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TrioCFD 1.9.8
TrioCFD documentation
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TrioCFD 1.9.8
TrioCFD documentation
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Generalities about the RANS (Reynolds-Averaged Navier-Stokes) turbulence models available in TrioCFD are given in this page. The recommended model is the \(k\)- \(\omega\) SST model.
The velocity \(\mathbf{u}(\mathbf{x},\,t)\) and the pressure \(p(\mathbf{x},\,t)\) of the flow field of an incompressible fluid are governed, independently of the temperature, by the continuity and momentum equations:
\[\begin{aligned} \boldsymbol{\nabla}\cdot\mathbf{u} & =0,\\ \rho_{0}\left[\frac{\partial\mathbf{u}}{\partial t}+\mathbf{u}\cdot\boldsymbol{\nabla}\mathbf{u}\right] & =-\boldsymbol{\nabla}p+\eta_{0}\boldsymbol{\nabla}^{2}\mathbf{u}+\rho_{0}\mathbf{F}_{v} \end{aligned} \]
The statistical treatment of the instantaneous Navier-Stokes equations leads to the decomposition of each field into a mean part and a fluctuating part (Reynolds decomposition):
\[\rho = \bar{\rho} + \rho', \quad \mathbf{u} = \overline{\mathbf{U}} + \tilde{\mathbf{u}}, \quad p = \overline{P} + \tilde{p} \]
where the symbol \(\overline{(\,)}\) denotes the statistical (or ensemble) average operator and the symbol \(\tilde{(\,)}\) the fluctuations (or deviations from these averages). The averaged mass and momentum equations are then written [18] :
\[\begin{aligned} \boldsymbol{\nabla}\cdot\overline{\mathbf{U}} & =0,\\ \rho_{0}\left[\frac{\partial\overline{\mathbf{U}}}{\partial t}+\overline{\mathbf{U}}\cdot\boldsymbol{\nabla}\overline{\mathbf{U}}\right] & =\boldsymbol{\nabla}\cdot\overline{\boldsymbol{\Sigma}}+\rho_{0}\overline{\mathbf{F}}_{v} \end{aligned} \]
with
\[\overline{\boldsymbol{\Sigma}}=-\overline{P}\mathbf{I}+2\eta_{0}\overline{\mathbf{S}}-\rho_{0}\overline{\tilde{\mathbf{u}}\tilde{\mathbf{u}}}\qquad\text{and}\qquad\overline{\mathbf{S}}=\frac{1}{2}(\boldsymbol{\nabla}\overline{\mathbf{U}}+\boldsymbol{\nabla}^{T}\overline{\mathbf{U}}) \]
The mean momentum balance is called the Reynolds equation. In this equation, the surface forces give rise to an additional term \(-\rho_{0}\overline{\tilde{\mathbf{u}}\tilde{\mathbf{u}}}\) that represents the turbulent agitation. The system of equations is open because of the presence of the fluctuating-velocity correlations \(\overline{\tilde{\mathbf{u}}\tilde{\mathbf{u}}}\).
Eddy-viscosity models are based on the analogy that turbulence plays for the flow the role viscosity plays for the fluid. Many models of the Reynolds tensor are possible; the most classical one, of interest here, is based on the Boussinesq hypothesis:
\[\overline{\tilde{\mathbf{u}}\tilde{\mathbf{u}}}=-2\nu_{T}\mathbf{\overline{S}}+\frac{2}{3}\overline{k}\mathbf{I} \]
where \(\nu_{T}\) is a scalar turbulent viscosity that expresses the effects of turbulent agitation. The \(\overline{k}\) term on the right-hand side is akin to a pressure due to turbulent agitation and is integrated into the pressure \(\overline{P}\). A closed equation for the mean velocity is thus obtained. This model introduces two new quantities, the turbulent viscosity \(\nu_{T}\) and the turbulent kinetic energy \(\overline{k}\). To close the system, the turbulent viscosity is defined by dimensional analysis as \(\nu_{T}=C_{\mu}f(\overline{k},X)\), with \(X\) a variable that can be related to the dissipation of the turbulent kinetic energy ( \(\overline{\epsilon}\) or \(\omega\), for instance) or the integral scale.
The standard \(k\)- \(\varepsilon\) model was introduced by [105]. Within the two-equation \(\overline{k}\)– \(\overline{\epsilon}\) framework, dimensional analysis gives for the turbulent viscosity:
\[\nu_{T}=C'_{\eta}\frac{\overline{k}^{2}}{\overline{\epsilon}} \]
The \(\overline{k}\)– \(\overline{\epsilon}\) model closes the system by solving two additional transport equations, one for the turbulent kinetic energy \(\overline{k}\) and the other for the dissipation rate \(\overline{\epsilon}\) [18] :
\[\begin{aligned} \frac{\partial\overline{k}}{\partial t}+\overline{U}_{j}\frac{\partial\overline{k}}{\partial x_{j}} & = \nu_{T}\left(\frac{\partial\overline{U}_{i}}{\partial x_{j}}+\frac{\partial\overline{U}_{j}}{\partial x_{i}}\right)\frac{\partial\overline{U}_{i}}{\partial x_{j}}+\frac{\partial}{\partial x_{j}}\left[\frac{\nu_{T}}{\sigma_{k}}\frac{\partial\overline{k}}{\partial x_{j}}\right]-\overline{\epsilon}\\ \frac{\partial\overline{\epsilon}}{\partial t}+\overline{U}_{j}\frac{\partial\overline{\epsilon}}{\partial x_{j}} & = C_{\epsilon_{1}}\nu_{T}\frac{\overline{\epsilon}}{\overline{k}}\left(\frac{\partial\overline{U}_{i}}{\partial x_{j}}+\frac{\partial\overline{U}_{j}}{\partial x_{i}}\right)\frac{\partial\overline{U}_{i}}{\partial x_{j}}+\frac{\partial}{\partial x_{j}}\left[\frac{\nu_{T}}{\sigma_{\epsilon}}\frac{\partial\overline{\epsilon}}{\partial x_{j}}\right]-C_{\epsilon_{2}}\frac{\overline{\epsilon}^{2}}{\overline{k}} \end{aligned} \]
These are advection-diffusion equations with source terms. The production term of turbulent kinetic energy (first term on the right-hand side) plays an important role in the wall modellings. The standard values of the five model constants \(C'_{\eta}\), \(C_{\epsilon_{1}}\), \(C_{\epsilon_{2}}\), \(\sigma_{k}\) and \(\sigma_{\epsilon}\) are set by default to: \(C'_{\eta}=0.09\), \(C_{\epsilon_{1}}=1.44\), \(C_{\epsilon_{2}}=1.92\), \(\sigma_{k}=1.0\) and \(\sigma_{\epsilon}=1.3\). Some of them can also vary depending on the type of flow considered (see the table below).
| Reference | \(C_{\eta}'\) | \(\sigma_{k}\) | \(\sigma_{\epsilon}\) | \(C_{\epsilon_{1}}\) | \(C_{\epsilon_{2}}\) | Flow |
|---|---|---|---|---|---|---|
| [84] | 0.09 | 1.0 | 1.3 | 1.55 | 2.00 | High Reynolds |
| [104] | 0.09 | 1.0 | 1.3 | 1.44 | 1.92 | Rotating |
| [20] | 0.09 | 1.0 | 1.3 | 1.35 | 1.92 | Low Reynolds |
| [50] | 0.09 | 1.0 | 1.3 | 1.39 | 1.80 | Low Reynolds |
| [121] | 0.09 | 1.0 | 1.3 | 1.60 | 1.92 | Jet |
| [9] | 0.09 | 1.0 | 1.3 | 1.40 | 1.92 | Buoyancy |
Synthesis of the parameter values of the \(\overline{k}\)– \(\overline{\epsilon}\) model (taken from [63]). One must keep in mind that the \(C_{\epsilon_{2}}\) constant was set to reproduce the decay of turbulent kinetic energy in homogeneous isotropic turbulence; those values must be modified with caution.
[158] proposed a modification of the \(k\)- \(\varepsilon\) model to ensure the realizability of the turbulent kinetic energy using the Schwarz inequality (its validations being ongoing). The two equations are written:
\[\begin{aligned} \frac{\partial\overline{k}}{\partial t}+\overline{U}_{j}\frac{\partial\overline{k}}{\partial x_{j}} & = \frac{\partial}{\partial x_{j}}\left(\frac{\nu_{T}}{\sigma_{k}}\frac{\partial\overline{k}}{\partial x_{j}}\right)+\left(2\nu_{T}S_{ij}-\frac{2}{3}\overline{k}\delta_{ij}\right)\frac{\partial\overline{U}_{i}}{\partial x_{j}}-\overline{\epsilon}\\ \frac{\partial\overline{\epsilon}}{\partial t}+\overline{U}_{j}\frac{\partial\overline{\epsilon}}{\partial x_{j}} & = \frac{\partial}{\partial x_{j}}\left[\frac{\nu_{T}}{\sigma_{\epsilon}}\frac{\partial\overline{\epsilon}}{\partial x_{j}}\right]+C_{1}S\epsilon-C_{2}\frac{\overline{\epsilon}^{2}}{\overline{k}+\sqrt{\nu\overline{\epsilon}}} \end{aligned} \]
with
\[S=\sqrt{2S_{ij}S_{ij}},\quad C_{1}=\max\left\{ 0.43,\,\frac{\eta}{5+\eta}\right\},\quad\eta=\frac{S\overline{k}}{\overline{\epsilon}} \]
\[C_{\eta}=\frac{1}{A_{0}+A_{s}U^{(*)}\frac{\overline{k}}{\overline{\epsilon}}},\quad A_{0}=4,\quad A_{s}=\sqrt{6}\cos\phi,\quad\phi=\frac{1}{3}\arccos\left(\sqrt{6}W\right),\quad W=\frac{S_{ij}S_{jk}S_{ki}}{(S_{ij}S_{ij})^{3/2}} \]
\[U^{(*)}=\sqrt{S_{ij}S_{ij}+\tilde{\Omega}_{ij}\tilde{\Omega}_{ij}},\quad\tilde{\Omega}_{ij}=\Omega_{ij}-2\epsilon_{ijk}\omega_{k},\quad\Omega_{ij}=\overline{\Omega}_{ij}\epsilon_{ijk}\omega_{k} \]
where \(\overline{\Omega}_{ij}\) is the mean rotation rate in a reference frame rotating with angular velocity \(\omega_{k}\). In TrioCFD, this realizable \(\overline{k}\)– \(\overline{\epsilon}\) model was developed and validated in [4].
Wall functions make it possible to avoid solving the Navier-Stokes equations and the turbulence model near the wall. The \(\overline{k}\)– \(\overline{\epsilon}\) model, coupled with a wall law, thus allows the core of the flow to be simulated while avoiding excessive computational cost due to a too-fine mesh near the wall. However, this type of model is unsuitable when the first mesh point is located in the viscous sublayer ( \(y^{+}<30\)). The higher the Reynolds number, the more negligible the thickness of this viscous sublayer, which makes the \(\overline{k}\)– \(\overline{\epsilon}\) model with a wall law well suited to high-Reynolds-number flows. This is why this model is also called the "high-Reynolds" \(\overline{k}\)– \(\overline{\epsilon}\) model.
When studying low-Reynolds-number flows, the viscous sublayer becomes more significant, making the use of a wall law inappropriate. It may then be preferable to use models called "low-Reynolds", which use damping functions and discretization-dependent terms to account for the numerical resolution of the viscous sublayer. The low-Reynolds \(\overline{k}\)– \(\overline{\epsilon}\) model leaves the transport equation for \(\overline{k}\) unchanged but modifies that of \(\overline{\epsilon}\) by adding attenuation terms in the near-wall zone. Because of the finer wall mesh, this model is more costly than the standard one. There are several low-Reynolds models in the literature. Their general form can be written (notation inspired by [84]):
\[\begin{aligned} \frac{\partial\overline{k}}{\partial t}+\overline{U}_{j}\frac{\partial\overline{k}}{\partial x_{j}} & = \nu_{T}\left(\frac{\partial\overline{U}_{i}}{\partial x_{j}}+\frac{\partial\overline{U}_{j}}{\partial x_{i}}\right)\frac{\partial\overline{U}_{i}}{\partial x_{j}}+\frac{\partial}{\partial x_{j}}\left[\left(\nu+\frac{\nu_{T}}{\sigma_{k}}\right)\frac{\partial\overline{k}}{\partial x_{j}}\right]-\overline{\epsilon}-\overline{\mathcal{K}}\\ \frac{\partial\overline{\epsilon}}{\partial t}+\overline{U}_{j}\frac{\partial\overline{\epsilon}}{\partial x_{j}} & = C_{\epsilon_{1}}\nu_{T}\frac{\overline{\epsilon}}{\overline{k}}f_{\epsilon_{1}}\left(\frac{\partial\overline{U}_{i}}{\partial x_{j}}+\frac{\partial\overline{U}_{j}}{\partial x_{i}}\right)\frac{\partial\overline{U}_{i}}{\partial x_{j}}+\frac{\partial}{\partial x_{j}}\left[\left(\nu+\frac{\nu_{T}}{\sigma_{\epsilon}}\right)\frac{\partial\overline{\epsilon}}{\partial x_{j}}\right]-C_{\epsilon_{2}}f_{\epsilon_{2}}\frac{\overline{\epsilon}^{2}}{\overline{k}}+\overline{\mathcal{E}}\\ \nu_{T} & = C_{\eta}f_{\eta}\frac{\overline{k}^{2}}{\overline{\epsilon}} \end{aligned} \]
For this model [105], the terms \(\overline{\mathcal{K}}\) and \(\overline{\mathcal{E}}\) are written:
\[\overline{\mathcal{K}}=2\nu\left(\frac{\partial\overline{k}^{1/2}}{\partial x_{j}}\right)^{2},\quad\overline{\mathcal{E}}=2.0\,\nu\,\nu_{T}\left(\frac{\partial^{2}\overline{U}_{i}}{\partial x_{j}\partial x_{l}}\right) \]
and the damping functions are:
\[f_{\epsilon_{1}}=1,\quad f_{\epsilon_{2}}=1.0-0.3e^{-Re_{t}^{2}},\quad f_{\eta}=e^{-2.5/(1+Re_{t}/50)} \]
where \(Re_{t}=\overline{k}^{2}/\nu\overline{\epsilon}\) is the turbulent Reynolds number. The empirical coefficients are \(C_{\eta}=0.09\), \(C_{\epsilon_{1}}=1.44\), \(C_{\epsilon_{2}}=1.92\), \(\sigma_{k}=1.0\) and \(\sigma_{\epsilon}=1.3\).
This model [84] is formulated in the original reference by considering only a single spatial derivative \(\partial/\partial y\). In that case the terms \(\overline{\mathcal{K}}\) and \(\overline{\mathcal{E}}\) are written:
\[\overline{\mathcal{K}}=2\nu\left(\frac{\partial\overline{k}^{1/2}}{\partial y}\right)^{2},\quad\overline{\mathcal{E}}=2.0\,\nu\,\nu_{T}\left(\frac{\partial^{2}\overline{U}_{i}}{\partial y^{2}}\right) \]
which, in 3D, are written identically to the two relations of the Launder & Spalding model. The damping functions are the same as for Launder & Spalding. The empirical parameters are \(C_{\eta}=0.09\), \(C_{\epsilon_{1}}=1.55\), \(C_{\epsilon_{2}}=2\), \(\sigma_{k}=1.0\) and \(\sigma_{\epsilon}=1.3\). This model differs from the previous one only by the values of the parameters \(C_{\epsilon_{1}}\) and \(C_{\epsilon_{2}}\).
In this model [103], the terms \(\overline{\mathcal{K}}\) and \(\overline{\mathcal{E}}\) are zero and the functions are given by:
\[f_{\epsilon_{1}}=1+\left(\frac{A_{c}}{f_{\eta}}\right)^{3},\quad f_{\epsilon_{2}}=1-e^{-Re_{t}^{2}},\quad f_{\eta}=(1-e^{-A_{\eta}Re_{y}})^{2}\left(1+\frac{A_{t}}{Re_{t}}\right) \]
where \(Re_{y}=\overline{k}^{1/2}y/\nu\) is the turbulent Reynolds number based on the wall distance \(y\). The coefficients \(A_{\eta}\), \(A_{t}\) and \(A_{c}\) are calibrated by comparing numerical results to experimental measurements; the obtained values are \(A_{\eta}=0.0165\), \(A_{t}=20.5\) and \(A_{c}=0.05\). The other coefficients are \(C_{\epsilon_{1}}=1.44\) and \(C_{\epsilon_{2}}=1.92\).
This model [104] is formulated for rotating flows (rotating disc), for which the independent coordinates are \(r\) (radial distance to the disc axis) and \(y\) (normal distance to the disc surface). In this case, additional terms involving the gradient of \(V_{\theta}/r\) appear in the \(\overline{k}\) and \(\overline{\epsilon}\) equations. The functions \(f_{\epsilon_{2}}\) and \(f_{\eta}\) and the constants are slightly different:
\[f_{\epsilon_{2}}=1.0-0.3e^{-Re_{t}^{2}},\quad f_{\eta}=e^{-3.4/(1+Re_{t}/50)^{2}} \]
and \(C_{\eta}=0.09\), \(C_{\epsilon_{1}}=1.44\), \(C_{\epsilon_{2}}=1.92\), \(\sigma_{k}=1.0\) and \(\sigma_{\epsilon}=1.3\).
In TrioCFD, the low-Reynolds \(\overline{k}\)– \(\overline{\epsilon}\) models of Jones & Launder [84] and Lam & Bremhorst [103] were implemented in [135]. The functions and parameters of Launder & Sharma [104] are also available.
To be documented.
Other \((\overline{k},\,\overline{\omega})\)-type models exist in the literature, such as the \((\overline{k},\,\overline{\omega})\) and \((\overline{k},\,\overline{\epsilon})\)-SST (Shear Stress Transport) models. The only recommended model is the SST one.
The standard \(k\)- \(\omega\) model is available for testing reasons only; it must not be used. The systems of equations of these models are presented in [8], from which the \((\overline{k},\,\omega)\) model is taken (originally from [184]), for which the turbulent viscosity is:
\[\nu_{T}=\frac{\overline{k}}{\tilde{\omega}},\qquad\tilde{\omega}=\max\left\{ \omega,\,C_{lim}\sqrt{\frac{2S_{ij}S_{ij}}{\beta^{*}}}\right\},\qquad\text{with }C_{lim}=\frac{7}{8} \]
The model is written:
\[\begin{aligned} \frac{\partial\overline{k}}{\partial t}+\overline{U}_{j}\frac{\partial\overline{k}}{\partial x_{j}} & = \frac{\partial}{\partial x_{j}}\left[\left(\nu+\sigma^{*}\frac{\overline{k}}{\omega}\right)\frac{\partial\overline{k}}{\partial x_{j}}\right]-\beta^{*}\overline{k}\omega+\tau_{ij}\frac{\partial\overline{U}_{i}}{\partial x_{j}}\\ \frac{\partial\omega}{\partial t}+\overline{U}_{j}\frac{\partial\omega}{\partial x_{j}} & = \frac{\partial}{\partial x_{j}}\left[\left(\nu+\sigma\frac{\overline{k}}{\omega}\right)\frac{\partial\omega}{\partial x_{j}}\right]-\beta\omega^{2}+\frac{\sigma_{d}}{\omega}\frac{\partial\overline{k}}{\partial x_{j}}\frac{\partial\omega}{\partial x_{j}}+a\frac{\omega}{\overline{k}}\tau_{ij}\frac{\partial\overline{U}_{i}}{\partial x_{j}} \end{aligned} \]
with:
\[\sigma_{d}=\begin{cases} 0 & \text{if }\frac{\partial\overline{k}}{\partial x_{j}}\frac{\partial\omega}{\partial x_{j}}\leq0\\ \sigma_{d0} & \text{if }\frac{\partial\overline{k}}{\partial x_{j}}\frac{\partial\omega}{\partial x_{j}}>0 \end{cases},\qquad f_{\beta}=\frac{1+85\chi_{\omega}}{1+100\chi_{\omega}},\qquad\chi_{\omega}=\left|\frac{\Omega_{ij}\Omega_{jk}S_{ki}}{(\beta^{*}\omega)^{3}}\right|,\qquad\Omega_{ij}=\frac{1}{2}\left(\frac{\partial\overline{U}_{i}}{\partial x_{j}}-\frac{\partial\overline{U}_{j}}{\partial x_{i}}\right) \]
The coefficients are: \(a=0.52\), \(\beta=\beta_{0}f_{\beta}\), \(\beta_{0}=0.0708\), \(\beta^{*}=0.09\), \(\sigma=0.5\), \(\sigma^{*}=0.6\), \(\sigma_{d0}=0.125\).
The baseline model is available for research purposes only; it must not be used.
The SST model is the recommended one.
Scalar (temperature) turbulence models — to be documented.
1.16.1