Patterns of steady axisymmetric supersonic compression flows with a Mach disk

Supersonic axisymmetric expansion flow is a typical and fundamental issue in gas dynamics. It plays a vital role in the high-speed external and internal flow fields regarding the contour design and performance evaluation of supersonic/hypersonic vehicles and their propulsion systems. The supersonic two-dimensional (2D) planar expansion flow is dominated by the well-known Prandtl–Meyer (P–M) theory. However, no similar explicit relation exists for the supersonic axisymmetric expansion flow, and only the computational fluid dynamics results could be employed at present. Therefore, this work focuses on developing the analytical solution of supersonic axisymmetric flow around a sharp convex corner on the basis of the generic gasdynamic functions in a newly established coordinate system for addressing the aforementioned issue. Theoretical derivations and numerical results prove that the flow deflection angle and Mach number in supersonic axisymmetric flow around a sharp convex corner obey the identical law to the 2D planar situation, that is, the P–M theory, while the local axisymmetric expansion fan is not the simple wave flow despite the conical flow. Meanwhile, the method of characteristics is employed to further explicate the intrinsic connection and difference between the 2D and axisymmetric sharp convex corner flow. The equivalence of sharp corner and curved surface flows with the identical deflection angle is discussed, and three limitations of the proposed analytical solution are clarified.

New node-centered finite-volume discretizations of advective and diffusive derivatives on structured meshes with quadrilateral cells are presented. They are applied to the solution of Euler and full Navier-Stokes equations using a pseudo time-dependent approach. Advective derivatives are split and upwind-biased. The most interes- ting aspect of the scheme lies in its ability to provide a good accuracy on meshes with severe skewness and stret- ching distortions. A very sensitive detector is presented which is capable of selectively identifying shock waves and unsufficiently resolved shear layers. It is used to automatically switch to a first-order upwind scheme in some regions and provides a sharp and monotone shock capture. Results obtained for plane and axisymmetric steady supersonic lamnar flows are discussed. They include blunt-body flows, a shock-boundary layer interac- tion and a flow over a compression comer. I. Introduction Most classical finite-volume schemes for the solution of Euler and Navier-Stokes equations require very regular and smooth meshes. Severe mesh skewness and stretching distortions can strongly deteriorate the accuracy and lead to significant parasitic diffusion or anti-diffusion effects. These effects can be responsible for relatively poor evaluations of important flow properties and eventually for numerical instabilities. The accuracy of several space- centered schemes using structured meshes has been theoretically and practically evaluated by Renard and Essersl. For that purpose, they considered a simple model governed by a linear advection-diffusion equation. Starting from an unsrretched rectangular mesh, more and more severe distortions were introduced in the latter using a special random grid distortion procedure. Different families of structured grids, each with different amounts of distortions measured by a parameter ranging from 0 to 1, were thus created and used to solve the model pro- blem. Errors obtained with finer and finer grids presen- ting the same amount of distortion were evaluated. All of the tested schemes, which are frequently thought of as

: A random choice method (RCM) is developed for obtaining fairly practical and efficient numerical solutions for two dimensional planar and axisymmetric steady supersonic flows, such as those for sharp edged planar airfoils, supersonic inlets of aircraft engines, pointed bodies of revolution, supersonic nozzles, and free jets. This method is based on the solution of a riemann problem, which is the elemental solution of the hyperbolic equations of two-dimensional steady supersonic flows. The Riemann problem consists of two waves separated by a slip stream, and each wave can be either an oblique shock wave or a Prandtl Meyer expansion wave. Advanced techniques are given for solving the Riemann problem iteratively, handling the boundary conditions along body and free jet surfaces, and computing only certain parts of flow fields of interest. Many interesting and practical numerical solutions are presented for different types of planar and axisymmetric flows, to demonstrate the applicability, capability, and limitations of the RCM. Numerical results are shown to be in excellent agreement with both known analytical solutions and results from the method of characteristics.

The paper is devoted to the control of supersonic flows past aerodynamic bodies under the action of external energy deposition. Mechanism of the interaction of a bow shock over a body with the oblique shock resulting from the refraction of the bow shock at the external energy source surface is revealed. Different types of shock waves intersection might appear in this process, among them are triple-shock configurations. The objectives of this paper are the investigations of unsteady triple-shock configurations in the vicinity of the surface of combined cylinder bodies “hemisphere-cylinder” and “hemisphere-cone-cylinder” under the action of external energy deposition and their effect on the body surface pressure. The studies have been conducted numerically using inviscid approach on a base of the Euler equations at M=4 for gaseous media with ratio of specific heats 1.4 and 1.2. Complex conservative difference schemes are used in the simulations. Dependences of the angles of triple-shock configuration on the rarefaction degree in an energy source and on the incident shock angle have been obtained. Generation of local space-time areas with increasing boundary pressure has been established. Mechanism of boundary pressure growth together with local front drag force increase is shown to be connected with a vortex action as well as the action of the arising shock segments in the vicinity of triple-shock configuration. Also, the comparison with the plane flow symmetry has been conducted. The results can be used for organization of flow control via external energy deposition by means of laser, microwave or electrical discharge.

Under the hypothesis that the initial perturbation has small BV norm, we prove that in any bounded domain the L1 norm of the difference between solutions to the isentropic Euler system of steady supersonic flow and the system of steady irrotational supersonic flow with the same initial data can be bounded by the cube of the total variation of the initial perturbation.

In this paper we study the waves generated over the slipline and their interactions with other waves for Mach reflection in steady two-dimensional supersonic flow. We find that a series of expansion and compression waves exist over the slip line, even in the region immediately behind the leading part of the reflected shock wave, previously regarded as a uniform flow. These waves make the leading part of the slipline, previously regarded as straight, deviate nonlinearly towards the reflecting surface. When the transmitted expansion waves from the upper corner first intersect the slipline, an inflexion point is produced. Downstream of this inflexion point, compression waves are produced over the slipline. By considering the interaction between the various expansion or compression waves, we obtain a Mach stem height, the shape and position of the slipline and reflected shock wave, compared well to computational fluid dynamics (CFD) results. We also briefly consider the case with a subsonic portion behind the reflected shock wave. The global flow pattern is obtained through CFD and the starting point of the sonic line is identified through a simple analysis. The sonic line appears to coincide with the first Mach wave from the upper corner expansion fan after transmitted from the reflected shock wave.

Shock reflection from an axial cylinder in axisymmetric supersonic steady flow

Influence of numerical and viscous dissipation on shock wave reflections in supersonic steady flows

A high-order accurate unstructured spectral difference lattice Boltzmann method for computing inviscid and viscous compressible flows

PurposeThe purpose of the present study is to develop a new numerical framework that can predict the supersonic base flow more accurately, including the development of axisymmetrically separated shear layer and recompression shock. To this end, two aspects are improved and combined, i.e. a newly self-adaptive turbulence eddy simulation (SATES) turbulence modeling method and a high-order discretization numerical scheme. Furthermore, the performance of the new numerical framework within a general-purpose PHengLEI software is assessed in detail.Design/methodology/approachSatisfactory prediction of the supersonic separated shear layer with unsteady wake flow is quite challenging. By using a unified turbulence model called SATES combining high-order accurate discretization numerical schemes, the present study first assesses the performance of newly developed SATES for supersonic axisymmetric separation flows. A high-order finite differencing-based compressible computational fluid dynamics (CFD) code called PHengLEI is developed and several different numerical schemes are used to investigate the effects on shock-turbulence interactions, which include the monotonic upstream-centered scheme for conservation laws (MUSCL), weighted compact nonlinear scheme (WCNS) and hybrid cell-edge and cell-node dissipative compact scheme (HDCS).FindingsCompared with the available experimental data and the numerical predictions, the results of SATES by using high-order accurate WCNS or HDCS schemes agree better with the experiments than the results by using the MUSCL scheme. The WCNS and HDCS can also significantly improve the prediction of flow physics in terms of the instability of the annular shear layer and the evolution of the turbulent wake.Research limitations/implicationsThe small deviations in the recirculation region can be found between the present numerical results and experimental data, which could be caused by the inaccurate incoming boundary layer condition and compressible effects. Therefore, a proper incoming boundary layer condition with turbulent fluctuations and compressibility effects need to be considered to further improve the accuracy of simulations.Practical implicationsThe present study evaluates a high-order discretization-based SATES turbulence model for supersonic separation flows, which is quite valuable for improving the calculation accuracy of aeronautics applications, especially in supersonic conditions.Originality/valueFor the first time, the newly developed SATES turbulence modeling method combining the high-order accurate WCNS or HDCS numerical schemes is implemented on the PHengLEI software and successfully applied for the simulations of supersonic separation flows, and satisfactory results are obtained. The unsteady evolutions of the supersonic annular shear layer are analyzed, and the hairpin vortex structures are found in the simulation.

The problem of supersonic streamlining of an aerodynamic (AD) body, “a plate blunted by a cylinder”, by a flow with the freestream Mach number M = 4 containing an external energy source has been studied, taking into account physicochemical transformations. The results of the effect of the ratio of specific heats γ changing in the range from 1.1 to 1.4 on the dynamics of triple-shock configurations and vortex-contact structures are presented for the interaction of an energy source with the bow shock wave. The energy source is modeled via the heated rarefied layer (filament). The angles in the triple-shock configurations, the stagnation pressure, together with the frontal drag force, have been studied dependent on the specific heats ratio γ, the characteristics of the energy source, and also on the angle of the incident shock. Vortex-contact structures have been researched for the Mach numbers 7, 8, 9, as well as the generation of the Richtmyer-Meshkov instability accompanying the formation of a triple-shock configuration. The results show a strong influence of the specific heats ratio of the gas medium and the parameters of the energy source on the triple-shock configuration and aerodynamic characteristics of the body. This conclusion can be useful for aerospace applications in the area of the design of nozzles, intakes, and high speed flying vehicles. Additionally, the results show the possibility of flow control in the atmospheres of other planets using external energy deposition.

A new Cartesian grid system approach is proposed for the numerical simulation of two-dimensional compressible viscous flows around an arbitrary shape body. The grid system is constructed of patched blocks which is composed of the nested-spacing regular grid. The numerical procedure is based on the method of lines. For the spatial discretization, the neighboring-point local collocation method is developed and introduced for the grid points near the body, and the 2nd-order central difference method is used for the points far from the body. The 2-stage Runge-Kutta method (RK 2) is used for the time integration scheme. For the unsteady solution the nesting time step technique is combined with RK 2. Numerical results are obtained for subsonic steady and unsteady flows around a circular cylinder and compared with the other results. The results for transonic and supersonic steady flows over NACA 0012 are compared with the GAMM workshop ones. The present approach is confirmed to be applicable for the arbitrary shape body, and the computational cost is about 50% that of the conventional boundary-fitted grid approach.

For a two-dimensional steady supersonic Euler flow past a convex cornered wall with right angle, a characteristic discontinuity (vortex sheet and/or entropy wave) is generated, which separates the supersonic flow from the quiescent gas (hence subsonic). We proved that such a transonic characteristic discontinuity is structurally stable under small perturbations of the upstream supersonic flow in BV. The existence of a weak entropy solution and Lipschitz continuous free boundary (i.e., characteristic discontinuity) is established. To achieve this, the problem is formulated as a free boundary problem for a nonstrictly hyperbolic system of conservation laws; and the free boundary problem is then solved by analyzing nonlinear wave interactions and employing the front tracking method.

Modelling of strain histories for memory integral fluids in steady axisymmetric flows

The problem of axisymmetric supersonic laminar flow separation over a compression corner has not been considered within the framework of triple-deck theory for several decades, despite significant advances in both theoretical methods and numerical techniques. In this study, we revisit the problem considered by Gittler & Kluwick (J. Fluid Mech., vol. 179, 1987, pp. 469–487), using the numerical method of Ruban (Zhurnal Vychislitel'noi Matematiki i Matematicheskoi Fiziki, vol. 18, issue 5, 1978, pp. 1253–1265) and Cassel et al. (J. Fluid Mech., vol. 300, 1995, pp. 265–285), termed the Ruban–Cassel method (RCM). The solution shows good agreement with the results of Gittler & Kluwick (J. Fluid Mech., vol. 179, 1987, pp. 469–487) for a scale external radius of 1 and scale angles from 1 to 6. However, for scale angles above 6.8, a wave packet appears. This wave packet is similar to that reported by Cassel et al. (J. Fluid Mech., vol. 300, 1995, pp. 265–285) for two-dimensional supersonic flow. As the external scale radius increases (from 1 to 10), the axisymmetric solution converges towards the two-dimensional solution for equivalent scale angle values. For a scale external radius of 10, the wave packet appears at a scale angle of 3.8, compared with the value of 3.9 by Cassel et al. (J. Fluid Mech., vol. 300, 1995, pp. 265–285). Inspection of the velocity profiles reveals that inflection points, while ubiquitous in shear flow, do not seem to play a relevant role in the appearance of the wave packet for the axisymmetric flow. Axisymmetric effects become more important as the scale external radius decreases below 0.5. A larger scale angle is necessary to produce a flow structure equivalent to that of the two-dimensional case. For scale external radius 0.1, the pressure gradient is substantially diminished and the solution is devoid of a second shear-stress minimum.