Analytical solution of supersonic axisymmetric flow around a sharp convex corner

Results of a numerical simulation of steady axisymmetric supersonic flows in convergent conical ducts and in overexpanded jets are presented. The characteristic feature of these compression flows is the formation of an initial longitudinally curved shock wave with intensity increasing downstream and toward the flow axis, which is finalized by the generation of a central Mach disk. Computations have demonstrated patterns of an irregular interaction of these shocks followed by the formation of a triple-shock configuration, including a reflected shock and a shear layer with entropy varying across the layer. The formation of triple-shock configurations is analogous to the configurations known for the steady inviscid two-dimensional flows where the irregular reflection of a wedge-generated shock from a wall with Mach stem formation occurs. Either a single triple-shock Mach configuration occurs or a triple-shock configuration corresponding to the von Neumann paradox condition is formed at the considered flow Mach numbers and initial angles of deflection to the axis of the flow behind the longitudinally curved shock wave.

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.

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.

As indicated in section 111 of Courant and Friedrichs's book [Supersonic Flow and Shock Waves, Interscience, New York, 1948], a supersonic flow around a sharp corner, one of the most important elementary flows, is effected by a rarefaction wave. This paper is concerned with the local structural stability problem on such a wave of a three-dimensional (3-D) steady full Euler system. More concretely, we establish the local existence and stability of a 3-D incomplete expansion centered rarefaction wave when the perturbed supersonic incoming flow moves around a sharp corner whose fixed wall beyond the corner is of a perturbation with respect to some oblique half-plane. © 2010 Society for Industrial and Applied Mathematics.

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

: High-performance graphics processing equipment consisting of a Silicon Graphics Origin 3400, 2 Silicon Graphics Octane SSEs, and other peripherals was purchased with funding from DURIP Grant No. DAADl9-OO-1-O1 16. This equipment significantly enhanced our research projects funded by DOD. For these research projects, Direct Numerical Simulations and Large-Eddy Simulations were employed to investigate complex transitional and turbulent flows for a wide range of applications and flow speeds, both subsonic and supersonic. Examples are: supersonic axisymmetric base flows (ARO), actuators for active flow control (AFOSR), control of separation using wall jets (AFOSR), and longitudinal vortices in turbulent boundary layers subjected to wall curvature and strong adverse pressure gradients (ONR). A common objective of these research projects was the physical understanding of the time-dependent dynamical behavior of the dominant turbulent structures. This understanding is essential for successful implementation of active flow control for DOD-relevant applications. The numerical simulations generate vast amounts of data that have to be post-processed in order to extract the essential physical mechanisms that govern the dynamical behavior of the large coherent structures. Turbulent flows characteristically exhibit a complicated time-space behavior, which is due to the dynamics of large coherent structures with a strongly non-periodic time behavior. In order to analyze this complicated spatio-temporal behavior, the flow-field data from the numerical simulations have to be visualized by rapid, interactive, time-dependent, graphics processing. The acquired graphics processing equipment with its flexible, scalable architecture enabled us to perform such interactive visualizations. The requested equipment filled a critical need for fully utilizing the computing power provided by DOD computers and thus for effectively carrying out the proposed research of our DOD grants.

The axisymmetric supersonic flow over passive, that is, nonablating, indented nosetips of reentry vehicles is determined using an unsteady implicit numerical algorithm which solves either the inviscid Euler equations or the 'thin-layer' Navier-Stokes equations. A nonorthogonal independent variable transformation is used to map the distorted physical region, containing multiple zones of embedded subsonic flow into a rectangular computational domain at whose boundaries the required permeable or impermeable boundary conditions are simulated. Use of the implicit algorithm results in faster convergence to the steady state because of a larger allowable time step over conventional explicit schemes. The numerical results obtained compare favorably with existing experimental data for very mildly and severely indented blunt nosetips.

Supersonic axisymmetric base flows are prototypical for flows behind projectiles and missiles. For these flows, drag reduction can be achieved by means of passive control of the near wake. Thereby, large (turbulent) coherent structures play a dominant role. The objective of the present investigation is to elucidate if and how successful passive flow control techniques modify these structures. To this end, first Direct Numerical Simulations (DNS) for a Reynolds number of ReD = 100,000 and Mach number of Ma =2.46 were performed using a high-order accurate and highly parallelized research code which was developed at the University of Arizona. Thereby, roughly 52 million grid points were employed. The DNS data serve to visualize typical structures of the unsteady flow field and to verify that the use of less computational costly RANS/LES methods is applicable for this flow. Two of these methods, the Flow Simulation Methodology (FSM) and Detached Eddy Simulations (DES), were then employed to investigate the supersonic base flow at ReD =3.3 × 106 and Ma = 2.46 using between 460,000 and seven million grid points. For the DES, the commercial CFD-code Cobalt was employed. This unstructured grid solver allowed then to perform simulations with boat-tailing. The obtained mean flow data are compared to available experimental results.

Shock reflection from an axial cylinder in axisymmetric supersonic steady flow

Altitude-adapted nozzles are designed to facilitate flow adaptation during rocket ascent in the atmosphere, without requiring mechanical activation. As a consequence, the performance of the nozzle is significantly improved. The aim of this study is to develop a new profile of axisymmetric supersonic nozzles adapted at altitude (Dual Bell Nozzle with Central Body), which is characterized by an E-D nozzle as a basic profile. The performances obtained for this nozzle (E-D Nozzle) are then compared to those of a Plug nozzle. The E-D nozzle shows significant performance advantages over the Plug nozzle, including a 13.02% increase in thrust, knowing that the length of the E-D nozzle is half that of the Plug nozzle under the same design conditions. Finally, viscous calculations using the k-ω SST turbulence model were conducted to compare the performance of the dual bell nozzle with central body (DBNCB) and the E-D nozzle with the same cross-sectional ratio, and to assess the impact of nozzle pressure ratio (NPR) variations on the operation mode of the DBNCB. The results obtained show that the DBNCB offers the best performance in most phases of flight.

In this paper, we construct a local supersonic flow in a 3-dimensional axis-symmetry nozzle when a uniform supersonic flow inserts the throat. We apply the local existence theory of boundary value problem for quasilinear hyperbolic system to solve this problem. The boundary value condition is set in particular to guarantee the character number condition. By this trick, the theory in quasilinear hyperbolic system can be employed to a large range of the boundary value problem.

Density measurements have been performed using Background Oriented Schlieren (BOS) and differential interferometry, in the case of supersonic flows around a cone-cylinder body, a hemisphere profile and two spike-tipped models for Mach numbers 2 and 3, and no angle of attack. Experiments have been driven in the blow-down wind tunnel of the FrenchGerman Research Institute of Saint-Louis (ISL). For BOS technique a specific Abel transform algorithm has been developed to rebuild density field from ray deflections. For differential interferometry an original and simple calibration and filtering process allows to obtain quantitative results in spite of experimental noise during the blow-down. Performances of these two very different techniques – accuracy, spatial resolution, limitations – have been evaluated and compared in the basic cases of cone and hemisphere. BOS measurements have been 2% accurate and spatial resolution has been 1 millimeter in a 100*100mm2 flowfield. In the same flowfield, differential interferometry has given 2% accurate measurements and spatial resolution has been 0.25 mm. Pretty good agreement has been found with RANS numerical simulations outside from boundary layers and bow shock regions. The latter is the main limitation to both of methods. In a bow shock region, BOS experimental set up is not “stigmatic”, while a “parasite” shadowgraph appears in differential interferometry. The lack of information leads to under-estimation of density in such a region. Having knowledge about performances and limitations of these two techniques, Background Oriented Schlieren and Differential Interferometry have been used to measure density in the case of a supersonic flow around a spike-tipped body: a disk-spike and a biconical-spike for Mach numbers 2 and 3 have been under study. Outside bow shock regions, a pretty good agreement has been found between measurements by BOS and by Differential Interferometry, which allow them to be used later for CFD validation.

The results of some conceptual studies on the development and aerodynamic design of lifting configurations feasible for hypersonic vehicles powered by airbreathing engines are presented and discussed. The configurations, including the waveriders, with three-dimensional (3D), so-called convergent inlets, were considered. These inlets are designed so that the supersonic flow over them has the streamlines converging in space. Besides, they have a compact throat cross-section shape close to a circle. The latter property facilitates the engine heat protection as against the conventional 2D (flat) inlets or axi-symmetric inlets with the centerbodv having the slotted throats. The research included, firstly, a class of convergent inlets and lifting configurations designed by combining the regions of supersonic inviscid uniform flows formed over swept wedges and V-shaped bodies. Another class of configurations is designed with the use of supersonic axisymmetric inviscid compression flows. Swept concave surfaces contoured to form 3D supersonic convergent compression flows were also studied. Under the said aerodynamic design a wide set of numerical methods from the simplest quadratic ones to the 2D and 3D ones was used. The results of parametric calculations of the configurations built are considered and, in particular, some questions concerning the integration of convergent inlets with the lifting surfaces are discussed.

In this paper, we study the expansion problem which arises in a two‐dimensional (2D) isentropic pseudo‐steady supersonic flow expanding into vacuum around a sharp corner. The problem catches interaction of an incomplete centered simple wave with a backward planar rarefaction wave as well as interaction of a simple wave with rigid wall boundary of the 2D self‐similar Euler equations. Using the methods of characteristic decompositions and invariant regions, we mainly deal with the interaction of two symmetric non‐planar simple waves of the 2D self‐similar Euler equations to obtain the global existence of the solution for the interaction of the simple wave with rigid wall boundary. It follows the global existence of the solution up to the interface of supersonic flow with vacuum of the expansion problem.

The aerodynamic parameters of an oscillating cone in unsteady axisymmetric supersonic flow are investigated on the basis of the inviscid perfect gas model both in the absence and in the presence of strong air injection from its flat end into the shock layer.