Compressible Euler Extension of a Massively-Parallel Spectral Element Solver

Computational performance of a parallelized three-dimensional high-order spectral element toolbox

A Legendre spectral element model for sloshing and acoustic analysis in nearly incompressible fluids

An efficient, second order accurate, universal generalized Riemann problem solver based on the HLLI Riemann solver

This study documents a new WIND validation case for computing low Mach number flows through nozzles and investigates the relative thrust performance of circular and non-circular (elliptical) nozzles. Threedimensional, steady-state, CFD simulations of flow through one circular and two elliptic nozzles (aspect ratios of 2:1 and 3:1) with the same inlet and exit areas were performed for three exit jet velocities; Mach 0.05, 0.075 and 0.1. First order implicit time integration scheme and the second order physical space differencing with the turbulence SST (viscous, two-equation Shear Stress Transport) model were used for WIND simulations. In separate open water experiments, a Jet Ski Personal Watercraft (PWC) was outfitted with matching nozzle geometries (circular and 2:1 elliptical nozzles) and was tested at similar jet exit velocities. A minimal drop in the relative thrust performance of elliptical nozzles compared to the circular nozzle at all three exit jet flow conditions was observed. CFD results are in good agreement with PWC experiments. 1.0 Introduction The motivation for this study was twofold. First, to investigate the use of a compressible flow solver WIND for application to low Mach number flows, and second, to provide a validation case in the study of relative thrust performance of axisymmetric (circular) versus non-axisymmetric (elliptical) nozzles. It is well known that the performance of compressible Navier-Stokes (N-S) codes, in terms of the solution-accuracy, as well as the iterative convergence, degrades as the Mach number approaches zero. This is attributed to the existence of two very different time scales associated with the acoustics and fluid motion. The disparity between the acoustic wave speed, u+a, and the convective wave speed, u, leads to a large (infinite) ratio of the largest-tosmallest eigenvalue of the compressible equations, creating a stiff system. This has been identified as a stumbling block in the use of compressible N-S equations for low Mach number flows. To address this problem, a technique knows as local preconditioning, developed in recent years, is currently being investigated on several fronts. This technique provides a means to alter the eigenvalues towards a favorable condition number, which reduces the * Director. Aerodynamics Group. Member AIAA † Physicist. Member AIAA ‡ President. Associate Fellow AIAA American Institute of Aeronautics and Astronautics 1 42nd AIAA Aerospace Sciences Meeting and Exhibit 5 8 January 2004, Reno, Nevada AIAA 2004-531 Copyright © 2004 by Orbital Research, Inc. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission. disparity in the wave speeds and also accelerates the convergence to steady state. The focus of this paper is not to develop or evaluate the performance of precond tioning techniques, but rather, to investigate the capabilities of a widely used Navier-Stokes compressible code WIND for low Mach number flow applications. A new validation case for WIND to study thrust performances of nozzles with low speed flows, i.e., exit jet velocities of Mach 0.05, 0.075 and 0.1, is presented. A good agreement between the CFD and experimental results is observed in the treatment of lowspeed incompressible flows; however, persistent problems encountered in reaching steady state convergence present an apparent need for improvements in the WIND code to address such flows. i This study also examines the relative thrust performances of circular versus elliptical nozzles (2:1 and 3:1 aspect ratio) using WIND. Results show that the thrust efficiency of both elliptical nozzles is minimally lower than the circular nozzle. The following sections present a brief description of the WIND code, design of the nozzles used in this validation study, computational grids, setting-up of the boundary conditions to simulate low Mach flow (medium: water) through nozzles, Jet Ski Personal Watercraft (PWC) experiments, and CFD results with experimental validation. 2.0 WIND Flow Solver WIND is a computational platform developed from the merger of three CFD codes: NASTD (the primary flow solver at McDonnell Douglas, now part of Boeing), NPARC (the original NPARC Alliance flow solver), and NXAIR (an AEDC code). It is a structured multi-zone flow solver with multiple turbulence models, and has been applied to a wide variety of complex configurations and flow regimes. An extensive validation effort is currently underway for the WIND code. This paper presents a new WIND validation case for applications to low Mach number flows. Thrust performance studies for three different nozzle designs with incompressible flow conditions at the nozzle exit are presented.

This paper presents development of hyperbolic Navier-Stokes (HNS) systems for three-dimensional viscous flow in edge-based cell-vertex unstructured solver, i.e. Stanford University Unstructured (SU2) platform. Numeric scheme and operation for isothermal wall and symmetry boundary are proposed. For validating the new solver, low Mach flows over two-dimensional NACA0012 airfoil and three-dimensional DLR-F6 wing-body are simulated. The results demonstrate the superiority of HNS over conventional Navier-Stokes (CNS) systems in prediction accuracy of friction coefficient and efficiency. In general, HNS can handle the low-Mach high-Reynolds real flow effectively and efficiently.

Moving overlapping grid methodology of spectral accuracy for incompressible flow solutions around rigid bodies in motion

In this paper, five different algorithms are presented for the simulation of low Mach flows with large temperature variations, based on second‐order central‐difference or fourth‐order compact spatial discretization and a pressure projection‐type method. A semi‐implicit three‐step Runge–Kutta/Crank–Nicolson or second‐order iterative scheme is used for time integration. The different algorithms solve the coupled set of governing scalar equations in a decoupled segregate manner. In the first algorithm, a temperature equation is solved and density is calculated from the equation of state, while the second algorithm advances the density using the differential form of the equation of state. The third algorithm solves the continuity equation and the fourth algorithm solves both the continuity and enthalpy equation in conservative form. An iterative decoupled algorithm is also proposed, which allows the computation of the fully coupled solution. All five algorithms solve the momentum equation in conservative form and use a constant‐ or variable‐coefficient Poisson equation for the pressure. The efficiency of the fourth‐order compact scheme and the performances of the decoupling algorithms are demonstrated in three flow problems with large temperature variations: non‐Boussinesq natural convection, channel flow instability, flame–vortex interaction. Copyright © 2010 John Wiley & Sons, Ltd.

Eulerian models with particle pressure for air-particle flows

Spectral element method in time for rapidly actuated systems

This paper studies different acceleration techniques for unsteady flow calculations. The results are compared with a non-accelerated, fully-explicit solution in terms of time-averaged pressure distributions, the unsteady pressure and entropy in the frequency domain and the skin friction factor. The numerical method solves the unsteady three-dimensional Navier-Stokes equations via an explicit time-stepping procedure. The flow in the first stage of a modern industrial gas turbine is chosen as a test case. After a description of the numerical method used for the simulation, the test case is introduced. The comparison of the different numerical algorithms for explicit schemes is intended to ease the decision about which acceleration technique to use for calculations as far as accuracy and computational time are concerned. The convergence acceleration methods under consideration are, respectively, explicit time-stepping with implicit residual averaging, explicit time-consistent multigrid and implicit dual time stepping. The investigation and comparison of the different acceleration techniques are applicable to all explicit unsteady flow solvers. As another point of interest, the influence of the stage blade count ratio on the flow field is investigated. For this purpose, a simulation with a stage pitch ratio of unity is compared with a calculation using the real ratio of 78:80, which requires a more sophisticated method for periodic boundary condition treatment. This paper should help to decide whether it is vital from the turbine designer’s point of view to model the real pitch ratio in unsteady flow simulations in turbine stages.

Mantle convection is the principal control on the thermal and geological evolution of the Earth. Mantle convection modeling involves solution of the mass, momentum, and energy equations for a viscous, creeping, incompressible non-Newtonian fluid at high Rayleigh and Peclet numbers. Our goal is to conduct global mantle convection simulations that can resolve faulted plate boundaries, down to 1 km scales. However, uniform resolution at these scales would result in meshes with a trillion elements, which would elude even sustained petaflops supercomputers. Thus parallel adaptive mesh refinement and coarsening (AMR) is essential. We present RHEA, a new generation mantle convection code designed to scale to hundreds of thousands of cores. RHEA is built on ALPS, a parallel octree-based adaptive mesh finite element library that provides new distributed data structures and parallel algorithms for dynamic coarsening, refinement, rebalancing, and repartitioning of the mesh. ALPS currently supports low order continuous Lagrange elements, and arbitrary order discontinuous Galerkin spectral elements, on octree meshes. A forest-of-octrees implementation permits nearly arbitrary geometries to be accommodated. Using TACC's 579 teraflops Ranger supercomputer, we demonstrate excellent weak and strong scalability of parallel AMR on up to 62,464 cores for problems with up to 12.4 billion elements. With RHEA's adaptive capabilities, we have been able to reduce the number of elements by over three orders of magnitude, thus enabling us to simulate large-scale mantle convection with finest local resolution of 1.5 km.

Mantle convection is the principal control on the thermal and geological evolution of the Earth. Mantle convection modeling involves solution of the mass, momentum, and energy equations for a viscous, creeping, incompressible non-Newtonian fluid at high Rayleigh and Peclet numbers. Our goal is to conduct global mantle convection simulations that can resolve faulted plate boundaries, down to 1 km scales. However, uniform resolution at these scales would result in meshes with a trillion elements, which would elude even sustained petaflops supercomputers. Thus parallel adaptive mesh refinement and coarsening (AMR) is essential. We present RHEA, a new generation mantle convection code designed to scale to hundreds of thousands of cores. RHEA is built on ALPS, a parallel octree-based adaptive mesh finite element library that provides new distributed data structures and parallel algorithms for dynamic coarsening, refinement, rebalancing, and repartitioning of the mesh. ALPS currently supports low order continuous Lagrange elements, and arbitrary order discontinuous Galerkin spectral elements, on octree meshes. A forest-of-octrees implementation permits nearly arbitrary geometries to be accommodated. Using TACC's 579 teraflops Ranger supercomputer, we demonstrate excellent weak and strong scalability of parallel AMR on up to 62,464 cores for problems with up to 12.4 billion elements. With RHEA's adaptive capabilities, we have been able to reduce the number of elements by over three orders of magnitude, thus enabling us to simulate large-scale mantle convection with finest local resolution of 1.5 km.

SUMMARY The spectral element method for the two-dimensional shallow water equations on the sphere is presented. The equations are written in conservation form and the domains are discretized using quadrilateral elements obtained from the generalized icosahedral grid introduced previously (Giraldo FX. Lagrange‐ Galerkin methods on spherical geodesic grids: the shallow water equations. Journal of Computational Physics 2000; 160: 336‐368). The equations are written in Cartesian co-ordinates that introduce an additional momentum equation, but the pole singularities disappear. This paper represents a departure from previously published work on solving the shallow water equations on the sphere in that the equations are all written, discretized, and solved in three-dimensional Cartesian space. Because the equations are written in a three-dimensional Cartesian co-ordinate system, the algorithm simplifies into the integration of surface elements on the sphere from the fully three-dimensional equations. A mapping (Song Ch, Wolf JP. The scaled boundary finite element method—alias consistent infinitesimal finite element cell method—for diffusion. International Journal for Numerical Methods in Engineering 1999; 45: 1403‐1431) which simplifies these computations is described and is shown to contain the Eulerian version of the method introduced previously by Giraldo (Journal of Computational Physics 2000; 160: 336‐368) for the special case of triangular elements. The significance of this mapping is that although the equations are written in Cartesian co-ordinates, the mapping takes into account the curvature of the high-order spectral elements, thereby allowing the elements to lie entirely on the surface of the sphere. In addition, using this mapping simplifies all of the three-dimensional spectral-type finite element surface integrals because any of the typical two-dimensional planar finite element or spectral element basis functions found in any textbook (for example, Huebner et al. The Finite Element Method for Engineers. Wiley, New York, 1995; Karniadakis GE, Sherwin SJ. Spectral:hp Element Methods for CFD. Oxford University ( ˘

Internal flows inside gravitationally stable astrophysical objects, such as the Sun, normal and compact stars, are rotating, highly compressed and extremely subsonic. Such low Mach number flows are usually encountered when studying, for example, the dynamo action in stars and planets or the nuclear burst on neutron stars and white dwarfs. Handling of such flows numerically on time-scales longer than the dynamical one is complicated and challenging. The aim of this paper is to address the numerical problems associated with the modelling of internal quasi-stationary, rotating low Mach number flows in stars and to discuss possible solution scenarios. It is shown that the quasi-symmetric approximate factorization method (AFM) as a pre-conditioner within a non-linear Newton-type defect-correction solution procedure is best suited for modelling quasi-stationary weakly compressible flows with moderate low Mach numbers. This method is robust as it can be applied to model time-dependent compressible flows without further modifications. The AFM-pre-conditioning techniques are shown to be extendable into three dimensions with an arbitrary equation of state. Classical dimensional splitting techniques, however, such as the alternating direction implicit or line-Gauss–Seidel methods are not suited for modelling compressible low Mach number flows. It is also argued that hot and low Mach number astrophysical flows cannot be considered as an asymptotic limit of incompressible flows, but rather as highly compressed flows with extremely stiff pressure terms. We show that, unlike the pseudo-pressure in incompressible fluids, a Poisson-like treatment for the pressure would smooth unnecessarily physically induced acoustic perturbations, thereby violating the conservation of the total energy. Results of several hydrodynamical calculations are presented, which demonstrate the capability of the solver to search for solutions, that correspond to stationary, viscous and rotating flows with a Mach number as small as <f><inline-fig> <link locator="mnras0400-0903-mu1"></inline-fig></f> as well as to fluid flows that are subject to ultra-strong Newtonian and general relativistic gravitational fields.

As an improved method of the lattice Boltzmann method (LBM), the regularized lattice Boltzmann method (RLBM) has been widely used to simulate fluid flow. For solving high Reynolds number problems, large eddy simulation (LES) and RLBM can be combined. The computation of fluid flow problems often requires a large number of computational grids and large-scale parallel clusters. Therefore, the high scalability parallel algorithm of RLBM with LES on a large-scale cluster has been proposed in this paper. The proposed parallel algorithm can solve complex flow problems with large-scale Cartesian grids and high Reynolds numbers. In order to achieve computational load balancing, the domain decomposition method (DDM) has been used in large-scale mesh generation. Three mesh generation strategies are adopted, namely 1D, 2D and 3D. Then, the buffer on the grid interface is introduced and the corresponding 1D, 2D and 3D parallel data exchange strategies are proposed. For the 3D lid-driven cavity flow and incompressible flow around a sphere under a high Reynolds number, the given parallel algorithm is analyzed in detail. Experimental results show that the proposed parallel algorithm has a high scalability and accuracy on hundreds of thousands of cores.