Three-dimensional Flow Problems Research Articles

Parallel finite element solution of three-dimensional Rayleigh-Bénard-Marangoni flows

A domain decomposition strategy and a parallel gradient-type iterative solution scheme have been developed and implemented for the computation of complex three-dimensional viscous flow problems involving heat transfer and surface tension effects. Special attention has been paid to the kernels for the computationally intensive matrix-vector products and dot products, to memory management, and to overlapping communication and computation. Details of these implementation issues are described together with associated performance and scalability studies. Representative Rayleigh-Benard and microgravity Marangoni flow calculations and performance results on the Cray T3D and T3E are presented. Performance studies have been recently carried out and sustained rates above 50 gigaflops and 100 gigaflops have been achieved on the 512-node T3E-600 and 1024-node T3E-900 configurations respectively. The work is currently being extended to tightly-coupled parallel 'Beowulf-type' PC clusters and some preliminary performance results on this platform are presented

Second-order time-accurate and geometrically conservative implicit schemes for flow computations on unstructured dynamic meshes

We consider the solution of two- and three-dimensional flow problems with moving boundaries using the Arbitrary Lagrangian Eulerian formulation or dynamic meshes. We focus on the case where spatial discretization is performed by unstructured finite volumes and/or finite elements. We formulate the consequence of the Geometric Conservation Law on the second-order implicit temporal discretization of the semi-discrete equations governing such problems, and use it as a guideline to construct a new family of second-order time-accurate and geometrically conservative implicit numerical schemes for flow computations on moving grids. We apply these new algorithms to the solution of three-dimensional flow problems with moving and deforming boundaries, demonstrate their superior accuracy and computational efficiency, and highlight their impact on the simulation of fluid/structure interaction problems.

An investigation of random choice method for three-dimensional steady supersonic flows

In this paper, an unsplit random choice method (RCM) is developed and applied to numerically solve three-dimensional supersonic steady flow problems. In order to keep the contacts (slip surfaces) crisply resolved, a new Lagrangian formulation is employed. Due to the lack of exact solutions to 3D Riemann problems, approximate Riemann solutions in the weak sense are adopted. The RCM is thus as efficient as the deterministic TVD schemes, and yields almost identical results in the model problems. Copyright © 1999 John Wiley & Sons, Ltd.

Fast multipole solvers for three-dimensional radiation and fluid flow problems

A number of physics problems can be modeled by a set of N elements, which have pair-wise interactions with one another. The use of such elements for the evolution of vorticity in fluid flows and the calculation of the velocity field from the evolving vorticity field is well known. Fast multipole methods for fluid flow problems have been developed in the past to reduce computational effort to something less than O(N 2). In this paper we develop a fast multipole solver with application to both 3-D radiation problems (calculation of the heat flux from the evolving temperature field in an absorbing medium) and 3-D fluid flow. This is accomplished by using a more general kernel for the associated volume integrals. This kernel also encompasses other applications such as gravitational fields, electrostatics, scattering, etc. The present algorithm has been designed to have a very high efficiency when used on massively parallel computers. This feature comes at the expense of computational effort, which is less than O(N 2) but greater than O(N) or O(NlnN).

Numerical investigation of cross-flow separation in three-dimensional supersonic flows around circular cones

The problem of supersonic three-dimensional flow over sharp cones is solved using the system of Navier-Stokes equations in a locally conical approximation. Numerical solutions are found using an implicit finite-difference scheme with second-order exponential approximation of the equations with respect to the cross-flow coordinate. The appearance and development of cross-flow separation on the leeward side of the cones in a laminar regime of gas flow in the shock layer are investigated.

Efficient parallel multigrid based solvers for large scale groundwater flow simulations

In this paper, we present parallel solvers for large linear systems arising from the finite-element discretization of three-dimensional groundwater flow problems. We have tested our parallel implementations on the Intel Paragon XP/S 150 supercomputer using up to 1024 parallel processors. Our solvers are based on multigrid and Krylov subspace methods. Our goal is to combine powerful algorithms and current generation high performance computers to enhance the capabilities of computer models for groundwater modeling. We show that multigrid can be a scalable algorithm on distributed memory machines. We demonstrate the effectiveness of parallel multigrid based solvers by solving problems requiring more than 64 million nodes in less than a minute. Our results show that multigrid as a stand alone solver works best for problems with smooth coefficients, but for rough coefficients it is best used as a preconditioner for a Krylov subspace method.

On the added mass of rippled discs

The problem of three-dimensional potential flow past a thin rigid screen is reduced to a hypersingular boundary integral equation. This equation is then projected onto a flat reference screen, which is taken to be a circular disc. Solutions are obtained for screens that are axisymmetric perturbations from the disc, so that the screen is rippled concentrically. The added mass is calculated for axisymmetric flow past such screens, correct to second order.

Modeling three-dimensional groundwater flows by the body-fitted coordinate (BFC) method : II. Free and moving boundary problems

This paper presents a methodology and solution procedure of the time-dependent body-fitted coordinate (BFC) method for the analysis of transient, three-dimensional groundwater flow problems characterized by free and moving boundaries. The technique consists of numerical grid generation, time-dependent body-fitted coordinate transformation, and application of the finite difference method (FDM) to the transformed partial differential equations. Based on the time-dependent BFC method, a three-dimensional finite-difference computer code, BFC3DGW, was developed and used to solve two unconfined flow problems. The code was verified by comparing numerical results with analytical solutions for a steady-state seepage problem. In order to demonstrate capability of the method in dealing with flow problems with irregular and moving boundary surfaces, an unconfined well-flow problem was solved by the developed code. Difficulties associated with the free and moving irregular boundary have been successfully overcome by employing this method.

Application of vector and parallel supercomputers to ground-water flow modeling

A new generation of high-performance computers offers geoscientists new and exciting tools for computationally-intensive scientific problems. This paper presents preliminary results on optimizing the performance for ground-water flow models on a Cray supercomputer. The traditional method of solving a tridiagonal matrix system, that of forward solution and backsubstitution (FB), is intrinsically data dependent, in that the equations at a given grid node depend on values from adjacent grid nodes. This feature effectively prohibits vectorizing and parallelizing the method for solution on vector and parallel processors. An alternative method of solution, that of reduction and backsubstitution (RB), has no data dependence and was implemented in Fortran for vector and parallel processors to solve a tridiagonal matrix system in a ground-water flow model. The vectorized code has an overall execution rate of 110 MFLOPs on a Cray Y-MP that is five to sixteen times faster than the scalar code. Implementing the code in vector-parallel mode using eight undedicated processors on a Cray Y-MP results in an overall additional speedup of 2.8 times in wall clock time for a two-dimensional flow problem and 2.2 times for a three-dimensional flow problem. In comparison with the FB code, the RB code requires 26% less CPU time and 66% less wall-clock time on the same flow problem. Demonstrative application of the model is included for two- and three-dimensional flow problems.

On a compact mixed-order finite element for solving the three-dimensional incompressible Navier-Stokes equations

Our work is an extension of the previously proposed multivariant element. We assign this refined element as a compact mixed-order element in the sense that use of this element offers a much smaller bandwidth. The analysis is implemented on quadratic hexahedral elements with a view to analysing a three-dimensional incompressible viscous flow problem using a method formulated within the mixed finite element context. The idea of constructing such a stable element is to bring the marker-and-cell (MAC) grid lay-out to the finite element context. This multivariant element can thus be classified as a discontinuous pressure element. We have several reasons for advocating the proposed multivariant element. The primary advantage gained is its ability to reduce the bandwidth of the matrix equation, as compared with its univariant counterparts, so that it can be effectively stored in a compressed row storage (CRS) format. The resulting matrix equation can be solved efficiently by a multifrontal solver owing to its reduced bandwidth. The coding is, however, complicated by the appearance of restricted degrees of freedom at mid-face nodes. Through analytic study this compact multivariant element has a marked advantage over the multivariant element of Gupta et al. in that both bandwidth and computation time have been drastically reduced. © 1997 John Wiley & Sons, Ltd.

COMPUTATION OF TURBULENT TWO-PHASE FLOW ON OVERLAPPED GRIDS

This article describes a numerical method of combining an overtopped grid technique with a recently developed two-fluid model for two-phase flow simulations. The combination of these two approaches provides a useful tool for numerical modeling of complex two-phase flows with irregular geometries. The turbulent gas-particle two-phase flow over a backward-facing step is considered to demonstrate the applicability and accuracy of the proposed method. The predictions show good agreement with experimental data, much better than the previously reported numerical solutions. The flexibility of the present numerical method is also demonstrated by simulating the two-phase flow in a T-junction channel The importance of this study is expected to be more apparent in dealing with three-dimensional and multiple-phase flow problems, where efficient numerical techniques and algorithms, such as multigrid methods and parallel computing on multiprocessor parallel computers, can be easily adapted to get a solution of complex flow problems within a reasonable computer time.

Three-Dimensional Reservoir Description From Multiwell Pressure Data and Prior Information

Summary Bayesian estimation techniques are applied to generate three-dimensional permeability and porosity fields conditioned to prior means, variograms, hard data and/or multiwell pressure data. A posteriori variances are generated to obtain a measure of the uncertainty in the rock property fields, or equivalently, a measure of the variability in realizations of the rock property fields. These a posteriori variances can also be used to quantify the value of data, i.e., to determine the reduction in uncertainty achieved by adding a particular type of data. A key ingredient of our methodology is the development and implementation of an efficient procedure to estimate sensitivity coefficients for three-dimensional single-phase flow problems.

Euler Transonic Solutions over Finite Wings Using the Clebsch Decomposition Method

The problem of steady three-dimensional Euler flow over finite wings has been solved by the method of Clebsch velocity decomposition. The velocity field is decomposed into an irrotational and a rotational part. The Euler equations are then recast into an elliptic-like equation and several convection equations. This approach provides a generalization of the full potential formulation to rotational Euler physics by allowing variations of convective quantities. Finite volume and finite difference schemes are used to derive numerical results. Different wings at different flow conditions including a case with multiple shocks occurrence were tested to demonstrate the ability of the proposed method. Comparisons with available experimental data and results from existing code are presented.

Three-Dimensional Vortex Simulation of Time Dependent Incompressible Internal Viscous Flows

A hybrid random vortex-boundary element method is developed for the solution of time-dependent incompressible three-dimensional internal flow problems. The numerical scheme is grid-free within the flow domain and is based on a combination of the Lagrangian vortex method to capture the convection and stretch of the vortical field, the random walk method to describe the diffusion process, and the boundary element method to superimpose a potential flow on the vortical field such that the normal flux boundary condition is satisfied. The no-slip boundary condition is satisfied by generating vorticity tiles on solid boundaries, which are subsequently diffused and convected into the flow interior. Additionally, a boundary condition is devised for the application of fully developed flow properties at the exit plane. In this paper, the formulation and the numerical scheme are presented, followed by a parametric study of the accuracy of the method using the model problem of the flow in a duct with square cross section at Re= 100. We show that the method converges to the analytical solution of the problem as the resolution of the time integration and the discretization are improved, and we discuss the impact of each resolution parameter on the accuracy. In addition, selected results from the simulation of an impulsively started flow over a cube at Re= 100 are presented. We use the results of this test case to demonstrate that the method captures the effect of sharp edges, parallel and normal to the streamwise flow direction, on the flow dynamics.

Modeling Flow and Transport in Drainage Areas with Shallow Ground Water

A new modified linked approach is introduced to solve the governing partial differential equations for subsurface flow and salt transport by finite difference. It combines the attractive features of the standard linked and continuous (composite) modeling approaches. The standard linked approach can be adopted to model irrigation-drainage systems that are traditionally based on a purely agricultural objective of maintaining crop productivity despite the presence of a saline shallow ground-water table. With current concerns on the water quality of agricultural drainage, a more rigorous modeling approach (i.e., continuous approach) is required, however, at the expense of a large increase in the computational requirement. By using one- (1D), two- (2D), and three-dimensional (3D) flow and salt transport problems, it is shown that the modified linked approach efficiently satisfies current modeling requirements of both crop productivity and water quality protection. Results of applying the model to estimate the ...

安定性解析に基づくMAC法の時間刻み幅の設定法

When the Marker and Cell (MAC) method is applied to analyze three-dimensional flow problem numerically, small time step (Δt) is usually taken to avoid divergence of the computations. However, taking small time step often results in consuming large amount of computational time. Therefore, a possible large time step should be selected provided that it ban give a convergent solution. In this paper, the equations determining the time step are derived based on the Von Neumann's stability condition for the threedimensional finite difference equations obtained by central difference, upstream difference, and Quadratic Upstream Interpolation for Convective Kinematics(QUICK) schemes. The proposed equations are adapted to determine the time steps for several flow problems and it is verified that they give reasonable and convergent results with reduced number of iteration calculations.

High performance solution of three-dimensional nonlinear aeroelastic problems via parallel partitioned algorithms: methodology and preliminary results

In this paper, we highlight some key elements of our approach to the solution of large-scale three-dimensional nonlinear aeroelastic problems on high performance computational platforms. These include a three-field arbitrary Lagrangian-Eulerian (ALE) finite volume/element formulation for the coupled fluid/structure problem, the derivation of a geometric conservation law for three-dimensional flow problems with moving boundaries and unstructured deformable meshes, and the solution of the corresponding coupled semi-discrete equations with a partitioned procedure. In particular, we present a family of mixed explicit/ implicit staggered solution algorithms, and discuss them with particular reference to accuracy, stability, subcycling, and parallel processing. We describe a general framework for the solution of coupled aeroelastic problems on heterogeneous and/or parallel computational platforms, and illustrate it with some preliminary numerical investigations on an iPSC-860, a Paragon XP S , and a Cray T3D massively parallel systems of the aerodynamics and aeroelastic transient response of a three-dimensional wing in transonic airstreams.

The application of a finite volume multigrid method to three-dimensional flow problems in a highly viscous fluid with a variable viscosity

Abstract In this paper we discuss the application of a finite-volume multigrid method to solve three-dimensional thermally driven convection in a highly viscous, incompressible fluid with a variable viscosity. The conservation laws are solved in the primitive variable formulation, A second-order control volume method is used as discretization. Two schemes are used for time stepping, a second-order implicit-explicit scheme based on the Crank-Nicolson and Adams-Bashforth method, and a fully-implicit θ-method. The implicit system of nonlinear equations are solved using multigrid iteration with the SIMPLER method as smoother. In this paper, we describe the implemented multigrid method and investigate its efficiency and the robustness for different viscosity contrasts. Convergence tests showed that with a small modification of the SIMPLER method, the multigrid method exhibits a satisfactory convergence rate even for viscosity contrasts up to 109. Three cases of time-dependent thermally driven convection with v...

Minimum induced power requirements for flapping flight

The Betz criterion for minimum induced loss is used to compute the optimal circulation distribution along the span of flapping wings in fast forward flight. In particular, we consider the case where flapping motion is used to generate both lift (weight support) and thrust. The Betz criterion is used to develop two different numerical models of flapping. In the first model, which applies to small-amplitude harmonic flapping motions, the optimality condition is reduced to a one-dimensional integral equation which we solve numerically. In the second model, which applies to large-amplitude periodic flapping motions, the optimal circulation problem is reduced to solving for the flow over an infinitely long wavy sheet translating through an inviscid fluid at rest at infinity. This three-dimensional flow problem is solved using a vortex-lattice technique. Both methods predict that the induced power required to produce thrust decreases with increasing flapping amplitude and frequency. Using the large-amplitude theory, we find that the induced power required to produce lift increases with flapping amplitude and frequency. Therefore, an optimum flapping amplitude exists when the flapping motion of wings must simultaneously produce lift and thrust.

Parallel benchmarks of turbulence in complex geometries

In this paper we present benchmark results from the parallel implementation of the three-dimensional Navier-Stokes solver Prism on different parallel platforms of current interest: IBM SP2 (all three types of processors), SGI Power Challenge XL and Cray C90. The numerical method is based on mixed spectral element-Fourier expansions in ( x − y) and z-directions, respectively. Each one (or a group) of the Fourier modes can be computed on a separate processor as the linear contributions in Navier-Stokes equations (Helmholtz solvers) are completely uncoupled. Coupling is obtained via the non-linear contributions (advection terms) and requires a global transpose of the data and one-dimensional multiple-point FFTs. We first analyze the computational complexity of Prism identifying basic computational kernels and communication bottlenecks. These kernels are benchmarked separately on several computer architectures. More specifically, we obtain timings for BLAS routines on all aforementioned processors but also on the SP1, Cray J90, the Intel Paragon, and the DEC AlphaServer 8400 5/300. A two-dimensional version of Prism is also benchmarked on most processors testing both direct and iterative methods of solution. Next, we define two three-dimensional benchmark flow problems in prototype complex geometries, and measure parallel scalability and performance using different message-passing libraries. Special emphasis is placed on runtime data processing, e.g. turbulence statistics computed during the simulation. Our results provide a measure of MFlop s sustained performance of individual processors, and indicate the limitations of the current communications software for typical problems in computational mechanics based on spectral or finite element discretizations.