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A nonlinear unified theory of type I and II irregularities is presented that explains their principal observed characteristics. The power spectrum is predicted by using Kolmogoroff-type conservation law for the power flow in cascading eddies.
A three-dimensional, viscous flow code was used to calculate the transonic flow about the forebody of the Convair CV-990 (Galileo II) research aircraft stationed at NASA Ames Research Center. The computations were used to determine the location for a differential pressure system. In addition, attitude sensor placements were verified. These instruments comprise a meteorological measurement system, which will be used for global determination of three-dimensional wind data. The code solves the thin layer form of the Reynolds-averaged Navier-Stokes equations using an implicit numerical procedure. The governing equations are written in a generalized, nonorthogonal coordinate system, and are cast in a strong conservation law form. Laminar boundary layer results are presented for free stream Mach number of 0.8 and angles of attack of zero and 2 deg. Use of this computational tool reduced the development time for the location of the sensors and aided in the optimal placement on the aircraft of these devices.
The application of a new implicit unconditionally stable high resolution total variation diminishing (TVD) scheme to steady state calculations. It is a member of a one parameter family of explicit and implicit second order accurate schemes developed by Harten for the computation of weak solutions of hyperbolic conservation laws. This scheme is guaranteed not to generate spurious oscillations for a nonlinear scalar equation and a constant coefficient system. Numerical experiments show that this scheme not only has a rapid convergence rate, but also generates a highly resolved approximation to the steady state solution. A detailed implementation of the implicit scheme for the one and two dimensional compressible inviscid equations of gas dynamics is presented. Some numerical computations of one and two dimensional fluid flows containing shocks demonstrate the efficiency and accuracy of this new scheme. Previously announced in STAR as N83-23085
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.
A new stream function formulation is developed for the solution of Euler's equations in the transonic flow region. The stream function and the density are the dependent variables in this method, while the governing equations for adiabatic flow are the momentum equations which are solved in the strong conservation law form. The application of this method does not require a knowledge of the vorticity. The algorithm is combined with the automatic grid solver (GRAPE) of Steger and Sorenson (1979) in order to study arbitrary geometries. Results of the application of this method are presented for the NACA 0012 airfoil at various Mach numbers and angles of attack, and cylinders. In addition, detailed comparisons are made with other solutions of the Euler equations.
Conservative dissipative difference schemes for computing one dimensional flow are introduced, and the recognition and representation of flow discontinuities are discussed. Multidimensional methods are outlined. Second order finite volume schemes are introduced. Conversion of difference schemes for a single linear convection equation into schemes for the hyperbolic system of the nonlinear conservation laws of ideal compressible flow is explained. Approximate Riemann solvers are presented. Monotone initial value interpolation; and limiters, switches, and artificial dissipation are considered.
A study of the flux vector splitting method of Steger and Warming for the solution of the time dependent Euler equations in strong conservation law form for arbitrary two-dimensional geometries is presented. The procedure employed here differs from that of Buning and Steger in that it uses a different algorithm and employs implicit boundary conditions. Moreover, the method, as implemented here, does not contain any explicit smoothing or any adjustable parameters. Calculations were carried out for an NACA 0012 airfoil at various Mach numbers and angles of attack, and cylinders. Steady symmetric solutions were obtained for the full cylinder at a freestream Mach number of .5 without imposing a symmetry condition. In general, good agreement with other methods was obtained.
The shock resolution of Harten's (1982) second-order explicit method for one-dimensional hyperbolic conservation laws is investigated for a two-dimensional gas-dynamic problem. The possible extension to a high resolution implicit method for both one- and two-dimensional problems is also investigated. Applications of Harten's method to the quasi-one-dimensional nozzle problem with two nozzle shapes (divergent and convergent-divergent) and the two-dimensional shock-reflection problem resulted in high shock resolution steady-state numerical solutions.
General features of the quantum noise problem expressed as the equations of motion for a particle coupled to a set of oscillators are investigated analytically. Account is taken of the properties of the companion oscillators by formulating quantum statistical correlation Langevin equations (QSLE). The frequency of the oscillators is then retained as a natural cut-off for the quantum noise. The QSLE is further extended to encompass the particle trajectory and is bounded by initial and final states of the oscillator. The states are expressed as the probability of existence at the moment of particle collision that takes the oscillator into a final state. Two noise sources then exist: a statistical uncertainty of the initial state and the quantum dynamical uncertainty associated with a transition from the initial to final state. Feynman's path-integral formulation is used to characterize the functional of the particle trajectory, which slows the particle. It is shown that the energy loss may be attributed to friction, which satisfies energy conservation laws.
A geometric approach, similar to Van Leer's MUSCL schemes, is used to construct a second-order accurate generalization of Godunov's method for solving scalar conservation laws. By making suitable approximations, a scheme is obtained which is easy to implement and total variation diminishing. The entropy condition is also investigated from the standpoint of the spreading of rarefaction waves. Quantitative information is obtained for Godunov's method on the rate of spreading which explain the kinks in rarefaction waves often observed at the sonic point.
The shock-layer flowfield is obtained with or without viscous and heat-conducting dissipations from the conservative laws of fluid dynamics equations using a shock-fitting implicity finite-difference technique. The governing equations are cast in curvilinear-orthogonal coordinates and transformed to the domain between the shock and the body. Another set of equations is used for the singular coordinate axis, which, together with a cone generator away from the stagnation point, encloses the computation domain. A time-dependent alternating direction implicit factorization technique is applied to integrate the equations with local-time increment until a steady solution is reached. The shock location is updated after the flowfield computation, but the wall conditions are implemented into the implicit procedure. Innovative procedures are introduced to define the initial flowfield, to treat both perfect and equilibrium gases, to advance the solution on a coarse-to-fine grid sequence, and to start viscous flow computations from their corresponding inviscid solutions. The results are obtained from a grid no greater than 28 by 18 by 7 and converged within 300 integration steps. They are of sufficient accuracy to start parabolized Navier-Stokes or Euler calculations beyond the nose region, to compare with flight and wind-tunnel data, and to evaluate conceptual designs of reentry spacecraft.
The construction and the analysis of nonoscillatory shock capturing methods for the approximation of hyperbolic conservation laws was begun. These schemes share many desirable properties with total variation diminishing schemes (TVD), but TVD schemes have at most first order accuracy, in the sense of truncation error, at extreme of the solution. A uniformly second order approximation was constucted, which is nonoscillatory in the sense that the number of extrema of the discrete solution is not increasing in time. This is achieved via a nonoscillatory piecewise linear reconstruction of the solution from its cell averages, time evolution through an approximate solution of the resulting initial value problem, and averaging of this approximate solution over each cell.
Numerical studies of time dependent cosmic ray shock structures in planar geometry are interesting because analytical time-independent solutions are available which include the non-linear reactions on the plasma flow. A feature of these time asymptotic solutions is that for higher Mach numbers (M approximately 5) and for a low cosmic ray upstream pressure the solution is not uniquely determined by the usual conservation laws of mass, momentum and energy. These numerical solutions clearly indicate that much work needs to be done before we understand shock acceleration as a time dependent process. The slowness of the process is possibly due to the fact that there is a diffusive flux into the downstream region in addition to the usual advective losses. Analytic investigations of thie phenomenon are required.
Subsonic viscous flows through planar cascades are examined by means of a semielliptic form of the Navier-Stokes equations, which is obtained by neglecting the streamwise diffusion of momentum and energy in their respective equations and providing upstream interaction through the pressure field. The influence of downstream conditions is propagated upstream in the subsonic flow, via a streamwise implicit determination of the pressure field, in order to permit departure-free solutions for the overall flow problem. The continuity equation is thereby satisfied directly. Governing equations are written in strong conservation law form, in terms of general, nonorthogonal transformed coordinates and Cartesian velocity components.
The Lax-Friedrichs scheme, approximating the scalar, genuinely nonlinear conservation law u sub t + f sub x (u) = O where f(u) is, say, strictly double dot f dot a sub asterisk O is studied. The divided differences of the numerical solution at time t do not exceed 2 (t dot a sub asterisk) to the -1. This one-sided Lipschitz boundedness is in complete agreement with the corresponding estimate one has in the differential case; in particular, it is independent of the initial amplitude in sharp contrast to linear problems. It guarantees the entropy compactness of the scheme in this case, as well as providing a quantitative insight into the large-time behavior of the numerical computation.
A Petrov-Galerkin finite element formulation is presented for first-order hyperbolic systems of conservation laws with particular emphasis on the compressible Euler equations. Applications of the methodology are made to one- and two-dimensional steady and unsteady flows with shocks. Results obtained suggest the potential of the type of methods developed.
Numerical solutions of the unsteady Euler equations are obtained using the classical fourth order Runge Kutta time marching scheme. This method is fully explicit and is applied to the governing equations in the finite volume, conservation law form. In order to determine the efficiency of this scheme for solving turbomachinery flows, steady blade-to-blade solutions are obtained for compressor and turbine cascades under subsonic and transonic flow conditions. Computed results are compared with other numerical methods and wind tunnel measurements. The present study also focuses on other important numerical aspects influencing the performance of the algorithm and the solution accuracy such as grid types, boundary conditions, and artificial viscosity. For this purpose, H, O, and C type computational grids as well as characteristic and extrapolation type boundary conditions are included in the solution procedure.
Computational aspects of modern high-resolution upwind finite-difference schemes for hyperbolic systems of conservation laws are examined. An operational unification is demonstrated for constructing a wide class of flux-difference-split and flux-split schemes based on the design principles underlying total variation diminishing (TVD) schemes. Consideration is also given to TVD scheme design by preprocessing, the extension of preprocessing and postprocessing approaches to general control volumes, the removal of expansion shocks and 'glitches', relaxation methods for implicit TVD schemes, and a new family of high-accuracy TVD schemes.