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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.

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.

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.

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 eN 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.

The stability of a hypersonic boundary layer on a flared cone was analysed for the same flow conditions as in earlier experiments (Zhang et al., Acta Mech. Sinica, vol. 29, 2013, pp. 48–53; Zhu et al., AIAA J., vol. 54, 2016, pp. 3039–3049). Three instabilities in the flared region, i.e. the first mode, the second mode and the Görtler mode, were identified using linear stability theory (LST). The nonlinear-parabolized stability equations (NPSE) were used in an extensive parametric study of the interactions between the second mode and the single low-frequency mode (the Görtler mode or the first mode). The analysis shows that waves with frequencies below 30 kHz are heavily amplified. These low-frequency disturbances evolve linearly at first and then abruptly transition to parametric resonance. The parametric resonance, which is well described by Floquet theory, can be either a combination resonance (for non-zero frequencies) or a fundamental resonance (for steady waves) of the secondary instability. Moreover, the resonance depends only on the saturated state of the second mode and is insensitive to the initial low-frequency mode profiles and the streamwise curvature, so this resonance is probably observable in boundary layers over straight cones. Analysis of the kinetic energy transfer further shows that the rapid growth of the low-frequency mode is due to the action of the Reynolds stresses. The same mechanism also describes the interactions between a second-mode wave and a pair of low-frequency waves. The only difference is that the fundamental and combination resonances can coexist. Qualitative agreement with the experimental results is achieved.

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.

Boundary layer transition over a lifting body of 1.6 m length at $2^\\circ$ angle of attack has been simulated at Mach 6 and a unit Reynolds number $1.0 \\times 10^7$ m$^{-1}$. The model geometry is the same as the Hypersonic Transition Research Vehicle designed by the China Aerodynamics Research and Development Center. Four distinct transitional regions are identified, i.e. windward vortex region, shoulder vortex region, windward cross-flow region and shoulder cross-flow region. Multi-dimensional linear stability analyses by solving the two-dimensional eigenvalue problem (spatial BiGlobal approach) and the plane-marching parabolized stability equations (PSE3D approach) are further carried out to uncover the dominant instabilities in the last three regions as well as the shoulder attachment-line region. The shoulder vortex is conducive to both inner and outer modes of shear-layer instability, of which the latter most likely trigger the vortex breakdown. A novel method is presented to substantially reduce the computational cost of BiGlobal and PSE3D in resolving the cross-flow instabilities in cross-flow regions. The peak frequencies of cross-flow modes lie between 15 and 45 kHz. Whereas oblique second Mack modes are marginally unstable in the windward cross-flow region, they could be strong enough to compete with the cross-flow modes in the shoulder cross-flow region. In the shoulder attachment-line region, there exists only one unstable mode of Mack instability, differing from previous studies that show a hierarchy of modes in the context of symmetrical attachment-line flows. Results of the numerical simulation and multi-dimensional stability analyses are compared when possible, showing a fair agreement between the two approaches and highlighting the necessity of considering non-parallel effects.

Direct numerical simulations (DNS) were carried out to investigate laminar-turbulent transition for a blunt (right) cone ($7^\\circ$ half-angle) at Mach 5.9 and zero angle of attack. First, (linear) stability calculations were carried out by employing a high-order Navier–Stokes solver and using very small disturbance amplitudes in order to capture the linear disturbance development. Contrary to standard linear stability theory (LST) results, these investigations revealed a strong ‘linear’ instability in the entropy-layer region for a very short downstream distance for oblique disturbance waves with spatial growth rates far exceeding those of second-mode disturbances. This linear instability behaviour was not captured with conventional LST and/or the parabolized stability equations (PSE). Secondly, a nonlinear breakdown simulation was performed using high-fidelity DNS. The DNS results showed that linearly unstable oblique disturbance waves, when excited with large enough amplitudes, lead to a rapid breakdown and complete laminar-turbulent transition in the entropy layer just upstream of the second-mode instability region.

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 eN prediction tool for the crossflow instability in swept-wing boundary-layer flows.

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.