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# The Hybrid Approach

In many hybrid approaches the main challenge resides in coupling the two different velocity fields arising from RANS (statistically averaged) and LES (filtered) . This is often done by applying a matching criteria, i.e. assuming the same turbulent viscosity at the interface, the same kinetic energy or dissipation etc. This poses a problem since these values represent totally different properties of different velocity fields. Instead of a single one velocity field to couple RANS and LES, the model presented here allows an overlap of both fields, with the RANS model driving the near wall RANS velocity field, without damping in any way the dissipation of resolved fluctuations. Many sub-grid models assume that the flow contains an inertial sub range and hence the sub-grid motions can be assumed to be isotropic. This is true only if the grid is small enough for the anisotropy introduced by the mean shear to be neglected. At high Reynolds numbers, the refinement of the grid becomes too costly, therefore restricting the LES method to low Reynolds numbers flows. As the solid boundary is approached, the mean shear becomes high enough to introduce anisotropy across a range of diminishing scales. It is then necessary for the model to represent at the same time subgrid-scale contributions to the mean shear stress and isotropic dissipation effects.

## Modelling

The instantaneous velocity can be decomposed as

%BEGINLATEX% \begin{equation*} U = \left\langle U \right\rangle +u' \end{equation*} %ENDLATEX%

where %$\left\langle U\right\rangle$% is the averaged velocity and %$u'$% is the fluctuating component.

Schumann %REFLATEX{Sch75}% proposed to split the residual stress tensor into a "locally isotropic" part and an "inhomogeneous" part. The isotropic part is proportional to the fluctuating strain and does not affect the mean flow equations but determines the rate of energy dissipation. The inhomogeneous part is proportional to the mean strain and controls the shear stress and mean velocity profile:

%BEGINLATEX% \begin{equation*} \tau^r_{ij}- \frac{2}{3} \tau_{kk}\delta_{ij} = -\underbrace{ 2\nu_r ( \overline{S}_{ij}-\langle \overline{S}_{ij} \rangle)}_{\mbox{\small locally isotropic}} -\underbrace{ 2\nu_a \langle \overline{S}_{ij} \rangle}_{\mbox{\small inhomogeneous}} \end{equation*} %ENDLATEX%

where %$\left\langle .\right\rangle$% denotes ensemble averaging of the filtered equations. The viscosities %$\nu_r$% and %$\nu_a$% are based on fluctuating and mean strains. The isotropic part of the residual stress tensor has a zero time mean value. By refining the grid the residual stresses must tend to zero, therefore the inhomogeneous part must have a grid dependence parameter in the turbulent viscosity %$\nu_a$%. Schumann %REFLATEX{Sch75}% used a mixing length model for %$\nu_a$% with the length scale computed as %$L=\min(\kappa y , C_{10} \Delta )$%, where %$C_{10}$% is a constant that is difficult to prescribe for all types of flows. %REFLATEX{Sch75}% and %REFLATEX{GroSch77}% tried to derive a theoretical value for the constant but they were forced to introduced corrective constants to agree with a range of experiments. %REFLATEX{MoiKim82}% used the same principle of splitting the residual stress but in their mixing length model, they use the spanwise size of the cell as the length scale. They argue that for the near wall region in a channel flow, the important structures are streaks that are finely spaced on the spanwise direction. Therefore a coarse resolution in the spanwise direction would lead to larger eddies and a thicker viscous sublayer.

Sullivan et al. %REFLATEX{SulMcwMoe94}% developed a similar approach for planetary boundary layer flows but chose %$\nu_a$% to match the Monin-Obukhov similarity theory %REFLATEX{BusWynIzuBra71}%. Bagget %REFLATEX{Bag98}% used a similar approach to compare two hybrid models, one "Schumann-like" and one "DES-like" but found excessive streamwise fluctuations leading to streaks that were much too large.

In the context of hybrid LES-RANS, a blending function, %$f_b$%, can be used to introduce a smooth transition between the resolved and the ensemble averaged turbulence parts. In the present study the total residual stress is written as:

%BEGINLATEX{label="eq:tau_hyb"}% \begin{equation*} \tau^r_{ij} - \frac{2}{3} \tau_{kk}\delta_{ij}= -2\nu_r f_b ( \overline{S}_{ij}-\langle \overline{S}_{ij} \rangle) - 2(1-f_b)\nu_a \langle \overline{S}_{ij} \rangle \label{eq:tau_hyb} \end{equation*} %ENDLATEX% In this way the averaged stress would be:

%BEGINLATEX% \begin{equation*} \left\langle \tau^r_{ij} - \frac{2}{3} \tau_{kk}\delta_{ij} \right\rangle = 2(1-f_b)\nu_a \langle \overline{S}_{ij} \rangle \end{equation*} %ENDLATEX%

which is just the RANS stress,and the total shear stress would be %$2(1-f_b)\nu_a \langle \overline{S}_{ij} \rangle + \left\langle u'v'\right\rangle$%. It is therefore necessary that the blending function %$f_b$% tends to one in the region where %$\left\langle u'v'\right\rangle$% is resolved correctly and to zero in the region near the wall where the shear stress is under resolved due to the coarse grid. The total rate of transfer of energy from the filtered motions to the residual scales is given by (assuming that %$\left\langle \nu_r \overline{S}_{ij}\overline{S}_{ij}\right\rangle \approx \nu_r\left\langle \overline{S}_{ij}\overline{S}_{ij}\right\rangle$% %REFLATEX{NicBagMoiCab01}%)

%BEGINLATEX% \begin{eqnarray*} -\left\langle \tau_{ij}\overline{S}_{ij} \right\rangle =& 2\left\langle \nu_r f_b (\overline{S}_{ij}-\left\langle \overline{S}_{ij} \right\rangle )\overline{S}_{ij} \right\rangle +2 (1-f_b)\left\langle \nu_a\left\langle \overline{S}_{ij}\right\rangle \overline{S}_{ij}\right\rangle \\ =& 2f_b\nu_r(\left\langle \overline{S}_{ij}\overline{S}_{ij}\right\rangle -\left\langle \overline{S}_{ij}\right\rangle \left\langle \overline{S}_{ij}\right\rangle )+2(1-f_b)\nu_a\left\langle \overline{S}_{ij}\right\rangle \left\langle \overline{S}_{ij}\right\rangle \label{eq:tauave} \end{eqnarray*} %ENDLATEX% which shows how the RANS viscosity contributes to dissipation in association with the mean velocity only, i.e. the resolved turbulent stresses are free to develop independently from the RANS viscosity.

## The turbulent viscosities models.

For the isotropic viscosity %$\nu_r$%, %REFLATEX{Sch75}% used a model based on the sub-grid energy. %REFLATEX{MoiKim82}% used the standard %REFLATEX{Sma63}% model based on the fluctuating strain. Here the later approach is used:

%BEGINLATEX% \begin{eqnarray*} \nu_r = (C_s \Delta)^2 \sqrt{2s'_{ij}s'_{ij}} \end{eqnarray*} %ENDLATEX% with %$s'_{ij} = \overline{S}_{ij} - \langle \overline{S}_{ij} \rangle$%. In the frame of unstructured codes, the filter width is taken as twice the cell volume (%$\Delta = 2 Vol$%).

In this study, the elliptic relaxation model $\varphi-f$ of %REFLATEX{LauUriUty04}% is used to calculate the RANS viscosity. This model solves for the ratio %$\varphi = \overline{v^2}/k$% used in the turbulent viscosity as:

%BEGINLATEX{label="eq:nuphi"}% \begin{equation*} \nu_a = C_{\mu}\varphi k T \label{eq:nuphi} \end{equation*} %ENDLATEX% where %$T = \max\left(\frac{k}{\varepsilon},C_T\sqrt{\frac{\nu}{\varepsilon}}\right)$%. For the channel flow calculations presented here, the choice of the RANS models hardly makes any difference but the elliptic relaxation method has been shown to perform well on separating and impinging flows and the aim of the present case is only to show the robustness of the coupling even with a sophisticated RANS model.

The blending function has been parametrised by the ratio of the turbulent length scale to the filter width:

%BEGINLATEX{label="eq:f_blen"}% \begin{equation*} f_b = \tanh \left( C_l \frac{L_t}{\Delta} \right)^n \label{eq:f_blen} \end{equation*} %ENDLATEX% Here %$C_l=1$ and $n=1.5$% are empirical constants. These values were chosen to match the shear stress profile based on channel flow results at %$Re_{\tau}=395$% with DNS data. When using the %$\varphi-f$% model (equation %REFLATEX{eq:nuphi}%), the wall distance is not a desirable parameter and the blending function can be formulated using %$L_t = \varphi k^{3/2}/\varepsilon$%. The blending function has been devised to connect the two length scales smoothly so its value is close to zero near the wall and unity far from it. Similar functions have been used in other hybrid approaches (See %REFLATEX{Abe05}%, %REFLATEX{Ham01}% or %REFLATEX{Spe98}%). Although the function in equation %REFLATEX{eq:f_blen}% is totally empirical, it has been tested for a range of Reynolds numbers and grids and gave satisfactory results (not presented here). The function allows a higher contribution from the LES part as the grid is refined. Different coefficients have been used in the optimisation of the blending function, but the results are not greatly affected but nevertheless are always better than standard LES on the same mesh. In equation (%REFLATEX{eq:tau_hyb}%) the averaged velocity has been calculated as a running average with an averaging window of about 10 times the eddy turnover time. Although it is possible also to use plane averaging in the case of the channel flow, this was not done in order to keep the formulation applicable for 3D flows where no plane averaging is possible.

## References

%BEGINLATEX{label="Sch75"}% \bibitem[Schumann(1975)]{Sch75} U.~Schumann. Subgrid scale model for finite difference simulations of turbulent flows in plane channels and annuli. \emph{Journal of Computational Physics}, 18:\penalty0 676--404, 1975. %ENDLATEX%

%BEGINLATEX{label="GroSch77"}% \bibitem[Grotzbach and Schumann(1977)]{GroSch77} G.~Grotzbach and U.~Schumann. \newblock Direct numerical simulation of turbulence velocity -pressure, and temperature fields in channel flows. \newblock In \emph{Symposium on turbulent shear flow}, pages 18--20, 1977. %ENDLATEX%

%BEGINLATEX{label="MoiKim82"}% \bibitem[Moin and Kim(1982)]{MoiKim82} P.~Moin and J.~Kim. \newblock Numerical investigation of turbulent channel flow. \newblock \emph{Journal of Fluid Mechanics}, 118:\penalty0 341--377, 1982. %ENDLATEX%

%BEGINLATEX{label="SulMcwMoe94"}% \bibitem[Sullivan et~al.(1994)Sullivan, McWilliams , and Moeng]{SulMcwMoe94} P.~Sullivan, J.~McWilliams, and C.~Moeng. \newblock A subgrid-scale model for large eddy simulation of planetary boundary layer. \newblock \emph{Boundary layer methodology}, 71\penalty0 (247--276), 1994. %ENDLATEX%

%BEGINLATEX{label="BusWynIzuBra71"}% \bibitem[Businger et~al.(1971)Businger, Wyngaard, Izumi, and Bradley]{BusWynIzuBra71} J.~A. Businger, J.~C. Wyngaard, Y.~Izumi, and E.~E. Bradley. \newblock Flux-profile relationships in the atmospheric surface layer. \newblock \emph{Journal of Atmospheric Science}, 28:\penalty0 181--189, 1971. %ENDLATEX%

%BEGINLATEX{label="Bag98"}% \bibitem[Baggett(1998)]{Bag98} J.S Baggett. \newblock On the feasibility of merging {LES} with {RANS} for the near-wall regions of attached turbulent flows. \newblock In \emph{Annual research Briefs}, pages 267--276. Center for turbulence research, Stanford, CA, 1998. %ENDLATEX%

%BEGINLATEX{label="NicBagMoiCab01"}% \bibitem[Nicoud et~al.(2001)Nicoud, Baggett, Moin, and Cabot]{NicBagMoiCab01} F.~Nicoud, J.~S. Baggett, P.~Moin, and W.~Cabot. \newblock Large eddy simulation wall-modeling based on suboptimal control theory and linear stochastic estimation. \newblock \emph{Physics of Fluids}, 13:\penalty0 2968--2984, 2001. %ENDLATEX%

%BEGINLATEX{label="LauUriUty04"}% \bibitem[Laurence et~al.(2004)Laurence, Uribe, and Utyuzhnikov]{LauUriUty04} D.~Laurence, J.C. Uribe, and S.~Utyuzhnikov. \newblock A robust formulation of the v2-f model. \newblock \emph{Flow, Turbulence and Combustion}, 73:\penalty0 169--185, 2004. %ENDLATEX%

%BEGINLATEX{label="Sma63"}% \bibitem[Smagorinsky(1963)]{Sma63} J.~Smagorinsky. \newblock General circulation experiments with the primitive equations: {I} the basic equations. \newblock \emph{Monthly Weather Review}, 91:\penalty0 99--164, 1963. %ENDLATEX%

%BEGINLATEX{label="Abe05"}% \bibitem[Abe(2005)]{Abe05} K~Abe. \newblock A hybrid {LES RANS} approach using an anisotropy-resolving algebraic turbulence model. \newblock \emph{International Journal of Heat and Fluid Flow}, 26:\penalty0 204--222, 2005. %ENDLATEX%

%BEGINLATEX{label="Ham01"}% \bibitem[Hamba(2001)]{Ham01} F.~Hamba. \newblock An attempt to combine large eddy simulation with the $k-\varepsilon$ model in a channel flow calculation. \newblock \emph{Theoretical and Computational Fluid Dynamics}, 14:\penalty0 323--336, 2001. %ENDLATEX%

%BEGINLATEX{label="Spe98"}% \bibitem[Speziale(1998)]{Spe98} C.G. Speziale. \newblock Turbulence modeling for time-dependant {RANS} and {VLES}: a review. \newblock \emph{AIAA Journal}, 26:\penalty0 179--184, 1998. %ENDLATEX%

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