*(by Valerio D’Alessandro v.dalessandro@univpm.it)*

In the last years Thermofluids research group of the Marche Polytechnic University started an active development of CFD solvers and models adopting OpenFOAM.

The main areas of interest are the following:

- development of low-dissipative solvers for incompressible flows based on high-order Runge-Kutta schemes;
- transition models for RANS equations;
- new hybrid RANS/LES models;

The previous work has been performed in collaboration with students, Ph.D. candidates, post-doc researches and professors of the Industrial Engineering and Mathematical Sciences Department of Marche Polytechnic University. Two Ph.D. thesis, involving significant OF work, were successfully defended.

Our work is constantly carried out two different small Linux clusters and on supercomputing facilities, likes those available at CINECA.

## Development of low-dissipative solvers for incompressible flows

This research activity was started in order to obtain low-dissipative solvers for DNS/LES of incompressible flows involving heat transfer. The OpenFOAM community is focusing on reducing the numerical dissipation for both incompressible and compressible flows. Various efforts are in progress, especially relating to low-dissipation Runge-Kutta (RK) methods for compressible flows, so we focused here on incompressible flows, with an emphasis on the time integration approach.

More specifically we have implemented different solvers adopting both implicit and explicit approaches for time integration. In the first case singly diagonally implicit Runge-Kutta (SDIRK) schemes were employed. It is worth noting that an iterated PISO-like procedure for handling the pressure-velocity coupling in each RK stage was used. For the explicit approach a projected-type RK methods was considered.

We found that the RK-based solution techniques, implemented using only OpenFOAM standard classes, completely canceled the numerical diffusion induced by the time integration approach, but demanded more computational resources than standard OpenFOAM solvers, [1]. The solvers proved reliable, robust and accurate on different computational setups, *i.e.* grids and spatial schemes, as well as in a large-eddy simulation. Implicit technique has proved computationally more efficient than explicit one. For this reason we have applied our implicit SDIRK solver within an ongoing research activity at Marche Polytechnic University. More in depth this research is aimed in the evaluation of dimples effect on low-Re number operating laminar airfoils. The NACA 64-014A airfoil has been considered. A row of dimples was placed at 55 % of chord length with one dimple diameter spacing in the span-wise direction. Our LES computations, validated with experimental tests performed at Environmental Wind Tunnel of our University, show that dimpled airfoil configuration experience a significant laminar separation bubble reduction, [2]. This result is particularly appealing in several applications, since dimples realization is objectively cheaper than other boundary layer control devices.

(a) Mesh representation (b) Vortical structure representation

Figure 1: NACA 64014A airfoil LES, [2]

## Transition models for RANS equations

In this research we have devised a transition approach for RANS based on the coupling between the γ-Re_{Θ,t} approach and the Spalart-Allmaras (SA) turbulence model. The motivation behind this activity is to obtain a suitable transition model requiring a limited computational cost. Indeed, DNS approach is still inapplicable to the study of engineering interest problems due to the significant computational resources required. On the other hand, the applicability of the LES technique is very limited for the same reason. Moreover standard RANS models are not always reliable to transitional flow predictions because they assume a fully turbulent regime.

It should be noted that γ-Re_{Θ,t} models were initially coupled with the SST k-ω turbulence model by its developers, but the γ-Re_{Θ,t} model could be applied to other models too. In this work, we have presented a γ-Re_{Θ,t} RANS model for transitional flows based on the SA equation. The main reasons behind this approach are as follows:

- the SA model offers very reliable results for external flow applications; and
- the SA model has lower computational costs than the SST k-ω equations.

For the analyzed cases, we thoroughly investigated the role of the different empirical correlations involved in the transition model. This is a crucial issue because a marked sensitivity of the γ-Re_{Θ,t} model to its empirical correlations in the k-ω context was apparent from the literature. Analyzing all the results collected in [3], we surmise that the Malan et al. and Langtry and Menter approaches produce better results in a wide range of cases.

It is worth noting that we obtained a good feedback from our model’s implementation in predicting airfoil performance (see Fig. 2), as well as in the computation of the natural transition on a zero pressure gradient flat plate.

(a) Velocity magnitude contour plot (b) Pressure distribution along the airfoil wall

Figure 2: E387 at Re = 2 10^{5} results, [3].

Lastly, the model briefly described here and a test case are freely downloadable on *gitHub *at the follwing address: https://github.com/vdalessa/gammaReThetatSA.

## New hybrid RANS/LES models

The Detached-Eddy Simulation (DES) technique introduced by Spalart et al. [4], is a hybrid RANS/LES approach that operates like RANS in the near-wall regions and like LES in separated flow zones. DES is probably the most popular hybrid RANS/LES method becauseit is simple to apply to a wide range of existing RANS models. It is also particularly attractive because it produces good results in several conditions with fewer computational resources than standard LES. The original DES formulation is based on the Spalart-Allmaras (SA), [5], eddy-viscosity RANS model, for which LES behavior is achieved by modifying the length scale used in the turbulence model.

DES models have also since been introduced that are based on two-equation RANS. In particular, Travin et al. [6] developed a DES model based on the SST k-ω model, and other modifications were proposed in various papers too (for this specific model), that involved replacing the length scale in the transport equations for turbulence.

It is only very recently that DES approaches based on the v^{2}-f RANS model have appeared in the literature, [7]. A first advantage of this model by comparison with other RANS methods lies in its ability to accurately predict the near-wall effects without specific treatments or expedients. DES approaches based on k-equation are very appealing because they use a length scale based on flow properties, not on grid size like the SA model. In LES mode, the v^{2}-f DES models also have a transport equation for the Sub-Grid Scales (SGS) kinetic energy that is less empirical than the SGS modified turbulent viscosity, ν, used in the standard SA-DES. The main drawback of these models lies in that they use four additional equations for turbulence modeling, while other DES techniques are less costly from this point of view.

In this research we have exploited the open-source features of OpenFOAM to implement the v^{2}-f DES model. We considered Dirichlet-type wall boundary condition for ε equation. From a theoretical point of view, this choice does not affect the quality of the results, while it provides a suitable equivalent boundary condition that is easier to be coded within OpenFOAM if compared with the Neumann condition proposed by Jee and Shariff.

The model was successfully validated computing the flow field past a cylinder at Re = 3900, Fig. 3. Ongoing work is devoted to the DDES development and to the application of this model to problems involving heat transfer.

Figure 3: Vortical structure, identified by the Q-criterion, isosurface colored by pressure.

Q = 0.5u^{2}_{∞}/D^{2} at T = 500D/u_{∞} , [8].

## Acknowledgements

We acknowledge the CINECA Award N. HP10CV8M72 YEAR 2015 under the ISCRA initiative, for providing high-performance computing resources and support.

We acknowledge the CINECA Award N. HP10CDBABZ YEAR 2016 under the ISCRA initiative, for providing high-performance computing resources and support.

## References

[1] V. D’Alessandro, L. Binci, S. Montelpare, and R. Ricci. On the development of OpenFOAM solvers based on explicit and implicit highorder RungeKutta schemes for incompressible ows with heat transfer. Computer Physics Communications , 222:14 30, 2018.[2] L. Binci, G. Clementi, V. D’Alessandro, S. Montelpare, and R. Ricci. Study of the ow eld past dimpled aerodynamic surfaces: numerical simulation and experimental verication. Journal of Physics: Conference Series , 923, 2017.

[3] Valerio D’Alessandro, Sergio Montelpare, Renato Ricci, and Andrea Zoppi. Numerical modeling of the ow over wind turbine airfoils by means of SpalartâAllmaras local correlation based transition model. Energy , 130:402 419, 2017.

[4] 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. In C. Liu and Z. Liu, editors, 1st AFOSR Int. Conf. on DNS/LES , Ruston, LA, August 48 1997.

[5] P.R. Spalart and S.R. Allmaras. A one-equation turbulent model for aerodynamic flows. La Recherche Aérospatiale , 1:521, 1994.

[6] A. Travin, M. Shur, M. Strelets, and P.R Spalart. Physical and numerical upgrades in the detached eddy simulation of complex turbulent ows. In Proceedings of EU-ROMECH Colloquium. , pages 239-254, 2002.

[7] K. Jee and K. Shari. Detachededdy simulation based on the v 2 f model. International Journal of Heat and Fluid Flow , 46:84101, 2014.

[8] V. D’Alessandro, S. Montelpare, and R. Ricci. Detachededdy simulations of the flow over a cylinder at Re = 3900 using OpenFOAM. Computers & Fluids , 136:152 169, 2016.