The Riemann problem for fluid flow of real materials is examined. An arbitrary equation of state is allowed, subject only to the physical requirements of thermodynamics. The properties of the isentropes and the shock Hugoniot loci that follow from conditions imposed on the equation of state are reviewed systematically. Important properties of these wave curves are determined by three dimensionless variables characterizing the equation of state: the adiabatic exponent $\ensuremath{\gamma}$, the Gr\"uneisen coefficient $\ensuremath{\Gamma}$, and the fundamental derivative $\mathcal{G}$. Standard assumptions on these variables break down near phase transitions. The result is an anomalous wave structure: either shock waves split into multiple waves, or composite waves form. Additional questions related to shock stability and nonuniqueness of the solution of the Riemann problem are discussed.

The Riemann problem for fluid flow of real materials

https://crd.lbl.gov/assets/pubs_presos/AMCS/ANAG/A17.pdf

Efficient Solution Algorithms for the Riemann Problem for Real Gases

In these lectures we present the basic ideas and recent development in the construction, analysis, and implementation of ENO (Essentially Non-Oscillatory) and WENO (Weighted Essentially Non-Oscillatory) schemes and their applications to computational fluid dynamics ENO and WENO schemes are high order accurate finite difference or finite volume schemes designed for problems with piecewise smooth solutions containing discontinuities The key idea lies at the approximation level, where a nonlinear adaptive procedure is used to automatically choose the locally smoothest stencil, hence avoiding crossing discontinuities in the interpolation procedure as much as possible ENO and WENO schemes have been quite successful in computational fluid dynamics and other applications, especially for problems containing both shocks and complicated smooth solution structures, such as compressible turbulence simulations and aeroacoustics

High Order ENO and WENO Schemes for Computational Fluid Dynamics

The properties that distinguish a shock wave from other waves are that its thickness is negligible compared with other characteristic lengths, and that the state of the medium is changed irreversibly by the passage of the wave. Shocks are therefore a highly nonlinear phenomenon, and parameter changes may be expected to lead to numerous bifurcations. 

The multitude of possibilities is compounded when more than one shock occurs, such as in the interaction of a shock with a solid surface or symmetry plane (i.e. in shock reflection). The subject of shock reflection is so complicated that it is necessary to introduce it at some length. In the interest of conciseness, this is done according to a logical rather than a historical sequence, with explicit reference to the important authors omitted in the text unless their work is relatively recent. 

Salient points in the early development of the subject should be mentioned here, however. These may be found in the experiments of Mach (1878), the theoretical work of von Neumann(1943), the experimental and theoretical work of the group around Bleakney at Princeton (e.g. Bleakney & Taub 1949, Smith 1945), the experiments of Kawamura & Saito (1956), Smith (1959), and Bryson & Gross (1961), and the theoretical work Lighthill (1949) and Jones et al. (195l). A review by Pack (1964) gives detailed references. Of the work since 1960, that of the group around Glass at Toronto (e.g. Law & Glass 1971, Ben-Dor & Glass 1979, 1980) and around Henderson at Sydney (e.g. Henderson & Lozzi 1975, 1979, Henderson & Gray 1981) stand out. A review by Griffith (1981) gives more detailed references. 

In this review the discussion is restricted to plane flow. This is because relatively little work has been done on three-dimensional situations, and because a compromise had to be made between depth and breadth of the field covered.

/pdf/regular-and-mach-reflection-of-shock-waves-3oxms92bbf.pdf

Regular and Mach Reflection of Shock Waves

Front tracking is an adaptive computational method in which a lower dimensional moving grid is fitted to and follows the dynamical evolution of distinguished waves in a fluid flow. The method takes advantage of known analytic solutions, derived from the Rankine-Hugoniot relations, for idealized discontinuities. In this paper the method is applied to the Euler equations describing compressible gas dynamics. The main thrust here is validation of the front tracking method: we present results on a series of test problems for which comparison answers can be obtained by independent methods.

Front Tracking for gas Dynamics

Interferometric data were obtained in the 10 cm × 18 cm hypervelocity shock tube of oblique shock-wave reflexions in argon at initial temperatures and pressures of nearly 300 °K and 15 Torr. The shock Mach-number range covered was 2 [les ] Ms [les ] 8 over a series of wedge angles 2° [les ] θw [les ] 60°. Dual-wavelength laser interferograms were obtained by using a 23 cm diameter field of view Mach-Zehnder interferometer. In addition to our numerous results, the available data for argon and helium obtained over the last two decades were also utilized. It is shown analytically and experimentally that in non-stationary flows six domains exist in the (Ms, θw) plane where regular reflexion (RR), single-Mach reflexion (SMR), complex-Mach reflexion (CMR) and double-Mach reflexion (DMR) can occur. The transition boundaries between these regions were all established analytically. The experimental results from different sources substantiate the present analysis, and areas of disagreement which existed in the literature are now clarified and resolved. It is shown that real-gas effects have a significant influence on the size of the regions and their boundaries. In addition, isopycnics (constant density lines) are given for the four types of reflexion, as well as the density distribution along the wedge surface. This data should provide a solid base for computational fluid dynamicists in comparing numerical techniques with actual experimental results.

Domains and boundaries of non-stationary oblique shock-wave reflexions. 2. Monatomic gas

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S002211207800004X

Effects of hydrogen impurities on shock structure and stability in ionizing monatomic gases. Part 1. Argon

Nonstationary oblique shock-wave reflections over compressive wedges in air and argon were investigated using infinite-fringe interferometric techniques. These allowed direct, continuous and accurate observations of the isopycnics (lines of constant density) of the flow field. The initial pressures for these experiments were made as high as possible (15 to 250 torr) in order to increase the number of isopycnics and to enhance their details and distribution along the wedge surface over a shock-Mach-number range 2.0 < Ms [les ] 8.7. Included in the study were two cases of regular reflection (RR) and one of each single Mach reflection (SMR), complex Mach reflection (CMR) and double-Mach reflection (DMR) for air, and one RR, SMR, CMR and DMR for argon. These particular cases, which we investigated previously in N2 and Ar using a finite-fringe technique, have been used by computational fluid dynamicists to check their finite-difference results against our experimental data. It will be shown that the isopycnic structure previously reported by us differs in detail, in most cases, from that of the present study. The major difference arises from the fact that it was only possible previously to obtain discrete points on isopycnics and along the wedge surface. Consequently, the results obtained before were not as accurate. Comparisons were made of actual wall-density distributions with numerical simulations of the density contours of the various flows obtained by a number of authors. Each numerical method displays its advantages and disadvantages in describing the details of the flow fields. The present experimental results for air are new. They are of great interest from a practical viewpoint. The experiments in argon were redone to provide better data for a gas free from real-gas effects in the range of initial conditions considered, in order to simplify the computations in the numerical simulations. Although the recent numerical simulations are better than those reported previously, additional efforts are required to improve the predictions of the shape, location and values of the isopycnics and other flow isolines in the various regions and along the wall, and to render the predictions free of computer ‘noise’. It is worth noting that real-gas effects did not play any significant role in determining the various wave systems in RR, SMR, CMR and DMR; a different claim was made in our previous work. Relaxation of nitrogen in air can be seen however, at the highest shock Mach numbers (Ms = 7.19 and 8.70), with relaxation lengths in good agreement with accepted predictions.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0022112083001226

An update on non-stationary oblique shock-wave reflections: actual isopycnics and numerical experiments

Details are given of an implicit six-point finite-difference scheme for solving two-temperature, laminar, boundary-layer flows not in chemical equilibrium in ionizing argon. The analysis extends previous work by considering the radiation-energy loss and the chemical reactions due to atom-atom and electron-atom collisions in the ionizing boundary-layer and free-stream flow. Also included are variations in transport properties based on known elastic-scattering cross-sections, effects of chemical reactions, radiation-energy loss and the electric-sheath wall boundary conditions. The results are compared with dual-wavelength interferometric boundary-layer data obtained by using a Mach-Zehnder interferometer 23 cm in diameter with the UTIAS 10 × 18 cm Hypervelocity Shock Tube for shocks of initial Mach numbers M s ∼ 13 and 16 moving into argon at a pressure p 0 ∼ 5 torr and temperature T 0 ∼ 297 °K. Considering the difficulties involved in solving such complex plasma flows, satisfactory agreement was obtained between analytic and experimental total-density profiles and electron-number-density profiles for the case M s ∼ 16 and good agreement for M s ∼ 13.

I. I. Glass

Papers

Domains and boundaries of non-stationary oblique shock-wave reflexions. 2. Monatomic gas

Effects of hydrogen impurities on shock structure and stability in ionizing monatomic gases. Part 1. Argon

An update on non-stationary oblique shock-wave reflections: actual isopycnics and numerical experiments

Ionizing argon boundary layers. Part 1. Quasi-steady flat-plate laminar boundary-layer flows

Ionizing argon boundary layers. Part 2. Shock-tube side-wall boundary-layer flows