Floquet analysis of fundamental,subharmonic and detuned secondary instabilities of Grtler vortices

Nonlinear parabolized stability equations are employed in this work to investigate the nonlinear development of the Grtler instability up to the saturation stage. The perturbed boundary layer is highly inflectional both in the normalwise and spanwise directions and receptive to the secondary instabilities. The Floquet theory is applied to solve the fundamental,subharmonic and detuned secondary instabilities. With the Grtler-vortices-distorted base flow,two classes of secondary disturbances,i.e. odd modes and even modes,are identified according to the eigenfunctions of the disturbances. These modes may result in diferent patterns in the late stages of the transition process. Li and Malik [1] have shown the sinuous and varicose types of breakdown originating from the odd and even modes. The current study focuses on the four most amplified modes termed the even modes Ⅰ Ⅱ and odd modes Ⅰ Ⅱ. Odd mode II was missing in the work of Li and Malik [1] probably due to their inviscid simplification. The detuned modes are confirmed to be less amplified than the fundamental(for the odd mode Ⅰ) and subharmonic modes(for even modes Ⅰ Ⅱ and the odd mode Ⅱ).     

Modelling flow transition in a hypersonic boundary layer with Reynolds-averaged Navier-Stokes approach

Based on Reynolds-averaged Navier-Stokes approach,a laminar-turbulence transition model is proposed in this study that takes into account the effects of different instability modes associated with the variations in Mach numbers of compressible boundary layer flows.The model is based on k-ω-γ three-equation eddy-viscosity concept with k representing the fluctuating kinetic energy,ωthe specific dissipation rate and the intermittency factorγ.The particular features of the model are that:1)k includes the non-tu...     

Experimental study on supersonic film cooling on the surface of a blunt body in hypersonic flow

The experimental study focuses on the heat flux on a double cone blunt body in the presence of tangential-slot super- sonic injection into hypersonic flow. The tests are conducted in a contoured axisymmetric nozzle with Mach numbers of 7.3 and 8.1, and the total temperature is about 900 K. The injection Mach number is 3.2, and total temperature is 300 K. A constant voltage circuit is developed to supply the temperature detectors instead of the normally used constant current circuit. The schlieren photographs are presented additionally to visualize the flow and help analyze the pressure relationship between the cooling flow and the main flow. The dependence of the film-cooling effectiveness on flow parameters, i.e. the blow ratio, the convective Mach number, and the attack angle, is determined. A semi-empirical formula is tested by the present data, and is improved for a better correlation.     

Interactions of a Projectile Charge with Two-Dimensional Dusty Plasmas

The interactions of a moving charge (namely, one additional dust particle) with a two-dimensional dusty plasma in gas discharge experiment are studied by means of the linearized hydrodynamic theory for the dusty plasma. Expressions are derived for the induced potential and the stopping power of the moving charge, when the charge flights parallel to and over the dust layer. The numerical results are obtained for different discharge pressures and different distances from the moving charge to the dust layer. The results show that the moving charge excites a V-shaped disturbance of induced potential or the so-called Mach cone in the dust layer, while the charge itself loses its energy.     

Numerical simulation and analysis of complex patterns in a two-layer coupled reaction diffusion system

The resonance interaction between two modes is investigated using a two-layer coupled Brusselator model. When two different wavelength modes satisfy resonance conditions, new modes will appear, and a variety of superlattice patterns can be obtained in a short wavelength mode subsystem. We find that even though the wavenumbers of two Turing modes are fixed, the parameter changes have influences on wave intensity and pattern selection. When a hexagon pattern occurs in the short wavelength mode layer and a stripe pattern appears in the long wavelength mode layer, the Hopf instability may happen in a nonlinearly coupled model, and twinkling-eye hexagon and travelling hexagon patterns will be obtained. The symmetries of patterns resulting from the coupled modes may be different from those of their parents, such as the cluster hexagon pattern and square pattern. With the increase of perturbation and coupling intensity, the nonlinear system will convert between a static pattern and a dynamic pattern when the Turing instability and Hopf instability happen in the nonlinear system. Besides the wavenumber ratio and intensity ratio of the two different wavelength Turing modes, perturbation and coupling intensity play an important role in the pattern formation and selection. According to the simulation results, we find that two modes with different symmetries can also be in the spatial resonance under certain conditions, and complex patterns appear in the two-layer coupled reaction diffusion systems.     

Application of Arnoldi method to boundary layer instability

The Arnoldi method is applied to boundary layer instability, and a finite difference method is employed to avoid the limit of the finite element method. This modus operandi is verified by three comparison cases, i.e., comparison with linear stability theory(LST) for two-dimensional(2D) disturbance on one-dimensional(1D) basic flow, comparison with LST for three-dimensional(3D) disturbance on 1D basic flow, and comparison with Floquet theory for 3D disturbance on 2D basic flow. Then it is applied to secondary instability analysis on the streaky boundary layer under spanwise-localized free-stream turbulence(FST). Three unstable modes are found, i.e., an inner mode at a high-speed center streak, a sinuous type outer mode at a low-speed center streak, and a sinuous type outer mode at low-speed side streaks. All these modes are much more unstable than Tollmien–Schlichting(TS) waves, implying the dominant contribution of secondary instability in bypass transition. The modes at strong center streak are more unstable than those at weak side streaks, so the center streak is ‘dangerous' in secondary instability.     

Numerical Investigation on Inviscid Instability of Streaky Structures in Incompressible Boundary Layer Flow

Numerical investigation is made on the effect of streaky structures in transition by inviscid linear disturbance equation with temporal mode. Several disturbances with different streamwise wave numbers were induced, and the evolutions with time step were received. It suggests that the exponential growth and periodic variation of the waves are in existence. As the streamwise wave number increases, the disturbance growth rate begins by increasing, reaches a maximum at around α=0.4 with a disturbance frequency of 0.2186 ＋ 0.001457i, and then decreases. Furthermore, the eigenfunctions of pressure disturbance are plotted.     

WG-Like Modes Selectivity in a Square Cavity with Posts

We investigate the characteristics of Whispering-Gallery（WG）-like modes in a square cavity with posts by employing the two-dimensional （2D） finite-difference time-domain （FDTD） technique combined with the effective index method. The results indicate that the posts can result in mode selection in the WG-like modes. The WG-like modes with odd mode numbers are not much sensitive to the sizes of the posts. However, the quality factor （i.e. Q-factor） of the WG-like modes with even mode numbers decreases sharply with the increasing size of the posts. The decreasing Q-factor is attributed to mode leakage and scattering loss due to the presence of the post. The mode selection increases the mode spacing of square cavity twice in an optimized structure.     

The quantum pressure correction to the excitation spectrum of the trapped superfluid Fermi gases in a BEC-BCS crossover

Using quantum hydrodynamic approaches, we study the quantum pressure correction to the collective excitation spectrum of the interacting trapped superfluid Fermi gases in the BEC-BCS crossover. Based on a phenomenological equation of state, we derive hydrodynamic equations of the system in the whole BEC-BCS crossover regime. Beyond the Thomas--Fermi approximation, expressions of the frequency corrections of collective modes for both spherical and axial symmetric traps excited in the BEC-BCS crossover are given explicitly. The corrections of the eigenfrequencies due to the quantum pressure and their dependence on the inverse interaction strength, anisotropic parameter and particle numbers of the condensate are discussed in detail.     

Heat transfer for boundary layers with cross flow

An analysis is presented to study the dual nature of solutions for the forced convective boundary layer flow and heat transfer in a cross flow with viscous dissipation terms in the energy equation. The governing equations are transformed into a set of three self-similar ordinary differential equations by similarity transformations. These equations are solved numerically using the very efficient shooting method. This study reveals that the dual solutions of the transformed similarity equations for velocity and temperature distributions exist for certain values of the moving parameter, Prandtl number, and Eckert numbers. The reverse heat flux is observed for larger Eckert numbers; that is, heat absorption at the wall occurs.     

Characteristics of nonlinear evolution of wavepackets in boundary layers

The nonlinear evolution of a finite-amplitude disturbance in a 3-D supersonic boundary layer over a cone was investigated recently by Liu et al.using direct numerical simulation(DNS).It was found that certain small-scale 3-D disturbances amplified rapidly.These disturbances exhibit the characteristics of second modes,and the most amplified components have a welldefined spanwise wavelength,indicating a clear selectivity of the amplification.In the case of a cone,the three-dimensionality of the base flow and the disturbances themselves may be responsible for the rapid amplification.In order to ascertain which of these two effects are essential,in this study we carried out DNS of the nonlinear evolution of a spanwise localized disturbance(wavepacket) in a flat-plate boundary layer.A similar amplification of small-scale disturbances was observed,suggesting that the direct reason for the rapid amplification is the three-dimensionality of the disturbances rather than the three-dimensional nature of the base flow,even though the latter does alter the spanwise distribution of the disturbance.The rapid growth of 3-D waves may be attributed to the secondary instability mechanism.Further simulations were performed for a wavepacket of first modes in a supersonic boundary layer and of Tollmien-Schlichting(T-S) waves in an incompressible boundary layer.The results show that the amplifying components are in the band centered at zero spanwise wavenumber rather than at a finite spanwise wavenumber.It is therefore concluded that the rapid growth of 3-D disturbances in a band centered at a preferred large spanwise wavenumber is the main characteristic of nonlinear evolution of second mode disturbances in supersonic boundary layers.     

The investigation of coherent structures in the wall region of a supersonic turbulent boundary layer based on DNS database

Through temporal mode direct numerical simulation, flow field database of a fully developed turbulent boundary layer on a flat plate with Mach number 4.5 and Reynolds number Reθ =1094 has been obtained. Commonly used detection meth- ods in experiments are applied to detecting coherent structures in the flow field, and it is found that coherent structures do exist in the wall region of a supersonic turbulent boundary layer. The detected results show that a low-speed streak is de- tected by using the Mu-level method, the rising parts of this streak are detected by using the second quadrant method, and the crossing regions from a low-speed streak to the high-speed one are detected by using the VITA method respectively. Notwithstanding that different regions are detected by different methods, they are all accompanied by quasi-stream-wise vortex structures.     

Guided modes in asymmetric metal-cladding left-handed material waveguides

We investigate guided modes in the asymmetric waveguide structure with a left-handed material (LHM) layer surrounded by air and metal. A graphical method is proposed to determine the guided modes. New properties of the oscillating and surface guided modes, such as absence of the fundamental mode, coexistence of the oscillating and surface guided modes, fast attenuation of the surface guided modes, and mode degeneracy, are analyzed in detail. We also investigate dispersive characteristics of the metal-LHM- air optical waveguide. The propagation constant increases with decreasing slab thickness for the first-order oscillating mode, which is different from that in traditional metal-cladding waveguides.     

Acoustic Rotation Modes in Complex Plasmas

Acoustic rotation modes in complex plasmas are investigated in a cylindrical system with an axial symmetry. The linear mode solution is derived. The mode in an infinite area is reduced to a classical dust acoustic wave in the region away from the centre. When the dusty plasma is confined in a finite region, the breathing and rotating-void behaviour are observed. Vivid structures of different mode number solutions are illustrated.     

Direct Numerical Simulation of Supersonic Turbulent Boundary Layer Flow

Direct numerical simulations of a spatially evolving supersonic flat-plate turbulent boundary layer flow with free Mach number M∞ = 2.25 and Reynolds number Re = 365000/in are performed. The transition process from laminar to turbulent flow is obtained by solving the three-dimensional compressible Navier-Stokes equations, using high-order accurate difference schemes. The obtained statistical results agree well with the experimental and theoretical data. From the numerical results it can be seen that the transition process under the considered conditions is the process which skips the Tollmien-Schlichting instability and the second instability through the instability of high gradient shear layer and becomes of laminar flow breakdown. This means that the transition process is a bypass-type transition process. The spanwise asymmetry of the disturbance locally upstream imposed is important to induce the bypass-type transition. Furthermore, with increasing the time disturbance frequency the transition will delay. When the time disturbance frequency is large enough, the transition will disappear.     

Flow characteristics of hypersonic inlets with different cowl-lip blunting methods

Under hypersonic flight conditions,the sharp cowl-lip leading edges have to be blunted because of the severe aerodynamic heating.This paper proposes four cowl-lip blunting methods and studies the corresponding flow characteristics and performances of the generic hypersonic inlets by numerical simulation under the design conditions of a flight Mach number of 6 and an altitude of 26 km.The results show that the local shock interference patterns in the vicinity of the blunted cowl-lips have a substantial influence on the flow characteristics of the hypersonic inlets even though the blunting radius is very small,which contribute to a pronounced degradation of the inlet performance.The Equal Length blunting Manner(ELM)is the most optimal in that a nearly even reflection of the ramp shock produces an approximately straight and weak cowl reflection shock.The minimal total pressure loss,the lowest cowl drag,maximum mass-capture and the minimal aeroheating are achieved for the hypersonic inlet.For the other blunting manners,the ramp shock cannot reflect evenly and produces more curved cowl reflection shock.The Type V shock interference pattern occurs for the Cross Section Cutting blunting Manner(CSCM)and the strongest cowl reflection shock gives rise to the largest flow loss and drag.The cowl-lip blunted by the other two blunting manners is subjected to the shock interference pattern that transits with an increase in the blunting radius.Accordingly,the peak heat flux does not fall monotonously with the blunting radius increasing.Moreover,the cowl-lip surface suffers from severe aerothermal load when the shear layer or the supersonic jet impinges on the wall.     

TM-like and TE-like Modes Coupling in a Two-Dimensional Photonic Crystal Slab Composed of Truncated Cone Silicon Rods

We numerically investigate the coupling of TE-like modes and TM-like modes in a two-dimensional （2D） photonic crystal （PC） slab composed of truncated cone silicon rods. In such structures, the classification of TE-like modes and TM-like modes is generally impossible and the coupling occurs due to vertical structural asymmetries. The frequency and wavevector dependences of the mode coupling are discussed by investigating the photonic band structures, and the coupling efficiency is studied by examining the transmittance. The results show that the efficiency of mode conversion is strengthened by the vertical asymmetry and weakened by the clear sinall gap. These structures could be used as polarization conversion devices in integrated optics.     

Generation of acoustic waves in the hypersonic boundary layer over a wavy wall

The effects of a wavy wall on a hypersonic boundary layer of a flared cone are investigated using experimental measurements and direct numerical simulations(DNSs). Non-contact optical measurements using a focused laser differential interferometer(FLDI) show that a wavy wall can significantly suppress the second mode, and multiple perturbations of new frequencies are generated over the wavy surface, which agrees well with numerical results. Using Lagrangian tracking of marked particles, it is demonstrated that the wavy wall geometry can induce mean flow oscillations while exciting acoustic waves. The frequencies of the excited disturbances over a wavy wall agree with the classical Rossiter model. The superposition of a disturbance propagating downstream and an acoustic wave propagating upstream at the same frequency but with different amplitudes and propagation velocities results in a spatial distribution with a streamwise-oscillatory pattern over the wavy surface. A simple two-wave superposition model that takes into account the phase velocities and wavenumbers of the convective disturbance and acoustic wave can well describe the modal behavior of excited disturbances over a wavy wall.     

Finite-Difference Lattice Boltzmann Scheme for High-Speed Compressible Flow： Two-Dimensional Case

Lattice Boltzmann （LB） modeling of high-speed compressible flows has long been attempted by various authors. One common weakness of most of previous models is the instability problem when the Mach number of the flow is large. In this paper we present a finite-difference LB model, which works for flows with flexible ratios of specific heats and a wide range of Mach number, from 0 to 30 or higher. Besides the discrete-velocity-model by Watari [Physica A 382 （2007） 502], a modified Lax Wendroff finite difference scheme and an artificial viscosity are introduced. The combination of the finite-difference scheme and the adding of artificial viscosity must find a balance of numerical stability versus accuracy. The proposed model is validated by recovering results of some well-known benchmark tests： shock tubes and shock reflections. The new model may be used to track shock waves and/or to study the non-equilibrium procedure in the transition between the regular and Mach reflections of shock waves, etc.     

The effect of the Taylor-Grtler vortex on Reynolds stress transport in the rotating turbulent channel flow

We investigates the effect of Taylor-Grtler vortex on the Reynolds stress transport in the rotating turbulent channel flow by direct numerical simulation. The Taylor-Grtler vortex is detected by longitudinal average of velocity fluctuation in the channel and defined as TG fluctuation. It has been found that turbulent diffusion is significant in the Reynolds stress transportation at the suction side of rotating turbulent channel in contrast with the turbulent channel flow without rotation and Taylor-Grtler vortex plays an important role in the turbulent diffusion in Reynolds stress transport. The paper focuses on the low and moderate rotation number, but the effect of the rotation number on the Reynolds stress transport is also reported.