Large Eddy Simulation of Turbulent Incompressible Flows

Turbulent flows occur in many processes in nature as well as in many industrial applications. A storm, for instance, is accompanied with a wind which has a high velocity and can change its direction and also its speed abruptly. The take-off of an aircraft leads to turbulent air above the runway which has to be calmed down before the start of the next aircraft. This necessary interval between the take-offs of the aircrafts limits the number of starts in a given time period. There are many more examples of turbulent flows, e.g., in oceanography and in wind channels for the design of cars. Often, turbulent flows go along with chemical reactions like in gas engines. A wide variety of other examples for turbulent flows can be found, e.g., in the book by Lesieur [Les97, Chapter I].

This chapter provides the mathematical tools which are used in this monograph and introduces basic notations. For a concise presentation of this preliminary material, theorems and inequality are given without proofs.

To compute the space averaged velocity ū and pressure p̄, equations for these quantities are needed. These equations have to be derived from the governing equations for u and p̄, i.e. from the Navier-Stokes equations. The simple approach consists in applying the filter which defines (ū, p̄) also to the Navier-Stokes equations. Then, under the assumption that differentiation and filtering commute, the basic equations of LES, the space averaged Navier-Stokes equations, are obtained. However, it turns out that an additional modelling step is necessary to derive equations for (ū, p̄) from the space averaged Navier-Stokes equations. This modelling step is discussed in Chapter 4.

In this chapter, models for the tensor uu ^T in the space averaged Navier-Stokes equations (3.11) are derived, where the aim is to model uu ^T in terms of (ū, p̄). As mentioned in Remark 3.1, the entries of this tensor are a priori not related to (ū, p̄).

We have seen in the previous chapters that it was not possible to derive equations for (ū, p̄) only from the Navier-Stokes equations since a modelling process was also necessary. Thus the quantities which will be computed using these models will not be (ū, p̄) but, hopefully good, approximations to (ū, p̄). To have a clear distinction between the large scale quantities (ū, p̄) and their approximations, we will denote the solution obtained by the LES models by (w, r).

This chapter presents analytical investigations of the existence and uniqueness of solutions of the LES models (5.1) in a bounded domain Ω. Since Ω is bounded, (5.1) has to be equipped with boundary conditions. The analysis presented in this chapter uses homogeneous Dirichlet boundary conditions. This is just for simplicity of presentation, extensions to other boundary conditions are possible, see Remark 6.16.

This chapter deals with the discretisation of the LES models of the Navier-Stokes equations where A is given by the approximation of the Fourier transform of the Gaussian filter, v _T is the turbulent viscosity and to simplify the notations we use f instead of \( \bar f \). System (7.1) has to be completed with boundary conditions. Depending on the boundary conditions, the additive constant of the pressure has to be fixed, see Section 5.3.

This chapter presents an error analysis of time-continuous finite element discretisations of the Smagorinsky model and the Taylor LES model.

The discretisation and linearisation of the LES models, described in Chapter 7, lead to linear systems of the abstract form with non-symmetric matrix A. These systems have to be solved in each step of the fixed point iteration (7.7) for each sub time step. The solution of these large number of systems of form (9.1) is the most time consuming part of the computations. That’s why, the solver applied to (9.1) is one of the most important components for the efficiency of the numerical simulations with the LES models.

In this section, we present a numerical study at a simple test problem which investigates if the LES models considered in this monograph fulfil or violate a condition which is in our opinion necessary for the acceptability of LES models.

Numerical studies of LES models which can be found in the literature try, in general, to simulate a turbulent flow as good as possible and to compare the numerical results with statistics of the flow field known from experiments or DNS data. However, the studies presented in this chapter address a different question. The main purpose of LES models is to provide an accurate approximation of (ū, p̄). A natural and very important question is : How good is the approximation of (ū, p̄) by the flow field computed with LES models? This is a fundamental question for each LES model. The performance of numerical tests studying this question requires reliable data for (ū, p̄) in time and space. Section 11.1 studies the above formulated question for the LES models considered in this monograph in a situation where reliable data for (ū, p̄) can be computed, namely a 2d mixing layer problem at Re = 10000. Section 11.2 presents numerical tests for a mixing layer problem in three dimensions and Re = 714. A comparison with filtered DNS data cannot be presented since it was not possible to perform a DNS in the three dimensional example.

This chapter contains a number of unresolved problems for the LES models considered in this monograph and topics for further investigations. The solution of each of the problems would improve the understanding of either the LES models or of the numerical algorithms used in the computations. The following list does not claim to be complete.