Steady Hypersonic Flow Research Articles

A constrained reduced-order method for fast prediction of steady hypersonic flows

A constrained reduced order model (ROM) based on proper orthogonal decomposition (POD) is proposed to achieve fast and accurate prediction of steady hypersonic flows. The proposed method addresses the convergence issue of the projection-based POD ROM which violates the boundary conditions by using a constrained Gauss-Newton iterative process. The constraints in the iteration are formulated by satisfying the physical boundary conditions. To achieve this, a weighting matrix constructed by an improved Gauss weighting function is adopted to determine the contribution of each cell to the constraint term. The proposed constrained reduced-order method is accelerated by a parallel algorithm based on message passing interface (MPI) and load balancing, which makes the method practical for prediction of complex flows with large memory cost. Fast predictions of hypersonic flows over the two-dimensional cylindrical blunt body and the three-dimensional reentry vehicle using the constrained reduced-order method show that the error is significantly smaller than that of the interpolation-based POD ROM and the projection-based POD ROM. Computing efficiency is increased by 2 ∼ 3 orders of magnitude compared to CFD.

An enthalpy‐preserving shock‐capturing term for residual distribution schemes

SummaryIn this contribution, we investigate strategies to perform shock‐capturing computation of steady hypersonic flow fields by means of residual distribution schemes. The ultimate objective is the computation of flow solutions for which the correct upstream enthalpy value is recovered in the postshock region. To this end, the parallelism existing between the classical Bx scheme and the stabilized finite element techniques is exploited. The simple Lax‐Friedrichs dissipation term is leveraged to build two new residual distribution schemes. Upon testing on both inviscid and viscous steady problems, solutions obtained with one of the two schemes are shown to recover the correct upstream total enthalpy level in the postshock region. This last scheme provides also improved wall pressure and skin friction predictions; heat transfer predictions are, unfortunately, similar to those offered by the Bx scheme. A conjecture for explaining this behavior is exposed.

Receptivity of the Boundary Layer over a Blunt Wedge with Distributed Roughness at Mach 6

A hypersonic flow field over a blunt wedge with or without roughness is simulated by a direct numerical simulation method. The effect of isolated and distributed roughnesses on the steady and unsteady hypersonic flow field and boundary layer is analyzed. The shape of roughness is controlled by cubic polynomial. The evolution of disturbance waves caused by slow acoustic wave in the boundary layer is investigated by fast Fourier spectrum analysis. The results show that there is a great influence of roughness on the evolution of disturbance waves in the hypersonic boundary layer. The disturbance waves are promoted in the upstream-half region of roughness while suppressed in the downstream-half region of roughness. There is always a mode competition among different modes both in the temporal domain and in the frequency domain in the boundary layer, and mode competition is affected by roughness. The location of the dominant mode which is changed to a second-order harmonic mode from the fundamental mode moves upstream. The vortices caused by roughness also impact the evolution of disturbance waves in the boundary layer. The fundamental mode is suppressed in the vortex region while other harmonic modes are promoted.

Stable Runge-Kutta discontinuous Galerkin solver for hypersonic rarefied gaseous flow based on 2D Boltzmann kinetic model equations

A stable high-order Runge-Kutta discontinuous Galerkin (RKDG) scheme that strictly preserves positivity of the solution is designed to solve the Boltzmann kinetic equation with model collision integrals. Stability is kept by accuracy of velocity discretization, conservative calculation of the discrete collision relaxation term, and a limiter. By keeping the time step smaller than the local mean collision time and forcing positivity values of velocity distribution functions on certain points, the limiter can preserve positivity of solutions to the cell average velocity distribution functions. Verification is performed with a normal shock wave at a Mach number 2.05, a hypersonic flow about a two-dimensional (2D) cylinder at Mach numbers 6.0 and 12.0, and an unsteady shock tube flow. The results show that, the scheme is stable and accurate to capture shock structures in steady and unsteady hypersonic rarefied gaseous flows. Compared with two widely used limiters, the current limiter has the advantage of easy implementation and ability of minimizing the influence of accuracy of the original RKDG method.

A novel simplified Bernoulli trials collision scheme in the direct simulation Monte Carlo with intelligence over particle distances

This work deals with the development of an intelligent inter-particle collision scheme using the Direct Simulation Monte Carlo (DSMC) method. Conventional DSMC collision schemes cannot perceive collision distances by their own routines and, consequently, must rely on other techniques to provide a choice of collision pairs with a smaller inter-particle space. Here, we propose a modification of the Simplified Bernoulli Trials (SBT) scheme, called the Intelligent Simplified Bernoulli Trials (ISBT) scheme, which can create pseudo-circular subcells that reduce approximately 25%–32% of the mean collision separation distance. The ISBT scheme tries to arrange the particle indexing and collision acceptance-rejection of the SBT scheme in a way that leads to the formation of virtual clusters. These inner-cell clusters then will cause the selection of the “near-neighbor” pair, which leads to smaller mean collision separation distances. Different low- and high-speed test cases, e.g., lid-driven cavity flow, steady hypersonic flow over a two-dimensional cylinder, and Mach 15.6 nitrogen flow over a 25°–55° axisymmetric biconic, are selected to assess the accuracy and efficiency of the ISBT collision scheme.

Hysteresis phenomena associated with shock waves interference in steady hypersonic flow

If an oblique shock wave encounters a wall, two reflection types are possible. According to the inclination of the shock, with respect to the incoming flow, the resulting configuration is either a regular reflection (RR) or a Mach reflection (MR). At Mach number higher than 2.2, both reflections are theoretically possible for some incidence angles. This range is known as the dual-solution domain, and its existence led to hypothesise an hysteresis in the RR-MR transition. In order to study this phenomenon, experimental investigations and numerical simulations have been performed. The first part of this article deals with the experimental and numerical tools which were used. The second part presents the results. According to the experimental configurations, the hysteresis may be obtained in two ways. The flow Mach number being kept constant (Mach 10) in the wind tunnel used (ONERA R3Ch WT), this phenomenon is obtained by variation of the aspect ratio or by variation of the incidence angle.

Critical ignition in rapidly expanding self-similar flows

The generic problem of ignition of a particle undergoing an expansion given by a power law rate of decay behind a decaying shock is addressed in the present study. It is demonstrated, using a one-step Arrhenius irreversible reaction, that a sufficiently strong expansion wave can quench the reaction. The critical conditions for extinction are obtained in closed form in terms of the time scale for the expansion process and the thermochemical properties of the gas, yielding a critical Damkohler number, i.e., the ratio of the expansion time scale to the homogeneous ignition time scale, given by (γ−1)(Ea/RT)−1/n, where n is the power law exponent of the self-similar expansion. The critical ignition criteria, which are valid in the asymptotic limit n(γ−1)(Ea/RT)=O(1), were found in excellent agreement with numerical results. The applicability of the results obtained are discussed for ignition in rapidly expanding flows which occur behind decaying shock waves, as encountered in problems of detonation initiation by a Taylor–Sedov blast wave, and reacting jet startup, and for reactions in steady hypersonic flows around projectiles.

Implementation of a transient adaptive sub-cell module for the parallel-DSMC code using unstructured grids

A method of transient adaptive sub-cells (TAS) suitable for unstructured grids that is modified from the existing one for the structured grids of DSMC is introduced. The TAS algorithm is implemented within the framework of a parallelized DSMC code (PDSC). Benchmarking tests are conducted for steady driven cavity flow, steady hypersonic flow over a two-dimensional cylinder, steady hypersonic flow over a cylinder/flare and the unsteady vortex shedding behind a two-dimensional cylinder. The use of TAS enables a reduction in the computational expense of the simulation since larger sampling cells and less simulation particles can be employed. Furthermore, the collision quality of the simulation is maintained or improved and the preservation of property gradients and vorticity at the scale of the sub-cells enables correct unsteady vortex shedding frequencies to be predicted. The use of TAS in a parallel-DSMC code allows simulations of unsteady processes at a level to be carried out efficiently, accurately and with acceptable computational time.

Effects of Numerics on Navier-Stokes Computations of Hypersonic Double-Cone Flows

A systematic study of the effects of the numerics on the simulation of a steady hypersonic flow past a sharp double cone is presented. Previous studies have shown that the double-cone flow is challenging to compute, making it useful for testing both numerical schemes and physical models. We focus on the numerical aspects only and show that the results are very sensitive to the numerical flux evaluation method and slope limiter used

Hypersonic flow past slender bodies in dispersive hydrodynamics

The problem of two-dimensional steady hypersonic flow past a slender body is formulated for dispersive media. It is shown that for the hypersonic flow, the original 2 + 0 boundary-value problem is asymptotically equivalent to the 1 + 1 piston problem for the fully nonlinear flow in the same physical system, which allows one to take advantage of the analytic methods developed for one-dimensional systems. This type of equivalence, well known in ideal Euler gas dynamics, has not been established for dispersive hydrodynamics so far. Two examples pertaining to collisionless plasma dynamics are considered.

Experimental investigation of the influence of downstream flow conditions on Mach stem height

An experimental work on the influence of downstream flow expansion process on Mach stem height has been carried out in steady hypersonic flows. The results showed clearly for the first time that the Mach stem height does not depend on the extent of expansion fan. These results contradict recent analytical findings from which a strong influence of downstream flow conditions on Mach stem height may be expected.

A Blunt-Nosed Thin Body in Hypersonic Flow

An inverse problem is posed to simulate the influence of the nose bluntness on the hypersonic inviscid steady flow field around a slender two-dimensional section. In this formulation a bow shock is given in advance whereas the body contour comes at the end of analysis being identified with a particular streamline. An equation for the shock shape involves two terms in the form of different powers of the distance measured along the direction of oncoming stream. The leading term derives from a classical similar solution for the strong viscous/inviscid interaction regime; a correction to it models bluntness effects at large distances downstream. Matched asymptotic expansions are used to solve the problem within the hypersonic small- disturbance theory. In the outer region bounded by the shock, two sets of ordinary differential equations control the pressure, density, and velocity distributions. The second-order approximation admits of an explicit integral obtainable from a consideration of the finite drag exerted on the blunted nose of a section. The use of the momentum conservation law allows us to predict the power of the exponent of the correction term entering the shock equation. The asymptotic behavior of both first- and second-order approximations is established and employed for providing conditions for a solution in the inner region occupied by a high-entropy layer. Governing equations here are solved, explicitly determining a dependence of the body shape on the correction in the shock representation. A thorough analysis of the Newtonian approach reveals certain limitations inherent in this sim- plified treatment of steady hypersonic flows.

An Evaluation of Roe's Scheme Generalizations for Equilibrium Real Gas Flows

The extension of Roe's approximate Riemann solver to equilibrium real gas is analyzed by means of a general formulation, allowing us to clarify the inherent nonuniqueness of the average state and the influence of the functional form of the equation of state. Several generalizations of Roe's scheme are then reviewed and their numerical performances are discussed by computing some 2D steady hypersonic flows. The flow solvers are coupled with a newly developed, efficient, and robust procedure for thermochemical air properties evaluation. All of the tested equilibrium solvers achieve very similar results. They are found of comparable numerical efficiency, the higher performances being associated with Vinokur's and Liou's solvers. It is concluded that equilibrium simulations in 2D are by no means less robust than the perfect gas ones, when coupled with the proposed procedure for properties evaluation.

Parallel computation of super-/hypersonic flows on workstation network and Transputer arrays

A numerical method for solving the Euler equations based on the method of lines has been implemented on networked engineering workstations and a multi Transputer system using a domain decomposition approach. The message passing is handled by PVM (Parallel Virtual Machine) software. The present approach is tested for both 2D and 3D test problems governed by the Euler equations. For dynamic load balancing and reduction of computational work a strategy of selective solutions of subdomains based on local residuals has been employed. For steady supersonic and hypersonic flow problems, a reduction of 40–50% computational work and 12-1% of CPU time has been achieved.

Singularities encountered in three-dimensional boundary layers under an adverse or favourable pressure gradient

Singularities in solutions of the classical boundary-layer equations are considered, numerically and analytically, in an example of steady hypersonic flow along a flat plate with three-dimensional surface roughness. First, a wide parametric study of the breakdown of symmetry-plane flow is performed for two particular cases of the surface geometry. Emphasis is put on the structural stability of the singularities’ development to local/global variation of the pressure distribution. It is found that, as usual, the solution behaviour under an adverse pressure gradient involves the Goldstein- or marginal-type singularity at a point of zero streamwise skin friction. As the main alternative, typical of configurations with favourable or zero pressure forcing, an inviscid breakdown in the middle of the flow is identified. Similarly to unsteady flows, the main features of the novel singularity include infinitely growing boundary-layer thickness and finite limiting values of the skin-friction components. Subsequent analytical extensions of the singular symmetry-plane solution then suggest two different scenarios for the global boundary-layer behaviour: one implies inviscid breakdown of the flow at some singular line, the other describes the development of a boundary-layer collision at a downstream portion of the symmetry plane. In contrast with previous studies of the collision phenomenon in steady flows, the present theory suggests logarithmic growth of boundary-layer thickness on both sides of the discontinuity. Finally, an example of numerical solution of the full three dimensional boundary layer equations is given. The flow régime chosen corresponds to inviscid breakdown of a centreplane flow under a favourable pressure gradient and development of the discontinuity/collision downstream. The numerical results near the origin of the discontinuity are found to be supportive, producing quantitative agreement with the local analytical description.

Solution of the steady Euler equations in a generalized Lagrangian formulation

A second-order Godunov-type shock-capturing scheme for solving the steady Euler equations in generalized Lagrangian coordinates has been developed and applied to compute steady supersonic and hypersonic flow problems. Following Hui and Zhao, the Lagrangian distance and a stream function are used as the coordinate lines that not only simplify the Riemann solution procedure but also have an intrinsic flow adaptive property embedded. Numerical examples for various supersonic flows involving strong flow discontinuities are given. Good agreement is obtained between computed results and shock expansion theory or available experimental data. It was found that the resolution of the slip line is almost exactly without smearing, the resolution of shock is always crisp even at increasing Mach number, and the Prandtl-Meyer expansion is adequately resolved with the second-order-accurate scheme.

Oscillating two-dimensional hypersonic airfoils at small angles of attack

Pitching oscillations of two-dimensional pointed-nose thin airfoils with small surface curvature are considered in this paper. The analysis relies on recently developed theories for steady and unsteady hypersonic flows past such airfoils. For small surface curvature T and small reduced frequency k, a double series in i and k is assumed here and shown to lead to very simple systems of linear equations having first- or second-degree polynomial solutions. Thus, simple closed-form formulas for the unsteady surface pressure and the stability derivatives of any curved nonsymmetric airfoil (with different upper and lower surface shapes), pitching at a small angle of attack, are obtained. Results for symmetric wedges at zero incidence are compared with other available analytical and experimental calculations and the agreement is found to be generally good.

High-resolution shock-capturing schemes for inviscid and viscous hypersonic flows

A class of high-resolution implicit total variation diminishing (TVD) type algorithms suitable for transonic multidimensional Euler and Navier-Stokes equations has been extended to hypersonic computations. The improved conservative shock-capturing schemes are spatially second- and third-order and are fully implicit. They can be first- or second-order accurate in time and are suitable for either steady or unsteady calculations. Enhancement of stability and convergence rate for hypersonic flows is discussed. With the proper choice of the temporal discretization and implicit linearization, these schemes are fairly efficient and accurate for very complex two-dimensional hypersonic inviscid and viscous shock interactions. This study is complemented by a variety of steady and unsteady viscous and inviscid hypersonic blunt-body flow computations. Due to the inherent stiffness of viscous flow problems, numerical experiments indicated that the convergence rate is in general slower for viscous flows than for inviscid steady flows.

翼付半円すい物体まわりの非粘性極超音速流の研究

The steady hypersonic flow around a slender conical flat-top wing-body has been solved by applying the equivalence principle, which is the theory that the problem of steady hypersonic flow past a slender body is equivalent to the problem of unsteady flow in a plane. Then, the problem of this conical flat-top wing-body becomes into the problem of an expanding motion of a hemi-cylinder with a plane wing. The flowfield of this expanding body has successfully been solved by means of a conformal transformation and a superposition technique. Moreover, the other problems of small angles of attack, vortices, shock shape etc. have been also solved. As for the experimental investigation, the flow around the expanding body has been visualized. The experimental results agree well with the theoretical ones.

Propagation of weak MHD waves in steady hypersonic flows with radiation

Propagation of weak MHD waves in steady hypersonic flows with radiation