Mach Independence of Entropy Layer Instabilities

This study discusses entropy layer instabilities and proposes a way to scale them with the free-stream Mach number. The entropy layer that develops over a blunt body in hypersonic flow is calculated in the inviscid limit, using a high-precision spectral shock-fitting method that solves the nonlinear Euler equations within the shock layer. Linear stability theory is employed to examine instabilities linked to the generalized inflection point within the entropy layer on a blunted 30-degree wedge, serving as a model geometry. These instabilities are found to be largely independent of the Mach number when appropriately normalized. Our findings suggest that, under the assumption of a calorically perfect gas, instabilities at different Mach numbers can be related to each other.

The effect of magnetogasdynamic forces is examined on the inviscid instability associated with a generalized inflection point by considering the specific case of an entropy layer. The non-linear Euler equations are solved with a very high-order algorithm in both space and time. Two different disturbance amplitudes are considered, a low value to verify the method by comparison with linear stability analysis and a high forcing value to explore magnetogasdynamic flow control. A finite-sized electrically conducting region straddling the generalized inflection point is assumed. The effect of the magnetic field is explored through direct interaction with the transverse velocity fluctuations, as well as through modification of the mean flow profile. The parallel magnetic field is effective in inhibiting the growth of the disturbance without significantly altering the mean flow. Transverse fields, applied at very low interaction parameters, generate new inflection points and localized shear layers at the edges of the conducting region, yielding disturbance growth through gradients of ponderomotive forces and Joule heating. Decelerating forces show the most significant increase in disturbance energy growth in all quantities examined. The effect of accelerating pondermotive forces is more nuanced, with competition between the effective favorable pressure gradient and Ohmic dissipation. In this case, although the mean square disturbance kinetic energy is diminished, Mack’s energy norm which also includes mean square temperature and density fluctuation terms shows growth. Overall, the results confirm that depending on magnetic and electric field configurations, the shear layer may be stabilized or destabilized, depending on need.

This paper describes the results of an experimental investigation on the effect of the MHD interaction in the plasma of the shock layer above a test body immersed into a hypersonic Argon flow. The hypersonic flow was obtained by means of the high-enthalpy arc-heated wind tunnel of Alta SpA at Mach 6 and Mach 15. An extensive activity was carried out in order to quantitatively characterize the weakly ionized free stream, for pressures in the range between 0.5 and 1.5 bar. Two different models, with embedded high strength magnets have been utilised: a sharp cone and a truncated-cone blunt body. The models are equipped with surface pressure measurement devices and electrostatic probes. Two geometrically identical, non magnetic models were used to compare results with and without MHD interaction. This paper will describe the experimental setup, the characterization activity, and the test campaign results. A short critical analysis on the obtained results will be also present.

In hypersonic flows, the location and extent of transition is a major issue particularly for applications such as aerospace planes, re-entry vehicles, and space shuttles. Therefore, the understanding and prediction of transition at hypersonic speeds is of both fundamental and practical significance. While the transition process is a complex one involving six serially occurring stages [318, 360, 467], rate-dominated disturbance amplification plays such a significant role that prediction methods may be based solely upon the linear stability theory when transition occurs in low disturbance environments. The e n -method [468], which is based upon linear stability theory, has enjoyed phenomenal success in transition prediction both in subsonic and supersonic flows [360, 469, 470]. It therefore makes sense to study the stability of hypersonic boundary layers and see if the theory could be linked to the onset of transition.

Recently, as aerodynamics was applied to flying vehicles with very high speed and flying at high altitude, the numerical simulation based on the Navier–Stokes (NS) equations was found that cannot correctly predict certain aero-thermo-dynamic properties in a certain range of velocity and altitude while the Knudsen number indicates that the flow is still in the continuum regime. As first noted by Zhou and Zhang (Science in China, 2015), the invalidity of NS equations for such flows might be attributed to an non-equilibrium effect originating from the combined effects of gas rarefaction and strong shear in the boundary-layer flows. In this paper, we present the scope, physical concept, mathematical model of this shear non-equilibrium effect in hypersonic flows, as well as the way of considering this effect in conventional computational fluid mechanics (CFD) for engineering applications. Several hypersonic flows over sharp bodies and blunt bodies are analyzed by the proposed new continuum model, named direct simulation Monte Carlo (DSMC) data-improved Navier–Stokes (DiNS) model. Recently, as aerodynamics was applied to flying vehicles with very high speed and flying at high altitude, the numerical simulation based on the Navier–Stokes (NS) equations was found that cannot correctly predict certain aero-thermo-dynamic properties in a certain range of velocity and altitude while the Knudsen number indicates that the flow is still in the continuum regime. As first noted by Zhou and Zhang (Science in China, 2015), the invalidity of NS equations for such flows might be attributed to an non-equilibrium effect originating from the combined effects of gas rarefaction and strong shear in the boundary-layer flows. In this paper, we present the scope, physical concept, mathematical model of this shear non-equilibrium effect in hypersonic flows, as well as the way of considering this effect in conventional computational fluid mechanics (CFD) for engineering applications. Several hypersonic flows over sharp bodies and blunt bodies are analyzed by the proposed new continuum model, named direct simulation Monte Carlo (DSMC) data-improved Navier–Stokes (DiNS) model..

Generation and development of disturbances in a hypersonic viscous shock layer on a flat plate is studied both experimentally and numerically. The study is performed at the Mach number M∞ = 21 and the Reynolds number ReL = 1.44 × 105 and is aimed at elucidating the physical mechanisms that govern the receptivity and instability of the shock layer at extremely high hypersonic velocities. The experiments are conducted in a hypersonic nitrogen-driven wind tunnel. An electron-beam fluorescence technique, a Pitot probe and a piezoceramic transducer are used to measure the mean density and Mach number contours, as well as density and pressure fluctuations, their spectra and spatial distributions in the shock layer. Direct numerical simulations are performed by solving the Navier–Stokes equations with a high-order shock-capturing scheme in a computational domain including the leading and trailing edges of the plate, so that the bow shock wave and the wake behind the plate are also simulated. It is demonstrated that computational and experimental data characterizing the mean flow field, intensity of density fluctuations and their spatial distributions in the shock layer are in close agreement. It is found that excitation of the shock layer by external acoustic waves leads to generation of entropy–vortex disturbances with two maxima of density fluctuations: directly behind the shock wave and on the external edge of the boundary layer. At the same time, the pressure fluctuations decay inward into the shock layer, away from the shock, which agrees with the linear theory of interaction of shock waves with small perturbations. Thus, the entropy–vortex disturbances are shown to dominate in the hypersonic shock layer at very high Mach numbers, in contrast with the boundary layers at moderate hypersonic velocities where acoustic modes are most important. A parametric numerical study of wave processes in the shock layer induced by external acoustic waves is performed with variations of frequency, amplitude and angle of propagation of external disturbances. The amplitude of generated disturbances is observed to grow and decay periodically along the streamwise coordinate, and the characteristics of these variations depend on the frequency and direction of incident acoustic waves. The hypersonic shock layer excited by periodic blowing and suction near the leading edge is also investigated; in the experiments, this type of excitation is obtained by using an oblique-cut whistle. It is shown that blowing/suction generates disturbances resembling those generated by external acoustic waves, with similar spatial distributions and phase velocities. This result paves the way for active control of instability development in the shock layer by means of destructive interference of two types of disturbances. Numerical simulations are performed to show that instability waves can be significantly amplified or almost entirely suppressed, depending on the relative phase of blowing/suction and acoustic disturbances. Wind-tunnel experiments completely confirm this numerical prediction. Thus, the feasibility of delaying instability development in the hypersonic shock layer has been demonstrated for the first time.

This paper investigates the linear stability of reacting mixing layers with special emphasis on the existence of multiple unstable modes. The governing equations for laminar flows are from two-dimensional compressible boundary-layer equations. The chemistry is a finite rate single step irreversible reaction with Arrhenius kinetics. For the incompressible reacintg mixing layer with variable density. A necessary condition for instability has been derived. The condition requires that the angular momentum, not the vorticity, to have a maximum in the flow domain. New inflectional modes of instability are found to exist in the outer part of the mixing layer. For the compressible reacting mixing layer, supersonic unstable modes may exist in the abscence of a generalized inflection point. The outer modes at high Mach numbers in the reacting mixing layer are continuations of the inflectional modes of low Mach number flows. However, the generalized inflection point is less important at supersonic flows.

The present paper discusses the base pressure and heat transfer of various blunt bodies at hypersonic speeds. It is shown that the base to stagnation pressure ratio of a blunt body is relatively independent of the Mach and Reynolds numbers in hypersonic flow. The base to stagnation heat transfer, however, shows a dependency on the latter, as it is found to increase with Reynolds numnber. Experimental results of surface heat transfer rate around a cylinder model are presented. These data were obtained in a free piston shock tunnel with total enthalpy of 3.98 MJ/kg. The free-stream unit Reynolds number per metre and Mach number were 1.47 × 106 and 10.9 respectively. The fore-body heat transfer results show good agreement with the theories of Lees and Kemp et al. The base region heat transfer data showed a local minimum at around 130° from the stagnation point followed by a gradual recovery towards the rear stagnation region. The magnitudes of heat transfer ratio in the base region were representative of those in the base region of planetary entry type configurations.

In the fundamental configuration studied here, a steady hypersonic free stream flows over a thin sharp aligned airfoil or flat plate with a leading-edge shock wave, and the flow field in the shock layer (containing a viscous and an inviscid layer) is steady laminar and two-dimensional, for a perfect gas without real and high-temperature gas effects. The viscous and inviscid layers are analysed and computed simultaneously in the region from the leading edge to the trailing edge, including the upstream-influence effect present, to determine the interactive flow throughout the shock layer and the positions of the shock wave and the boundary-layer edge, where matching is required. Further theoretical analysis of the shock layer helps to explain the computational results, including the nonlinear breakdown possible when forward marching against enhanced upstream influence, for example as the wall enthalpy increases towards its insulated value. Then the viscous layer is computed by sweeping methods, for higher values of wall enthalpies, to prevent this nonlinear breakdown for airfoils including the flat plate. Thin airfoils in hypersonic viscous flow are treated, for higher values of the wall enthalpies and with the upstream-influence effect, as are hypersonic inviscid flows, by modifying the computational methods used for the flat plate. Also, the behaviour of the upstream influence for bodies of relatively large thickness, and under wall velocity slip and enthalpy jump for flat plates, is discussed briefly from a theoretical point of view.Subsequent to the present work, computations based on the Navier–Stokes and on the parabolized Navier–Stokes equations have yielded excellent and good agreement respectively with the present predictions for large Mach and Reynolds numbers.

Corresponding to Oswatitsch’s Mach number independence principle the Mach number of hypersonic inviscid flows, , does not affect components of various non-dimensional formulations such as velocity and density, pressure coefficients and Mach number behind a strong shock. On this account, the principle is significant in the development process for hypersonic vehicles. Oswatitsch deduced a system of partial differential equations which describes hypersonic flow. These equations are the basic gasdynamic equations as well as Crocco’s theorem which are reduced for the case of very high Mach numbers, . Their numerical solution can not only result in simplified algorithms prospectively utilized to describe the flow around bodies flying mainly in the lower stratosphere with very high Mach numbers. It also offers a deeper understanding of similarity effects for hypersonic flows. In this paper, a solution method for Oswatisch’s equations for perfect gas, based on a 4-step Runge-Kutta-algorithm, is presented including a fast shock-fitting procedure. An analysis of numerical stability is followed by a detailed comparison of results for different Mach numbers and ratios of the specific heats.

The receptivity phenomenon, which is the process of environmental disturbances initially entering the boundary layers and generating disturbance waves, is one of the important but not well understood mechanisms involving laminar–turbulent transition of hypersonic flows. This paper presents a numerical simulation study of the receptivity to weak free-stream fast acoustic waves for a Mach 7.99 axisymmetric flow over a $7^\circ$ half-angle blunt cone. In hypersonic boundary-layer flow over a blunt cone, the process of receptivity to free-stream disturbances is altered considerably by the presence of a bow shock and an entropy layer. In the present study, both steady and unsteady flow solutions are obtained by computing the full Navier–Stokes equations with a fifth-order shock-fitting finite-difference scheme, which is able to account for the effects of bow-shock/free-stream-disturbance interaction accurately. The current numerical results for the steady base flow are compared with previous experimental and numerical results. In addition, a normal-mode linear stability analysis is used to identify the main components of boundary-layer disturbances generated by forcing free-stream fast acoustic waves. It is found that neither the first mode nor the second-mode instability waves are excited by free-stream fast acoustic waves in the early region along the cone surface, although the Mack modes can be unstable there. Instead, the second mode is excited downstream of the second-mode Branch I neutral stability point. The delay of the second-mode excitation is because the hypersonic boundary-layer receptivity is governed by a two-step resonant interaction process: (i) resonant interactions between the forcing waves and a stable boundary-layer wave mode I near the leading-edge region; and (ii) resonant interactions between the induced stable mode I and the unstable second Mack mode downstream. The same receptivity mechanism also explains the results that no first Mack mode components are generated by the current receptivity process because there is no resonant interaction between fast acoustic waves and the first Mack mode.

An experimental study of the mean wake flow field and its stability has been carried out in the far wake of circular cylinders at a Mach number of 6. The Reynolds numbers ranged from 200 to 4000 with a few measurements at higher Reynolds number. Pitot pressure, static pressure and mean flow hot wire measurements were done at many axial stations behind cylinders up to (x/d) = 2400. The inner wake formed from the cylinder boundary layers is laminar and loses its identity within the first 60 diameters or less depending on the Reynolds numbers so that only the outer wake, caused by the bow shock, has to be considered. Within a certain region the experimental results compare well with linear laminar theory, but the wake profiles are not similar up to the farthest downstream station (x/d = 2400). At four Reynolds numbers strong deviations from steady laminar behavior were observed far behind the cylinder, indicating breakdown of the flow because of non-linear instability effects. In the instability study hot wire fluctuation measurements were made over the whole frequency range (f = 1 - 320 KC) and also at particular frequencies in a band width of 1 KC up to x/d = 12000 at the lowest Reynolds number. Two instability regions were found and investigated: the linear growth region and the non-linear region. In the linear region there is quite a close correspondence with linear stability theory. The onset of non-linearity is characterized by the simultaneous strong deviation of the mean flow from laminar steady behavior, the increase of the fundamental frequency fluctuation component on the wake axis and the sudden rise of the first harmonic frequency component. The non-linear region is compared with the non-linear wake region at low speeds behind a flat plate. On the basis of these measurements a tentative picture is given of the onset of non-linearity and/or transition in the inner and outer wake behind blunt bodies at hypersonic speeds.

An extraneous shock impinging on a blunt body in hypersonic flow is observed to alter the flow around the body and increase the local heat-transfer rate near the impingement point. A novel, quasi-static technique is developed to study this phenomenon. A hemispherical, glass model, equipped with platinum thin-film thermometers, is injected into a hypersonic tunnel through a slot in a variable-incidence flat plate. Analog networks provide graphs of the heat-transfer rates at various points on the model as a function of the model's position relative to the extraneous shock. Peaks in local heat-transfer rates up to 10 times the local, unperturbed, freestream values are recorded as the model traverses the shock. The peak heating is severest on the side of the model nearest the plate and increases with increasing shock strength, occurring over a narrow region where a shear layer or jet, originating at the intersection of the bow shock and the impinging shock, meets the model surface. A physical model is set up which predicts variations in shock interference patterns, in surface pressure distributions, and in the intensity and extent of the peak heating in accordance with experiment. Nomenclature M = Mach number P = pressure normalized with respect to freestream pressure q = heat-transfer rate, cal-cm~2-sec~1 r = radius of hemisphere, mm t = time, sec T = temperature, °C U = velocity normalized with respect to freestream velocity V = injection speed, m-sec1 x = distance along surface measured from stagnation point, mm

A shock tunnel is used to perform tests at hypersonic flow conditions including weak real gas effects. Pressure and heat flux distributions are measured around typical re-entry configurations. From these data cp values and Stanton numbers are deduced. For constant Mach and Reynolds number the experimental results achieved indicate a strong influence of the total temperature on the Stanton number distribution. For the results presented this behavior is mainly based on entropy layer and viscous interaction effects. A correlation function which takes into account these effects correlates the Stanton numbers achieved for different flow conditions and in different wind tunnels fairly well.

Effect of leading edge bluntness on the interaction of ramp induced shock wave with laminar boundary layer at hypersonic speed