Numerical studies of finite element variational multiscale methods for turbulent flow simulations

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Abstract

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.

Introduction

The simulation of many processes in nature and industry requires as a subtask the simulation of turbulent incompressible flows. Incompressible flows are modeled by the incompressible Navier–Stokes equations, which read in dimensionless formut-2ν·D(u)+(u·)u+p=fin(0,T)×Ω,·u=0in[0,T]×Ω,u(0,x)=u0inΩ,where u(t,x) denotes the fluid velocity, D(u)=(u+uT)/2 is the velocity deformation tensor, p(t,x) is the fluid pressure, T is the simulation time, and ΩR3 is a spatial domain. The parameter ν>0 is the kinematic viscosity, and f(t,x) represents the external body force (e.g. gravity). The initial velocity field u0 is assumed to be divergence-free. The equations (1) have to be closed with appropriate boundary conditions.

Turbulent flows are characterized by a wide spectrum of sizes of the flow structures (scales) ranging from large ones to very small ones. In general, most of the small scales cannot be even represented on grids of the underlying discretization of the Navier–Stokes equations. Consequently, these scales cannot be simulated. However, they are essential for the turbulent character of the flow (energy cascade) and neglecting them in the simulations would lead to laminar results (which are of course wrong). The difficult task consists in modeling the influence of these unresolved (small, fine) scales onto the resolved scales, which is called turbulence modeling. There are many ideas and approaches for turbulence modeling, to mention only two popular ones, the kε model [1] and the (traditional) Large Eddy Simulation (LES) [2]. A comparatively new approach are variational multiscale (VMS) methods which posses similarities but also fundamental differences to the traditional LES.

VMS methods are based on general ideas for the simulation of multiscale phenomena from [3], [4]. The first presentation of these ideas in connection with turbulent flows can be found in [5] and the first numerical results were published around the same time in [6], [7]. Meanwhile, one can distinguish several classes of VMS methods. However, they are all based on two fundamental features:

  • VMS methods use a variational form of the underlying equation which is formulated in appropriate function spaces.

  • The scales are defined by projections into subspaces.

These two features distinguish VMS methods from traditional LES approaches. Traditional LES methods are based on a strong form of the underlying equation and the large scales are defined by spatial averaging (filtering). This way of defining the large scales leads to difficulties in the rigorous mathematical analysis of LES models for flows in bounded domains since always additional terms arise from the necessary commutation of the filter operator and the spatial differential operators. These additional terms are neglected in practice resulting in so-called commutation errors. It has been shown analytically that these errors are not negligible on the boundary and in a vicinity of the boundary [8], [9], [10], [11]. We think that the definition of scales in the way of VMS methods is more appropriate, at least from the mathematical point of view. For instance, in [12], [13] the commutation between the projection operator defining the large scales in certain VMS methods and spatial differential operators has been shown.

Apart from the two fundamental properties given above, the available proposals of VMS methods are quite different. We will concentrate here on finite element VMS (FEVMS) methods. VMS methods based on other discretizations can be found, for instance, in [6], [7], [13], [14], [15], [16]. The FEVMS method proposed in the pioneering paper [5] is based on a two-scale decomposition of the flow field into large scales (u¯,p¯) and small scales (uˆ,pˆ). A finite dimensional space is used for defining the large scales and consequently, the dimension of the space for the small scales is infinite. After the decomposition of the flow field, a variational form of the Navier–Stokes equations (1) can be written as a coupled system of two sets of equations, one with the test functions from the large scale space and the other one with the test functions from the small scale space. The second set consists of infinitely many equations. In [5], for finite element methods, it was proposed to approximate the infinite dimensional small scale space with local, higher order polynomials, so-called bubble functions, which model the so-called resolved small scales. In addition, the influence of the scales which are not resolved by the bubble functions is modeled with an eddy viscosity model. This eddy viscosity model acts directly only on the bubble functions. To our knowledge, numerical simulations with this form of a FEVMS method cannot be found so far in the literature. This method will be studied in the present paper and its shortcomings will be addressed.

In [17], [18], [19], a FEVMS method based on the initial proposal from [5] was developed and studied. In addition to using bubble functions for the resolved small scale velocity and an eddy viscosity model which accounts for the influence of the unresolved scales onto the resolved small scales, the resolved small scale pressure is modeled by the residual of the large scale continuum equation, see [20] for a motivation of this modeling in the context of the Stokes equations. This model leads to a well-known stabilization term in the large scale equations, the so-called grad–div stabilization [21], [22]. FEVMS methods of this type are included in the numerical studies presented in this paper. In Sections 2.2 The bubble-based FEVMS method, 2.3 The bubble-based FEVMS method with grad–div stabilization a derivation of the bubble-based FEVMS methods is presented which uses the bubble finite element spaces from the beginning. This way, the bubble-based FEVMS methods are derived on the basis of a three-scale decomposition of the flow field into large, resolved small and unresolved scales [23].

There is another class of FEVMS methods in which the scale separation relies on the polynomial degree of finite element functions [24], [25]. In [24], continuous finite elements were used with a hierarchical basis whereas in [25] discontinuous finite elements were applied. In both FEVMS methods, the large scales are defined by the low order polynomials and the resolved small scales by the higher order polynomials.

A proposal of a FEVMS method based on a three-scale decomposition of the flow field can be found in [12]. A main feature of this method is the use of standard finite element spaces for all resolved scales. The separation of the large scales and the resolved small scales is achieved with an additional large scale space. An equation defining the projection into this space is explicitly contained in this method. This projection-based three-scale FEVMS method adds an additional viscous term for the resolved small scales to the finite element momentum equation of the Navier–Stokes equations. The parameter in this term is generally chosen to be an eddy viscosity model of Smagorinsky-type [12], [26], [27]. The ideas of this projection-based VMS method can also be applied in the framework of finite volume discretizations [13]. Some variants of the projection-based FEVMS method are considered in the numerical studies, see Section 2.4.

The philosophy of the three-scale VMS methods resembles the philosophy of the dynamic Smagorinsky LES model [28], [29]. In this LES model, the parameter in the Smagorinsky eddy viscosity term is adjusted dynamically to reduce the viscosity of the model in regions were it is not needed. In the three-scale VMS methods, the reduction of the viscosity is achieved by applying the Smagorinsky model only to the resolved small scales, whose definition depends on the current solution.

The last VMS method which will be mentioned in some detail further on in the introduction was proposed quite recently in [30]. It is based on a two-scale decomposition of the flow field achieved via a projection operator [31]. By writing a variational form of the Navier–Stokes equations as a coupled system with test functions from the large and the small scale subspaces, respectively, it is readily shown that the small scales can be formally represented as an unknown functional of the residual of the large scales. To approximate this functional, the small scales are expressed as a perturbation series where the perturbation parameter is an appropriate norm of the large scale residual. For the unknown terms of this perturbation series, a recursive system of equations can be derived where the left hand side in all equations has the form of a linearized Navier–Stokes equations. Thus, the solution of these equations can be formally expressed with an appropriate Green’s operator, the so-called fine-scale Green’s operator. In [31], it has been shown that the fine-scale Green’s operator can be expressed using the classical Green’s operator and the projection defining the scale separation. To obtain a numerical method, in [30] it is proposed to truncate the perturbation series after the first term and to approximate the fine-scale Green’s operator. Altogether, the small scales are modeled in [30] to be proportional to the residual of the large scales. Inserting this model into the equations with the large scale test functions, one obtains a generalization of the well-known streamline-upwind Petrov–Galerkin (SUPG) method for the Navier–Stokes equations. Besides the SUPG term and the grad–div term, additional terms arise which can be interpreted as the cross stresses and the Reynolds stresses known from the traditional LES. The two-scale VMS method from [30] does not use an eddy viscosity model. However, there is some freedom in choosing the parameters in the additional terms. This quite new method is not yet included in the numerical studies presented in this paper.

This paper will present numerical studies at turbulent channel flows at Reτ=180 and Reτ=395. Numerical studies of turbulence models at academic test examples may have different goals. One goal might be to use resolutions (grids) which are sufficiently fine compared with the Reynolds number such that an underresolved Direct Numerical Simulation (DNS) can be performed and to show that statistics of the flows are captured better if a turbulence model is applied. Another goal might be to apply turbulence models on grids where a DNS blows up in finite time. In this situation, the use of a turbulence model is necessary in order to perform any simulations at all. The latter case is more likely in applications and it will be considered in this paper. Thus, simulations on rather coarse grids, compared with the Reynolds number, will be presented, which study and compare bubble-based FEVMS methods and some variants of the projection-based FEVMS method. To our best knowledge, a comparison of these methods is so far not available.

The paper is organized as follows. Section 2 contains a detailed description of all considered FEVMS methods. The numerical tests for the channel flow problems are presented in Section 3. Finally, Section 4 summarizes the conclusions and gives an outlook.

Section snippets

Finite element variational multiscale methods

This section contains a rather detailed presentation of the studied methods. It has been shown for turbulent channel flow problems that even tiny changes in the data or algorithms eventually lead to large changes in the instantaneous flow fields [32]. The changes in mean values and statistics are less significant. Nevertheless, in view of the sensitivity of the turbulent channel flow to small perturbations, it seems important to us to give at least a detailed description of the main algorithms

Numerical studies

We studied the different types of FEVMS methods at the benchmark problems of the turbulent channel flows at Reτ=180 and Reτ=395. The setup of the problems and the reference values have been taken from [45].

Summary and outlook

Two principal different approaches of FEVMS methods were applied at turbulent channel flow simulations on rather coarse grids. Several variants for both approaches were assessed, details of the assessment can be found in Section 3.4.

A main conclusion of the numerical studies is that the application of bubble-based FEVMS methods is quite complicated. It has to be decided which simplifying assumptions for the resolved small scales are permissible, how fine should the local meshes be and which

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    The research of A. Kindl was supported by the Deutsche Forschungsgemeinschaft (DFG) by Grants No. Jo 329/7-1 and No. Jo 329/7-2.

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