Two-dimensional Hypersonic Flow Research Articles

Implicit unified gas-kinetic scheme for steady state solution of hypersonic thermodynamic non-equilibrium flows

The unified gas-kinetic scheme (UGKS), designed to solve Boltzmann equation and its model equations, aims to accurately resolve multi-scale flow problems within a single computation framework. In this study, an implicit UGKS is proposed for the steady state solution of diatomic gas flows based on a multiple-temperature Boltzmann model equation with non-equilibrium among translational, rotational and vibrational modes (Zhang et al., 2023), utilizing a novel hybrid Cartesian-unstructured discrete velocity space (DVS) mesh approach. The multiple-temperature macroscopic equations corresponding to the Boltzmann model equation are simultaneously solved implicitly to address the stiffness and nonlinearity of the collision operator encountered when solving the Boltzmann model equation in isolation. Both macroscopic equations and Boltzmann model equation are solved by using the point relaxation symmetric Gauss–Seidel method to achieve a fast convergence rate. The efficiency of the implicit UGKS can be improved by about two orders of magnitude compared to the explicit one. Moreover, the hybrid Cartesian-unstructured DVS achieves a noteworthy reduction in both CPU-hours and memory consumption, requiring only 15%∼23% of the advanced unstructured DVS. It effectively alleviates the dimensional crisis of UGKS in solving three-dimensional hypersonic flows, and can be extended to other DVS-based methods. As a result, UGKS extends its applicable regime to simulations of hypersonic thermodynamic non-equilibrium flows with Mach numbers up to 30, achieving three-dimensional flow simulations of complex geometrical configurations with relatively low computational resources. A series of test cases, including normal shock structures, two-dimensional hypersonic flows around cylinder, blunt wedge and flat plate, and three-dimensional hypersonic flows around a sphere and an X38-like configuration vehicle, are conducted to validate the implicit UGKS with hybrid Cartesian-unstructured DVS. Overall, the proposed method shows advantages of unified multi-scale methods in terms of computational efficiency and capturing multi-scale solution of hypersonic thermodynamic non-equilibrium flows.

General synthetic iterative scheme for polyatomic rarefied gas flows

Recently, the general synthetic iterative scheme (GSIS) has been proposed to solve the Boltzmann equation efficiently and accurately in the whole range of gas rarefaction. Here, the GSIS is extended and further improved to find the steady-state solutions in nonlinear polyatomic gas flows. The numerical method is developed within the unstructured finite volume framework, where the kinetic equation and synthetic equations are solved alternately. The synthetic equations, which are derived from the modified Rykov kinetic model, reduce to the two-temperature (translational and rotational) Navier–Stokes–Fourier equations in the continuum flow regime, and include high-order constitutive relations for rarefaction effects in other flow regimes. These high-order terms are obtained by solving the kinetic model with the discrete velocity method. When the synthetic equations are solved, their macroscopic quantities are used to guide the evolution of velocity distribution functions in the kinetic model. As a consequence of this two-way coupling, GSIS not only finds steady-state solutions quickly but also relieves the constraints on spatial cell size. Furthermore, the robustness of GSIS is improved by modifying the boundary macroscopic fluxes in the macroscopic implicit iteration. The efficiency and multiscale property of GSIS is demonstrated through several challenging test cases, including the one-dimensional normal shock wave with large internal-translational relaxation times, the two-dimensional hypersonic flow passing a cylinder, and the pump-driven flow in Europe’s demonstration fusion power plant.

A multiscale discrete velocity method for diatomic molecular gas

In the previous study, the multiscale discrete velocity method (MDVM) has been developed for monatomic gas with particle translational motion only. Unlike the unified gas-kinetic scheme (UGKS) and discrete unified gas-kinetic scheme, which are the typical representative of multiscale kinetic methods, MDVM achieves multiscale property by mixing the solution of macroscopic control equations and the Boltzmann equation, without the need to calculate complex interface flux. Therefore, MDVM has a higher computational efficiency. To broaden the application scope of MDVM, the Rykov model, which elucidates the exchange of energy between molecular translational and rotational energies, is introduced into MDVM in this paper. Numerical simulations are conducted for various cases, including one-dimensional shock tube, one-dimensional nitrogen shock structure, two-dimensional lid-driven cavity flow, and two-dimensional hypersonic flows around a flat plate and a blunt circular cylinder. The present results agree well with those from the diatomic UGKS method, demonstrating the developed diatomic MDVM can simulate multi-scale, strongly non-equilibrium, diatomic molecular gas flow while exhibiting certain efficiency improvements compared to the diatomic UGKS.

A time-relaxed Monte Carlo method preserving the Navier-Stokes asymptotics

In this paper, a new time-relaxed Monte Carlo (TRMC) method is proposed for the inhomogeneous Boltzmann equation. Compared to the standard TRMC scheme, the proposed method performs the same convection operator, however, divides the collision operator by a micro-macro decomposition. The continuous part of the collision operator is constructed based on the first-order Chapman-Enskog expansion and solved by an explicit second-order scheme, while the numerical solution of the rest nonequilibrium part is still provided by the standard TRMC scheme. In this way, the new TRMC method demonstrates the same accuracy as the standard TRMC scheme in the kinetic limit, however, preserves Navier-Stokes asymptotics and the second-order accuracy in the fluid limit. Several numerical cases of inhomogeneous flows, such as the one-dimensional Poiseuille flow, Sod tube flow, the shock wave and two-dimensional hypersonic flow past a cylinder, are calculated and compared with direct simulation Monte Carlo (DSMC) or Navier-Stokes solutions. It is noted that the new TRMC scheme is more accurate and efficient than the standard TRMC and DSMC methods for simulations of multi-scale gas flows.

Data-driven nonlinear constitutive relations for rarefied flow computations

To overcome the defects of traditional rarefied numerical methods such as the Direct Simulation Monte Carlo (DSMC) method and unified Boltzmann equation schemes and extend the covering range of macroscopic equations in high Knudsen number flows, data-driven nonlinear constitutive relations (DNCR) are proposed first through the machine learning method. Based on the training data from both Navier-Stokes (NS) solver and unified gas kinetic scheme (UGKS) solver, the map between responses of stress tensors and heat flux and feature vectors is established after the training phase. Through the obtained off-line training model, new test cases excluded from training data set could be predicated rapidly and accurately by solving conventional equations with modified stress tensor and heat flux. Finally, conventional one-dimensional shock wave cases and two-dimensional hypersonic flows around a blunt circular cylinder are presented to assess the capability of the developed method through various comparisons between DNCR, NS, UGKS, DSMC and experimental results. The improvement of the predictive capability of the coarse-graining model could make the DNCR method to be an effective tool in the rarefied gas community, especially for hypersonic engineering applications.

Analytical method of nonlinear coupled constitutive relations for rarefied non-equilibrium flows

It is well known that Navier-Stokes equations are not valid for those high-Knudsen and high-Mach flows, in which the local thermodynamically non-equilibrium effects are dominant. To extend the non-equilibrium describing the ability of macroscopic equations, Nonlinear Coupled Constitutive Relation (NCCR) model was developed from Eu’s generalized hydrodynamic equations to substitute linear Newton’s law of viscosity and Fourier’s law of heat conduction in conservation laws. In the NCCR model, how to solve the decomposed constitutive equations with reasonable computational cost is a key ingredient of this scheme. In this paper, an analytic method is proposed firstly. Compared to the iterative procedure in the conventional NCCR model, the analytic method not only obtains exact roots of the decomposed constitutive polynomials, but also preserves the nonlinear constitutive relations in the original framework of NCCR methods. Numerical tests to assess the efficiency and accuracy of the proposed method are conducted for argon shock structures, Couette flows, two-dimensional hypersonic flows over a cylinder and three-dimensional supersonic flows over a three-dimensional sphere. These superior advantages of the current method are expected to render itself a powerful tool for simulating the hypersonic rarefied flows and microscale flows of high Knudsen number for engineering applications.

Quasi-classical trajectory-based non-equilibrium chemical reaction models for hypersonic air flows

Phenomenological models, such as Park’s widely used two temperature model, overpredict the reaction rate coefficients at vibrationally cold conditions and underpredict it at vibrationally hot conditions. To this end, two new chemical reaction models, the nonequilibrium total temperature (NETT) and nonequilibrium piecewise interpolation models for the continuum framework are presented. The focus is on matching the reaction rate coefficients calculated using a quasiclassical trajectory based dissociation cross section database. The NETT model is an intuitive model based on physical understanding of the reaction at a molecular level. A new nonequilibrium parameter and the use of total temperature in the exponential term of the Arrhenius fit ensure the NETT model has a simple and straightforward implementation. The efficacy of the new model was investigated for several equilibrium and nonequilibrium conditions in the form of heat bath simulations. Additionally, two-dimensional hypersonic flows around a flat blunt-body were simulated by employing various chemical reaction models to validate the new models using experimental shock tube data. Park’s two temperature model predicted higher dissociation rates and a higher degree of dissociation leading to lower peak vibrational temperatures compared to those predicted by the new nonequilibrium models. Overall, the present work demonstrates that the new nonequilibrium models perform better than Park’s two temperature model, especially in simulations with a high degree of nonequilibrium, particularly as observed in re-entry flows.

A new direct simulation Monte Carlo implementation for more efficient simulation of hypersonic flow over arbitrarily shaped bodies using dynamic cells

Methods are reported for less computationally expensive and more accurate implementations of the direct simulation Monte Carlo (DSMC) method for the simulation of high speed gas flows over arbitrarily shaped bodies. A new particle-tracking algorithm with a saving of computational time of up to 10% is reported in which tracking of particles is done with the help of big triangles having vertices lying on the boundary curves. An algorithm has been developed to generate DSMC cells for collision and sampling that contain a prescribed number of molecules. This algorithm is able to generate over 90% cells having the optimum number of seven or eight molecules for simulating collisions. Sampling for macroscopic properties is done on dynamic cells that contain a number of particles varying spatially as a function of the local number density. A criterion for finding the number of particles in sampling cells is presented. This criterion has been found to result in accurate and fast simulation of two-dimensional hypersonic flows of argon over a wedge, and argon and nitrogen over a circular cylinder.

Aerothermal characteristics of bleed slot in hypersonic flows

Two types of flow configurations with bleed in two-dimensional hypersonic flows are numerically examined to investigate their aerodynamic thermal loads and related flow structures at choked conditions. One is a turbulent boundary layer flow without shock impingement where the effects of the slot angle are discussed, and the other is shock wave boundary layer interactions where the effects of slot angle and slot location relative to shock impingement point are surveyed. A key separation is induced by bleed barrier shock on the upstream slot wall, resulting in a localized maximum heat flux at the reattachment point. For slanted slots, the dominating flow patterns are not much affected by the change in slot angle, but vary dramatically with slot location relative to the shock impingement point. Different flow structures are found in the case of normal slot, such as a flow pattern similar to typical Laval nozzle flow, the largest separation bubble which is almost independent of the shock position. Its larger detached distance results in 20% lower stagnation heat flux on the downstream slot corner, but with much wider area suffering from severe thermal loads. In spite of the complexity of the flow patterns, it is clearly revealed that the heat flux generally rises with the slot location moving downstream, and an increase in slot angle from 20° to 40° reduces 50% the heat flux peak at the reattachment point in the slot passage. The results further indicate that the bleed does not raise the heat flux around the slot for all cases except for the area around the downstream slot corner. Among all bleed configurations, the slot angle of 40° located slightly upstream of the incident shock is regarded as the best.

Parallel DSMC method using dynamic domain decomposition

A general parallel direct simulation Monte Carlo method using unstructured mesh is introduced, which incorporates a multi-level graph-partitioning technique to dynamically decompose the computational domain. The current DSMC method is implemented on an unstructured mesh using particle raytracing technique, which takes the advantages of the cell connectivity information. In addition, various strategies applying the stop at rise (SAR) (IEEE Trans Comput 1988; 39:1073-1087) scheme is studied to determine how frequent the domain should be re-decomposed. A high-speed, bottom-driven cavity flow, including small, medium and large problems, based on the number of particles and cells, are simulated. Corresponding analysis of parallel performance is reported on IBM-SP2 parallel machine up to 64 processors. Analysis shows that degree of imbalance among processors with dynamic load balancing is about 1 / 6 - 1 / 2 of that without dynamic load balancing. Detailed time analysis shows that degree of imbalance levels off very rapidly at a relatively low value with increasing number of processors when applying dynamic load balancing, which makes the large problem size fairly scalable for processors more than 64. In general, optimal frequency of activating SAR scheme decreases with problem size. At the end, the method is applied to compute two two-dimensional hypersonic flows, a three-dimensional hypersonic flow and a three-dimensional near-continuum twin-jet gas flow to demonstrate its superior computational capability and compare with experimental data and previous simulation data wherever available.

Asymptotic defect boundary layer theory applied to thermochemical non-equilibrium hypersonic flows

Viscous flow computations are required to predict the heat flux or the viscous drag on an hypersonic re-entry vehicle. When real gas effects are included, Navier–Stokes computations are very expensive, whereas the use of standard boundary layer approximations does not correctly account for the ‘entropy layer swallowing’ phenomenon. The purpose of this paper is to present an extension of a new boundary layer theory, called the ‘defect approach’, to two-dimensional hypersonic flows including chemical and vibrational non-equilibrium phenomena. This method ensures a smooth matching of the boundary layer with the inviscid solution in hypersonic flows with strong entropy gradients. A new set of first-order boundary layer equations has been derived, using a defect formulation in the viscous region together with a matched asymptotic expansions technique. These equations and the associated transport coefficient models as well as thermochemical models have been implemented. The prediction of the flow field around the blunt-cone wind tunnel model ELECTRE with non-equilibrium free-stream conditions has been done by solving first the inviscid flow equations and then the first-order defect boundary layer equations. The numerical simulations of the boundary layer flow were performed with catalytic and non-catalytic conditions for the chemistry and the vibrational mode. The comparison with Navier–Stokes computations shows good agreement. The wall heat flux predictions are compared to experimental measurements carried out during the MSTP campaign in the ONERA F4 wind tunnel facility. The defect approach improves the skin friction prediction in comparison with a classical boundary layer computation.

Stabilization of the Burnett equations and application to hypersonicflows

Numerical solutions of the Burnett for hypersonic flow at high altitudes in the continuum transitional regime were not possible except for some one-dimensional flows. It is shown from both analytical investigation and numerical computations that the Burnett are unstable to disturbances of small wavelengths. This fundamental instability arises in numerical computations when the grid spacing is less than the order of a mean free path and precludes Burnett flowfield computations above a certain maximum altitude for any given aerospace vehicle. A new set of termed the augmented Burnett equations has been developed and shown to be stable both by a linearized stability analysis and by direct numerical computations for one-dimensional and plane-two-dimensional flows. The latter represents the first known Burnett solutions for two-dimensional hypersonic flow over blunt leading edges. The comparison of these solutions with the conventional Navier-Stokes solutions reveals that the difference is small in low-altitude low-speed flows but significant in high-altitude hypersonic flows.

Thermo-chemical models for hypersonic flows

Abstract Investigations on the effects of high-temperature gas models have been performed by computing two-dimensional hypersonic flows. Thermal nonequilibrium effects are investigated by flow simulations around a vehicle of the Orbital Experiment along re-entry trajectory ranging from altitudes of 50 to 100 km, with the one-/two-temperature gas results being compared regarding the shock standoff distance, the heating rate and chemical species' concentrations. Additionally, calculations are performed for shock-tube flows in order to clarify the effects of the modelings included in the two-temperature model: the choice of chemical reaction constants, preferential dissociation and diffusive vibrational relaxation. Finally, a vibrational relaxation model for high-temperature gas without the need for information about immediately behind the shock wave is presented.

Homogeneous and Heterogeneous Vectored-Injection Cooling at Hypersonic Velocities

A theoretical study of surface mass transfer effects in two-dimensional hypersonic flow over a flat plate with heat transfer is described. The binary-diffusion laminar bounndary-layer equations are...

Quantitative Space and Time Resolved Laser Schlieren System for the Study of Hypersonic Flow

A space and time resolved quantitative schlieren system has been developed for the measurement of density gradient distributions in a two-dimensional hypersonic flow field. The device uses a narrow beam helium-neon continuous laser which is scanned across the flow field by a fast revolving multifaced mirror. A knife edge-photomultiplier assembly monitors the deflection of the laser beam caused by the presence of density gradients in the test gas. Spatial resolution which is determined by the diameter of the laser beam in the flow field is less than 0.6 mm. Time resolution is determined by the speed of rotation of the mirror and the number of mirror faces but in the present setup it is limited to approximately 2×10−3 sec by the frequency response of the data recording system. Quantitative performance has been checked for a known density gradient which was simulated by the use of an accurately specified glass wedge.

Two-Dimensional Hypersonic Flow of an Ideal Gas with Infinite Electric Conductivity Past a Two-Dimensional Magnetic Dipole

Two-Dimensional Hypersonic Flow of an Ideal Gas with Infinite Electric Conductivity Past a Two-Dimensional Magnetic Dipole

Two-Dimensional Airfoils at Moderate Hypersonic Velocities

Expressions for lift, drag,' and pitching moment coefficients are derived for a large number of airfoil sections in two-dimensional hypersonic flow. These expressions, differing from those obtained using linearized, second-order, and third-order theories, can be used throughout a moderate range of hypersonic Mach Numbers. The fundamental parameter of these formulas is the two-dimensional hypersonic flow similarity parameter, K = M0d. To obtain these expressions, the shock-wave and expansion-wave equations for pressure ratios across such waves are expanded in powers of K. Following this, it is shown that the first three terms so obtained for both wave types are practically identical for values of K less than about 1.0. Further, it is shown that they closely approach the exact pressure-ratio expressions. The application of the expressions thus derived to various airfoil sections is then demonstrated, and a table of airfoil coefficients is presented. It is shown that the airfoil formulas can be applied to finite wings with reasonable accuracy for design estimations.