Strong and auxiliary forms of the semi-Lagrangian method for incompressible flows

We present a review of the semi-Lagrangian method for advection-diusion and incompressible Navier-Stokes equations discretized with high-order methods. In particular, we compare the strong form where the departure points are computed directly via backwards integration with the auxiliary form where an auxiliary advection equation is solved instead; the latter is also referred to as Operator Integration Factor Splitting (OIFS) scheme. For intermediate size of time steps the auxiliary form is preferrable but for large time steps only the strong form is stable

Spectro-consistent discretization of Navier-Stokes: a challenge to RANS and LES

In this paper, we discuss the results of a fourth-order, spectro-consistent discretization of the incompressible Navier-Stokes equations. In such an approach the discretization of a (skew-)symmetric operator is given by a (skew-)symmetric matrix. Numerical experiments with spectro-consistent discretizations and traditional methods are presented for a one-dimensional convection-diffusion equation. LES and RANS are challenged by giving a number of examples for which a fourth-order, spectro-consistent discretization of the Navier-Stokes equations without any turbulence model yields better (or at least equally good) results as large-eddy simulations or RANS computations, whereas the grids are comparable. The examples are taken from a number of recent workshops on complex turbulent flows.

A fully semi-Lagrangian discretization for the 2D Navier--Stokes equations in the vorticity--streamfunction formulation

A numerical method for the two-dimensional, incompressible Navier--Stokes equations in vorticity--streamfunction form is proposed, which employs semi-Lagrangian discretizations for both the advection and diffusion terms, thus achieving unconditional stability without the need to solve linear systems beyond that required by the Poisson solver for the reconstruction of the streamfunction. A description of the discretization of Dirichlet boundary conditions for the semi-Lagrangian approach to diffusion terms is also presented. Numerical experiments on classical benchmarks for incompressible flow in simple geometries validate the proposed method

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

Spectral methods for CFD

One of the objectives of these notes is to provide a basic introduction to spectral methods with a particular emphasis on applications to computational fluid dynamics. Another objective is to summarize some of the most important developments in spectral methods in the last two years. The fundamentals of spectral methods for simple problems will be covered in depth, and the essential elements of several fluid dynamical applications will be sketched

Arbitrary Lagrangian-Eulerian form of flowfield dependent variation (ALE-FDV) method for moving boundary problems

Flowfield Dependent Variation (FDV) method is a mixed explicit-implicit numerical scheme that was originally developed to solve complex flow problems through the use of so-called implicitness parameters. These parameters determine the implicitness of FDV method by evaluating local gradients of physical flow parameters, hence vary across the computational domain. The method has been used successfully in solving wide range of flow problems. However it has only been applied to problems where the objects or obstacles are static relative to the flow. Since FDV method has been proved to be able to solve many complex flow problems, there is a need to extend FDV method into the application of moving boundary problems where an object experiences motion and deformation in the flow. With the main objective to develop a robust numerical scheme that is applicable for wide range of flow problems involving moving boundaries, in this study, FDV method was combined with a body interpolation technique called Arbitrary Lagrangian-Eulerian (ALE) method. The ALE method is a technique that combines Lagrangian and Eulerian descriptions of a continuum in one numerical scheme, which then enables a computational mesh to follow the moving structures in an arbitrary movement while the fluid is still seen in a Eulerian manner. The new scheme, which is named as ALE-FDV method, is formulated using finite volume method in order to give flexibility in dealing with complicated geometries and freedom of choice of either structured or unstructured mesh. The method is found to be conditionally stable because its stability is dependent on the FDV parameters. The formulation yields a sparse matrix that can be solved by using any iterative algorithm. Several benchmark stationary and moving body problems in one, two and three-dimensional inviscid and viscous flows have been selected to validate the method. Good agreement with available experimental and numerical results from the published literature has been obtained. This shows that the ALE-FDV has great potential for solving a wide range of complex flow problems involving moving bodies

Combination of WENO and Explicit Runge–Kutta Methods for Wind Transport in the Meso-NH Model

This paper investigates the use of the weighted essentially nonoscillatory (WENO) space discretization methods of third and fifth order for momentum transport in the Meso-NH meteorological model, and their association with explicit Runge–Kutta (ERK) methods, with the specific purpose of finding an optimal combination in terms of wall-clock time to solution. A linear stability analysis using von Neumann theory is first conducted that considers six different ERK time integration methods. A new graphical representation of linear stability is proposed, which allows a first discrimination between the ERK methods. The theoretical analysis is then completed by tests on numerical problems of increasing complexity (linear advection of high wind gradient, orographic waves, density current, large eddy simulation of fog, and windstorm simulation), using a fourth-order-centered scheme as a reference basis. The five-stage third-order and fourth-order ERK combinations appear as the time integration methods of choice for coupling with WENO schemes in terms of stability. An explicit time-splitting method added to the ERK temporal scheme for WENO improves the stability properties slightly more. When the spatial discretizations are compared, WENO schemes present the main advantage of maintaining stable, nonoscillatory transitions with sharp discontinuities, but WENO third order is excessively damping, while WENO fifth order provides better accuracy. Finally, WENO fifth order combined with the ERK method makes the whole physics of the model 3 times faster compared to the classical fourth-order centered scheme associated with the leapfrog temporal scheme

Large-eddy simulation of the flow in a lid-driven cubical cavity

Large-eddy simulations of the turbulent flow in a lid-driven cubical cavity have been carried out at a Reynolds number of 12000 using spectral element methods. Two distinct subgrid-scales models, namely a dynamic Smagorinsky model and a dynamic mixed model, have been both implemented and used to perform long-lasting simulations required by the relevant time scales of the flow. All filtering levels make use of explicit filters applied in the physical space (on an element-by-element approach) and spectral (modal) spaces. The two subgrid-scales models are validated and compared to available experimental and numerical reference results, showing very good agreement. Specific features of lid-driven cavity flow in the turbulent regime, such as inhomogeneity of turbulence, turbulence production near the downstream corner eddy, small-scales localization and helical properties are investigated and discussed in the large-eddy simulation framework. Time histories of quantities such as the total energy, total turbulent kinetic energy or helicity exhibit different evolutions but only after a relatively long transient period. However, the average values remain extremely close