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At least 199 records · Page 11

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NASA NTRS1977

The artificial compression method for computation of shocks and contact discontinuities. I - Single conservation laws

The paper discusses the use of the artificial compression method for the computation of discontinuous solutions of a single conservation law by finite difference methods. The single conservation law has either a shock or a contact discontinuity. Any monotone finite difference scheme applied to the original equation smears the discontinuity, while the same scheme applied to the equation modified by an artificial compression flux produces steady progressing profiles. If L is any finite difference scheme in conservation form and C is an artificial compressor, the split flux artificial compression method CL is a corrective scheme: L smears the discontinuity while propagating it; C compresses the smeared transition toward a sharp discontinuity. Numerical implementation of artificial compression is described.

Harten, A.↗

NASA NTRS1978

Coefficient matrices for implicit finite difference solution of the inviscid fluid conservation law equations

Although the Navier-Stokes equations describe most flows of interest in aerodynamics, the inviscid conservation law equations may be used for small regions with viscous forces. Thus, Euler equations and several time-accurate finite difference procedures, explicit and implicit, are discussed. Although implicit techniques require more computational work, they permit larger time steps to be taken without instability. It is noted that the Jacobian matrices for Euler equations in conservation-law form have certain eigenvalue-eigenvector properties which may be used to construct conservative-form coefficient matrices. This reduces the computation time of several implicit and semiimplicit schemes. Extensions of the basic approach to other areas are suggested.

Steger, J. L.↗

NASA NTRS1978

Simulation study of the effect of fuel-conservative approaches on ATC procedures and terminal area capacity

Fuel-conservative procedures have been investigated using real-time air traffic control simulations linked to two piloted simulators. The fuel-conservative procedures studied were profile descents and two types of landing approaches. The investigation determined the effect of these procedures on the ATC system and terminal area capacity. It examined the mixing of aircraft executing fuel-conservative approaches with those executing conventional approaches. The results indicate a systems fuel savings for the landing approaches under all tested conditions except at, or near, maximum system capacity. Also, there is a fuel savings and reduced controller workload for the profile descent procedures.

Tobias, L.↗

NASA NTRS1979

Energy conserving norms for the solution of hyperbolic systems of partial differential equations

The hyperbolic system of partial differential equations with a real constant square coefficient matrix A is considered. The problem of finding an energy conserving norm for the solution of the system is reduced to the problem of characterizing those matrices appearing in the boundary conditions which satisfy two specific matrix equations. Necessary and sufficient conditions on the coefficient matrix A and the matrices appearing in boundary conditions are derived for an energy conserving norm. The conditions serve as criteria on a given system which determine whether or not the solution will have its energy conserved in some norm. Examples of specific systems and boundary conditions are also provided.

Gunzburger, M. D.↗

NASA NTRS1979

Simulation study of the operational effects of fuel-conservative approaches

Fuel-conservative procedures have been investigated using real-time air traffic control simulations linked to two piloted simulators. The fuel-conservative procedures studied were profile descents and two types of landing approaches, delayed flap and IATA. The investigation determined the effect of these procedures on the ATC system operation. It examined the mixing of aircraft executing fuel-conservative approaches with those executing conventional approaches. The most difficult approach type mix of traffic was found to be 50% conventional and 50% delayed flap. However, for the test scenario chosen, arrival rates of at least 30 aircraft per hour were feasible and resulted in a net average fuel saving, even for the most difficult mix. Also, there is a fuel savings and reduced controller workload for the profile descent procedures.

Tobias, L.↗

NASA NTRS1981

Conservative full-potential calculations for axisymmetric, transonic flow

A conservative, finite-difference, full-potential relaxation code has been developed to solve transonic flow around axisymmetric or nonlifting, planar, two-dimensional bodies. The program utilizes the artificial compressibility method to provide an upwind bias in supersonic regions. Calculated examples include a wide variety of axisymmetric and planar two-dimensional shapes with various blunt, pointed and open ends in subsonic, transonic, and low supersonic free streams. Comparisons between conservative and nonconservative full-potential calculations show perfect agreement at convergence when the flow is entirely subsonic. Noticeable differences exist, however, between the conservative and nonconservative solutions for transonic flows with shocks.

Green, L. L.↗

NASA NTRS1982

Development of a locally mass flux conservative computer code for calculating 3-D viscous flow in turbomachines

The VANS successive approximation numerical method was extended to the computation of three dimensional, viscous, transonic flows in turbomachines. A cross-sectional computer code, which conserves mass flux at each point of the cross-sectional surface of computation was developed. In the VANS numerical method, the cross-sectional computation follows a blade-to-blade calculation. Numerical calculations were made for an axial annular turbine cascade and a transonic, centrifugal impeller with splitter vanes. The subsonic turbine cascade computation was generated in blade-to-blade surface to evaluate the accuracy of the blade-to-blade mode of marching. Calculated blade pressures at the hub, mid, and tip radii of the cascade agreed with corresponding measurements. The transonic impeller computation was conducted to test the newly developed locally mass flux conservative cross-sectional computer code. Both blade-to-blade and cross sectional modes of calculation were implemented for this problem. A triplet point shock structure was computed in the inducer region of the impeller. In addition, time-averaged shroud static pressures generally agreed with measured shroud pressures. It is concluded that the blade-to-blade computation produces a useful engineering flow field in regions of subsonic relative flow; and cross-sectional computation, with a locally mass flux conservative continuity equation, is required to compute the shock waves in regions of supersonic relative flow.

Walitt, L.↗

NASA NTRS1982

Conservative supra-characteristics method for splitting the hyperbolic systems of gasdynamics with computed boundaries for real and perfect gases

Implicit methods developed by Beam and Warming (1978 and Briley and McDonald (1977) make it possible to overcome the hyperbolic stiffness of the conservative compressible Navier-Stokes equations in the fine wall region computational mesh for high Reynold's number flow. Certain difficulties related to the use of these methods could be overcome by employing an approach reported by Roe (1981). In the present investigation Roe's conceptual framework has been adopted for constructing globally conservative finite difference methods. A globally conservative upwind finite difference method (CSCM) consisting of both implicit interior point and boundary point equations is constructed from a new characteristics based flux difference splitting. It is found that the employed upwind eigenvector split scheme which combines fully coupled implicit interior point and boundary point approximations has the desired properties of robust stability and accuracy.

Lombard, C. K.↗

NASA NTRS1983

Conservation laws of wave action and potential enstrophy for Rossby waves in a stratified atmosphere

The evolution of wave energy, enstrophy, and wave motion for atmospheric Rossby waves in a variable mean flow are discussed from a theoretical and pedagogic standpoint. In the absence of mean flow gradients, the wave energy density satisfies a local conservation law, with the appropriate flow velocity being the group velocity. In the presence of mean flow variations, wave energy is not conserved, but wave action is, provided the mean flow is independent of longitude. Wave enstrophy is conserved for arbitrary variations of the mean flow. Connections with Eliassen-Palm flux are also discussed.

Straus, D. M.↗

NASA NTRS1984

Second- and third-order upwind difference schemes for hyperbolic conservation laws

Second- and third-order two time-level five-point explicit upwind-difference schemes are described for the numerical solution of hyperbolic systems of conservation laws and applied to the Euler equations of inviscid gas dynamics. Nonliner smoothing techniques are used to make the schemes total variation diminishing. In the method both hyperbolicity and conservation properties of the hyperbolic conservation laws are combined in a very natural way by introducing a normalized Jacobian matrix of the hyperbolic system. Entropy satisfying shock transition operators which are consistent with the upwind differencing are locally introduced when transonic shock transition is detected. Schemes thus constructed are suitable for shockcapturing calculations. The stability and the global order of accuracy of the proposed schemes are examined. Numerical experiments for the inviscid Burgers equation and the compressible Euler equations in one and two space dimensions involving various situations of aerodynamic interest are included and compared.

Yang, J. Y.↗

NASA NTRS1985

Planning fuel-conservative descents in an airline environmental using a small programmable calculator: Algorithm development and flight test results

A simple, airborne, flight-management descent algorithm was developed and programmed into a small programmable calculator. The algorithm may be operated in either a time mode or speed mode. The time mode was designed to aid the pilot in planning and executing a fuel-conservative descent to arrive at a metering fix at a time designated by the air traffic control system. The speed model was designed for planning fuel-conservative descents when time is not a consideration. The descent path for both modes was calculated for a constant with considerations given for the descent Mach/airspeed schedule, gross weight, wind, wind gradient, and nonstandard temperature effects. Flight tests, using the algorithm on the programmable calculator, showed that the open-loop guidance could be useful to airline flight crews for planning and executing fuel-conservative descents.

Knox, C. E.↗

NASA NTRS1985

On the convergence of difference approximations to scalar conservation laws

A unified treatment of explicit in time, two level, second order resolution, total variation diminishing, approximations to scalar conservation laws are presented. The schemes are assumed only to have conservation form and incremental form. A modified flux and a viscosity coefficient are introduced and results in terms of the latter are obtained. The existence of a cell entropy inequality is discussed and such an equality for all entropies is shown to imply that the scheme is an E scheme on monotone (actually more general) data, hence at most only first order accurate in general. Convergence for total variation diminishing-second order resolution schemes approximating convex or concave conservation laws is shown by enforcing a single discrete entropy inequality.

Osher, S.↗

NASA NTRS1983

Conservation laws of wave action and potential enstrophy for Rossby waves in a stratified atmosphere

Straus, D. M.↗

NASA NTRS1986

An analysis of finite-difference and finite-volume formulations of conservation laws

Finite-difference and finite-volume formulations are analyzed in order to clear up the confusion concerning their application to the numerical solution of conservation laws. A new coordinate-free formulation of systems of conservation laws is developed, which clearly distinguishes the role of physical vectors from that of algebraic vectors which characterize the system. The analysis considers general types of equations--potential, Euler, and Navier-Stokes. Three-dimensional unsteady flows with time-varying grids are described using a single, consistent nomeclature for both formulations. Grid motion due to a non-inertial reference frame as well as flow adaptation is covered. In comparing the two formulations, it is found useful to distinguish between differences in numerical methods and differences in grid definition. The former plays a role for non-Cartesian grids, and results in only cosmetic differences in the manner in which geometric terms are handled. The differences in grid definition for the two formulations is found to be more important, since it affects the manner in which boundary conditions, zonal procedures, and grid singularities are handled at computational boundaries. The proper interpretation of strong and weak conservation-law forms for quasi-one-dimensional and axisymmetric flows is brought out.

Vinokur, Marcel↗

NASA NTRS1988

On the convergence of difference approximations to scalar conservation laws

A unified treatment is given for time-explicit, two-level, second-order-resolution (SOR), total-variation-diminishing (TVD) approximations to scalar conservation laws. The schemes are assumed only to have conservation form and incremental form. A modified flux and a viscosity coefficient are introduced to obtain results in terms of the latter. The existence of a cell entropy inequality is discussed, and such an equality for all entropies is shown to imply that the scheme is an E scheme on monotone (actually more general) data, hence at most only first-order accurate in general. Convergence for TVD-SOR schemes approximating convex or concave conservation laws is shown by enforcing a single discrete entropy inequality.

Osher, Stanley↗

NASA NTRS1988

Properties of finite difference models of non-linear conservative oscillators

Finite-difference (FD) approaches to the numerical solution of the differential equations describing the motion of a nonlinear conservative oscillator are investigated analytically. A generalized formulation of the Duffing and modified Duffing equations is derived and analyzed using several FD techniques, and it is concluded that, although it is always possible to contstruct FD models of conservative oscillators which are themselves conservative, caution is required to avoid numerical solutions which do not accurately reflect the properties of the original equation.

Mickens, R. E.↗

NASA NTRS1988

Universal limiter for transient interpolation modeling of the advective transport equations: The ULTIMATE conservative difference scheme

A fresh approach is taken to the embarrassingly difficult problem of adequately modeling simple pure advection. An explicit conservative control-volume formation makes use of a universal limiter for transient interpolation modeling of the advective transport equations. This ULTIMATE conservative difference scheme is applied to unsteady, one-dimensional scalar pure advection at constant velocity, using three critical test profiles: an isolated sine-squared wave, a discontinuous step, and a semi-ellipse. The goal, of course, is to devise a single robust scheme which achieves sharp monotonic resolution of the step without corrupting the other profiles. The semi-ellipse is particularly challenging because of its combination of sudden and gradual changes in gradient. The ULTIMATE strategy can be applied to explicit conservation schemes of any order of accuracy. Second-order schemes are unsatisfactory, showing steepening and clipping typical of currently popular so-called high resolution shock-capturing of TVD schemes. The ULTIMATE third-order upwind scheme is highly satisfactory for most flows of practical importance. Higher order methods give predictably better step resolution, although even-order schemes generate a (monotonic) waviness in the difficult semi-ellipse simulation. Little is to be gained above ULTIMATE fifth-order upwinding which gives results close to the ultimate for which one might hope.

Leonard, B. P.↗

NASA NTRS1989

An analysis of finite-difference and finite-volume formulations of conservation laws

Finite-difference and finite-volume formulations are analyzed in order to clear up the confusion concerning their application to the numerical solution of conservation laws. A new coordinate-free formulation of systems of conservation laws is developed, which clearly distinguishes the role of physical vectors from that of algebraic vectors which characterize the system. The analysis considers general types of equations: potential, Euler, and Navier-Stokes. Three-dimensional unsteady flows with time-varying grids are described using a single, consistent nomenclature for both formulations. Grid motion due to a non-inertial reference frame as well as flow adaptation is covered. In comparing the two formulations, it is found useful to distinguish between differences in numerical methods and differences in grid definition. The former plays a role for non-Cartesian grids, and results in only cosmetic differences in the manner in which geometric terms are handled. The differences in grid definition for the two formulations is found to be more important, since it affects the manner in which boundary conditions, zonal procedures, and grid singularities are handled at computational boundaries. The proper interpretation of strong and weak conservation-law forms for quasi-one-dimensional and axisymmetric flows is brought out.

Vinokur, Marcel↗