Supersonic axisymmetric flow past a blunt cone executing longitudinal low-frequency oscillations

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.

General formulation and solution of Navier-Stokes and energy equations are sought in the study of threedimensional axisymmetric unsteady stagnation-point flow and heat transfer impinging on a flat plate when the plate is moving with variable velocity and acceleration towards the main stream or away from it. As an application, among others, this accelerated plate can be assumed as a solidification front which is being formed with variable velocity. An external fluid, along z - direction, with strain rate a impinges on this flat plate and produces an unsteady three-dimensional axisymmetric flow in which the plate moves along z direction with variable velocity and acceleration in general. A reduction of Navier-Stokes and energy equations is obtained by use of appropriate similarity transformations, for the first time. The obtained ordinary differential equations are solved by using finite-difference numerical techniques. Velocity and pressure profiles along with temperature profiles are presented for different examples of the plate velocity functions and selected Prandtl numbers. According to the results obtained, the velocity and thermal boundary layers feel the effect of variations of the plate velocity more than the plate acceleration. It means that the minimum boundary layer thickness happens at the maximum value of the plate velocity and acceleration effect plays a secondary role.

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.

Compressible flow past thin wings and slender bodies are studied. First, subsonic flow past thin wings are analyzed by means of potential flow theory. The kernel function method is introduced for arbitrary planforms undergoing simple harmonic oscillations. The spanwise polynomial approximation and the chordwise trigonometric function approximations which automatically satisfy the Kutta condition and inherit the leading edge singularity are used. The Doublet-Lattice method as a more general numerical approach is given for the analysis of the flow past additional surfaces like tail or store surfaces, which are not necessarily in plane with the wing surface or surfaces having spanwise deflections. Airfoil response to the arbitrary unsteady motion is also given for subsonic flows. A brief review of shock waves and Mach waves in a supersonic flow is given. Afterwards, unsteady supersonic potential flow is studied for a simple harmonically oscillating point source. First, unsteady flow past an airfoil is considered. Then, supersonic flow about thin wings are analyzed using the Mach box technique. Introduction to supersonic kernel function method is briefly provided. Arbitrary unsteady motion of an airfoil in a supersonic flow is presented. Slender body theory is introduced to analyze the cross flow past bodies of considerable fineness where the cross flow is shown to be approximately incompressible under certain conditions. In connection with the slender body approximation, the Munk’s airship theory is utilized to predict the stability derivatives of missile like bodies.

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.

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.

A computerized analysis of supersonic nonuniform flows over sharp and spherically blunted cones at angle of attack

The axisymmetric interaction between a supersonic jet with a finite expansion ratio and a barrier is accompanied by the formation of complex sub- and supersonic flow in a shock layer whose thickness depends on the parameters of the jet and the position of the barrier. The main relationships of the interaction process have been established experimentally ([1–3] and others) and individual results of numerical calculations of such flows are known [4]. An analytical investigation of the parameters in the shock layer formed ahead of a plane barrier when an underexpanded jet impinges on it is presented below. The results of [5], where the region near the axis of a shock layer of arbitrary thickness is analyzed within the framework of a model of flow with a constant density, is placed at the basis of the analysis.

Supersonic flows about conical bodies

Unsteady, axisymmetric stagnation flow about a circular cylinder is examined when the far-field flow is a periodic function of time with a fixed time average and an oscillatory part of prescribed amplitude and frequency. Solutions are computed for arbitrary values of the Reynolds number, quantifying the effects of surface curvature, and a frequency parameter based on the period of the far-field flow. It is found that solutions remain regular and periodic provided that the far-field amplitude lies below a critical value. Above this value, solutions terminate in a finite-time singularity. The blow-up time is delayed by increasing the curvature of the surface. These results are corroborated by asymptotic predictions valid in the limits of small and large amplitude and frequency. For large Reynolds number, the problem reduces to the two-dimensional stagnation-point flow against a plane wall studied by previous authors.

The aim of this work is to develop a new numerical calculation program to determine the effect of the stagnation temperature on the calculation of the supersonic flow around a pointed airfoils using the equations for oblique shock wave and the Prandtl Meyer expansion, under the model at high temperature, calorically imperfect and thermally perfect gas, lower than the dissociation threshold of the molecules. The specific heat at constant pressure does not remain constant and varies with the temperature. The new model allows making corrections to the perfect gas model designed for low stagnation temperature, low Mach number, low incidence angle and low airfoil thickness. The stagnation temperature is an important parameter in our model. The airfoil should be pointed at the leading edge to allow an attached shock solution to be seen. The airfoil is discretized into several panels on the extrados and the intrados, placed one adjacent to the other. The distribution of the flow on the panel in question gives a compression or an expansion according to the deviation of the flow with respect to the old adjacent panel. The program determines all the aerodynamic characteristics of the flow and in particular the aerodynamic coefficients. The calculation accuracy depends on the number of panels considered on the airfoil. The application is made for high values of stagnation temperature, Mach number and airfoil thickness. A comparison between our high temperature model and the perfect gas model is presented, in order to determine an application limit of the latter. The application is for air.

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.

A simple but very efficient numerical method based on the finite difference technique has been developed for solving time-dependent non-linear flow problems. The governing equations of motion are discretized by a backward-time central-space scheme, whereby only the variables in the non-linear terms other than the main variable of the transport equation are replaced with the corresponding values calculated previously. The resulting algebraic equations for each main variable are solved by directly applying the Gaussian algorithm altered by considering the sparse structure of the diagonal-banded coefficient matrix adequately. The method is applied here to study the unsteady axisymmetric isotherm flow of an incompressible viscous fluid in a spherical shell with a stationary inner sphere and a rotating outer sphere. The description given in literature of the flow under consideration concentrates analytically on the asymptotic behaviour for very large Reynolds number Re starting from the almost rigid rotation. The case of small or moderate Reynolds numbers could be studied numerically only for Re 1000. Therefore, no data are available for the large intermediate region at high Reynolds numbers. In contrast to literature, consistent solutions for a large range of Reynolds number from 10 to 20000 are obtained with the method described here. A comparison of the results with those in literature shows a good agreenment up to Re=1000. At high Re the flow field confirms certain features such as the Stewartson shear layers as predicted by the asymptotic theory. With the results presented, a contribution is made for filling the gap between the asymptotic theory and numerical results in literature.

Unsteady axisymmetric stagnation-point flow of a viscous fluid on a cylinder

Pulsed water jets at the velocities of 300m/s to 500m/s produced by piston impact are examined by numerical analysis. Straight nozzles are used with the outlet diameters of 5 and 10mm. The two-step Lax-Wendroff method is employed for the unsteady axisymmetric two-phase flow in the nozzles using a Flux-Corrected Transport algorithm to limit artificial diffusion. After short transition, the flow in the nozzles arrives at a quasi-steady state. In the nozzle outlets, the contract flow and the separation with strong cavitation are formed. The vapor cavities also spread into the water column. Because of the short transition and contract flow, the jet speed of the straight nozzles is higher than that of the taper nozzles with the same nozzle outlet diameter. On the assumption that the flow in the nozzles is steady, the maximum jet speed can be approximated. It is illustrated that the nozzle shape affects the jet speed.