A high-order semi-explicit discontinuous Galerkin solver for 3D incompressible flow with application to DNS and LES of turbulent channel flow

We present an efficient discontinuous Galerkin scheme for simulation of the incompressible Navier-Stokes equations including laminar and turbulent flow. We consider a semi-explicit high-order velocity-correction method for time integration as well as nodal equal-order discretizations for velocity and pressure. The non-linear convective term is treated explicitly while a linear system is solved for the pressure Poisson equation and the viscous term. The key feature of our solver is a consistent penalty term reducing the local divergence error in order to overcome recently reported instabilities in spatially under-resolved high-Reynolds-number flows as well as small time steps. This penalty method is similar to the grad-div stabilization widely used in continuous finite elements. We further review and compare our method to several other techniques recently proposed in literature to stabilize the method for such flow configurations. The solver is specifically designed for large-scale computations through matrix-free linear solvers including efficient preconditioning strategies and tensor-product elements, which have allowed us to scale this code up to 34.4 billion degrees of freedom and 147,456 CPU cores. We validate our code and demonstrate optimal convergence rates with laminar flows present in a vortex problem and flow past a cylinder and show applicability of our solver to direct numerical simulation as well as implicit large-eddy simulation of turbulent channel flow at $Re_{\tau}=180$ as well as $590$.Comment: 28 pages, in preparation for submission to Journal of Computational Physic

Partitioned solution to fluid-structure interaction problems in application to free-surface flows

International audienceIn this work we discuss a way to compute the impact of free-surface flow on nonlinear structures. The approach chosen rely on a partitioned strategy that allows to solve strongly coupled fluid-structure interaction problem. It is then possible to re-use existing and validated strategy for each sub-problem. The structure is formulated in a Lagrangian way and solved by the finite element method. The free-surface flow approach considers a Volume-Of-Fluid (VOF) strategy formulated in an Arbitrary Lagrangian-Eulerian (ALE) framework, and the finite volume are used to discrete and solve this problem. The software coupling is ensured in an efficient way using the Communication Template Library (CTL). Numerical examples presented herein concern 2D validations case but also 3D problems with a large number of equations to be solved

Symmetry-preserving discretization of Navier-Stokes on unstructured grids: collocated vs staggered

The essence of turbulence are the smallest scales of motion. They result from a subtle balance between convective transport and diffusive dissipation. Mathematically, these terms are governed by two differential operators differing in symmetry: the convective operator is skew-symmetric, whereas the diffusive is symmetric and positive-definite. On the other hand, accuracy and stability need to be reconciled for numerical simulations of turbulent flows around complex configurations. With this in mind, a fully-conservative discretization method for general unstructured grids was proposed [Trias et al., J.Comp.Phys. 258, 246-267, 2014]: it exactly preserves the symmetries of the underlying differential operators on a collocated mesh. However, any pressure-correction method on collocated grids suffer from the same drawbacks: the cell-centered velocity field is not exactly incompressible and some artificial dissipation is inevitable introduced. On the other hand, for staggered velocity fields, the projection onto a divergence-free space is a well-posed problem: given a velocity field, it can be uniquely decomposed into a solenoidal vector and the gradient of a scalar (pressure) field. This can be easily done without introducing any dissipation as it should be from a physical point-of-view. In this work, we explore the possibility to build up staggered formulations based on collocated discrete operators.F.X.T., N.V. and A.O. have been financially supported by the Ministerio de Economía y Competitividad, Spain, ANUMESOL project (ENE2017-88697-R). F.X.T. and A.O. are supported by the Generalitat de Catalunya RIS3CAT-FEDER, FusionCAT project (001-P-001722). N.V. was supported by an FI AGAUR-Generalitat de Catalunya fellowship (2017FI B 00616). Calculations were performed on the IBM MareNostrum 4 supercomputer at the BSC. The authors thankfully acknowledge these institutions.Peer ReviewedPostprint (published version

Nonlinear fluid-structure interaction problem. Part II: space discretization, implementation aspects, nested parallelization and application examples

International audienceThe main focus of the present article is the development of a general solution framework for coupled and/or interaction multi-physics problems based upon re-using existing codes into software products. In particular, we discuss how to build this software tool for the case of fluid-structure interaction problem, from finite element code Feap for structural and finite volume code OpenFOAM for fluid mechanics. This is achieved by using the Component Template Library (CTL) to provide the coupling between the existing codes into a single software product. The present CTL code-coupling procedure accepts not only different discretization schemes, but different languages, with the solid component written in Fortran and fluid component written in \Cpp. Moreover, the resulting CTL-based code also accepts the nested parallelization. The proposed coupling strategy is detailed for explicit and implicit fixed-point iteration solver presented in the Part I of this paper, referred to Direct Force-Motion Transfer/Block-Gauss-Seidel. However, the proposed code-coupling framework can easily accommodate other solution schemes. The selected application examples are chosen to confirm the capability of the code-coupling strategy to provide a quick development of advanced computational tools for demanding practical problems, such as 3D fluid models with free-surface flows interacting with structures

Nonlinear fluid-structure interaction problem. Part I: implicit partitioned algorithm, nonlinear stability proof and validation examples

International audienceIn this work we consider the fluid-structure interaction in fully nonlinear setting, where different space discretization can be used. The model problem considers finite elements for structure and finite volume for fluid. The computations for such interaction problem are performed by implicit schemes, and the partitioned algorithm separating fluid from structural iterations. The formal proof is given to find the condition for convergence of this iterative procedure in the fully nonlinear setting. Several validation examples are shown to confirm the proposed convergence criteria of partitioned algorithm. The proposed strategy provides a very suitable basics for code-coupling implementation as discussed in Part II

Solution of the 2D Navier-Stokes equations by a new FE fractional step method.

In this work, a mathematical and numerical approach for the solution of the 2D Navier-Stokes equations for incompressible fluid flow problems is investigated. A new flux conservative technique for the solution of the elliptic part of the equations is formulated. In the new model, the non linear convective terms of the momentum equations are approximated by means of characteristics and the spatial approximations, of equal order, are obtained by polynomials of degree two. The advancing in time is afforded by a fractional step method combined with a suitable stabilization technique so that the Inf-Sup condition is respected. In order to keep down the computational cost, the algebraic systems are solved by an iterative solver (Bi-CGSTAB) preconditioned by means of Schwarz additive scalable preconditioners. The properties of the new method are verified carrying out several numerical tests. At first, some elliptic, parabolic and convective-diffusive problems are solved and discussed, then the results of some time dependent and stationary 2D Navier-Stokes problems (in particular the well known benchmark problem of the natural convection in a square cavity) are discussed and compared to those found in the literature. Another, potentially very important application of the numerical tools developed, regards the solution of 1D Shallow-Water equations. In fact the use of the fractional steps scheme for advancing in time and the finite elements (of different polynomial degrees) for the spatial approximation, makes the above mentioned approach computationally profitable and convenient for real applications. The efficiency and accuracy of the numerical model have been checked by solving a theoretical test. Finally, a brief description of the software suitably developed and used in the tests conclude the thesis

Solution of the 2D Navier-Stokes equations by a new FE fractional step method.

In this work, a mathematical and numerical approach for the solution of the 2D Navier-Stokes equations for incompressible fluid flow problems is investigated. A new flux conservative technique for the solution of the elliptic part of the equations is formulated. In the new model, the non linear convective terms of the momentum equations are approximated by means of characteristics and the spatial approximations, of equal order, are obtained by polynomials of degree two. The advancing in time is afforded by a fractional step method combined with a suitable stabilization technique so that the Inf-Sup condition is respected. In order to keep down the computational cost, the algebraic systems are solved by an iterative solver (Bi-CGSTAB) preconditioned by means of Schwarz additive scalable preconditioners. The properties of the new method are verified carrying out several numerical tests. At first, some elliptic, parabolic and convective-diffusive problems are solved and discussed, then the results of some time dependent and stationary 2D Navier-Stokes problems (in particular the well known benchmark problem of the natural convection in a square cavity) are discussed and compared to those found in the literature. Another, potentially very important application of the numerical tools developed, regards the solution of 1D Shallow-Water equations. In fact the use of the fractional steps scheme for advancing in time and the finite elements (of different polynomial degrees) for the spatial approximation, makes the above mentioned approach computationally profitable and convenient for real applications. The efficiency and accuracy of the numerical model have been checked by solving a theoretical test. Finally, a brief description of the software suitably developed and used in the tests conclude the thesis

Efficient upwind algorithms for solution of the Euler and Navier-stokes equations

An efficient three-dimensionasl tructured solver for the Euler and Navier-Stokese quations is developed based on a finite volume upwind algorithm using Roe fluxes. Multigrid and optimal smoothing multi-stage time stepping accelerate convergence. The accuracy of the new solver is demonstrated for inviscid flows in the range 0.675 :5M :5 25. A comparative grid convergence study for transonic turbulent flow about a wing is conducted with the present solver and a scalar dissipation central difference industrial design solver. The upwind solver demonstrates faster grid convergence than the central scheme, producing more consistent estimates of lift, drag and boundary layer parameters. In transonic viscous computations, the upwind scheme with convergence acceleration is over 20 times more efficient than without it. The ability of the upwind solver to compute viscous flows of comparable accuracy to scalar dissipation central schemes on grids of one-quarter the density make it a more accurate, cost effective alternative. In addition, an original convergencea cceleration method termed shock acceleration is proposed. The method is designed to reduce the errors caused by the shock wave singularity M -+ 1, based on a localized treatment of discontinuities. Acceleration models are formulated for an inhomogeneous PDE in one variable. Results for the Roe and Engquist-Osher schemes demonstrate an order of magnitude improvement in the rate of convergence. One of the acceleration models is extended to the quasi one-dimensiona Euler equations for duct flow. Results for this case d monstrate a marked increase in convergence with negligible loss in accuracy when the acceleration procedure is applied after the shock has settled in its final cell. Typically, the method saves up to 60% in computational expense. Significantly, the performance gain is entirely at the expense of the error modes associated with discrete shock structure. In view of the success achieved, further development of the method is proposed

Stratified shallow flow modelling

Environmental hydraulics covers a very wide range of applications including free surface flows in rivers. estuaries and lakes. To find engineering solutions to environmental hydraulics problems. 3D numerical modelling is nowadays widely used. However. the computation of sharp spatial gradients (such as found in stratified estuaries and lakes. around plumes near outfalls along rivers and coasts or in exchange areas of high shear). and the modelling of these processes along steep bathymetric slopes (such as found at the edge of dredged channels or of the continental shelf) remains a challenge. In addition. crude assumptions (such as the hydrostatic assumption) are often made to the primary differential equations in order to simplify the problem and enable long term prediction of environmental hydraulic changes. In this thesis. a robust adaptive mesh displacement (AMD) method is implemented and validated against the lock exchange case in particular. The AMD method aims at vertically focusing nodes within each water column to capture sharp gradients. while reducing the number of nodes or requiring prior knowledge of the flow structure. Second. a direct computation of dynamic pressure is introduced based on the equation of vertical momentum and validated against the analytical potential flow theory solution of a source-sink pair. Dynamic pressure is necessary to model destratification recirculation devices. or flow over dredge channel. or solitary waves. for instance. This direct computation method makes the hydrostatic assumption redundant. Third. a new advection scheme is implemented. whose main advantage is simplicity averaging over Riemann problems without solving them. while excessive numerical viscosity is compensated for by using high-resolution MUSCL type reconstruction. Recommendations are made in this thesis to extend the advection scheme developed herein for tracer advection to the non-linear shallow water equations. to the diffusion terms and to turbulence closure laws within the same finite element framework