Passive Scalar Transport Modeling for Hybrid RANS/LES Simulation

Abstract

A transport model for hybrid RANS/LES simulation of passive scalars is proposed. It invokes a dynamically computed subgrid Prandtl number. The method is based on computing test-filter fluxes. The formulation proves to be especially effective on coarse grids, as occur in DES. After testing it in a wall resolved LES, the present formulation is applied to the Adaptive DDES model of Yin et al. (Phys. Fluids 27, 025105 2015). It is validated by turbulent channel flow and turbulent boundary layer computations.

Appendix: Adaptive DDES model

The adaptive-DDES formulation of Yin et al. [4] is summarized here. The shielding function for Delayed Detached Eddy Simulation [3] is adopted:

$$\begin{array}{@{}rcl@{}} f_{d} &=& 1- tanh([C_{d1}r_{d}]^{C_{d2}}) \qquad C_{d1}=8, C_{d2}=3\\ r_{d} &=& \frac{k/\omega+\nu}{\kappa^{2} {d_{w}^{2}} \sqrt{S^{2}+{\Omega}^{2}}} \end{array} $$

(A1)

where k/ω is the RANS eddy viscosity formula, ν is the molecular viscosity, κ the vonKarman constant, d _w the wall distance, S and Ω rate of strain and rate of rotation, respectively.

It is incongruous to refer to the flow as, simultaneously, being Reynolds averaged and an instantaneous realization: reference to ‘RANS’ and ‘Eddy Simulation’ regions is loose terminology. Detached eddy simulation is best described as a length scale formulation, used to simulate realizations. The length scales and eddy viscosity are defined by:

$$\begin{array}{@{}rcl@{}} \left. \begin{array}{lll} \ell_{DDES} &=& \ell_{RANS} - f_{d}\max(0, \ell_{RANS}-\ell_{LES})\\ \ell_{RANS} &=& \frac{\sqrt{k}}{\omega}\\ \ell_{LES} &=& C_{DES}{\Delta}\\ {\Delta} &=& f_{d}V^{1/3} + (1-f_{d})h_{max}\\ \nu_{T} &=& \ell_{DDES}^{2}\,\omega \end{array}\right\} \end{array} $$

(A2)

This ν _T defines the production term of the k equation in the k−ω RANS model [20], leaving all the other terms unaltered.

$$\begin{array}{@{}rcl@{}} \frac{Dk}{Dt} &=& 2\nu_{T}|S|^{2} - C_{\mu}k\omega + \nabla\cdot[(\nu + \sigma_{k}(k/\omega))\nabla{}k] \\ \frac{D\omega}{Dt} &=& 2C_{\omega1}|S|^{2} - C_{\omega2}\omega^{2} + \nabla\cdot[(\nu + \sigma_{\omega}(k/\omega))\nabla\omega] \end{array} $$

(A3)

The standard constants are invoked,

$$ C_{\mu} = 0.09, \quad \sigma_{k} = 0.5, \quad \sigma_{\omega} = 0.5, \quad C_{\omega1} = 5/9, \quad C_{\omega2} = 3/40 $$

(A4)

It was shown by [4] that an adaptive procedure can improve predictions. In their method, the dynamic procedure of LES [24] is applied to the eddy viscosity (A2) to evaluate C _{D E S} locally within the flow. Following that method, define the test filter stresses,

$$\begin{array}{@{}rcl@{}} L_{ij} &=& \widehat{{u}_{i}{u}_{j}} -\hat{{u}}_{i}\hat{{u}}_{j} \\ M_{ij} &=& ({\Delta}^{2}\widehat{{\omega}{S}_{ij}}-\hat{\Delta}^{2}\hat{{\omega}}\hat{{S}}_{ij} ) \end{array} $$

(A5)

These are used to determine C _{D E S} ; but, in order for the test filter to be valid, a significant portion of the inertial range needs to be resolved. The coarse meshes that sometimes are used in DES may not capture enough of the small scales. For this reason, a bound is placed on the computed value of C _{D E S}

$$\begin{array}{@{}rcl@{}} C_{dyn}^{2} &=& \max\left( 0, 0.5\frac{L_{ij}M_{ij}}{M_{ij}M_{ij}}\right) \\ C_{DES} &=& \max(C_{lim}, C_{dyn}) \end{array} $$

(A6)

where the lower limit is determined by:

$$\begin{array}{@{}rcl@{}} C_{lim} &=& C_{DES}^{0}\left[1-\tanh\left( \alpha\exp\left( \frac{-\beta{}h_{max}}{L_{k}}\right)\right)\right]\\ C_{DES}^{0} &=& 0.12, \quad \alpha = 25,\quad \beta = 0.05, \quad L_{k} = \left( \frac{\nu^{3}}{\epsilon}\right)^{1/4} \\ \epsilon &=& 2(C_{DES}^{0}h_{max})^{2}\omega{}|S|^{2} + C_{\mu}k\omega \end{array} $$

(A7)

Equation A7 gagues the mesh resolution by comparing mesh size to the Kolmogoroff scale, L _k . If the grid is coarse, C _{D E S} reverts to a default value of 0.12.