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Chapter 3 Large Eddy Simulation Method Chapter 3 Large Eddy Simulation method In this chapter a short view on the present state of the Large Eddy Simulation technique is given, and some of the most used Large Eddy Simulation models are described. 3.1 Introduction Large Eddy Simulation (LES) is a method of predicting turbulent flows which involves the solution of fully time-dependent, three-dimensional flow fields by use of the Navier-Stokes equations. In a LES the flow solution will become physically unstable in the same manner as a real turbulent fluid. The large-scale turbulence as such is not modelled: only eddies smaller than the mesh size need to be represented by a so-called subgrid scale model. LES is capable of producing an enormous amount of potentially valuable information. Since the simulations contain the time-dependent evolution of eddies in three dimensions, statistics are generated that are difficult to obtain from more traditional methods of turbulence prediction, such as closure modelling. At the same time, structural information about the formation, evolution, and the dynamical role of coherent turbulent structures is also present in the simulations ,- information that is necessarily lacking in predictions based on statistical modelling. Thus, LES has the potential to inform of the physical mechanisms arising from the statistics of turbulence. Accurate prediction of the turbulent flows encountered in engineering practice remains the principal challenge of computational fluid dynamics. One of the primary difficulties involves with simulation and modelling is that turbulence is comprised in a wide range of length and time scales. The largest scales of motion are responsible for most of the momentum and energy transport and are strongly dependent on the flow configuration, while the smallest eddy motions tend to depend only on viscosity and are more universal than the large eddies. Large eddy simulation (LES) is a technique rapidly emerging as a viable approach for the prediction of complex turbulent flows. In LES the contribution of the large, energy- containing scales of motion is computed directly, and only the effect of the smallest scales of turbulence is modelled. Since the small scales are more homogeneous and less affected by the boundary conditions 25 CHAPTER 3 LARGE EDDY SIMULATION METHOD than the largest eddies, it is possible to model them by using simpler models than those required in other techniques for turbulence modelling. LES was originally developed because a direct solution of the equations describing the transport of mass, momentum, and energy, the well-known Navier-Stokes equations, is beyond the capacity of even the largest supercomputers for most flows of engineering interest. It is possible to characterize the computational cost of a direct simulation in terms of the Reynolds number, Re, which is the ratio of inertial to viscous forces and which characterizes the state of fluid motion. This was already depicted in the previous chapter. In LES transport equations are derived for the large eddies by spatially filtering the Navier-Stokes equations. The application of a filter to a representative turbulent variable is shown in Fig. 3.1. As shown in the figure, variations of the fluctuating variable occurring on long wavelengths (i.e., larg scales) are preserved, while the shorter wavelength variation is removed by the filter. The mean value of the velocity fluctuations are retained by following the filtering operation, but other statistics, e.g., cross correlations, will differ between the filtered and velocity fluctuations due to the absence of the small-scale contribution to the filtered velocities. However, in turbulent flows many correlations of interest for engineering prediction are mostly influenced by the large scales of motion and the loss of information arising from the filtering operation is typically not large. u(x) _ _ u ui+1 i-1 _ _ ui+2 ui (x) u i-1 i i+1 i+2 x xxxxi-1 i i+1 i+2 Figure 3.1: Filtered velocity (grey steps) computed by spatially filtering the velocity fluctuations spatial (solid black line) on a fine uniform grid. The filtering procedure will lead to turbulent stresses which must be modelled. In LES the model must account only for the effect of those motions not resolved by the computational mesh, known as the subgrid scales. Since the small eddies tend to be more homogeneous and respond more rapidly to external perturbations than the large scales, it will be possible to model their effect on the resolved motions by use of relatively simple approaches. In fact, the vast majority of models used in LES are simple algebraic expressions in which a so-called `eddy viscosity' is introduced to relate the turbulent stresses to the strain rate of the large eddies. 26 CHAPTER 3 LARGE EDDY SIMULATION METHOD The main limitation of LES has traditionally been the inability of models to account for changes in turbulence structure without arbitrary adjustment. Dynamic modelling of the turbulent stresses, introduced in 1991 by Germano, represents a new approach since rather than being prescribed as input in advance the model coefficients will be computed during the course of the calculation. The dynamic models are fundamentally different from other approaches, since the models reflect local properties of the flow and are sensitive to changes in external conditions. LES and dynamic models have now been applied to prediction of a wide array of geometrically simple turbulent flows, yielding agreement with experimental measurements as being good as or better than other approaches requiring arbitrary adjustment. The focus of current applications of LES is given to flows subject to the complicating effects encountered in practice. These include strong pressure gradients as in variable-area ducts, streamline curvature, flow separation from a surface, and flows in which the mean velocity profile is three-dimensional. These effects severely distort the turbulence structure and are a stringent test for models. In addition to complications arising from these physical effects, recent applications have also addressed numerical issues important for advancing LES as a predictive technique for complex flows. For further information on these subjects, see Breuer (1998), Chollet et al. (1997), Liu and Liu (1997), Piomelli (1999), Piomelli (1996), Rodi et al.(1996), Voke et al. (1996) and Tafti (1996), to name a few. Turbulent flows of engineering interest exhibit spatial growth, and it is therefore necessary to prescribe the inlet boundary of the computational domain with a realistic, time-dependent description of the dependent variables. Recent work has shown that it is optimal to prescribe turbulent inlet conditions which are themselves are solutions to the Navier-Stokes equations. This can be accomplished by performing a separate calculation in which the results are subsequently fed into the inlet boundary of the main simulation. In the next section the filtering operation for the Navier-Stokes equations, followed by the subgrid scale models will be discussed. 3.2 The filtered Navier-Stokes equations Historically, a general definition of the LES filtering was first introduced by Leonard (1974). Specifically, the flow variable is decomposed into large (filtered) and sub-grid (residual) scales as follows: (3.1) The large (resolve) scale is defined as: (3.2) k where the integral is applied for the whole flow domain, D, x i and x i are space vectors, and G j is the general filter function in three dimensions. Apparently, the effect of G j is to remove the small scale 27 CHAPTER 3 LARGE EDDY SIMULATION METHOD fluctuations from u i in forming the large scale and then is the resolved subgrid scale field variable. If the variable is constant, the filter function must satisfy the constraint of (3.3) as can been seen Eqn.(3.2). Meanwhile, if the filter function is homogenous, i.e. the following convolution relation holds after transforming Eqn.(3.2) into Fourier wave space, (3.4) where k j is the wave number in three dimensions, and is the transformed value of the filtered variable in the wave space, etc. For a uniform filter function G j (if the filter width is constant) with properties of piecewise, continuously differentiable and bounded, it can be shown that the filtering and differentiation operators are commutative. This means that . But in general, the filtered variable is not necessarily constant. This is in contracts to the Reynolds averaging approach where the averaged (filtered) variable is constant (independent of time) thus resulting from the infinite time averaging procedure. The averaged variable in this approach carries all the turbulence information in a single scale. Usually, any form of filter function that is homogenous can be used to filter the Navier-Stokes equations. However, only three different forms of the filter functions have been used: namely the spectral cut-off, the Gaussian and the top hat filter function, because of the constraints of the numerical schemes employed. The Spectral cut-off filter: The spectral cut-off filter is defined in the physical space as: (3.5) F _ where the filter width, j is / k c and k c is the cut-off wave number. The filter’s counterpart in the Fourier space is: (3.6) 28 CHAPTER 3 LARGE EDDY SIMULATION METHOD From its definition in Fourier space, this filter will translate all information with a wave number greater than k c. By substituting Eqn. (3.6) into Eqn. (3.4) it will easy to deduce that using the spectral method with a finite node resolution is equivalent to employing a spectral cut-off filter function. This filter is therefore is implicitly used in the spectral method, and the cut-off wave number k c is the largest wave number resolved. The Gaussian filter function: The Gaussian filter function which has a Gaussian distribution by definition is given as: (3.7) and its Fourier transformed equivalent is: (3.8) In contrast to the cut-off filter which has a discontinous spectrum, the Gaussian filter is continous across the whole wave space (but it diminishes exponentially to zero).
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