The Large-Eddy Simulation (LES) approach to turbulence consists in obtaining, by direct resolution of the Navier-Stokes equations, the large-scale characteristics of the turbulence, so that only the "small-scale" motions need to be modelled. The contributions of the large scales are isolated by introducing a filtered spatial averaging operator: \(\tilde{f}(\mathbf{x},\,t)=\int_{V}G(\mathbf{x},\,\mathbf{x}')f(\mathbf{x}',\,t)dV\), and any field function of the flow is decomposed as \(f(\mathbf{x},\,t)=\tilde{f}(\mathbf{x},\,t)+f'(\mathbf{x},\,t)\), where \(f'(\mathbf{x},\,t)\) is the subgrid fluctuation.

Filtered equations

Applying the filtered averaging operator to the equations of motion gives:

\[\begin{aligned} \frac{\partial\tilde{u}_{j}}{\partial x_{j}} & = 0\\ \frac{\partial\tilde{u}_{i}}{\partial t}+\frac{\partial}{\partial x_{j}}(\widetilde{u_{i}u_{j}}) & = \frac{1}{\rho_{0}}\frac{\partial\tilde{p}}{\partial x_{i}}+\nu\frac{\partial^{2}\tilde{u}_{i}}{\partial x_{j}\partial x_{j}} \end{aligned} \]

The filtered spatial average of the product \(\widetilde{u_{i}u_{j}}\) is rewritten as:

\[\widetilde{u_{i}u_{j}}=\tilde{u}_{i}\tilde{u}_{j}+\tilde{L}_{ij}+\tilde{R}_{ij} \]

where:

\[\begin{aligned} \tilde{L}_{ij} & = \widetilde{\tilde{u}_{i}\tilde{u}_{j}}-\tilde{u}_{i}\tilde{u}_{j}\\ \tilde{R}_{ij} & = \widetilde{\tilde{u}_{i}u_{j}'}+\widetilde{\tilde{u}_{j}u_{i}'}+\widetilde{u_{i}'u_{j}'} \end{aligned} \]

The first relation characterises the Leonard stresses, while the second characterises the subgrid Reynolds stresses. The closure problem of the LES procedure consists in determining a relation for \(\tilde{R}_{ij}\) in order to obtain the solution of a realisation of the flow.

Smagorinsky model

The subgrid stress tensor \(\tilde{R}_{ij}\) can be written:

\[\tilde{R}_{ij}=\tilde{T}_{ij}+\frac{1}{3}\tilde{R}_{kk}\delta_{ij} \]

One of the most widely used subgrid models is that of Smagorinsky, which assumes a linear relation of the anisotropic tensor \(\tilde{T}_{ij}\) with the filtered strain field \(\tilde{S}_{ij}\):

\[\tilde{T}_{ij}=-2\nu_{T}\tilde{S}_{ij} \]

The subgrid eddy viscosity \(\nu_{T}\) is chosen such that:

\[\nu_{T}=(C_{S}\Delta)^{2}\sqrt{\sum_{ij}\tilde{S}_{ij}\tilde{S}_{ij}} \]

where \(\Delta\) is the filter width and \(C_{S}\) is a positive constant that can vary depending on the application. The authors of [181] (p. 2203) state that the value can vary from \(C_{S}=0.05\) to \(C_{S}=0.16\). In TrioCFD, the value can be specified in the input data file; by default, it is set to \(C_{S}=0.18\) if it is not.

WALE model

The alternative model is LES-WALE (Wall-Adaptative Local Eddy-viscosity) [125] :

\[\nu_{T}=(C_{W}\Delta)^{2}\frac{(\tilde{S}_{ij}^{d}\tilde{S}_{ij}^{d})^{3/2}}{(\tilde{S}_{ij}\tilde{S}_{ij})^{5/2}+(\tilde{S}_{ij}^{d}\tilde{S}_{ij}^{d})^{5/4}} \]

with

\[\tilde{S}_{ij}^{d}=\tilde{S}_{ik}\tilde{S}_{kj}+\tilde{\Omega}_{ik}\tilde{\Omega}_{kj}-\frac{1}{3}\delta_{ij}\left(\tilde{S}_{mn}\tilde{S}_{mn}-\tilde{\Omega}_{mn}\tilde{\Omega}_{mn}\right) \]

where \(\tilde{\Omega}_{ij}\) is defined by

\[\tilde{\Omega}_{ij}=\frac{1}{2}\left(\frac{\partial\tilde{u}_{i}}{\partial x_{j}}-\frac{\partial\tilde{u}_{j}}{\partial x_{i}}\right) \]

Here, \(\Delta\) is chosen equal to the grid size. When \(C_{S}\) is equal to 0.18, an appropriate value of \(C_{w}\) is between \(0.55\leq C_{w}\leq0.6\) [125] (p. 170). Under certain flow conditions, the best-suited value is \(C_{w}=0.5\), and this is the one chosen in the simulations.

Thermal LES with Algo-QC

Solved equations

The Algo_QC module in TrioCFD solves the low Mach number equations in the context of quasi-compressible equations, as well as many Thermal Large-Eddy Simulation (T-LES) models. This approximation aims to narrow the gap between the incompressible and compressible Navier-Stokes equations and is suitable for the computation of convection in a fluid in the presence of large density gradients [132]. The low Mach number equations consider internal wave propagation but do not take acoustic waves into account; thus, the numerical methods developed for incompressible flow can be used. This approximation leads to dividing the pressure into two terms: the thermodynamic pressure, which is homogeneous, and the mechanical pressure. The Stokes hypothesis is assumed to be true [133]. The solved equations are the following:

Mass conservation equation:
\[\frac{\partial \rho}{\partial t} + \frac{\partial \rho U_j}{\partial x_j} = 0 \]
Momentum conservation equation:
\[\frac{\partial \rho U_i}{\partial t} = - \frac{\partial \rho U_j U_i}{\partial x_j} - \frac{\partial P}{\partial x_i} + \frac{\partial \Sigma_{ij}(U, T)}{\partial x_j} \]
Energy conservation equation:
\[\frac{\partial U_j}{\partial x_j} = - \frac{1}{\gamma P_0} \left[ (\gamma -1) \left( \frac{\partial Q_j }{\partial x_j} \right) + \frac{\mathrm{d} P_0}{\mathrm{d} t}\right] \]
Ideal gas law:
\[T = \frac{P_0}{\rho r} \]

with \(\rho\) the density, \(T\) the temperature, \(\gamma\) the heat capacity ratio, \(r=287\ \text{J}\cdot\text{kg}^{-1}\cdot\text{K}^{-1}\), \(t\) the time, \(P\) the mechanical pressure, \(P_0\) the thermodynamic pressure, \(U_i\) the \(i\)-th component of velocity, and \(x_i\) the Cartesian coordinate in the \(i\)-th direction. The Einstein summation convention is employed. The thermodynamic pressure is obtained by integrating the energy conservation equation over the volume of the studied geometry:

\[\frac{\mathrm{d} P_0}{\mathrm{d} t} = -(\gamma -1) \left( \frac{1}{V} \int Q_j(T)\, \mathrm{d} S_j \right) \]

with \(V\) the volume of the computational domain and \(S_j\) the boundaries of the investigated domain. The shear-stress tensor and the conductive heat flux are computed respectively with the functions \(\Sigma_{ij}(U, T)\) and \(Q_j(T)\), assuming a Newtonian fluid and Fourier's law:

\[\Sigma_{ij}(U, T) = \mu(T) \left( \frac{\partial U_i}{\partial x_j} + \frac{\partial U_j}{\partial x_i}\right) - \frac{2}{3}\mu(T) \frac{\partial U_k}{\partial x_k} \delta_{ij} \]

\[Q_j(T) = - \lambda(T)\frac{\partial T}{\partial x_j} \]

where \(\mu(T)\) is the dynamic viscosity, \(\lambda(T)\) the thermal conductivity, and \(\delta_{ij}\) the Kronecker symbol. The fluid passing through the channel is air. Sutherland's law [167] is used to compute the viscosity from the temperature:

\[\mu(T) = \mu_0 \left(\frac{T}{T_0}\right)^{3/2} \frac{T_0 + S}{T + S} \]

where \(\mu_0 = 1.716 \times 10^{-5}\ \text{Pa}\cdot\text{s}\), \(S = 110.4\ \text{K}\), and \(T_0 = 273.15\ \text{K}\). The Prandtl number is assumed constant, and the heat capacity at constant pressure \(C_p\) is obtained from the average of the wall temperatures. The conductivity is deduced from the Prandtl number, the heat capacity at constant pressure and the viscosity:

\[\lambda(T) = \frac{C_p}{P_r} \mu(T) \]

We suppose two boundary temperatures \(T_{\text{hot}}\) and \(T_{\text{cold}}\), where the temperature difference is of the order of \(300K\) to \(400K\). The streamwise ( \(x\)) and spanwise ( \(z\)) directions are periodic. A hyperbolic tangent mesh is used in the wall-normal direction ( \(y\)). The wall-normal grid coordinates are symmetrical with respect to the plane \(y=h\). In the first half of the channel, they are given by

\[y_k = h \left( 1 + \frac{1}{a} \tanh\left[ \left(\frac{k-1}{N_y-1} - 1\right)\tanh^{-1}(a)\right] \right), \]

with \(a\) the mesh dilatation parameter.

Filtered low Mach number equations

We consider the large-eddy simulation of the low Mach number equations in two formulations as introduced in [42]. The Velocity formulation expresses the filtered low Mach number equations in terms of variables filtered with the unweighted classical filter ( \(\overline{\,\cdot\,}\)). The Favre formulation expresses them using Favre-filtered variables, based on the density-weighted Favre filter ( \(\widetilde{\,\cdot\,}\)) defined for any field \(\psi\) as \(\widetilde{\psi} = \overline{\rho \psi} / \overline{\rho}\). The two formulations involve a different set of subgrid terms. However, the two most significant subgrid terms are similar in the two formulations [40], [41], [42]. In both cases, one subgrid term is related to the nonlinearity of momentum convection and another to the correlation of density and velocity. Excluding all other subgrid terms, the filtered low Mach number equations are given in the Velocity formulation by:

Mass conservation equation:
\[\frac{\partial \overline{\rho}}{\partial t} + \frac{\partial}{\partial x_j}\left(\overline{\rho}\, \overline{U}_{\!j} + F_{\rho U_j}\right) = 0, \]
Velocity transport equation:
\[\frac{\partial \overline{U}_{\!i}}{\partial t} = - \frac{\partial \left(\overline{U}_{\!j}\, \overline{U}_{\!i} + F_{U_j U_i}\right)}{\partial x_j} + \overline{U}_{\!i}\, \frac{\partial \overline{U}_{\!j}}{\partial x_j} - \frac{1}{\overline{\rho}}\frac{\partial \overline{P}}{\partial x_i} + \frac{1}{\overline{\rho}} \frac{\partial \varSigma_{ij}(\boldsymbol{\overline{U}}, \overline{T})}{\partial x_j}, \]
Energy conservation equation:
\[\frac{\partial \overline{U}_{\!j}}{\partial x_j} = - \frac{1}{\gamma P_{0}}\left[ (\gamma - 1)\frac{\partial Q_j(\overline{T})}{\partial x_j} + \frac{\partial P_{0}}{\partial t} \right], \]
Ideal gas law:
\[\overline{T} = \frac{P_{0}}{r\, \overline{\rho}}, \]

and in the Favre formulation by:

Mass conservation equation:
\[\frac{\partial \overline{\rho}}{\partial t} + \frac{\partial \overline{\rho}\, \widetilde{U_{j}}}{\partial x_j} = 0, \]
Momentum conservation equation:
\[\frac{\partial \overline{\rho}\, \widetilde{U}_{\!i}}{\partial t} = - \frac{\partial \left(\overline{\rho}\, \widetilde{U}_{\!j} \widetilde{U}_{\!i} + \overline{\rho}\, G_{U_j U_i}\right)}{\partial x_j} - \frac{\partial \overline{P}}{\partial x_i} + \frac{\partial \varSigma_{ij}(\boldsymbol{\widetilde{U}},\widetilde{T})}{\partial x_j}, \]
Energy conservation equation:
\[\frac{\partial}{\partial x_j}\left(\widetilde{U}_{\!j} + \overline{\rho}\, G_{U_j/\rho}\right) = - \frac{1}{\gamma P_{0}}\left[ (\gamma - 1)\frac{\partial Q_j(\widetilde{T})}{\partial x_j} + \frac{\partial P_{0}}{\partial t} \right], \]
Ideal gas law:
\[\widetilde{T} = \frac{P_{0}}{\overline{\rho}\, r}, \]

with the same physical quantities as above; the Einstein summation convention is used. The functions \(\varSigma_{ij}(\boldsymbol{U}, T)\) and \(Q_j(T)\) compute the shear-stress tensor and the conductive heat flux associated with a given velocity and temperature, assuming a Newtonian fluid and Fourier's law:

\[\begin{aligned} \varSigma_{ij}(\boldsymbol{U}, T) &= \mu(T) \left(\frac{\partial U_i}{\partial x_j} + \frac{\partial U_j}{\partial x_i}\right) - \frac{2}{3} \mu(T) \frac{\partial U_k}{\partial x_k} \delta_{ij}, \\ Q_j(T) &= - \lambda(T) \frac{\partial T}{\partial x_j}, \end{aligned} \]

The momentum-convection subgrid term is defined as \(F_{U_j U_i} = \overline{U_j U_i} - \overline{U}_{\!j}\, \overline{U}_{\!i}\) in the Velocity formulation and \(G_{U_j U_i} = \widetilde{U_j U_i} - \widetilde{U}_{\!j} \widetilde{U}_{\!i}\) in the Favre formulation. The density-velocity correlation subgrid term is defined as \(F_{\rho U_j} = \overline{\rho U_j} - \overline{\rho}\, \overline{U}_{\!j}\) in the Velocity formulation and \(G_{U_j/\rho} = \widetilde{U_j/\rho} - \widetilde{U}_{\!j}/\overline{\rho}\) in the Favre formulation. The two formulations are related by

\[\frac{F_{\rho U_j}}{\overline{\rho}} = - \overline{\rho}\, G_{U_j/\rho}. \]

The fluid is air. We use Sutherland's law [167] to compute the viscosity,

\[\mu(T) = \mu_0 \left(\frac{T}{T_0}\right)^{\frac{3}{2}} \frac{T_0 + S}{T + S}, \]

with \(\mu_0 = 1.716\cdot 10^{-5}\) Pa·s, \(S=110.4\) K and \(T_0 = 273.15\) K. We assume a constant Prandtl number \(Pr = 0.76\) and a heat capacity at constant pressure \(C_p = 1005\) J·kg⁻¹·K⁻¹. The conductivity is deduced from the Prandtl number, the heat capacity at constant pressure and the viscosity, \(\lambda(T) = (C_p/Pr)\,\mu(T)\). The ideal gas specific constant is \(r=287\) J·kg⁻¹·K⁻¹.

These equations can be solved through the keyword large_eddy_simulation_formulation, with either the favre or velocity value.

The following section is directly taken from [40], [41] and [42].

Subgrid-scale models

The subgrid terms of the Velocity and Favre formulations are formally similar; the same modelling procedure is used in both cases. The subgrid-scale models may be expressed as a function of the filter length scales and of the filtered velocity and density in the two formulations:

\[\begin{aligned} F_{U_j U_i} &\approx \tau_{ij}^{\mathrm{mod}}(\boldsymbol{\overline{U}}, \boldsymbol{\overline{\Delta}}), & G_{U_j U_i} &\approx \tau_{ij}^{\mathrm{mod}}(\boldsymbol{\widetilde{U}}, \boldsymbol{\overline{\Delta}}), \\ F_{\rho U_j} &\approx \pi_{j}^{\mathrm{mod}}(\boldsymbol{\overline{U}}, \overline{\rho}, \boldsymbol{\overline{\Delta}}), & G_{U_j/\rho} &\approx \pi_{j}^{\mathrm{mod}}(\boldsymbol{\widetilde{U}}, 1/\overline{\rho}, \boldsymbol{\overline{\Delta}}), \end{aligned} \]

where the functions \(\tau_{ij}^{\mathrm{mod}}\) and \(\pi_j^{\mathrm{mod}}\) are model-dependent but do not depend on the formulation. Eddy-viscosity models for the subgrid term associated with momentum convection may be written in the form

\[\tau_{ij}^{\mathrm{mod}}(\boldsymbol{U}, \boldsymbol{\overline{\Delta}}) = - 2 \nu_e^{\mathrm{mod}}(\boldsymbol{g}, \boldsymbol{\overline{\Delta}})\, S_{ij}, \]

with \(S_{ij} = \tfrac{1}{2}\left( g_{ij} + g_{ji} \right)\) the rate-of-deformation tensor and \(\boldsymbol{g}\) the velocity gradient, defined by \(g_{ij} = \partial_j U_i\). The term \(\tau_{ij}^{\mathrm{mod}}\) may be considered traceless without loss of generality, even in the incompressible case, since the trace can be included in the filtered pressure \(\overline{P}\). The eddy viscosity \(\nu_e^{\mathrm{mod}}\) is given by the model used. The following models from the literature are investigated using a priori tests:

Smagorinsky model [161] : \(\nu_e^{\mathrm{Smag.}} = \left( C^{\mathrm{Smag.}} \overline{\Delta} \right)^2 \left|\boldsymbol{S}\right|\)
WALE model [125] : \(\nu_e^{\mathrm{WALE}} = \left( C^{\mathrm{WALE}} \overline{\Delta} \right)^2 \dfrac{\left(\mathcal{S}^d_{ij} \mathcal{S}^d_{ij}\right)^{3/2}}{\left(S_{mn} S_{mn}\right)^{5/2} + \left(\mathcal{S}^d_{mn} \mathcal{S}^d_{mn}\right)^{5/4}}\)
Vreman model [176] : \(\nu_e^{\mathrm{Vreman}} = C^{\mathrm{Vreman}} \sqrt{\dfrac{\mathrm{II}_G}{g_{mn}g_{mn}}}\)
Sigma model [126] : \(\nu_e^{\mathrm{Sigma}} = \left( C^{\mathrm{Sigma}} \overline{\Delta} \right)^2 \dfrac{\sigma_3\left(\sigma_1 - \sigma_2\right)\left(\sigma_2 - \sigma_3\right)}{\sigma_1^2}\)
AMD model [149] : \(\nu_e^{\mathrm{AMD}} = C^{\mathrm{AMD}} \dfrac{\max(0, - G_{ij} S_{ij})}{g_{mn}g_{mn}}\)
VSS model [151] : \(\nu_e^{\mathrm{VSS}} = \left( C^{\mathrm{VSS}} \overline{\Delta} \right)^2 \dfrac{\left(R_{ij} R_{ij}\right)^{3/2}}{\left(S_{mn}S_{mn}\right)^{5/2}}\)
Kobayashi model [91] : \(\nu_e^{\mathrm{Koba.}} = C^{\mathrm{Koba.}} \overline{\Delta}^2 \left|F_g\right|^{3/2} (1-F_g) \left|\boldsymbol{S}\right|\)

where

\(\left|\boldsymbol{S}\right|=\sqrt{2 S_{ij} S_{ij}}\) is a norm of \(\boldsymbol{S}\),
\(\mathcal{S}^d_{ij} = \tfrac{1}{2}\left( g_{ik}g_{kj} + g_{jk}g_{ki} \right) - \tfrac{1}{3}g_{kp}g_{pk} \delta_{ij}\) is the traceless symmetric part of the squared velocity gradient tensor,
\(\sigma_1 \geq \sigma_2 \geq \sigma_3\) are the three singular values of \(\boldsymbol{g}\),
\(G_{ij} = \overline{\Delta}_k^2 g_{ik} g_{jk}\) is the gradient model for the subgrid term associated with momentum convection [106],
\(\mathrm{II}_G = \tfrac{1}{2}\left(\operatorname{tr}^2(G) - \operatorname{tr}(G^2)\right)\) is its second invariant,
\(R_{ij}=\beta_i g_{jj}\) is the volumetric strain-stretching, with \(\beta=\left(S_{23}, S_{13}, S_{12}\right)\),
\(F_g = \left(\varOmega_{ij}\varOmega_{ij} - S_{ij}S_{ij}\right)/\left(\varOmega_{mn}\varOmega_{mn} + S_{mn}S_{mn}\right)\) is the coherent structure function, with \(\varOmega_{ij} = \tfrac{1}{2}\left( g_{ij} - g_{ji} \right)\) the rate-of-rotation tensor.

Only constant-coefficient versions of the eddy-viscosity and eddy-diffusivity models are considered. The typical values of the coefficients from the literature are \(C^{\mathrm{Smag.}} = 0.10\), \(C^{\mathrm{WALE}} = 0.55\), \(C^{\mathrm{Vreman}}=0.07\), \(C^{\mathrm{Sigma}}=1.5\), \(C^{\mathrm{AMD}}=0.3\), \(C^{\mathrm{VSS}}=1.3\) and \(C^{\mathrm{Koba.}}=0.045\). The corresponding dynamic versions of these models are not considered, in order to assess the relevance of the models before any dynamic correction [64], [109], [134]. The filter length scale is computed following [32] as \(\overline{\Delta}=(\overline{\Delta}_x\overline{\Delta}_y\overline{\Delta}_z)^{1/3}\). A review of alternative possible definitions may be found in [173].

Following the same rationale, eddy-diffusivity models for the density-velocity correlation subgrid term may be written in the form

\[\pi_{j}^{\mathrm{mod}}(\boldsymbol{U}, \phi, \boldsymbol{\overline{\Delta}}) = - 2 \kappa_e^{\mathrm{mod}}(\boldsymbol{g}, \boldsymbol{d}, \boldsymbol{\overline{\Delta}})\, d_j, \]

with \(\boldsymbol{d}\) the scalar gradient, defined by \(d_{j} = \partial_j \phi\). It is common to express the eddy diffusivity using the constant subgrid-scale Prandtl or Schmidt number assumption,

\[\kappa_e^{\mathrm{mod}} = \frac{1}{Pr_t} \nu_e^{\mathrm{mod}}, \]

where \(Pr_t\) is the subgrid-scale Prandtl or Schmidt number. This provides a corresponding eddy-diffusivity model for each eddy viscosity. The number \(Pr_t\) corresponds to a subgrid-scale Schmidt number in the Velocity formulation and a subgrid-scale Prandtl number in the Favre formulation. Given the formal similarity between the density-velocity correlation subgrid term in the two formulations and the ideal gas law relating density and temperature, it is presumed that the same value may be used in the two formulations. Alternatively, some specific eddy-diffusivity models have been suggested in the literature [67], [2]. The following specific model is also investigated:

Scalar AMD model [2] : \(\kappa_e^{\mathrm{SAMD}} = C^{\mathrm{SAMD}} \dfrac{\max(0, - D_j d_j)}{d_m d_m}\), with \(D_j = \overline{\Delta}_k^2 g_{jk} d_k\) the gradient model for the density-velocity correlation subgrid term.

In addition, two new eddy-viscosity and eddy-diffusivity models were devised. First, the Anisotropic Smagorinsky model is a modified version of the Smagorinsky model, devised to involve the three filter length scales (improving the anisotropy of the model). It is obtained by substituting the velocity gradient \(\boldsymbol{g}\) and the scalar gradient \(\boldsymbol{d}\) by the scaled velocity gradient \(\boldsymbol{g^a}\), defined by \(g^a_{ij} = (\overline{\Delta}_j/\overline{\Delta}) \partial_j U_i\), and the scaled scalar gradient \(\boldsymbol{d^a}\), defined by \(d^a_j = (\overline{\Delta}_j/\overline{\Delta}) \partial_j \phi\):

\[\begin{aligned} \tau_{ij}^{\mathrm{An.\,Smag.}} &= - 2 \nu_e^{\mathrm{Smag.}}(\boldsymbol{g^a}, \boldsymbol{\overline{\Delta}})\, S^a_{ij}, \\ \pi_{j}^{\mathrm{An.\,Smag.}} &= - 2 \kappa_e^{\mathrm{Smag.}}(\boldsymbol{g^a}, \boldsymbol{d^a}, \boldsymbol{\overline{\Delta}})\, d^a_j, \end{aligned} \]

with \(S^a_{ij} = \tfrac{1}{2}( g^a_{ij} + g^a_{ji} )\) the scaled rate-of-deformation tensor. A similar procedure could be applied to obtain anisotropic versions of the WALE, Sigma, VSS and Kobayashi models.

The multiplicative mixed model based on the gradient model (MMG model) is a functional model whose magnitude is determined by the gradient model [106] and whose orientation is aligned with the rate-of-deformation tensor or the scalar gradient depending on the subgrid term. The eddy viscosity and eddy diffusivity are given by:

MMG model: \(\nu_e^{\mathrm{MMG}} = - C^{\mathrm{MMG}} \dfrac{G_{kk}}{\left|\boldsymbol{S}\right|}\)
Scalar MMG model: \(\kappa_e^{\mathrm{SMMG}} = - C^{\mathrm{SMMG}} \dfrac{\sqrt{D_i D_i}}{\sqrt{d_m d_m}}\)

Using the Smagorinsky model and the isotropic-part modelling of [189], \(\tau_{mm}^{\mathrm{Yosh.}} = 2 C^{\mathrm{Yosh.}} \overline{\Delta}^2 \left|\boldsymbol{S}\right|^2\), the MMG model \(\tau_{ij}^{\mathrm{MMG}} = - 2 \nu_e^{\mathrm{MMG}} S_{ij}\) can be reformulated as

\[\tau_{ij}^{\mathrm{MMG}} = G_{kk}\frac{ \tau_{ij}^{\mathrm{Smag.}}}{\tau_{mm}^{\mathrm{Yosh.}}} \]

emphasising that the MMG model combines the magnitude of the gradient model and the structure of the Smagorinsky model. By identification, \(C^{\mathrm{MMG}} = (C^{\mathrm{Smag.}})^2/(2C^{\mathrm{Yosh.}})\). Note that the Vreman, AMD and scalar AMD models also directly involve the gradient model [106].

Keywords

The LES models are enabled and selected through the keywords of the dns_qc_double data block. The keywords that enable or select a model are summarised below; the model constants, anisotropic coefficients, dynamic-procedure types, filter kernel and two-layer tuning parameters are listed with their types in the dns_qc_double keyword reference.

Functional (eddy-viscosity / eddy-diffusivity) models:

turbulent_viscosity: enable the functional model \(\tau^{mod}(\overline{U}, \overline{\Delta}) \approx R_{ij}\) for the momentum subgrid stress
turbulent_diffusivity: enable the functional model \(\pi^{mod}\) for the scalar subgrid flux
turbulent_viscosity_model (and turbulent_diffusivity_model for the scalar): functional model. Model keywords include constant, unsrho ( \(constant/\rho\)), smagorinsky, vreman, sigma, wale, amd, amd_comp, amdnoclip, amdscalar, amdscalarnoclip, rds, vss, kobayashi.
type_velocity_turbulent_diffusion: type of turbulent velocity diffusion, for computing the shear-stress tensor. Values are:
- simple: \(\mu_{turb}\nabla u\)
- simple_with_transpose: \(\mu_{turb}(\nabla u + \nabla^T u)\)
- full: \(\mu_{turb}(\nabla u + \nabla^T u - 2/3\, \nabla \cdot u\, \delta_{ij})\)
- simple_anisotropic: \(\mu_{turb}^a \nabla^a u, \quad \nabla^a_i = \Delta_i \nabla_i\)
- simple_with_transpose_anisotropic: \(\mu_{turb}^a (\nabla^a u + \nabla^{a,\,T} u), \ \nabla^a_i = \Delta_i \nabla_i\)
- full_anisotropic: \(\mu_{turb}^a (\nabla^a u + \nabla^{a,\,T} u - 2/3\, \nabla^a \cdot u\, \delta_{ij}), \quad \nabla^a_i = \Delta_i \nabla_i\)
type_scalar_turbulent_diffusion: type of turbulent scalar diffusion for computing the heat flux. Values are:
- normal: \(\lambda \nabla T\)
- anisotropic: \(\lambda^a \nabla^a T, \quad grad^a_i=\Delta_i \nabla_i\)

Structural models:

structural_uu: enable the structural model \(\tau^{mod}(\overline{U}, \overline{\Delta}) \approx R_{ij}\) for the momentum subgrid stress
structural_uscalar: enable the structural model \(\pi^{mod}\) for the scalar subgrid flux
structural_uu_model (and structural_uscalar_model for the scalar): structural model

Since functional and structural models can also be mixed, the model constants can be varied as a function of the height in the channel:

\[\tau_{ij}=\alpha\, \tau_{ij}^{func} + \beta\, \tau_{ij}^{struct} \]

where \(\alpha\) and \(\beta\) are applied through a hyperbolic tangent law, as proposed in [164] :

\[C_i^{func}=C^{func}+\left(0.5+0.5\, \tanh \left(\frac{y_i-s_c}{s_f}\right) \right) (C_c-C^{func}) \]

where \(i\) is the number of the \(i\)-th cell in the wall-normal direction and \(y\) the distance to the boundary, \(s_f=0.00016252\), \(s_c=0.00023217\) and \(C_c=0\) (values directly taken from [164]). The constant decreases the further one is from the boundary.

variation_cste_modele_fonctionnel: enable the two-layered mixed model

The smoothing parameters \(s_c\) and \(s_f\), the expected friction Reynolds numbers and the per-model weightings of the two-layer model (smoothing_center_fr, smoothing_factor_fr, Re_tau_fr, Re_tau_ch, ponderation_fr, ponderation_ch, center_constant) are documented with their types in the dns_qc_double keyword reference.