Three-dimensional effects on regular reflection in steady supersonic flows

The reflection of shock waves between two symmetrical wedges is investigated for the case of three-dimensional flows. Oblique shadowgraphs at various optical angles of yaw and pitch were used to examine the nature of fully three-dimensional flows, with wedge aspect ratios as low as 0.25 being considered. These images were used to construct surface models of the overall flow field for various reflection patterns and aspect ratios, which provides a visual indication of the flow field shape. For a Mach number of 3.1, and suitable wedge angles, the flow field with regular reflection on the tunnel centreline and Mach reflection further out is examined. The point of transition from regular reflection to the peripheral Mach surfaces is identified for various wedge angles and aspect ratios. It is shown that the transition points move outwards from the central plane as the aspect ratio decreases. This shows that three-dimensional flows favor regular reflection, because of the increasing curvature of the incident shock as the wedge becomes narrower, causing a decrease in the local angle of incidence. The height of the Mach stem is shown to be highly dependent on the geometry of the test wedge models. The Mach stem height decreases with aspect ratio due to the three-dimensional relieving effect, where the increase in lateral flow relieves the pressure over the surfaces of the wedges. Experimental evidence of the existence of the strong oblique shock solution in steady flows is presented.Received: 7 July 2003, Revised: 20 October 2003, Accepted: 6 November 2003, Published online: 10 February 2004PACS: 47.40.Nm Correspondence to: B.W. Skews     

Development of pulsation regimes in one-dimensional unsteady MHD flows with switching off of the electrical conductivity

This paper investigates the gas flow in an electromagnetic field when the conductivity, being a function of the thermodynamic gas parameters, vanishes during the flow (switching off of the conductivity). In the case of steady supersonic flows in an expanding nozzle it was first shown analytically [1] and then confirmed by numerical experiment [2] that stable steady flow is not possible for all the problem parameters (for example, the values of the magnetic field at the exit). Instead of a steady flow a periodic regime is realized when narrow regions of conducting gas with currents flowing through them detach from the conducting region and propagate down the channel. In these papers the conductivity was assumed to be a function of only the temperature, such that for T T* (T) = 0. In [3, 4] the flows of conducting gas in the channels were calculated both with the given dependence of the gas conductivity on the temperature and on the basis of a three-component model by means of the Saha equation. At the same time, the development of periodic regimes in the flow in the nozzle was observed in both cases, but the mechanism of the origin of the current layers was not explained. The self-similar problem of the withdrawal of a nonconducting piston from a half-space occupied by a conducting gas with a magnetic field was investigated in [5] in a linear formulation. At the same time, regions of the problem parameters (the velocity of the piston and the magnetic field on it) were found when, in spite of the self-similar formulation of the problem, there is no self-similar solution. At the same time, regions exist where several solutions are possible. The possibility of the formation of isothermal rarefaction zones with low electrical conductivity when the Joule heating is balanced by the cooling of the gas on expansion (Butler waves) [6] was not taken into account in this paper, since they are unstable with respect to superheating. However, in the case of flow in a nozzle it was shown [2] that precisely the development of instabilities in these zones leads to the formation of the periodic regime. In the present paper the solution of the self-similar problem is constructed in a nonlinear formulation. The reason for the occurrence of regions in which the solution is multiply valued, which is associated with the process of arrival at self-similar boundary conditions, is explained. It is shown that a quasiperiodic regime can arise in the solution, occurring, in particular, in the regions of the problem parameters where there is no self-similar solution.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 115–122, July–August, 1986.     

Shock-fitting method for two-dimensional inviscid,steady supersonic flows in ducts

Summary A numerical technique to solve the Euler equations for two-dimensional, steady, inviscid, supersonic flows in ducts is presented. The technique is second-order accurate and is based on the-scheme plus shock-fitting method. Many cases have been tested. The method used and the results obtained demonstrate the versatility of the technique specially if confronted with analogous methods that use shock-capturing techniques.

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Sommario Viene presentato un metodo numerico per l'integrazione delle equazioni di Eulero per flussi supersonici, bidimensionali, stazionari, non-viscosi in condotti. La tecnica, che possiede una accuratezza del secondo ordine, è basata sui metodi del-scheme e dello shock-fitting. Molti sono stati i casi provati. Il metodo usato ed i risultati ottenuti dimostrano la versatilità della tecnica, specialmente se confrontata con metodi analoghi che fanno uso di tecniche shock-capturing.

     

An investigation of random choice method for three‐dimensional steady supersonic flows

In this paper, an unsplit random choice method (RCM) is developed and applied to numerically solve three‐dimensional supersonic steady flow problems. In order to keep the contacts (slip surfaces) crisply resolved, a new Lagrangian formulation is employed. Due to the lack of exact solutions to 3D Riemann problems, approximate Riemann solutions in the weak sense are adopted. The RCM is thus as efficient as the deterministic TVD schemes, and yields almost identical results in the model problems. Copyright © 1999 John Wiley & Sons, Ltd.     

Extension of Bernoulli's theorem on steady flows of inviscid fluids to steady flows of plastic solids

A differential equation is derived which is valid along any streamline in a steady flow of a general continuum where the streamline is also a trajectory of principal stress. For special materials and flow conditions this equation can be integrated to give algebraic relations between variables along the streamline. For inviscid fluids this leads to Bernoulli's famous theorem relating pressure and velocity along any streamline. For three-dimensional ideal flows of Tresca plastic solids and planar ideal flows of general rigid/perfectly plastic solids, it also leads to known results along any streamline. For other special constitutive materials in rigid/plastic solids additional streamline relations are obtained.     

Investigation of supersonic flows in conical nozzles

The article gives the results of an investigation of flows in supersonic axisymmetric conical nozzles with the presence of shock waves in the flow. The method of straight-through calculation [1] and the method of small perturbations [2] are used. An investigation is made of the effect of various geometric parameters and of the adiabatic index on the flow of a gas in conical nozzles. A comparison is made with experimental data.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 101–107, May–June, 1974.In conclusion, the authors thank N. V. Drozdov for his participation in carrying out the calculations.     

Motion of tracer particles in supersonic flows

 Factors that may act on particle motion in high-speed flow are investigated. The classical expressions of drag coefficient C _D for a sphere are reviewed. Then, a drag expression is proposed, extending Cunningham’s method to higher velocities and Knudsen numbers. This law, valid from continuum to free molecule conditions, for Re≲200 and M≲1 (where Re and M are, respectively, the Reynolds and Mach numbers based on relative velocity), is used to compare calculated and experimental values of the drag coefficient, as well as the particle velocities across an oblique shock wave. Calculated results are found to be in agreement with experiments. Received: 3 June 1997/Accepted: 16 August 1998     

Transition of unsteady flows of evaporation to steady state

We investigate the half-space problem of evaporation and condensation in the scope of discrete kinetic theory. Exact solutions are found to the boundary value problem and the initial boundary value problems of the flow in the half space for a discrete velocity model. The results are used to analyze the transition of the unsteady solutions towards steady states. To cite this article: A. d'Almeida, C. R. Mecanique 336 (2008).     

Two-layer flows of gas in supersonic axisymmetric nozzles

This article describes two methods for calculating two-layer flows. The first is a generalization of a numerical method for solving the inverse problem [1] for the case of two-layer flows, without taking mixing into account. The second is a method of characteristics, for calculating a two-layer flow in a supersonic nozzle. In this case, the usual method of characteristics is changed in such a way that it is possible to calculate a point on the interface between two layers having different adiabatic indices, and different total pressures and temperatures. This article also gives the results of calculation of two-layer flows in nozzles with different adiabatic indices and different ratios of the mass flow rates of the gas in the layers.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 76–81, July–August, 1970.The calculations were programmed and carried out by G. D. Vladimirova and M. F. Tamarovskii, to whom the author expresses his thanks.     

Existence of supersonic zones in three-dimensional separation flows

Up till now the region of three-dimensional separation flows which occur with supersonic flow past obstacles has received insufficient study. Supersonic flow with a Mach number of 2.5 past a cylinder mounted on a plate was studied in [1]. A local zone with supersonic velocities was found in the reverse subsonic flow region ahead of the cylinder. Its presence is explained by the three-dimensional nature of the flow. Similar supersonic zones are not observed in the case of supersonic flow over plane and axisymmetric steps.The present paper presents the results of experimental studies whose objective was refinement of the flow pattern ahead of a cylinder on a plate and the study of the local supersonic zones.The experiments were performed in a supersonic wind tunnel with a freestream Mach number M₁=3.11. The 24-mm-diameter cylinder with pressure taps along the generating line was mounted perpendicular to the surface of a sharpened plate. The distance from the plate leading edge to the cylinder axis wasl ₀=140 mm. The plate was pressure tapped along the flow symmetry axis. The Reynolds number was Rl ₀=u₀ l ₀/v ₁, Rl ₀=1.87.10⁷, where u₁ andv ₁ are the freestream velocity and the kinematic viscosity, respectively. The pressures were measured using a Pilot probe with internal and external diameters of 0.15 and 0.9 mm, respectively.The probe was displaced in the flow symmetry plane at a distance of 1.6 mm from the plate surface and at a distance of 1.1 mm along the leading generator of the cylinder. The flow on the surface of the plate and cylinder was studied with the aid of a visualization composition and the flow past the model was photographed with a schlieren instrument. Typical patterns of the visualization composition distribution and the pressure distribution curves over the plate surface, and also photographs of the flow past the model, are shown in [1].     

Lp-approach to steady flows of viscous compressible fluids in exterior domains

We investigate steady compressible flows in three-dimensional exterior domains for small data and for both zero and nonzero (but constant) velocity at infinity. We prove existence and uniqueness of solutions in L ^p -spaces, p>3, and study their regularity as well as their decay at infinity.     

Stabilization of non-premixed flames in supersonic reactive flows

A Lagrangian framework is set out to describe turbulent non-premixed combustion in high speed coflowing jet flows. The final aim is to provide a robust computational methodology to simulate, in various conditions, the underexpanded GH2/GO2 torch jet that is used to initiate combustion in an expander cycle engine. The proposed approach relies on an early modelling proposal of Borghi and his coworkers. The model is well suited to describe finite rate chemistry effects and its recent extension to high speed flows allows one to take the influence of viscous dissipation phenomena into account. Indeed, since the chemical source terms are highly temperature sensitive, the influence of viscous phenomena on the thermal runaway is likely to be all the more pronounced since the Mach number values are high. The validation of the extended model has been recently performed through the numerical simulation of two distinct well-documented experimental databases. Only a brief summary of this preliminary validation step is provided here. The main purpose of the present work is to proceed with the numerical simulation of geometries that bring together the essential peculiarities of the underexpanded GH2/GO2 torch. The behavior of the corresponding supersonic coflowing jet flames for various conditions is discussed in the light of computational results. To cite this article: J.-F. Izard, A. Mura, C. R. Mecanique 337 (2009).     

Selection of steady regimes of a one-parameter family in the problem of plane convective flow through a porous medium

Convective flow through a porous medium in a rectangular vessel with a linear temperature profile steadily maintained on the boundary is investigated. On the basis of numerical experiments the realization of steady regimes that belong to a globally stable, one-parameter family is studied at different vessel dimensions and initial temperature distributions. The regime selection is shown to strongly depend on the initial fluid temperature: a vicinity of two regimes is realized from initial data similar with the state of rest; at high initial fluid heating the regimes are selected from a vicinity of two other regimes; and in intermediate situations any of infinite number of steady regimes can be attained.     

Numerical simulation of gap effect in supersonic flows

The gap effect is a key factor in the design of the heat sealing in supersonic vehicles subjected to an aerodynamic heat load. Built on S-A turbulence model and Roe discrete format, the aerodynamic environment around a gap on the surface of a supersonic aircraft was simulated by the finite volume method. As the presented results indicate, the gap effect depends not only on the attack angle, but also on the Mach number.     

Calculation of Reynolds stresses in turbulent supersonic flows

The baseline numerical procedure of interest in this study combines flux vector splitting, flux difference splitting and an explicit treatment of the diffusion terms of the flow equations. The viscous terms are treated explicitly to preserve the wave propagation properties of the Euler fluxes and permit splitting. The experience with this scheme has been limited to laminar or, at best, ‘eddy viscosity’ flows. In this paper the applicability of the scheme is extended to include the calculation of turbulent Reynolds stresses in supersonic flows. The schemes and our implementation are discussed. Both laminar and turbulence subsets of the Reynolds/Favre-averaged equations are tested, with a discussion of relative performance. The test problem for turbulence consists of a zero-pressure-gradient supersonic boundary layer as well as a supersonic boundary layer experiencing the combined effects of adverse pressure gradient, bulk compression and a concave streamline curvature. Excellent agreement with experimental measurements is observed for most of the quantities compared, which suggests that the numerical procedures presented in this paper are potentially very useful.     

On the steady solution of linear advection problems in steady recirculating flows

Numerical modelling of non-Newtonian flows typically involves the coupling between equations of motion characterized by an elliptic behaviour, and the fluid constitutive equation, which is an advection equation linked to the fluid history. In this paper we prove that linear steady advection problems in steady recirculating flows have only one solution when the kinematics differs from a rigid motion. We also give a numerical procedure to determine this steady solution. We will describe this numerical procedure for two linear models the first will be the SFRT flow model and the second will be a simplified linear formulation of the Pom–Pom viscoelastic model.