Calculation of Reynolds stress field in OpenFOAM


Keywords
Reynolds stress, shear flow, fieldAverage, variance, covariance

In the simulations of wall-bounded turbulent flows, such as turbulent channel flow [1], the distributions of the Reynolds stress components in the wall-normal direction are matters of interest to researchers and engineers.

For a statistically steady turbulent flow, a prime-squared mean of the velocity field (UPrime2Mean in OpenFOAM) gives the Reynolds stress field

\begin{eqnarray}
\frac{1}{N} \sum_{k=1}^{N} \left( U_i^{(k)} – \overline{U_i} \right)^2
&=& \frac{1}{N} \sum_{k=1}^{N} {u_{i}^{(k)}}^2 \\
&=& \frac{1}{N} \sum_{k=1}^{N}
\left[
\begin{array}{ccc}
u_{1}^{(k)}u_{1}^{(k)} & u_{1}^{(k)}u_{2}^{(k)} & u_{1}^{(k)}u_{3}^{(k)} \\
u_{2}^{(k)}u_{1}^{(k)} & u_{2}^{(k)}u_{2}^{(k)} & u_{2}^{(k)}u_{3}^{(k)} \\
u_{3}^{(k)}u_{1}^{(k)} & u_{3}^{(k)}u_{2}^{(k)} & u_{3}^{(k)}u_{3}^{(k)}
\end{array}
\right] \\
&=&
\left[
\begin{array}{ccc}
\overline{u_{1}u_{1}} & \overline{u_{1}u_{2}} & \overline{u_{1}u_{3}} \\
\overline{u_{2}u_{1}} & \overline{u_{2}u_{2}} & \overline{u_{2}u_{3}} \tag{1} \\
\overline{u_{3}u_{1}} & \overline{u_{3}u_{2}} & \overline{u_{3}u_{3}}
\end{array}
\right], \\
\end{eqnarray}
where \(U_i^{(k)}\) denotes the instantaneous velocity field \(U_i(\boldsymbol{x}, k\Delta t)\) and it is decomposed into the time-averaged mean \(\overline{U_i}\) and fluctuating components \({u_{i}}^{(k)}\),
\begin{equation}
{U_{i}}^{(k)} = \overline{U_i} + {u_{i}}^{(k)}. \tag{2}
\end{equation}

The diagonal components of the Reynolds stress tensor are the variances of the velocity components and the off-diagonal elements are the covariances of them.

Function Object – fieldAverage

Turbulent Channel Flow

Turbulent channel flow is one of the most fundamental wall-bounded shear flows and it has been widely used to study the structure of near-wall turbulence. Many DNS calculations have been carried out and produced a lot of informative data, which has contributed considerably to the improvements of other turbulence models, such as RANS and LES.

Moser et al. have released the statistical data obtained from their DNS of turbulent channel flow on their web site [2]. The data set contains, among other information, the components of the Reynolds stress tensor at three Reynolds numbers
\begin{equation}
Re_{\tau} = \frac{u_{\tau}h}{\nu} \approx 180,\; 395,\; 590, \tag{3}
\end{equation}
where \(u_{\tau}\) is the friction velocity, \(h\) is the channel half-height and \(\nu\) is the kinematic viscosity.

I will post my DNS results, which will help us to understand the flow behaviors in the near-wall region. The channel395 tutorial in OpenFOAM is the case of large eddy simulation (LES) instead of DNS at \(Re_{\tau}=395\).

References

[1] R. D. Moser, J. Kim and N. N. Mansour, Direct numerical simulation of turbulent channel flow up to \({\rm Re}_{\tau}\)=590. Physics of Fluids 11(4), 943-945, 1999.
[2] DNS Data for Turbulent Channel Flow. Available at: http://turbulence.ices.utexas.edu/data/MKM/ [Accessed: 9 April 2017].
[3] turbulence.ices.utexas.edu file server. Available at: http://turbulence.ices.utexas.edu/ [Accessed: 4 May 2017].

Temperature calculation from energy variables in OpenFOAM


The energy conservation equation is expressed in terms of internal energy \(e\) or enthalpy \(h\) in OpenFOAM and the temperature field is calculated from these solution variables. More precisely, we can specify the energy variable in the energy entry in the thermophysicalProperties file and the available options are the followings:

  • specific sensible enthalpy \(h {\rm [J/kg]}\) (sensibleEnthalpy)
  • specific sensible internal energy \(e {\rm [J/kg]}\) (sensibleInternalEnergy)

If we select the hePsiThermo thermophysical model, the temperature field \(T\) is calculated from the solved energy variable in the following function, where the compressibility \(\psi\), dynamic viscosity \(\mu\) and thermal diffusivity \(\alpha\) are also calculated.

Source Code

The calculation procedure of the temperature field depends on the selected energy variable, so the following THE function accordingly switches the called method.

If we choose the sensible enthalpy as the energy variable, the following THs function is called to calculate the temperature from the sensible enthalpy.

The calculation of the temperature is done iteratively using the Newton-Raphson method.

If the specific heat capacity at constant pressure \(c_p\) is expressed in the form of temperature polynomial function (hPolynomial)
\begin{equation}
c_p(T) = \sum_{i=0}^7 c_i T^i, \tag{1}
\end{equation}
the temperature in the j-th cell \(T_j\) is calculated from the following equation
\begin{equation}
\displaystyle \int_{T_{std}}^{T_j} \left( \sum_{i=0}^7 c_i T^i \right) dT = h_j, \tag{2}
\end{equation}
where \(h_j\) is the sensible enthalpy value in the j-th cell. In general the equation will be nonlinear, the iterative solution technique is implemented.

Compressible Flow Solvers

The above function calculate() is called by rhoPimpleFoam, rhoSimpleFoam and sonicFoam etc. from the line “thermo.correct()” after solving the energy conservation equation EEqn:

After updating the compressibility field \(\psi\), the pressure Poisson equation pEqn is constructed and solved.

The density field \(\rho\) is calculated with updated pressure and compressibility fields.

Summary
Energy variable Function used to calculate temperature
sensible enthalpy THs
absolute enthalpy THa
sensible internal energy TEs
absolute internal energy TEa

Study with great curiosity and let’s see the physical phenomena from a HIGHER point of view!

Thermal Boundary Conditions in OpenFOAM


Many thermal boundary conditions are available in OpenFOAM. I will upload some basic cases that explain the usage of these boundary conditions.

Source Code
src/TurbulenceModels/compressible/turbulentFluidThermoModels/derivedFvPatchFields/

  • convectiveHeatTransfer

    It calculates the heat transfer coefficients from the following empirical correlations for forced convection heat transfer:
    \begin{eqnarray}
    \left\{
    \begin{array}{l}
    Nu = 0.664 Re^{\frac{1}{2}} Pr^{\frac{1}{3}} \left( Re \lt 5 \times 10^5 \right) \\
    Nu = 0.037 Re^{\frac{4}{5}} Pr^{\frac{1}{3}} \left( Re \ge 5 \times 10^5 \right) \tag{1} \label{eq:NuPlate}
    \end{array}
    \right.
    \end{eqnarray}
    where \(Nu\) is the Nusselt number, \(Re\) is the Reynolds number and \(Pr\) is the Prandtl number.

  • externalCoupledTemperature
  • externalWallHeatFluxTemperature

    This boundary condition can operate in the following two modes:
    Mode#1 Specify the heat flux \(q\)
    \begin{equation}
    -k \frac{T_p – T_b}{\vert \boldsymbol{d} \vert} = q + q_r \tag{2} \label{eq:fixedHeatFlux}
    \end{equation}
    * \(k\): thermal conductivity
    * \(q_r\): radiative heat flux
    * \(T_b\): temperature on the boundary

    Mode#2 Specify the heat transfer coefficient \(h\) and the ambient temperature \(T_a\) (Fig. 1)
    \begin{equation}
    -k \frac{T_p – T_b}{\vert \boldsymbol{d} \vert} = \frac{T_a – T_b}{R_{th}} + q_r \tag{3} \label{eq:fixedHeatTransferCoeff}
    \end{equation}
    * \(R_{th}\): total thermal resistance of convective and conductive heat transfer
    \begin{equation}
    R_{th} = \frac{1}{h} + \sum_{i=1}^{n} \frac{l_i}{k_i} \tag{4} \label{eq:Rth}
    \end{equation}

    Figure 1
  • fixedIncidentRadiation
  • lumpedMassWallTemperature

    There is a dimensionless quantity called the Biot number, which is defined as
    \begin{equation}
    Bi = \frac{l/k}{1/h} = \frac{hl}{k}, \tag{5} \label{eq:Biot}
    \end{equation}
    where \(h\) is the heat transfer coefficient, \(k\) is the thermal conductivity of a solid and \(l\) is the characteristic length of the solid. As the definition in Eq. \eqref{eq:Biot} indicates, it represents the ratio of the internal conduction resistance \(l/k\) and the external convection resistance \(1/h\). If the Biot number is small (\(Bi \ll 1\)), the solid may be treated as a simple lumped mass system of an uniform temperature. This boundary condition calculates the uniform temperature variation \(\Delta T\) on the boundary from the following equation:
    \begin{equation}
    m c_p \Delta T = Q \Delta t. \tag{6} \label{eq:lumpedmass}
    \end{equation}
    * \(m\): total mass [kg]
    * \(c_p\): specific heat capacity [J/(kg.K)]
    * \(Q\): net heat flux on the boundary [W]
    * \(\Delta t\): time step [s]

  • outletMappedUniformInletHeatAddition
  • totalFlowRateAdvectiveDiffusive
  • wallHeatTransfer
  • compressible::thermalBaffle1D
  • compressible::turbulentHeatFluxTemperature
  • compressible::turbulentTemperatureCoupledBaffleMixed
  • compressible::turbulentTemperatureRadCoupledMixed
  • compressible::alphatJayatillekeWallFunction
  • compressible::alphatPhaseChangeWallFunction
  • compressible::alphatWallFunction

Filter Width – maxDeltaxyz


Several options are available in OpenFOAM for calculation of the filter width used in the large eddy simulation (LES) and detached eddy simulation (DES). In this blog post, the maxDeltaxyz option is covered in some detail.

OpenFOAM Version: OpenFOAM-dev, OpenFOAM v1612+

Implementation in OpenFOAM

The maxDeltaxyz option calculates the filter width of the \(i-\)th cell \(\Delta_i\) by taking the maximum distance between the cell center \(P_i\) and each face center \(F_j\)
\begin{equation}
\Delta_i = {\rm deltaCoeff} \times \max_{1 \le j \le n_i} \left\{ \overline{P_iF_j} \right\}, \tag{1} \label{eq:deltaxyz}
\end{equation}
where \({\rm deltaCoeff}\) is a constant of proportion (user input) and \(n_i\) is the number of the faces of the \(i-\)th cell.

Fig. 1 Graphical illustration for the case of a regular hex cell.

For a regular hexahedral cell shown in Fig. 1, the computed \(\Delta\) using Eq. \eqref{eq:deltaxyz} equals one-half of the maximum cell width \(\Delta x\), so the deltaCoeff coefficient should be set to 2 in the turbulenceProperties file as shown below.

The filter width is calculated in the following function.

Implementation in OpenFOAM v1612+

In OpenFOAM v1612+, the face normal vectors are considered in order to take into account the mesh non-orthogonality.

Some comments

This definition \eqref{eq:deltaxyz} is often used in the detached eddy simulation (DES) where some anisotropic grid cells such as “book”, “pencil” and “ribbon”-shaped cells exist in a boundary layer mesh. The position where switching between the RANS and LES modes occurs in the DES97 model [1] depends on how to calculate the filter width \(\Delta\)
\begin{equation}
\tilde{d} \equiv {\rm min}\left( d, C_{DES}\Delta \right), \tag{2} \label{eq:dTilda}
\end{equation}
where \(d\) is the distance to the closest wall and \(C_{DES}\) is a calibration constant.
Several modified length scales have been developed to prevent a delay of the Kelvin-Helmholtz instability in free and separated shear layers [2].

References

[1] P. R. Spalart, W.-H. Jou, M. Strelets and S. R. Allmaras, Comments on the feasibility of LES for wings, and on a hybrid RANS/LES approach. 1st AFOSR Int. Conf. on DNS/LES, Aug. 4-8, 1997, Ruston, LA. In “Advances in DNS/LES”, C. Liu and Z. Liu Eds., Greyden Press, Columbus, OH.
[2] P. R. Spalart, Detached-Eddy Simulation. Annu. Rev. Fluid Mech. 41, 181-202, 2009.

WALE SGS model in OpenFOAM


OpenFOAM Version: OpenFOAM-dev, OpenFOAM-1612+

Implementation in OpenFOAM

In the WALE (Wall-Adapting Local Eddy-viscosity) model, the subgrid scale viscosity is computed as
\begin{equation}
\nu_{sgs} = C_{k} \Delta \sqrt{k_{sgs}}, \tag{1} \label{eq:nusgs}
\end{equation}
where \(C_{k}\) is a model constant and \(k_{sgs}\) is the subgrid scale kinetic energy.

The traceless symmetric part of the square of the velocity gradient tensor \(S^d\) is calculated in the following function.
\begin{eqnarray}
S_{ij}^d = \frac{1}{2} \left( \frac{\partial \overline{u}_k}{\partial x_i}\frac{\partial \overline{u}_j}{\partial x_k} + \frac{\partial \overline{u}_k}{\partial x_j}\frac{\partial \overline{u}_i}{\partial x_k} \right) – \frac{1}{3} \delta_{ij} \frac{\partial \overline{u}_k}{\partial x_l}\frac{\partial \overline{u}_l}{\partial x_k}, \tag{2} \label{eq:sd}
\end{eqnarray}
where \(\delta_{ij}\) is the Kronecker delta.

The subgrid-scale kinetic energy is
\begin{equation}
k_{sgs} = \left( \frac{C_w^2 \Delta}{C_k} \right)^2 \frac{\left( S_{ij}^d S_{ij}^d \right)^3}{\left( \left( \overline{S}_{ij} \overline{S}_{ij} \right)^{5/2} + \left( S_{ij}^d S_{ij}^d \right)^{5/4} \right)^2}, \tag{3} \label{eq:ksgs}
\end{equation}
where
\begin{equation}
\overline{S}_{ij} = \frac{1}{2} \left( \frac{\partial \overline{u}_{i}}{\partial x_{j}} + \frac{\partial \overline{u}_{j}}{\partial x_{i}}\right), \tag{4} \label{eq:s}
\end{equation}
is the resolved-scale strain rate tensor.

Finally, we can get the following expression by substituting Eq. \eqref{eq:ksgs} into Eq. \eqref{eq:nusgs}:
\begin{eqnarray}
\nu_{sgs} = \left( C_w \Delta \right)^2 \frac{\left( S_{ij}^d S_{ij}^d \right)^{3/2}}{\left( \overline{S}_{ij} \overline{S}_{ij} \right)^{5/2} + \left( S_{ij}^d S_{ij}^d \right)^{5/4}}, \tag{5} \label{eq:nusgs2}
\end{eqnarray}
It is the same as Eq. (13) in [1].

Features
  • Algebraic eddy viscosity model (0-equation model)
  • The rotation rate is taken into account in the calculation of \(\nu_{sgs}\)
  • To be able to handle transition
  • Damping is Not necessary for \(\nu_{sgs}\) in the near-wall region
References

[1] F. Nicoud and F. Ducros, Subgrid-scale stress modelling based on the square of the velocity gradient tensor. Flow, Turbulence and Combustion, 62(3), 183-200, 1999.

Final expression of the SGS eddy viscosity is relatively simple and the implementation into OpenFOAM is not so complicated but the process of deriving the expression described in [1] is not easy to understand. I want to develop my SGS model someday 🙂

I will upload some animations next year.

Tensor Operations in OpenFOAM


Tensor operations are frequently used in turbulence modeling because the strain rate and vorticity tensors can be expressed in terms of the velocity gradient tensor.

In this blog post, I will pick out some typical operations and give brief explanations of them with some usage examples in OpenFOAM.

Keywords
strain rate tensor, vorticity tensor, Q-criterion, Hodge dual

OpenFOAM Version
OpenFOAM-dev

Gradient of a Vector Field | fvc::grad(u)

The gradient of a velocity vector \(\boldsymbol{u}\) returns a velocity gradient tensor (second rank tensor).
\begin{eqnarray}
\nabla \boldsymbol{u} &\equiv& \partial_{i} u_j \\
&=& \left(
\begin{matrix}
\partial u_1/\partial x_1 & \partial u_2/\partial x_1 & \partial u_3/\partial x_1 \\
\partial u_1/\partial x_2 & \partial u_2/\partial x_2 & \partial u_3/\partial x_2 \\
\partial u_1/\partial x_3 & \partial u_2/\partial x_3 & \partial u_3/\partial x_3
\end{matrix} \right)
\end{eqnarray}

Symmetric Part of a Second Rank Tensor | symm(T)

\begin{eqnarray}
{\rm symm}(\boldsymbol{T}) &\equiv& \frac{1}{2} (\boldsymbol{T} + \boldsymbol{T}^T) \\
&=& \frac{1}{2} \left(
\begin{matrix}
2T_{11} & T_{12} + T_{21} & T_{13} + T_{31} \\
T_{21} + T_{12} & 2T_{22} & T_{23} + T_{32} \\
T_{31} + T_{13} & T_{32} + T_{23} & 2T_{33}
\end{matrix} \right)
\end{eqnarray}

Twice the Symmetric Part of a Second Rank Tensor | twoSymm(T)

\begin{eqnarray}
{\rm twoSymm}(\boldsymbol{T}) &\equiv& \boldsymbol{T} + \boldsymbol{T}^T \\
&=& \left(
\begin{matrix}
2T_{11} & T_{12} + T_{21} & T_{13} + T_{31} \\
T_{21} + T_{12} & 2T_{22} & T_{23} + T_{32} \\
T_{31} + T_{13} & T_{32} + T_{23} & 2T_{33}
\end{matrix} \right)
\end{eqnarray}

Skew-symmetric Part of a Second Rank Tensor | skew(T)

\begin{eqnarray}
{\rm skew}(\boldsymbol{T}) &\equiv& \frac{1}{2} (\boldsymbol{T} – \boldsymbol{T}^T) \\
&=& \frac{1}{2} \left(
\begin{matrix}
0 & T_{12} – T_{21} & T_{13} – T_{31} \\
T_{21} – T_{12} & 0 & T_{23} – T_{32} \\
T_{31} – T_{13} & T_{32} – T_{23} & 0
\end{matrix} \right)
\end{eqnarray}

The symm and skew operations of the velocity gradient tensor field frequently appear in the source code. The following is a typical example.

In the above code, the symmetric and antisymmetric parts of the velocity gradient tensor \(\partial u_j/\partial x_i\) are defined as follows:
\begin{eqnarray}
S_{ij} &=& \frac{1}{2} \left( \frac{\partial u_j}{\partial x_i} + \frac{\partial u_i}{\partial x_j} \right), \\
\Omega_{ij} &=& \frac{1}{2} \left( \frac{\partial u_j}{\partial x_i} – \frac{\partial u_i}{\partial x_j} \right),
\end{eqnarray}
where \(S_{ij}\) is the strain rate tensor and \(-\Omega_{ij}\) is the vorticity (spin) tensor.

Hodge Dual | *T

\begin{equation}
*T = \left( T_{23},\;-T_{13},\;T_{12} \right)
\end{equation}

The vorticity vector \(\boldsymbol{\omega}\) is calculated as the Hodge dual of the skew-symmetric part of the velocity gradient tensor.
\begin{eqnarray}
\boldsymbol{\omega} &=& 2 \times \left( * \Omega_{ij} \right) \\
&=& 2 \times \left( \Omega_{23},\;-\Omega_{13},\;\Omega_{12} \right) \\
&=& \left( \frac{\partial u_3}{\partial x_2}-\frac{\partial u_2}{\partial x_3},\;\frac{\partial u_1}{\partial x_3}-\frac{\partial u_3}{\partial x_1},\;\frac{\partial u_2}{\partial x_1}-\frac{\partial u_1}{\partial x_2} \right)
\end{eqnarray}

The operator * in front of the skew is the Hodge dual operator.

Inner Product of Two Second Rank Tensors | T & S

\begin{equation}
P_{ij} = T_{ik} S_{kj}
\end{equation}

Double Inner Product of Two Second Rank Tensors | T && S

\begin{align}
s = T_{ij}S_{ij} &= T_{11}S_{11} + T_{12}S_{12} + T_{13}S_{13} \\
&+ T_{21}S_{21} + T_{22}S_{22} + T_{23}S_{23} \\
&+ T_{31}S_{31} + T_{32}S_{32} + T_{33}S_{33}
\end{align}

Trace of a Second Rank Tensor | tr(T)

\begin{equation}
{\rm tr}(\boldsymbol{T}) \equiv T_{11} + T_{22} + T_{33} = \boldsymbol{T} {\rm \&}{\rm \&} \boldsymbol{I}
\end{equation}

The Q-criterion can be used to identify vortex cores:
\begin{align}
Q &= \frac{1}{2} \left[ (\nabla \cdot \boldsymbol{u})^2 – {\rm tr}(\nabla \boldsymbol{u}^2)\right] \\
&= \frac{1}{2} \left[ (\nabla \cdot \boldsymbol{u})^2 + \Omega_{ij}\Omega_{ij} – S_{ij}S_{ij} \right].
\end{align}
For incompressible flows, it can be simplified as follows:
\begin{equation}
Q = \frac{1}{2} \left[ \Omega_{ij}\Omega_{ij} – S_{ij}S_{ij} \right].
\end{equation}

References

[1] OpenFOAM Programmer’s Guide
[2] CFD Direct | Tensor Mathematics

One equation eddy-viscosity SGS model in OpenFOAM


OpenFOAM Version: OpenFOAM-dev, OpenFOAM-1606+

Implementation in OpenFOAM

Assumption 1: Eddy viscosity approximation
As in the case of the Smagorinsky SGS model, the one equation eddy viscosity SGS model uses the eddy viscosity approximation as the name suggests, so the subgrid scale stress tensor \(\tau_{ij}\) is approximated as follows:
\begin{equation}
\tau_{ij} \approx \frac{2}{3} k_{sgs} \delta_{ij} – 2 \nu_{sgs} dev(\overline{D})_{ij}, \tag{1} \label{eq:tauij}
\end{equation}
where \(\nu_{sgs}\) is the subgrid scale eddy viscosity, \(\overline{D}\) is the resolved-scale strain rate tensor defined as
\begin{equation}
\overline{D}_{ij} = \frac{1}{2} \left( \frac{\partial \overline{u}_{i}}{\partial x_{j}} + \frac{\partial \overline{u}_{j}}{\partial x_{i}}\right), \tag{2} \label{eq:Dij}
\end{equation}
and the subgrid scale kinetic energy \(k_{sgs}\) is
\begin{equation}
k_{sgs} = \frac{1}{2} \tau_{kk} = \frac{1}{2} \left( \overline{u_{k}u_{k}} – \overline{u}_{k}\overline{u}_{k} \right). \tag{3} \label{eq:ksgs}
\end{equation}
The subgrid scale eddy viscosity \(\nu_{sgs}\) is computed using \(k_{sgs}\)
\begin{equation}
\nu_{sgs} = C_{k} \sqrt{k_{sgs}} \Delta, \tag{4} \label{eq:nusgs}
\end{equation}
where \(C_{k}\) is a model constant whose default value is \(0.094\).

The procedure so far is the same as the Smagorinsky SGS model but there is a difference in what follows. These models are different in terms of how to compute \(k_{sgs}\). The Smagorinsky model assumes the local equilibrium to compute \(k_{sgs}\) but the one equation eddy viscosity model solves a transport equation of \(k_{sgs}\) [1].

The second category of SGS model is one-equation eddy viscosity models. The main reason to develop the one-equation SGS models is to overcome the deficiency of local balance assumption between the SGS energy production and dissipation adopted in algebraic eddy viscosity models. Such a phenomenon may occur in high Reynolds number flows and/or in the cases of coarse grid resolution. The first one-equation eddy viscosity SGS model was theoretically derived by Yoshizawa and Horiuti [2], in which the SGS kinetic energy is defined as \(k_{sgs} = \frac{1}{2}(\overline{u_k^2} – \overline{u}_k^2)\), and the SGS eddy viscosity, \(\nu_{sgs}\), is computed using \(k_{sgs}\) as \(\nu_{sgs} = C_{k} k_{sgs}^{1/2} \Delta\). A transportation equation is derived to account for the historic effect of \(k_{sgs}\) due to production, dissipation and diffusion:

Transport Equation of \(k_{sgs}\)
\begin{align}
&\frac{\partial (\rho k_{sgs})}{\partial t} + \frac{\partial (\rho \overline{u}_j k_{sgs})}{\partial x_{j}} – \frac{\partial}{\partial x_{j}} \left[ \rho \left(\nu + \nu_{sgs}\right) \frac{\partial k_{sgs}}{\partial x_{j}} \right] \\
&= \; – \rho \tau_{ij} : \overline{D}_{ij} \; – \; C_{\epsilon} \frac{\rho k_{sgs}^{3/2}}{\Delta}, \tag{5} \label{eq:ksgsEqn}
\end{align}
where the operator : is a double inner product, \(\rho\) is the density, \(\nu\) is the kinetic viscosity and \(C_{\epsilon}\) is another model constant. The terms in Eq. \eqref{eq:ksgsEqn} are, starting from the left, the time derivative term, convective term, diffusion term, production term and dissipation term. In the case of the Smagorinsky SGS model, only the production and dissipation terms are taken into account with the assumption of the local equilibrium.

The production term in Eq. \eqref{eq:ksgsEqn} can be rearranged to yield the expression used in the source code as follows:
\begin{align}
– \rho \tau_{ij} : \overline{D}_{ij} &= \left( – \frac{2}{3} \rho k_{sgs} \delta_{ij} + 2 \rho \nu_{sgs} dev(\overline{D})_{ij} \right) : \overline{D}_{ij} \\
&=\;- \frac{2}{3} \rho k_{sgs} \frac{\partial \overline{u}_{k}}{\partial x_{k}} \\
&\;\;\;+ \rho \nu_{sgs} \frac{\partial \overline{u}_i}{\partial x_{j}} \left( 2\overline{D}_{ij} – \frac{1}{3} tr(2\overline{D}) \delta_{ij} \right). \tag{6} \label{eq:production}
\end{align}
Please see the appendix below for more detailed process of this deformation.

References

[1] S. Huang and Q. S. Li, A new dynamic one-equation subgrid-scale model for large eddy simulations. INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, 81, 835-865, 2010.
[2] A. Yoshizawa and K. Horiuti, A Statistically-Derived Subgrid-Scale Kinetic Energy Model for the Large-Eddy Simulation of Turbulent Flows. Journal of the Physical Society of Japan, 54(8), 2834-2839, 1985.

Appendix: Deformation of the Production Term

\begin{align}
&\; 2 \rho \nu_{sgs} dev(\overline{D})_{ij} : \overline{D}_{ij} \\
=&\; \rho \nu_{sgs} \frac{1}{2} \left( \frac{\partial \overline{u}_i}{\partial x_j} + \frac{\partial \overline{u}_j}{\partial x_i} – \frac{2}{3} tr(\overline{D}) \delta_{ij} \right) \left( \frac{\partial \overline{u}_i}{\partial x_j} + \frac{\partial \overline{u}_j}{\partial x_i} \right) \\
=&\; \rho \nu_{sgs} \frac{1}{2} \left( 2 \frac{\partial \overline{u}_i}{\partial x_j}\frac{\partial \overline{u}_i}{\partial x_j} + 2 \frac{\partial \overline{u}_i}{\partial x_j}\frac{\partial \overline{u}_j}{\partial x_i} – \frac{4}{3} tr(\overline{D}) \delta_{ij} \frac{\partial \overline{u}_i}{\partial x_j} \right) \\
=&\; \rho \nu_{sgs} \frac{\partial \overline{u}_i}{\partial x_j} \left( \frac{\partial \overline{u}_i}{\partial x_j} + \frac{\partial \overline{u}_j}{\partial x_i} – \frac{2}{3} tr(\overline{D}) \delta_{ij} \right) \\
=&\; \rho \nu_{sgs} \frac{\partial \overline{u}_i}{\partial x_j} \left( 2 \overline{D}_{ij} \; – \frac{1}{3} tr(\overline{2D}) \delta_{ij} \right) \\
=&\; \rho G \tag{7} \label{eq:productionTerm}
\end{align}

Smagorinsky SGS model in OpenFOAM


The Smagorinsky subgrid scale (SGS) model was developed by Joseph Smagorinsky in the meteorological community in the 1960s. It is based on the eddy viscosity assumption, which postulates a linear relationship between the SGS shear stress and the resolved rate of strain tensor. This model serves as a base for other SGS models.

OpenFOAM Version: OpenFOAM-dev, OpenFOAM-1606+

Implementation in OpenFOAM

The Smagorinsky SGS model is the oldest and best known subgrid scale model:
\begin{align}
\tau_{ij} &= \overline{u_{i}u_{j}} – \overline{u}_{i}\overline{u}_{j} \tag{1a}\\
&= \frac{1}{3} \tau_{kk} \delta_{ij} + \left( \tau_{ij} \; – \frac{1}{3} \tau_{kk} \delta_{ij} \right) \tag{1b} \label{eq:smg2}\\
&\approx \frac{1}{3} \tau_{kk} \delta_{ij} – 2 \nu_{sgs} dev(\overline{D})_{ij} \tag{1c} \label{eq:smg3}\\
&= \frac{2}{3} k_{sgs} \delta_{ij} – 2 \nu_{sgs} dev(\overline{D})_{ij}, \tag{1d} \label{eq:smg4}
\end{align}
where \(\nu_{sgs}\) is the subgrid scale eddy viscosity and the resolved-scale strain rate tensor \(\overline{D}_{ij}\) is defined as
\begin{equation}
\overline{D}_{ij} = \frac{1}{2} \left( \frac{\partial \overline{u}_{i}}{\partial x_{j}} + \frac{\partial \overline{u}_{j}}{\partial x_{i}}\right), \tag{2} \label{eq:D}
\end{equation}
and the subgrid scale kinetic energy \(k_{sgs}\) is
\begin{equation}
k_{sgs} = \frac{1}{2} \tau_{kk} = \frac{1}{2} \left( \overline{u_{k}u_{k}} – \overline{u}_{k}\overline{u}_{k} \right). \tag{3} \label{eq:ksgs}
\end{equation}
The subgrid scale stress tensor \(\tau_{ij}\) is split into an isotropic part \(\frac{1}{3} \tau_{kk} \delta_{ij}\) and an anisotropic part \(\tau_{ij} \; – \frac{1}{3} \tau_{kk} \delta_{ij}\) in Eq. \eqref{eq:smg2}. I think \(dev(\overline{D})\) is used instead of \(\overline{D}\) because the anisotropic part is a traceless tensor in Eq. \eqref{eq:smg3}.

Assumption 1: Eddy viscosity approximation
In analogy with the molecular viscous stress in laminar flows, the anisotropic part is approximated by relating it to the resolved rate of strain tensor \(\overline{D}_{ij}\) as in Eq. \eqref{eq:smg3}
\begin{equation}
\tau_{ij} \; – \frac{1}{3} \tau_{kk} \delta_{ij} \approx \; – 2 \nu_{sgs} dev(\overline{D})_{ij}. \tag{1c}
\end{equation}
In OpenFOAM, the subgrid scale viscosity is computed as
\begin{equation}
\nu_{sgs} = C_{k} \Delta \sqrt{k_{sgs}}, \tag{4} \label{eq:nusgs}
\end{equation}
where \(C_{k}\) is a model constant whose default value is \(0.094\) and \(\Delta\) is the grid size that defines the subgrid length scale. The method for calculating \(\Delta\) is specified in the turbulenceProperties file and the available options are as follows:

  • cubeRootVol
  • maxDeltaxyz
  • maxDeltaxyzCubeRoot
  • smooth
  • vanDriest
  • Prandtl
  • IDDESDelta

For the Smagorinsky SGS model, the vanDriest option is the first choice.

Assumption 2: Local equilibrium
The SGS kinetic energy \(k_{sgs}\) is computed with the assumption of the balance between the subgrid scale energy production and dissipation (local equilibrium)
\begin{equation}
\overline{D} : \tau_{ij} + C_{\epsilon} \frac{k_{sgs}^{1.5}}{\Delta} = 0, \tag{5} \label{eq:equilibrium}
\end{equation}
where the operator : is a double inner product of two second-rank tensors that can be evaluated as the sum of the 9 products of the tensor components. We can compute \(k_{sgs}\) by solving Eq. \eqref{eq:equilibrium} as shown below:
\begin{align}
&\overline{D} : \left( \frac{2}{3} k_{sgs} I -2 \nu_{sgs} dev(\overline{D}) \right) + C_{\epsilon} \frac{k_{sgs}^{1.5}}{\Delta} = 0 \\
\Leftrightarrow &\quad \overline{D} : \left( \frac{2}{3} k_{sgs} I -2 C_{k} \Delta \sqrt{k_{sgs}} dev(\overline{D}) \right) + C_{\epsilon} \frac{k_{sgs}^{1.5}}{\Delta} = 0 \\
\Leftrightarrow &\quad \sqrt{k_{sgs}} \left( \frac{C_{\epsilon}}{\Delta} k_{sgs} + \frac{2}{3} tr(\overline{D}) \sqrt{k_{sgs}} -2 C_{k} \Delta \left( dev(\overline{D}) : \overline{D} \right) \right) = 0 \\
\Leftrightarrow &\quad a k_{sgs} + b \sqrt{k_{sgs}} – c = 0 \\
\Rightarrow &\quad k_{sgs} = \left( \frac{-b + \sqrt{b^2 + 4ac}}{2a} \right)^2, \tag{6} \label{eq:ksgs2}
\end{align}
where
\begin{eqnarray}
\left\{
\begin{array}{l}
a = \frac{C_{\epsilon}}{\Delta} \\
b = \frac{2}{3} tr(\overline{D}) \tag{7} \label{eq:coeffs}\\
c = 2 C_{k} \Delta \left( dev(\overline{D}) : \overline{D} \right).
\end{array}
\right.
\end{eqnarray}

In the case of incompressible flows, Eq. \eqref{eq:coeffs} reduces to
\begin{eqnarray}
\left\{
\begin{array}{l}
b = \frac{2}{3} tr(\overline{D}) = 0 \\
c = 2 C_{k} \Delta \left( dev(\overline{D}) : \overline{D} \right) = C_{k} \Delta \vert \overline{D} \vert^2, \tag{8} \label{eq:coeffsInc}
\end{array}
\right.
\end{eqnarray}
where
\begin{equation}
\vert \overline{D} \vert = \sqrt{2 \overline{D} : \overline{D}}. \tag{9} \label{eq:abs}
\end{equation}
By substituting these relations into Eq. \eqref{eq:ksgs2}, we can get
\begin{equation}
k_{sgs} = \frac{c}{a} = \frac{C_{k} \Delta^2 \vert \overline{D} \vert^2}{C_{\epsilon}}. \tag{10} \label{eq:ksgsInc}
\end{equation}
We can get the following expression for the SGS eddy viscosity in the case of incompressible flows by substituting the above equation into Eq. \eqref{eq:nusgs}
\begin{equation}
\nu_{sgs} = C_{k} \sqrt{\frac{C_{k}}{C_{\epsilon}}} \Delta^2 \vert \overline{D} \vert. \tag{11} \label{eq:nusgsInc}
\end{equation}
By comparing it with the following common expression in the literature
\begin{equation}
\nu_{sgs} = \left( C_{s} \Delta \right)^2 \vert \overline{D} \vert, \tag{12} \label{eq:nusgstext}
\end{equation}
we can get the following relation for the Smagorinsky constant \(C_{s}\):
\begin{equation}
C_{s}^2 = C_{k} \sqrt{\frac{C_{k}}{C_{\epsilon}}}. \tag{13} \label{eq:smgConstInc}
\end{equation}

Features
  • Algebraic eddy viscosity model (0-equation model)
  • The rotation rate is not taken into account in the calculation of \(\nu_{sgs}\)
  • Based on the local equilibrium assumption
  • Appropriate value of the Smagorinsky constant is problem dependent
  • Not to be able to handle transition
  • Not to be able to deal with energy backscatter
  • Van-Driest damping is necessary for the SGS eddy viscosity in the near-wall region

The last three items arise from the fact that \(\nu_{sgs} \ge 0\) in Eq. \eqref{eq:nusgs}.

References

[1] (Free AccessJ. Smagorinsky, General circulation experiments with the primitive equations: I. The basic experiment*. Monthly weather review, 91(3), 99-164, 1963.

Large Eddy Simulation (LES)


Large Eddy Simulation (LES) is one of the most promising methods for computing industry-relevant turbulent flows. It is used to predict unsteady flow behaviors with lower computational cost as compared to Direct Numerical Simulation (DNS).

The essential idea of LES dates back to Joseph Smagorinsky (1963) in meteorology and to James W. Deardorff (1970) in engineering.

Keywords
Inertial subrange, Wall-modeled LES, Wall-resolved LES, Smagorinsky, Taylor microscale

Nomenclature – Abbreviations
  • LES: Large Eddy Simulation
  • ILES: Implicit Large Eddy Simulation
  • MILES: Monotone Integrated Large Eddy Simulation
  • VLES: Very Large Eddy Simulation
  • WMLES: Wall-Modeled Large Eddy Simulation
  • SGS: subgrid-scale
  • WALE: Wall-Adapting Local Eddy-Viscosity
What is Large Eddy Simulation?

Professor Parviz Moin of Stanford University briefly explains what LES is in his interview. I’m surprised that he received his Masters and Ph.D. degrees in Mathematics and Mechanical Engineering from Stanford in the same year (1978)!

The verification of LES simulations is difficult because both the error induced by the SGS model and the numerical discretization error are dependent on the grid resolution. In the following presentation (37:40-42:06), he introduces one method using explicit filtering to distinguish the errors [6]:

If the grid-independent solution of the explicit filtered LES equations fails to accurately predict the filtered DNS flow field, its failure can be solely attributed to the capability of the subgrid stress model employed.

Categorization of LES

LES can be classified in terms of the following points as shown in Fig. 1:

  • How the effect of the SGS stress is modeled
  • How the spatial filtering is applied in the solution procedure.
Fig. 1

In OpenFOAM, LES with the implicit filtering is implemented, in which only the filter width is specified and the filter shape is not. There exists this ambiguity in the definition of the filter. Lund [7] provides a clear description of the implicit filtering in LES:

The nearly universal approach is to simply write down the filtered Navier-Stokes equations together with an assumed model for the subgrid-scale stresses and then apply the desired spatial discretization to this “filtered” system. Although it is rarely mentioned, what one is doing by adopting this procedure is to imagine that the finite support of the computational mesh together with the low-pass characteristics of the discrete differentiating operators act as an effective filter. One then directly associates the computed velocity field with the filtered velocity. This procedure will be referred to as implicit filtering since an explicit filtering operation never appears in the solution procedure.

The explicit filtering methodology aims to damp the error-prone length scales smaller than the filter width that can be generated by the nonlinear convective term. In the case of the implicit filtering, we hope that this error has little effect on the resolved scales.

Subgrid-scale (SGS) model
Fig. 2
SGS models in OpenFOAM
  • Smagorinsky

    Smagorinsky SGS model

  • kEqn

    One equation eddy-viscosity model

  • dynamicLagrangian

    Dynamic SGS model with Lagrangian averaging

  • dynamicKEqn

    Dynamic one equation eddy-viscosity model

  • WALE

    Wall-adapting local eddy-viscosity (WALE) SGS model

  • DeardorffDiffStress

    Differential SGS Stress Equation Model

Other SGS models
  • Vreman SGS model [3]

    \begin{equation}
    \nu_{sgs} = c \sqrt{\frac{B_{\beta}}{\alpha_{ij}\alpha_{ij}}}
    \end{equation}

Calculation of filter width in OpenFOAM

The method for calculating the filter width \(\Delta\) is specified in the turbulenceProperties file. The available options in OpenFOAM are as follows:

  • cubeRootVol
  • maxDeltaxyz
  • maxDeltaxyzCubeRoot
  • smooth
  • vanDriest
  • Prandtl
  • IDDESDelta
Grid resolution measures

The choice of the computational grid size has sensible impact on the quality of the LES simulations and there are several techniques to assess the grid resolution in the LES computations [4].

  • Estimations based on prior RANS results
  • Single-grid estimators
  • Multi-grid estimators
Principal use
  • Aeroacoustics
  • Aerodynamics
  • Combustion
Refereces

[1] W. Rodi, J. H. Ferziger, M. Breuer and M. Pourquié, Status of Large Eddy Simulation: Results of a Workshop. Journal of Fluids Engineering, 119(2), 248-262, 1997.
[2] P. Comte, Large eddy simulations and subgrid scale modelling of turbulent shear flows
[3] A. W. Vreman, An eddy-viscosity subgrid-scale model for turbulent shear flow: Algebraic theory and applications. Physics of Fluids, 16(10), 2004.
[4] S. E. Gant, Reliability Issues of LES-Related Approaches in an Industrial Context. Flow, turbulence and combustion, 84(2), 325-335, 2010.
[5] U. Piomelli, Large eddy simulations in 2030 and beyond. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 372(2022), 2014.
[6] S. T. Bose, P. Moin and D. You, Grid-independent large-eddy simulation using explicit filtering. Center for Turbulence Research Annual Research Briefs 2008.
[7] T. S. Lund, The Use of Explicit Filters in Large Eddy Simulation. Computers and Mathematics with Applications 46, 603-616, 2003.

Refereces (Japanese)

[8] N. Fukushima et al., A Scale Self-Recognition Mixed SGS Model in Large Eddy Simulation of Homogeneous Isotropic Turbulence. Nagare, 34, 419-422, 2015.

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