Piecewise Parabolic Method Research Articles

APPLICATION OF THE PIECEWISE PARABOLIC FINITE ANALYTIC METHOD TO THE THREE-DIMENSIONAL CAVITY FLOW

Abstract The three-dimensional piecewise parabolic finite analytic method (PPFAM3D), with vector potential and vorticity as dependent variables, was used to solve the steady three-dimensional (3-D) laminar cavity flow. Predictions were obtained for Reynolds numbers between 100 and 2000. Results for Re = 100, 400, and 1000 are compared with those available in the literature. The overall agreement is excellent. Results are also compared with the two-dimensional PPFAM cavity flow, showing the effect of three-dimensionality on the velocity predictions at the symmetry plane. The PPFAM3D is a robust numerical algorithm requiring no relaxation to produce physically meaningful and numerically converged solutions. It is a promising tool for solving a variety of flow phenomena governed by unsteady second-order partial differential convection-diffusion equations

A FCT Method for Staggered Mesh

A finite difference method on a staggered mesh for shock hydrodynamics with diffusion controlled via flux corrected transport is described. The algorithm is conservative, free streaming invariant and well behaved around shocks. This method is second order in accuracy in the worse case with several third- and fourth-order features. The algorithm is designed in Lagrangian coordinates and an arbitrary mesh can be used when remapping with piecewise parabolic method.

The piecewise parabolic finite analytic method Part 2: Application

In this second part, the piecewise parabolic finite analytic method ( PPFAM) is used to discretize the stream function and vorticity equations. The numerical scheme thus obtained is used to simulate a laminar cavity flow and a laminar backward-facing step flow. The numerical simulation shows that the PPFAM is the least affected by numerical diffusion when compared with more traditional numerical schemes and that it can provide a stable and accurate numerical solution for real flow problems. Furthermore, compared with other numerical schemes, the PPFAM produces a stable, convergent, and more accurate solution for the cavity flow with the largest flow rate value, ∥Ψ∥ max, for Reynolds numbers ranging from 10 to 2000. When compared with available experimental and numerical results, the PPFAM produces the largest and most accurate reattachment length for the backward-facing step channel flow.

New insights into large eddy simulation

Fluid dynamic turbulence is one of the most challenging computational physics problems because of the extremely wide range of time and space scales involved, the strong nonlinearity of the governing equations, and the many practical and important applications. While most linear fluid instabilities are well understood, the nonlinear interactions among them makes even the relatively simple limit of homogeneous isotropic turbulence difficult to treat physically, mathematically, and computationally. Turbulence is modeled computationally by a two-stage bootstrap process. The first stage, direct numerical simulation, attempts to resolve the relevant physical time and space scales but its application is limited to diffusive flows with a relatively small Reynolds number (Re). Using direct numerical simulation to provide a database, in turn, allows calibration of phenomenological turbulence models for engineering applications. Large eddy simulation incorporates a form of turbulence modeling applicable when the large-scale flows of interest are intrinsically time dependent, thus throwing common statistical models into question. A promising approach to large eddy simulation involves the use of high-resolution monotone computational fluid dynamics algorithms such as flux-corrected transport or the piecewise parabolic method which have intrinsic subgrid turbulence models coupled naturally to the resolved scales in the computed flow. The physical considerations underlying and evidence supporting this monotone integrated large eddy simulation approach are discussed.

A numerical study of supersonic turbulence

We report on simulations of nonstationary supersonic isotropic turbulent flow using the Piecewise-Parabolic Method on uniform grids of 20482 in two dimensions and 2563 in three dimensions. Intersecting shock waves initiate the transfer of energy from long to short wavelengths. Weak shocks survive for many acoustic times τac. In two dimensions eddies merge over many τac. In three dimensions vortex sheets break-up into short vortex filaments within two τac. Entropy fluctuations, produced by strong shocks, are stretched into filaments over several τac. These filaments persist and are mirrored in the density. We observe three temporal phases: onset, with the initial formation of shocks; quasi-supersonic, with strong density contrasts; and post-supersonic, with a slowly decaying root mean square Mach number. Compressive modes quickly establish a k−2 velocity power spectrum. In three dimensions solenoidal modes build-up during the supersonic phase delineating through time a Kolmogorov k−5/3 envelope and leaving a self-similarly decaying ∼k−0.9 spectrum at lower wave numbers.

The piecewise parabolic finite analytic method. Part I: Theory

To meet both the qualitative and quantitative requirements of the convective diffusion process, a numerical scheme, called the piecewise parabolic finite analytic method, is established by using a piecewise parabolic function as a boundary condition. This scheme incorporates the local analytic solution of the convective diffusion equation into the numerical scheme. It is free from numerical diffusion and automatically incorporates the convective diffusion effects. In this first part, a detailed study of the piecewise parabolic finite analytic scheme is presented. Initially, the maximum and minimum principle of the convective diffusion equation is used to determine the properties of the piecewise parabolic finite analytic coefficients. Then, based upon the truncation error and properties of the coefficients, the existence, uniqueness, stability, convergence, and error estimates of the numerical solution are obtained. For nonlinear problems the existence and error estimates are also obtained.

Semi-Lagrangian Piecewise Biparabolic Scheme for Two-Dimensional Horizontal Advection of a Passive Scalar

Abstract A conservative semi-Lagrangian algorithm with a snall computational diffusion is presented that may be applied to advection of passive scalars in numerical models of the atmosphere. The technique is preferable for the horizontal semistaggered grids where a scalar point is surrounded by four velocity points. It is a coupling of the semi-Lagrangian approach and the piecewise parabolic method (PPM). Unlike the original PPM when applied to the advection of passive scalars, the new scheme is a fully two-dimensional algorithm. Also, it is not restricted by the linear stability condition. This paper describes the two steps comprising the two-dimensional algorithm. The first one is the interpolation pressure for getting the piecewise biparabolic function, and the second is a conservative remapping from the original grid to the grid made by the departure domains. Several test integrations are presented in which the described scheme performs very successfully.

The Performance of Classical versus Modern Finite-Volume Advection Schemes for Atmospheric Modeling in a One-Dimensional Test-bed

Abstract The numerical solution of the transport (i.e., the continuity) equation for trace species, particularly in three-dimensional circulation models, has recently received great attention. Two different approaches are common. First, the classical numerical methods employed in circulation models for the solution of other continuity equations are used for the transport problem. In these methods, filters are commonly applied to control undesirable features in the solution. Second, methods were developed that were specially designed to obviate the problems arising when the transport equation is numerically solved. Here, both approaches are investigated and compared in a simple one-dimensional test-bed. The methods discussed encompass leapfrog time stepping, with both discrete and spectral spatial resolution, various filters, and four Eulerian finite-volume advection schemes: the Prather scheme, the Bott scheme, the piecewise parabolic method (PPM), and four versions of the multidimensional positive-defini...

Three-dimensional hydrodynamic simulations of narrow-angle-tail radio sources. I - The Begelman, Rees, and Blandford model

view Abstract Citations (45) References (27) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS Three-dimensional Hydrodynamic Simulations of Narrow-Angle--Tail Radio Sources. I. The Begelman, Rees, and Blandford Model Balsara, Dinshaw S. ; Norman, Michael L. Abstract Narrow-angle-tail radio sources have been explained by various authors as being the result of ram-pressure-induced bending of supersonic radio jets by a transonic crosswind. The details of the models differ, but their common element is the necessity of a supersonic jet and a crosswind. We have, therefore, made a computational study of the bending of light supersonic jets by a wind flowing normally to it using the highly accurate RIEMANN code run at the highest resolutions that are currently feasible. RIEMANN is a three-dimensional hydrocode that implements the piecewise parabolic method of Colella and Woodward. An attempt is made to identify those characteristics of narrow-angle-tail sources that might have reasonably general physical explanations and then to verify whether these characteristics can be reproduced by simulations. In this work we have resolved the internal structure of the jet including primary and secondary oblique shocks. We have elucidated their role in jet bending and deceleration and the formation of knots. We have shown the effect of Rayleigh-Taylor instabilities on jet cross-sectional evolution. We have also resolved the Kelvin-Helmholtz instability through which the jet eventually disrupts. Our interpretations of the numerical results are supported with quantitative analyses whenever possible. We have also made detailed comparisons with the analytical work of Begelman, Rees, & Blandford. Moreover, we have applied our results to the interpretation of the archetypal narrow-angle-tail radio source NGC 1265. Publication: The Astrophysical Journal Pub Date: July 1992 DOI: 10.1086/171531 Bibcode: 1992ApJ...393..631B Keywords: Astronomical Models; Kelvin-Helmholtz Instability; Radio Jets (Astronomy); Supersonic Flow; Three Dimensional Models; Radio Sources (Astronomy); Space Density; Taylor Instability; Astrophysics; GALAXIES: INDIVIDUAL NGC NUMBER: NGC 1265; GALAXIES: JETS; METHODS: NUMERICAL; VIDEOTAPES full text sources ADS | data products SIMBAD (1) NED (1)

Three-dimensional supersonic homogeneous turbulence: A numerical study.

A turbulent homogeneous perfect gas with no forcing is computed using the piecewise-parabolic method on a uniform grid of 256{sup 3} zones. The initial rms Mach number is one. The main results are (i) persistence of a quasisupersonic phase with strong density contrasts over several shock formation times; (ii) late appearance of a post-supersonic self-similarly decaying regime in which the total velocity spectrum varies with wave number as {ital k}{sup {minus}1}; and (iii) both shock curvature and velocity gradients are important mechanisms of vorticity production in the first half of the supersonic phase.

Theoretical and Numerical Structure for Unstable One-Dimensional Detonations

The spatio-temporal structure of unstable detonations in a single space dimension is studied through a combination of numerical and asymptotic methods. A new high resolution numerical method for computing unstable detonations is developed. This method combines the piecewise parabolic method (PPM) with conservative shock tracking and adaptive mesh refinement. A new nonlinear asymptotic theory for the spatio-temporal growth of instabilities is also developed. This asymptotic theory involves a nonclassical “Hopf bifurcation”, because resonant acoustic scattering states with exponential growth in space cross the imaginary axis and become nonlinear eigenmodes in a complex free-boundary problem for a nonlinear hyperbolic equation. An interplay between the asymptotic theory and numerical simulation is used to elucidate the spatio-temporal mechanisms of nonlinear stability near the transition boundary; in particular, a quantitative-qualitative explanation is developed for the experimentally observed instabilities...

Numerical simulations of the nonlinear kink modes in linearly stable supersonic slip surfaces

We have performed high-resolution numerical simulations of supersonic slip surfaces to confirm and illuminate earlier analytic nonlinear stability calculations of such structures. This analytic work was in turn inspired by earlier computer simulations reported in Woodward (1985) and Woodward et al. (1987). In particular Artola & Majda (1987) examined the response of a supersonic slip surface to an incident train of small-amplitude nonlinear sound waves. They found analytic solutions which indicate that nonlinear resonance occurs at three angles of incidence which depend upon the Mach number of the relative motion. The two-dimensional simulations described here numerically solve this problem for a Mach-4 flow using the piecewise-parabolic method (Colella & Woodward 1984; Woodward & Colella 1984). The simulations show that sound waves incident at a predicted resonance angle excite nonlinear behaviour in the slip surface. At these angles the amplitude of the reflected waves is much greater than the incident wave amplitude (i.e. a shock forms). The observed resonance is fairly broad, but the resonance narrows as the strength of the incident waves is reduced.The nature of the nonlinear kink modes observed in the simulations is similar to that discussed by Artola & Majda. Most of the modes move in either direction with speeds near the predicted value. Speeds of other than this value are observed, but the disagreement is not serious in view of the strongly nonlinear behaviour seen in the simulations but not treated in the analytic work. The stationary modes seen in the analytic results are perhaps observed as transient structures. They may eventually dominate the flow at late times (Woodward et al. 1987).The role of the kink modes in the stability of slab jets is discussed, and it is argued that the stationary modes are more disruptive than the propagating modes.

Application of the Piecewise Parabolic Method (PPM) to Meteorological Modeling

Abstract The Piecewise Parabolic Method (PPM), a numerical technique developed in astrophysics for modeling fluid flows with strong shocks and discontinuities is adapted for treating sharp gradients in small-scale meteorological flows. PPM differs substantially from conventional gridpoint techniques in three ways. First, PPM is a finite volume scheme, and thus represents physical variables as averages over a grid zone rather than single values at discrete points. Second, a unique, monotonic parabola is fit to the zone average of each dependent variable using information from neighboring zone averages. As shown in a series of one- and two-dimensional linear advection experiments, the use of parabolas provides for extremely accurate advection, particularly of sharp gradients. Furthermore, the monotonicity constraint renders PPM's solutions free from Gibbs oscillations. PPM's third attribute is that each zone boundary is treated as a discontinuity. Using the method of characteristic the nonlinear flux of qua...

AN OBSERVATIONAL AND NUMERICAL STUDY OF EXTRAGALACTIC RADIO SOURCES

The Very Large Array is used to make radio continuum observations of a sample of 12 distant (z = 1) high-luminosity 3CR galaxies. Images with arc-second resolution are produced at X20 cm and X6 cm. The total and polarized intensity images are used to create maps of the fractional polarization, depolarization, rotation measure, and spectral-index distributions. Additional arc-second-resolution observations at Â.2 cm, higher-resolution A.6 cm and X2 cm maps, and optical R -band continuum and emission-line images of [O n] X3727 are presented for 3C 337. The results show that a galaxy scale, two-phase Faraday medium capable of depolarizing the radio emission exists in the environments of the sample galaxies. The warm phase is observed in emission lines, while the hot gas can presently be seen only through its Faraday effects. Suggestive evidence for dynamically significant interactions between the emission-line gas and the radio sources is presented. Numerical simulations of the interaction of a Mach 4 relative velocity slip surface with several trains of incident nonlinear sound waves are performed using the Piecewise-Parabolic method. A resonance in the response of the slip surface to the waves is demonstrated, confirming analytical predictions. Characteristic combinations of nonlinear waves associated with finite kinks which propagate along the slip surface are generated only when the waves are incident at nearly the predicted resonant angle. The width of the resonance decreases as the incident wave strength is reduced. PPM simulations of the supersonic slip surface between 10: 1 density ratio fluids are made at M = 1, 1.9, and 4. Further simulations test the effects of grid resolution, the SLIC interface tracking method, and the initial transverse perturbation. The slip surface has a different characteristic response on either side of the linear stability of M = 1.77. Below this limit the slip surface promptly rolls up into large-scale vortices. At Mach 4 the evolution is dominated by the propagating kink modes noted above, and the interface is much more stable. Transition behavior is observed at M = 1.9, with neither response dominant. The mass and momentum entrainment efficiencies measured in the simulations decrease with increasing Mach number, and scaling arguments suggest that astrophysically-significant amounts of ambient material can be entrained by extragalactic radio

Errors for calculations of strong shocks using an artificial viscosity and an artificial heat flux

The artificial viscosity ( Q) method of von Neumann and Richtmyer is a tremendously useful numerical technique for following shocks wherever and whenever they appear in the flow. We show that it must be used with some caution, however, as serious Q-induced errors (on the order of 100%) can occur in some strong shock calculations. We investigate three types of Q errors: 1. Excess Q heating, of which there are two types: (a) excess wall heating on shock formation and (b) shockless Q heating; 2. Q errors when shocks are propagated over a nonuniform mesh; and 3. Q errors in propagating shocks in spherical geometry. As a basis of comparison, we use as our standard the Lagrangian formulation with Q = C 0 2 ϱl 2( u x ) 2. This standard Q is compared with Noh's (Q & H) shock-following method, which employs an artificial heat flux ( H) in addition to Q, and with the (non- Q) piecewise-parabolic method (PPM) of Colella and Woodward. Both the (Q & H) method and PPM (particularly when used with an adaptive shock-tracking mesh) give superior results for our test problems. In spherical geometry, Schulz's and Whalen's tensor Q formulations of the hydrodynamic equations prove to be more accurate than the standard Q formulation, and when Schulz's formulation is combined with Noh's (Q & H) method, superior results are achieved.

An implicit-explicit hybrid method for Lagrangian hydrodynamics

We describe a new implicit-explicit hybrid method for solving the equations of hydrodynamics. The scheme is an extension of the explicit second-order piecewise-parabolic method (PPM) which is unconditionally stable. The scheme is thus of the Godunov type. It is conservative, accurate to second order in both space and time, and makes use of a nonlinear Riemann solver to obtain fluxes of the conserved quantities. The hybrid character of the method provides increased accuracy and computational efficiency. Switching between implicit and explicit formulations occurs smoothly and in a natural way and is performed separately for each characteristic family of waves. The method provides high resolution with shocks spread over only one zone and can produce accurate answers to most reasonable problems without the use of an artificial viscosity. 1986 Academic Press, Inc.

Shock formation and decay in hydrodynamic flows with mass loading

Hydrodynamic flows with mass loading are expected to occur in the environment of several bodies of the solar system, in particular near comets. We extend the so-called “piecewise parabolic method” (PPM), originally developed for ordinary hydrodynamics, in such a way that time-dependent flows with mass accretion can be treated. Numerical examples illustrating the influence of the mass loading effect on the formation and decay of shocks in the flow are presented and discussed.