Equations In Conservation-law Form Research Articles

Solving the fully nonlinear weakly dispersive Serre equations for flows over dry beds

AbstractWe describe a numerical method for solving the Serre equations that can simulate flows over dry bathymetry. The method solves the Serre equations in conservation law form with a finite volume method. A finite element method is used to solve the auxiliary elliptic equation for the depth‐averaged horizontal velocity. The numerical method is validated against the lake at rest analytic solution, demonstrating that it is well‐balanced. Since there are currently no known nonstationary analytical solutions to the Serre equation that involve bathymetry, a nonstationary forced solution, involving bathymetry was developed. The method was further validated and its convergence rate established using the developed nonstationary forced solution containing the wetting and drying of bathymetry. Finally, the method is also validated against experimental results for the run‐up of a solitary wave on a sloped beach. The finite‐volume finite‐element approach to solving the Serre equation was found to be accurate and robust.

3D Compressible Navier-Stokes Equations Solve Optical Properties of Forced Shear Layers

The optical properties of shear layers is a topic of research with applications in high power lasers and optical imaging systems. In recent years, it has been shown that large-scale structures are intrinsic features in a plane turbulent mixing layer. While there has been some research on the effects of turbulence on laser beams, there has been no study of the optical effects of large-scale structures as they exist in shear layer development, nor how these effects may be controlled. Since the susceptibility of a free shear layer to a forcing input increases as the location of the forcing moves closer to the trailing edge of the splitter plate. Disturbances can be imposed on shear layer using very small-amplitude displacements of the splitter-plate trailing edge in a direction perpendicular to the flow direction. We are studying the optical effects of large-scale structures in forced and unforced situations using periodic motion of a small flap at the trailing edge of the splitter plate. Three-dimensional Navier-Stokes equations in conservation law form are solved directly using the WENO scheme for the simulation of a spatially-developing free shear layer.

Local isometric immersions of pseudospherical surfaces described by evolution equations in conservation law form

For the pseudospherical surfaces described by a class of second order evolution equations, of the form zt=A(x,t,z)z2+B(x,t,z,z1), we consider the problem of local isometric immersion into the 3-dimensional Euclidean space E3 with a second fundamental form depending on finite-order jets of solutions z of the considered equations. We also provide an extension of our analysis to the case of k-th order evolution equations in conservation law form. Examples of equations admitting such local isometric immersions, are provided by equations like Burgers, Murray, Svinolupov–Sokolov, Kuramoto–Sivashinsky, Sawada–Kotera, Kaup–Kupershmidt, as well as hierarchies of evolution equations in conservation law form like Burgers, mKdV and KdV.

Numerical flowfield analysis of the next generation Vulcan nozzle

For numerical simulations of reacting, viscous nozzle flows in advanced rocket nozzles, a numerical method has been developed. The code is based on the well-established DLR Euler/thin layer Navier Stokes code NSHYP that was originally written for hypersonic re-entry flow simulations. The code solves the three-dimensional, time-dependent Euler and thin-layer Navier- Stokes equations in conservation-law form. Hydrogen and oxygen are considered as fuel and oxidizer, respectively. The passive scalar approach accounts for the multicomponent diffusion effects. Turbulent transport is described with an algebraic eddy-viscosity, and a &-e two-equations model. Calculations of conventional bell nozzles were performed. Results of Vulcan Mark 1 nozzle flow simulations are shown and compared in detail with another wellestablished numerical approach, based on the method of characteristi cs. A modified Vulcan engine is proposed, where the gas generator exhaust gases are injected into the main nozzle. Results of an engine analysis are presented. Based on this engine analysis, numerical flowfield calculations of the modified nozzle were performed and are presented.

A generalized Lagrangian method for solving the steady shallow water equations

A generalized Lagrangian formulation of the shallow water equations in conservation law form using the Lagrangian distance and stream function as independent variables is described. The resulting system is fully hyperbolic and renders the flux vector continuous at tangential discontinuities. A Godunov-type shock capturing method is developed through the solution of the steady Riemann problem. A second-order nonoscillatory extension is also devised to enhance the resolution. Numerical experiments indicate that the generalized Lagrangian method attains good resolution of oblique standing waves and is excellent in resolving tangential discontinuities in high-speed steady shallow water flows.

Shall4 — an implicit compact fourth-order FORTRAN program for solving the shallow-water equations in conservation-law form

A FORTRAN IV computer program is documented implementing a compact fourth-order accurate finite difference scheme in a spatially factored form, for solving the nonlinear shallow-water equations on a limited domain. In contrast to the usual fourth-order schemes this compact fourth-order scheme requires the solution of only either block-tridiagonal or cyclic block-tridiagonal coefficient matrices. Moreover this compact fourth-order scheme is related to the finite-element method and has a smaller truncation error than the usual fourth-order schemes. The integral invariants of the shallow-water equations are calculated at each time-step and were determined to be well conserved during the numerical integration, ensuring that a realistic nonlinear structure is obtained. A Schumann-Wallington low-pass filtering procedure was incorporated in the program to overcome the increased aliasing due to the higher accuracy method. A third-order boundary condition is imposed, preserving the overall fourth-order convergence rate of the interior approximation.

Boundary Approximations for Implicit Schemes for One-Dimensional Inviscid Equations of Gasdynamics

The applicability to practical calculations of recent theoretical developments in the stability analysis of difference approximations is examined for initial boundary-value problems of the hyperbolic type. For the experiments the one-dimensional inviscid gasdynamic equations in conservation law form are selected. A class of implicit schemes based on linear multistep methods for ordinary differential equations is chosen and the use of space or space-time extrapolations as implicit or explicit schemes is emphasized. Some examples with various inflow-outflow conditions highlight the commonly discussed issues: explicit vs implicit schemes, and unconditionally stable schemes. HEN finite-difference schemes are used to solve initial boundary-value problems for the equations of fluid dynamics, it is well known that most methods require more conditions than those required by the governing partial differential equations. These additional conditions for the finite-difference equations are often called numerical conditions. The conditions cannot be imposed arbitrarily but are determined, in general, using interior information, for example, by extrapolation or uncentered approximations. In this paper, any procedure used to provide a condition will be called a boundary scheme/' Whatever schemes are used for the conditions, it is a common practice to assume that the scheme has a local effect and will not affect the solution globally. During the early 1970s, Kreiss,1'2 Osher,3 Gustafsson et al.,4 Varah,5 and Gustafsson6 published a series of papers establishing methods for checking the stability and accuracy of difference approximations with schemes included. Since then, further progress has been made in the theory of linear difference approximations for initial boundary-value problems of the hyperbolic and parabolic type.7'12 Because improper treatment of the conditions can lead to instability and inaccuracy, even though we start with a stable interior scheme (i.e., scheme for the interior points), it is appropriate to adopt an approach that includes the stability and accuracy of the combined interior and schemes. Surveys of recent developments and extensive bibliographies are included in papers by Coughran13 and Yee.14 The purpose of this paper is to examine the applicability to practical calculations (for nonlinear gasdynamic problems) of recent theoretical stability analyses of implicit difference approximations for initial boundary-value problems of the hyperbolic type. As computations have progressed, the use of the conservative form of the gasdynamic equations has gained popularity. For physical reasons it is sometimes desirable to specify conditions in the nonconservative variables and to compute with conservative variables in the interior. We will consider the additional complications introduced by this procedure.

Semi-implicit Hopsco tch-Type Met hods for the Time-Dependent Naiver-Stokes Equations

Treatment of the time-dependent Navier-Stokes equatipns for the solution of the flowfield about airfoil configurations is generally performed using either explicit or implicit finite-difference methods. As a viable alternative, hopscotch-type methods combine the speed of the explicit and the favorable stability criteria of the implicit methods. Such methods are constructed here for the time-dependent Navier-Stokes equations in conservation law form; their implementation to a number of test cases (one of Euler's equations), including the stringent case of shock-wave/bo undary-layer interaction with separation, indicates the possibility of competitive accuracy and more rapid computing time than is required by currently popular, fully implicit methods. , •' e = total enthalpy F,F',G,G' = defined in Eqs. (3) and (20), respectively ij = index / = identity matrix L = (nonlinear) finite-difference operator M = Mach number n = index P = pressure Pr = Prandtl number Q,Q',R,R' = defined in Eqs. (4) and (21), respectively Re = Reynolds number T - temperature T' = source term

An Implicit Compact Fourth-Order Algorithm for Solving the Shallow-Water Equations in Conservation-Law Form

Abstract An alternating-direction implicit finite-difference scheme is developed for solving the nonlinear shallow-water equations in conservation-law form. The algorithm is second-order time accurate, while fourth-order compact differencing is implemented in a spatially factored form. The application of the higher order compact Pade differencing scheme requires only the solution of either block-tridiagonal or cyclic block-tridiagonal coefficient matrices, and thus permits the use of economical block-tridiagonal algorithms. The integral invariants of the shallow-water equations, i.e., mass, total energy and enstrophy, are well conserved during the numerical integration, ensuring that a realistic nonlinear structure is obtained. Largely in an experimental way, two methods are investigated for determining stable approximations for the extraneous boundary conditions required by the fourth-order method. In both methods, third-order uncentered differences at the boundaries are utilized, and both preserve the o...

Computation of the Viscous Supersonic Flow over External Axial Corners

The viscous supersonic flowfield surrounding symmetric and asymmetric swept external axial corners is computed using existing second-order, implict and explicit finite-difference computer codes. The governing partial differential equations in conservation-law form are hyperbolic with respect to time and are assumed to be conical; that is, any variation of the radial convective or diffusive terms is assumed to be negligible. An inviscid numerical study for symmetric 90 deg dihedral rounded corners reveals both single and triple crossflow stagnation point flows, depending on the corner radius of curvature. For asymmetrical 90 deg dihedral configurations (i.e., those obtained by varying one of the wedge angles), both single and triple stagnation point flows are also observed. For certain highly asymmetrical configurations, the inviscid flow spills over the corner and in so doing generates a local embedded supersonic bubble. Viscous numerical solutions are obtained for both symmetric and highly asymmetric axial corners for which experimental surface oil flow and vapor screen data are available. For the symmetric configuration, both the laminar and turbulent numerical solutions result in flow away from the corner, which agrees with the experimental data. For the highly asymmetrical configuration, the viscous numerical solution and the experimental data result in flow around the corner. The computed shock locations for both configurations agree fairly well with the experimental vapor screen results.

Diffraction of a Shock Wave by a Compression Corner: I. Regular Reflection

The unsteady, two-dimensional flowfield resulting from the interaction of a moving planar shock wave with a compression corner is determined using a second-order, discontinuity-fitting, finite-difference approach. The time-dependent Euler equations are transformed to normalize the distance between the body and peripheral shock and to include the existing self-similar property of the flow. The resulting set of partial differential equations in conservation-law form is then solved in a time-dependent fashion using MacCormack's scheme. The vortical singularity, which lies on the body surface, and the single reflected shock are both treated as discontinuities in the numerical procedure. The results of the numerical simulation compare quite favorably with existing experimental interferograms and yield better flowfield resolution than previous first-order, shock-capturing, numerical solutions.

Transonic Flow Around Rotor Blade Elements

An analytical/numerical method for calculating blade-to-blade transonic flow fields around blade elements of an axial compressor rotor of a turbomachine is presented. Radius variation and stream sheet convergence are included. The analysis considers the time-dependent Navier-Stokes equations in conservation-law form which allows for viscous shock wave formation. A discretized method of characteristics is used to determine the boundary conditions along the blade surface and exit plane. An explicit, time-dependent, second-order, finite difference technique is utilized to numerically solve these partial differential equations in a transformed computational plane. Comparison of numerical results with cascade and rotor data indicate good agreement for blade surface pressure distributions and mixed-out exit conditions.

Three-Dimensional, Shock-on-Shock Interaction Problem

The unsteady, three-dimensional flowfield resulting from the interaction of a plane shock with a cone-shaped vehicle traveling supersonically is determined, using a second-order, shock-capturing, finite-difference approach. The time-dependent, inviscid gasdynamic equations are transformed to include the self-similar property of the flow, to align various coordinate surfaces with known shock waves, and to cluster points in the vicinity of the intersection of the transmitted incident shock and the surface of the vehicle. The governing partial differential equations in conservation-law form are then solved iteratively using MacCormack's (1969) algorithm.

Multishocked, Three-Dimensional Supersonic Flowfields with Real Gas Effects

This program determines the supersonic flowfield surrounding three-dimensional wing-body configurations of a delta wing. It was designed to provide the numerical computation of three dimensional inviscid, flowfields of either perfect or real gases about supersonic or hypersonic airplanes. The governing equations in conservation law form are solved by a finite difference method using a second order noncentered algorithm between the body and the outermost shock wave, which is treated as a sharp discontinuity. Secondary shocks which form between these boundaries are captured automatically. The flowfield between the body and outermost shock is treated in a shock capturing fashion and therefore allows for the correct formation of secondary internal shocks . The program operates in batch mode, is in CDC update format, has been implemented on the CDC 7600, and requires more than 140K (octal) word locations.