Large Eddy Simulation

Large eddy simulations solve the larger structures of the turbulent ow, while the small eddies are modeled. This approach relies on the fact that most of the turbulent kinetic energy is contained in the large structures, while the eects of the small eddies can be modeled due to their universal and nearly isotropic character. The cut-o size of the resolved eddies can be dened according to the accuracy requirements. Therefore, LES permits the resolution of unsteady turbulent ows of many technical systems at com-putational costs that are acceptable to research purposes and within current computing capabilities. Following the development of computing technologies and the emergence of skilled users, temporally resolved calculations are on the way to becoming an engineering tool.

Spatial Filtering

The separation between the large resolved and the small modeled scales is achieved by applying a spatial low-pass lter. Thus, the ltered eld of a quantity φ is determined by convolution with a LES lter, G, by:

φ(x, t) = Z

φ (x− x⁰, t) G (x⁰) dx⁰. (3.13) In general, any normalized low-pass lter can be applied. The most frequently used consists of a volume-averaged top-hat ltering introduced by Deardor [42] and extended

3 Modeling Approaches

for non-uniform grids by Schumann [151].

φ(x, y, z, t) = 1 The LES lter size, ∆, represents the averaged lter length, ∆ = (∆x∆y∆z)^1/3.

Transport of Mass

Applying Favre-ltering to the equation of mass conservation (Eq. 2.1) yields the con-tinuity equation for LES (Eq. 3.15), which can be solved in terms of ltered quantities.

∂ρ

∂t + ∂(ρue_i)

∂xi

= 0 (3.15)

Transport of Momentum

The Navier-Stokes equations for LES (Eq. 3.16) are obtained by ltering the momentum equations 2.2:

Thereby, fτ_ij represents the resolved deviatoric stress tensor:

f

where fS_ij represents the ltered strain rate tensor:

Sf_ij = 1

The last term on the right-hand-side of Eq. 3.16 is the residual stress tensor. It accounts for the momentum ux due to sub-grid motions and is called the sub-grid scale (SGS) stress tensor:

τ_ij^sgs =ue_iue_j− gu_iu_j. (3.19) Sub-grid Stress Modeling

The SGS stress tensor τij^sgs appears in an unclosed form and needs to be modeled. The following paragraphs introduce the sub-grid scale modeling approaches applied within this work.

3 Modeling Approaches

Eddy Viscosity

Boussinesq's eddy viscosity approximation assumes that the eects of the small scales on the ow are similar to those of the molecular viscosity. Accordingly, the sub-grid term τ^sgs, which accounts for these eects can be retrieved by dening a turbulent viscosity, ν_t. Hence, the sub-grid stress is modeled as:

τ_ij^sgs− 1 Introducing the eective viscosity νef f = ν + ν_t, the ltered Navier-Stokes equations (Eq. 3.16) can be rewritten as:

∂ ¯ρeui

The pressure parameter P is introduced to incorporate the trace of the sub-grid stress tensor (Eq. 3.22). The pressure correction scheme is applied to determine the value of the pressure parameter, P , so that the conservation of mass is satised, whilst the physical pressure, p, remains unknown.

P = p− 1

3ρτ_kk^sgs (3.22)

Smagorinsky Model

In order to solve the momentum equation (Eq. 3.21), the turbulent viscosity, νt, needs to be closed. The closure introduced by Smagorinsky [152] relates the turbulent viscosity to the LES lter length ∆ and to the strain rate tensor fS_ij as:

ν_t = (C_s∆)²| eS|, (3.23)

with  eS =

q

2 fS_ijSf_ij. (3.24)

The the constant Cs is a model parameter, which is referred to as the Smagorinsky con-stant. A major drawback of this approach is the dependency of the constant on the ow region. While Lilly [115] proposed a value of Cs = 0.173, Piomelli et al. [134] suggested a value of Cs = 0.065 for a channel ow. Indeed, signicant variations in the considered

ow region have been identied. In particularly, a reasonable value for Cs in a channel

ow is 0.2, while a value of 0.065 is more appropriate near walls.

Germano Dynamic Procedure

The dependency of the Smagorinsky constant on the location within the ow eld is

3 Modeling Approaches

overcome by applying an additional test ltering to determine the local constant value.

This method, referred to as the Germano Dynamic Procedure [65], relies on the fact that the correct value of Cs is expected to yield the same amount of turbulence, no matter what the lter size. Thus, Germano applied a test lter to the already ltered eld , b∗, using a lter size Γ larger than the lter size ∆. Similar to the decomposition of the correlation term gu_iu_j (Eq. 3.19), introduced for the ltered Navier-Stokes equations 3.16, a ne structure τ_ij^sgs,Γ resulting from the double ltering is dened as a residual term between the unknown correlation term dug_iu_j and its resolved part beu_ibeuj:

τ_ij^sgs,Γ = bue_ibeuj− dug_iu_j. (3.25) Applying the test lter to the sub-grid stress τij^sgs,∆ yields:

τ\_ij^sgs,∆ = due_iue_j− dug_iu_j. (3.26) Combining Equations 3.25 and 3.26 gives the Germano identity, Eq. 3.27:

L_ij = τ_ij^sgs,Γ− \τ_ij^sgs,∆ = bue_ibeu^j − due_iue_j. (3.27) Applying the eddy viscosity approach, Eq. 3.20, and the Smagorinsky approximation of turbulent viscosity, Eq. 3.23, to both sub-grid stresses yields:

τ_ij^sgs,∆−1

3τ_kk^sgs,∆δ_ij = 2C_g∆²| eS|

Sf_ij − 1 3Sf_kkδ_ij



= 2C_gm^sgs,∆_ij , (3.28)

τ_ij^sgs,Γ−1

3τ_kk^sgs,Γδ_ij = 2C_gΓ²| eS|

Sf_ij − 1 3Sf_kkδ_ij



= 2C_gm^sgs,Γ_ij , (3.29) where the Germano parameter, Cg = C_s², has been substituted with the Smagorinsky constant, Cs. Thus, the Germano identity, Eq. 3.27, can be rewritten as:

L_ij − 1

3L_kkδ_ij = 2C_gM_ij, (3.30) where

M_ij = \m^sgs,∆_ij − m^sgs,Γij . (3.31)

Equation 3.30 is an overdetermined system of equations, with ve independent compo-nents since the trace is zero. Lilly [116] proposed solving this system by a least-square approach, which leads to the following formula for the Germano parameter:

3 Modeling Approaches

Accordingly, the Germano procedure may lead to negative values of Cg. This is incon-sistent with the Smagorinsky model and a clipping is applied (Eq. 3.33) to restrict Cg to positive values.

Filtering the scalar transport equation (Eq. 2.5) yields the transport equation for LES:

∂( ¯ρ eφ)

where the convection term, fφui, is separated into its resolved part eφeui and a ne-structure scalar ux term F_φ,i^sgs.

φuf_i = eφue_i− F_φ,i^sgs (3.35)

Modeling of the Sub-grid Scale Scalar Flux

The ne structure scalar ux term, F_φ,i^sgs, in Eq. 3.34 is unknown and must be modeled.

Similar to the modeling of the SGS momentum ux, turbulence is assumed to contribute to mixing in the same way as an additional diusion. Thus, the sub-grid scalar transport due to turbulence is approximated with an eddy diusivity approach, which is similar to the eddy viscosity approach. Therefore, a turbulent diusivity, Dφ,t, is introduced to relate the sub-grid ux to the scalar gradient, _∂x^{∂ ˜φ}_i, as:

F_φ,i^sgs = D_φ,t∂ ˜φ

∂x_i. (3.36)

Introducing Eq. 3.36 into Eq. 3.34 yields:

∂( ¯ρ eφ) Within the CFD code, the diusion coecient Dφ,t is retrieved through the Schmidt number, Sc, which denes the ratio of momentum transport due to viscosity to the scalar transport due to diusion. In the same sense, a turbulent Schmidt number, Sct, is

intro-3 Modeling Approaches

duced for comparing the uxes due to turbulence:

Sc = ν

D_f ≈ ν˜

Df_f ; Sc_t= ν_t

D_f,t. (3.38)

The laminar Schmidt number, Sc, is accepted as 0.7. A range of values were applied for the turbulent Schmidt number of mixture fraction, from Sct = 0.4 in [137] Sct = 0.7 in [22], [94] and [45]. Following [94, 45] a constant value of Sct = 0.7 has been used throughout this work. The source term, f˙ω_φ, represents the spatially ltered source term of scalar, φ and needs to be modeled for chemical reactions. The next section presents the modeling approaches used within this work.