On the implementation of the k − ε turbulence model in incompressible flow solvers based on a finite element discretization (original) (raw)
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In this work we present a stabilized equal order finite element formulation of incompressible Navier-Stokes equations augmented by a κ− turbulence model. The aim of this paper is to evaluate the main numerical difficulties associated with the solution of this kind of problems, mainly the possitiveness of the mathematical operators involved and the rate of convergence of the whole system. We propose a particular way to circumvent these drawbacks using
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Journal of the Brazilian Society of Mechanical Sciences and Engineering, 2013
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We present a robust and accurate discretization approach for incompressible turbulent flows based on highorder discontinuous Galerkin methods. The DG discretization of the incompressible Navier-Stokes equations uses the local Lax-Friedrichs flux for the convective term, the symmetric interior penalty method for the viscous term, and central fluxes for the velocity-pressure coupling terms. Stability of the discretization approach for under-resolved, turbulent flow problems is realized by a purely numerical stabilization approach. Consistent penalty terms that enforce the incompressibility constraint as well as inter-element continuity of the velocity field in a weak sense render the numerical method a robust discretization scheme in the under-resolved regime. The penalty parameters are derived by means of dimensional analysis using penalty factors of order 1. Applying these penalty terms in a postprocessing step leads to an efficient solution algorithm for turbulent flows. The proposed approach is applicable independently of the solution strategy used to solve the incompressible Navier-Stokes equations, i.e., it can be used for both projection-type solution methods as well as monolithic solution approaches. Since our approach is based on consistent penalty terms, it is by definition generic and provides optimal rates of convergence when applied to laminar flow problems. Robustness and accuracy are verified for the Orr-Sommerfeld stability problem, the Taylor-Green vortex problem, and turbulent channel flow. Moreover, the accuracy of high-order discretizations as compared to low-order discretizations is investigated for these flow problems. A comparison to state-ofthe-art computational approaches for large-eddy simulation indicates that the proposed methods are highly attractive components for turbulent flow solvers.
International Journal for Numerical Methods in Fluids, 1999
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International Journal for Numerical Methods in Fluids, 2015
This paper presents the second validation step of a compressible discontinuous Galerkin solver with symmetric interior penalty (DGM/SIP) for the direct numerical simulation (DNS) and the large eddy simulation (LES) of complex flows. The method has already been successfully validated for DNS of an academic flow and has been applied to flows around complex geometries (e.g. airfoils and turbomachinery blades). During these studies, the advantages of the dissipation properties of the method have been highlighted, showing a natural tendency to dissipate only the under-resolved scales (i.e the smallest scales present on the mesh), leaving the larger scales unaffected. This phenomenon is further enhanced as the polynomial order is increased. Indeed, the order increases the dissipation at the largest wave numbers, while its range of impact is reduced. These properties are spectrally compatible with a subgrid-scale model, and hence DGM may be well suited to be used for an implicit LES (ILES) approach. A validation of this DGM/ILES approach is here investigated on canonical flows, allowing to study the impact of the discretisation on the turbulence for under-resolved computations. The first test case is the LES of decaying homogeneous isotropic turbulence (HIT) at very high Reynolds number. This benchmark allows to assess the spectral behaviour of the method for implicit LES. The results are in agreement with theory and are even slightly more accurate than other numerical results from literature, obtained using a pseudo-spectral (PS) method with a state-of-the-art subgrid-scale model. The second benchmark is the LES of the channel flow. Three Reynolds numbers are considered: Re D 395, 590 and 950. The results are compared with DNS of Moser et al. and Hoyas et al., also using PS methods. Both averaged velocity and fluctuations are globally in good agreement with the reference, showing the ability of the method to predict equilibrium wall-bounded flow turbulence. To show that the method is able to perform accurate DNS, a DNS of HIT at Re D 64 and a DNS of the channel flow at Re D 180 are also performed. The effects of the grid refinement are investigated on the channel flow at Re D 395, highlighting the improvement of the results when refining the mesh in the spanwise direction. Finally, the modification of the ILES parameters, that is the Riemann solver and of the SIP coefficient, is studied on both cases, showing a significant influence on the choice of the Riemann solver.
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Advances in Fluid Mechanics VIII, 2010
In the present article we were testing our flow solver for turbulent channel flow. Velocity-vorticity formulation of Navier-Stokes equations is applied, thus governing equations are given for the kinematic and kinetic aspects of flow instead of mass and momentum equations. The solution algorithm first solves the kinematics equation for unknown boundary vorticity values using the single domain boundary element numerical method. The next step of the solution algorithm is calculation of the domain velocity field, which is also achieved by solving the kinematics equation. In this and later cases we use the sub-domain boundary element method. After the velocity field is known, we calculate the turbulent kinetic energy and turbulent dissipation fields to obtain the turbulent viscosity. Finally, the vorticity field redistribution is calculated via the kinetics equation. For laminar solutions it was shown that the use of the boundary domain integral method accuracy of solutions for benchmark test cases is very high on coarse meshes. However, since this method is still limited, with high CPU and memory requirements, parallelization of the algorithm is a must for calculating turbulent flows. This was achieved with the use of a MPI (message passing interface) standard.
2018
Turbulence modelling is still evolving, and efforts are on to improve and develop numerical methods to simulate the real turbulence structures by using the empirical and experimental information. The monotonically integrated large eddy simulation (MILES) is an attractive approach for modelling turbulence in high Re flows, which is based on the solving of the unfiltered flow equations with no explicit sub-grid scale (SGS) model. In the current work, this approach has been used, and the action of the SGS model has been included implicitly by intrinsic nonlinear high-frequency filters built into the convection discretization schemes. The MILES solver is developed using the opensource CFD OpenFOAM libraries. The role of flux limiters schemes namely, Gamma, superBee, van-Albada and van-Leer, is studied in predicting turbulent statistical quantities for a fully developed channel flow with a friction Reynolds number, Re<sub>T</sub> = 180, and compared the numerical predictions ...
International Journal for Numerical Methods in Fluids
Both compressible and incompressible Navier-Stokes solvers can be used and are used to solve incompressible turbulent flow problems. In the compressible case, the Mach number is then considered as a solver parameter that is set to a small value, M ≈ 0.1, in order to mimic incompressible flows. This strategy is widely used for high-order discontinuous Galerkin discretizations of the compressible Navier-Stokes equations. The present work raises the question regarding the computational efficiency of compressible DG solvers as compared to a genuinely incompressible formulation. Our contributions to the state-of-the-art are twofold: Firstly, we present a high-performance discontinuous Galerkin solver for the compressible Navier-Stokes equations based on a highly efficient matrix-free implementation that targets modern cache-based multicore architectures. The performance results presented in this work focus on the node-level performance and our results suggest that there is great potential for further performance improvements for current state-of-the-art discontinuous Galerkin implementations of the compressible Navier-Stokes equations. Secondly, this compressible Navier-Stokes solver is put into perspective by comparing it to an incompressible DG solver that uses the same matrix-free implementation. We discuss algorithmic differences between both solution strategies and present an in-depth numerical investigation of the performance. The considered benchmark test cases are the threedimensional Taylor-Green vortex problem as a representative of transitional flows and the turbulent channel flow problem as a representative of wall-bounded turbulent flows.