimetry planes, elucidate the development of a momentum deficit wake which rapidly decays downstream of the element followed by mild growth, representing the first experimental evidence of transient behaviour in swept wing boundary layers. The outcome of this work identifies a strong scalability of the transition dynamics to roughness amplitude and location, warranting the upscaling of roughness elements to more accessible, measurable and spatially resolved configurations in future experiments.

REPEATED KERNELS OF THE GREEN’S FUNCTION OF PARABOLIC SHILOV EQUATIONS WITH VARIABLE COEFFICIENTS AND NEGATIVE GENUS

The concept of parabolicity by Shilov generalizes the concept of parabolicity by Petrovsky of equations with partial derivatives and leads to a significant expansion of the known Petrovsky class with those parabolic equations, the order of which may not coincide with the parabolicity index. Generally speaking, such an extension deprives of the parabolic stability сoncerning the change of the coefficients of parabolic Shilov equations, which is inherent to the Petrovsky class equations. As a result, significant difficulties arise in the study of the Cauchy problem for parabolic Shilov equations with variable coefficients. In the 60s of the last century, Y.I. Zhytomyrsky defined a special class of parabolic Shilov equations, which extends the Shilov class and at the same time is parabolically resistant to changes in the junior coefficients. For this class, by the method of successive approximations, he established the correct solvability of the Cauchy problem in the class of bounded initial functions of finite smoothness. However, to obtain more general results, it is important to know the Green’s function of the Cauchy problem. In this publication, for parabolic Shilov equations with bounded smooth variable coefficients and negative genus, estimates of repeated kernels of the Green’s function of the Cauchy problem are established, which allow us to investigate the properties of the density of volume potential of this function. These results are important for the development of the Cauchy problem theory for parabolic Shilov equations by classical means of the Green’s function.

Effects of phase difference between instability modes on boundary-layer transition

Phase effect on the modal interaction of flow instabilities is investigated for laminar-to-turbulent transition in a flat-plate boundary-layer flow. Primary and secondary three-dimensional (3-D) oblique waves at various initial phase differences between these two instability modes. Three numerical methods are used for a systematic approach for the entire transition process, i.e. before the onset of transition well into fully turbulent flow. Floquet analysis predicts the subharmonic resonance where a subharmonic mode locally resonates for a given basic flow composed of the steady laminar flow and the fundamental mode. Because Floquet analysis is limited to the resonating subharmonic mode, nonlinear parabolised stability equation analysis (PSE) is conducted with various phase shifts of the subharmonic mode with respect to the given fundamental mode. The application of PSE offers insights on the modal interaction affected by the phase difference up to the weakly nonlinear stage of transition. Large-eddy simulation (LES) is conducted for a complete transition to turbulent boundary layer because PSE becomes prohibitively expensive in the late nonlinear stage of transition. The modulation of the subharmonic resonance with the initial phase difference leads to a significant delay in the transition location up to $\Delta Re_{x, tr} \simeq 4\times 10^5$ as predicted by the current LES. Effects of the initial phase difference on the spatial evolution of the modal shape of the subharmonic mode are further investigated. The mechanism of the phase evolution is discussed, based on current numerical results and relevant literature data.

Wavepacket modelling of broadband shock-associated noise in supersonic jets

Abstract

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.

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.

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.

Toward Automatic Parabolized Stability Equation-Based Transition-to-Turbulence Prediction for Aerodynamic Flows

The inclusion of transition-to-turbulence effects in computational fluid dynamics simulations is essential to accurately predict drag reduction from the use of laminar flow technologies. The parabolized stability equation (PSE) method takes into account nonlocal and nonparallel effects on boundary-layer dynamics. Its computational cost compares to that of linear stability theory (LST) analysis, which does not account for these effects. However, difficulties related to the robustness of PSE have prevented its application to industrial cases, where the more straightforward LST approach has been adopted because of its relative ease of use. When using PSE with an transition method, it is necessary to determine the stability modes that trigger transition and their neutral points (NPs). A robust PSE-based transition framework is proposed that includes a boundary-layer solver, a database method, and an LST solver that provides the required stability modes and the corresponding NPs so that a robust PSE calculation is automatically performed. The current approach leverages an automatic framework for the application of PSE-based transition prediction to aerodynamic flow analysis.

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.

From primary instabilities to secondary instabilities in G\xf6rtler vortex flows

We have studied the transformation process from primary instabilities to secondary instabilities with direct numerical simulations and stability theories (Spatial Biglobal and plane-marching parabolized stability equations) in detail. First Mack mode and second Mack mode are shown to be able to evolve into the sinuous mode and the varicose mode of secondary instability, respectively. Although the characteristics of second Mack mode eventually lose in the downstream due to the synchronization with the continuous spectrum, second Mack mode is found to be able to trigger a sequence of mode resonations which in turn give birth to highly unstable secondary instabilities. In contrast, first Mack mode does not involve in any mode synchronization. Nevertheless, it can still “jump" to a sinuous mode of secondary instability with a much larger growth rate than that of the first Mack mode. Therefore, secondary instabilities of Görtler vortices are highly receptive to the primary instabilities in the upstream, so that one should consider the primary instability in the upstream and the secondary instability in the downstream as a whole in order to get an accurate prediction of the boundary layer transition.

Semistability of Lazarsfeld–Mukai bundles via parabolic structures

Our aim in this article is to produce new examples of semistable Lazarsfeld–Mukai bundles on smooth projective surfaces X using the notion of parabolic vector bundles. In particular, we associate natural parabolic structures to any rank two (dual) Lazarsfeld–Mukai bundle and study the parabolic stability of these parabolic bundles. We also show that the orbifold bundles on Kawamata coverings of X corresponding to the above parabolic bundles are themselves certain (dual) Lazarsfeld–Mukai bundles. This gives semistable Lazarsfeld–Mukai bundles on Kawamata covers of the projective plane and of certain K3 surfaces.

Control of oblique-type breakdown in a supersonic boundary layer employing streaks

The effectiveness of streak modes in controlling the oblique-type breakdown in a supersonic boundary-layer at Mach 2.0 is investigated using direct numerical simulations. Investigations in the literature have shown the effectiveness of streak modes in delaying the onset of transition dominated by two-dimensional waves, but in oblique breakdown, three-dimensional waves and a strong streak mode dominate the transition process. Paredes et al. (J. Fluid Mech., vol. 831, 2017, pp. 524–553) discussed the possible stabilization of supersonic boundary layers by optimally growing streaks using parabolized stability equations. However, no study has as yet been reported regarding direct nonlinear control of oblique breakdown. This study deals with the effects of large-amplitude decaying streak modes generated by a blowing–suction strip at the wall to control full breakdown in a reference case. Modes with four to five times the fundamental wavenumber are found to be beneficial for controlling the transition. In the first region after the control-mode forcing, the beneficial mean-flow distortion (MFD), generated by inducing the control mode, is solely responsible for hampering the growth of the fundamental-mode. On the whole, the MFD and the three-dimensional part of the control contribute equally towards controlling the oblique breakdown. The results show significant suppression of transition, and substantial improvements have been observed in the levels of the skin-friction coefficient and wall-temperature in comparison to the uncontrolled case. Moreover, refreshing the control using an additional downstream control strip increases the gain. However, the forcing amplitude must be carefully chosen in order not to introduce a generalized inflection point in the spanwise averaged mean flow invoking enhanced disturbance growth.

Parabolized stability analysis of chemically reacting boundary-layer flows in equilibrium conditions

Stability of self-similar hypersonic flows over a flat plate in local thermodynamic equilibrium is studied by means of linear parabolized stability equations. A spectral collocation numerical method together with a finite-difference-based marching scheme are used to solve the equations, while the gas mixture and transport properties are computed with a table look-up procedure. High-temperature effects are found to destabilize the second-Mack-mode regardless of the adiabatic/isothermal wall condition considered. On the other hand, nonlocal effects show a Mach-dependence influence: stabilizing at lower Mach, while destabilizing at higher ones. Strong oscillations in the growth rate of perturbations in local thermodynamic equilibrium are found to be caused by damped supersonic modes.

New stability results for explicit Runge–Kutta methods

The theory of polar forms of polynomials is used to provide sharp bounds on the radius of the largest possible disc (absolute stability radius), and on the length of the largest possible real interval (parabolic stability radius), to be inscribed in the stability region of an explicit Runge–Kutta method. The bounds on the absolute stability radius are derived as a consequence of Walsh’s coincidence theorem, while the bounds on the parabolic stability radius are achieved by using Lubinsky–Ziegler’s inequality on the coefficients of polynomials expressed in the Bernstein bases and by appealing to a generalized variation diminishing property of Bezier curves. We also derive inequalities between the absolute stability radii of methods with different orders and number of stages.