SEARCH · Search NASA

Results for “Euler equations”

Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

SourceResearch connectionFrom yearThrough year

Quote a phrase for an exact phrase match. Source license links do not imply unrestricted reuse.

At least 37 records · Page 2

All sources NASA NTRS DOE OSTI

NASA NTRS1985

Finite difference solutions of the Euler equations in the vicinity of sharp edges

Attempts have been made to explain why finite difference solutions of the Euler equations can describe flows with large vortical structures around sharp-edged bodies. The present paper is concerned with the influence of a singular sharp edge on the truncation error for a set of discretized Euler equations. An analysis is conducted of the distribution of the truncation error of one finite difference approximation of the Euler equations near a sharp edge of a thin plate. The analysis leads to a determination of the size of the region of the neighborhood of such a singularity. Attention is given to the consistency of a discretization of the Euler equations, and numerical experiments.

Hartwich, P.-M.↗

NASA NTRS1992

Design of optimally smoothing multistage schemes for the Euler equations

A recently derived local preconditioning of the Euler equations is shown to be useful in developing multistage schemes suited for multigrid use. The effect of the preconditioning matrix on the spatial Euler operator is to equalize the characteristic speeds. When applied to the discretized Euler equations, the preconditioning has the effect of strongly clustering the operator's eigenvalues in the complex plane. This makes possible the development of explicit marching schemes that effectively damp most high-frequency Fourier modes, as desired in multigrid applications. The technique is the same as developed earlier for scalar convection schemes: placement of the zeros of the amplification factor of the multistage scheme in locations where eigenvalues corresponding to high-frequency modes abound.

Van Leer, Bram↗

NASA NTRS1986

Unsteady transonic flows past airfoils using the Euler equations

An explicit finite difference solver for the Euler equations was developed and used to calculate the flow past two of the AGARD standard aeroelastic configurations, an NACA 64A010 and an NLR 7301. The algorithm employed uses a modified four step Runge-Kutta time stepping scheme, boundary conditions determined from the time-dependent theory of characteristics and an unsteady automatic grid generation procedure. In general, the calculated results are in good agreement with available experiment. Moreover, they demonstrate the importance of using the Euler equations for super-critical airfoils and in situations where the transonic small disturbance theory results in poor agreement with experiment.

Smith, G. E.↗

NASA NTRS1983

Design and implementation of a multigrid code for the Euler equations

The steady-state equations of inviscid fluid flow, the Euler equations, are a nonlinear nonelliptic system of equations admitting solutions with discontinuities (for example, shocks). The efficient numerical solution of these equations poses a strenuous challenge to multigrid methods. A multigrid code has been developed for the numerical solution of the Euler equations. In this paper some of the factors that had to be taken into account in the design and development of the code are reviewed. These factors include the importance of choosing an appropriate difference scheme, the usefulness of local mode analysis as a design tool, and the crucial question of how to treat the nonlinearity. Sample calculations of transonic flow about airfoils will be presented. No claim is made that the particular algorithm presented is optimal.

Jespersen, D. C.↗

NASA NTRS1991

Three dimensional unstructured multigrid for the Euler equations

The three-dimensional Euler equations are solved on unstructured tetrahedral meshes using a multigrid strategy. The driving algorithm consists of an explicit vertex-based finite-element scheme, which employs an edge-based data-structure to assemble the residuals. The multigrid approach employs a sequence of independently generated coarse and fine meshes to accelerate the convergence to steady-state of the fine grid solution. Variables, residuals and corrections are passed back and forth between the various grids of the sequence using linear interpolation. The addresses and weights for interpolation are determined in a preprocessing stage using an efficient graph traversal algorithm. The preprocessing operation is shown to require a negligible fraction of the CPU time required by the overall solution procedure, while gains in overall solution efficiencies greater than an order of magnitude are demonstrated on meshes containing up to 350,000 vertices. Solutions using globally regenerated fine meshes as well as adaptively refined meshes are given.

Mavriplis, D. J.↗

NASA NTRS1991

Uniform high order spectral methods for one and two dimensional Euler equations

Uniform high order spectral methods to solve multi-dimensional Euler equations for gas dynamics are discussed. Uniform high order spectral approximations with spectral accuracy in smooth regions of solutions are constructed by introducing the idea of the Essentially Non-Oscillatory (ENO) polynomial interpolations into the spectral methods. The authors present numerical results for the inviscid Burgers' equation, and for the one dimensional Euler equations including the interactions between a shock wave and density disturbance, Sod's and Lax's shock tube problems, and the blast wave problem. The interaction between a Mach 3 two dimensional shock wave and a rotating vortex is simulated.

Cai, Wei↗

NASA NTRS1989

Transonic flutter calculations using the Euler equations

In transonic flutter problems where shock motion plays an important part, it is believed that accurate predictions of the flutter boundaries will require the use of codes based on the Euler equations. Only Euler codes can obtain the correct shock location and shock strength, and the crucially important shock excursion amplitude and phase lag. The present study is based on the finite volume scheme developed by Jameson and Venkatakrishnan for the 2-D unsteady Euler equations. The equations are solved in integral form on a moving grid. The variable are pressure, density, Cartesian velocity components, and total energy.

Bendiksen, Oddvar O.↗

NASA NTRS1986

An embedding method for the steady Euler equations

Certain difficulties arise in connection with the numerical solution of a direct finite difference representation of the steady Euler equations. Johnson (1979, 1981, 1982) has, therefore, proposed a surrogate-equation technique, in which the first-order steady Euler equations are embedded in a certain second-order system of equations. The present paper is concerned with the theoretical justification for such an embedding approach. For the numerical solution of the two-dimensional steady Euler equations, it is shown that, under a continuity restriction, it is possible to solve a second-order embedded system together with appropriate additional boundary conditions. The result indicates that a more direct and potentially more efficient approach to the steady solutions exists than the alternative of solving the unsteady equations.

Chang, S.-H.↗

NASA NTRS1993

Uniform high-order spectral methods for one- and two-dimensional Euler equations

Uniform high order spectral methods to solve multi-dimensional Euler equations for gas dynamics are discussed. Uniform high order spectral approximations with spectral accuracy in smooth regions of solutions are constructed by introducing the idea of the Essentially Non-Oscillatory (ENO) polynomial interpolations into the spectral methods. The authors present numerical results for the inviscid Burgers' equation, and for the one-dimensional Euler equations including the interactions between a shock wave and density disturbance, Sod's and Lax's shock tube problems, and the blast wave problem. The interaction between a Mach 3 two-dimensional shock wave and a rotating vortex is simulated.

Cai, Wei↗

NASA NTRS2005

Solving Nonlinear Euler Equations with Arbitrary Accuracy

A computer program that efficiently solves the time-dependent, nonlinear Euler equations in two dimensions to an arbitrarily high order of accuracy has been developed. The program implements a modified form of a prior arbitrary- accuracy simulation algorithm that is a member of the class of algorithms known in the art as modified expansion solution approximation (MESA) schemes. Whereas millions of lines of code were needed to implement the prior MESA algorithm, it is possible to implement the present MESA algorithm by use of one or a few pages of Fortran code, the exact amount depending on the specific application. The ability to solve the Euler equations to arbitrarily high accuracy is especially beneficial in simulations of aeroacoustic effects in settings in which fully nonlinear behavior is expected - for example, at stagnation points of fan blades, where linearizing assumptions break down. At these locations, it is necessary to solve the full nonlinear Euler equations, and inasmuch as the acoustical energy is of the order of 4 to 5 orders of magnitude below that of the mean flow, it is necessary to achieve an overall fractional error of less than 10-6 in order to faithfully simulate entropy, vortical, and acoustical waves.

Dyson, Rodger W.↗

NASA NTRS2004

Textbook Multigrid Efficiency for the Steady Euler Equations

A fast multigrid solver for the steady incompressible Euler equations is presented. Unlike time-marching schemes, this approach uses relaxation of the steady equations. Application of this method results in a discretization that correctly distinguishes between the advection and elliptic parts of the operator, allowing efficient smoothers to be constructed. Solvers for both unstructured triangular grids and structured quadrilateral grids have been written. Computations for channel flow and flow over a nonlifting airfoil have computed. Using Gauss-Seidel relaxation ordered in the flow direction, textbook multigrid convergence rates of nearly one order-of-magnitude residual reduction per multigrid cycle are achieved, independent of the grid spacing. This approach also may be applied to the compressible Euler equations and the incompressible Navier-Stokes equations.

Roberts, Thomas W.↗

NASA NTRS1993

Comparison of coordinate-invariant and coordinate-aligned upwinding for the Euler equations

A floating-shock fitting method for the Euler equations has been developed that uses one-sided spatial differences along and across streamlines. This method has been applied to unsteady shocked flow in a duct with a ramp, to supercritical flow over a circular cylinder, and to transonic flow over airfoils. Compared to methods using coordinate-aligned upwind differencing, the coordinate-invariant upwinding generally required fewer grid points to produce crisp shocks and shears. For transonic airfoils, coordinate-invariant upwind differencing advances the agreement between computations and experiment, and it reduces the grid dependency of the computed results for strong shocks. The computational expenditure is comparable to that of coordinate-aligned upwind methods for the Euler equations.

Hartwich, Peter M.↗

NASA NTRS1985

Implicit finite-difference methods for the Euler equations

The present paper is concerned with two-dimensional Euler equations and with schemes which are in use of the time of this writing. Most of the development presented carries over directly to three dimensions. The characteristics of the two-dimensional Euler equations in Cartesian coordinates are considered along with generalized curvilinear coordinate transformations, metric relations, invariants of the transformation, flux Jacobian matrices and eigensystems, numerical algorithms, flux split algorithms, implicit and explicit nonlinear control (smoothing), upwind differencing in supersonic regions, unsteady and steady-state computation, the diagonal form of implicit algorithm, metric differencing and invariants, boundary conditions, geometry and mesh generation, and sample solutions.

Pulliam, T. H.↗

NASA NTRS1998

Perfectly Matched Layer for Linearized Euler Equations in Open and Ducted Domains

Recently, perfectly matched layer (PML) as an absorbing boundary condition has widespread applications. The idea was first introduced by Berenger for electromagnetic waves computations. In this paper, it is shown that the PML equations for the linearized Euler equations support unstable solutions when the mean flow has a component normal to the layer. To suppress such unstable solutions so as to render the PML concept useful for this class of problems, it is proposed that artificial selective damping terms be added to the discretized PML equations. It is demonstrated that with a proper choice of artificial mesh Reynolds number, the PML equations can be made stable. Numerical examples are provided to illustrate that the stabilized PML performs well as an absorbing boundary condition. In a ducted environment, the wave mode are dispersive. It will be shown that the group velocity and phase velocity of these modes can have opposite signs. This results in a confined environment, PML may not be suitable as an absorbing boundary condition.

Tam, Christopher K. W.↗

NASA NTRS1983

High resolution applications of the Osher upwind scheme for the Euler equations

The 'Osher' scheme was introduced by Osher (1981). It represents an upwind finite-difference method for hyperbolic systems of conservation laws, including the Euler equations. In studies conducted by Osher (1981) and Osher and Solomon (1982), the method was applied to the nonisentropic form of the Euler equations in one dimension and the isentropically restricted form in two spatial dimensions, both in Cartesian coordinates. Chakravarthy and Osher (1982) have shown an approach for extending the Osher scheme to the Euler equations written for general geometries, taking into account the use of mappings to arbitrary curvilinear coordinate systems. The present investigation is concerned with the high resolution extension of the Osher scheme to second-order accuracy. Results are presented for several example problems, giving attention to quasi-one-dimensional Laval nozzle flow, a one-dimensional shock tube problem, and supersonic flow over a cylinder.

Chakravarthy, S. R.↗

NASA NTRS1995

On absorbing boundary conditions for linearized Euler equations by a perfectly matched layer

Recently, Berenger introduced a Perfectly Matched Layer (PML) technique for absorbing electromagnetic waves. In the present paper, a perfectly matched layer is proposed for absorbing out-going two-dimensional waves in a uniform mean flow, generated by linearized Euler equations. It is well known that the linearized Euler equations support acoustic waves, which travel with the speed of sound relative to the mean flow, and vorticity and entropy waves, which travel with the mean flow. The PML equations to be used at a region adjacent to the artificial boundary for absorbing these linear waves are defined. Plane waves solutions to the PML equations are developed and wave propagation and absorption properties are given. It is shown that the theoretical reflection coefficients at an interface between the Euler and PML domains are zero, independent of the angle of incidence and frequency of the waves. As such, the present study points out a possible alternative approach for absorbing out-going waves of the Euler equations with little or no reflection in computation. Numerical examples that demonstrate the validity of the proposed PML equations are also presented.

Hu, Fang Q.↗

NASA NTRS1994

Multiblock, Multigrid Solution Of Euler Equations

Method of numerical solution of Euler equations of three-dimensional flow of compressible fluid involves combination of multiblock and multigrid strategies. In multiblock strategy, flow field divided, into multiple smaller, more computationally-convenient zones and computational grid fitted to applicable flow boundaries generated in each block. In multigrid strategy used here, different quantities computed, variously, on finer or coarser grids. Minimizing cost of computation by using fewest grid points yielding acceptably accurate values of affected variable. Multigrid strategy found effective in accelerating convergence to steady state, while multiblock strategy provides geometric flexibility.

Melson, N. Duane↗

NASA NTRS1991

Characteristic time-stepping or local preconditioning of the Euler equations

A derivation is presented of a local preconditioning matrix for multidimensional Euler equations, that reduces the spread of the characteristic speeds to the lowest attainable value. Numerical experiments with this preconditioning matrix are applied to an explicit upwind discretization of the two-dimensional Euler equations, showing that this matrix significantly increases the rate of convergence to a steady solution. It is predicted that local preconditioning will also simplify convergence-acceleration boundary procedures such as the Karni (1991) procedure for the far field and the Mazaheri and Roe (1991) procedure for a solid wall.

Van Leer, Bram↗