scholarly journals A Massively Parallel Hybrid Finite Volume/Finite Element Scheme for Computational Fluid Dynamics

Mathematics ◽  
2021 ◽  
Vol 9 (18) ◽  
pp. 2316
Author(s):  
Laura Río-Martín ◽  
Saray Busto ◽  
Michael Dumbser
Keyword(s):  
Fluid Dynamics ◽  
Computational Fluid Dynamics ◽  
Finite Element ◽  
Mach Number ◽  
Finite Volume ◽  
Degrees Of Freedom ◽  
Stokes Equations ◽  
Massively Parallel ◽  
Navier Stokes ◽  
Navier Stokes Equations

In this paper, we propose a novel family of semi-implicit hybrid finite volume/finite element schemes for computational fluid dynamics (CFD), in particular for the approximate solution of the incompressible and compressible Navier-Stokes equations, as well as for the shallow water equations on staggered unstructured meshes in two and three space dimensions. The key features of the method are the use of an edge-based/face-based staggered dual mesh for the discretization of the nonlinear convective terms at the aid of explicit high resolution Godunov-type finite volume schemes, while pressure terms are discretized implicitly using classical continuous Lagrange finite elements on the primal simplex mesh. The resulting pressure system is symmetric positive definite and can thus be very efficiently solved at the aid of classical Krylov subspace methods, such as a matrix-free conjugate gradient method. For the compressible Navier-Stokes equations, the schemes are by construction asymptotic preserving in the low Mach number limit of the equations, hence a consistent hybrid FV/FE method for the incompressible equations is retrieved. All parts of the algorithm can be efficiently parallelized, i.e., the explicit finite volume step as well as the matrix-vector product in the implicit pressure solver. Concerning parallel implementation, we employ the Message-Passing Interface (MPI) standard in combination with spatial domain decomposition based on the free software package METIS. To show the versatility of the proposed schemes, we present a wide range of applications, starting from environmental and geophysical flows, such as dambreak problems and natural convection, over direct numerical simulations of turbulent incompressible flows to high Mach number compressible flows with shock waves. An excellent agreement with exact analytical, numerical or experimental reference solutions is achieved in all cases. Most of the simulations are run with millions of degrees of freedom on thousands of CPU cores. We show strong scaling results for the hybrid FV/FE scheme applied to the 3D incompressible Navier-Stokes equations, using millions of degrees of freedom and up to 4096 CPU cores. The largest simulation shown in this paper is the well-known 3D Taylor-Green vortex benchmark run on 671 million tetrahedral elements on 32,768 CPU cores, showing clearly the suitability of the presented algorithm for the solution of large CFD problems on modern massively parallel distributed memory supercomputers.

Development of SIMPLE-Like Algorithms for Incompressible Navier-Stokes Equations in Liquid Processing of Materials

2012 ◽  
Vol 184-185 ◽  
pp. 944-948 ◽  
Author(s):  
Hai Jun Gong ◽  
Yang Liu ◽  
Xue Yi Fan ◽  
Da Ming Xu
Keyword(s):  
Fluid Dynamics ◽  
Computational Fluid Dynamics ◽  
Stokes Equations ◽  
Simple Algorithm ◽  
Navier Stokes ◽  
Incompressible Fluid Flow ◽  
Numerical Accuracy ◽  
Simple Scheme ◽  
Navier Stokes Equations ◽  
Computational Fluid Dynamics Cfd

For a clear and comprehensive opinion on segregated SIMPLE algorithm in the area of computational fluid dynamics (CFD) during liquid processing of materials, the most significant developments on the SIMPLE algorithm and its variants are briefly reviewed. Subsequently, some important advances during last 30 years serving as increasing numerical accuracy, enhancing robustness and improving efficiency for Navier–Stokes (N-S) equations of incompressible fluid flow are summarized. And then a so-called Direct-SIMPLE scheme proposed by the authors of present paper introduced, which is different from SIMPLE-like schemes, no iterative computations are needed to achieve the final pressure and velocity corrections. Based on the facts cited in present paper, it conclude that the SIMPLE algorithm and its variants will continue to evolve aimed at convergence and accuracy of solution by improving and combining various methods with different grid techniques, and all the algorithms mentioned above will enjoy widespread use in the future.

On the Immersed Boundary Method: Finite Element Versus Finite Volume Approach

2012 ◽  
Author(s):  
Angelo Frisani ◽  
Yassin A. Hassan
Keyword(s):  
Finite Element ◽  
Analytical Solution ◽  
Finite Volume ◽  
Stokes Equations ◽  
Numerical Results ◽  
Navier Stokes ◽  
Immersed Boundary ◽  
Order Accuracy ◽  
Navier Stokes Equations ◽  
Two Dimensional Flow

A projection approach is presented for the coupled system of time-dependent incompressible Navier-Stokes equations in conjunction with the Immersed Boundary Method (IBM) for solving fluid flow problems in the presence of rigid objects not represented by the underlying mesh. The IBM allows solving the flow for geometries with complex objects without the need of generating a body fitted mesh. The no-slip boundary constraint is satisfied applying a boundary force at the immersed body surface. Using projection and interpolation operators from the fluid volume mesh to the solid surface mesh (i.e., the “immersed” boundary) and vice versa, it is possible to impose the extra constraint to the incompressible Navier-Stokes equations as a Lagrange multiplier in a fashion very similar to the effect pressure has on the momentum equations to satisfy the divergence-free constraint. The projection operation removes the immersed boundary surface slip and non-divergence-free components of the velocity field. The boundary force is determined implicitly at the inner iterations of the fractional step method implemented. No constitutive relations for the immersed boundary objects fluid interaction are required, allowing the formulation introduced to use larger CFL numbers compared to previous methodologies. An overview of the immersed boundary approach is presented showing third order accuracy in space and second order accuracy in time when the simulation results for the Taylor-Green decaying vortex are compared to the analytical solution using the Immersed Finite Element Method (IFEM). For the Immersed Finite Volume Method (IFVM) a ghost-cell approach is used. Second order accuracy in space and first order accuracy in time are obtained when the Taylor-Green decaying vortex test case is compared to the analytical solution. The numerical results are compared with the analytical solution also for adaptive mesh refinement (for the IFEM) showing an excellent error reduction. Computations were performed using IFEM and IFVM approaches for the time-dependent incompressible Navier-Stokes equations in a two-dimensional flow past a stationary circular cylinder at Re = 20, and 40, where shedding effects are not present. The drag coefficient and the recirculation length error compared to the experimental data is less than 3–4%. Simulations for the two-dimensional flow past a stationary circular cylinder at Re = 100 were also performed. For Re numbers above 46, unsteadiness generates vortex shedding, and an unsteady flow regime is present. The results shown are in excellent quantitative and qualitative agreement with the flow pattern expected. The numerical results obtained with the discussed IFEM and IFVM were also compared against other immersed boundary methodologies available in literature and simulation performed with the commercial computational fluid dynamics code STAR-CCM+/V5.02.009 for which a body fitted finite volume numerical discretization was used. The benchmark showed that the numerical results obtained with the implemented immersed boundary methods are very close to those obtained from STAR-CCM+ with a very fine mesh and in a good agreement with the other IBM techniques. The IBM based of finite element approach is numerically more accurate than the IBM based on finite volume discretization. In contrast, the latter is computationally more efficient than the former.

A New Finite Volume Element Formulation for the Non-Stationary Navier-Stokes Equations

2014 ◽  
Vol 6 (5) ◽  
pp. 615-636 ◽  
Author(s):  
Zhendong Luo
Keyword(s):  
Finite Element ◽  
Finite Volume ◽  
Stokes Equations ◽  
Volume Element ◽  
Navier Stokes ◽  
Element Formulation ◽  
Navier Stokes Equations ◽  
Fully Discrete ◽  
Finite Volume Element ◽  
Stationary Navier Stokes Equations

AbstractA semi-discrete scheme about time for the non-stationary Navier-Stokes equations is presented firstly, then a new fully discrete finite volume element (FVE) formulation based on macroelement is directly established from the semi-discrete scheme about time. And the error estimates for the fully discrete FVE solutions are derived by means of the technique of the standard finite element method. It is shown by numerical experiments that the numerical results are consistent with theoretical conclusions. Moreover, it is shown that the FVE method is feasible and efficient for finding the numerical solutions of the non-stationary Navier-Stokes equations and it is one of the most effective numerical methods among the FVE formulation, the finite element formulation, and the finite difference scheme.

scholarly journals Solving Galbrun's Equation with a Discontinuous galerkin Finite Element Method

2019 ◽  
Vol 105 (6) ◽  
pp. 1149-1163 ◽  
Author(s):  
Marcus Maeder ◽  
Andrew Peplow ◽  
Maximilian Meindl ◽  
Steffen Marburg
Keyword(s):  
Finite Element ◽  
Discontinuous Galerkin ◽  
Degrees Of Freedom ◽  
Stokes Equations ◽  
Early Stage ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Linearized Euler Equations ◽  
Displacement Based ◽  
Spurious Modes

Over many years, scientists and engineers have developed a broad variety of mathematical formulations to investigate the propagation and interactions with flow of flow-induced noise in early-stage of product design and development. Beside established theories such as the linearized Euler equations (LEE), the linearized Navier–Stokes equations (LNSE) and the acoustic perturbation equations (APE) which are described in an Eulerian framework, Galbrun utilized a mixed Lagrange–Eulerian framework to reduce the number of unknowns by representing perturbations by means of particle displacement only. Despite the advantages of fewer degrees of freedom and the reduced effort to solve the system equations, a computational approach using standard continuous finite element methods (FEM) suff ers from instabilities called spurious modes that pollute the solution. In this work, the authors employ a discontinuous Galerkin approach to overcome the difficulties related to spurious modes while solving Galbrun's equation in a mixed and pure displacement based formulation. The results achieved with the proposed approach are compared with results from previous attempts to solve Galbrun's equation. The numerical determination of acoustic modes and the identification of vortical modes is discussed. Furthermore, case studies for a lined-duct and an annulus supporting a rotating shear-flow are investigated.

scholarly journals Spectral Wave Explicit Navier-Stokes Equations for wave-structure interactions using two-phase Computational Fluid Dynamics solvers

2021 ◽  
Vol 221 ◽  
pp. 108513
Author(s):  
Zhaobin Li ◽  
Benjamin Bouscasse ◽  
Guillaume Ducrozet ◽  
Lionel Gentaz ◽  
David Le Touzé ◽  
...  
Keyword(s):  
Fluid Dynamics ◽  
Computational Fluid Dynamics ◽  
Stokes Equations ◽  
Wave Structure ◽  
Navier Stokes ◽  
Two Phase ◽  
Navier Stokes Equations

Computational fluid dynamics prediction of hydrodynamic forces on a manoeuvring ship including effects of dynamic sinkage and trim

2017 ◽  
Vol 233 (1) ◽  
pp. 251-266 ◽  
Author(s):  
Yi Liu ◽  
Lu Zou ◽  
Zao-Jian Zou
Keyword(s):  
Fluid Dynamics ◽  
Experimental Data ◽  
Computational Fluid Dynamics ◽  
Stokes Equations ◽  
Hydrodynamic Forces ◽  
Numerical Error ◽  
Navier Stokes ◽  
Container Ship ◽  
Navier Stokes Equations ◽  
Sinkage And Trim

Understanding the manoeuvring performance of a ship requires accurate predictions of the hydrodynamic forces and moments on the ship. In the present study, the hydrodynamic forces and moments on a manoeuvring container ship at various rudder and drift angles are numerically predicted by solving the unsteady Reynolds-averaged Navier–Stokes equations. The effects of dynamic sinkage and trim on the hydrodynamic forces are first investigated together with a grid dependency study to estimate the numerical error and uncertainty caused by grid discretization, and with a validation study combining the experimental data. The results show that the effect of dynamic sinkage and trim is non-negligible, since including it improves the hydrodynamic force predictions and reduces the numerical error and uncertainty, and the averaged error and uncertainty are smaller than the other computational fluid dynamics results where sinkage and trim were fixed with given values from model tests. Therefore, it is included in the subsequent systematic simulations regarding the influence of rudder and drift angles. The computed forces, moments and rudder coefficients at different rudder and drift angles on the container ship are compared with the benchmark model test data. From the computations, all the predicted quantities are in satisfactory agreement with the experimental data. The details of the flow filed and hydrodynamic forces, such as pressure distributions, transverse force distributions along the hull, velocity contours, streamlines and wave patterns are presented and discussed, and a deep insight into the physical mechanism in the hydrodynamic forces on a manoeuvring ship is obtained.

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