Geometrical Shock Dynamics Research Articles

The dynamics of shock wave propagation far downstream of an abrupt area expansion

Predicting the transient flow fields that develop when a shock wave passes through an area expansion is a fundamental problem in compressible fluid mechanics and significant in many engineering applications. Experiments, large eddy simulations and geometrical shock dynamics are used to study the mechanism by which a normal shock wave that expands across an area expansion evolves into a uniform normal shock far downstream of it. This study analyses shock waves with moderate Mach numbers of 1.1–1.8 that expand at area ratios of up to 5. As the shock wave propagates into the expanded region, it experiences rapid deceleration, forming a non-uniform shock front. Impinging on the walls of the larger cross-section region, the shock wave reflects and generates a complex and highly transient shock pattern near the expansion region. We have found that as the shock front propagates further downstream, a laterally moving shock wave that intersects the shock front at a triple point reverberates laterally between the walls. This process effectively evens out the flow behind the incident shock front, thus reducing the variation of properties behind it. The extended duration of this process leads to significant pressure fluctuation behind the shock front. The results show that the evolution of the shock front can be scaled using the expanded region height and the velocity of the shock wave far downstream of the expansion. The results enabled the formulation of a simple empirical relation, allowing us to predict the shock velocity far downstream of gradual and abrupt area expansions.

Kinks on elliptical convergent shock waves in hypersonic flow

Kinks commonly appear on the convergent shock surface when an internal conical flow deviates from the axisymmetric state. In this paper, the formation mechanisms of kinks on internal conical shocks (ICSs) generated by elliptical ring wedges with typical entry aspect ratios ( $AR{\rm s}$ ) in a Mach 6 flow are revealed using a theoretical method, in which the spatial evolution of the three-dimensional elliptical ICS is converted into a temporal evolution of a two-dimensional elliptical moving shock (EMS) using the hypersonic equivalence principle. To simultaneously track the shock front of the EMS and the disturbances propagating along it, a front-disturbance tracking method (FDTM) based on geometrical shock dynamics is proposed. It is found that the shock–compression disturbances from the same family initially near the major axis catch up with the disturbance initially emitted from the major axis to form kinks on the EMS. The equivalent kink formation positions predicted by the FDTM always lag behind the real kink formation positions on the elliptical ICS because the applicability of the hypersonic equivalence principle decays as the shock strengthens along the incoming flow direction. The accuracy of the equivalent kink formation positions predicted by the FDTM gradually declines with the reduction in $AR$ , but it can be significantly improved for all $AR{\rm s}$ after a modification of the equivalent relationship using the shock angle in the major plane of the elliptical ICS, which provides a new way to solve the kinks on the elliptical ICS.

Two-Dimensional Geometrical Shock Dynamics for Blast Wave Propagation and Post-Shock Flow Effects

Geometrical shock dynamics (GSD) is a model capable of efficiently predicting the position, shape, and strength of a shock wave. Compared to the traditional Euler method that solves the inviscid Euler equations, GSD is a reduced-order model derived from the method of characteristics which results in a more computationally efficient approach since it only considers the motion of the shock front instead of the entire flow field. Here, a study of post-shock flow effects in two dimensions has been performed. These post-shock flow effects become increasingly important when modeling blast wave propagation over extended times or distances, i.e., a shock front that decays in speed and that has decaying properties behind it. A comparison between the first-order complete, fully complete and point-source GSD (PGSD) models reveals the importance of preserving an intact post-shock flow term, which is truncated by the original GSD model, in predicting blast motion. Lagrangian simulations were performed for the case of interaction between two cylindrical blast waves and the results were compared to prior experimental work. The results showed an agreement in attenuation of the maximum pressure at the Mach stem, but an overestimation of the Mach stem growth at its early stage was observed using PGSD. To address this issue, another model was developed that combines the PGSD model with shock–shock approximate theory (PGSDSS), but it excessively attenuates Mach stem evolution.

Diffraction of shock waves through a non-quiescent medium

An investigation of shock diffraction through a non-quiescent background medium is presented using both experimental and numerical techniques. Unlike diffracting shocks in quiescent media, a spatial distortion of the shock front occurs, producing a region of constant shock angle. An example of this process arises in the exhaust from a pulse-detonation combustor. As the background velocity is increased, such as through the inclusion of a converging nozzle at the exhaust, the spatial distortion becomes more apparent. Numerical simulations using a compressible Euler solver demonstrate that the distortion is not due to the geometrical influence of the nozzle, but rather is a function of the magnitude of the background flow velocity. The distortion is studied using a modified geometrical shock dynamics formulation which includes the background flow and is validated against experiments. A simple model is presented to predict the shock distortion angle in the weak-shock limit. Finally, the axial decay behaviour of the shock is investigated and it is shown that the advection of the shock by the background flow delays the arrival of the head and tail of the expansion characteristic at the centreline. This leads to an increase in the rate of decay of the shock Mach number as the background flow velocity is increased.

Ray-tracking methods for characterizing the dynamics of curved detonation

Current analyses of curved detonations are mostly limited to the dynamics along the wall or the symmetry axis due to the lack of efficient approaches for reliably tracking stream tubes with curved shock fronts. To address this lack, the present work proposes a novel curved ray-tracking algorithm with two implementation methods. The curved ray is characterized by a specific arc of constant curvature perpendicularly intersecting both successive fronts. The methods were validated against Whitham's exact geometrical shock dynamics solutions of the self-similar shock diffraction problem. As compared to the typical forward straight-ray method, which is of the first order, the proposed methods demonstrate convergence rates greater by more than one order of magnitude and tend to be of the second order. The convergence analysis enabled to determine the range of time resolution required for the proposed methods to provide reliable results. This range is comparable to the acquisition rate of a modern high-speed camera that is commonly used in detonation visualization experiments. Differences between the proposed curved ray-tracking methods and the typical forward straight-ray method have been further examined by analyzing the ray dynamics during the weakly unstable hydrogen–oxygen–argon detonation diffraction. The ray-tube-based velocity-curvature relationship showed satisfactory agreement with those already well-established in quasi-steady experiments and predicted by the generalized Zeldovich–von Neumann–Doering model. This suggests the limited role of unsteadiness and cellular structure in the macro-scale dynamics of weakly unstable detonations.

Numerical and experimental evaluation of shock dividers

Mitigation of pressure pulsations in the exhaust of a pulse detonation combustor is crucial for operation with a downstream turbine. For this purpose, a device termed the shock divider is designed and investigated. The intention of the divider is to split the leading shock wave into two weaker waves that propagate along separated ducts with different cross sections, allowing the shock waves to travel with different velocities along different paths. The separated shock waves redistribute the energy of the incident shock wave. The shock dynamics inside the divider are investigated using numerical simulations. A second-order dimensional split finite volume MUSCL-scheme is used to solve the compressible Euler equations. Furthermore, low-cost simulations are performed using geometrical shock dynamics to predict the shock wave propagation inside the divider. The numerical simulations are compared to high-speed schlieren images and time-resolved total pressure recording. For the latter, a high-frequency pressure probe is placed at the divider outlet, which is shown to resolve the transient total pressure during the shock passage. Moreover, the separation of the shock waves is investigated and found to grow as the divider duct width ratio increases. The numerical and experimental results allow for a better understanding of the dynamic evolution of the flow inside the divider and inform its capability to reduce the pressure pulsations at the exhaust of the pulse detonation combustor.

Evolution of a shock generated by an impulsively accelerated, sinusoidal piston

Abstract

On the shock dynamics of weak converging shock waves in solid materials

We presented a geometrical shock dynamics model to predict the behavior of weak converging shock waves in solid materials. Taking into consideration Mie–Gruneisen equation of state the analytical solution is obtained for the flow behind the converging shock-front propagating in solids. The analytical formaluas are also obtained for the shock velocity, pressure, density, particle velocity, temperature, speed of sound, adiabatic bulk modulus, and change-in-entropy behind or across the weak converging shock wave. For this it was assumed that the medium is a homogeneous and isotropic, and the disturbances behind the front do not overtake the converging shock waves. The effects due to an increase in (i) the propagation distance from the axis or centre of convergence, (ii) the Gruneisen parameter, and (iii) the material parameter, are explored on the shock velocity and quantities in the shocked titanium, brass, tantalum, iron, stainless steel 304, aluminum 6061-T6 and OFHC copper. The geometrical shock dynamics model provided a clear picture of whether and how the properties of solids are affected due to the passage of converging shock inside the solid materials.

Extension of geometrical shock dynamics for blast wave propagation

The direct numerical simulation of blast waves is a challenging task due to the wide range of spatial and temporal scales involved. Moreover, in a real environment (topography, urban area), the blast wave interacts with geometrical obstacles, resulting in reflection, diffraction, and wave recombination phenomena. The shape of the front becomes complex, which limits the efficiency of simple empirical methods. This work aims at contributing to the development of a fast running method for blast waves propagating in the presence of obstacles. This is achieved through an ad hoc extension of the simplified hyperbolic geometrical shock dynamics (GSD) model, which leads to a drastic reduction in the computational cost in comparison with the full Euler system. The new model, called geometrical blast dynamics, is able to take into account any kind of source and obstacle. It relies on a previous extension of GSD for diffraction over wedges to obtain consistent physical behavior, especially in the limit of low Mach numbers. The new model is fully described. Its numerical integration is straightforward. Results compare favorably with experiments, semiempirical models from the literature, and Eulerian simulations, over a wide range of configurations.

An immersed boundary method for geometrical shock dynamics

A new algorithm is presented for the resolution of the Geometrical Shock Dynamics model in presence of obstacles in an Eulerian framework. The numerical method relies on a Fast-Marching like algorithm on a Cartesian grid, which allows dealing with complex geometrical configurations and shock waves interactions at a reduced computational cost. The application of an homogeneous Neumann condition at the border of rigid obstacles, not aligned with the mesh, is based on the Immersed Boundary Method. For a given obstacle, a set of ghost points is defined in the solid area and the corresponding unknown flow parameters are updated using a compatibility condition. A good agreement is observed between numerical results, experimental data and CFD simulations, both in 2D and 3D, which demonstrates the validity and the capabilities of this method.

Beyond the limitation of geometrical shock dynamics for diffraction over wedges

Geometrical shock dynamics (GSD) is a simplified model for nonlinear shock wave propagation for which the front evolution is governed by a local relation between the geometry of the shock and its velocity, the so-called A–M rule. Numerous studies have proven the ability of the GSD model to estimate correctly the leading shock front in interactions with obstacles. Nevertheless, a solution for the problem of diffraction over a convex corner does not always exist, especially for weak shocks. To overcome this limitation, we propose an ad hoc modification of the A–M relation for two-dimensional configurations: an extra term based on the transverse variation of the Mach number is added. This new closure is fitted against experimental observations, which ensures, by construction, a correct behaviour for expansive shocks. A Lagrangian numerical solver is developed, for which this new term is activated only on specific parts of the front. Results of this new model are compared with the original GSD model, experiments, and Eulerian simulations for several cases of increasing complexity. A noticeable improvement in the solution is observed.

Spontaneous singularity formation in converging cylindrical shock waves

We develop a nonlinear, Fourier-based analysis of the evolution of a perturbed, converging cylindrical strong shock using the approximate method of geometrical shock dynamics (GSD). This predicts that a singularity in the shock-shape geometry, corresponding to a change in Fourier-coefficient decay from exponential to algebraic, is guaranteed to form prior to the time of shock impact at the origin, for arbitrarily small, finite initial perturbation amplitude. Specifically for an azimuthally periodic Mach-number perturbation on an initially circular shock with integer mode number q and amplitude proportional to epsilon > 1. The constant of proportionality is obtained as a function of gamma and is independent of the initial shock Mach number M-0. Singularity formation corresponds to the transition from a smooth perturbation to a faceted polygonal form. Results are qualitatively verified by a numerical GSD comparison.

Singularity formation on perturbed planar shock waves

We present an analysis that predicts the time to development of a singularity in the shape profile of a spatially periodic perturbed, planar shock wave for ideal gas dynamics. Beginning with a formulation in complex coordinates of Whitham’s approximate model geometrical shock dynamics (GSD), we apply a spectral treatment to derive the asymptotic form for the leading-order behaviour of the shock Fourier coefficients for large mode numbers and time. This is shown to determine a critical time at which the coefficients begin to decay, with respect to mode number, at an algebraic rate with an exponent of $-5/2$, indicating loss of analyticity and the formation of a singularity in the shock geometry. The critical time is found to be inversely proportional to a representative measure of perturbation amplitude $\unicode[STIX]{x1D716}$ with an explicit analytic form for the constant of proportionality in terms of gas and shock parameters. To leading order, the time to singularity formation is dependent only on the first Fourier mode. Comparison with results of numerical solutions to the full GSD equations shows that the predicted critical time somewhat underestimates the time for shock–shock (triple-point) formation, where the latter is obtained by post-processing the numerical GSD results using an edge-detection algorithm. Aspects of the analysis suggest that the appearance of loss of analyticity in the shock surface may be a precursor to the first appearance of shock–shocks, which may account for part of the discrepancy. The frequency of oscillation of the shock perturbation is accurately predicted. In addition, the analysis is extended to the evolution of a perturbed planar, fast magnetohydrodynamic shock for the case when the external magnetic field is aligned parallel to the unperturbed shock. It is found that, for a strong shock, the presence of the magnetic field produces only a higher-order correction to the GSD equations with the result that the time to loss of analyticity is the same as for the gas-dynamic flow. Limitations and improvements for the analysis are discussed, as are comparisons with the analogous appearance of singularity formation in vortex-sheet evolution in an incompressible inviscid fluid.

Comparison of geometrical shock dynamics and kinematic models for shock-wave propagation

Geometrical Shock Dynamics (GSD) is a simplified model for nonlinear shock wave propagation. It is based on the decomposition of the shock front into elementary ray tubes with a simple relation linking its local curvature and velocity. This relation is obtained under the assumption of strong shock in order to neglect the effect of the post-shock flow on the front. More recently, a new simplified model, referenced as the Kinematic model, was proposed. This model is obtained by combining the three-dimensional Euler equations and the Rankine-Hugoniot relations at the front, which leads to an equation for the normal variation of the shock Mach number at the wave front. In the same way as GSD, the Kinematic model is closed by neglecting the post-shock flow effects. Although each model's approach is different, we prove here their structural equivalence: the Kinematic model can be rewritten under the form of GSD with a specific A − M relation. Both models are thus compared through a wide variety of examples including experimental data or Eulerian simulations results when available. Attention is drawn to the simple cases of compression ramps and convex corners' diffraction. The analysis is completed by the more complex cases of the diffraction over a cylinder, a sphere, a mound and a trough.

Geometrical shock dynamics applied to condensed phase materials

Taylor blast wave (TBW) theory and geometrical shock dynamics (GSD) theory describe a radially expanding shock wave front through an inert material, typically an ideal gas, in the strong blast wave limit and weak acoustic limit respectively. We simulate a radially expanding blast shock in air using a hydrodynamic simulation code and numerically describe the intermediate region between these two limits. We test our description of the intermediate shock phase through a two-dimensional simulation of the Bryson and Gross experiment. We then apply the principles of GSD to materials that follow the Mie–Gruneisen equation of state, such as plastics and metals, and derive an equation that accurately relates the acceleration, velocity and curvature of the shock through these materials. Along with detonation shock dynamics (DSD), which describes detonation shock propagation through high explosive fluids, we develop a hybrid DSD/GSD model for the simulation of heterogeneous explosives. This model enables computationally efficient simulation of the shock front in high explosive/inert mixtures consisting of simple or complex geometric configurations. We simulate an infinite two-dimensional slab consisting of one half explosive, PBXN-9, and one half aluminium and model the boundary angle conditions using shock polar analysis. We also simulate a series of high explosive unit cells embedded with aluminium spherical particles, and we compare the propagation of the detonation shock front with a direct numerical simulation performed with the ALE3D code.

Geometrical shock dynamics for magnetohydrodynamic fast shocks

We describe a formulation of two-dimensional geometrical shock dynamics (GSD) suitable for ideal magnetohydrodynamic (MHD) fast shocks under magnetic fields of general strength and orientation. The resulting area–Mach-number–shock-angle relation is then incorporated into a numerical method using pseudospectral differentiation. The MHD-GSD model is verified by comparison with results from nonlinear finite-volume solution of the complete ideal MHD equations applied to a shock implosion flow in the presence of an oblique and spatially varying magnetic field ahead of the shock. Results from application of the MHD-GSD equations to the stability of fast MHD shocks in two dimensions are presented. It is shown that the time to formation of triple points for both perturbed MHD and gas-dynamic shocks increases as $\unicode[STIX]{x1D716}^{-1}$, where $\unicode[STIX]{x1D716}$ is a measure of the initial Mach-number perturbation. Symmetry breaking in the MHD case is demonstrated. In cylindrical converging geometry, in the presence of an azimuthal field produced by a line current, the MHD shock behaves in the mean as in Pullin et al. (Phys. Fluids, vol. 26, 2014, 097103), but suffers a greater relative pressure fluctuation along the shock than the gas-dynamic shock.

Parallel implementation of geometrical shock dynamics for two dimensional converging shock waves

Geometrical shock dynamics (GSD) theory is an appealing method to predict the shock motion in the sense that it is more computationally efficient than solving the traditional Euler equations, especially for converging shock waves. However, to solve and optimize large scale configurations, the main bottleneck is the computational cost. Among the existing numerical GSD schemes, there is only one that has been implemented on parallel computers, with the purpose to analyze detonation waves. To extend the computational advantage of the GSD theory to more general applications such as converging shock waves, a numerical implementation using a spatial decomposition method has been coupled with a front tracking approach on parallel computers. In addition, an efficient tridiagonal system solver for massively parallel computers has been applied to resolve the most expensive function in this implementation, resulting in an efficiency of 0.93 while using 32 HPCC cores. Moreover, symmetric boundary conditions have been developed to further reduce the computational cost, achieving a speedup of 19.26 for a 12-sided polygonal converging shock.

Converging cylindrical magnetohydrodynamic shock collapse onto a power-law-varying line current

We investigate the convergence behaviour of a cylindrical, fast magnetohydrodynamic (MHD) shock wave in a neutrally ionized gas collapsing onto an axial line current that generates a power law in time, azimuthal magnetic field. The analysis is done within the framework of a modified version of ideal MHD for an inviscid, non-dissipative, neutrally ionized compressible gas. The time variation of the magnetic field is tuned such that it approaches zero at the instant that the shock reaches the axis. This configuration is motivated by the desire to produce a finite magnetic field at finite shock radius but a singular gas pressure and temperature at the instant of shock impact. Our main focus is on the variation with shock radius $r$, as $r\rightarrow 0$, of the shock Mach number $M(r)$ and pressure behind the shock $p(r)$ as a function of the magnetic field power-law exponent ${\it\mu}\geqslant 0$, where ${\it\mu}=0$ gives a constant-in-time line current. The flow problem is first formulated using an extension of geometrical shock dynamics (GSD) into the time domain to take account of the time-varying conditions ahead of the converging shock, coupled with appropriate shock-jump conditions for a fast, symmetric MHD shock. This provides a pair of ordinary differential equations describing both $M(r)$ and the time evolution on the shock, as a function of $r$, constrained by a collapse condition required to achieve tuned shock convergence. Asymptotic, analytical results for $M(r)$ and $p(r)$ are obtained over a range of ${\it\mu}$ for general ${\it\gamma}$, and for both small and large $r$. In addition, numerical solutions of the GSD equations are performed over a large range of $r$, for selected parameters using ${\it\gamma}=5/3$. The accuracy of the GSD model is verified for some cases using direct numerical solution of the full, radially symmetric MHD equations using a shock-capturing method. For the GSD solutions, it is found that the physical character of the shock convergence to the axis is a strong function of ${\it\mu}$. For $0\leqslant {\it\mu}<4/13$, $M$ and $p$ both approach unity at shock impact $r=0$ owing to the dominance of the strong magnetic field over the amplifying effects of geometrical convergence. When ${\it\mu}\geqslant 0.816$ (for ${\it\gamma}=5/3$), geometrical convergence is dominant and the shock behaves similarly to a converging gas dynamic shock with singular $M(r)$ and $p(r)$, $r\rightarrow 0$. For $4/13<{\it\mu}\leqslant 0.816$ three distinct regions of $M(r)$ variation are identified. For each of these $p(r)$ is singular at the axis.

A fast-marching like algorithm for geometrical shock dynamics

We develop a new algorithm for the computation of the Geometrical Shock Dynamics (GSD) model. The method relies on the fast-marching paradigm and enables the discrete evaluation of the first arrival time of a shock wave and its local velocity on a Cartesian grid. The proposed algorithm is based on a first order upwind finite difference scheme and reduces to a local nonlinear system of two equations solved by an iterative procedure. Reference solutions are built for a smooth radial configuration and for the 2D Riemann problem. The link between the GSD model and p-systems is given. Numerical experiments demonstrate the efficiency of the scheme and its ability to handle singularities.

Converging cylindrical shocks in ideal magnetohydrodynamics

We consider a cylindrically symmetrical shock converging onto an axis within the framework of ideal, compressible-gas non-dissipative magnetohydrodynamics (MHD). In cylindrical polar co-ordinates we restrict attention to either constant axial magnetic field or to the azimuthal but singular magnetic field produced by a line current on the axis. Under the constraint of zero normal magnetic field and zero tangential fluid speed at the shock, a set of restricted shock-jump conditions are obtained as functions of the shock Mach number, defined as the ratio of the local shock speed to the unique magnetohydrodynamic wave speed ahead of the shock, and also of a parameter measuring the local strength of the magnetic field. For the line current case, two approaches are explored and the results compared in detail. The first is geometrical shock-dynamics where the restricted shock-jump conditions are applied directly to the equation on the characteristic entering the shock from behind. This gives an ordinary-differential equation for the shock Mach number as a function of radius which is integrated numerically to provide profiles of the shock implosion. Also, analytic, asymptotic results are obtained for the shock trajectory at small radius. The second approach is direct numerical solution of the radially symmetric MHD equations using a shock-capturing method. For the axial magnetic field case the shock implosion is of the Guderley power-law type with exponent that is not affected by the presence of a finite magnetic field. For the axial current case, however, the presence of a tangential magnetic field ahead of the shock with strength inversely proportional to radius introduces a length scale \documentclass[12pt]{minimal}\begin{document}$R=\sqrt{\mu _0/p_0}\,I/(2\,\pi )$\end{document}R=μ0/p0I/(2π) where I is the current, μ0 is the permeability, and p0 is the pressure ahead of the shock. For shocks initiated at r ≫ R, shock convergence is first accompanied by shock strengthening as for the strictly gas-dynamic implosion. The diverging magnetic field then slows the shock Mach number growth producing a maximum followed by monotonic reduction towards magnetosonic conditions, even as the shock accelerates toward the axis. A parameter space of initial shock Mach number at a given radius is explored and the implications of the present results for inertial confinement fusion are discussed.