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.
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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)
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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). This feature has led some researchers to believe that it might offer an advantage over the cut-off filters. However, Piomelli et al. (1988) used numerical experiments which indicated that the filter performance might actually depend on the subgrid scale model employed. It therefore seems that no definite conclusion can be made in this case.
The Top-Hat Filter function: The top-hat filter functions are used implicitly in many finite difference schemes (Rogallo and Moin (1984)) expressed as
(3.9)
And its equivalent form in Fourier space is:
(3.10)
As in the Gaussian filter case, the spectrum of the top hat filter is continue across the whole wave number spectrum. If Eqn.(3.8) is inserted into Eqn. (3.2) the following will be obtained:
(3.11) which is a finite-volume averaging procedure. Here F is the characteristic filter width. In almost all the codes using finite difference, finite volume or finite element the top hat filter functions are used. And the
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other filter function are restricted to spectral codes. If the filter function is applied to the incompressible Navier-Stokes equations the following will be obtained:
(3.12)
(3.13) where the effect of the small scale upon the resolved scales of the turbulence appears in the subgrid scale stress term, , which must be modelled. The unresolved subgrid scale stress tensor can also be rewritten as:
(3.14)
g As expected, i j needs to be modelled in order to close the system of equations, so that the filtered Navier-Stokes equations can be solved numerically. An accurate subgrid scale model can provide a way to account for effects of the unresolved small scale eddies through interactions of the resolved velocity scales. This is the closure problem present in any turbulence modelling approach due to the non-linearity of the Navier-Stokes equations. Traditionally, the three terms on the right hand side of Eqn. (3.13) can be identified as the Leonard stresses (Leonard 1974), the subgrid scale cross stresses, and the subgrid scale Reynolds stresses, respectively. The Leonard term, which signifies the interactions within the resolvable turbulence scales, can easily be calculated, if the spectral method is used, since only a second filtering operation will need to be performed. (Unfortunately, this will not be quite as simple if the finite difference method is employed instead.) Therefore, only the cross term and the residual stress terms need to be modelled in general. But Speziale (1985) has shown that while modelling, the three terms together would satisfy the Galilean invariance property that all physical laws should be invariant with respect to an inertial coordinate transformation, and special care will be needed if separate models are to be used. For instance, the eddy viscosity model of Moin and Kim (1982) is not Galilean invariant if the Leonard term is calculated explicitly. In the past, different strategies have been used to model the physics inherent for the two separate terms and the Leonard term was calculated explicitly with the spectral method, or approximated with a Taylor series expansion by use of the finite difference method (Leonard, 1974). Nonetheless, the advantages of separating the subgrid scale stresses [i.e., Eqn. (3.13)] over the single term treatment
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[i.e. Eqn. (3.12)] are not yet unequivocally proven. In addition, while Leonard (1974) observed that the Leonard term accounts for a significants energy drain from the large to the small scales in isotropic turbulence, Antonopoulos-Domis (1981) came to the opposite conclusion, namely that the Leonard term represents the reverse of the energy cascade process, i.e. a turbulent energy source instead of a drain, in a homogeneous turbulence simulation. It is not clear why the discrepancy exists. But for these reasons, the present research has used only the one model approach, i.e., to parameterize Eqn. (3.12) with a single model.
3.3 The Smagorinsky subgrid scale model As mentioned in Chapter 1, the motivation for LES arise from the experimental observations that the small scales which need to be modelled are more isotropic and universal. Therefore, a simple closure model, such as the eddy viscosity type should suffice to provide adequate information on the small scale physics under different flow conditions. Unfortunately, the past experience has indicated that a suitable choice of the subgrid scale model is still critical to ensure a successful LES performance, and the generality of simple models is not as great as initially hoped. In his simulation of the channel flow, Deardorff (1970) used the Smagorinsky eddy viscosity model (Smagorinsky, 1963) and found that although the model was adequate for the channel flow simulation, the model constant needed to be adjusted to achieve optimum results for sustaining the turbulence energy. Ever since then, most research efforts have focused on improving the eddy viscosity SGS model (Schumann 1975; Clark et al. 1979; Bardina et al. 1983), or going to a more complicated closure like the one-equation model (Horiuti 1985). Deardorff (1973) even tried to solve the SGS model transport equation involving 10 partial differential equations, similar to the procedure used for the Reynolds stress transport model. This 10-equation model still did not easily produce “input-free” results, at least for the channel flow case. Besides, resorting to complicated models would seem to defeat the purpose of LES, not to mention the extra computational costs involved. Therefore, in the present work only the Smagorinsky model and its variants will be discussed. The SGS eddy viscosity model of Smagorinsky (1963) is given as
(3.15) i where T is the subgrid scale eddy viscosity and the strain rate tensor is given as:
(3.16)
g The term k k in the left hand side of Eqn. (3.14) is the unresolved subgrid-scale turbulent kinetic energy. g Since the rate of strain tensor Si j is trace-free in the incompressible case, i.e., S 11 +S 22 +S 33 = 0, the kk term is needed so that when the indices in Eqn. (3.14) are contracted, both sides of the equation will decline to zero. In incompressible flow simulations, the kinetic energy term can be combined with the
31 CHAPTER 3 LARGE EDDY SIMULATION METHOD
pressure and will have no dynamic effect; no parameterization will therefore be needed for the term. To close the model, the subgrid scale turbulent energy production and dissipation are equated to obtain
(3.16)
and the subgrid scale length scale given by . Cs is the model constant yet to be determined, and F is the filter width which is a function of the grid resolution. The filter width is typically taken as the width of the numerical grid spacing in the finite difference/volume context so that when the grid is refined, the effect of the SGS model will diminish accordingly.
(3.17)
The model constant, C s, can vary in different flow regimes. For homogeneous isotropic turbulence,
Lily calculated C s to be 0.23 (see Germano et al., 1991). For the plane channel flow, Deardorff (1970) found 0.1 to be the optimal representation. However, in the inner wall region (i.e. the viscous sublayer and the buffer layer), the turbulent energy budget is no longer dominated by the dissipation and production alone, due to the highly anisotropic nature of the wall-bounded turbulence. Further, the turbulent kinetic energy will decrease towards the wall until a viscous region is approached (where the flow is no longer turbulent). The eddy viscosity model is consequently not valid in that region. In an effort to account for the anisotropy, Moin and Kim (1982) have proposed to modify the length scale by including a van Driest type wall damping function as
(3.18)
F F F + where x , y and z are the filter width in the x, y and z directions, respectively, and A is the equivalent van Drist constant (= 25). Another modified form of the length scale was proposed by Piomelli et al. (1988) as:
(3.19)
A closer look at this modification of the wall function reveals that the length scale near the wall 3 behaves like y+ which was particularly chosen to match the asymptotic scale of the Reynolds shear stress. This observation is supported by DNS results (Kim et al., 1987). With the above modifications, the length scale will diminish towards zero as the wall is approached. Nevertheless, numerical experiments have indicated that often the modified length scales are too dissipative in the near-wall region, thus suppressing the kinetic energy level and causing under-prediction of the wall shear stress. In addition,
32 CHAPTER 3 LARGE EDDY SIMULATION METHOD modifications of this kind are expected to be somewhat problem dependent, especially when the flow geometry is complex.
3.4 The Scale Similarity subgrid scale model Scale-similar models were introduced by Bardina et al. (1982), who assumed that the component of the subgrid scale which most active in the energy transfer from large to small scales can be estimated with sufficient accuracy from the smallest resolved scales, which can be obtained by filtering the subgrid scale velocity given in Eqn.(3.2) as:
(3.20)
They then made the following assumptions:
(3.21)
(3.22) to yield a model of the form (3.23)
where Cs s is a model coefficient that must be set to one to recover the Galilean invariance (Special, 1995).
3.5 The Dynamic subgrid scale model The aforementioned deficiency in the Smagorinsky SGS model (and its variants) has led researchers to look for better alternatives. Among them, the dynamic SGS model proposed by Germano et al. (1991) has sparked renewed interest in LES research due to its ease of use and dynamic nature. In other words, the dynamic model attempts to remedy some of the shortcomings in the Smagorinsky model by allowing the model coefficient to be calculated locally by use of two levels of spectral information already present in the numerical calculation. Since the model coefficient formulated becomes a function of space and time and may become negative, it can account for the backscatter of the turbulent energy. Additionally, the coefficient will automatically decrease towards a solid wall and achieve a zero value when the flow is laminar, thus eliminating the need for an ad hoc treatment of the model constant to adapt to the flow situations (e.g. the transitional flow). So far, the dynamic model and its extension have been successfully applied to many different types of turbulent flows, e.g., transitional channel flow (Piomelli et al., 1991), compressible isotropic turbulence (Moin et al., 1991), passive scalar channel flow (Cabot and Moin, 1993),
33 CHAPTER 3 LARGE EDDY SIMULATION METHOD
the backward facing-step flow (Akselvoll and Moin, 1993), the flow over a 2D obstacle inside the channel (Yang and Ferziger, 1993), 3D cavity flow (Zang et al., 1993), a co-swirling jet (Olsson and Fuchs, 1994), a longitudinal vortex (Sreedhar and Ragab, 1994), and the compressible transitional boundary layer (El-Hady et al., 1994). Among these works, it is worth to mentioning that reasonably good results were also obtained by use of the dynamic model for the backward-facing step flow by Akselvoll and Moin (1993) whereas coarse-grid DNS fails to reproduce the same quality of results. Similar good results were obtained for the transitional channel by use of the dynamic model, whereas the results obtained by use of the traditional Smagorinsky model were not quite satisfactory. For a recent review on the progress of the dynamic SGS model, see Moin and Jimenéz (1993), Piomelli and Chasnov (1996), Piomelli (1998) and Piomelli (1999). To derive the dynamic SGS model (following that of Germano et al., 1991), besides the filtering operation of Eqn. (3.2), we define the “test” filter variables are defined in a similar fashion.
(3.24) where (™) defines the test filter operator, and is any test filter function that operates at any scale larger than the original grid filter function G j. Similar to the procedure used for Eqn. (3.13), the test-filtered turbulent stress tensor resulting from the test-filtered Navier-Stokes equations is represented by
(3.25)
g So far, both i j and T i j are unknown, but they are related by the algebraic identity of Germano (1992): (3.26) where , identified as the test-level Leonard term. It can be easily shown that the term is computable from the resolved field once the test filter is defined. At last, it is assumed that the eddy viscosity model applies to both the grid filter and test filter levels, resulting in the following expressions:
(3.27) and
(3.28)
g where the terms k k and T k k are the turbulent kinetic energy at two different filter levels, respectively. A substitution of Eqn’s. (3.23) and (3.24) into the algebraic identity of Eqn. (3.21), will gives
(3.29)
34 CHAPTER 3 LARGE EDDY SIMULATION METHOD where
(3.30) and is the test grid filter ratio.
Since the only unknown factor in Eqn. (3.24) is the dynamic coefficient C d , it can be determined by solving the tensor equation. But because the above equation is of a symmetric tensor type, i.e., L ij and
M ij are symmetric (and additionally Mkk = 0), there are in fact five independent equations to determine a single unknown, the model coefficient C d . Apparently, the problem is over-specified. Lilly (1992) used a least-squares approach to minimize the error in calculating the coefficient. The result were
(3.31)
The above formula is fully space dependent. However, numerical experiments have indicated that often, the model coefficient has large positive or negative spikes numerically distributed around the flow domain (i.e., the denominator term can become small), thus making the numerical calculations unstable. To remedy this situation, a suitable averaging is further defined. In the present calculations, the following modified formula was found acceptable:
(3.32)
À Ä where Z denotes plane averaging in the homogeneous direction(s). As a result, the coefficient is reduced to a function of inhomogeneous direction(s), only. As readily seen, the dynamic SGS model is basically also an eddy viscosity type, but with the model coefficient computed locally instead of given empirically. The only input needed is the grid ratio of . Germano et al. (1991) found that the ratio of 2 will give the best results. This value will also be employed in the present simulations at a later point. For a channel flow, the model coefficient after plane averaging will be a function of the single inhomogeneous direction only; this usually makes the coefficient all positive and consequently a dissipative. On the other hand, the coefficient in a square duct flow is two-dimensional and thus allowing the appearance of negative values of viscosity. If the flow does not have a homogenous direction to which plane-averaging can be applied, averaging along the characteristics or over a small number of neighbouring cells may in some cases be sufficient. However, this was not confirmed for the present simulations.
35 CHAPTER 3 LARGE EDDY SIMULATION METHOD
Breuer (1998) suggested under-relaxation in time of C d like :
(3.33)
With a filter parameter of I of the order 10 - 3, this will dampen out all high-frequency oscillations and only leave the low-frequency variations. This was unfortunately not enough to ensure numerical stability. Breuer (1998) found that it was necessary to perform additional plane averaging for the flow past a circular cylinder. Piomelli and Liu (1994) proposed a similar filtering in time in order to stabilize the coefficient C. They found that by estimating the value of the present time step by some backward extrapolation schemes:
(3.34) after which they applied a first order approximation scheme for the and evaluated it by using an explicit Euler scheme. The result was:
(3.35)
Sagaut et al. (1994) found that it would help to set additional limits for the C of 0.2, which corresponds to the Smagorinsky constant leading to an over-dissipative model and they let the lower limits be defined as:
(3.36)
This will impose a constraint on the total viscosity to be positive, where Y is the molecular viscosity. Olsson and Fuchs (1994) chose an upper limit of 0.5 and a lower limit of -0.01, permitting a little room for backscatter effects. They showed that to set these constraints on the dynamic coefficient would only affect a limited number of nodes, but this will be necessary to ensure numerical stability. In this thesis both the dynamic subgrid scale model with the plane-averaging and the modification by providing an under-relaxation in time with Eqn. (3.33) and (3.36) as additional constraints will provide sufficient numerical stability to perform the simulations shown in chapters 5 and 6. An upper limit of C was set to 0.25. Additionally the number of nodes which has to be corrected by Eqn. (3.36) was in all cases less than 4% . To overcome the need for additional methods in order to preserve numerical stability, Ghosal et al. (1995) tried to optimize the equation for C globally, but still with the constraint that C should be greater than 0. This optimization leads to an integral equation (Fredholm’s integral equation of the second kind) which is expensive to solve. Ghosal et al. (1995) reported an increase in the CPU time of up til
36 CHAPTER 3 LARGE EDDY SIMULATION METHOD
50%. Which are a very high cost to pay for additional numerical stability.
3.6 The Dynamic one-equation subgrid scale model Davidson (D) (1997), Davidson (A) (1997), Davidson(B) (1997) introduced a one-equations subgrid scale model. One-equation models will solve a transport equation for the subgrid-scale energy ksgs to obtain the velocity scale. The model will only be described in brief; further information is given in the paper mentioned above.
The equation for the transport of the subgrid kinetic energy, ksgs, reads:
(3.37)
where
(3.38) and
(3.39)
L The subgrid scale kinetic energy at the test level is defined as K Ti j and .
The coefficient C* has the following form:
(3.40)
where production of subgrid scale kinetic energy is given as:
(3.41)
To ensure numerical stability, a constant value of C in space,
37 CHAPTER 3 LARGE EDDY SIMULATION METHOD
(3.42)
All local dynamic information will be included through the source terms. This is physically sounder as large local variations in C only appear in the source term and the effect of the large fluctuations in the dynamic coefficient has been smoothed out. The coefficients in the one-equation model only affect the stresses indirectly. In the standard dynamic model, the C coefficient is linearly proportional to the stresses. One-equation models have been used by several researchers. Among their advantages is the fact that the independent definition of the velocity scale results in a more accurate prescription of the SGS time scale compared to algebraic eddy-viscosity models. In order to evaluate variables on the test grid filter, the procedure by Zhang et al. (1993) was used, which carried out by the integration over the test cell assuming linear variation of the variable (Fig 3.2). This method is used for the code described later in Chapter 4.
(i,j+1) Test Cell
(i,j+½)
(i-½,j) (i+½,j) Grid Cell (i-1,j)(i,j) (i+1,j)
(i,j-½) half way nodes
j (i,j-1)
i
Figure 3.2: Test filter volume in two dimensions and the variable locations on the cell centre grid.
Next, a description of the implementation of the most frequently used code in this thesis will be given.
38