# Comparison of wind pressure coefficients on a cube between experimental results and some CFD approaches with code_saturne

## Abstract

This article is an investigation about the CFD modeling of strain effects generated by wind on buildings. This analysis has been done with the open- source software code_saturne^{[1]} with which several turbulence models have been tested. The objective is to highlight strengths and weaknesses of tested models thanks to a comparison with experimental data. Final goal is to select a good compromise between precision and speed for industrial studies.

*Keywords : CFD, wind engineering, code_saturne*

## Introduction

### Context

Currently, in the civil engineering world, wind impact on buildings is mostly evaluated with prescriptive methodology as Eurocode^{[2]}. However, it’s sometimes necessary to have more detailed results, and CFD is one way of achieving this. But there is a large set of possibilities to model a given situation. Compromises have then to be made, between the required precision and the simulation costs in terms of time and ressources. These compromises especially concern :

- The turbulence modeling. Historically, RANS models with eddy viscosity hypothesis, such as k-ε or k-ω approaches, are often used thanks to their low numerical cost and their ease of use. With years, lots of variants have been developped, each adapted to a given set of situations.

In the RANS family, models with second moment closures have also been developped, named as Reynolds Stress Models (RSM). These are more complete, because they propose to solve transport equations for the Reynolds stresses.

More recently, the growth of computing ressources allow to use LES. However, for large industrial cases, this approach remain very expensive and is not realistic. - The meshing approach, depending of available numerical schemes and chosen turbulent model. Semi-automatic meshes, with boundary layers generation, have made huge progress these two last decades. However, in most CFD software, some approaches (LES, …), still require a high quality mesh with hexahedral structure.

### Objective

Pros and cons of the different approaches are described in a lot of publications. In this work, we propose to realise an overview of different possibilities proposed by the open-source software code_saturne^{[1]}, developped by EDF. The objective of the analysis is to compare results of some modeling approaches with two experimental data concerning wind pressure coefficients on a simple cube. We will focus on turbulence, with three sets of models :

- Models based on a high-Reynolds approach
- Models based on a low-Reynolds approach
- LES

Results difference between models and experimental data, as well as between models themselves, will allow to select best approaches in terms of precision but also in terms of numerical costs. Specifically, we will focus on the relevance of the high-Reynolds approach, far less demanding in mesh size and mesh quality compared to the others.

## Used experimental data

Two sets of experimental data are considered in this analysis :

- Data from a full-scale cube
- Data from a wind tunnel cube

Complementarity of these two data sets allow to give credit to obtained results, as they are pretty close (cf. results section). Below is a quick description of the two configurations.

### Full-scale Silsoe cube

The full-scale experimental configuration is the Silsoe Experimental Building (SEB) shown in Figure 1. It’s a cube of dimension [6m×6m×6m], situated at the Silsoe Research Institute (SRI), specifically constructed to allow full-scale experiments. This cube has been built in a flat terrain with short grass, so the effective roughness length is constant and well-known about 0.01m, and the boundary layer profile can be easily modeled in a numerical approach.

Pressure sensors are located on some cube’s faces along lines at mid-width or mid-depth (grey squares in Figure 1). More details concerning these tapping points and the realised measures can be found in Richards and Hoxey, 2008^{[5]}. Finally, available experimental data include mean, fluctuating and spectral properties (cf. Richards et al., 2007^{[3]}, Richards and Hoxey, 2008^{[5]}).

### Wind-tunnel cube

The second set of experimental data used in this work is from a wind-tunnel cube of dimension [0.2m×0.2m×0.2m] by Irtaza et al., 2013^{[4]}. It’s a scale of 1:30 compared to the full-scale cube, as shown in Figure 2.

The average wind speed at eave height of the cube was 4.81m/s. This value allows to have a Reynolds number in the range of 0.72×10^{5} to 1.09×10^{5}, a similar range compared to the full-scale experiment. The location of pressure sensors are shown in Figure 3 for the top face. All details concerning the wind-tunnel experiments and the data assimilation are available in Irtaza et al., 2013^{[4]}. The selected configuration is a headwind without angle of attack.

### Obtained experimental profiles

The velocity profile and the turbulent intensity on the full-scale cube were measured by Richards et al., 2007^{[3]} and are respectively shown in Figure 4 and Figure 5 for an headwind, with zero angle of attack. The first one is quite similar to a logarithmic profile, while the longitudinal turbulence intensity is around 18%.

Similar data are available for the wind-tunnel cube with Irtaza et al., 2013^{[4]}, also shown in Figure 4 and Figure 5. Mean velocity profile as well as turbulence intensity are quite similar to those of the full-scale experiment.

## CFD modeling

### code_saturne software

Computational simulations were realised with code_saturne. It’s a free, open-source software developed by EDF R&D. As written on code-saturne website, “It solves the Navier-Stokes equations with a finite volume approach for 2D, 2D-axisymmetric and 3D flows, steady or unsteady, laminar or turbulent, incompressible or weakly dilatable, isothermal or not, with scalar transport”. More information is available in the publication from Frédéric Archambeau et al., 2004^{[1]}, as well as on website https://www.code-saturne.org/cms/web/. The used version for this work was the v7.0.5.

### Model description

#### Geometry and boundary conditions

The modeled cube in simulations is the same as the wind-tunnel cube of dimension [H×H×H] where H=0.2m. The used computational domain is presented in Figure 6, Figure 7 and Figure 8 :

- The stream wise at inlet is
*X*direction, without angle of attack. - The domain size is 29H along stream wise direction
*X*, 13H along lateral direction*Y*and 5H along vertical direction*Z* - The cube is located 6H downstream of the inlet, 22H upstream of the outlet, and is centered along lateral direction
*Y*with 6H on both sides of the cube.

With such dimensions, flow around obstacle doesn’t have effect on inflow or outflow boundary conditions, which is a requirement for most CFD applications.

Inflow boundary condition is configured with the following log law for the velocity :

U(z) = \cfrac{U_{\tau}}{\kappa} ln \Big(\cfrac{z+z_0}{z_0}\Big)

where U_{τ} = 0.648m/s is the friction velocity, κ=0.41 is Von Karman’s constant and Z_{0} =0.01m is the surface roughness length parameter. The obtained profile, shown in Figure 9, ensures the value U(z=H)=4.81m/s, as in the wind-tunnel experiment previously described.

From the Figure 5, the turbulence intensity of full-scale and wind-tunnel experiments are decreasing for z/H < 2, with a value around 18% at z/H=1. A simplified profile, shown in Figure 10, is used in our simulations, with a linear decrease from I(z/H=0)=20% to I(z/H=2)=16%, and a constant turbulence intensity of 16% for z/H > 2.

Turbulent parameters to specify at the inlet section depends on the chosen approach (cf. following section), but the used relations are always the same :

- The kinetic energy of turbulence is calculated as follows :

k(z) = \cfrac{3}{2} \big(U_{avg} I(z)\big)^2

where U_{avg} is the mean wind speed at inlet and I(z) is the turbulence intensity.

- The dissipation rate is defined according to the following equation :

ε(z) = C_{\mu}^{3/4}\cfrac{k^{3/2}}{l_s}

where C_{μ}=0.09 and l_{s} is the turbulence integral length scale.

- The specific rate of dissipation is linked to previous definitions with this relation :

ω(z) = \cfrac{ε}{C_{\mu}k}

- In the case of RSM, the Reynolds tensor is defined as following :

R_{11}=\cfrac{2k}{3}; R_{22}=\cfrac{2k}{3}; R_{33}=\cfrac{2k}{3}; R_{12}=0; R_{13}=0; R_{23}=0

- Concerning LES, default code_saturne settings are conserved.

#### Numerical settings

Following hypotheses are adopted for simulations :

- Air is considered incompressible with ρ=1.2kg/m
^{3}. - We focus on wind dynamic effects. Consequently, gravity isn’t taken into account in simulations.
- There is no thermal resolution.
- For all RANS simulations, a steady state approach is adopted, with a local adaptive time step ensuring a CFL below 1.
- For LES simulation, an unsteady approach is adopted, with a constant time step of 5×10
^{-4}s. - A second order centered scheme is adopted for the velocity resolution, whereas a first order upwind scheme is used for turbulence variables concerning RANS simulations.

#### Mesh arrangement

Two meshes are used, depending on the chosen turbulence model (cf. following section). In any case, they are unstructured but hexa-dominant, with a mesh size from 40mm to 2.5mm, the main refinement being around the cube and for the height ratio z/H < 2, as shown in Figure 11. They were generated with a semi-automatic tool and an Octree type algorithm. The difference between them concerns discretisation of the viscous sub-layer :

- The first mesh “
*HR_Mesh*” is configured to not resolve viscous sub-layer. - The second one “
*LR_Mesh*” includes a boundary layer, with a first cell height of 0.5mm.

In addition, usual convergence tests were realised to validate the relevance of these meshes.

### Selected approaches for turbulence

Lots of turbulence models are available on code_saturne (cf. the theory guide^{[6]} and the user guide^{[7]}). Six are used in this work, including :

- First order RANS models : k-ε linear production, k-ω SST and v
^{2}f BL v^{2}k - Second order RANS models (RSM) : SSG and EBRSM
- LES model : WALE

Depending on their properties, these models were categorized in three approaches, as described below.

#### High-Reynolds approach

In the high-Reynolds approach, the viscous sub-layer is not resolved : a scalable two-scale wall function (log law) is used on cube faces and on the ground, with the mesh “*HR_Mesh*“. This way, we ensure y+ around 30 at walls. This strategy is used for the two following turbulence models :

- k-ε linear production
- RSM-SSG

#### Low-Reynolds approach

In the low-Reynolds approach, the viscous sub-layer is resolved. The mesh “*LR_Mesh*” is used, without wall function. This allows to obtain y+ around 1 at walls. This strategy is used for the four following turbulence models :

- k-ε linear production
- k-ω SST
- v
^{2}f BL v^{2}k - RSM-EBRSM

It should be noted that this approach is not recommended with the k-ε linear production model. This will nevertheless allow to compare results of the two approaches with it.

#### LES approach

For our LES simulation, the mesh “*LR_Mesh*” is used with a one scale log law. It ensures y+ around 1, as well as Δx+ and Δz+ < 20.

- The chosen LES model is : LES-WALE

## Results

### Convergence and post-processed variables

To ensure simulations convergence, sensors were located in some areas of interest :

- Close to the inlet section, to control inlet profiles.
- Around the cube, to control consistency of results.
- Close to the outlet section, to control any backflow or numerical instability coming from this boundary condition.

All realised simulations converged. Once the convergence reached, averaged values were built during the end of simulations, as shown in Figure 12 and Figure 13 for the LES calculation. Results analyse presented in following sections only concerns these mean values.

In addition, wind pressures measured are expressed in a non-dimensional form, with the definition of the pressure coefficient C_{p} :

C_p = \cfrac{p_{mean}-p_0}{\cfrac{1}{2} \rho U_h^2}

where p_{mean} is the built averaged pressure as previously described, p_{0} is the spatial mean pressure at the inlet section, and U_{h} = 4.81m/s is the inlet velocity at the cube height H=0.2m.

### Discussion

Full-scale results from Richards et al.^{[3]} and wind-tunnel results from Irtaza et al.^{[4]} match well, so firstly, analysis is based on comparison with these experimental data. Figure 14 and Figure 15 respectively show pressure coefficients along the mid-width and the mid-depth of the cube for all realised CFD simulations, and the comparison with mentioned papers.

Firstly, from Figure 14, we can say that along the main stream wise :

- k-ε linear production model gives the worst trends, both high and low Reynolds approaches. It overestimates pressure coefficients on most of the windward face, with Cp twice as important as experimental results at the top of this face (distance ratio between 0.8 and 1). In contrast, negative pressure coefficients on the top face aren’t as pronounced as in Richards et al.
^{[3]}and Irtaza et al.^{[4]}, apart from the presence of a strong minimum at the beginning of the face (distance ratio between 1 and 1.2). - Other models give results close to experimental data on the windward face (distance ratio between 0 and 1) but only the LES simulation is near these data on the top face (distance ratio between 1 and 2), although there is some differences between k-ε/v2f simulations on one side (Cp around 50% higher than experimental data), and k-ω SST/SSG/EBRSM simulations on the other side (values and trends closer to experimental data).
- Finally, on the leeward face (distance ratio between 2 and 3), all simulations are close enough, with pressure coefficients a little higher than those of experimental data.

From Figure 15, it’s apparent that along the mid-depth of the cube :

- Only the LES simulation provides results close enough to experimental data, although there is still a difference from 10% to 20%.
- All other tested models predict higher pressure coefficients, with substantial difference compared to experimental data. More specifically, the behaviour around the two edges of the top face (distance ratio = 1 and distance ratio = 2) is significantly different between simulations and experiments.
- As previously observed with Figure 14, k-ε linear production and v2f models propose the worst results compared to both experimental data.

Then, Figure 16 to Figure 18 show now pressure coefficient map and contours of three faces of the cube (respectively the windward, the top and one of the two side faces) for all realised simulations. Assuming that the LES simulation is the closest from experimental data, comparison between simulation results is mainly based on this one.

From these figures, it’s possible to notice that :

- On the windward face, k-ε linear production model (both approaches) seems to be outside the box, with pressure coefficients far more higher than in every other models. Others are more similar, especially in the localisation of the flow stagnation point, although Cp values are lower in the LES simulation.
- On the top face, in accordance with observations from Figure 14 and Figure 15, none of the RANS models predict similar results to LES. From the very negative values of pressure coefficients near to the windward edge of this top face, this difference is potentially caused by an overestimation of the wake recirculation just behind this edge and a corresponding lack of velocity on the remaining part of the top face in the tested RANS models , especially concerning the k-ε LP and the v2f models (RSM-EBRSM and k-ω SST models are closer to LES results). It can be seen in Figure 19, which shows streamlines along a mid-width slice around the cube for LES, k-ε LP and k-ω SST simulations.
- On the side face, observations are quite similar to those observed on the top face : RSM-EBRSM and k-ω SST models are the closest to LES results, although the difference remains substantial.

## Conclusion & perspective

This work allowed to test some turbulence models of code_saturne in a configuration with two available experimental data. It clearly highlights benefits of LES in wind engineering, given that none of the tested RANS models can accurately predict pressure coefficient on this basic configuration. More specifically, observed results show that k-ε linear production and v2f models should not be used for these applications. In view of the other tested models, it appears that the k-ω SST model with a low Reynolds approach could be a correct strategy, given that RSM models can be fairly unstable in more complex configuration.

To complete this approach, some points need to be considered, as the impact of a fully unstructured mesh (tetra-dominant with a prismatic boundary layer, for example), and the computation stability in a more complex configuration. Finally, although this study primarily focused on average pressure coefficients, it should be kept in mind that one of the desirable contributions of CFD in civil engineering would be to have information concerning unsteady fluctuations (peak pressure coefficients).

**Thank you for reading this article, and for visiting our website! 🙂**

## Nomenclature

** CFD** : Computational Fluid Dynamics

*: Large Eddy Simulation*

**LES***: Reynolds Averaged Navier Stokes*

**RANS***: Reynolds Stress Models*

**RSM***: Silsoe Experimental Building*

**SEB***: Silsoe Research Institute*

**SRI**## References

[1] Frédéric Archambeau, Namane Méchitoua and Marc Sakiz, ‘Code_Saturne: a Finite Volume Code for the Computation of Turbulent Incompressible Flows’, International Journal on Finite Volumes, volume 1, 2004.

[2] Eurocode 1: Actions on structures — Part 1-4: General actions — Wind actions, NF EN 1991-1-4, 2005.

[3] Richards P.J., Hoxey R.P., Connell B.D., Lander D.P. 2007. ‘Wind-tunnel modelling of Silsoe Cube’, Journal of Wind Engineer-

ing and Industrial Aerodynamics , pp. 1384-1399.

[4] H. Irtaza, R.G. Beale, M.H.R. Godley, A. Jameel, ‘Comparison of wind pressure measurements on Silsoe experimental building fro full-scale observation, wind-tunnel experiments and various CFD techniques’, International Journal of Engineering, Science and Technology, Vol. 5, No. 1, 2013, pp. 28-41.

[5] Richards P.J., Hoxey R.P. 2008. ‘Wind loads on the roof of a 6 m cube’, Journal of Wind Engineering and Industrial Aerodynam-

ics , pp. 984-993, 96.

[6] code_saturne 7.0 Theory Guide, 2021.

[7] code_saturne 7.0 practical user’s guide, 2021.