Inviscid Shock Research Articles

Measurements in the annular shear layer of high subsonic and under-expanded round jets

An experimental study has been undertaken to document compressibility effects in the annular shear layers of axisymmetric jets. Comparison is made of the measured flow development with the well-documented influence of compressibility in planar mixing layers. High Reynolds number (~106) and high Mach number jets issuing from a convergent nozzle at nozzle pressure ratios (NPRs) from 1.28 to 3.0 were measured using laser Doppler anemometry instrumentation. Detailed radial profile data are reported, particularly within the potential core region, for mean velocity, turbulence rms, and turbulence shear stress. For supercritical NPRs the presence of the pressure waves in the inviscid shock cell region as the jet expanded back to ambient pressure was found to exert a noticeable effect on shear layer location, causing this to shift radially outwards at high supercritical NPR conditions. After a boundary layer to free shear layer transition zone, the turbulence development displayed a short region of similarity before adjustment to near-field merged jet behaviour. Peak turbulence rms reduction due to compressibility was similar to that observed in planar layers with radial rms suppression much stronger than axial. Comparison of the compressibility-modified annular shear layer growth rate with planar shear layer data on the basis of the convective Mach number (M C) showed notable differences; in the annular shear layer, compressibility effects began at lower M C and displayed a stronger reduction in growth. For high Mach number aerospace propulsion applications involving round jets, the current measurements represent a new data set for the calibration/validation of compressibility-affected turbulence models.

Free Surface Axially Symmetric Flows and Radial Hydraulic Jumps

Free surface, axially symmetric shallow flow is analysed in both the centrifugal and centripetal directions. Referring to the inviscid steady flow over a flat plate characterised by a unique value of specific energy, the analytical sub- and supercritical solutions are determined. Furthermore, the sub- and supercritical steady solutions for the flow with friction over a flat plate are determined, provided that inertial terms are important compared with frictional terms. As the inviscid case, the sub- and supercritical solutions over a bottom topography are determined, provided that the bottom unevenness is compatible with a continuous solution. In the fundamental case of inviscid flow over a flat plate, the discontinuous solution is also analysed for a direct hydraulic jump imposed by proper boundary conditions. In the simple scheme of an inviscid shock of zero length, the jump position and the sequent depths are analytically derived, thus indicating that all results are uniquely functions of one dimensionless number, i. e. the specific energy ratio between the sub- and supercritical flows.

Convex Entropy, Hopf Bifurcation, and Viscous and Inviscid Shock Stability

We consider by a combination of analytical and numerical techniques, some basic questions regarding the relations between inviscid and viscous stability and the existence of a convex entropy. Specifically, for a system possessing a convex entropy, in particular for the equations of gas dynamics with a convex equation of state, we ask: (1) can inviscid instability occur? (2) can viscous instability not detected by inviscid theory occur? (3) can there occur the—necessarily viscous—effect of Hopf bifurcation, or “galloping instability”? and, perhaps most important from a practical point of view, (4) as shock amplitude is increased from the (stable) weak-amplitude limit, can there occur a first transition from viscous stability to instability that is not detected by inviscid theory? We show that (1) does occur for strictly hyperbolic, genuinely nonlinear gas dynamics with certain convex equations of state, while (2) and (3) do occur for an artifically constructed system with convex viscosity-compatible entropy. We do not know of an example for which (4) occurs, leaving this as a key open question in viscous shock theory, related to the principal eigenvalue property of Sturm Liouville and related operators. In analogy with, and partly proceeding close to, the analysis of Smith on (non-)uniqueness of the Riemann problem, we obtain convenient criteria for shock (in)stability in the form of necessary and sufficient conditions on the equation of state.

Numerical assessment of a shock‐detecting sensor for low dissipative high‐order simulation of shock–vortex interactions

SUMMARYConsidering the importance of high‐order schemes implementation for the simulation of shock‐containing turbulent flows, the present work involves the assessment of a shock‐detecting sensor for filtering of high‐order compact finite‐difference schemes for simulation of this type of flows. To accomplish this, a sensor that controls the amount of numerical dissipation is applied to a sixth‐order compact scheme as well as a fourth‐order two‐register Runge–Kutta method for numerical simulation of various cases including inviscid and viscous shock–vortex and shock–mixing‐layer interactions. Detailed study is performed to investigate the performance of the sensor, that is, the effect of control parameters employed in the sensor are investigated in the long‐time integration. In addition, the effects of nonlinear weighting factors controlling the value of the second‐order and high‐order filters in fine and coarse non‐uniform grids are investigated. The results indicate the accuracy of the nonlinear filter along with the promising performance of the shock‐detecting sensor, which would pave the way for future simulations of turbulent flows containing shocks. Copyright © 2014 John Wiley & Sons, Ltd.

Underexpanded Jet Development from a Rectangular Nozzle with Aft-Deck

An experimental study is reported of underexpanded supersonic jet plumes issuing from a high-aspect-ratio convergent rectangular nozzle. Schlieren visualization, Pitot probe, and Laser Doppler Anemometry measurements are carried out to capture the plume development in the near field, and in particular the effect on the plume flow of a finite-length extended shelf or aft-deck attached to the lower nozzle wall. This creates asymmetry in the inviscid shock cell pattern and the entrainment characteristics, both of which influence shear-layer growth and plume trajectory. A net pressure force is induced on the aft-deck wall, which leads to transverse deflection of the jet plume once it leaves the aft-deck, both upward and downward, depending on aft-deck length and nozzle pressure ratio. For sufficiently high nozzle pressure ratio and a sufficiently long aft-deck, separation and reattachment of the plume from the aft-deck is observed. Detailed mapping of both mean velocity and turbulence in the plume near field has been carried out, enabling comparison of flow behavior for a clean nozzle and a nozzle with aft-deck. The data provided are proposed as a suitable benchmark validation test case for computational fluid dynamics studies of rectangular nozzle plumes with aft-deck interaction effects.

On Type VI–V transition in hypersonic double-wedge flows with thermo-chemical non-equilibrium effects

The transition from Type VI to V of inviscid shock interactions on a double-wedge geometry is investigated theoretically and numerically for hypersonic non-equilibrium gas flows. The shock polar method valid for the non-equilibrium gas is developed by introducing a non-equilibrium relaxation length. A preliminary rule for choosing the suitable non-equilibrium relaxation length in different regions of the double-wedge flow is proposed by analyzing the relaxing characteristics. Following this proposal, the shock polar method valid for the non-equilibrium gas can be easily applied to other hypersonic shock interactions. Numerical simulation is used to capture the complete transition from Type VI to V and to obtain the second wedge angle span for this process. The comparison between the theory and the computation indicates that the shock polar method for the non-equilibrium gas can well predict the regional pressure and the critical angle. It is found that the non-equilibrium gas effects lead to a larger second wedge angle for the shock interaction transition and these effects become more significant with a larger Mach number of the inflow. It is interesting that the variation of the transition wedge angle with the Mach number of inflow behaves oppositely for the calorically perfect gas and the non-equilibrium gas.

Periodic mass loss from viscous accretion flows around black holes

We investigate the behaviour of low angular momentum viscous accretion flows around black holes using Smooth Particle Hydrodynamics (SPH) method. Earlier, it has been observed that in a significant part of the energy and angular momentum parameter space, rotating transonic accretion flow undergoes shock transition before entering in to the black hole and a part of the post-shock matter is ejected as bipolar outflows, which are supposed to be the precursor of relativistic jets. In this work, we simulate accretion flows having injection parameters from the inviscid shock parameter space, and study the response of viscosity on them. With the increase of viscosity, shock becomes time dependent and starts to oscillate when the viscosity parameter crosses its critical value. As a result, the in falling matter inside the post-shock region exhibits quasi-periodic variations and causes periodic ejection of matter from the inner disc as outflows. In addition, the same hot and dense post-shock matter emits high energy radiation and the emanating photon flux also modulates quasi-periodically. Assuming a ten solar mass black hole, the corresponding power density spectrum peaks at the fundamental frequency of few Hz followed by multiple harmonics. This feature is very common in several outbursting black hole candidates. We discuss the implications of such periodic variations.

Cauchy–Gelfand problem for quasilinear conservation law

We obtain the precise asymptotic (t→∞) for solution f(x,t) of Cauchy–Gelfand problem for quasilinear conservation law ∂f∂t+φ(f)∂f∂x=0 with initial data of bounded variation f(x,0)=f0(x). The main theorem develops results of Liu (1981) [22], Kruzhkov, Petrosjan (1987) [20], Henkin, Shananin (2004) [10], Henkin (2012) [9]. Proofs are based on vanishing viscosity estimates and localized Maxwell type conservation laws. The main application consists in the reconstruction of parameters of initial data responsible for location of inviscid shock waves in the solution f(x,t).

Three-Dimensional Shock Wave Detection Based on the Theory of Characteristics

This paper describes a method to detect three-dimensional, steady inviscid shock waves from computational fluid dynamics data. The method is based on the principle that the collision of the characteristics of the same family causes the generation of a shock wave in a two-dimensional flow. In a three-dimensional flowfield, however, there is an infinite number of characteristics. Therefore, the plane on which the streamline can be treated locally as a planar curve is introduced, and the characteristic as an intersection between the Mach cone and this plane is defined. As a result, the shock waves in a three-dimensional, steady flowfield can be treated by a similar methodology as that in a two-dimensional flowfield.

Shock waves in dense hard disk fluids

Media composed of colliding hard disks (2D) or hard spheres (3D) serve as good approximations for the collective hydrodynamic description of gases, liquids and granular media. In the present study, the compressible hydrodynamics and shock dynamics are studied for a two-dimensional hard-disk medium at both the continuum and discrete particle level descriptions. For the continuum description, closed form analytical expressions for the inviscid hydrodynamic description, shock Hugoniot, isentropic exponent and shock jump conditions were obtained using the Helfand equation of state. The closed-form analytical solutions permitted us to gain physical insight into the role of the material’s density on its compressibility, i.e. how the medium compresses under mechanical loadings and sustains wave motion. Furthermore, the predictions were found in excellent agreement with calculations using the event driven molecular dynamics method involving 30,000 particles over the entire range of compressibility spanning the dilute ideal gas and liquid phases. In all cases, it was found that the energy imparted by the piston motion to the thermalized medium behind the propagating shock was quasi-independent of the medium’s packing fraction, with a correction vanishing with increasing shock Mach numbers.

Weakly nonlinear simple waves in Hertzian chains

The discrete system of equations for a granular chain consisting of a large number of spheres interacting via the hertz force is cast as an effective medium. In the long wavelength limit, the second order equation of motion for the effective medium possess a subset of simple waves obeying a first order equation of reduced nonlinear index. Simple waves are those in which knowledge of one dependent variable determines all the rest. For a given initial strain, the simple wave solution prescribes initial mass velocity. Strain and velocity profiles from the first order equation are used as initial conditions in simulations for the second order discrete system. Results for viscous and inviscid shock formation compare very well between the second order system and the reduced first order equation. Second order simulation of colliding waves reveals the ability of waves to pass through each other, with a phase advance accruing during the collision process. Results may be related to explosions in granular media. [Work supported by the Office of Naval Research.]

An approach to solving the boundary free edge difficulties in SPH modelling: application to a viscous accretion disc in close binaries

In this work, we propose a SPH interpolating Kernel reformulation suitable also to treat free edge boundaries in the computational domain. Application to both inviscid and viscous stationary low compressibility accretion disc models in Close Binaries (CB) are shown. The investigation carried out in this paper is a consequence of the fact that a low compressibility modelling is crucial to check numerical reliability. Results show that physical viscosity supports a well-bound accretion disc formation, despite the low gas compressibility, when a Gaussian-derived Kernel (from the Error Function) is assumed, in extended particle range - whose Half Width at Half Maximum (HWHM) is fixed to a constant $h$ value - without any spatial restrictions on its radial interaction (hereinafter GASPHER). At the same time, GASPHER ensures adequate particle interpolations at the boundary free edges. Both SPH and adaptive SPH (hereinafter ASPH) methods lack accuracy if there are not constraints on the boundary conditions, in particular at the edge of the particle envelope: Free Edge (FE) conditions. In SPH, an inefficient particle interpolation involves a few neighbour particles; instead, in the second case, non-physical effects involve both the boundary layer particles themselves and the radial transport. Either in a regime where FE conditions involve the computational domain, or in a viscous fluid dynamics, or both, a GASPHER scheme can be rightly adopted in such troublesome physical regimes. Despite the applied low compressibiity condition, viscous GASPHER model shows clear spiral pattern profiles demonstrating the better quality of results compared to SPH viscous ones. Moreover a successful comparison of results concerning GASPHER 1D inviscid shock tube with analytical solution is also reported.

Stability of noncharacteristic boundary layers in the standing-shock limit

We investigate one- and multi-dimensional stability of noncharacteristic boundary layers in the limit approaching a standing planar shock wave U(x1), x 1 > 0, obtaining necessary conditions of (i) weak stability of the limiting shock, (ii) weak stability of the constant layer u≡ U - := lim z→-∞ Ū(z), and (iii) nonnegativity of a modified Lopatinski determinant similar to that of the inviscid shock case. For Lax 1-shocks, we obtain equally simple sufficient conditions; for p-shocks, p > 1, the situation appears to be more complicated. Using these results, we determine the stability of certain gas dynamical boundary layers, generalizing earlier work of Serre—Zumbrun and Costanzino-Humphreys—Nguyen—Zumbrun.

Reconsideration of inviscid shock interactions and transition phenomena on double-wedge geometries in a M ∞ = 9 hypersonic flow

Shock polar analysis as well as 2-D numerical computation technique are used to illustrate a diverse flow topology induced by shock/shock interaction in a M∞ = 9 hypersonic flow. New flow features associated with inviscid shock wave interaction on double-wedge-like geometries are reported in this study. Transition of shock interaction, unsteady oscillation, and hysteresis phenomena in the RR ↔ MR transition, and the physical mechanisms behind these phenomena are numerically studied and analyzed.

Baroclinic vortex influence on wave drag reduction induced by pulse energy deposition

We present the results of numerical analysis of wave drag reduction by a single-pulse energy deposition in a supersonic flow field around a sphere. The wave drag for the sphere was reduced as a result of the interaction between a low-density core following the blast wave produced by the energy deposition and the bow shock developed in front of the sphere. We investigated the drag reduction mechanism in terms of the unsteady flow field induced by the interaction. The effects of deposited energy and deposition location on energy reduction were examined by parametric study. From the obtained results, we refined the parameters, utilizing the baroclinic source term that produced vorticity in the vortex equation when the gradients of density and pressure were not parallel. The baroclinic vortex driven by Richtmyer–Meshkov-like instability was strong enough to contribute to the temporary low-entropy shock formation that caused low wave drag for the supersonic object. We determined that the reduced energy had a linear dependence on the radius of the low-density core formed in the blast wave and was proportional to the square of the freestream Mach number. Such dependencies could be predicted with the assumption that the energy was consumed by the baroclinic vortex generation and advected downward without thermalization in an inviscid shock layer.

Nonclassical multidimensional viscous and inviscid shocks

Extending our earlier work on Lax-type shocks of systems of conservation laws, we establish existence and stability of curved multidimensional shock fronts in the vanishing viscosity limit for general Lax- or undercompressive-type shock waves of nonconservative hyperbolic systems with parabolic regularization. The hyperbolic equations may be of variable multiplicity and the parabolic regularization may be of “real”, or partially parabolic, type. We prove an existence result for inviscid nonconservative shocks that extends to multidimensional shocks a one-dimensional result of X. Lin proved by quite different methods. In addition, we construct families of smooth viscous shocks converging to a given inviscid shock as viscosity goes to zero, thereby justifying the small viscosity limit for multidimensional nonconservative shocks. In our previous work on shocks we made use of conservative form especially in parts of the low frequency analysis. Thus, most of the new analysis of this paper is concentrated in this area. By adopting the more general nonconservative viewpoint, we are able to shed new light on both the viscous and inviscid theory. For example, we can now provide a clearer geometric motivation for the low frequency analysis in the viscous case. Also, we show that one may, in the treatment of inviscid stability of nonclassical and/or nonconservative shocks, remove an apparently restrictive technical assumption made by Mokrane and Coulombel in their work on, respectively, shocktype nonconservative boundary problems and conservative undercompressive shocks. Another advantage of the nonconservative perspective is that Lax and undercompressive shocks can be treated by exactly the same analysis.

Existence and Stability of Multidimensional Shock Fronts in the Vanishing Viscosity Limit

In this paper we present a new approach to the study of linear and nonlinear stability of inviscid multidimensional shock waves under small viscosity perturbation, yielding optimal estimates and eventually an extension to the viscous case of the celebrated theorem of Majda on existence and stability of multidimensional shock waves. More precisely, given a curved Lax shock solution u0 to a hyperbolic system of conservation laws, we construct nearby viscous shock solutions u ε to a parabolic viscous perturbation of the hyperbolic system which converge to u0 as viscosity ε→0 and satisfy an appropriate (conormal) version of Majda’s stability estimate. The main new feature of the paper is the derivation of maximal and optimal estimates for the linearization of the parabolic problem about a highly singular approximate solution. These estimates are more robust than the singular estimates obtained in our previous work, and permit us to remove an earlier assumption limiting how much the inviscid shock we start with can deviate from flatness. The key to the new approach is to work with the full linearization of the parabolic problem, that is, the linearization with respect to both u ε and the unknown viscous front, and to allow variation of the front at all stages – not only in the construction of the approximate solution as done in previous work, but also in the final error equation. After reformulating the problem as a transmission problem, we show that the linearized problem can be desingularized and optimal estimates obtained by adding an appropriate extra boundary condition involving the front. The extra condition determines a local evolution rule for the viscous front.

Two-phase viscous modeling of compaction of granular materials

An inviscid model for deflagration-to-detonation transition in granular energetic materials is extended by addition of explicit intraphase momenta and energy diffusion so as to (1) enable the use of a straightforward numerical scheme, (2) avoid prediction of structures with smaller length scales than the component grains, and (3) have a model prepared to describe long time scale transients that are present in some slow processes which can lead to detonation. The model is shown to be parabolic, frame invariant, and dissipative. Consideration of the characteristics for cases with and without intraphase diffusion indicate what boundary conditions are necessary for a well posed problem. A simple numerical method, based on a method of lines applied to the nonconservative form of the equations, is shown to predict convergence at the proper rate to unique solutions which agree well with known solutions for an unsteady inviscid shock tube and a steady piston-driven viscous shock. A series of simulations of inert piston-driven subsonic compaction waves in which the additional mechanisms of interphase compaction, drag, and heat transfer are systematically introduced at an order of magnitude suggested by experiments reveals that interphase drag and heat transfer equilibrate velocities and temperatures, and that compaction equilibrates solid and configurational stresses. At higher piston velocities, supersonic shock and compaction waves are induced; comparison of predictions with and without viscosity demonstrate some of the computational advantages of explicit inclusion of diffusion. The local dissipation rates for each mechanism are quantified, and it is seen that dissipation due to compaction dominates that due to intraphase and interphase transport of linear momenta and energy, suggesting that compaction is the key mechanism in inducing the transition to detonation in piston-driven experiments.

Nonet-Cartesian Grid Method for Shock Flow Computations

A nonet-Cartesian grid method, based on anisotropic/isotropic refinement, is presented for solving the Euler equations in gas dynamic problems. Grids are generated automatically, by the recursive subdivision of a single cell into nine subcells for isotropic nonet-Cartesian grids and into three subcells independently in each direction for anisotropic nonet-Cartesian grids, encompassing the entire flow domain. The grid generation method is applied here to steady inviscid shock flow computation. A finite difference formulation for the Euler equation using nonet-Cartesian grids is used to treat complex two-dimensional configuration. Results using this approach are shown to be competitive with other methods. Further, it is demonstrated that this method provides a simple and accurate procedure for solving flow problems involving multielement airfoils.

Multidimensional viscous shocks II: The small viscosity limit

AbstractIn this paper we prove the existence of curved, multidimensional viscous shocks and also justify the small‐viscosity limit.Starting with a curved, multidimensional (inviscid) shock solution to a system of hyperbolic conservation laws, we show that the shock can be obtained as a small‐viscosity limit of solutions to an associated parabolic problem (viscous shocks). The two main hypotheses are a natural Evans function assumption on the viscous profile, together with a restriction on how much the shock can deviate from flatness. The main tools are a conjugation lemma that removes xN/ϵ dependence from the linearization of the parabolic problem about the viscous profile, new degenerate Kreiss‐type symmetrizers used to prove an L2 estimate for the linearized problem, and a finite‐regularity calculus of semiclassical and mixed type (classical‐semiclassical) pseudodifferential operators. © 2003 Wiley Periodicals, Inc.