A higher-order Godunov method for the hyperbolic equations modelling shock dynamics

sented in these previous studies. Because the linearized Euler equations in conservative form are used in the present work, new source terms have to be derived for the conservative set of equations. A formal derivation of source terms for the linearized Euler equations in conservative form is presented, and the system of equations is compared to those of Goldstein and Bailly et al. Simplified versions of the derived source terms are also developed. To test the derived source terms, a direct simulation of a forced two-dimensional mixing layer is carried out. The direct simulation provides the mean flow and source field for the inhomogeneous linearized Euler equations. The solutions to the linearized Euler equations with source terms are compared to the solution of the direct simulation and show good agreement. All simulations are performed using Tam and Webb’s fourth-order dispersion-relation-preserving scheme and a four-step fourth-order Runge‐Kutta time-marching technique. Artificial selective damping introduced through the numerical scheme is used to avoid spurious waves. Absorbing boundary conditions based on characteristic variables are used at the free boundaries and a buffer layer is added at the outflow.

A simple numerical scheme for the calculation of the motion of shock waves in gases based on Whitham's theory of geometrical shock dynamics is developed. This scheme is used to study the propagation of shock waves along walls and in channels and the self-focusing of initially curved shockfronts. The numerical results are compared with exact and numerical solutions of the geometrical-shock-dynamics equations and with recent experimental investigations.

A nonlinear aerodynamic analysis technique based on the full potential equation in conservative form has been modified to permit treatment of supersonic flows with embedded subsonic regions (typically near the fuselage-canopy juncture and the wing leading edge). Solution procedures for the equations do not require any specific form of geometry or physical grid system. This results in the capability to analyze easily very complex geometries provided the posed problem lies within the isentropic restrictions of the full potential theory. Characteristic signal propagation theory is used to monitor the type dependent flow and a conservative switching scheme is employed to transition from the supersonic marching algorithm to a subsonic relaxation procedure and vice versa. An implicit approximate factorization scheme is used to solve the finite-difference equations. These modifications now permit analysis of fully three-dimensional flowfields including the interference effects due to lifting surface wakes. Improved grid generation capability allows analysis of complete complex aircraft geometries (fuselage, wing, tail, wing wake, and tail wake). Results are presented showing very good correlations with experimental surface pressure data and aerodynamic force data at both design and off-design operating points. Configurations examined include several waverider concepts, an arrow wing-body with wake, an advanced tactical fighter concept, and a fighter forebody-canard configuration.

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.

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.

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

This paper presents a numerical simulation method developed for separated flow in cascades using the Euler equations and demonstrates the feasibility of this method. MacCormack’s two-steps explicit finite difference scheme is used to discretize the equations in conservation form, and the artificial viscosity is added to the discretized inviscid equations by means of the self-adapted filter technique. The initial separation boundary is given according to simple experimental results. The numerical simulation results including subsonic and transonic turbine cascades flow with or without separation show that the fundamental idea of this numerical method is reasonable and simple. The present study indicates that for solving certain engineering problems it is a simple and effective tool for adding some viscosity corrections to inviscid flow model, especially the current when the Navier-Stokes equations have not been solved very effectively for various complicated flows in turbomachinery.

We construct a new first-order central-upwind numerical method for solving systems of hyperbolic equations in conservative form. It applies in multidimensional structured and unstructured meshes. The proposed method is an extension of the UFORCE method developed by Stecca, Siviglia and Toro, in which the upwind bias for the modification of the staggered mesh is evaluated taking into account the smallest and largest wave of the entire Riemann fan. The proposed first-order method is shown to be identical to the Godunov upwind method in applications to a 2 x 2 linear hyperbolic system. The method is then extended to non-linear systems and its performance is assessed by solving the two-dimensional inviscid shallow water equations. Extension to second-order accuracy is carried out using an ADER-WENO approach in the finite volume framework on unstructured meshes. Finally, numerical comparison with current competing numerical methods enables us to identify the salient features of the proposed method.

Calculation of external-internal flow fields for mixed-compression inlets

This paper describes a Godunov method applicable to tactical missiles in steady supersonic flow. The scheme is cast in a finite-volume formulation using a mesh generated by a multiple-zone procedure. The bow shock is fitted using the information contained in the Riemann problem. Surface pressures are determined using a method of characteristics analysis that degenerates near shocks or expansions to the oblique shock or PrandtlMeyer expansion relations. The scheme is applied to body-alone and finned configurations and calculations are completed without the use of artificial viscosity or special procedures.

The present work studies upwind schemes applied to the solution of aeronautical and aerospace problems. The Harten, the Frink, Parikh and Pirzadeh, the Liou and Steffen and the Radespiel and Kroll algorithms, all first order accurate in space, are studied. The Euler equations in conservative form, employing a finite volume formulation and a structured spatial discretization, in the two-dimensional space, are solved. A time splitting method and a Runge-Kutta method of five stages are used to perform the time march of the numerical schemes. The steady state physical problems of the supersonic flow along a ramp and around a blunt body configuration are studied. All algorithms are accelerated to the steady state solution using a spatially variable time step. This technique has proved excellent gains in terms of convergence ratio as reported in Maciel. The results have demonstrated that the Liou and Steffen scheme has presented the most critical solutions, in both example-cases, in relation to the others schemes and a more accurate solution, in terms of the determination of the stagnation pressure in the blunt body case, than the Harten and the Radespiel and Kroll schemes. In the ramp problem, the Harten scheme predicts the best pressure distribution along the ramp wall in comparison with theoretical results. In the blunt body problem, the Liou and Steffen scheme presents the highest value of Cp at the configuration nose in relation to the other schemes. Values of cL and cD have been accurately predicted by all schemes, except by the Harten scheme.

This paper describes a numerical method for designing closed aerofoils that correspond to given surface speed distributions including flows with shock waves. The prescribed surface speed distribution in general contains three parameters whose values are found as part of the solution. The inverse aerofoil design problem is not well posed unless the prescribed speed satisfies three constraints that can be used to fix the value of the three adjustable parameters. The computational method is based on the numerical solution of the full potential equation in conservation form by an approximate factorization–multigrid scheme. The nature of the constraints and several strategies for introducing the necessary freedom in the prescribed speed are discussed along with typical solutions illustrating the method.

A beginner in the field of computing is hardly aware of the importance of formulations responsible for the accuracy in scientific computing. In theoretical fluid mechanics it is not important as to which form of Navier–Stokes equation is solved. But, in computing this is paramount, and there is always ongoing debate among practitioners about the superiority of different formulations and numerical methods employed by different schools of thought.

Physics-informed neural networks for high-speed flows

An adaptive finite volume method for 2D steady Euler equations with WENO reconstruction