Instability Of Shock Waves Research Articles

Design and Computational Analysis of an Axisymmetric Contoured Supersonic Propulsion Nozzle

The design of supersonic axisymmetric nozzles to a required exit Mach number necessitates applying the Method of Characteristics (MoC) that reduces the partial differential equations system of hyperbolic form governing such internal flow-fields to two ordinary differential equations known as the characteristic and the compatibility equations. These relationships are then transformed into finite difference equations and solved using an appropriate predictor-corrector algorithm.The present study applies the MoC to the design of a contoured nozzle that accelerates the combustion gases to supersonic velocities. The selected configuration has a throat constituted of two circular arcs with the one downstream attached to a 2nd-order polynomial profile at the attachment point. The solution, in terms of the static pressure along the centerline and the wall is obtained. It was found that the flow expands gradually without any instability or possibility of shock waves. A gradual expansion represented by a decrease in pressure is noticed immediately downstream of the throat, the flow tending to expand gradually from the throat to the exit. Along the wall, the expansion is important immediately downstream of the throat while it tends to stabilize downstream to reach the ambient pressure at the outlet. In terms of performance, the thrust coefficient developed shows the great expansion process within the divergent supersonic section. The configuration is also found to develop a great specific thrust.

Instabilities of Transonic Turbulent Flow over a Flat-Sided Wedge

The transonic turbulent two-dimensional airflow over a symmetric flat-sided double wedge is studied numerically. Solutions of the Reynolds-averaged Navier-Stokes equations are obtained with ANSYS-18.2 CFX finite-volume solver of second order accuracy on a fine mesh. The solutions demonstrate an extreme sensitivity of the flow field and lift coefficient to variation of the angle of attack α or free-stream Mach number M∞. Non-unique flow regimes and hysteresis in certain bands of α and M∞ are identified. Interaction of shock waves and local supersonic regions is discussed. The study confirms a concept of shock wave instability due to a coalescence/rupture of supersonic regions. In addition to the instability of shock wave locations, the numerical simulation shows a buffet onset, i.e., self-exciting oscillations due to instability of a boundary layer separation at the rear of wedge. Curious flow regimes with positive lift at negative angles α and, vice versa, with negative lift at positive angles α, are pointed out. A piecewise continuous dependence of the lift coefficient on two free-stream parameters, α and M∞, is discussed.

Investigation of counter-rotating shock wave phenomenon and instability mechanisms of rotating detonation engine with hollow combustor and Laval nozzle

The counter-rotating shock wave phenomenon and instability mechanisms in the hollow rotating detonation combustor with a Laval nozzle are investigated. The in-house solver BYRFoam based on the OpenFOAM platform and the detailed chemical reaction kinetic mechanism are used. The coupling of the detonation wave and the shock wave is revealed, with the continuous presence of the counter-rotating shock wave in the combustor that propagates in the opposite direction to the detonation wave and oscillates in intensity. The intensities of both the detonation wave and the counter-rotating shock wave are found to be constantly oscillating, and this instability is referred to as the collision-oscillation instability. It is caused by the complex interaction between the detonation wave, the fresh gas and the counter-rotating shock wave. The velocity difference between the detonation wave and the counter-rotating shock wave is found to lead to the migration of the collision point, which in turn leads to the peak periodic undulation phenomenon of the pressure curve, that is, the wave packet. This instability is called the counter-rotating instability, which co-exists with the collision-oscillation instability for a long time. The results of this study are compared with the experimental data, and the reasons for the oscillation of the experimental pressure signal are explained.

The stability of expanding reactive shocks in a van der Waals fluid

Despite the extensive literature accumulated since the pioneering works of D'yakov and Kontorovich in the 1950s, the stability of steady shocks is still an open question when realistic boundary conditions are accounted. The consideration of a supporting mechanism, which is indeed a necessary condition for shock steadiness, modifies the perturbation shock dynamics in the unstable range. The Noh problem is a suitable example to form steady expanding shocks. This configuration is of great interest to the high-energy-density-physics community because of its direct application to inertial confinement fusion and astrophysics, for which the stagnation of a supersonically converging material via an accretion shock front is ubiquitous. In this work, we extend the generalized Noh problem, both base-flow solution and linear stability analysis, to conditions where endothermic or exothermic transformations undergo across the shock. Within the spontaneous acoustic emission conditions found for a van der Waals gas [J. W. Bates and D. C. Montgomery, “The D'yakov-Kontorovich instability of shock waves in real gases,” Phys. Rev. Lett. 84, 1180 (2000)], we find that cylindrical and spherical expanding shocks become literally unstable for sufficiently high mode numbers. Counterintuitively, the effect of exothermicity or endothermicity across the shock is found to be stabilizing or destabilizing, respectively.

Metastable shock waves in the region of their ambiguous representation

The behavior of perturbed shocks belonging to Hugoniot segments with ambiguous representation of the shock-wave discontinuity is considered. The segments are adjacent to the region of the shock wave instability L > 1 + 2M from the side of greater and lower pressure. Here, L is the D’yakov parameter and M is the post-shock Mach number. The model equation of state is used. For the constructed Hugoniot curve, the ambiguity sections are overlapped by the shock wave neutral stability regions. The three-dimensional simulations with varying post-shock parameters and perturbation intensity are performed. It is shown that shock waves considered are metastable: for any intensity of the shock wave, there are intervals of the perturbation threshold values inside of which the shock is neutrally stable or unstable. Neutral stability is characterized by the weak damping of secondary waves occurring as result of the perturbation.

Studies on unsteady mode transition of a turbine based combined cycle (TBCC) inlet with multiple movable panels

Using the overset grid method, this paper numerically studies the unsteady mode transition from the turbojet mode to the ramjet mode of a 2D TBCC inlet with multiple movable panels, for the first time. The flow field during mode transition is highly unsteady so that the steady simulation is not appropriate. Under the baseline mode transition strategy, the instability of shock waves near the throat causes the oscillation of inlet exit performance parameters. In addition, mode transitions under different turbojet back pressure are investigated, and the oscillation occurs in most cases. In order to suppress the oscillation and increase the total pressure recovery, the Unsteady Panel to Panel Optimization Method (UPPOM) is proposed to optimize the final rotation angles of movable panels. In UPPOM, the movable panels operate in a designed motion pattern, where the mode transition process is divided into two steps, namely geometry adaption and flowpath switching. After geometry adaption, the oscillation is successfully removed, and the total pressure recovery at the turbojet flowpath exit is increased by 4.24%. In the second step of optimization, a new optimization index of Weighted Total Pressure Recovery (WTPR) is employed to increase the overall performance of TBCC inlet. In the sense of the WTPR, the performance of the optimization result is increased by 5.92%.

Interaction of metastable shock waves with perturbations

The shock wave metastable behavior in equilibrium media is detected in numerical experiments. This phenomenon is directly related to the ambiguous representation of a shock wave discontinuity when the shock belongs to the S-shaped fragment of the pressure-velocity Hugoniot. After interaction with weak perturbations the metastable shock wave recovers its original form and parameters (like stable shock). If the amplitude of the perturbations exceeds some critical value, the shock wave instability is developed. The instability is accompanied by formation of the transverse secondary waves switching local post-shock parameters between two Hugoniot segments separated from each other by the absolute instability region. The problem of the metastable shock wave interaction with local entropy perturbation passing through the shock front is studied in the framework of three-dimensional problem formulation. The shock wave transition to the limiting self-oscillating mode is described. Some features of this mode are completely determined by the shape of the Hugoniot and isentropes and does not depend on the shock wave structure. Namely, the shape of the shock front corrugations and flow structure evolution are analyzed. These features may be served as marker of the corresponding thermodynamic anomalies.

Dissipative Instability of Shock Waves

A new condition for the linear dissipative instability of the strong plane shock wave in an arbitrary medium is obtained. The instability of the shock is realized due to the flow instability behind its front, which is similar to the known dissipative instability in the boundary layer for the Tollmien-Schlichting waves. It is found that within the limit of low viscosity the one-dimensional longitudinal disturbances grow much faster than the two-dimensional corrugation ones. It points to a better correspondence to experiment of the new condition for the absolute instability of the shock in comparison with theory, which does not take viscosity into account.

A modified HLLEM scheme and shock stability analysis

Reliable numerical simulations for hypersonic flows require an accurate, robust and efficient numerical scheme. The low-dissipation shock-capturing methods often suffer various forms of shock wave instabilities when used to simulate hypersonic flow problems numerically. For the two-dimensional(2D) inviscid compressible Euler equations, the stability analysis of the low-dissipation HLLEM scheme is conducted. The odd and even perturbations are added to the initial state in the streamwise direction and the transverse direction respectively, and the evolution equations of perturbations are deduced to explore the mechanism of instability inherent in the HLLEM scheme. The results of stability analysis show that the perturbations of density and shear velocity in the flux transverse to the shock wave front are undamped. Due to the symmetry, the 2D Sedov blast wave problem is computed to prove the multidimensionality of the shock instability. In the one-dimensional case which is free from the instability, the undamped property of density perturbation is also existent but no shear velocity is found. The conclusion can be drawn as follows: the shock instability of HLLEM scheme is triggered by the perturbation growth of shear velocity in the flux transverse to the shock wave front. Based on the conclusion of stability analysis, the instability of HLLEM scheme is cured by adding the shear viscosity to the transverse flux. In order to avoid affecting the resolution of the shear layer due to the introduction of too high shear viscosity, two functions to detect the shock wave and the subsonic regimes are defined, so that the shear viscosity is only added to the transverse flux in the subsonic regime of the shock layer, while the rest of numerical fluxes are still computed by the original HLLEM scheme. The results of stability analysis and some challenging numerical test problems show that the modified HLLEM scheme not only retains the merits of the original HLLEM, such as, resolving contact discontinuity and shear wave accurately, but also has greatly improved its robustness, inhibiting the unstable phenomena from occurring effectively when computing the strong shock wave problems.

Построение гибридных численных потоков, обеспечивающих подавление развития ударно-волновой неустойчивости

Построение гибридных численных потоков, обеспечивающих подавление развития ударно-волновой неустойчивости

Corrugation Instability of Rarefaction Shock Waves and Filament Generation in Plasma Focus

Corrugation instability of rarefaction shock waves (RSWs) is studied in the simple plasma model with London current. Filament generation in the plasma focus (PF) due to the RSW corrugation instability is considered. To achieve an optimum structure of filaments in the PF current sheath, it is proposed to use special metal inserts near an insulator.

Instability of relativistic shock waves: numerical study on the basis of model equation of state

Instability of relativistic shock waves: numerical study on the basis of model equation of state

Instability of relativistic shock waves: Numerical study on the basis of model equation of state

The behavior of unstable relativistic shock waves is studied with the use of specially developed model equation of state (EOS). The EOS admits the Taub-Hugoniot adiabats with segments on which the criteria of the relativistic shock wave stability are violated. The instability segments are overlapped by the regions with ambiguous representation of the shock-wave discontinuity. The simulations are fulfilled for L < − 1 and L > (1 + 2M + v0v1)/(1 − v0v1) instability conditions, where L is relativistic analog of Dyakov parameter, M is post-shock Mach number, v0 and v1 are pre- and post-shock velocities in the shock attached reference frame. Under the condition of ambiguous representation of the shock-wave discontinuity in the former case the splitting of the unstable shock with formation of a composite compression wave with Lorentz factor dependent structure is observed. It is shown that the latter condition leads to two-dimensional non-stationary solutions characterized by presence of strong transverse waves.

Primary cause and mechanisms of structural instability of strong shock waves in gases

This paper is aimed at finding the primary cause and mechanisms of flow structure rearrangement (stable/unstable) behind the front of strong shock waves (SW) in gases. Results of experimental and calculated investigations of strong shock waves in atomic gases obtained by the authors of this work and other researchers for the last several decades are analyzed. A unique analytical approach, referred to by the authors as the source of electrons method, is used, which makes it possible to determine the relationship between the kinetic processes of ionization of particles and the energy exchange in a non-equilibrium flow (avalanche ionization region) behind a shock wave in argon in a wide range of flow conditions: M = 10–30, pressure ahead of the shock wave front p1 = 1–25 torr. In addition to the basic processes, the following additional mechanisms are considered: (a) associative ionization in interatomic collisions with participation of highly excited atoms, (b) integral radiation losses, (c) convective energy transfer, and (d) electrons heating in the polarization electric field. It is found that the primary cause of flow structure instability in shock waves is the presence of flow boundaries responsible for (i) weak gas-dynamic perturbations and (ii) local (near-wall) inhomogeneity of the medium ahead of the shock wave front and behind it. The mechanism of structural instability of the shock wave in atomic gases at low Mach numbers is caused by a cardinal change in the relationship of the basic and additional kinetic and energy exchange processes between the ionized flow components behind the front, even in the case of a weak perturbation of the atomic–ionic component temperature. In the range of high Mach numbers, the instability mechanism is due to the thermal effect induced by wall and near-wall gas layers heating ahead of the SW front by precursor radiation and also due to possible energy release initiated by polarization of the electric field near the wall behind the shock wave front.

Transonic flow hysteresis in a twin intake model

ABSTRACTTurbulent airflow in a supersonic intake model of rectangular cross-section is studied numerically. Instability of shock waves formed in the intake and ahead of the entrance is examined. Solutions of the Reynolds-averaged Navier–Stokes equations are obtained with a finite-volume solver of second-order accuracy. The expulsion and the swallowing of the shocks with a variation of free-stream parameters are studied at both subsonic and supersonic conditions prescribed at the outlet. Hysteresis in the dependence of 2D flow on the free-stream velocity and angle-of-attack is documented. An influence of 3D effects on the flow is examined.

A Shock Wave Instability Induced on a Periodically Disturbed Interface With Plasma

The shock wave instability induced when interacting with a small waviness on an interface was investigated analytically and numerically. The perturbation to the shock was phenomenologically treated assuming this as the consequence of the shock refraction. The instability develops in the form of wave-like stretchings into the lower density medium followed with the loss of stability in the flow behind it, and eventually evolving into an intense vortex structure. The instability mode is aperiodical and unconditional, and either a transition to another stable state or continuous development as a secondary flow is possible. Among other interesting features are: a similarity law in the spatial and temporal evolution of the perturbations with respect to the interface curvature; the instability locus independence of the gas density distribution thus identifying the interface conditions as the sole triggering factor; the role of the density gradient in the instability evolution discriminating between qualitatively different outcomes; and the possibility of decay via non-viscous dumping mechanisms. The phenomenological connection between the shock and the interface stability is discussed.

Transonic flow instability in the entrance region of a channel with breaks of walls

We study numerically turbulent flow of the air in a channel with breaks of walls. The flow is supersonic downstream of the break section and in the free stream. Instability of shock waves formed in the channel and in front of the entrance is examined. Solutions of the Reynolds-averaged Navier–Stokes equations are obtained with a finite-volume solver of second-order accuracy. The solutions demonstrate an expulsion/swallowing of the shock waves with variation of the free-stream Mach number. Effects of the upper wall slopes on the shock wave positions and flow bifurcation are examined.

Shock wave bifurcation in channels with a bend

The study addresses 2D and 3D turbulent transonic flows in divergent channels with a bend, where a shock wave is formed upstream of the sonic line/surface arisen over an expansion corner of the lower wall. Solutions of the Reynolds-averaged Navier–Stokes equations are obtained with a finite-volume solver of the second-order accuracy on fine meshes. Numerical simulations reveal a considerable hysteresis in the shock wave position versus the supersonic Mach number given at the inlet. A dependence of the hysteresis on the slopes of walls and length of channel is analyzed. The bifurcation band persists when unsteady perturbations are imposed at the inlet. A physical interpretation of the shock wave instability is suggested.

Unsteady transitions of separation patterns in single expansion ramp nozzle

The single expansion ramp nozzle is one of the optimal configurations for a planar rocket-based combined cycle engine because of its good integration and self-adaptability at off-design operation. The single expansion ramp nozzle is seriously overexpanded when the vehicle is at low speed, resulting in complex flow separation phenomena. Several separation patterns have been found in the single expansion ramp nozzle. Numerical simulations have shown that the transition between these separation patterns occurs in the nozzle startup and shutdown processes. However, only a few relevant experimental studies have been reported. This study reproduces the nozzle startup and shutdown processes using wind tunnel experiments. Two restricted shock separation patterns are observed in the experiment, namely, a separation bubble either forms on the ramp or the flap. The detailed flow fields in the transition processes are captured using a high-speed camera. The shock wave structures in the two separation patterns, influences of the nozzle pressure ratio (NPR) on the separation patterns and changes of the shock waves in the transition processes are discussed in detail. Shock wave instabilities accompany the separation transition, which usually takes less than 5 ms. The nozzle pressure ratios corresponding to the separation pattern transition are different in the startup and shutdown processes, which leads to a hysteresis effect.

Behavior of relativistic shock waves in nuclear matter

An equation of state of nuclear matter describing the quark–hadron phase transition is derived within a variant of the MIT bag model. This equation of state is analyzed in order to test the criteria of instability and neutral stability of relativistic shock waves. It is shown that Taub adiabats passing through the phase transition region contain segments with an ambiguous representation of the shock-wave discontinuity, which means the possibility of splitting a shock wave with the formation of a combined compression wave. Isentropes passing through the mixed phase region have bends and are not completely convex (in the p–T plane, where T is the generalized specific volume); as a result, shock and combined rarefaction waves can appear. Numerical simulation has confirmed the appearance of these singularities of wave processes in nuclear matter in the region of the phase transition from the hadronic state to the quark–gluon plasma.