The PANS k–ε model in a zonal hybrid RANS–LES formulation

doi:10.1016/j.ijheatfluidflow.2014.01.002

International Journal of Heat and Fluid Flow

Volume 46, April 2014, Pages 112-126

https://doi.org/10.1016/j.ijheatfluidflow.2014.01.002 Get rights and content

Highlights

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Evaluated in fully developed channel flow and embedded LES in a hump flow.
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$f_{k}$ is either constant or computed.
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For channel flow it gives good results for a large span of Reynolds numbers $(4000 ⩽ {Re}_{τ}$ ⩽ 32,000).
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Three grids are used in the wall-parallel planes ( $32^{2}, 64^{2}$ and $128^{2}$ ); the model gives grid-independent results.
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The turbulent viscosity is also grid-independent (same on the three grids $32^{2}, 64^{2}$ and $128^{2}$ ).

Abstract

A new approach to use the partially averaged Navier–Stokes (PANS) model as a hybrid RANS–LES model is presented. It is evaluated in fully developed channel flow and embedded LES in a hump flow. For the channel flow, the two RANS–LES interfaces are parallel to the walls. In the URANS region, $f_{k}$ is set to one. In the LES region, $f_{k}$ is set to a constant value (the baseline value is $f_{k} = 0.4$ ) or it is computed. It is found that the new model gives good results for channel flow for a large span of Reynolds numbers ( $4000 ⩽ {Re}_{τ}$ ⩽ 32,000). In the channel flow simulations, three different grids are used in the wall-parallel planes, $32^{2}, 64^{2}$ and $128^{2}$ , and the model yields virtually grid-independent flow fields and turbulent viscosities. Embedded LES is used for the hump flow which is well predicted. The RANS–LES interface is normal to the flow from the inlet. RANS is used upstream of the interface. Downstream this interface, RANS is used near the wall and LES is used away from the wall.

Introduction

Wall-bounded Large Eddy Simulation (LES) is affordable only at low Reynolds number. At high Reynolds number, the LES must be combined with a URANS treatment of the near-wall flow region. There are different methods for bridging this problem such as Detached Eddy Simulation (DES) (Spalart et al., 1997, Spalart, 2000, Shur et al., 2008), hybrid LES/RANS (Davidson and Peng, 2003, Temmerman et al., 2005) and Scale-Adapted Simulations (SAS) (Menter and Egorov, 2010, Egorov et al., 2010); for a review, see Fröhlich and von Terzi (2008). The two first classes of models take the SGS length scale from the cell size whereas the last (SAS) involves the von Kármán lengthscale.

The DES, hybrid LES/RANS and the SAS models have one thing in common: in the LES region, the turbulent viscosity is reduced. This is achieved in different ways. In some models, the turbulent viscosity is reduced indirectly by increasing the dissipation term in the k equation as in two-equation DES (Travin et al., 2000). In other models, such as in the two-equation X-LES (Kok et al., 2004) and in the one-equation hybrid LES–RANS (Davidson and Billson, 2006, Temmerman et al., 2005), it is accomplished by reducing the length scale in both the expression for the turbulent viscosity as well as for the dissipation term in the k equation.

In the partially averaged Navier–Stokes (PANS) model (Girimaji, 2006a) and the Partially Integrated Transport Model (PITM) (Schiestel and Dejoan, 2005, Chaouat and Schiestel, 2005), the turbulent viscosity is reduced by decreasing the destruction term in the dissipation ( $ε$ ) equation which increases $ε$ . This decreases the turbulent viscosity in two ways: first, the turbulent viscosity is reduced because of the enhancement of $ε$ , and, second, the turbulent kinetic energy, k, decreases because of the increased dissipation term, $ε$ .

In the SAS model based on the $k – ω$ model, the turbulent viscosity is reduced by an additional source term, $P_{SAS}$ , in the $ω$ equation. The source term is activated by resolved turbulence; in steady flow it is inactive. When the momentum equations are in turbulence-resolving mode, $P_{SAS}$ increases which increases $ω$ . This decreases the turbulent viscosity in two ways: first, directly, because $ω$ appears in the denominator in the expression for the turbulent viscosity, $ν_{t}$ , and, second, because k is reduced due to its increased dissipation term $β^{*} k ω$ .

The PANS model and the PITM models are very similar to each other although their derivations are completely different. The only difference in the models is that in the PANS model the turbulent diffusion coefficients in the k and $ε$ equations are modified. These two models do not use the filter width, and can hence be classified as URANS models. On the other hand, a large part of the turbulence spectrum is usually resolved which is in contrast to standard URANS models. PANS and PITM models have by Fröhlich and von Terzi (2008) been classified as second-generation URANS models, or 2G-URANS models.

The PANS model is used in the present work. In PANS, two new coefficients are introduced, $f_{k}$ and $f_{ε}$ . The former denotes the ratio of the modeled to the total turbulent kinetic energy, and the latter denotes the corresponding ratio of the dissipation. Since small turbulent scales affected by dissipation are not resolved in the present study, the coefficient $f_{ε}$ is set to one. The coefficient $f_{k}$ may vary from one (RANS) to zero (DNS). For LES it should take values somewhere in between. When using PANS in turbulence-resolving mode, $f_{k}$ is usually set to a constant value (both in space and time). In Girimaji (2006b) they used a constant $f_{k}$ , mostly 0.4 and 0.5. Frendia et al. (2006) used a constant $f_{k}$ coefficient of $0.2$ and set $f_{ε} = 0.5$ , 0.667 or 1.0. They found that $f_{ε} = 0.667$ , i.e. $f_{k} / f_{ε} = 0.3$ , gave the best results on the chosen grid. In Ji et al., 2011, Ji et al., 2012 they used constant values of $f_{k}$ for simulating cavitating flow around an hydrofoil and a cavitating propeller. They found that the smaller $f_{k}$ , the higher the shedding frequency. They recommend a value of $f_{k} = 0.2$ . In Girimaji and Abdol-Hamid (2005) they used either a constant $f_{k}$ in space (in the range $0.4 < f_{k} < 1$ ) or they let $f_{k}$ vary. They compute it as $3 {(Δ_{\min} / L_{t})}^{2 / 3}$ where $Δ_{\min}$ is the smallest grid cell size and $L_{t} = {(k + k_{res})}^{3 / 2} / ε$ where $k + k_{res}$ denotes modeled plus resolved turbulent kinetic energy. They obtained the space-varying $f_{k}$ from a pre-cursor steady RANS simulation and then kept it constant in time in the PANS simulations. The lowest constant value, $f_{k} = 0.4$ , gave the best results. Lakshmipathy et al. (2011) used PANS for vortex shedding around a circular cylinder at high Reynolds number. The evaluated different constant $f_{k}$ values (0.5, 0.6, 0.7, 1.0) and they found that $f_{k} = 0.5$ gave best agreement with experiments. Basu et al. (2007) propose a new form of varying $f_{k}$ . They present results also for constant $f_{k}$ using 0.3, 0.75 and 0.85. They show that the varying $f_{k}$ and $f_{k} = 0.3$ give good results. Furthermore, it is shown that when the varying $f_{k}$ method is used, $f_{k}$ takes values between 0.2 and 0.4 in the turbulence-resolving regions. An extension of PANS, based on a four-equation $k – ε – ζ – f$ model, was recently proposed (Basara et al., 2011). They compute $f_{k}$ as $C_{μ}^{- 1 / 2} {(Δ / L_{t})}^{2 / 3}$ where $Δ = V^{1 / 3}$ (V denotes cell volume). A near-wall low-Reynolds number capability was added to PANS so that the equations can be integrated all the way up to the wall (Ma et al., 2011). In that work, it was furthermore shown that the PANS model is a good SGS model for wall-resolved LES at low Reynolds numbers. It was found that a constant value of $f_{k} = 0.4$ was appropriate. Davidson and Peng, 2011, Davidson and Peng, 2013 present embedded LES applied to channel flow and the flow over a hump. Davidson (2012) used PANS in LES mode of a developing boundary layer and the flow over a backstep. Different constant $f_{k}$ values were evaluated in these works. A value of $f_{k} = 0.4$ was recommended.

In the present work, the PANS model is used as a zonal hybrid LES/RANS model to simulate wall-bounded flow at high Reynolds number. $f_{k} = 1$ in the near-wall region, and $f_{k} < 1$ in the LES region. Different constant values of $f_{k}$ have been used in the literature, see above. A value of $0.4 ⩽ f_{k} ⩽ 0.5$ has been shown to be best in most of the works, except in cavitating flow where a value of 0.2 was found to be optimal. Based on the work in the literature, a baseline value of $f_{k} = 0.4$ is chosen for the LES region in the present work. Constant $f_{k}$ values in the range $0.2 ⩽ f_{k} ⩽ 0.6$ are also evaluated. Although most investigations in the literature have used a constant $f_{k}$ , it should conceptionally be dependent on the grid size. A smaller $f_{k}$ should be used when the grid is refined and vice versa. Hence, simulations are also carried out in the present study using a variable $f_{k}$ in space where $f_{k} = C_{μ}^{- 1 / 2} {(Δ / L_{t})}^{2 / 3}$ as proposed by Basara et al. (2011).

The paper is organized as follows: the equations and the modeling are presented in the next section followed by a description of the numerical method. The following section presents and discusses the results and conclusions are drawn at the end of the paper.

Section snippets

Mean flow equations

The momentum equations with an added turbulent viscosity reads $\frac{\partial {\bar{u}}_{i}}{\partial t} + \frac{\partial {\bar{u}}_{j} {\bar{u}}_{i}}{\partial x_{j}} = δ_{1 i} - \frac{1}{ρ} \frac{\partial \bar{p}}{\partial x_{i}} + \frac{\partial}{\partial x_{j}} ((ν + ν_{t}) \frac{\partial {\bar{u}}_{i}}{\partial x_{j}})$ where the first term on the right side is the driving pressure gradient in the streamwise direction, which is used only in the channel flow simulations.

The LRN PANS $k – ε$ turbulence model

The low-Reynolds number partially averaged Navier–Stokes (LRN PANS) turbulence model reads (Ma et al., 2011) $\begin{matrix} \frac{\partial k}{\partial t} + \frac{\partial {\bar{u}}_{j} k}{\partial x_{j}} = \frac{\partial}{\partial x_{j}} [(ν + \frac{ν_{t}}{σ_{ku}}) \frac{\partial k}{\partial x_{j}}] + P_{k} - ε \\ \frac{\partial ε}{\partial t} + \frac{\partial {\bar{u}}_{j} ε}{\partial x_{j}} = \frac{\partial}{\partial x_{j}} [(ν + \frac{ν_{t}}{σ_{ε u}}) \frac{\partial ε}{\partial x_{j}}] + C_{ε 1} f_{1} P_{k} \frac{ε}{k} - C_{ε 2}^{*} \frac{ε^{2}}{k} \\ P_{k} = ν_{t} (\frac{\partial {\bar{u}}_{i}}{\partial x_{j}} + \frac{\partial {\bar{u}}_{j}}{\partial x_{i}}) \frac{\partial {\bar{u}}_{i}}{\partial x_{j}}, C_{ε 2}^{*} = C_{ε 1} + \frac{f_{k}}{f_{ε}} (C_{ε 2} f_{2} \end{matrix}$

Numerical method

An incompressible, finite volume code is used (Davidson and Peng, 2003). The numerical procedure is based on an implicit, fractional step technique with a multigrid pressure Poisson solver and a non-staggered grid arrangement. For the momentum equations, central differencing is used in space in the channel flow simulations; in the hump flow simulations 5% upwinding is employed using the second-order van Leer (1974) scheme.

For both flows, the Crank–Nicolson scheme is used in the time domain and

Channel flow

Fully developed channel flow is computed for Reynolds numbers ${Re}_{τ} = u_{τ} δ / ν = 4000$ , 8000, 16,000 and 32,000, where $δ$ denotes half channel width. The baseline mesh has $64 \times 64$ cells in the streamwise (x) and spanwise (z) directions, respectively. The size of the domain is $x_{\max} = 3.2, y_{\max} = 2$ and $z_{\max} = 1.6$ . A simulation with twice as large domain in the $x – z$ plane ( $x_{\max} = 6.4$ and $z_{\max} = 3.2$ ) with $128 \times 128$ cells was also made for ${Re}_{τ} = 4000$ , and identical results were obtained as for the smaller domain. The number of

Conclusions

A new approach for using PANS as a zonal hybrid RANS–LES model has been presented. It has been evaluated for channel flow at different Reynolds numbers ( $4000 ⩽ {Re}_{τ}$ ⩽ 32,000) and gives good agreement with the log-law. Furthermore, it was found that the model gives virtually grid-independent results when refining the grid in the wall-parallel planes ( $N_{x} \times N_{z} = 32 \times 32, 64 \times 64$ and $128 \times 128$ ). When using a constant $f_{k}$ , the turbulent viscosities obtained on these three grid are nearly the same. An SGS model using

Acknowledgements

The financial support of SNIC (the Swedish National Infrastructure for Computing) for computer time at C3SE (Chalmers Center for Computational Science and Engineering) is gratefully acknowledged. This project was financed by the EU project ATAAC (Advanced Turbulence Simulation for Aerodynamic Application Challenges), Grant Agreement No. 233710. http://cfd.mace.manchester.ac.uk/ATAAC/WebHome

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The PANS k–ε model in a zonal hybrid RANS–LES formulation

Highlights

Abstract

Introduction

Section snippets

Mean flow equations

The LRN PANS k–ε turbulence model

Numerical method

Channel flow

Conclusions

Acknowledgements

Int. J. Heat Fluid Flow

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Int. J. Heat Mass Transfer

Int. J. Heat Fluid Flow

Int. J. Heat Fluid Flow

Int. J. Heat Fluid Flow

Int. J. Heat Fluid Flow

Int. J. Heat Fluid Flow

Int. J. Heat Fluid Flow

J. Comput. Phys.

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The PANS $k – ε$ model in a zonal hybrid RANS–LES formulation

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