Structure Of Normal Shock Waves Research Articles

The Mott-Smith solution to the regular shock reflection problem

The classical Mott-Smith solution for one-dimensional normal shock wave structure is extended to the two-dimensional regular shock reflection problem. The solution for the non-equilibrium molecular velocity distribution function along the symmetry-plane streamline is obtained as a weighted sum of four Maxwellians. An analysis of applicability of the solution has been performed using the results of direct simulation Monte Carlo calculations for a range of incident shock wave intensities. Accuracy of the solution improves with increasing$Ma_n$, the Mach number normal to the shock front, so that the solution becomes rather accurate for strong shocks with$Ma_n>8$.

Mechanism of shock-train/boundary-layer interaction in spanwise concave isolator channels

A series of spanwise concave channel models of isolators with different radii were investigated to reveal the mechanism of shock train/boundary layer interaction in the inward-turning inlet. Numerical simulations of the shock train flow under different incoming Mach numbers, incoming static pressures, and backpressure ratios were performed. The results show that the shock train leading edge exhibits a “λ” shape in the concave channel, and that a Mach stem is present at the intersection of the shock waves on the top and bottom walls. As the radius of the concave wall is increased, the streamwise force generated by the wall decreases and the upward force perpendicular to the wall increases, which leads to an increase in the thickness of the boundary layer and weakening of the ability of the boundary layer fluid to resist the pressure gradient. Owing to the increase in the radius of the concave channel, the shock train moves upstream, the Mach stem in the shock train leading edge becomes shorter, the distance between the reflection shock waves increases, and the shock train becomes longer. Compared with the concave bottom wall, the separation flow at the flat top wall extends further streamwise. As the incoming Mach number increases, the inertial effect is strengthened, and the shock train evolves towards a normal shock wave structure. As the incoming static pressure decreases, the viscous effect is enhanced, and the flow field of the shock train tends to evolve to reflect more shock waves with compression over longer streamwise distances.

Shock wave structure in non-ideal dilute gases under variable Prandtl number

This paper investigates the structure of normal shock waves for a planar steady flow of non-ideal dilute gases under variable viscosity and thermal conductivity using the Navier–Stokes–Fourier approach to the continuum model. The gas is assumed to follow the simplified van der Waals equation of state along with the power-law temperature-dependent coefficients of shear viscosity, bulk viscosity, and thermal conductivity. A closed system of nonlinear differential equations having a variable Prandtl number ( $$\Pr $$ ) is formulated. Exact analytical solutions of the shock wave structure in non-ideal gases are derived for $$\Pr \rightarrow \infty $$ and $$\Pr \rightarrow 0$$ limits, and the corresponding profiles for velocity and temperature are obtained. For $$\Pr \rightarrow 0$$ , an isothermal shock is encountered for high Mach numbers. It appears sooner in non-ideal gases. The solution profiles for $$\Pr =2/3$$ are obtained numerically and compared with the corresponding profiles for $$\Pr \rightarrow 0$$ , 3/4, and $$\infty $$ under the same initial conditions. Qualitative agreement is obtained with the theoretical and experimental results for the shock wave structure. The inverse shock thickness is computed for different values of $$\Pr $$ , and it is found that the inverse shock thickness increases with an increase in the Prandtl number. The bulk viscosity, the non-idealness parameter, the specific heat ratio, the power-law index, and the pre-shock Mach number have a significant effect on the shock wave structure.

Parametric study of wind tunnel test section configurations for stabilizing normal shock wave structure

Presented are results from a parametric investigation of wind-tunnel test-section configurations with a goal of stabilization of normal-shock-wave structure. The test section includes a two-flow-passage arrangement, where each passage is separated by a shock-wave-holding plate. The top wall for the top passage is contoured relative to the streamwise-flow direction, and a choking flap is located at the downstream portion of the bottom flow passage. Altered are the streamwise and spanwise positions of the shock-wave-holding plate, angle of the choking flap, and amount of venting. Of interest are shadowgraph flow visualization images, grayscale spectral energy variations, and integrated grayscale spectral energy levels. Higher static-pressure ratio downstream of the shock wave (caused by higher choking-flap angle, lower shock-wave-holding-plate position, and less venting) is associated with greater shock-wave-standoff distance (relative to the shock-wave-holding plate) and decreased flow unsteadiness. The most optimal arrangement includes a stabilized normal shock wave and lambda foot, which are largely two dimensional over the shadowgraph visualization volume.

Large-scale molecular dynamics simulations of normal shock waves in dilute argon

Large-scale molecular dynamics (MD) simulations using the Lennard-Jones potential are performed to study the structure of normal shock waves in dilute argon. Nonperiodic boundary conditions in the flow direction are applied by coupling the MD domain with a two-dimensional finite-volume computational fluid dynamics (CFD) solver to correctly generate the inflow and outflow particle reservoirs. Detailed comparisons are made with direct simulation Monte Carlo (DSMC) solutions using the variable-hard-sphere (VHS) collision model. By performing realistic MD simulations of full shock waves, this article presents a more sensitive evaluation of the VHS model parameters (via temperature and velocity distribution functions) than is possible using available experimental density measurements. In the high temperature range (300–8000 K), where the Chapman–Enskog theory supports the VHS model assumptions, near-perfect agreement between MD and DSMC solutions is demonstrated and inverse shock thickness predictions reproduce experimental measurements. In the low temperature range (16–300 K), theory predicts and MD simulation confirms that the VHS collision model becomes less valid.

One-Dimensional Multiple-Temperature Gas-Kinetic Bhatnagar-Gross-Krook Scheme for Shock Wave Computation

Accurately computing the inner structure of normal shock waves or oblique shock waves is crucial for many hypersonic applications. As such, it will improve the prediction accuracy of aerodynamics properties and aerothermal effects on hypersonic vehicles and spacecraft during atmospheric entries. Because a shock wave usually has a thickness of a few mean free paths, it is very difficult to accurately compute the detailed nonequilibrium inner structure across a shock wave with a continuum method. This paper reports a gas-kinetic Bhatnagar-Gross-Krook scheme for computations of one-dimensional vibrationally nonequilibrium nitrogen flows through a planar shock wave. The present gas-kinetic Bhatnagar-Gross-Krook scheme solves for the shock structure with multiple temperatures, including two translational temperatures, one rotational temperature, and one vibrational temperature. The salient features of the present gas-kinetic Bhatnagar-Gross-Krook method are multifold. Its applicability covers a wide simulation regime, extending that of continuum flows to the transition flows; it is more computationally efficient in time than the traditional direct simulation Monte Carlo method for shock wave simulation. To provide proper downstream subsonic boundary conditions for very strong shock waves, it is required to determine a proper postshock equilibrium state in which all temperatures have accomplished relaxation processes to a common equilibrium temperature. Analytical expressions of a complete set of generalized Rankine-Hugoniot relations across a planar shock wave are obtained to account for the variant specific heat ratio due to inner energy excitations. Numerical simulation results by the present gas-kinetic Bhatnagar-Gross-Krook scheme and the direct simulation Monte Carlo method are found to be in good agreement.

The Efficiency of Inelastic Collision Models with the Variable Sphere Molecular Model in the Direct Simulation Monte Carlo Method

The variable sphere (VS) molecular model for the inverse power law (IPL) potential is combined with the statistical inelastic cross section (SICS) and Borgnakke-Larsen (BL) models and the transport coefficients are estimated using the Monte Carlo integration technique for Wang-Chang-Uhlenbeck (WCU) theory. The VS model with SICS (VS-SICS) and BL (VS-BL) models are applied to the simulation of the normal shock wave structure in a nitrogen gas using the direct simulation Monte Carlo (DSMC) method, and then compared with the results of VSS model combined with the SICS model (VSS-SICS) and BL model (VSS-BL). Structures of number density, rotational temperature profiles, and rotational energy distribution in the VS-SICS and VS-BL models shock wave agree well with those of the VSS-SICS and VSS-BL models. The VS model demonstrated applicability to the inelastic molecular collision model.

Rotational Collision Number and Rotationally Collision Cross Section of Inelastic Collision Model for Nitrogen Molecule in the Direct Simulation Monte Carlo Method

Transport coefficients and rotational collision number of nitrogen molecule were evaluated using the Monte Carlo integration with the classical trajectory calculation (CTC) for a set of equations in the Wang-Chang-Uhlenbeck (WCU) theory, and then the rotationally inelastic collision cross section for the statistical inelastic cross section (SICS) model was made so as to reproduce the rotational collision number of CTC. Transport coefficients and rotational collision number of SICS model with the rotationally inelastic cross section presented in this study were in reasonable agreement with those of CTC. SICS model was applied to the simulation of nitrogen normal shock wave structure and compared with CTC and experimental results. The density and translational structure in the shock wave were in reasonable agreement with those of CTC results. However, the structures of rotational temperature in the highly non-equilibrium state were in poor agreement with those of CTC.

Variable sphere molecular model for inverse power law and Lennard-Jones potentials in Monte Carlo simulations

The variable sphere (VS) molecular model for the Monte Carlo simulation of rarefied gas flow is introduced to provide consistency for diffusion and viscosity cross-sections with those of any realistic intermolecular potential. It is then applied to the inverse power law (IPL) and Lennard-Jones (LJ) potentials. The VS model has a much simpler scattering law than either the variable hard sphere (VHS) or variable soft sphere (VSS) models; also, it has almost the same computational efficiency as the VHS and VSS models. A simulation of velocity relaxation in a homogeneous space and two comparative simulations of molecular diffusion in a homogeneous heat-bath gas and normal shock wave structure in a monatomic gas are made to examine VS model validity. The relaxation to a Maxwellian distribution function and equipartition between all degrees of freedom are well established; good agreement is shown in the molecular diffusion and shock wave structure between the VS model and the IPL and LJ potentials. The VS model is combined with the statistical inelastic cross-section (SICS) model and applied to simulation of translational and rotational energy relaxation in a homogeneous space. The VS model shows the relaxation of Maxwellian and Boltzmann distribution functions and equipartition between all degrees of freedom. Comparative calculation between the VS model with the SICS (VS-SICS) model and the VSS model with the SICS (VSS-SICS) model is made for rotational relaxation in a nitrogen normal shock wave. Good agreement is shown in the shock wave structure and rotational energy distribution function between the VS-SICS model and the VSS-SICS model. This study demonstrates that diffusion and viscosity cross-sections, rather than the scattering law of each molecular collision, affect macroscopic transport phenomena.

Parallel computing of diatomic molecular rarefied gas flows

A parallel algorithm for direct simulation Monte Carlo calculation of diatomic molecular rarefied gas flows is presented. For reliable simulation of such flow, an efficient molecular collision model is required. Using the molecular dynamics method, the collision of N 2 molecules is simulated. For this molecular dynamics simulation, the parameter decomposition method is applied for parallel computing. By using these results, the statistical collision model of diatomic molecule is constructed. For validation this model is applied to the direct simulation Monte Carlo method to simulate the energy distribution at equilibrium condition and the structure of normal shock wave. For this DSMC calculation, the domain decomposition is applied. It is shown that the collision process of diatomic molecules can be calculated precisely and the parallel algorithm can be efficiently implemented on the parallel computer.

Test of Efficiency of Variable Soft Sphere and Variable Hard Sphere Molecular Models in the Direct Simulation Monte Carlo Method.

The efficiency of the Variable Soft Sphere (VSS) and Variable Hard Sphere (VHS) molecular models is investigated by estimating the transport coefficients, and simulating a free jet in a binary gas mixture, and a normal shock wave structure in a binary gas mixture, using the Lennard-Jones (LJ) potential and LJ potential type VSS and VHS models. It is confirmed that the transport coefficients using the VSS model are consistent with those of the LJ potential, and calculated results using the VSS model for free jet and normal shock wave structures are in good agreement with those of the LJ potential. There are, however, some discrepancies in transport coefficients and calculated results using the VHS model and the LJ potential

Test of Efficiency of Variable Hard Sphere(VHS) and Variable Soft Sphere(VSS) Models in the Direct Simulation Monte Carlo Method.

The efficiency of the variable hard sphere (VHS) and variable soft sphere (VSS) molecular models is investigated by estimating the transport coefficients, and simulating a free jet in a binary gas mixture and a normal shock wave structure in a binary gas mixture using the Lennard-Jones (6-12) (LJ) potential and LJ potential (type) VHS and VSS models. It is confirmed that the transport coefficients using the VSS model are consistent with those of the LJ potential and calculated results using the VSS model for free jet and shock wave structures are in good agreement with those of LJ potential. There are, however, some discrepancies in transport coefficients and results calculated using the VHS model and LJ potential.

Structure of normal shock waves in a gas-particle mixture

Structure of normal shock waves in a gas-particle mixture

Lennard-Jones-Potential-Type Isotropic Scattering Model for the Direct-Simulation Monte Carlo Method. 1st Report.

The Lennard-Jones (6-12)-potential-type isotropic scattering model for the direct-simulation Monte Carlo (DSMC) method is presented. This model is an extention of the variable hard sphere (VHS) model in which diameter (collision cross section) varies with relative collision velocity (energy) and deflection angle is isotropic. The validity of the Lennard-Jones (LJ)-potential-type isotropic scattering model is studied by the simulation of argon (Ar) normal shock wave structure. Calculation results, such as number density, velocity, temperature, and heat flux are in good agreement with those of LJ potential.

Lennard-Jones-Potential-Type Isotropic Scattering Model for the Direct-Simulation Monte Carlo Method. (2nd Report).

The efficiency of the Lennard-Jones (6-12) -potential-type isotropic scattering model for the direct-simulation Monte Carlo (DSMC) method is studied by the simulation of normal shock wave structure in helium (He) and argon (Ar) binary gas mixture. The Lennard-Jones (LJ)-type isotropic scattering model has a varying diameter (collision cross section) with relative collision energy (velocity) which is consistent with the diffusion or viscosity coefficient of LJ potential. Calculation results of this model, such as number density, velocity, and translational temperature, are in reasonable agreement with those of LJ potential, although profiles of this model are slightly thinner than those of LJ potential.

Structure of normal shock waves: Direct numerical analysis of the Boltzmann equation for hard-sphere molecules

The structure of normal shock waves is investigated on the basis of the standard Boltzmann equation for hard-sphere molecules. This fundamental nonlinear problem in rarefied gas dynamics is analyzed numerically by a newly developed finite-difference method, where the Boltzmann collision integral is computed directly without using the Monte Carlo method. The velocity distribution function, as well as the macroscopic quantities, is accurately obtained. The numerical results are compared with the Mott-Smith and the direct simulation Monte Carlo results in detail. The analytical solution for a weak shock wave based on the standard Boltzmann equation is also presented up to the second order of the shock strength together with its explicit numerical data for hard-sphere molecules.

Accuracy of the Burnett equations for hypersonic real gas flows

Burnett solutions for the structure of normal shock waves for Mach numbers ranging from 1.1 to 50 have been obtained by using the time-dependent equation set in conjunction with modern computational fluid dynamic techniques. The current work seeks to provide corrected and improved high-resolution direct simulation Monte Carlo (DSMC) results for hypersonic shocks in monatomic gases. These DSMC results will serve as a reliable test base for assessing the accuracy of various continuum equations more advanced than Navier-Stokes, such as the Burnett equations. The main conclusion of Fisko and Chapman (1988) that the Burnett equations are more accurate than the Navier-Stokes equations for all gases and at all Mach numbers in matching DSMC shock wave results is reassesed by comparing the improved and corrected DSMC results with continuum Burnett results. Various Burnett results for shocks in a diatomic gas, nitrogen, which include improvements to the work of Lumpkin and Chapman (1989) are presented. It is concluded that the Burnett equations represent a significant improvement over Navier-Stokes equations for flows exhibiting translational nonequilibrium.

Variable soft sphere molecular model for inverse-power-law or Lennard-Jones potential

The variable soft sphere (VSS) molecular model is introduced for both the viscosity and diffusion cross sections (coefficients) to be consistent with those of the inverse-power-law (IPL) or Lennard-Jones (LJ) potential. The VSS model has almost the same analytical and computational simplicity (computation time) as the variable hard sphere (VHS) model in the Monte Carlo simulation of rarefied gas flows. The null-collision Monte Carlo method is used to make comparative calculations for the molecular diffusion in a heat-bath gas and the normal shock wave structure in a simple gas. For the most severe test of the VSS model for the IPL potential, the softest practical model corresponding to the Maxwell molecule is chosen. The agreement in the molecular diffusion and shock wave structure between the VSS model and the IPL or LJ potential is remarkably good.

Normal shock-wave structure in two-phase vapour-droplet flows

A study of the structure of stationary, fully and partly dispersed, normal shock waves in steady vapour–droplet, two-phase flow is presented. Pure substances only are considered, but, unlike most previous work, the droplet population is allowed to be polydispersed. It is shown how the effects of thermal relaxation for such a mixture can be elegantly incorporated into the analysis.Three types of fully dispersed wave are identified. Type I waves are dominated by thermal relaxation and an approximate analytical solution is presented which gives results in close agreement with accurate numerical solutions of the governing equations. The analysis predicts some unexpected behaviour of the thermodynamic variables and demonstrates the correct scaling parameters for such flows. An approximate analysis is also presented for Type II waves, dominated by both velocity and thermal relaxation. Type III waves, where all three relaxation processes are important, are of little practical significance and are only briefly discussed. Partly dispersed waves are also considered and the results of a numerical simulation of the relaxation zone are presented. A linearized solution of this problem is possible but, unlike other relaxing gas flows, does not give good agreement with the more exact numerical calculations.The apparent discontinuity in the speed of sound in a vapour–droplet mixture as the wetness fraction tends to zero has been responsible for some confusion in the literature. This problem is reconsidered and it is shown that the transition from the two-phase equilibrium to the single-phase frozen shock wave speed is continuous.

Numerical study of discrete-velocity gases

A finite-difference method for solving the discrete Boltzmann equations, which are the governing equations for a model gas in which molecules have many discrete velocities, is developed. The method is applied to three fundamental problems in rarefied gas flow to study the features of discrete-velocity gases: normal shock wave structure, heat transfer between two parallel plates, and two-dimensional vapor deposition. Two different discrete-velocity gas models are used.