Heat flux measurement over a cone in a shock tube flow

This paper is the part 2 of our previous thin film heat transfer measurements. In the first report we measured time variations of heat flux over a cylinder placed in a shock tube flow and compared experimental results with CFD results, Saito et al. (Shock Waves 14:327–333, 2004). We report a result of heat transfer measurements over an 86° apex angle cone surface impinged by a Ms = 2.38 shock wave in air with distributed thin film transfer gauges along cone surface and its comparison with results of numerical simulations. We performed double exposure holographic interferometric observation, and also from the heat transfer measurement and numerical simulation, confirmed the presence of delayed transition from regular to Mach reflection over the cone. The numerical estimation of delayed transition distance from the apex agreed very well with experimental one.      

Mach reflection of a plane shock wave passing a mountain of 45° inclination

The transition from regular reflection (RR) to Mach reflection (MR) as a plane shock wave diffracts around a triangular mountain of 45° inclination is analysed in this paper, both by optical measurement in a shock tube and by numerical simulation the numerical method developed by Li Yingfan^[1] is of the FLIC type with triangular mesh. The dependence of the critical transition point Lk ofRR→MR on shock Mach numberM _i is analyzed and the variations of the incidence angle ω _i of the impinging shock and the reflection angle ω _r with the distanceL ^* are investigated. Our experimental and numerical results agree well with the theoretical results of Iton and Italya.     

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     

Study on the Fuel Air Mixing Induced by a Shock Wave Propagating into a H2-Air Interface

2nd-order upwind TVD scheme was used to solve the laminar, fully Navier-Stokes equations. The numerical simulations were done on the propagation of a shock wave with Ma _S = 2 and 4 into a hydrogen and air mixture in a duct and a duct with a rearward step. The results indicate that a swirling vortex may be generated in the lopsided interface behind the moving shock. Meanwhile, the complex shock system is also formed in this shear flow region. A large swirling vortex is produced and the fuel mixing can be enhanced by a shock wave at low Mach number. But in a duct with a rearward step, the shock almost disappears in hydrogen for Ma _S = 2. The shock in hydrogen will become strong if Ma _S is large. Similar to the condition of a shock moving in a duct full of hydrogen and air, a large vortex can be formed in the shear flow region. The large swirling vortex even gets through the reflected shock and impacts on the lower wall. Then, the distribution of hydrogen behind the rearward step is divided into two regions. The transition from regular reflection to Mach reflection was observed as well in case Ma _S = 4.     

Numerical simulation of receptivity of a hypersonic boundary layer to acoustic disturbances

Direct numerical simulations of the evolution of disturbances in a viscous shock layer on a flat plate are performed for a free-stream Mach number M _∞ = 21 and Reynolds number Re _L = 1.44 · 10⁵. Unsteady Navier-Stokes equations are solved by a high-order shock-capturing scheme. Processes of receptivity and instability development in a shock layer excited by external acoustic waves are considered. Direct numerical simulations are demonstrated to agree well with results obtained by the locally parallel linear stability theory (with allowance for the shock-wave effect) and with experimental measurements in a hypersonic wind tunnel. Mechanisms of conversion of external disturbances to instability waves in a hypersonic shock layer are discussed. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 48, No. 3, pp. 84–91, May–June, 2007.     

Assessment of WENO-extended two-fluid modelling in compressible multiphase flows

The two-fluid modelling based on an advection-upwind-splitting-method (AUSM)-family numerical flux function, AUSM⁺-up, following the work by Chang and Liou [Journal of Computational Physics 2007;225: 840–873], has been successfully extended to the fifth order by weighted-essentially-non-oscillatory (WENO) schemes. Then its performance is surveyed in several numerical tests. The results showed a desired performance in one-dimensional benchmark test problems: Without relying upon an anti-diffusion device, the higher-order two-fluid method captures the phase interface within a fewer grid points than the conventional second-order method, as well as a rarefaction wave and a very weak shock. At a high pressure ratio (e.g. 1,000), the interpolated variables appeared to affect the performance: the conservative-variable-based characteristic-wise WENO interpolation showed less sharper but more robust representations of the shocks and expansions than the primitive-variable-based counterpart did. In two-dimensional shock/droplet test case, however, only the primitive-variable-based WENO with a huge void fraction realised a stable computation.     

Interaction of a shock wave with a two-phase interface

The acceleration by an incident shock of a planar interface between a gas and a particle-gas mixture has been investigated experimentally and numerically. The experiments were conducted in a newly developed vertical shock tube in which the planar interface of the particle-gas mixture was generated and its particle concentration history was measured. Polydisperse corn starch particles with a mean diameter of 10m were used. We recorded the motion of the interface, as well as of the incident and reflected shock by using a 4 channel spark shadowgraph. The experimental conditions were Mach numberM _s=5.15 and initial pressurep ₁=50kPa for various particle concentrations in nitrogen. The reflected shock appears with a delay after the incident shock enters the particle-gas mixture. Numerical methods were employed to solve the two-phase governing equations. Experiments and numerical solutions are in good agreement.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.     

Wave processes in the shock layer on a flat plate at an angle of attack

A numerical and experimental study of receptivity of the viscous shock layer on a flat plate aligned at an angle of attack to external acoustic perturbations is performed. Density and pressure fluctuations are measured in experiments at the free-stream Mach number M _∞ = 21 and Reynolds number Re ₁ = 6·10 ⁵ m ⁻¹ . Direct numerical simulations of receptivity of the viscous shock layer to external acoustic perturbations in wide ranges of the governing parameters are performed by solving the Navier-Stokes equations with the use of high-order shock-capturing schemes. The calculated intensities of density and pressure fluctuations are found to be in good agreement with experimental data. Results of the study show that entropy-vortex disturbances dominate in the shock layer at small angles of attack, whereas acoustic perturbations prevail at angles of attack above 20°.     

Effect of sound-absorbing materials on intensity of disturbances in the shock layer on a flat plate aligned at an angle of attack

Results of a numerical and experimental study of characteristics of disturbances in a hypersonic shock layer on a flat plate covered by a sound-absorbing coating and aligned at an angle of attack are presented. Experiments and computations are performed for the free-stream Mach number M _∞ = 21 and Reynolds number Re _L = 6 · 10⁴. A possibility of suppressing pressure fluctuations in the shock layer at frequencies of 20–40 kHz with the use of tubular and porous materials incorporated into the plate surface is demonstrated. Results of numerical simulations are found to be in good agreement with experimental data.     

Estimation of shock induced vorticity on irregular gaseous interfaces: a wavelet-based approach

We study the interaction of a shock with a density-stratified gaseous interface (Richtmyer–Meshkov instability) with localized jagged and irregular perturbations, with the aim of developing an analytical model of the vorticity deposition on the interface immediately after the passage of the shock. The jagged perturbations, meant to simulate machining errors on the surface of a laser fusion target, are characterized using Haar wavelets. Numerical solutions of the Euler equations show that the vortex sheet deposited on the jagged interface rolls into multiple mushroom-shaped dipolar structures which begin to merge before the interface evolves into a bubble-spike structure. The peaks in the distribution of x-integrated vorticity (vorticity integrated in the direction of the shock motion) decay in time as their bases widen, corresponding to the growth and merger of the mushrooms. However, these peaks were not seen to move significantly along the interface at early times i.e. t < 10 τ, where τ is the interface traversal time of the shock. We tested our analytical model against inviscid simulations for two test cases – a Mach 1.5 shock interacting with an interface with a density ratio of 3 and a Mach 10 shock interacting with a density ratio of 10. We find that this model captures the early time (t/τ ∼ 1) vorticity deposition (as characterized by the first and second moments of vorticity distributions) to within 5% of the numerical results. PACS 47.40.Nm; 47.20.Ma     

Experimental study and direct numerical simulation of the evolution of disturbances in a viscous shock layer on a flat plate

The evolution of disturbances in a hypersonic viscous shock layer on a flat plate excited by slow-mode acoustic waves is considered numerically and experimentally. The parameters measured in the experiments performed with a free-stream Mach number M _∞ = 21 and Reynolds number Re _L = 1.44 · 10⁵ are the transverse profiles of the mean density and Mach number, the spectra of density fluctuations, and growth rates of natural disturbances. Direct numerical simulation of propagation of disturbances is performed by solving the Navier-Stokes equations with a high-order shock-capturing scheme. The numerical and experimental data characterizing the mean flow field, intensity of density fluctuations, and their growth rates are found to be in good agreement. Possible mechanisms of disturbance generation and evolution in the shock layer at hypersonic velocities are discussed. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 47, No. 5, pp. 3–15, September–October, 2006.     

Experimental and numerical analysis of the structure of pseudo-shock systems in laval nozzles with parallel side walls

Detailed numerical and experimental investigations of pseudo-shock systems in a Laval nozzle with parallel side walls are carried out. The location of the pseudo-shock system is defined in this system of two choked Laval nozzles by the ratio of the critical cross sections A₂^*/A₁^*{{A}_{2}^*/{A}_{1}^*} , the stagnation pressure loss across the shock system and viscous losses. The wall pressure distributions and high-speed schlieren videos recorded in the experiments are compared to the results of a steady and an unsteady numerical simulation. For the steady case, good agreement is found between the calculated and measured shock structure and pressure distribution along the primary nozzle wall, except for a remaining slight deviation in the shock position. For the unsteady case, in which asymmetric shock configurations are observed, deviations of the results with respect to the stochastic wall attachment of the shock system are given which indicate the necessity of further investigations on that topic.     

Numerical simulation of a single rising bubble by VOF with surface compression

The capability of the direct volume of fluid method for describing the surface dynamics of a free two‐dimensional rising bubble is evaluated using quantities of a recently published benchmark. The model equations are implemented in the open source computational fluid dynamics library OpenFOAM®. Here, a main ingredient of the numerical method is the so‐called surface compression that corrects the fluxes near the interface between two phases. The application of this method with respect to two test cases of a benchmark is considered in the main part. The test cases differ in physical properties, thus in different surface tension effects. The quantities centre of mass position, circularity and rise velocity are tracked over time and compared with the ones given in the benchmark. For test case one, where surface tension effects are more pre‐eminent, deviations from the benchmark results become more obvious. However, the flow features are still within reasonable range. Nevertheless, for test case two, which has higher density and viscosity ratios and above all a lower influence of the surface tension force, good agreement compared with the benchmark reference results is achieved. This paper demonstrates the good capabilities of the direct volume of fluid method with surface compression with regard to the preservation of sharp interfaces, boundedness, mass conservation and low computational time. Some limitation regarding the occurrence of parasitic currents, bad pressure jump prediction and bad grid convergence have been observed. With these restrictions in mind, the method is suitable for the simulation of similar two‐phase flow configurations. Copyright © 2012 John Wiley & Sons, Ltd.     

Shock wave reflection over wedges: a benchmark test for CFD and experiments

In the Shock Wave Journal Vol. 2, No. 4 a benchmark test for shock wave reflection over wedges was announced. International scientists who are interested in shock wave research were invited to participate. The benchmark test aimed at comparison of various advanced numerical schemes as well as experimental results. During the last three years more than twenty results, including both CFD and experiments, were collected from all over the world. Efforts contributed by these scientists made the present benchmark test reach to a standard of the state of the art of the computational fluid dynamics applied to the shock wave research. However, it was regrettable not to publish all the results collected due to limitation on the available page number. Received 5 November 1994 / Accepted 9 September 1996     

Shock waves and turbulence in a hypersonic inlet

A numerical investigation for an axisymmetric hypersonic turbulent inlet flow field of a perfect gas is presented for a three-shock configuration consisting of a biconic and a cowl. An upwind parabolized Navier-Stokes solver based on Roe's scheme is used to compute an oncoming flow Mach numberM =8, temperatureT =216 K, and pressureP =5.5293×10³ N/m². In order to assess the flow quantities, the interaction between shock and turbulence, and the inlet efficiency, three different flow calculations — laminar, turbulent with incompressible and compressible two-equationk- turbulence models — have been performed in this work.Computational results show that turbulence is markedly enhanced across an oblique shock with step-like increases in turbulence kinetic energy and dissipation rate. This enhancement is at the expense of the mean kinetic energy of the flow. Therefore, the velocity behind the shock is smaller in turbulent flow and hence the shock becomes stronger. The entropy increase through a shock is caused not only by the amplification of random molecular motion, but also by the enhancement of the chaotic turbulent flow motion. However, only the compressiblek- turbulence model can properly predict a decrease in turbulence length scale across a shock. Our numerical simulation reveals that the incompressiblek- turbulence model exaggerates the interaction between shock and turbulence with turbulence kinetic energy and dissipation rate remaining high and almost undissipated far beyond the shock region. It is shown that proper modeling of turbulence is essential for a realistic prediction of hypersonic inlet flowfield. The performed study shows that the viscous effect is not restricted in the boundary layer but extends into the main flow behind a shock wave. The loss of the available energy in the inlet performance therefore needs to be determined from the shock-turbulence interaction. The present study predicts that the inlet efficiency becomes relatively lower when turbulence is taken into account.     

Slow Eigenvalues of Self-similar Solutions of the Dafermos Regularization of a System of Conservation Laws: an Analytic Approach

The Dafermos regularization of a system of n hyperbolic conservation laws in one space dimension has, near a Riemann solution consisting of n Lax shock waves, a self-similar solution u = u _ε(X/T). In Lin and Schecter (2003, SIAM J. Math. Anal. 35, 884–921) it is shown that the linearized Dafermos operator at such a solution may have two kinds of eigenvalues: fast eigenvalues of order 1/ε and slow eigenvalues of order one. The fast eigenvalues represent motion in an initial time layer, where near the shock waves solutions quickly converge to traveling-wave-like motion. The slow eigenvalues represent motion after the initial time layer, where motion between the shock waves is dominant. In this paper we use tools from dynamical systems and singular perturbation theory to study the slow eigenvalues. We show how to construct asymptotic expansions of eigenvalue-eigenfunction pairs to any order in ε. We also prove the existence of true eigenvalue-eigenfunction pairs near the asymptotic expansions.     

Vorticities in a LES Model for 3D Periodic Turbulent Flows

This paper is devoted to the study of a LES model to simulate turbulent 3D periodic flow. We focus our attention on the vorticity equation derived from this LES model for small values of the numerical grid size δ. We obtain entropy inequalities for the sequence of corresponding vorticities and corresponding pressures independent of δ, provided the initial velocity u₀ is in L_x² while the initial vorticity ω₀ = ∇ × u₀ is in L_x¹. When δ tends to zero, we show convergence, in a distributional sense, of the corresponding equations for the vorticities to the classical 3D equation for the vorticity.     

Temperature measurement of reflected shock wave by using chemical indicator

This report describes a new method for measuring the temperature of the gas behind the reflected shock wave in shock tube, corresponding to the reservoir temperature of a shock tunnel, based on the chemical reaction of small amount of CF₄ premixed in the test gas. The final product C₂F₄ is used as the temperature indicator, which is sampled and detected by a gas chromatography in the experiment. The detected concentration of C₂F₄ is correlated to the temperature of the reflected shock wave with the initial pressureP ₁ and test time τ as parameters in the temperature range 3 300 K<T<5 600 K, pressure range 5 kPa<P ₁<12 kPa and τ≅0.4 ms. The project supported by the China Aerodynamics Project for Basic Researches (J13.5.2 ZK04)     

Experimental study of density fluctuations in a hypersonic laminar wake behind a cone

The characteristics of natural fluctuations of density in a laminar near wake behind a sharp cone in a hypersonic flow of nitrogen at zero incidence are studied by the method of electronbeam fluorescence at Mach numberM=21 and unit Reynolds numberRe ₁=6·10⁵ m⁻¹. The distributions of the mean density, integral fluctuations, and spectra of density fluctuations are obtained, the longitudinal and azimuthal phase velocities of perturbations are determined, and the growth rates of perturbations in the wake are found. The results are compared with the measurement data in the shock layer on a flat plate. Institute of Theoretical and Applied Mechanics, Siberian Division, Russian Academy of Sciences, Novosibirsk 630090. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 41, No. 3, pp. 111–117, May–June, 2000     

Numerical prediction of turbulent thermocapillary convection in superposed fluid layers with a free interface

The present paper introduces a new numerical method for predicting the characteristics of thermocapillary turbulent convection in a differentially-heated rectangular cavity with two superposed and immiscible fluid layers. The unsteady Reynolds form of the Navier–Stokes equations and energy equation are solved by using the control volume approach on a staggered grid system using SIMPLE algorithm. The turbulence quantities are predicted by applying the standard k–ε turbulence model. The level set formulation is applied for predicting the topological changes of the interface separating the two fluid layers and to provide an accurate and robust modeling of the interfacial normal and tangential stresses. The computational results obtained showed good agreement when compared with the previous experimental, numerical and analytical benchmark data for different validation cases in both laminar and turbulent regimes. The present numerical method is then applied to predict the velocity and temperature distribution in two immiscible liquid layers with undeformable interface for a wide range of Marangoni numbers. The laminar-turbulent transition is demonstrated by obtaining the turbulence features at high interfacial temperature gradient which is characterized by high Marangoni number. The effect of increasing Marangoni number on the interface dynamics in turbulent regime is also investigated.