Verification of parabolized stability equations for its application to compressible boundary layers

Parabolized stability equations （PSE） were used to study the evolution of disturbances in compressible boundary layers. The results were compared with those obtained by direct numerical simulations （DNS）, to check if the results from PSE method were reliable or not. The results of comparison showed that no matter for subsonic or supersonic boundary layers, results from both the PSE and DNS method agreed with each other reasonably well, and the agreement between temperatures was better than those between velocities. In addition, linear PSE was used to calculate the neutral curve for small amplitude disturbances in a supersonic boundary layer. Compared with those obtained by linear stability theory （LST）, the situation was similar to those for incom- pressible boundary layer.

Improved nonlinear parabolized stability equations approach for hypersonic boundary layers

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.

CALCULATION OF TWO-DIMENSIONAL PARABOLIC STABILITY EQUATIONS

Two dimensional parabolic stability equations (PSE) are numerically solved using expansions in orthogonal functions in the normal direction.The Chebyshev polynomials approximation,which is a very useful form of orthogonal expansions, is applied to solving parabolic stability equations. It is shown that results of great accuracy are effectively obtained.The availability of using Chebyshev approximations in parabolic stability equations is confirmed.

Self-consistent parabolized stability equation(PSE) method for compressible boundary layer

The parabolized stability equation （PSE） method has 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, for nonlinear problems, the complex wave number of each Fourier mode is determined by the so-called phase-locked rule, which results in non-self-consistency in the wave numbers. In this paper, a modification is proposed to make it self-consistent. The main idea is that, instead of allowing wave numbers to be complex, all wave numbers are kept real, and the growth or decay of each mode is simply manifested in the growth or decay of the modulus of its shape function. The validity of the new formulation is illustrated by comparing the results with those from the corresponding direct numerical simulation （DNS） as applied to a problem of compressible boundary layer with Mach number 6.

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 i EPSE is validated in several compressible and incompressible boundary layers. The computational results show that the prediction accuracy of the i EPSE 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 i EPSE. As a local non-parallel stability analysis tool, the i EPSE has great potential application in the eNtransition prediction in general three-dimensional boundary layers.

ANALYSIS AND APPLICATION OF ELLIPTICITY OF STABILITY EQUATIONS ON FLUID MECHANICS

By using characteristic analysis of the linear and nonlinear parabolic stability equations ( PSE), PSE of primitive disturbance variables are proved to be parabolic intotal. By using sub- characteristic analysis of PSE, the linear PSE are proved to be elliptical and hyperbolic-parabolic for velocity U, in subsonic and supersonic, respectively; the nonlinear PSE are proved to be elliptical and hyperbolic-parabolic for relocity U + u in subsonic and supersonic., respectively . The methods are gained that the remained ellipticity is removed from the PSE by characteristic and sub-characteristic theories , the results for the linear PSE are consistent with the known results, and the influence of the Mach number is also given out. At the same time , the methods of removing the remained ellipticity are further obtained from the nonlinear PSE .

NONPARALLEL BOUNDARY LAYER STABILITY IN HIGH SPEED FLOWS

The parabolized stability equations （PSEs） for high speed flows, especially supersonic and hypersonic flows, are derived and used to analyze the nonparallel boundary layer stability. The proposed numerical techniques for solving PSE include the following contents： introducing the efficiently normal transformation of the boundary layer, improving the computational accuracy by using a high-order differential scheme near the wall, employing the predictor-corrector and iterative approach to satisfy the important normalization condition, and implementing the stable spatial marching. Since the second mode dominates the growth of the disturbance in high Mach number flows, it is used in the computation. The evolution and characteristics of the boundary layer stability in the high speed flow are demonstrated in the examples. The effects of the nonparallelizm, the compressibility and the cooling wall on the stability are analyzed. And computational results are in good agreement with the relevant data.

PSE as applied to problems of secondary instability in supérsonic boundary layers

Parabolized stability equations （PSE） approach is used to investigate problems of secondary instability in supersonic boundary layers. The results show that the mechanism of secondary instability does work, whether the 2-D fundamental disturbance is of the first mode or second mode T-S wave. The variation of the growth rates of the 3-D sub-harmonic wave against its span-wise wave number and the amplitude of the 2-D fundamental wave is found to be similar to those found in incompressible boundary layers. But even as the amplitude of the 2-D wave is as large as the order 2%, the maximum growth rate of the 3-D sub-harmonic is still much smaller than the growth rate of the most unstable second mode 2-D T-S wave. Consequently, secondary instability is unlikely the main cause leading to transition in supersonic boundary layers.

Linear stability analysis of supersonic axisymmetric jets

Stabilities of supersonic jets are examined with different velocities, momentum thicknesses, and core temperatures. Amplification rates of instability waves at inlet are evaluated by linear stability theory （LST）. It is found that increased velocity and core temperature would increase amplification rates substantially and such influence varies for different azimuthal wavenumbers. The most unstable modes in thin momentum thickness cases usually have higher frequencies and azimuthal wavenumbers. Mode switching is observed for low azimuthal wavenumbers, but it appears merely in high velocity cases. In addition, the results provided by linear parabolized stability equations show that the mean-flow divergence affects the spatial evolution of instability waves greatly. The most amplified instability waves globally are sometimes found to be different from that given by LST.

On Nonlinear Evolution of C-type Instability in Nonparallel Boundary Layers

The process of evolution, especially that of nonlinear evolution, of C-type instability of laminar-turbulent flow transition in nonparallel boundary layers are studied by means of a newly developed method called parabolic stability equations （PSE）. Initial conditions, which are very important for the nonlinear problem, are investigated by computing initial solution of the harmonic waves, modifying the mean-flow-distortion, and giving initial value of TS wave and its subharmonic waves at initial station by solving linear PSE. A numerical method with high-order accuracy are developed in the text, the key normalization conditions in the PSE are satisfied, and nonlinear PSE are solved efficiently and implemented stably by the spatial marching. It has been shown that the computed process of nonlinear evolution of C-type instability in Blasius boundary layer is in good agreement with the experimental results.

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.

Linear spatial instability analysis in 3D boundary layers using plane-marching 3D-LPSE

It is widely accepted that a robust and efficient method to compute the linear spatial amplified rate ought to be developed in three-dimensional （3D） boundary layers to predict the transition with the e^N method, especially when the boundary layer varies significantly in the spanwise direction. The 3D-linear parabolized stability equation （3D- LPSE） approach, a 3D extension of the two-dimensional LPSE （2D-LPSE）, is developed with a plane-marching procedure for investigating the instability of a 3D boundary layer with a significant spanwise variation. The method is suitable for a full Mach number region, and is validated by computing the unstable modes in 2D and 3D boundary layers, in both global and local instability problems. The predictions are in better agreement with the ones of the direct numerical simulation （DNS） rather than a 2D-eigenvalue problem （EVP） procedure. These results suggest that the plane-marching 3D-LPSE approach is a robust, efficient, and accurate choice for the local and global instability analysis in 2D and 3D boundary layers for all free-stream Mach numbers.

Interfacial instability of ferrofluid flow under the influence of a vacuum magnetic field

This study is to numerically test the interfacial instability of ferrofluid flow under the presence of a vacuum magnetic field.The ferrofluid parabolized stability equations(PSEs)are derived from the ferrofluid stability equations and the Rosensweig equations,and the characteristic values of the ferrofluid PSEs are given to describe the ellipticity of ferrofluid flow.Three numerical models representing specific cases considering with/without a vacuum magnetic field or viscosity are created to mathematically examine the interfacial instability by the computation of characteristic values.Numerical investigation shows strong dependence of the basic characteristic of ferrofluid Rayleigh-Taylor instability(RTI)on viscosity of ferrofluid and independence of the vacuum magnetic field.For the shock wave striking helium bubble,the magnetic field is not able to trigger the symmetry breaking of bubble but change the speed of the bubble movement.In the process of droplet formation from a submerged orifice,the collision between the droplet and the liquid surface causes symmetry breaking.Both the viscosity and the magnetic field exacerbate symmetry breaking.The computational results agree with the published experimental results.

Applications of EPSE method for predicting crossflow instability in swept-wing boundary layers

The nth-order expansion of the parabolized stability equation （EPSEn） is obtained from the Taylor expansion of the linear parabolized stability equation （LPSE） in the streamwise direction. The EPSE together with the homogeneous boundary conditions forms a local eigenvalue problem, in which the streamwise variations of the mean flow and the disturbance shape function are considered. The first-order EPSE （EPSE1） and the second-order EPSE （EPSE2） are used to study the crossflow instability in the swept NLF（2）-0415 wing boundary layer. The non-parallelism degree of the boundary layer is strong. Compared with the growth rates predicted by the linear stability theory （LST）, the results given by the EPSE1 and EPSE2 agree well with those given by the LPSE. In particular, the results given by the EPSE2 are almost the same as those given by the LPSE. The prediction of the EPSE1 is more accurate than the prediction of the LST, and is more efficient than the predictions of the EPSE2 and LPSE. Therefore, the EPSE1 is an efficient ey prediction tool for the crossflow instability in swept-wing boundary-layer flows.

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 fiat 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.

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 A vortex.

EVOLUTION ANALYSIS OF TS WAVE AND HIGH-ORDER HARMONIC WAVES IN BOUNDARY LAYERS

Linear and nonlinear evolutions of TS wave and high-order harmonic waves in boundary layers are studied based on the parabolic stability equation （PSE）. Initial conditions are derived by the local method with the Landau expansion. The evolution process and characteristics of the disturbance amplitude and the velocity profile, etc. , especially stronger nonlinear effects, are computed by an efficient numerical method. Effects and regulations of different initial amplitudes, frequencies and pressure gradients on the evolution of disturbances are explored, which are directly relative to the stability and the transition in boundary layers. Simulation results are in good agreement with the data of the accuracy direct numerical simulation （DNS） using full Navier-Stokes equations.

NONLINEAR EVOLUTION ANALYSIS OF T-S DISTURBANCE WAVE AT FINITE AMPLITUDE IN NONPARALLEL BOUNDARY LAYERS

The nonlinear evolution problem in nonparallel boundary layer stability was studied. The relative parabolized stability equations of nonlinear nonparallel boundary layer were derived. The developed numerical method, which is very effective, was used to study the nonlinear evolution. of T-S disturbance wave at finite amplitudes. Solving nonlinear equations of different modes by using predictor-corrector and iterative approach, which is uncoupled between modes, improving computational accuracy by using high order compact differential scheme, satisfying normalization condition I determining tables of nonlinear terms at different modes, and implementing stably the spatial marching, were included in this method. With different initial amplitudes, the nonlinear evolution of T-S wave was studied. The nonlinear nonparallel results of examples compare with data of direct numerical simulations (DNS) using full Navier-Stokes equations.

Nonlinear Stability of Supersonic Nonparallel Boundary Layer Flows

This article studies the nonlinear evolution of disturbance waves in supersonic nonparallel boundary layer flows by using nonlinear parabolic stability equations （NPSE）. An accurate numerical method is developed to solve the equations and march the NPSE in a stable manner. Through computation,are obtained the curves of amplitude and disturbance shape function of harmonic waves. Especially are demonstrated the physical characteristics of nonlinear stability of various harmonic waves,including instantaneous stream wise vortices,spanwise vortices and Λ structure etc,and are used to study and analyze the mechanism of the transition process. The calculated results have evidenced the effectiveness of the proposed NPSE method to research the nonlinear stability of the supersonic boundary layers.