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 τ = 180 as well as 590.
SummaryWe present a novel approach to wall modeling for the Reynolds-averaged Navier-Stokes equations within the discontinuous Galerkin method. Wall functions are not used to prescribe boundary conditions as usual, but they are built into the function space of the numerical method as a local enrichment, in addition to the standard polynomial component. The Galerkin method then automatically finds the optimal solution among all shape functions available. This idea is fully consistent and gives the wall model vast flexibility in separated boundary layers or high adverse pressure gradients. The wall model is implemented in a high-order discontinuous Galerkin solver for incompressible flow complemented by the Spalart-Allmaras closure model. As benchmark examples, we present turbulent channel flow starting from Re = 180 and up to Re = 100000 as well as flow past periodic hills at Reynolds numbers based on the hill height of Re H = 10595 and Re H = 19000.
A novel approach to wall modeling for the incompressible Navier-Stokes equations including flows of moderate and large Reynolds numbers is presented. The basic idea is that a problem-tailored function space allows prediction of turbulent boundary layer gradients with very coarse meshes. The proposed function space consists of a standard polynomial function space plus an enrichment, which is constructed using Spalding's law-of-the-wall. The enrichment function is not enforced but "allowed" in a consistent way and the overall methodology is much more general and also enables other enrichment functions. The proposed method is closely related to detached-eddy simulation as near-wall turbulence is modeled statistically and large eddies are resolved in the bulk flow. Interpreted in terms of a threescale separation within the variational multiscale method, the standard scale resolves large eddies and the enrichment scale represents boundary layer turbulence in an averaged sense. The potential of the scheme is shown applying it to turbulent channel flow of friction Reynolds numbers from Re τ = 590 and up to 5, 000, flow over periodic constrictions at the Reynolds numbers Re H = 10, 595 and 19, 000 as well as backward-facing step flow at Re h = 5, 000, all with extremely coarse meshes. Excellent agreement with experimental and DNS data is observed with the first grid point located at up to y + 1 = 500 and especially under adverse pressure gradients as well as in separated flows.
We extend the approach of wall modeling via function enrichment to detached-eddy simulation. The wall model aims at using coarse cells in the near-wall region by modeling the velocity profile in the viscous sublayer and log-layer. However, unlike other wall models, the full Navier-Stokes equations are still discretely fulfilled, including the pressure gradient and convective term. This is achieved by enriching the elements of the high-order discontinuous Galerkin method with the law-of-the-wall. As a result, the Galerkin method can "choose" the optimal solution among the polynomial and enrichment shape functions. The detached-eddy simulation methodology provides a suitable turbulence model for the coarse near-wall cells. The approach is applied to wallmodeled LES of turbulent channel flow in a wide range of Reynolds numbers. Flow over periodic hills shows the superiority compared to an equilibrium wall model under separated flow conditions.
A zone of increasingly stretched grid is a robust and easy-to-use way to avoid unwanted reflections at artificial boundaries in wave propagating simulations. In such a buffer zone there are two main damping mechanisms, dissipation and under-resolution that turns a traveling wave into an evanescent wave. We present analysis in one and two space dimensions showing that evanescent decay through under-resolution is a very efficient way to damp waves. The analysis is supported by numerical computations.
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