# Numerical Analysis and Simulation of Turbulent Flows, Large Eddy Simulation (LES), Variational Multiscale Methods (VMS)

Reviews

 Review of VMS methods Naveed Ahmed, Tomas Chacon Rebollo, Volker John, Samuele Rubino, Review of Variational Multiscale Methods for the Simulation of Turbulent Incompressible Flows ,Arch. Comput. Methods Engrg. 24, 115 - 164, 2017 Various realizations of variational multiscale (VMS) methods for simulating turbulent incompressible flows have been proposed in the past fifteen years. All of these realizations obey the basic principles of VMS methods: They are based on the variational formulation of the incompressible Navier-Stokes equations and the scale separation is defined by projections. However, apart from these common basic features, the various VMS methods look quite different. In this review, the derivation of the different VMS methods is presented in some detail and their relation among each other and also to other discretizations is discussed. Another emphasis consists in giving an overview about known results from the numerical analysis of the VMS methods. A few results are presented in detail to highlight the used mathematical tools. Furthermore, the literature presenting numerical studies with the VMS methods is surveyed and the obtained results are summarized. Short review of some aspects in LES and VMS Volker John, On Large Eddy Simulation and Variational Multiscale Methods in the Numerical Simulation of Turbulent Incompressible Flows , Applications of Mathematics 51, 321 - 353, 2006 The numerical simulation of turbulent flows is one of the great challenges in Computational Fluid Dynamics (CFD). In general, a Direct Numerical Simulation (DNS) is not feasible due to limited computer resources (performance and memory) and the use of a turbulence model becomes necessary. The paper will discuss several aspects of two approaches of turbulent modeling - Large Eddy Simulation (LES) and Variational Multiscale (VMS) models. Topics which will be addressed are the detailed derivation of these models, the analysis of commutation errors in LES models as well as other results from the mathematical analysis.

Variational Multiscale Methods (VMS)

 Tetrahedral Meshes V.John, A. Kindl, C. Suciu "Finite Element LES and VMS Methods on Tetrahedral Meshes", J. Comp. Appl. Math. 233, 3095 - 3102, 2010 Finite element methods for problems given in complex domains are often based on tetrahedral meshes. This paper demonstrates that the so-called rational Large Eddy Simulation model and a projection-based Variational Multiscale method can be extended in a straightforward way to tetrahedral meshes. Numerical studies are performed with an inf-sup stable second order pair of finite elements with discontinuous pressure approximation. Variational Multiscale Method with Adaptive Large Scale Space V.John, A. Kindl "A Variational Multiscale Method for Turbulent Flow Simulation with Adaptive Large Scale Space" J. Comput. Phys. 229, 301 - 312, 2010 In turbulent flows it is only feasible to simulate large flow structures. Variational multiscale (VMS) methods define these flow structures by projections onto appropriate function spaces. This paper presents a finite element VMS method which chooses the large scale projection space adaptively. The adaption controls the influence of an eddy viscosity model and it is based on the size of the so--called resolved small scales. The adaptive procedure is described in detail. Numerical studies at a turbulent channel flow and a turbulent flow around a cylinder are presented. It is shown that the method selects the large scale space in a reasonable way and that appropriately chosen parameters improve the results compared to the basic method, which uses the same local large scale space in the whole domain and for all times. Finite element error estimates for turbulent viscosity of Smagorinsky type V.John, S.Kaya and A. Kindl "Finite Element Error Analysis for a Projection-Based Variational Multiscale Method with Nonlinear Eddy Viscosity " J. Math. Anal. Appl., 344, 627 - 641, 2008 The paper presents a finite element error analysis for a projection-based variational multiscale (VMS) method for the incompressible Navier-Stokes equations. In the VMS method, the influence of the unresolved scales onto the resolved small scales is modeled by a Smagorinsky-type turbulent viscosity. Finite element error estimates for constant turbulent viscosity V.John and S.Kaya "Finite Element Error Analysis of a Variational Multiscale Method for the Navier-Stokes Equations" Adv. Comput. Math. 28, 43 - 61, 2008 The paper presents finite element error estimates of a variational multiscale method (VMS) for the incompressible Navier-Stokes equations. The constants in these estimates do not depend on the Reynolds number but on a reduced Reynolds number or on the mesh size of a coarse mesh. Bubble VMS and projection based VMS Volker John and Adela Tambulea, On finite element variational multiscale methods for incompressible turbulent flows , Proceedings of ECCOMAS CFD, ISBN 90-9020970-0, 2006 Two realizations of finite element variational multiscale (VMS) methods for the simulation of incompressible turbulent flows are studied. The difference between the two approaches consists in the way the spaces for the large scales and the resolved small scales are chosen. The paper addresses issues of the implementation of these methods, the treatment of the additional terms and equations in the temporal discretization, and the additional costs of these methods. Implementation and numerical results Volker John and Songul Kaya, " A Finite Element Variational Multiscale Method for the Navier-Stokes Equations", SIAM J. Sci. Comp. 26, 1485 - 1503, 2005 The paper presents a variational multiscale method (VMS) for the incompressible Navier-Stokes equations which is defined by a large scale space $L^H$ for the velocity deformation tensor and a turbulent viscosity $\nu_T$. The connection of this method to the standard formulation of a VMS is explained. It is studied under which conditions on $L^H$, the VMS can be implemented easily and efficiently into an existing finite element code for solving the Navier-Stokes equations. Numerical tests with the Smagorinsky LES model for $\nu_T$ are presented.

Analysis of Time Averaged Statistics of Finite Element Approximations

 Volker John, William Layton and Carolina Manica, Convergence of time averaged statistics of finite element approximations of the Navier-Stokes equations , SIAM J. Numer. Anal. 46, 151 - 179, 2008 When discussing numerical solutions of the Navier-Stokes equations, especially when turbulent flows are concerned, there are at least two questions that can be raised. What is meaningful to compute? How to determine the fidelity of the computed solution with respect to the true solution? This paper takes a step towards the answer of these questions for turbulent flows. We consider long time averages of weak solutions of the Navier-Stokes equations, rather than strong solutions. We present error estimates for the time averaged energy dissipation rate, drag and lift, most of which are optimal for small Reynolds/generalized Grashoff numbers. For shear flows, we address the question of fidelity of the computed solution with respect to the true solution, in view of Kolmogorov's energy cascade theory.

Analysis of Finite Element Discretizations for LES Models

 Smagorinsky model with minimal regularity assumptions V. John and W. J. Layton , " Analysis of Numerical Errors in Large Eddy Simulation", SIAM J. Numer. Anal. 40 (3), 995 - 1020, 2002 We consider the question of numerical errors'' in large eddy simulation. It is often claimed that straightforward discretization and solution using centered methods of models for large eddy motion can simulate the motion of turbulent flows with complexity independent of the Reynolds number and depending only on the resolution $\delta$'' of the eddies sought. This report considers precisely this question analytically: is it possible to prove error estimates for discretizations of {\it actually used} large eddy models whose error constants depend only on $\delta$ but not $Re$? We consider the most common, simplest and most mathematically tractable model and the most mathematically clear discretization. In two cases, we prove such an error estimate and try to explain why our technique of proof fails in the most general case. Our analysis aims to assume as little time regularity on the true solution as possible. Taylor LES model T. Iliescu, V. John and W. J. Layton, "Convergence of Finite Element Approximations of Large Eddy Motion" , Num. Meth. Part. Diff. Equ., 18: 689 - 710, 2002 Fluid motion in many applications occurs at higher Reynolds numbers. In these applications dealing with turbulent flow is thus inescapable. One promising approach to the simulation of the motion of the large structures in turbulent flow is large eddy simulation in which equations describing the motion of local spatial averages of the fluid velocity are solved numerically. This report considers numerical-errors'' in LES. Specifically, for one family of space filtered flow models, we show convergence of the finite element approximation of the model and give an estimate of the error.

Commutation Errors

 Navier-Stokes equations, filter with non-constant filter width Luigi C. Berselli, Carlo R. Grisanti, Volker John, "Analysis of commutation errors for functions with low regularity" , J. Comp. Appl. Math. 206, 1027 - 1045, 2007 Commutation errors arise in the derivation of the space averaged Navier-Stokes equations, the basic equations for the Large Eddy Simulation of turbulent flows, if the filter is non-uniform or asymmetric (skewed). These errors need to be analyzed for turbulent flow fields, where one expects a limited regularity of the solution. This paper studies the order of convergence of commutation errors, as the filter width tends to zero, for functions with low regularity. Several convergence results are proved and also that convergence may fail (or its order decreases) if the functions become less smooth. The main results are those dealing with H\"older-continuous functions and with functions having singularities. The sharpness of the analytic results is confirmed with numerical illustrations. Navier-Stokes equations, filter with non-constant filter width, comparison to the size of the Reynolds stress tensor for the turbulent channel flow Luigi C. Berselli, Volker John, "Asymptotic behavior of commutation errors and the divergence of the Reynolds stress tensor near the wall in the turbulent channel flow" , Math. Methods Appl. Sci. 29, 1709 - 1719, 2006 The derivation of the space averaged Navier-Stokes equations for the Large Eddy Simulation (LES) of turbulent incompressible flows introduces two groups of terms which do not depend only on the space averaged flow field variables: the divergence of the Reynolds stress tensor and commutation errors. Whereas the former is studied intensively in the literature, the latter terms are usually neglected. This note studies the asymptotic behavior of these terms for the turbulent channel flow at a wall in the case that the commutation errors arise from the application of a non-uniform box filter. To perform analytical calculations, the unknown flow field is modeled by a couple of wall laws (Reichardt law and $1/\alpha$-th power law) for the mean velocity profile and highly oscillating functions model the turbulent fluctuations. The asymptotics which are derived show that near the wall the commutation errors are at least as important as the divergence of the Reynolds stress tensor. Navier-Stokes equations, filter with constant filter width A. Dunca, V. John, W.J. Layton, "The Commutation Error of the Space Averaged Navier-Stokes Equations on a Bounded Domain", in G.P. Galdi, J.G. Heywood, R. Rannacher (Eds.), Contributions to Current Challenges in Mathematical Fluid Mechanics, Advances in Mathematical Fluid Mechanics 3, Birkhäuser Verlag Basel, 53 - 78, 2004 In Large Eddy Simulation of turbulent flows, the Navier-Stokes equations are convolved with a filter and differentiation and convolution are interchanged, introducing an extra commutation error term, which is nearly universally dropped from the resulting equations. We show that the commutation error is asymptotically negligible in $L^p(\mathbb R^d)$ (i.e., it vanishes as the averaging radius $\delta \to 0$) if and only if the fluid and the boundary exert exactly zero force on each other. Next, we show that the commutation error tends to zero in $H^{-1}(\Omega)$ as $\delta\to 0$. Convergence is proven also for a weak form of the commutation error. The order of convergence is studied in both cases. Last, we study the influence of the commutation error on the energy balance of the filtered equations.

A Priori and A Posteriori Error Estimates for the Large Eddies

 Differential filter A. Dunca, V. John "Finite element error analysis of space averaged flow fields defined by a differential filter" , Math. Mod. Meth. Appl. Sci. (M3AS) 14, 603 - 618, 2004 This paper analyses finite element approximations of space averaged flow fields which are given by filtering, i.e. averaging in space, the solution of the steady state Stokes and Navier-Stokes equations with a differential filter. It is shown that $\|\overline{\bu} -\overline{\bu^h}\|_{L^2}$, the error of the filtered velocity $\overline{\bu}$ and the filtered finite element approximation of the velocity $\overline{\bu^h}$, converges under certain conditions of higher order than $\|{\bu} -{\bu^h}\|_{L^2}$, the error of the velocity and its finite element approximation. It is also proved that this statement stays true if the $L^2$-error of finite element approximations of $\overline{\bu}$ and $\overline{\bu^h}$ is considered. Numerical tests in two and three space dimensions support the analytical results. Equilibrium Navier-Stokes equations A. Dunca, V. John, W.J. Layton, "Approximating Local Averages of Fluid Velocities: the equilibrium Navier-Stokes equations" , Appl. Numer. Math. 49, 187 - 205, 2004 In the approximation of higher Reynolds number flow problems, the usual approach is to seek to approximate suitable velocity averages rather than the pointwise fluid velocity itself. We consider an approach to this question wherein the averages are local, spatial averages computed with the Gaussian filter (as in large eddy simulation) and the averages are approximated without using either turbulent closure models or wall laws. The approach we consider is a (underresolved) direct numerical simulation followed by postprocessing to extract accurate flow averages. A priori and a posteriori estimates are given for $\| g_\delta\ast(\bu-\bu^h)\|_0$ which can give guidance for the coupling between the averaging radius $\delta$ and the mesh width $h$. Numerical experiments support the error estimates and illustrate the adaptive grid refinement procedure. < Stokes equations V. John, W.J. Layton, Approximating Local Averages of Fluid Velocities: The Stokes Problem" , Computing 66, 269 - 287, 2001 As a first step to developing mathematical support for finite element approximation to the large eddies in fluid motion we consider herein the Stokes problem. We show that the local average of the usual approximate flow field $\1^h$ over radius $\delta$ provides a very accurate approximation to the flow structures of $O(\delta)$ or greater. The extra accuracy appears for quadratic or higher velocity elements and degrades to the usual finite element accuracy as the averaging radius $\delta \rightarrow h$ (the local meshwidth). We give both a priori and a posteriori error estimates incorporating this effect.

Boundary Conditions

 Volker John, Anastasios Liakos, "Time dependent flow across a step: the slip with friction boundary condition" , Int. J. Numer. Meth. Fluids 50, 713 - 731, 2006 The paper studies numerically the slip with friction boundary condition in the time-dependent incompressible Navier-Stokes equations. Numerical tests on two- and three-dimensional channel flows across a step using this boundary condition on the bottom wall are performed. The influence of the friction parameter on the flow field is studied and the results are explained according to the physics of the flow. Due to the stretching and tilting of vortices, the three-dimensional results differ in many respects from the two-dimensional ones. V. John, W.J. Layton, N. Sahin. "Derivation and Analysis of Near Wall Models for Channel and Recirculating Flows" , Comput. Math. Appl. 48, 1135 - 1151, 2004 The problem of predicting features of turbulent flows occurs in many applications such as geophysical flows, turbulent mixing, pollution dispersal and even in the design of artificial hearts. One promising approach is large eddy simulation (LES), which seeks to predict local spacial averages $\ov{\bu}$ of the fluid's velocity $\bu$. There are several core difficulties in LES. Closure models are very important in applications in which the equations must be integrated over a long time interval. In engineering applications, however, often the equations are solved over moderate time intervals and the core difficulty is associated with modeling near wall turbulence in complex geometries. Thus, one important problem in LES is to find appropriate boundary conditions for the flow averages which depend on the behavior of the unknown flow near the wall. Inspired by early works of Navier and Maxwell, we develop such boundary conditions of the form $$\ov{\bu} \cdot \bn = 0 \mbox{ and } \beta(\delta,Re,|\ov{\bu} \cdot \btau|) \ov{\bu}\cdot \btau +2Re^{-1}\bn \cdot \D(\ov\bu) \cdot \btau = 0$$ on the wall. We derive effective friction coefficients $\beta$ appropriate for both channel flows and recirculating flows and study their asymptotic behavior as the averaging radius $\delta\to 0$ and as the Reynolds number $Re \to \infty$. In the first limit, no-slip conditions are recovered. In the second, free-slip conditions are recovered. V. John "Slip with friction and penetration with resistance boundary conditions for the Navier-Stokes equations - numerical tests and aspects of the implementation" , J. Comp. Appl. Math. 147, 287 - 300, 2002 We consider slip with friction and penetration with resistance boundary conditions in the steady state Navier-Stokes equations. This paper describes some aspects of the implementation of these boundary conditions for finite element discretizations. Numerical tests on two and three dimensional channel flows across a step using the slip with friction boundary condition study the influence of the friction parameter on the position of the reattachment point and the reattachment line, respectively, of the recirculating vortex.

Computational Results

 Philipp W. Schroeder, Volker John, Philip L. Lederer, Christoph Lehrenfeld, Gert Lube, Joachim Schöberl On reference solutions and the sensitivity of the 2d Kelvin-Helmholtz instability problem, Computers and Mathematics with Applications 77, 1010 - 1028, 2019 Two-dimensional Kelvin--Helmholtz instability problems are popular examples for assessing stabilized discretizations or turbulence models for incompressible flows. 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 are obtained for several 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 is provided based on the theory of self-organization of 2d turbulence. Volker John, Adela Kindl Numerical Studies of Finite Element Variational Multiscale Methods for Turbulent Flow Simulations , Comput. Meth. Appl. Mech. Engrg. 199, 841 - 852, 2010 Different realizations of variational multiscale (VMS) methods within the framework of finite element methods are studied in turbulent channel flow simulations. One class of VMS methods uses bubble functions to model resolved small scales whereas the other class contains the definition of the resolved small scales by an explicit projection in its set of equations. All methods are employed with eddy viscosity models of Smagorinsky type. The simulations are performed on grids for which a Direct Numerical Simulation blows up in finite time. Volker John and Adela Kindl, Variants of Projection-Based Finite Element Variational Multiscale Methods for the Simulation of Turbulent Flows , Int. J. Numer. Meth. Fluids 56, 1321 - 1328, 2008 Some variants of a three-scale projection-based finite element variational multiscale (VMS) method are studied for turbulent channel flow computations at $Re_\tau = 180$. Different spaces for the large scales, two eddy viscosity models and two ways of discretizing the projection terms in time are explored. The results obtained with the resolved small scales in the definition of the eddy viscosity are very sensitive to the temporal discretization of the projection terms. The computations were performed on three grids commonly used in turbulent channel flow simulations. Volker John and Michael Roland, Simulations of the Turbulent Channel Flow at $Re_{\tau} = 180$ with Projection-Based Finite Element Variational Multiscale Methods , Int. J. Numer. Meth. Fluids 55, 407 - 429, 2007 Projection-based variational multiscale (VMS) methods, within the framework of an inf-sup stable second order finite element method for the Navier-Stokes equations, are studied in simulations of the turbulent channel flow problem at $Re_\tau = 180$. For comparison, the Smagorinsky Large Eddy Simulation (LES) model with van Driest damping is included into the study. The simulations are performed on very coarse grids. The VMS methods give often considerably better results. For second order statistics, however, the differences to the reference values are sometimes rather large. The dependency of the results on parameters in the eddy viscosity model is much weaker for the VMS methods than for the Smagorinsky LES model with van Driest damping. It is shown that one uniform refinement of the coarse grids allows an underresolved Direct Numerical Simulations (DNS). V. John. "An assessment of two models for the subgrid scale tensor in the rational LES model" , J. Comp. Appl. Math. 173, 57 - 80, 2005 LES models seek to approximate the large scales of a flow which are defined by a space average $(\fil{\bu},\fil{p})$ of the velocity $\bu$ and the pressure $p$ of the flow. A natural question which arises is~: Given reliable data for $(\fil{\bu},\fil{p})$, how accurate is the approximation of $(\fil{\bu},\fil{p})$ by the solution computed with a LES model~? This paper presents numerical studies of this question at a 2d and 3d mixing layer problem for the rational LES model with two types of models for the subgrid scale tensor: the Smagorinsky model and a model proposed by Iliescu and Layton. Whereas in the 2d mixing layer problem the model by Iliescu and Layton showed better results, the behaviour of both models was similar in the 3d mixing layer problem. movies of the numerical simulations T. Iliescu, V. John, W. J. Layton, G. Matthies and L. Tobiska , "A numerical study of a class of LES models ", Int. J. Comput. Fluid Dyn. 17(1), 75 - 85, 2002 This paper tests if two related types of LES models satisfy some simple necessary conditions for acceptability: replication of laminar flows and boundedness of total kinetic energy. The considered LES models are based on the approximation of the Fourier transform of the Gaussian filter by a simpler function. One uses a Taylor polynomial approximation (Taylor LES model), whereas the other model is obtained by a rational approximation (rational LES model). The numerical experiments at high Reynolds number 2D and 3D driven cavity flows show a blow up of the total kinetic energy of the solutions computed with the Taylor LES model. The details of the calculations and the review of this model's derivation point to this blow up being clearly a shortcoming of the model. In contrast, the rational LES model gives solutions with bounded total kinetic energy. In addition, the large eddies are well captured on a coarse grid.