Computational Fluid Dynamics

The following sections are included:

• Preface

• Committees

• Contents

The steadily increasing power of supercomputing systems is enabling very high resolution simulations of compressible, turbulent flows in the high Reynolds number limit, which is of interest in astrophysics as well as in several other fluid dynamical applications. This paper discusses two such simulations, using grids of up to 8 billion cells. In each type of flow, convergence in a statistical sense is observed as the mesh is refined. The behavior of the convergent sequences indicates how a subgrid-scale model of turbulence could improve the treatment of these flows by high-resolution Euler schemes like PPM. The best resolved case, a simulation of a Richtmyer–Meshkov mixing layer in a shock tube experiment, also points the way toward such a subgrid-scale model. Analysis of the results of that simulation indicates a proportionality relationship between the energy transfer rate from large to small motions and the determinant of the deviatoric symmetric strain as well as the divergence of the velocity for the large-scale field.

Direct numerical simulations of the three-dimensional Hall–MHD equations and of a long-wavelength asymptotic model are used to study the instabilities and the nonlinear dynamics of a circularly polarized Alfvén wave subject to a weak random noise. The evolution is shown to be strongly sensitive to the spectral extension of the initial noise, due to the presence of competing instabilities. The formation of magnetic filaments is usually observed when only large-scale modulational perturbations are permitted, while a more turbulent picture is obtained when small-scale unstable modes are initially excited. A filamentary dynamics nevertheless develops in the presence of a broad initial spectrum in the case of a right-hand polarized pump of long wavelength.

The following sections are included:

• Introduction

• The "yguazú-a" code

• Validation of the code

• Observations and models of a jet from a young star

• Summary

• Acknowledgments

• References

In this paper are presented the main results obtained by the adaptation of the nonhydrostatic meteorological model MésoNH, developed at CNRM (Centre National des Recherches Météorologiques) in Toulouse (France) to high angular resolution astronomical observations. Challenges and perspectives for future progress are discussed.

We present preliminary results of a series of numerical simulations, in one and two dimensions with different resolutions (1024 and 3072 zones for the 1D case, and of 256× 128 and 512 × 256 zones for the 2D case), of interplanetary shock waves using the magnetohydrodynamic (MHD) numerical code ZEUS-3D. Interplanetary shocks are produced by different perturbations associated with solar activity and propagate in the interplanetary medium throughout the solar wind. The objective of this study is to understand different physical characteristics of the origin and propagation of interplanetary shocks. The results of the numerical simulations will be compared with in-situ observations of interplanetary shocks by different spacecraft. The code ZEUS-3D has been tested and is used efficiently on the CRAY Y-MP and SGI Origin 2000 computers at UNAM's Supercomputer Center.

A fast radiative shock in the interstellar medium (ISM) is a powerful source of ionizing photons. These photons are produced in the hot postshock cooling gas and can propagate both upstream and downstream. The photons travelling upstream encounter preshock gas and may produce an extensive precursor ionized region, while those travelling downstream influence the ionization and temperature structure of the recombination region of the shock. Supernova remnants (SNR) are a source of fast shocks in the ISM. In this work we investigate the influence a SNR has on the ionization of the ISM during its lifetime by means of hydrodynamic simulations of the evolution of remnants that include a detailed calculation of the nonequilibrium ionization state of the gas and radiative transfer of the radiation field produced in the postshock cooling region.

We present results of numerical simulations performed to study the interaction between jets and Supernova Remnants (SNRs). Our aim is to explain the strange morphologies some SNRs exhibit in the radio-continuum, such as the case of SNR W50 in which a shell is interacting with the jets from the SS 433 source.

We investigate numerically the role of thermal instability (TI) as a generator of density structures in the interstellar medium (ISM), both by itself and in the context of a globally turbulent medium. Simulations of flows in the presence of the instability only show that the condensation process which forms a dense phase ("clouds") is highly dynamical. Final static situations, characterized by a bimodal or single peaked with a slope change density histogram (PDF), may be established, but the equilibrium is very fragile. Simulations containing the instability and various types of turbulent energy injection show that large-scale turbulent forcing is incapable of erasing the signature of TI in the density PDFs, but small-scale, stellar-like forcing causes the PDFs to transit from bimodal to a single-slope power law, erasing the signature of the instability. The third group of simulations are models of the ISM including the magnetic field, the Coriolis force, self-gravity and stellar energy injection. These simulations show no significant difference between the PDFs of stable and unstable cases, and reach stationary regimes, suggesting that the combination of the stellar forcing and the extra effective pressure provided by the magnetic field and the Coriolis force overwhelm TI as a density-structure generator in the ISM, TI becoming a second-order effect.

We explore the dissipative ability of turbulent compressible flows that model the interstellar medium (ISM) at intermediate-to-large scales. The main feature of our simulations is the (realistic) way in which the turbulent kinetic energy is injected to the fluid: around the star formation sites, the gas is accelerated radially away from the "stars", acquiring a well-defined final velocity difference u_f over a characteristic length scale l_f. We study the dependence of turbulent dissipation on these two quantities. The spatially scattered, small-scale nature of this forcing gives rise to the coexistence of both forced and decaying turbulent regimes within the same flow. In the forced case, the global dissipation time is proportional to (l_f/u_f)/u_rms, where u_rms is the rms velocity dispersion of the flow. The kinetic energy injection and dissipation rates are very close, implying that most of the turbulence is dissipated near the localized input sources. In the decaying regime, the kinetic energy decays as a power law in time, with an exponent ~ -0.8. Our results, if applicable to the vertical direction in the Galactic disk, are consistent with models of galaxy evolution in which large-scale star formation is self-regulated by an energy balance in the vertical component of the gaseous disk. On the other hand, our results do not support galaxy formation models in which the stellar energy injection in the disk is required to self-regulate star formation and reheat the gas at the level of the whole cosmological halo (of typical sizes 15–20 times larger than the optical galaxy).

We investigate the behavior of the magnetic pressure, b², in fully turbulent MHD flows in "1+2/3" dimensions by means of its effect on the probability density function (PDF) of the density field. We start by reviewing our previous results for general polytropic flows, according to which the value of the polytropic exponent determines the functional shape of the PDF. A lognormal density PDF appears in the isothermal (γ = 1) case, but a power-law tail at either large or small densities appears for large Mach numbers when γ > 1 and γ < 1, respectively. In the isothermal magnetic case, the relevant parameter is the field fluctuation amplitude, δB/B. A lognormal PDF still appears for small field fluctuations (generally the case for large mean fields), but a significant low-density excess appears at large fluctuation amplitudes (weak mean fields), similar to the behavior at γ > 1 of polytropic flows. We interpret these results in terms of simple nonlinear MHD waves, for which the magnetic pressure behaves linearly with the density in the case of the slow mode, and quadratically in the case of the fast wave. Finally, we discuss some implications of these results, in particular the fact that the effect of the magnetic field in modifying the PDF is strongest when the mean field is weak.

Multifractal analysis is a powerful technique that allows categorization of the structure of complex objects. We present preliminary results on the multifractal spectrum f(α) of observational and numerically simulated interstellar medium (ISM) data. We consider numerical simulations in two and three dimensions of the ISM at intermediate scales (hundreds of pc), as well as molecular gas data. For purely fractal objects, the multifractal spectrum is reduced to a single point. When applied to observational and simulated ISM data, the technique shows well-defined multifractal spectra in all cases, including velocity-channel projections of 3D density cubes, implying that the ISM has a multifractal structure, rather than being a simple fractal. For density data, the quantity α corresponds to the scaling exponent between mass and size. The fact that a range of values of α is found in all cases supports previous claims (Vázquez-Semadeni, Ballesteros-Paredes and Rodríguez 1997) that this exponent is not unique. Additionally, for density data, a significant fraction of the f(α) curve lies on values of α smaller than the dimension of the embedding space, indicating hierarchical structuring of the field (larger structures have smaller average densities). This property is also satisfied by projected velocity-channel data. Three- and two-dimensional simulations show significantly different f(α) curves, which suggests intrinsically different geometrical properties.

Stratiform low clouds have important effects as weather events, particularly for aircraft terminal operations, and as a major influence on regional and global climatology. The structure and modeling of such clouds is reviewed, with emphasis on their microphysics and radiative iniuences. Large eddy simulation has been applied to stratiform clouds with some success, though limited by resolution.

We have implemented in Mexico the Advanced Regional Prediction System (ARPS) developed by the Center for Analysis and Prediction of Storms (CAPS) of the University of Oklahoma, to study deep convective clouds that develop almost daily over Mexico City during the rainy season (from June to November). Mexico City is located in an elevated basin surrounded by mountains on three sides (East, South and West), that reach up to 1.5 km above mean basin level. The orographic forcing is crucial to the development of convection, but the larger scale conditions determine the location and strength of the convective activity. This convection is more frequently observed in the SW corner of the basin, where climatological precipitation values are a factor of 2 larger than in the NE sector. In this study we have simulated a storm that occurred on August 18, 1997, and gave way to generalized flooding in the WSW area of the basin.

The results indicate that the main storm characteristics are fairly well reproduced by the model, when comparing with the precipitation measured at the surface by the city's raingauge network. The dynamics of a long-lasting precipitating cloud appears to be the interaction of a daughter cell with its parent cells. The main mechanism for precipitation production is the interaction of cloud ice with hail in the initial 2 cells, although some warm rain conversion is also active.

The exponential instability of the Legendre polynomial flows, Rossby–Haurwitz waves, Wu–Verkley waves and monopole, dipole and quadrupole modons on a sphere is considered. These flows are exact solutions to the barotropic vorticity equation on a rotating sphere. We have obtained a conservation law for disturbances and a condition necessary for the normal mode instability of each such flow, and estimated the maximum growth (and decay) rate of the modes. We have also shown that the amplitude of any unstable, decaying and nonstationary mode is orthogonal to the basic solution in the energy inner product. Any mode not satisfying the new instability condition is neutral. The condition localizes the unstable modes in phase space, characterizes the spectral structure of growing perturbations, and is helpful in testing a computational algorithm developed for the numerical linear stability study of a flow on a sphere. Some properties of the conditions for different types of modons are discussed, too.

A domain decomposition approach and finite element formulation are developed for parallel distributed simulation of surface tension driven viscous flow and transport processes. The scheme is implemented in a finite element code MGFLO and is used to study performance on CRAY T3E and SGI Origin 2000 and 3000 parallel supercomputers as well as several PC clusters. The parallel algorithm implementation is briefly discussed, and scaled speedup studies are presented. Representative simulation results for surfactant and thermocapillary driven surface tension flows are also presented.

Recent breakthroughs in CFD technology have led to a high degree of parallel scaling of CFD simulations in a distributed shared-memory environment. A variety of research and commercial CFD software demonstrate scalable levels that are linear beyond 256 processors on the ccNUMA system architecture developed by SGI. Parallel algorithms typical of contemporary CFD software offer efficient strategies that overcome bottlenecks for moderate levels of parallelism. However, to achieve efficient parallelism on 100's of processors, additional consideration must be given to topics such as performance of system software and awareness of communication architectures. This paper examines the requirements for highly scalable CFD simulations on an assortment of industrial application examples.

Numerical simulations of the atmospheric motions require the most powerful machines and a high performance computer technology. The French meteorological model MésoNH is able to perform several simultaneous simulations on a nested grid to focus on specific regions described by a higher spatial resolution. This paper describes software and techniques used for implementing the MésoNH model on parallel processor computers especially for the grid nesting part.

A numerical study has been performed to solve the problem of 3D natural convection in a cubical cavity containing an isotropic saturated porous medium with internal heat generation. The governing equations were transformed to a vector potential formulation and the resulting model was discretized using orthogonal collocation with Legendre polynomials. The resulting sets of nonlinear algebraic equations were solved by nonlinear relaxation. This study considers the influence of Rayleigh number and internal heat generation on isotherms and Nusselt number. The accuracy of the numerical method was verified via an energy balance with maximum errors in the order of 2 %.

The exact numerical solutions of nonlinear direct interaction approximation equation or set of equations (DIA equations for Green's functions), describing the propagation of scalar impurities and magnetic field in a turbulent medium, are presented. It is assumed that the turbulence is stationary, isotropic and homogeneous. The steady state values of turbulent diffusivities and alpha coefficient are calculated. It is shown that such approach allows us to obtain the satisfactory values of turbulent transport coefficients for all the values of turbulent Strouhal numbers.

Smoothed Particle Hydrodynamics (SPH) was originally developed to handle multidimensional problems in astrophysics. Like all particle methods it makes use of the fact that advection follows the motion of the particles. As a consequence there are never problems with advection, and interfaces are automatically followed because particles define interfaces. In addition, the resolution of SPH varies naturally and automatically in space and time. These advantages carry over to a wide variety of fluid dynamical problems involving more than one phase and more than one material. In this paper I will describe how SPH can be extended to deal with problems involving solid bodies impacting fluids, debris flows hitting water, and a recent extension of Benz and Asphaug's¹ work on the fracture of brittle solids using SPH.

The lattice-Boltzmann method is a relatively new approach for the simulation of complex fluids which has a mesoscopic character, intermediate between continuum fluid dynamics approach and the atomistic approach based on molecular dynamics. Major advantages of the method include its ability to deal with the complex dynamics of interfaces and complicated boundaries, and the inherently parallel nature of the underlying algorithm. We review a novel lattice-Boltzmann scheme for simulation of binary and ternary fluids involving surfactant^1,2, discuss a parallel implementation of the algorithm and present preliminary results of large-scale simulations of the complex dynamics of ternary amphiphilic fluids performed on massively parallel platforms.

Natural convection in a slender cylindrical cavity with square cross section and aspect ratio (side to height) of 0.5 has been studied with the aim of understanding the structure of the multiplicity of steady-state solutions and its transit to unsteady flow. Prandtl number is considered equal to 5. The numerical code used for the analysis is based on the control volume method and the integration strategy is similar to SIMPLE. A continuation method is used to determine that a single branch of solutions (trivially symmetric solutions are considered the same solution) is found for Ra ≤ 6 × 10⁵; then, two branches are found for 6 × 10⁵ ≤ Ra ≤ 7.5 × 10⁵. Calculations with Ra > 8 × 10⁵ indicate that the system presents a bifurcation leading to irregular oscillations.

A finite element code is used to obtain information regarding the two-dimensional, steady-state flow field and heat transfer in a channel formed by a pair of equal amplitude, equal wave length, sinusoidal plates at different isothermal temperatures. A sufficiently long channel with several corrugations is considered so that periodic boundary conditions over one wave length are assumed in the flow inside the inner corrugations and the flow is assumed to be laminar. The nondimensional parameters governing the problem are similar to those encountered in compact heat exchangers. The primary objective of this study is to determine the nature of the hydrodynamics and the heat transfer characteristics as the phase angle between the geometry of the two plates is varied for a constant mean separation parameter. Quantitatively, the dimensionless pressure drop, ΔP, and Nusselt number, Nu, is found. An optimum configuration for which is the largest is determined. The results are explained in light of the hydrodynamic and thermal characteristics indicated by the numerical simulation.

Permanent-molding casting of aluminum-based pistons is affected by defects that cause rejection in quality-check control. Current scientific and technical literature plus in-situ experiences have allowed the identification of the causes of such problems. One of these causes is a direct consequence of a poor feeding system, which produces excessive velocities and critical conditions that generate turbulence and detachment of the thin film oxide lining of the feeding system, as well as bubble trapping. In this work, by means of numerical simulation, different feeding systems currently in use were studied (the analysis does not account for bubble trapping.) Modifications are suggested based on the analysis of these results. These modifications have a direct impact in the geometry of the feeding system, avoiding stagnation points as well as excessive velocities, giving as a result important reductions in the rejection of defective pistons.

In this work we analyze the quasisteady state problem of heat transfer in an impermeable solid by considering a background, constant, vertical temperature gradient, G, and the existence of a tilted slot either solid-filled or fluid-filled and located at its middle part. We study cases when a solid medium in the fracture has smaller thermal conductivity than the solid matrix and when a fluid medium has a thermal conductivity smaller than the surrounding solid. In this latter case, which is the most important from the practical point of view such as in the study of oil reservoirs, the heat diffusion in the solid is almost independent of the fluid part and therefore the boundary conditions on the fracture walls are obtained only as a solution of the heat transfer problem along the solid matrix. The energy heat equation is solved numerically for each part of the solid (besides the fracture) in a two-dimensional domain, by employing a paralellized finite differences code in order to find the temperature profiles (isotherms) around the fracture. Finally, these results are also employed in the study of the thermal convection in the fluid when the fracture is very long compared with the aperture size. We have obtained analytical solutions for this particular case.

Relativistic magnetohydrodynamics is discussed within the framework of irreversible thermodynamics. This is done using Kaluza's ideas about unifying external fields in terms of the corresponding space–time curvatures for a given metric. The outcome of this approach is rewarding. The conservation equations follow in a direct way as well as the entropy balance equation with an entropy production whose form suggests the type of constitutive equations that are consistent with its semipositive definite property. Further, the resulting transport equations are of a hyperbolic type in agreement with causality. Specific numerical examples will be presented by way of illustration.

The main interest of this work is the Chapman–Enskog method for solving the Boltzmann equation. The calculation of the diffusion coefficient using the classical trajectory method for three realistic potentials for He–N₂ is reported. A comparison of the calculated diffusion coefficients with experimental data and the extended corresponding states principle is also performed. Limitations of the Navier–Stokes equations for some problems are mentioned and a more detailed analysis of this point for the shock wave problem is considered.