Statistical behavior of the orthogonal subgrid scale stabilization terms in the finite element large eddy simulation of turbulent flows

Highlights

• The OSS stabilization terms may be an alternative to large eddy simulation models.

• Their statistical behavior has been analyzed for isotropic turbulence.

• Velocity and pressure correlations from statistical fluid mechanics have been used.

• The stabilization terms show the appropriate behavior in the inertial subrange.

Abstract

Numerical simulations have proved that Variational Multiscale Methods (VMM) perform well as pure numerical large eddy simulation (LES) models. In this paper we focus on the orthogonal subgrid scale (OSS) finite element method and make an analysis of the statistical behavior of its stabilization terms in the quasi static approximation. This is done by resorting to results from classical statistical fluid mechanics concerning two point velocity, pressure and combined correlation functions of various orders. Given a fine enough mesh with characteristic element size h in the inertial subrange of a turbulent flow, it is shown that the rate of transfer of subgrid kinetic energy provided by the OSS stabilization terms does not depend on h and that it equals the molecular physical dissipation rate (up to a dimensionless constant that only depends on the finite element shapes) for a proper redesign of the standard parameters of the formulation. This is a noteworthy fact taking into account that the subgrid stabilization terms do not arise from physical considerations, but from the mathematical necessity to allow equal interpolation for the pressure and velocity fields, as well as to control convection. Therefore, the obtained results contribute somehow to the line of reasoning supporting that pure numerical approaches (i.e., without introducing additional physical models) could probably suffice in the LES simulation of turbulent flows.

Introduction

Two parallel lines have been followed in the past years to simulate incompressible turbulent flows that can be of engineering interest. On the one side, the drawbacks of RANS (Reynolds Averaged Navier–Stokes) models combined with the impossibility to perform DNS (Direct Numerical Simulation) computations for large Reynolds number problems led to the development of LES (large eddy simulation) strategies (see e.g., [1]). On the other side, the numerical problems that arise when trying to solve the discrete differential or weak versions of CDR (Convection–Diffusion-Reaction) equations have motivated the development of several stabilization strategies to mitigate them. A landmark in the development of these stabilization methods was the appearance of the subgrid scale stabilization approach or, as originally termed, the variational multiscale method (VMM), in the framework of finite element methods [2], [3]. Both approaches, LES and VMM applied to fluid dynamics, share some features like being based on a scale decomposition of the continuous velocity and pressure fields of the Navier–Stokes equations. However, in the former case this scale separation is performed at the continuous level, whilst in the latter case it is inherently carried out in the discretization process. The relation between both methods is not fully understood at present and it is not clear whether they should be used together or independently in the simulation of turbulent flows. As it will be detailed below, in this paper we aim at gaining some further insight on this connection through the analysis of the statistical behavior of the VMM stabilization terms. In particular, the subgrid scale terms of the quasi-static approximation in the Orthogonal Subgrid Scale (OSS) finite element method will be considered [4].

In conventional LES the scale decomposition between large and small flow scales has been traditionally performed by means of a filtering process (see e.g., [5], [6], [7]) defined through a convolution operation. The filter is applied to the Navier–Stokes equations usually assuming that it commutes with the differential operators and a new equation for the filtered velocity and pressure fields is derived. However, this equation contains the divergence of the so-called residual stress tensor that depends on the exact velocity field. This term has to be modeled somehow to obtain a closed system of equations only depending on filtered quantities. Once a physical LES model is chosen, the resulting filtered equation is finally discretized and solved.

This standard LES approach suffers from several mathematical difficulties such as knowing the error introduced when the filtering and differentiation operators are assumed to commute [8], knowing which should be the appropriate choice for the LES boundary conditions and, what is probably more important, knowing which is the relation between the errors introduced by the physical LES model and by the discretization procedure. Some of these subjects have received recent attention both from analytical (see e.g., [9], [10]) and numerical points of view (see e.g., [11], [12], [13], [14], [15]). In [16] a review of several LES models was performed and some interesting conclusions were drawn out, such as the fact that filtering is not indispensable to achieve LES models, that aiming at an exact closure for the residual stress tensor is a paradoxic program and that some LES models have the remarkable propriety of being more regular than the original Navier–Stokes equations, leading to problems for which uniqueness of solutions can be proved. In this sense, it was concluded that a LES model should fulfill with two main requisites, namely, it should regularize the Navier–Stokes equations yielding to well-posed problems and it should lead to suitable weak solutions (i.e., physically acceptable solutions). In an attempt to provide a first step towards a mathematical definition of LES, the notion of suitable approximations to the Navier–Stokes equations was then introduced in [17]. In this context, it is worthwhile mentioning that a DNS using the Galerkin method with low order finite elements constitutes a suitable approximation to the Navier–Stokes equations, which may justify the fact that sometimes better results are achieved for low-order methods when no LES model is employed [18].

In the VMM or subgrid scale approach to solve turbulent flows, the scale separation is carried out by means of a projection onto the finite element space. Two equations are then obtained respectively governing the dynamics of the large and small scales. Large scales are those that can be captured by the computational mesh, while small or subgrid scales are those not captured by the mesh. Modeling takes place when giving an approximated solution for the subgrid scales equation, which is to be inserted in the large scale equation to account for its effects. One could then view the VMM method as an alternative approach to simulate large eddies in turbulent flows, giving place to pure numerical LES in contrast to traditional or physical LES. However, in what concerns terminology and unless otherwise specified, throughout this work we will generally use the plain acronym LES to refer to conventional or physical LES, which involves filtering and finding a closure for the residual stress tensor. As far as the authors know, the possibility of using a VMM model as a numerical LES approach was originally proposed in [4] and applied successfully for the first time in [19].

It should be noted that the initial motivation of the VMM method was to solve some of the numerical problems associated with the simulation of the discrete Navier–Stokes equations, such as the necessity to satisfy the inf-sup condition (which implies the use of different interpolation spaces for the velocity and pressure fields) or the numerical instabilities appearing for convective dominated flows. Consequently, when the VMM was first applied to the simulation of turbulent flows a physical LES model (Smagorinsky’s model) was still included, although solely acting on the subgrid scale equation [20], [21], [22]. As mentioned above, the idea that the stabilization terms in the VMM approach could be sufficient to simulate turbulent flows was already pointed out in the framework of orthogonal subgrid scale (OSS) stabilization methods [4] (see also [23]) as a natural extension to that work. Later it was re-introduced in [24], [25] and further elaborated in [26]. A quite definite step supporting the approach has been the very good results obtained in the simulation of various turbulence benchmark problems such as isotropic turbulence, turbulent channel flows or surface mounted objects, with the sole use of numerical stabilization (see [19] and also [27], [28], [29], [30]). Actually, to the best of our knowledgment, this “numerical” line of thinking initiated with the MILES (Monotone Integrated LES) approach [31] c.f. [15] (see also [1] and references therein). We shall come back to this point in Section 5.

In this paper we aim at analyzing the relation between pure numerical LES based on the VMM formulation and physical LES from a different perspective. We would like to make use of existing statistics [32] for isotropic turbulence and check how the stabilization terms of the quasi-static OSS method [33], [34], [4] behave according to them. For a fine enough computational mesh so that its characteristic element size h lies in the inertial subrange of a turbulent flow, we make use of the two point correlation functions of various orders for the pressure and velocity fields, as well as for their combination. To do so we assume that the finite element OSS solution is close at the nodes to the interpolant, which is reasonable, so that the statistics for the exact velocity and pressure fields also apply to the OSS solution. The analysis will reveal that the contribution to the energy balance equation from the OSS stabilization terms are proportional to the physical dissipation rate and independent of h to dominant order, for an appropriate redesign of the formulation parameters. Therefore, the OSS method satisfies an important point a closure LES model requires, namely that the rate of kinetic energy transferred from the filtered large scales to the small ones is proportional to the physical dissipation rate at the Kolmogorov length scale (see e.g., [35], [36]). The property described is a noteworthy fact given that the stabilization terms arise in the discrete weak Navier–Stokes equations from purely numerical considerations regarding stability issues, so there is no a priori need for them to behave in the appropriate physical way. Besides, we would like to remark that the forthcoming results are not to be confused with a mere scaling analysis. Many numerical strategies include stabilization terms, which are obviously dimensionally correct, but this does not imply that they can be related to existing turbulence statistics that support their behavior in the inertial subrange.

The paper is organized as follows. In Section 2 the energy balance equations for the continuous Navier–Stokes and LES problems are presented together with their discrete counterparts using the Galerkin and OSS finite element methods. The problem we would like to address is established and the OSS terms accounting for the transfer of kinetic energy to subscales that should be proportional to the physical dissipation rate are identified. In Section 3 we proceed to the explicit discretization of these terms, showing that their ensemble average can be written as products of geometrical factors multiplying two point second and fourth-order nodal velocity correlations, second order pressure correlations and triple-order velocity–pressure correlations. In Section 4, results from statistical fluid mechanics are used to relate these correlations to the physical dissipation rate, which is the main goal of the paper. Some general comments and remarks, together with references to numerical experiments supporting the pure numerical approach to solve turbulent flows are given in Section 5. Conclusions are finally drawn in Section 6.