Parabolized Stability Equation Method Research Articles

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).

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.

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.

Improvement for expansion of parabolized stability equation method in boundary layer stability analysis

An improved expansion of the parabolized stability equation (iEPSE) method is proposed for the accurate linear instability prediction in boundary layers. It is a local eigenvalue problem, and the streamwise wavenumber α and its streamwise gradient dα/dx are unknown variables. This eigenvalue problem is solved for the eigenvalue dα/dx with an initial α, and the correction of α is performed with the conservation relation used in the PSE. The iEPSE is validated in several compressible and incompressible boundary layers. The computational results show that the prediction accuracy of the iEPSE is significantly higher than that of the ESPE, and it is in excellent agreement with the PSE which is regarded as the baseline for comparison. In addition, the unphysical multiple eigenmode problem in the EPSE is solved by using the iEPSE. As a local non-parallel stability analysis tool, the iEPSE has great potential application in the eN transition prediction in general three-dimensional boundary layers.

Discrete- and finite-bandwidth-frequency distributions in nonlinear stability applications

A new “wave packet” formulation of the parabolized stability equations method is presented. This method accounts for the influence of finite-bandwidth-frequency distributions on nonlinear stability calculations. The methodology is motivated by convolution integrals and is found to appropriately represent nonlinear energy transfer between primary modes and harmonics, in particular nonlinear feedback, via a “nonlinear coupling coefficient.” It is found that traditional discrete mode formulations overestimate nonlinear feedback by approximately 70%. This results in smaller maximum disturbance amplitudes than those observed experimentally. The new formulation corrects this overestimation, accounts for the generation of side lobes responsible for spectral broadening, and results in disturbance representation more consistent with the experiment than traditional formulations. A Mach 6 flared-cone example is presented.

Parabolized Stability Equations Code with Automatic Inflow for Swept Wing Transition Analysis

This paper aims at developing a parabolized stability equation transition analysis code coupled to the industrial in-house accurate Reynolds-averaged Navier–Stokes code developed at Israel Aerospace Industries for stability analysis of three-dimensional boundary layers over swept wings. In the first step, the parabolized stability equation derivation in a body-fitted coordinate system is exposed, as well as its classical initialization, using the solution procedure of the global Chebyshev eigenvalue problem for the linear stability equations. This initialization procedure demands large user interactions, and it prevents the parabolized stability equation method to be of practical use in an industrial environment. Therefore, one exposes in the second step a new way to automatically generate the inflow solution of the linear stability equations needed by the parabolized stability equations. This new robust approach paves the way for innovative automatic industrial use of the parabolized stability equation technology without the need to involve the user. This approach supplies the inflow conditions required for the parabolized stability equation marching procedure without the need to solve the standalone linear stability global eigenvalue problem. The solution of the linear stability equations at initialization is now obtained by iteratively converging the local linear stability solver starting from the new inflow empirical approximation obtained in two special coordinate systems for crossflow and Tollmien–Schlichting, respectively. These formulas represent some invariant functions of the stability characteristics of the exact Navier–Stokes laminar boundary-layer profiles instead of self-similar boundary-layer profiles like the Falkner–Skan–Cooke. Standard academic validation test cases and industrial results of the parabolized stability equation/Reynolds-average Navier-Stokes codes on a generic unmanned aerial vehicle swept wing are exposed.

Stability analysis method considering non-parallelism: EPSE method and its application

The e-N method is widely used in transition prediction. The amplitude growth rate used in the e-N method is usually provided by the linear stability theory (LST) based on the local parallel hypothesis. Considering the non-parallelism effect, the parabolized stability equation (PSE) method lacks local characteristic of stability analysis. In this paper, a local stability analysis method considering non-parallelism is proposed, termed as EPSE since it may be considered as an expansion of the PSE method. The EPSE considers variation of the shape function in the streamwise direction. Its local characteristic is convenient for stability analysis. This paper uses the EPSE in a strong non-parallel flow and mode exchange problem. The results agree well with the PSE and the direct numerical simulation (DNS). In addition, it is found that the growth rate is related to the normalized method in the non-parallel flow. Different results can be obtained using different normalized methods. Therefore, the normalized method must be consistent.

Receptivity of Hypersonic Boundary Layers over Straight and Flared Cones

The effects of adverse pressure gradients on the receptivity and stability of hypersonic boundary layers were numerically investigated. Simulations were performed for boundary-layer flows over a straight cone and two flared cones. The steady and the unsteady flowfields were obtained by solving the two-dimensional Navier–Stokes equations in axisymmetric coordinates using the fifth-order-accurate weighted essentially nonoscillatory scheme for space discretization and using a third-order total-variation-diminishing Runge–Kutta scheme for time integration. The mean boundary-layer profiles were analyzed using local stability and nonlocal parabolized stability equations methods. After the most amplified disturbances were identified, two-dimensional plane acoustic waves were introduced at the outer boundary of the computational domain and time-accurate simulations were performed. The adverse pressure gradient was found to affect the boundary-layer stability in two important ways. First, the frequency of the most amplified second-mode disturbance was increased relative to the zero-pressure gradient case. Second, the amplification of first- and second-mode disturbances was increased. Although an adverse pressure gradient enhances instability wave growth rates, small nose-tip bluntness was found to delay transition due to the low receptivity coefficient and the resulting weak initial amplitude of the instability waves. The computed and measured amplitude-frequency spectra in all three cases agree very well in terms of frequency and the shape except for the amplitude.

Swept wing boundary-layer receptivity to localized surface roughness

AbstractThe receptivity to localized surface roughness of a swept-wing boundary layer is studied by direct numerical simulation (DNS) and computations using the parabolized stability equations (PSEs). The DNS is laid out to reproduce wind tunnel experiments performed by Saric and coworkers, where micron-sized cylinders were used to trigger steady crossflow modes. The amplitudes of the roughness-induced fundamental crossflow wave and its superharmonics obtained from nonlinear PSE solutions agree excellently with the DNS results. A receptivity model using the direct and adjoint PSEs is shown to provide reliable predictions of the receptivity to roughness cylinders of different heights and chordwise locations. Being robust and computationally efficient, the model is well suited as a predictive tool of receptivity in flows of practical interest. The crossflow mode amplitudes obtained based on both DNS and PSE methods are 40 % of those measured in the experiments. Additional comparisons between experimental and PSE data for various disturbance wavelengths reveal that the measured disturbance amplitudes are consistently larger than those predicted by the PSE-based receptivity model by a nearly constant factor. Supplementary DNS and PSE computations suggest that possible natural leading-edge roughness and free-stream turbulence in the experiments are unlikely to account for this discrepancy. It is more likely that experimental uncertainties in the streamwise location of the roughness array and cylinder height are responsible for the additional receptivity observed in the experiments.

Applications of parabolized stability equation for predicting transition position in boundary layers

The phenomenon of laminar-turbulent transition exists universally in nature and various engineering practice. The prediction of transition position is one of crucial theories and practical problems in fluid mechanics due to the different characteristics of laminar flow and turbulent flow. Two types of disturbances are imposed at the entrance, i.e., identical amplitude and wavepacket disturbances, along the spanwise direction in the incompressible boundary layers. The disturbances of identical amplitude are consisted of one two-dimensional (2D) wave and two three-dimensional (3D) waves. The parabolized stability equation (PSE) is used to research the evolution of disturbances and to predict the transition position. The results are compared with those obtained by the numerical simulation. The results show that the PSE method can investigate the evolution of disturbances and predict the transition position. At the same time, the calculation speed is much faster than that of the numerical simulation.

The reliability of the improved e N method for the transition prediction of boundary layers on a flat plate

The transition criterion in the improved eN method is that transition would occur whenever the velocity amplitude of disturbance reaches 1%–2% of the free stream velocity, while in the conventional eN method, the N factor is an empirical factor. In this paper the reliability of this key assumption in the improved eN method is checked by results of transition prediction by using the Parabolized Stability Equations (PSE). Transition locations of an incompressible boundary layer and a hypersonic boundary layer at Mach number 6 on a flat plate are predicted by both the improved eN method and the PSE method. Results from both methods agree fairly well with each other, implying that the transition criterion proposed in the improved eN method is reliable.

Tollmien-schlichting wave amplitudes on a semi-infinite flat plate and a parabolic body: comparison of a parabolized stability equation method and direct numerical simulations

In this paper, the interaction of free-stream acoustic waves with the leading edge of an aerodynamic body is investigated and two different methods for analysing this interaction are considered. Results are compared for a method which incorporates Orr–Sommerfeld calculations using the parabolized stability equation to those of direct numerical simulations. By comparing the streamwise amplitude of the Tollmien–Schlichting wave, it is found that non-modal components of the boundary layer response to an acoustic wave can persist some distance downstream of the lower branch. The effect of nose curvature on the persisting non-modal eigenmodes is also considered, with a larger nose radius allowing the non-modal eigenmodes to persist farther downstream.

Vortex structure simulation for supersonic mixing layers using nonlinear PSE method

The method of nonlinear parabolized stability equations (PSE) is applied in the simulation of vortex structures in compressible mixing layer. The spatially-evolving unstable waves, which dominate the vortex structure, are investigated through spatial marching method. The instantaneous flow field is obtained by adding the harmonic waves to basic flow. The results show that T-S waves do not keep growing exponentially as the linear evolution, the energy transfer to high order harmonic modes, and that finally all harmonic modes get saturated due to nonlinear interaction. The mean flow distortion induced by the nonlinear interaction between the harmonic modes and their conjugate harmonic ones, makes great change of the average flow and increases the thickness of mixing layer. PSE methods can well capture the two- and three-dimensional large scale nonlinear vortex structures in mixing layers such as vortex roll-up, vortex pairing, and Λ vortex.

SIMULATION OF VORTEX STRUCTURE IN SUPERSONIC FREE SHEAR LAYER USING PSE METHOD

The method of parabolized stability equations (PSE) are applied in the analysis of nonlinear stability and the simulation of flow structure in supersonic free shear layer. High accuracy numerical techniques including self-similar basic flow, high order differential method, appropriate transformation and decomposition of nonlinear terms are adopted and developed to solve the PSE effectively for free shear layer. The spatial evolving unstable waves which dominate the flow structure are investigated through nonlinear coupling spatial marching methods. The nonlinear interactions between harmonic waves are further analyzed and instantaneous flow field are obtained by adding the harmonic waves into basic flow. Relevant data agree well with that of DNS. The results demonstrate that T-S wave does not keeping growing exponential as the linear evolution, the energy transfer to high order harmonic modes and finally all harmonic modes get saturation due to the nonlinear interaction; Mean flow distortion is produced by the nonlinear interaction between the harmonic and its conjugate harmonic, makes great change to the average flow and increases the thickness of shear layer; PSE methods can well capture the large scale nonlinear flow structure in the supersonic free shear layer such as vortex roll-up, vortex pairing and nonlinear saturation.

High-Speed Boundary-Layer Instability: Old Terminology and a New Framework

The discrete spectrum of disturbances in high-speed boundary layers is discussed with emphasis on singularities caused by synchronization of the normal modes. Numerical examples illustrate different spectral structures and jumps from one structure to another with small variations of basic flow parameters. It is shown that this singular behavior is due to branching of the dispersion curves in the synchronization region. Depending on the locations of the branch points, the spectrum contains an unstable mode or two. In connection with this, the terminology used for instability of high-speed boundary layers is clarified. It is emphasized that the spectrum branching may cause difficulties in stability analyses based on traditional linear stability theory and parabolized stability equations methods. Multiple-mode considerations and direct numerical simulations are needed to clarify this issue.

A nonlinear PSE method for two-fluid shear flows with complex interfacial topology

A nonlinear stability method is developed for laminar two-fluid shear flows which undergo changes in the interface topology. The method is based on the nonlinear parabolized stability equations (PSE) and incorporates a scalar-based interface capturing (IC) scheme in order to track complex deformations of the fluid interface. In doing so, the formulation retains the flexibility and physical insight of instability-wave based methods, while providing hydrodynamic modeling capabilities similar to direct numerical calculations: the new formulation, referred to as the IC-PSE, can capture the nonlinear physical mechanisms responsible for generating large-scale, two-fluid structures, without incurring heavy computational costs. This approach is valid for spatially developing, laminar two-fluid shear flows which are convectively unstable, and can naturally account for the growth of finite amplitude interfacial waves, along with changes to the interfacial topology. We demonstrate the accuracy of the IC-PSE against direct Navier–Stokes calculations for two-fluid mixing layers with density and viscosity stratification. The comparisons show that the IC-PSE can predict the dynamics of the instability waves and capture the formation of Kelvin–Helmholtz vortex rolls and large scale liquid structures, at an order of magnitude less computational cost than direct calculations. The role of surface tension in the IC-PSE formulation is shown to be valid for flows in which Re/We ≲ 1, and the method accurately predicts the formation and non-linear evolution of flow structures in this limit. This is demonstrated for spatially developing mixing layers which lead to vortex roll-up and ligaments, prior to droplet formation. The pinch-off process itself is a high surface tension phenomenon and in not considered herein. The method also accurately captures the effect of interfacial waves on the mean flow, and the topology changes during the non-linear evolution of the two-fluid structures.

Computational Modeling of Roughness-Based Laminar Flow Control on a Subsonic Swept Wing

A combination of parabolized stability equations and secondary instability theory has been applied to a low-speed swept airfoil model with a chord Reynolds number of 7.15 million, with the goal of evaluating this methodology in the context of transition prediction for a known configuration for which roughness-based crossflow transition control has been demonstrated under flight conditions. Nonlinear parabolized stability equations computations indicate that progressive reduction in the growth of the linearly most amplified stationary crossflow mode can be achieved via increasingly stronger control input corresponding to the first harmonic of the target mode. The reduction in the target mode amplitude is accompanied by reduced linear growth rates of the high-frequency secondary instabilities that lead to rapid breakdown of the laminar flow. The secondary instability predictions based on secondary instability theory are shown to agree well with those based on the parabolized stability equations. The possibility of overcontrol is also assessed, and it is found that premature transition due to excessive control can be avoided by keeping the control amplitude below a certain threshold. The nonlinear development of the most unstable Z-mode secondary instability is traced using the parabolized stability equation method, so as to yield physics-based prediction of crossflow-dominated transition.

A new method for computing laminar-turbulent transition and turbulence in compressible boundary layers—PSE+DNS

A new method for computing laminar-turbulent transition and turbulence in compressible boundary layers is proposed. It is especially useful for computation of laminar-turbulent transition and turbulence starting from small-amplitude disturbances. The laminar stage, up to the beginning of the breakdown in laminar-turbulent transition, is computed by parabolized stability equations (PSE). The direct numerical simulation (DNS) method is used to compute the transition process and turbulent flow, for which the inflow condition is provided by using the disturbances obtained by PSE method up to that stage. In the two test cases including a subsonic and a supersonic boundary layer, the transition locations and the turbulent flow obtained with this method agree well with those obtained by using only DNS method for the whole process. The computational cost of the proposed method is much less than using only DNS method.

PSE as applied to problems of transition in compressible boundary layers

A new idea of using the parabolized stability equation (PSE) method to predict laminar-turbulent transition is proposed. It is tested in the prediction of the location of transition for compressible boundary layers on flat plates, and the results are compared with those obtained by direct numerical simulations (DNS). The agreement is satisfactory, and the reason for this is that the PSE method faithfully reproduces the mechanism leading to the breakdown process in laminar-turbulent transition, i.e., the modification of mean flow profile leads to a remarkable change in its stability characteristics.

Boundary-Layer Stability Analysis of the Hypersonic International Flight Research Transition Experiments

Boundary-layer stability analysis is performed by computational fluid dynamics simulation of experiments conducted in theCalspan–University at BuffaloResearchCenter Large EnergyNational ShockTunnel in support of the first flight of the Hypersonic International Flight Research Experimentation program. From the laminar flow solutions, disturbances are calculated using the linear parabolized stability equations method and instability is quantified by integrating the resulting disturbance growth rates. Comparisons aremade between the experimentally measured transition locations and the results of the parabolized stability equations analysis. The results show that for the cases tested, the e transition correlation works better than the commonly usedRe =Me engineering criterion for predicting the onset of boundary-layer transition from laminar to turbulent flow.