4940686
On reference solutions and the sensitivity of the 2D Kelvin-Helmholtz instability problem
Abstract Two-dimensional Kelvin–Helmholtz instability problems are popular examples for assessing discretizations for incompressible flows at high Reynolds number. Unfortunately, the results in the literature differ considerably. This paper presents computational studies of a Kelvin–Helmholtz instability problem with high order divergence-free finite element methods. Reference results in several quantities of interest are obtained for three different Reynolds numbers up to the beginning of the final vortex pairing. A mesh-independent prediction of the final pairing is not achieved due to the sensitivity of the considered problem with respect to small perturbations. A theoretical explanation of this sensitivity to small perturbations is provided based on the theory of self-organization of 2D turbulence. Possible sources of perturbations that arise in almost any numerical simulation are discussed.
123612939
The local structure of turbulence in incompressible viscous fluid for very large Reynolds numbers
§1. We shall denote by uα(P) = uα (x1, x2, x3, t), α = 1, 2, 3, the components of velocity at the moment t at the point with rectangular cartesian coordinates x1, x2, x3. In considering the turbulence it is natural to assume the components of the velocity uα (P) at every point P = (x1, x2, x3, t) of the considered domain G of the four-dimensional space (x1, x2, x3, t) are random variables in the sense of the theory of probabilities (cf. for this approach to the problem Millionshtchikov (1939) Denoting by Ᾱ the mathematical expectation of the random variable A we suppose that ῡ2α and (duα /dxβ)2― are finite and bounded in every bounded subdomain of the domain G.
122060992
Dissipation of energy in the locally isotropic turbulence
In my note (Kolmogorov 1941a) I defined the notion of local isotropy and introduced the quantities Bdd(r)=[ud(M′)−ud(M)]2,¯[un(M′)−un(M)¯]2, where r denotes the distance between the points M and M', ud(M) and ud(M') are the velocity components in the direction MM'¯¯ at the points Mand M', and un(M) and un(M') are the velocity components at the points M andM' in some direction, perpendicular to MM'.
124966644
DIRECT NUMERICAL SIMULATION OF TURBULENT CHANNEL FLOWS UP TO RE=590
Numerical simulations of fully developed turbulent channel flow at three Reynolds numbers up to Reτ=590 are reported. It is noted that the higher Reynolds number simulations exhibit fewer low Reynolds number effects than previous simulations at Reτ=180. A comprehensive set of statistics gathered from the simulations is available on the web at http://www.tam.uiuc.edu/Faculty/Moser/channel.
59464895
A pseudo-compressible variational multiscale solver for turbulent incompressible flows
In this work, we design an explicit time-stepping solver for the simulation of the incompressible turbulent flow through the combination of VMS methods and artificial compressibility. We evaluate the effect of the artificial compressibility on the accuracy of the explicit formulation for under-resolved LES simulations. A set of benchmarks have been solved, e.g., the 3D Taylor–Green vortex problem in turbulent regimes. The resulting method is proven to be an effective alternative to implicit methods in some application ranges (in terms of problem size and computational resources), providing comparable results with very low memory requirements. As an example, with the explicit approach, we are able to solve accurately the Taylor-Green vortex benchmark in a fine mesh with $$512^3$$5123 cells on a 12 cores 64 GB ram machine.
123763923
Large eddy simulation of turbulent incompressible flows by a three‐level finite element method
The variational multiscale method provides a methodical framework for large eddy simulation of turbulent flows. In this work, a particular implementation in the form of a three-level finite element method separating large resolved, small resolved, and unresolved scales is proposed. Residual-free bubbles are used for the numerical approximation of the small-scale momentum equation. A stabilizing term is added, in order to take into account the effect of the small-scale continuity equation. This implementation guarantees the stability of the method without further provisions and offers substantial computational savings on the small-scale level. Furthermore, it is accounted for the unresolved scales by a specific dynamic modelling procedure. The method is tested for two different turbulent flow situations.
4644714
Divergence-Free H(div)-FEM for Time-Dependent Incompressible Flows with Applications to High Reynolds Number Vortex Dynamics
In this article, we consider exactly divergence-free H(div)-conforming finite element methods for time-dependent incompressible viscous flow problems. This is an extension of previous research concerning divergence-free $$H^1$$H1-conforming methods. For the linearised Oseen case, the first semi-discrete numerical analysis for time-dependent flows is presented whereby special emphasis is put on pressure- and Reynolds-semi-robustness. For convection-dominated problems, the proposed method relies on a velocity jump upwind stabilisation which is not gradient-based. Complementing the theoretical results, H(div)-FEM are applied to the simulation of full nonlinear Navier–Stokes problems. Focussing on dynamic high Reynolds number examples with vortical structures, the proposed method proves to be capable of reliably handling the planar lattice flow problem, Kelvin–Helmholtz instabilities and freely decaying two-dimensional turbulence.
37631134
On the Divergence Constraint in Mixed Finite Element Methods for Incompressible Flows
The divergence constraint of the incompressible Navier--Stokes equations is revisited in the mixed finite element framework. While many stable and convergent mixed elements have been developed throughout the past four decades, most classical methods relax the divergence constraint and only enforce the condition discretely. As a result, these methods introduce a pressure-dependent consistency error which can potentially pollute the computed velocity. These methods are not robust in the sense that a contribution from the right-hand side, which influences only the pressure in the continuous equations, impacts both velocity and pressure in the discrete equations. This article reviews the theory and practical implications of relaxing the divergence constraint. Several approaches for improving the discrete mass balance or even for computing divergence-free solutions will be discussed: grad-div stabilization, higher order mixed methods derived on the basis of an exact de Rham complex, $H(div)$-conforming finite ...
11232710
TIME-ASYMPTOTICS AND THE SELF-ORGANIZATION HYPOTHESIS FOR 2D NAVIER-STOKES EQUATIONS
In this paper we study the long-time behaviour of solutions with a two-dimensional structure of the Navier-Stokes equations on a periodic grid. From a rigorous investigation of the decrease of the energy E and the enstrophy W, it follows that the Rayleigh quotient Q=WE is monotonically decreasing on solutions. This is shown to imply that the spatial structure of any solution tends to a critical point of Q, which is the structure of some planar vortex, and that all these structures are unstable except for the fundamental one which has the longest wavelength. The time dependence of the approach towards this self-organized state is investigated in some detail. In the spectral plane, the mean-squared wave-numbers of both the spectral energy- and enstrophy density are shown to decrease as a function of time. For the enstrophy this implies that the normal cascade does not match the discriminating effect of the viscous dissipation.
121109769
Turbulence in fluids
to Turbulence in Fluid Mechanics.- Basic Fluid Dynamics.- Transition to Turbulence.- Shear Flow Turbulence.- Fourier Analysis of Homogeneous Turbulence.- Isotropic Turbulence: Phenomenology and Simulations.- Analytical Theories and Stochastic Models.- Two-Dimensional Turbulence.- Beyond Two-Dimensional Turbulence in GFD.- Statistical Thermodynamics of Turbulence.- Statistical Predictability Theory.- Large-Eddy Simulations.- Towards "Real World Turbulence".
123186076
On the inviscid instability of the hyperbolictangent velocity profile
The Rayleigh stability equation of inviscid linearized stability theory was integrated numerically for amplified disturbances of the hyperbolic-tangent velocity profile. The evaluation of the eigenvalues and eigenfunctions is followed by a discussion of the streamline pattern of the disturbed flow. Here no qualitative distinction is found between an amplified and the neutral disturbance. But considering the vorticity distribution of the disturbed flow it is shown that in the case of amplified disturbances two concentrations of vorticity occur within a disturbance wavelength, while in the neutral case only one maximum of vorticity exists. The results are discussed with respect to the instability mechanism of free boundary-layer flow.
122770250
Local Grid Refinement in Large-Eddy Simulation
A method for performing nested-grid calculations with a Large-Eddy Simulation (LES) code is described. The grid consists of a coarse mesh and a fine mesh which overlaps the coarse in some region. A standard finite-volume method is used on both meshes. By means of grid communication, the velocity and pressure at both meshes are matched. To check that large eddies which are already resolved on the coarse grid are not affected by the nesting procedure, a simple two-dimensional mixing layer is simulated. Several simulations of this flow have been carried out with a different number of grid points on the nested grid. It is found that, without much extra computational effort, the grid-nesting improves the turbulent statistics with respect to the results found on the coarse mesh. This improvement occurs first of all in the region where grid refinement is applied, but better results are also found on the coarse mesh outside the grid-refinement region. Furthermore, it is shown that the large-scale structures in this flow are not influenced by the boundary between the coarse and fine grid.
21071333
Analysis of a full space-time discretization of the Navier-Stokes equations by a local projection stabilization method
A finite element error analysis of a local projection stabilization (LPS) method for the time-dependent Navier–Stokes equations is presented. The focus is on the high-order term-by-term stabilization method that has one level, in the sense that it is defined on a single mesh, and in which the projection-stabilized structure of standard LPS methods is replaced by an interpolation-stabilized structure. The main contribution is on proving, theoretically and numerically, the optimal convergence order of the arising fully discrete scheme. In addition, the asymptotic energy balance is obtained for slightly smooth flows. Numerical studies support the analytical results and illustrate the potential of the method for the simulation of turbulent flows. Smooth unsteady flows are simulated with optimal order of accuracy.
122625823
Large-eddy simulation of the turbulent mixing layer
Six subgrid models for the turbulent stress tensor are tested by conducting large-eddy simulations (LES) of the weakly compressible temporal mixing layer: the Smagorinsky, similarity, gradient, dynamic eddy-viscosity, dynamic mixed and dynamic Clark models. The last three models are variations of the first three models using the dynamic approach. Two sets of simulations are performed in order to assess the quality of the six models. The LES results corresponding to the first set are compared with filtered results obtained from a direct numerical simulation (DNS). It appears that the dynamic models lead to more accurate results than the non-dynamic models tested. An adequate mechanism to dissipate energy from resolved to subgrid scales is essential. The dynamic models have this property, but the Smagorinsky model is too dissipative during transition, whereas the similarity and gradient models are not sufficiently dissipative for the smallest resolved scales. In this set of simulations, at moderate Reynolds number, the dynamic mixed and Clark models are found to be slightly more accurate than the dynamic eddy-viscosity model. The second set of LES concerns the mixing layer at a considerably higher Reynolds number and in a larger computational domain. An accurate DNS for this mixing layer can currently not be performed, thus in this case the LES are tested by investigating whether they resemble a self-similar turbulent flow. It is found that the dynamic models generate better results than the non-dynamic models. The closest approximation to a self-similar state was obtained using the dynamic eddy-viscosity model.
17688553
Adaptive Wavelet Solvers for the Unsteady Incompressible Navier-Stokes Equations
In this paper we describe adaptive wavelet-based solvers for the Navier-Stokes equations. Our approach employs a Petrov-Galerkin scheme with tensor products of Interpolet wavelets as ansatz functions. We present the fundamental algorithms for the adaptive evaluation of differential operators and non-linear terms. Furthermore, a simple but efficient preconditioning technique for the resulting linear systems is introduced. For the Navier-Stokes equations a Chorin-type projection method with a stabilized pressure discretization is used. Numerical examples demonstrate the efficiency of our approach.
119865192
Interior penalty variational multiscale method for the incompressible Navier-Stokes equation: Monitoring artificial dissipation
In this paper, we apply the interior penalty method analyzed in [E. Burman and M.--A. Fernndez, Numer. Math. 107 (2007), no. 1, 39--77; MR2317827 (2008c:65317)] to flows at high Reynolds number. As a possible measure of solution quality we propose to monitor the ratio between the artificial dissipation induced by the numerical method and the computed physical dissipation. For smooth flows we prove that for our method the artificial dissipation serves as an a posteriori error estimator and also that it vanishes at an optimal rate. In the case of flows with multiscale features, we discuss a heuristic approach relating the decay of the artificial dissipation to the decay rate of the power spectrum. Some numerical results in two space dimensions are presented examining the relation between the numerical dissipation and solution quality.
121378763
A deconvolution‐based fourth‐order finite volume method for incompressible flows on non‐uniform grids
This paper is concerned with the development of a new high-order finite volume method for the numerical simulation of highly convective unsteady incompressible flows on non-uniform grids. Specifically, both a high-order fluxes integration and the implicit deconvolution of the volume-averaged field are considered. This way, the numerical solution effectively stands for a fourth-order approximation of the point-wise one. Moreover, the procedure is developed in the framework of a projection method for the pressure–velocity decoupling, while originally deriving proper high-order intermediate boundary conditions. The entire numerical procedure is discussed in detail, giving particular attention to the consistent discretization of the deconvolution operation. The present method is also cast in the framework of approximate deconvolution modelling for large-eddy simulation. The overall high accuracy of the method, both in time and space, is demonstrated. Finally, as a model of real flow computation, a two-dimensional time-evolving mixing layer is simulated, with and without sub-grid scales modelling. Copyright © 2003 John Wiley & Sons, Ltd.
51993512
Fractional-step methods and finite elements with symmetric stabilization for the transient Oseen problem
This paper deals with the spatial and time discretization of the transient Oseen equations. Finite elements with symmetric stabilization in space are combined with several time-stepping schemes (monolithic and fractional-step). Quasi-optimal (in space) and optimal (in time) error estimates are established for smooth solutions in all flow regimes. We first analyze monolithic time discretizations using the Backward Differentation Formulas of order 1 and 2 (BDF1 and BDF2). We derive a new estimate on the time-average of the pressure error featuring the same robustness with respect to the Reynolds number as the velocity estimate. Then, we analyze fractional-step pressure-projection methods using BDF1. The stabilization of velocities and pressures can be treated either implicitly or explicitly. Numerical results illustrate the main theoretical findings.
123852979
Computation of turbulent flows using a finite calculus–finite element formulation
We present a formulation for analysis of turbulent incompressible flows using a stabilized finite element method (FEM) based on the finite calculus (FIC) procedure. The stabilization terms introduced by the FIC approach allow to solve a wide range of fluid flow problems at different Reynolds numbers, including turbulent flows, without the need of a turbulence model. Examples of application of the FIC/FEM formulation to the analysis of 2D and 3D incompressible flows at large Reynolds numbers exhibiting turbulence features are presented.
209814350
Applied analysis of the Navier-Stokes equations
The Navier–Stokes equations are a set of nonlinear partial differential equations comprising the fundamental dynamical description of fluid motion. They are applied routinely to problems in engineering, geophysics, astrophysics, and atmospheric science. This book is an introductory physical and mathematical presentation of the Navier–Stokes equations, focusing on unresolved questions of the regularity of solutions in three spatial dimensions, and the relation of these issues to the physical phenomenon of turbulent fluid motion. Intended for graduate students and researchers in applied mathematics and theoretical physics, results and techniques from nonlinear functional analysis are introduced as needed with an eye toward communicating the essential ideas behind the rigorous analyses.
119628458
High Reynolds number Navier–Stokes solutions and boundary layer separation induced by a rectilinear vortex
Abstract We compute the solutions of Prandtl’s and Navier–Stokes equations for the two dimensional flow induced by a rectilinear vortex interacting with a boundary in the half plane. For this initial datum Prandtl’s equation develops, in a finite time, a separation singularity. We investigate the different stages of unsteady separation for Navier–Stokes solution at different Reynolds numbers Re = 103–105, and we show the presence of a large-scale interaction between the viscous boundary layer and the inviscid outer flow. We also see a subsequent stage, characterized by the presence of a small-scale interaction, which is visible only for moderate-high Re numbers Re = 104–105. We also investigate the asymptotic validity of boundary layer theory by comparing Prandtl’s solution to Navier–Stokes solutions during the various stages of unsteady separation.
119584609
Maximum palinstrophy growth in 2D incompressible flows
Abstract In this study we investigate vortex structures which lead to the maximum possible growth of palinstrophy in two-dimensional incompressible flows on a periodic domain. The issue of palinstrophy growth is related to a broader research program focusing on extreme amplification of vorticity-related quantities which may signal singularity formation in different flow models. Such extreme vortex flows are found systematically via numerical solution of suitable variational optimization problems. We identify several families of maximizing solutions parameterized by their palinstrophy, palinstrophy and energy and palinstrophy and enstrophy. Evidence is shown that some of these families saturate estimates for the instantaneous rate of growth of palinstrophy obtained using rigourous methods of mathematical analysis, thereby demonstrating that this analysis is in fact sharp. In the limit of small palinstrophies the optimal vortex structures are found analytically, whereas for large palinstrophies they exhibit a self-similar multipolar structure. It is also shown that the time evolution obtained using the instantaneously optimal states with fixed energy and palinstrophy as the initial data saturates the upper bound for the maximum growth of palinstrophy in finite time. Possible implications of this finding for the questions concerning extreme behaviour of flows are discussed.
118570790
Vortices, Maximum Growth and the Problem of Finite-Time Singularity Formation
In this work we are interested in extreme vortex states leading to the maximum possible growth of palinstrophy in 2D viscous incompressible flows on periodic domains. This study is a part of a broader research effort motivated by the question about the finite-time singularity formation in the 3D Navier?Stokes system and aims at a systematic identification of the most singular flow behaviors. We extend the results reported in Ayala and Protas (2014 J. Fluid Mech. 742 340?67) where extreme vortex states were found leading to the growth of palinstrophy, both instantaneously and in finite time, which saturates the estimates obtained with rigorous methods of mathematical analysis. Here we uncover the vortex dynamics mechanisms responsible for such extreme behavior in time-dependent 2D flows. While the maximum palinstrophy growth is achieved at short times, the corresponding long-time evolution is characterized by some nontrivial features, such as vortex scattering events.
85511424
Dissipation of coherent structures in confined two-dimensional turbulence
Since the seminal article “Inertial ranges in two-dimensional turbulence” by Kraichnan in 1967, our understanding of the dynamics and transport properties of two-dimensional turbulence is largely built on the assumption of homogeneity and isotropy of statistically steady or decaying turbulence. In the last two decades, more attention has been paid to the presence of lateral walls, either with stress-free or no-slip boundary conditions, and also considering a variety of geometries such as square, rectangular, or circular domains. The impact of confining boundaries on the dynamics of two-dimensional turbulence is important. This is in sharp contrast with three-dimensional turbulence, where homogeneity and isotropy are locally restored due to the cascade process in the inertial range. The impact of confining boundaries is therefore limited in three-dimensional turbulence. The presence of an inverse energy cascade in two-dimensional turbulence, however, will continuously generate large-scale energy-containing...
122468402
Finite Element Methods for Viscous Incompressible Flows: A Guide to Theory, Practice, and Algorithms
Discretization of the Primitive Variable Formulation: A Primitive Variable Formulation. The Finite Element Problem and the Div-St abi lity Condition. Finite Element Spaces. Alternate Weak Forms, Boundary Conditions and Numerical Integration. Penalty Methods. Solution of the Discrete Equations: Newton's Method and Other Iterative Methods. Solving the Linear Systems. Solution Methods for Large Reynolds Numbers. Time Dependent Problems: A Weak Formulation and Spatial Discretizations. Time Discretizations. The Streamfunction-Vorticity Formulation: Algorithms for the Streamfunction-Vorticity Equations. Solution Techniques for Multiply Connected Domains. The Streamfunction Formulation: Algorithms for Determining Streamfunction Approximations. Eigenvalue Problems Connected with Stability Studies for Viscous Flows: Energy Stability Analysis of Viscous Flows. Linearized Stability Analysis of Stationary Viscous Flows. Exterior Problems: Truncated Domain-Artificial Boundary Condition Methods. Nonlinear Constitutive Relations: A Ladyzhenskaya Model and Algebraic Turbulence Models. Bingham Fluids. Electromagnetically or Thermally Coupled Flows: Flows of Liquid Metals. The Boussinesq Equations. Remarks on Some Topics That Have Not Been Considered: Problems, Formulations, Algorithms, and Other Issues That Have Not Been Considered. Bibliography. Glossary of Symbols. Index.
117037950
Theory and practice of finite elements
I Theoretical Foundations.- 1 Finite Element Interpolation.- 2 Approximation in Banach Spaces by Galerkin Methods.- II Approximation of PDEs.- 3 Coercive Problems.- 4 Mixed Problems.- 5 First-Order PDEs.- 6 Time-Dependent Problems.- III Implementation.- 7 Data Structuring and Mesh Generation.- 8 Quadratures, Assembling, and Storage.- 9 Linear Algebra.- 10 A Posteriori Error Estimates and Adaptive Meshes.- IV Appendices.- A Banach and Hilbert Spaces.- A.1 Basic Definitions and Results.- A.2 Bijective Banach Operators.- B Functional Analysis.- B.1 Lebesgue and Lipschitz Spaces.- B.2 Distributions.- B.3 Sobolev Spaces.- Nomenclature.- References.- Author Index.
14881937
Implicit-explicit Runge-Kutta methods for time-dependent partial differential equations
Abstract Implicit-explicit (IMEX) linear multistep time-discretization schemes for partial differential equations have proved useful in many applications. However, they tend to have undesirable time-step restrictions when applied to convection-diffusion problems, unless diffusion strongly dominates and an appropriate BDF-based scheme is selected (Ascher et al., 1995). In this paper, we develop Runge-Kutta-based IMEX schemes that have better stability regions than the best known IMEX multistep schemes over a wide parameter range.
125900888
Nonlinear Functional Analysis and its Applications: III: Variational Methods and Optimization
As long as a branch of knowledge offers an abundance of problems, it is full of vitality. David Hilbert Over the last 15 years I have given lectures on a variety of problems in nonlinear functional analysis and its applications. In doing this, I have recommended to my students a number of excellent monographs devoted to specialized topics, but there was no complete survey-type exposition of nonlinear functional analysis making available a quick survey to the wide range of readers including mathematicians, natural scientists, and engineers who have only an elementary knowledge of linear functional analysis. I have tried to close this gap with my five-part lecture notes, the first three parts of which have been published in the Teubner-Texte series by Teubner-Verlag, Leipzig, 1976, 1977, and 1978. The present English edition was translated from a completely rewritten manuscript which is significantly longer than the original version in the TeubnerTexte series. The material is organized in the following way: Part I: Fixed Point Theorems. Part II: Monotone Operators. Part III: Variational Methods and Optimization. Parts IV jV: Applications to Mathematical Physics. The exposition is guided by the following considerations: (a) What are the supporting basic ideas and what intrinsic interrelations exist between them? (/3) In what relation do the basic ideas stand to the known propositions of classical analysis and linear functional analysis? ( y) What typical applications are there? Vll Preface viii Special emphasis is placed on motivation.
6134302
High order exactly divergence-free Hybrid Discontinuous Galerkin Methods for unsteady incompressible flows
Abstract In this paper we present an efficient discretization method for the solution of the unsteady incompressible Navier–Stokes equations based on a high order (Hybrid) Discontinuous Galerkin formulation. The crucial component for the efficiency of the discretization method is the distinction between stiff linear parts and less stiff non-linear parts with respect to their temporal and spatial treatment. Exploiting the flexibility of operator-splitting time integration schemes we combine two spatial discretizations which are tailored for two simpler sub-problems: a corresponding hyperbolic transport problem and an unsteady Stokes problem. For the hyperbolic transport problem a spatial discretization with an Upwind Discontinuous Galerkin method and an explicit treatment in the time integration scheme is rather natural and allows for an efficient implementation. The treatment of the Stokes part involves the solution of linear systems. In this case a discretization with Hybrid Discontinuous Galerkin methods is better suited. We consider such a discretization for the Stokes part with two important features: H ( div ) -conforming finite elements to guarantee exactly divergence-free velocity solutions and a projection operator which reduces the number of globally coupled unknowns. We present the method, discuss implementational aspects and demonstrate the performance on two and three dimensional benchmark problems.
119614579
Polynomial robust stability analysis for $H(\textrm{div})$-conforming finite elements for the Stokes equations
In this work we consider a discontinuous Galerkin method for the discretization of the Stokes problem. We use $H(\textrm{div})$-conforming finite elements as they provide major benefits such as exact mass conservation and pressure-independent error estimates. The main aspect of this work lies in the analysis of high order approximations. We show that the considered method is uniformly stable with respect to the polynomial order $k$ and provides optimal error estimates $ \| \boldsymbol{u} - \boldsymbol{u}_h \|_{1_h} + \| \Pi^{Q_h}p-p_h \| \le c \left( h/k \right)^s \| \boldsymbol{u} \|_{s+1} $. To derive those estimates, we prove a $k$-robust LBB condition. This proof is based on a polynomial $H^2$-stable extension operator. This extension operator itself is of interest for the numerical analysis of $C^0$-continuous discontinuous Galerkin methods for $4^{th}$ order problems.
55031263
Towards computable flows and robust estimates for inf-sup stable FEM applied to the time-dependent incompressible Navier–Stokes equations
Inf-sup stable FEM applied to time-dependent incompressible Navier–Stokes flows are considered. The focus lies on robust estimates for the kinetic and dissipation energies in a twofold sense. Firstly, pressure–robustness ensures the fulfilment of a fundamental invariance principle and velocity error estimates are not corrupted by the pressure approximability. Secondly, Re-semi-robustness means that constants appearing on the right-hand side of kinetic and dissipation energy error estimates (including Gronwall constants) do not explicitly depend on the Reynolds number. Such estimates rely on the essential regularity assumption $$\nabla u \in L^1(0,T;L^\infty (\varOmega ))$$∇u∈L1(0,T;L∞(Ω)) which is discussed in detail. In the sense of best practice, we review and establish pressure- and Re-semi-robust estimates for pointwise divergence-free $$H^1$$H1-conforming FEM (like Scott–Vogelius pairs or certain isogeometric based FEM) and pointwise divergence-free H(div)-conforming discontinuous Galerkin FEM. For convection-dominated problems, the latter naturally includes an upwind stabilisation for the velocity which is not gradient-based.
118432058
Mixed Finite Element Methods and Applications
Preface.- Variational Formulations and Finite Element Methods.- Function Spaces and Finite Element Approximations.- Algebraic Aspects of Saddle Point Problems.- Saddle Point Problems in Hilbert spaces.- Approximation of Saddle Point Problems.- Complements: Stabilisation Methods, Eigenvalue Problems.- Mixed Methods for Elliptic Problems.- Incompressible Materials and Flow Problems.- Complements on Elasticity Problems.- Complements on Plate Problems.- Mixed Finite Elements for Electromagnetic Problems.- Index.
118746172
Mathematical Aspects of Discontinuous Galerkin Methods
Basic concepts.- Steady advection-reaction.- Unsteady first-order PDEs.- PDEs with diffusion.- Additional topics on pure diffusion.- Incompressible flows.- Friedhrichs' Systems.- Implementation.
30758051
Implicit-explicit methods for time-dependent partial differential equations
Implicit-explicit (IMEX) schemes have been widely used, especially in conjunction with spectral methods, for the time integration of spatially discretized partial differential equations (PDEs) of diffusion-convection type. Typically, an implicit scheme is used for the diffusion term and an explicit scheme is used for the convection term. Reaction-diffusion problems can also be approximated in this manner. In this work we systematically analyze the performance of such schemes, propose improved new schemes, and pay particular attention to their relative performance in the context of fast multigrid algorithms and of aliasing reduction for spectral methods.For the prototype linear advection-diffusion equation, a stability analysis for first-, second-, third-, and fourth-order multistep IMEX schemes is performed. Stable schemes permitting large time steps for a wide variety of problems and yielding appropriate decay of high frequency error modes are identified. Numerical experiments demonstrate that weak decay...
123466565
On the accuracy of high-order discretizations for underresolved turbulence simulations
In this paper, we investigate the accuracy of a high-order discontinuous Galerkin discretization for the coarse resolution simulation of turbulent flow. We show that a low-order approximation exhibits unacceptable numerical discretization errors, whereas a naive application of high-order discretizations in those situations is often unstable due to aliasing. Thus, for high-order simulations of underresolved turbulence, proper stabilization is necessary for a successful computation. Two different mechanisms are chosen, and their impact on the accuracy of underresolved high-order computations of turbulent flows is investigated. Results of these approximations for the Taylor–Green Vortex problem are compared to direct numerical simulation results from literature. Our findings show that the superior discretization properties of high-order approximations are retained even for these coarsely resolved computations.
52917991
Robust and efficient discontinuous Galerkin methods for under-resolved turbulent incompressible flows
Abstract We present a robust and accurate discretization approach for incompressible turbulent flows based on high-order 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-of-the-art computational approaches for large-eddy simulation indicates that the proposed methods are highly attractive components for turbulent flow solvers.
124084100
On the Influence of Polynomial De-aliasing on Subgrid Scale Models
In this work we investigate the interplay of polynomial de-aliasing and sub-grid scale models for large eddy simulations based on discontinuous Galerkin discretizations. It is known that stability is a major concern when simulating underresolved turbulent flows with high order nodal collocation type discretizations. By changing the interpolatory character of the nodal collocation type discretization to a projection based discretization by increasing the number of quadrature points (polynomial de-aliasing), one is able to remove the aliasing induced stability problems. We focus on this effect and on the consequence for large eddy simulations with explicit subgrid scale models. Often, subgrid scale models have to achieve two possibly conflicting tasks in a single simulation: firstly stabilizing the numerics and secondly modeling the physical effect of the missing scales. Within a discontinuous Galerkin approach, it is possible to use either a fast (but potentially aliasing afflicted) nodal collocation discretization or a projection-based (but computationally costly) variant in combination with an explicit subgrid scale model. We use this framework to investigate the effect on the appropriate model parameter of a standard Smagorinsky subgrid scale model and of a Variational Multiscale Smagorinsky formulation. For this we first consider the 3-D viscous Taylor-Green vortex example to investigate the impact on the stability of the method and second the turbulent flow past a circular cylinder to investigate and compare the accuracy of the results. We show that the aliasing instabilities of collocative discretizations severely limit the choice of the model constant, in particular for high order schemes, while for de-aliased DG schemes, the closure model parameters can be chosen independently from the numerical scheme. For the cylinder flow, we also find that for the same model settings, the projection-based results are in better agreement with the reference DNS than those of the collocative scheme.
14932396
NETGEN An advancing front 2D/3D-mesh generator based on abstract rules
Abstract. In this paper, the algorithms of the automatic mesh generator NETGEN are described. The domain is provided by a Constructive Solid Geometry (CSG). The whole task of 3D mesh generation splits into four subproblems of special point calculation, edge following, surface meshing and finally volume mesh generation. Surface and volume mesh generation are based on the advancing front method. Emphasis is given to the abstract structure of the element generation rules. Several techniques of mesh optimization are tested and quality plots are presented.

195820347
Stability of Overintegration Methods for Nodal Discontinuous Galerkin Spectral Element Methods
We perform stability analyses for discontinuous Galerkin spectral element approximations of linear variable coefficient hyperbolic systems in three dimensional domains with curved elements. Although high order, the precision of the quadratures used are typically too low with respect to polynomial order associated with their arguments, which introduces aliasing errors that can destabilize an approximation, especially when the solution is underresolved. We show that using a larger number of points in the volume quadrature, often called "overintegration", can eliminate the aliasing term associated with the volume, but introduces new aliasing errors at the surfaces that can destabilize the solution. Increased quadrature precision on both the volume and surface terms, on the other hand, leads to a stable approximation. The results support the findings of Mengaldo et al. [Dealiasing techniques for high-order spectral element methods on regular and irregular grids. Journal of Computational Physics, 299:56 -- 81, 2015] who found that fully consistent integration was more robust for the solution of compressible flows than the volume only version.