Parabolized Stability Equations Research Articles

Flight Effects on Turbulent-Jet Wave Packets

An investigation is carried out to evaluate how an external uniform stream affects turbulent-jet coherent structures. The presence of this uniform stream outside the jet represents the effect of forward flight on the velocity field. Coherent structures are modelled as wave packets, and the flight effect is studied using the parabolized stability equations (PSEs). Measurements of mean base flows and axial velocity fluctuations on the jet centerline at assumed flight conditions are described. The velocity fields were supplied to a PSE model to retrieve the wave packet signatures of axial velocity fluctuations. Overall results are in good agreement with experimental measurements. There is an observed increase in wave packet wavelengths and phase velocities (with respect to the nozzle) as flight velocity increases. Also, axial amplification rates are determined and show stabilization in the near-nozzle region, which is confirmed by experimental power spectral densities on the jet centerline. This tendency holds as the freestream velocity increases.

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.

Effect of Shape of Two-Dimensional Smooth Hump on Boundary Layer Instability

Effect of hump shape on boundary layer instability is investigated for incompressible, supersonic, and hypersonic speed regimes. Flat-plate boundary layers over a two-dimensional smooth hump of five different geometries are considered. The linear evolution of instability waves is analyzed by using the parabolized stability equation. The resulting growth rate and N-factor are compared for several frequencies. The destabilization effect of the hump is evaluated in terms of the change in N-factor. The overall influence of the hump is also assessed by comparisons of N-factor envelopes and predicted transition locations based on eN-method. Results show that the destabilization effect is the greatest for the incompressible case. The degree of destabilization appears to depend not only on the hump area but also the other geometric parameters. The variation of the predicted transition Reynolds number depending on the hump shape is pronounced only in a certain range of hump heights. Although the supersonic boundary layer also exhibits similar characteristics, the effectiveness of hump is rather low, and relatively insensitive to the hump shape. In the case of the hypersonic boundary layer, the hump exhibits either destabilization or stabilization effect, depending on the relative location of a synchronization point. The degree of destabilization depends strongly on the both hump area and frequency. Either the transition delay or early transition could be predicted depending on the hump location and critical N-factor value employed. Unlike the incompressible case, the dependency on the hump shape is not confined to a certain range of hump heights.

Improvement of the parabolized stability equation to predict the linear evolution of disturbances in three-dimensional boundary layers based on ray tracing theory

A new method to predict the linear evolution of disturbances in three-dimensional inhomogeneous boundary layers is proposed, named RTPSE, in which the line-marching parabolized stability equation (PSE) is improved by applying ray tracing (RT) theory. Results show that RTPSE can accurately predict the spanwise wave number and amplitude ratio for both stationary and traveling crossflow waves, while the traditional PSE could not.

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.

On the influence of two-dimensional hump roughness on laminar–turbulent transition

The presence of large surface irregularities such as humps, where the height is similar to the local boundary-layer (BL) displacement thickness, introduces regions of localized strong streamwise gradients in the base flow quantities. These large gradients can significantly modify the spatial development of incoming disturbances that lead to laminar–turbulent transition in wall-bounded flows [e.g., Tollmien–Schlichting (TS) waves]. Techniques such as Parabolized Stability Equations (PSE) are not suited for BL instability analysis in such regions: their formulation assumes that streamwise variations of base flow and disturbance quantities are small, allowing a marching procedure for their resolution. On the other hand, the Adaptive Harmonic Linearized Navier–Stokes (AHLNS) equations can handle these large streamwise gradients by using a fully elliptic system of equations, similar to Linearized Navier–Stokes (LNS), Harmonic LNS (HLNS), or Direct Numerical Simulation (DNS). Moreover, in AHLNS (as in PSE), a wave-like character of the instabilities is assumed, leading to a significant reduction in the number of streamwise grid points required compared with LNS, HLNS, or DNS computations. In the present study, an efficient combination of PSE and AHLNS is used to investigate the effect of height, length, and shape of a single hump placed on a flat plate in a two-dimensional flow field at Ma∞ = 0.5 without pressure gradient. The effect of this hump on the spatial evolution of TS waves, in terms of N-factors, is presented. An expected laminar–turbulent transition onset, via the eN methodology, is also described. It is shown that the shape of the surface irregularity, together with the height and length, plays an important role for the location of laminar–turbulent transition onset in convectively unstable flows.

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.

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.

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.

A critical assessment of the parabolized stability equations

The parabolized stability equations (PSE) are a ubiquitous tool for studying the stability and evolution of disturbances in weakly nonparallel, convectively unstable flows. The PSE method was introduced as an alternative to asymptotic approaches to these problems. More recently, PSE has been applied with mixed results to a more diverse set of problems, often involving flows with multiple relevant instability modes. This paper investigates the limits of validity of PSE via a spectral analysis of the PSE operator. We show that PSE is capable of accurately capturing only disturbances with a single wavelength at each frequency and that other disturbances are not necessarily damped away or properly evolved, as often assumed. This limitation is the result of regularization techniques that are required to suppress instabilities arising from the ill-posedness of treating a boundary value problem as an initial value problem. These findings are valid for both incompressible and compressible formulations of PSE and are particularly relevant for applications involving multiple modes with different wavelengths and growth rates, such as problems involving multiple instability mechanisms, transient growth, and acoustics. Our theoretical results are illustrated using a generic problem from acoustics and a dual-stream jet, and the PSE solutions are compared to both global solutions of the linearized Navier–Stokes equations and a recently developed alternative parabolization.

Large-eddy simulation with parabolized stability equations for turbulent transition using OpenFOAM

Laminar-to-turbulent transition is simulated with large-eddy simulation (LES) coupled with parabolized stability equations (PSE). A canonical transition on a flat plate is computed. The PSE-coupled approach accurately reproduces the subharmonic resonance and subsequent turbulent boundary layer using the solver OpenFOAM. Various LES inlet locations are simulated to reduce the computational domain. Nonlinear PSE provides appropriate instabilities at various inlet locations. The computational cost and domain are reduced by 30% with the LES inlet close to the final stage of the nonlinear transition region, compared to the baseline, upstream LES inlet. Current LES results are compared with relevant data in the literature.

High-Reynolds number transitional flow simulation via parabolized stability equations with an adaptive RANS solver

The accurate prediction of transition is relevant for aerodynamic analysis and design applications. Extending the laminar flow region over airframes is a potential way to reduce the skin friction drag, which in turn reduces fuel burn and greenhouse gas emissions. This paper introduces a numerical framework that includes the modeling of transition effects for high Reynolds number flows in a high-fidelity, Reynolds–averaged Navier–Stokes (RANS) aerodynamic design framework. The CFD solver uses a discontinuous Galerkin (DG) finite element approach and includes goal-oriented adaptation. The Spalart–Allmaras (SA) turbulence model is used for the closure of the governing equations. In the flow stability analysis, the nonlocal, nonparallel effects that characterize boundary layers are accounted for by using the parabolized stability equations (PSE). Transition onset is obtained through an eN method based on the PSE computations, while a smooth intermittency function includes the transition region length. Numerical results for the NLF(1)-0416 airfoil present good agreement with experimental data, improving the computations when compared to fully-turbulent ones.

Stabilization of crossflow instability with plasma actuators: Linearized Navier–Stokes simulations

This paper describes work carried out within the European Union (EU)-Russia Buterfli project to look at the control of transition-causing “target” stationary cross flow vortices, by the use of distributed plasma actuation to generate sub-dominant “killer” modes. The objective is to use the “killer” modes to control the “target” modes through a non-linear stabilizing mechanism. The numerical modelling and results are compared to experimental studies performed at the TsAGI T124 tunnel for a swept plate subject to a favorable pressure gradient flow. A mathematical model for the actuator developed at TsAGI was implemented in a linearized Navier–Stokes (LNS) solver and used to model and hence predict “killer” mode amplitudes at a measurement plane in the experiment. The LNS analysis shows good agreement with experiment, and the results are used as input for non-linear parabolized stability equation (PSE) analysis to predict the effect of these modes on crossflow transition. Whilst the numerical model indicates a delay in transition, experimental results indicated an advance in transition rather than delay. This was determined to be due to actuator-induced unsteadiness arising in the experiment, resulting in the generation of travelling crossflow disturbances which tended to obscure and thus dominate the plasma stabilized stationary disturbances.

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

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

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

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

History and Progress of Boundary-Layer Transition on a Mach-6 Flared Cone

Results from experimental, theoretical, and computational efforts are combined in a comprehensive review to summarize the improved understanding of the underlying physical mechanisms causing second-mode dominated laminar–turbulent transition on a 3 m circular-arc flared cone for the Boeing/U.S. Air Force Office of Scientific Research Mach 6 Quiet Tunnel (BAM6QT) conditions. Qualitatively similar patterns of streamwise streaks have been observed in experiments, using temperature sensitive paint, and in high-resolved “controlled” breakdown direct numerical simulations (DNS). The experimental and computational results using nonlinear parabolized stability equations (NPSEs) and DNS revealed that the maximum second-mode pressure fluctuation reaches extremely large values (30–60% of the surface pressure) before the breakdown to turbulence. These values are significantly higher as compared to what has been observed in incompressible flow, therefore providing an explanation for an elongated transition/breakdown regime in high-speed compressible flow. A sudden spectral broadening in the disturbance content has been observed in the three different approaches once the primary wave saturates nonlinearly, indicating a sudden nonlinear amplification of a broad spectrum of frequencies and wave numbers when transition begins. Good agreement was found between NPSE calculations and experiments when comparing the spectral broadening onset location, the total disturbance amplitude, and the amplitude ratio between the primary disturbances and the first harmonics, which shows that NPSEs can accurately reproduce/predict the experimental results. The DNS results revealed that the streaks on the surface of the cone are caused by steady streamwise vortices that are generated by nonlinear interaction of the primary and secondary disturbance wave. The similarities in the streak patterns suggest that the mechanisms observed in the controlled breakdown DNS are likely the same mechanisms leading to laminar–turbulent transition in the natural transition scenario observed in the BAM6QT. The more in-depth understanding of the transition process, and especially the understanding of the mechanisms leading to the streak patterns, can be used to explore potential flow control strategies to mitigate the detrimental effects of the massive heat overshoots over the turbulent values arising as a consequence of these streaks.

Development of an improved algorithm for solving nonlinear parabolized stability equations for the purpose of relaxing the initial conditions

Nonlinear parabolized stability equations (NPSE) have been proven to be a useful and convenient tool for the investigation of the stability and transition problems of boundary layers. However, in its original formulation, as initial conditions, the complex wave number and shape function of each Fourier mode are required for solving nonlinear problems, which result ambiguity and difficulty to set the initial values. In this paper, I propose a modified algorithm to remove its ambiguity and difficulty. The main idea is that, instead of solving all wave numbers with auxiliary condition, all wave numbers except primary mode are kept real and decided by phase-locked rule, and the shape function of each mode is simply solved by using nonlinear SOR method. The validity of the new formulation is illustrated by comparing the results with those from the corresponding original NPSE simulations as applied to a problem of Goertler instability and H-type transition of a flat plate boundary layer.

Analysis of Second-Mode Amplitudes on Sharp Cones in Hypersonic Wind Tunnels

This research effort coordinated by the NATO Science and Technology Organization Applied Vehicle Technology 240 specialists’ group compiles and analyzes second-mode amplitudes on sharp slender cones. The analysis focuses on pressure fluctuations measured with piezoelectric sensors in 11 hypersonic wind tunnels operated by nine organizations located in three NATO countries (Belgium, Germany, and the United States) and Japan. The measurements are at freestream Mach numbers between 5 and 14, unit Reynolds numbers between 1.5 and , and wall-to-total temperature ratios between 0.1 and 0.8. The study shows that second-mode growth rates can be predicted with parabolized stability equations (PSEs) over the wide range of conditions. The maximum second-mode amplitudes vary weakly at edge Mach number greater than but significantly decrease at lower . The maximum factor envelope from PSE and the measured amplitudes are used to estimate the initial amplitudes . At each Mach number, varies approximately as Reynolds number. This leads to transition factors that increase with Reynolds number. This behavior is consistent with the results from Marineau (“Prediction Methodology for Second-Mode-Dominated Boundary-Layer Transition in Wind Tunnels,” AIAA Journal, Vol. 55, No. 2, 2017, pp. 484–499). This study indicates that amplitude-based methods for second-mode-dominated hypersonic boundary-layer transition predictions should be applicable across a wide range of conditions.

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

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