An accurate shock-capturing finite-difference method to solve the Savage-Hutter equations in avalanche 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.

Investigations of flutter in transonic turbine cascades have shown that the movement of unsteady normal shocks has an important effect on the excitation of blades. In order to predict this phenomenon correctly, detailed studies concerning the response of unsteady blade pressures versus different parameters of an oscillating shock wave should be performed, if possible isolated from other flow effects in cascades. In the present investigation the correlation between an oscillating normal shock wave and the response of wall-mounted time-dependent pressure transducers was studied experimentally in a nozzle with fluctuating back pressure. Excitation frequencies between 0 Hz and 180 Hz were investigated. For the measurements, various measuring techniques were employed. The determination of the unsteady shock position was made by a line scan camera using the Schlieren flow visualization technique. This allowed the simultaneous use of unsteady pressure transducers to evaluate the behavior of the pressure under the moving shock. A numerical code, based on the fully unsteady Euler equations in conservative form, was developed to simulate the behavior of the shock and the pressures. The main results of this work were: (1) The boundary layer over an unsteady pressure transducer has a quasi-steady behavior with respect to the phase lag. The pressure amplitude depends on the frequency of the back pressure. (2) For the geometry investigated the shock amplitude decreased with increasing excitation frequency. (3) The pressure transducer sensed the arriving shock before the shock had reached the position of the pressure transducer. (4) The computed unsteady phenomena agree well with the results of the measurements.

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 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.

Physics-informed neural networks for high-speed 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.

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.

The transport of energetic particles in a spatially varying magnetic field is described by the focused transport equation. In the past two versions of this equation were investigated. The more commonly used standard form described a pitch-angle isotropization process but does not conserve the norm. In the current paper we consider the focused transport equation in conservative form also called modified focused transport equation. This equation conserves the norm but does not describe pitch-angle isotropization. We use the previously developed subspace method to solve the focused transport equation analytically and numerically. For a pure analytical treatment we employ the two-dimensional subspace approximation. Furthermore, we consider a higher dimensionality for which one needs to evaluate occurring matrix exponentials numerically. This type of semi-numerical approach is much faster than traditional solvers and, therefore, it is very useful.

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.

Numerical simulation of the instability of an inviscid liquid jet in a coflowing compressible airstream

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.

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.

The three-dimensional incompressible Navier-Stokes equations in conservative form are discretized with the finite volume method, and are solved in a time-accurate manner, using the pseudo-compressibility technique. A third-upwind differencing scheme, based on the Roe’s flux-splitting, is used to compute the convective term in the MUSCL scheme way which satisfies the TVD condition so as to avoid excessive numerical diffusivity. The viscous term is evaluated by a Gauss integration in the finite volume method way, resulting in the center differencing. An implicit factorization approximate method, based on the Euler implicit scheme, is used for the discretization of time derivative. In the time-accurate formulation, the time derivatives of velocity components in the momentum equations are differenced using a first-order, two-point, backward-difference formula. To satisfy the equation of continuity at each physical time step, the artificial compressibility relation is also introduced and several subiterations in pseudotime are required. The cell-centered layout is adopted, in which flow variable node is placed at the center of grid cell and the grid cell is used as control volume.

A mixed analysis inverse procedure based on the full potential equation in conservation form was developed to recontour a given base wing to produce density linearization scheme in applying the pressure boundary condition in terms of the velocity potential. The FL030 finite volume analysis code was modified to include the inverse option. The new surface shape information, associated with the modified pressure boundary condition, is calculated at a constant span station based on a mass flux integration. The inverse method is shown to recover the original shape when the analysis pressure is not altered. Inverse calculations for weakening of a strong shock system and for a laminar flow control (LFC) pressure distribution are presented. Two methods for a trailing edge closure model are proposed for further study.

A rather complete mathematical model for a common-rail injection-system dynamics numerical simulation was developed to support experimentation, layout, and control design, as well as performance optimization. The thermofluid dynamics of the hydraulic-system components, including rail, connecting pipes, and injectors was modeled in conjunction with the solenoid-circuit electromagnetics and the mechanics of mobile elements. One-dimensional flow equations in conservation form were used to simulate wave propagation phenomena throughout the high-pressure connecting pipes, including the feeding pipe of the injector nozzle. In order to simulate the temperature variations due to the fuel compressibility, the energy equation was used in addition to mass conservation and momentum balance equations. Besides, the possible cavitation phenomenon effects on the mass flow rate through the injector bleed orifice and the nozzle holes were taken into account. A simple model of the electromagnetic driving circuit was used to predict the temporal distribution of the force acting on the pilot-valve anchor. It was based on the experimental time histories of the current through the solenoid and of the associated voltage that is provided by the electronic control unit to the solenoid. The numerical code was validated through the comparison of the prediction results with experimental data, that is, pressure, injected flow rate, and needle lift time histories, taken on a high performance test bench Moehwald-Bosch MEP2000-CA4000. The novel injection-system mathematical model was applied to the analysis of transient flows through the hydraulic circuit of a commercial multijet second-generation common-rail system, paying specific attention to the wave propagation phenomena, to their dependence on solenoid energizing time and rail pressure, as well as to their effects on system performance. In particular, an insight was also given into the model capability of accurately predicting the wave dynamics effects on the rate and mass of fuel injected when the dwell time between two consecutive injections is varied.