Three-dimensional effects on regular reflection in steady supersonic flows

The reflection of shock waves between two symmetrical wedges is investigated for the case of three-dimensional flows. Oblique shadowgraphs at various optical angles of yaw and pitch were used to examine the nature of fully three-dimensional flows, with wedge aspect ratios as low as 0.25 being considered. These images were used to construct surface models of the overall flow field for various reflection patterns and aspect ratios, which provides a visual indication of the flow field shape. For a Mach number of 3.1, and suitable wedge angles, the flow field with regular reflection on the tunnel centreline and Mach reflection further out is examined. The point of transition from regular reflection to the peripheral Mach surfaces is identified for various wedge angles and aspect ratios. It is shown that the transition points move outwards from the central plane as the aspect ratio decreases. This shows that three-dimensional flows favor regular reflection, because of the increasing curvature of the incident shock as the wedge becomes narrower, causing a decrease in the local angle of incidence. The height of the Mach stem is shown to be highly dependent on the geometry of the test wedge models. The Mach stem height decreases with aspect ratio due to the three-dimensional relieving effect, where the increase in lateral flow relieves the pressure over the surfaces of the wedges. Experimental evidence of the existence of the strong oblique shock solution in steady flows is presented.Received: 7 July 2003, Revised: 20 October 2003, Accepted: 6 November 2003, Published online: 10 February 2004PACS: 47.40.Nm Correspondence to: B.W. Skews     

Unsteady convective surface heat flux measurements on cylinder for CFD code validation

The surface convective heat transfer rates to a cylinder have been measured using platinum thin film gauges in a shock tube and the results have been used to validate two numerical codes. The investigations have been carried out at different incident shock Mach numbers and flow Reynolds numbers. The measured and simulated results give an insight into the transient flow fields around the model in the shock tube and are a valuable means of complementing the numerical and experimental techniques used in this study.Received: 27 August 2003, Accepted: 11 November 2003, Published online: 17 February 2004Correspondence to: T. Saito     

A soap film shock tube to study two-dimensional compressible flows

 A new experimental approach to the study of the two-dimensional compressible flow phenomena is presented. In this technique, a variety of compressible flows were generated by bursting plane vertical soap films. An aureole and a “shock wave” preceding the rim of the expanding hole were clearly observed using traditional high-speed flash photography and a fast line-scan charge coupled device (CCD) camera. The moving shock wave images obtained from the line-scan CCD camera were similar to the x–t diagrams in gas dynamics. The moving shock waves cause thickness jumps and induce supersonic flows. Photographs of the supersonic flows over a cylinder and a wedge are presented. The results suggest clearly the feasibility of the “soap film shock tube”. Received: 11 May 2000/Accepted: 2 November 2000     

Effect of boundary layer on Mach reflection over a wedge surface

Abstract. In this paper, we consider the phenomenon of unsteady Mach reflection generated by a plane shock wave advancing over a straight wedge surface, with particular attention to the deviation of the flow field from the self-similar nature. We examine the observed change in angle between incident and reflected shocks, which is in contrast to the fact that the angle should remain constant with time in a self-similar flow. The effect of the boundary layer behind the advancing shock wave over the surface of the wedge is considered to cause this, and boundary layer theory is utilized to estimate the thickness of the layer. It is found that the thickness increases as to the time t compared with t by the overall expansion in the self-similar flow. Assuming that the thicker boundary layer is effectively equivalent to a change in wedge angle, the effect of the boundary layer on the flow field should be less in later stages with larger t values in accordance with the observation above. Received 6 March 2000 / Accepted 23 April 2001     

The effect of an unsteady drag force on the structure of a non-equilibrium region behind a shock wave in a gas-particle mixture

For numerical analysis of shock wave propagation in gas-particle mixtures, drag coefficients of a sphere in steady flows are generally used. However, it is shown both experimentally and numerically that a shock loaded solid sphere experiences unsteady drag forces. The paper describes a model of unsteady drag force and its effect on the structure of the non-equilibrium region behind a shock front traveling in a dusty gas. The results are compared with those obtained by using a steady drag coefficient and are discussed. It is demonstrated that the large drag force at the early stage of the interaction between shock-wave induced flow and a solid particle affects the flow structure that is obtained with a steady drag force.      

The Falkner–Skan Wedge Flows of Power-Law Fluids Embedded in a Porous Medium

The non-Darcy flow characteristics of power-law non-Newtonian fluids past a wedge embedded in a porous medium have been studied. The governing equations are converted to a system of first-order ordinary differential equations by means of a local similarity transformation and have been solved numerically, for a number of parameter combinations of wedge angle parameter m, power-law index of the non-Newtonian fluids n, first-order resistance A and second-order resistance B, using a fourth-order Runge–Kutta integration scheme with the Newton–Raphson shooting method. Velocity and shear stress at the body surface are presented for a range of the above parameters. These results are also compared with the corresponding flow problems for a Newtonian fluid. Numerical results show that for the case of the constant wedge angle and material parameter A, the local skin friction coefficient is lower for a dilatant fluid as compared with the pseudo-plastic or Newtonian fluids.     

Weak shock diffraction

The diffraction of a weak shock by a rigid wedge is analyzed theoretically via the theory of weakly nonlinear geometrical acoustics, which is the same as Whitham's nonlinearization technique. First linear weakly nonlinear geometrical acoustics is explained. Then the linear acoustics results for weak shock diffraction by a wedge are presented. Next these results are modified according to the principles of weakly nonlinear geometrical acoustics. The results show that the compressive diffracted wavefronts of linear acoustics are actually shocks, and their positions and strengths are found. The infinite gradients of the linear acoustics rarefaction waves are found to be finite but discontinuous gradients. Finally the results are specialized to a shock hitting a right-angled wedge, a shock coming off a right-angled wedge, and a shock hitting a thin semi-infinite screen.     

The isotope effect of gaseous hydrogen under shock compression

The shock compression method has been used to measure the Hugoniot data and shock temperature for gaseous hydrogen samples, covering the pressure range of 55-140 MPa and the temperature range of 3400-4500 K and with the initial conditions of P ₀ = 0.6 MPa, 1.2 MPa and T ₀ at room temperature. Spectral radiance histories emitted from shocked D ₂ and H ₂ + D ₂ (equimolar mixture) are monitored by a pyrometer system with seven wavelength channels. Theoretical calculations based on the Saha model with Debye-Hückel correction for the shock compression behavior of shocked gaseous samples are in good agreement with the measured Hugoniot data, but show slightly higher values for the shock temperature when comparing with experiments. An isotope effect relevant to these shocked hydrogen species has been found in the linear shock velocity vs particle velocity relation, in which the correlation factor between these hydrogen isotopes or hydrogen mixtures is simply of initial density dependence.Received: 8 December 2002, Accepted: 8 May 2003, Published online: 2 September 2003PACS: 62.50 + p, 31.30.GS, 51.90. + r     

Mixed Convection of Water at 4°C Along a Wedge with Variable Surface Flux in a Porous Medium

In this study, the mixed convection of water at 4°C along a wedge in a porous medium is investigated numerically using finite difference method. In order to explore the effect of mixed convection, both forced and free convection-dominated regimes are considered. Non-similarity solutions are obtained for the variable wall flux boundary condition. Velocity and temperature profiles as well as local dimensionless skin friction and Nusselt number are obtained and compared with the available numerical results for various values of different parameters. The wedge angle geometry parameter m and mixed convection parameter ξ are ranged from 0 to 1 in both regimes whereas different values of λ are considered for the purpose of comparison of heat transfer results.     

Two-dimensional numerical study of planar shock-wave/moving-body interactions

The interaction of a planar shock wave with a body moving at supersonic speed has been considered with particular focus on shock-on-shock interactions. Three interaction types were previously identified by Smyrl (1964). This work adds a fourth type to these interactions and restates the interaction type classification and the transitions between the various interaction types. A pseudo-steady analysis of key interaction points and flow features is used to predict the various transitions. The criteria presented here are compared to results from a numerical Euler model of the interactions. A comparison of results from the numerical model with experimental results shows good agreement and thus the existence of the various interaction types and transition criteria have been confirmed using the numerical model.Received: 28 November 2002, Accepted: 23 August 2003, Published online: 12 November 2003Correspondence to: C. LawL.T. Felthun: Present address: Fluent Inc., 10 Cavendish Court, Centerra Resource Park, Lebanon, NH 03766, USAAspects of this work were presented at the 23 ISSW by Law et al. (2001)     

Interaction of a shock with a sphere suspended in a vertical shock tube

Shock wave interaction with a sphere is one of the benchmark tests in shock dynamics. However, unlike wind tunnel experiments, unsteady drag force on a sphere installed in a shock tube have not been measured quantitatively. This paper presents an experimental and numerical study of the unsteady drag force acting on a 80 mm diameter sphere which was vertically suspended in a 300 mm x 300 mm vertical shock tube and loaded with a planar shock wave of M _s = 1.22 in air. The drag force history on the sphere was measured by an accelerometer installed in it. Accelerometer output signals were subjected to deconvolution data processing, producing a drag history comparable to that obtained by solving numerically the Navier-Stokes equations. A good agreement was obtained between the measured and computed drag force histories. In order to interpret the interaction of shock wave over the sphere, high speed video recordings and double exposure holographic interferometric observations were also conducted. It was found that the maximum drag force appeared not at the time instant when the shock arrived at the equator of the sphere, but at some earlier time before the transition of the reflected shock wave from regular to Mach reflection took place. A negative value of the drag force was observed, even though for a very short duration of time, when the Mach stem of the transmitted shock wave relfected and focused at the rear stagnation point of the sphere.Received: 31 March 2003, Accepted: 7 July 2003, Published online: 2 September 2003     

Experimental investigation of the asymptotically approached Mach reflection over the second surface in a double wedge reflection

The assumption that the Mach reflection which is formed over the second surface of a double wedge with angles _w ¹ and kw/2 approaches asymptotically the Mach reflection which would have been obtained by an identical incident shock wave over a single wedge with an angle _w = _w ² was verified experimentally. The verification of this assumption supports the shock polar analysis suggested by Ben-Dor et al. (1987) for the study of the reflection process of a planar shock wave over a double wedge. Measurements of the rate of approach to the asymptotic value are also provided.     

Modification of Whitham's ray shock theory for analyzing the reflection of weak shock waves over small wedge angles

The weak Mach reflection phenomenon has been analyzed by applying both the shock dynamics approach and the disturbance propagation concept. The analysis which is based on modified Whitham's ray shock theory results in analytical expressions for the triple point trajectory angle,, and the shape of the curved Mach stem, which are functions of the incident shock wave Mach number,M _i, and the reflecting wedge angle, _w. The analytical results were found to be in good agreement with experimental results.This article was processed using Springer-Verlag T_EX Shock Waves macro package 1.0 and the AMS fonts, developed by the American Mathematical Society.     

Attenuation of shock waves by granular filters

Attenuation of shock waves in granular filters has been studied. Both pressurized air and solid explosives have been used for generating shock waves in a shock tube. The shock tube had a total length of m, and an internal diameter of 355 mm. Two large scale experiments have also been carried out in a tunnel with a cross-sectional area of 6.5 m². The results are compared with results found in the literature (Zloch, 1976; Medvedev et al., 1990; Britan et al., 2001) and previous experiments in a smaller scale by Slungaard (2002). A simple correlation based on the work of Zloch (1976) for shock wave attenuation in tube bundles and an extensive amount of experiments, is proposed. The correlation can be used to estimate the attenuation of the shock wave through a granular filter with filter characteristic . Setting B=6 will give a conservative estimate of the attenuation, while setting B=3 will give the best fit to all the results from this study and the results found in the literature. The correlation is independent of the type of driver (pressurised air or solid explosives) and upstream shock strength.Received: 30 September 2002, Accepted: 19 December 2002, Published online: 6 March 2003     

A study of the interaction between two triple points

In the Mach reflection of plane shock on a concave double wedge, after two triple points collide with each other, the wave pattern is usually complicated. In this paper, firstly, a shock dynamic approach is used for studying this problem. In this approach, the method of shock-shock polar is used for better understanding the pattern of disturbance propagation. A downward-traveling shock-shock disturbance on the Mach stem is predicted theoretically. Secondly, based on the idea of shock dynamic approach, a gas dynamic model is built for studying the same problem. A similar result is obtained and the formation of the downward-traveling triple point is analyzed. This downward-traveling disturbance propagates and reflects between the upward-traveling shock-shock locus and the wall surface, causing the Mach number of Mach stem to increase and making the wave configuration approach to the one in the Mach reflection on a single wedge.Received: 19 April 2004, Accepted: 2 December 2004, Published online: 16 March 2005[/PUBLISHED]Correspondence to: P. Xie, Z.Y. Han     

Visualization of shock-wave formation processes during shock reflection at obstacles with multiple steps

According to standard textbooks on compressible fluid dynamics, a shock wave is formed by an accumulation of compression waves. However, the process by which an accumulated compression wave grows into a shock wave has never been visualized. In the present paper, the authors tried to visualize this process using a model wedge with multiple steps. This model is useful for generating a series of compression waves and can simulate a compression process that occurs in a shock tube. By estimating the triple-point trajectory angle, we demonstrated visually that an accumulated compression wave grows into a shock wave. Further reflection experiments over a rough-surface wedge confirmed the tendency for the triple point trajectory angle to reach the asymptotic value _s in the end.This work was first presented at the Symposium on Shock Waves, Japan 2002     

Shock-unsteadiness model applied to oblique shock wave/turbulent boundary-layer interaction

Reynolds-averaged Navier–Stokes prediction of shock wave/turbulent boundary layer interactions can yield significant error in terms of the size of the separation bubble. In many applications, this can alter the shock structure and the resulting surface properties. Shock-unsteadiness modification of Sinha et al. (Physics of Fluids, Vol.15, No.8, 2003) has shown potential in improving separation bubble prediction in compression corner flows. In this article, the modification is applied to oblique shock wave interacting with a turbulent boundary layer. The challenges involved in the implementation of the shock-unsteadiness correction in the presence of multiple shock waves and expansion fans are addressed in detail. The results show that a robust implementation of the model yields appreciable improvement over standard k–ω turbulence model predictions.     

On the interference of a weak and a strong shock wave in a dissociating nitrogen flow

The influence of the nitrogen dissociation on the interactions due to the interference of two planar shock waves in a hypersonic high enthalpy flow is theoretically investigated for infinite reaction rates. The two limiting cases of infinitely slow and infinitely fast reactions are modelled as a perfect gas and an ideal dissociating gas in chemical equilibrium.To investigate the influence of finite reaction rates on the interactions of shock waves, experiments are performed in the high enthalpy shock tunnel Göttingen (HEG) with a wind tunnel model consisting of a wedge type shock generator and a transversally mounted cylinder. The pressure and heat transfer loads resulting from the shock wave interferences are measured and the flow field is visualized by means of interferograms. The experimental results are compared with the results of a numerical simulation for a dissociating nitrogen flow and with the experimental results for a perfect gas flow.     

A numerical study of oblique shock wave reflections over wedges in an ideal quantum gas

The various oblique shock wave reflection patterns generated by a moving incident shock on a planar wedge using an ideal quantum gas model are numerically studied using a novel high resolution quantum kinetic flux splitting scheme. With different incident shock Mach numbers and wedge angles as flow parameters, four different types of reflection patterns, namely, the regular reflection, simple Mach reflection, complex Mach reflection and the double Mach reflection as in the classical gas can be classified and observed. Both Bose–Einstein and Fermi–Dirac gases are considered.      

Experimental and numerical studies of underwater shock wave attenuation

The attenuation of an underwater shock wave by a thin porous layer is studied both experimentally and numerically. The shock waves are generated by exploding 10 mg silver azide pellets and the pressures at different distances from the explosion center are measured. Measurements are also carried out with a gauze layer placed between the explosion source and the pressure gauge. The results with and without the gauze layer are compared evaluating the shock wave attenuation. Numerical simulations of the phenomenon are also carried out for a simple wave attenuation model. The results are compared with the experimental data. Despite the simple mathematical model of wave attenuation, the agreement between the experimental and numerical results is reasonable.Received: 22 October 2002, Accepted: 17 June 2003, Published online: 5 August 2003PACS: 47.11.+j, 47.40.Nm, 47.55.Mh