A Godunov method for supersonic tactical missiles

: 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.

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.

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

A numerical method of solution is developed for steady–state hyperbolic equations in conservation form. An upwind direction is identified and a smooth mapping is introduced to align the equations with that direction. A flux F(u) in the upwind direction is regarded as the ‘state’ variable and advanced numerically on a grid according to a second–order extension of Godunov' method. The solution u is determined from the computed flux, and a constraint on the mapping is that the Jacobian, det(∂F∂u), is of one sign. The method is applied to the equations modelling shock–wave propagation according to Whitham' theory of geometrical shock dynamics. In this context the solution vector u gives the components of the gradient of a function whose level curves determine the shock positions at successive times. Shock propagation in both uniform and non–uniform gases is considered and several example calculations are presented. A further application of the numerical method is inviscid supersonic flow, as governed by the steady Euler equations. The steady supersonic flow in a converging–diverging duct is computed to illustrate the numerical method for this latter application.

For Mach reflection in steady supersonic flow, the slipline and reflected shock wave from the triple point are disturbed by secondary Mach waves generated over the slipline and by the expansion fan from the rear wedge corner. Analytical expressions for the shape of the curved slipline and reflected shock wave are derived in this paper. It is found that, due to transmitted expansion waves from the expansion fan, the slipline has a slope discontinuity at the turning point, i.e., the intersection point of the slipline and the leading characteristics of the transmitted expansion wave. The hypothetical shock wave calculated by considering this slope discontinuity as flow deflection angle matches a similar wave observed in numerical results by computational fluid dynamics, suggesting the existence of a weak shock wave from this turning point. The effects of the secondary Mach waves upstream of the turning point and of the turning point weak shock wave mutually cancel out approximately so that the transmitted Mach waves can be approximated as straight characteristic lines. This simplification leads to a fast analytical model which can predict the Mach stem height and shape of the slipline and reflected shock wave with increasing accuracy for the decreasing deflection angle of the slipline at the triple point. The slipline slope discontinuity at the turning point and the hypothetical turning point weak shock wave are new phenomena found in this work.

This study investigates the effect of energy deposition on pressure and thermal loads generated by an Edney IV interaction. This complex supersonic jet impingement problem is formed by the intersection of an oblique shock generated by a 15-deg wedge with a Mach 3.45 bow shock in front of a 0.0254-m diam sphere. The full three-dimensional Reynolds-averaged Navier-Stokes equations are solved with the k-w turbulence model. Mesh-resolved simulations show that surface pressure without energy addition is in good agreement with experiments. The peak value is 1.8 times that observed without the impinging shock. Results from a grid resolution study confirm that the surface pressure is less sensitive than the heat flux. For flow control, a spherical energy pulse with a volume of 3 mm 3 and energy of 283 mJ is deposited upstream of the primary triple point. The unsteady interaction of the energy spot and its induced blast wave with the oblique shock, the distorted bow shock, and the impinging supersonic jet is elucidated in the context of the lensing phenomenon. The simulations indicate significant impact of the energy deposition on the surface pressure and heat flux. The instantaneous surface pressure and heat flux rise when the blast wave and high-energy spot hit the surface and fall when the expansion waves reach the surface. However, the overall integrated stress and thermal loads are reduced, mainly due to the effect of the expansion waves.

The dynamic characteristic as one key feature determines the earthquake response and seismic performance of structures.Curved box beams occur coupling of flexural and torsional modes,which companied with warping.Based on the precise transfer matrix method,precise transfer matrixs for solving the natural frequencies and modes of curved box beam out plane have been derivated.The analysis of vibration characteristics of single-span curved box beam with pinned-pinned ends has been done. Example shows that the precise transfer matrix method is a simple and effective method for dynamic characteristic analysis of curved box beams.

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.

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

The difficulties inherent in converting the zone‐centered Godunov method into a 3D Lagrangian/ALE scheme have led us to propose an SMG scheme. The SMG/Q version presented here solves internal energy and momentum equations by using only zone‐centered “collision” Riemann problems. It is formulated in a dual Godunov/Classical‐Lagrange way. A limited‐slope approximation of zone‐centered velocity gradients produces a second‐order extension of this method. Basic test cases, both 1D and 3D, demonstrate the SMG/Q features.

The exact Riemann Solver to the Shallow Water equations for natural channels with bottom steps

A characteristic analysis of an ultrasonic motor (USM) at the design stage has thus farbeen impossible. Therefore, a characteristic analysis method is suggested on the basis of aproposed model describing the complex nonlinear contact condition between therotor and stator. The proposed contact model and analysis method can guidetheoretical research on the minimization of the main disadvantages of the USM, whichmainly result from the contact mechanism. The validity and usefulness of thesuggested analysis method is verified by experimental data from a prototyped USM.

Well-posedness for two-dimensional steady supersonic Euler flows past a Lipschitz wedge

Edney III type shock-shock interaction over a hemisphere cylinder