Benchmark on the Aerodynamics of a Rectangular 5:1 Cylinder (BARC)

Flows Around Bodies

Underlying Flow Regime 2-15

Evaluation

Comparison of CFD Calculations with Experiments

Bulk parameters

The main flow bulk parameters obtained in the different wind tunnel and numerical studies are reported in Tables 7 and 8: ${\left.t-avg(C_{D})\right.}$ and ${\left.t-avg(C_{L})\right.}$ are the time- and spanwise-averaged drag and lift coefficients per unit length, respectively; ${\left.t-std(C_{L})\right.}$ is the standard deviation of the time variation of the lift coefficient; ${\left.St_{D}=f_{s}D/U\right.}$ is the Strouhal number, where the shedding frequency ${\left.f_{s}\right.}$ is evaluated from the time fluctuations of the lift coefficient or from pressure or velocity time signals (we refer to the single cited articles for more details).

First of all, we remark that at present, among the different wind tunnel tests carried out in the framework of the BARC benchmark, bulk parameters are available only from Schewe [‌53, 54] and from Bartoli et al. [‌5] (only the Strouhal number). In general, several wind tunnel data are available in the literature for the flow around the same body geometry as far as the Strouhal number is concerned and only a few for the mean drag coefficient; these data are also reported in Table 7 for comparison. Conversely, bulk-parameter values computed in 25-36 simulations of the BARC configuration are available. The histograms of the bulk parameters obtained by computational simulations are plotted in Fig. 4. For the sake of brevity, detailed values are not given herein (we refer to the cited papers which may be made available upon request to the interested readers) and only the range of the results obtained in all the simulations carried out in each single contribution is reported in Table 8. The ensemble average over the available data and the standard deviation are also reported in Table 8. The data of the 2D LES in [‌1] and of the simulations in [‌71] have been excluded from the computation of the ensemble average and of the standard deviations, since they deviate significantly from the other data (see also the discussion below). Moreover, 2D LES is a priori expected to give unreliable results, while the simulations in [‌71] are probably affected by a too small size of the computational domain.

Figure 4: Computational results: histograms of the bulk parameters; (a) ${\left.t-avg(C_{D})\right.}$ over 36 realizations, (b) ${\left.t-avg(C_{L})\right.}$ over 36 realizations, (c) ${\left.t-std(C_{L})\right.}$ over 30 realizations, (d) ${\left.t\right.}$ over 25 realizations.

The values of ${\left.t-avg(C_{D})\right.}$ obtained in most of the simulations are very close to 1 and this is in good agreement with the available wind tunnel data. In particular, [‌54] obtains ${\left.t-avg(C_{D})\right.}$ equal to ${\left.1.029\right.}$ at ${\left.{\text{Re}}_{D}=26400\right.}$ , while the measured values vary very little ${\left.(0.94\leq t-avg(C_{D})\leq 1.05)\right.}$ for ${\left.2\times 10^{4}\leq {\text{Re}}_{D}\leq 2\times 10^{6}\right.}$ . Moreover, an overall good agreement is observed between the predictions obtained in the various numerical studies, in spite of the previously outlined differences in numerics, modelling and simulation set-up. Indeed, the standard deviation of the data remains lower than 5% of the ensemble-averaged prediction. It will be shown in the following that, although the characteristics of flow on the cylinder sides significantly vary among the different simulations, the near-wake structure and, consequently, both the base mean pressure (see Fig. 5b) and the mean drag coefficient, show only small differences. Only a few data deviate significant from the others: the 2D LES in [‌1] shows a discrepancy with the experimental value of [‌54] of 35% and the simulations in [‌71] a maximum discrepancy of 28%. Therefore, it appears that the choice of the computational domain may significantly affect also the prediction of quantities which are rather insensitive to modeling and to the other simulation parameters. By excluding these data, the LES contributions to BARC give ${\left.t-avg(C_{D})\in [0.96,1.04]\right.}$ , while the URANS and hybrid simulations give ${\left.t-avg(C_{D})\in [0.965,1.295]\right.}$ . The Strouhal number is another quantity for which the predictions given in the different simulations are rather close to each other and in good agreement with the available wind tunnel data. Since ${\left.St_{D}\right.}$ gives the dimensionless frequency of the vortex shedding behind the cylinder, this is a further confirmation that the dynamics of the near wake is satisfactorily captured in all the simulations in spite of the differences in the flow features on the cylinder lateral sides.

Conversely, the oscillations in time of the lift coefficient are very sensitive to the complex dynamics of the flow on the lateral cylinder sides. Indeed, a large spread of the numerical predictions is observed for the standard deviation of the lift coefficient. Note how the ensemble average of ${\left.t-std(C_{L})\right.}$ obtained in the different simulations is significantly larger than the only available wind tunnel value (Tab. 7). In this case, Arslan et al. [‌1] does not observe a significant difference between 2D and 3D simulations. However, in general, this quantity seems to be sensitive to many numerical, modeling and simulation parameters. Mannini and Schewe [‌28] show a significant impact of numerical dissipation, [‌10, 11] and [‌17, 18] point out a decrease of ${\left.t-std(C_{L})\right.}$ with increasing grid resolution, [‌17, 18] a decrease with increasing Reynolds number and [‌10, 11, 27] a decrease with increasing the spanwise extent of the computational domain. Finally, turbulence modeling has also a significant impact on the predictions of ${\left.t-std(C_{L})\right.}$ : lower values are obtained in [‌27] in DES simulations than in URANS, while [‌71] generally find lower values in LES than in IDDES. Note, however, that the IDDES predictions of ${\left.t-std(C_{L})\right.}$ of [‌71] are significantly larger than those of the DES in [‌27]. Finally, [‌17, 18] and [‌1] also observe remarkable effects of the SGS model in the predictions of this quantity obtained in LES simulations.

Finally, the mean lift coefficient is a priori expected to be zero. Although values of ${\left.t-avg(C_{L})\right.}$ close to zero are obtained in most of the simulations, there are a few cases in which its absolute value is significant (Bruno et al. [‌10] and Wei and Kareem [‌71]). This might be due to the fact that the time interval used to compute the averaged quantities is not large enough to obtain statistical convergence. Nonetheless, in Bruno et al. [‌10], a careful check of the convergence of the averaged quantities is made, and, hence, at least in that case, the statistical sample may be assumed to be adequate. Therefore, it may be argued that a ${\left.t-avg(C_{L})\right.}$ value significantly different from zero is an indication of an asymmetry of the mean flow {which may be triggered by very small perturbations of different nature}. This point will be more deeply analysed in the following.

**Table 7: Bulk parameters: wind tunnel data**
${\left.\right.}$	${\left.t-avg(C_{D})\right.}$	${\left.t-avg(C_{L})\right.}$	${\left.t-std(C_{L})\right.}$	${\left.St_{D}\right.}$
Bartoli et al. [‌5]	—	—	—	${\left.0.12\right.}$
Schewe [‌53, 54]	${\left.1.029\right.}$	${\left.\sim 0\right.}$	${\left.\sim 0.4\right.}$	${\left.0.111\right.}$
Nakamura and Mizota [‌33]	${\left.\sim 1\right.}$	—	—	—
Nakamura and Yoshimura [‌34]	${\left.\sim 1\right.}$	—	—	—
Nakamura and Nakashima [‌35]	—	—	—	${\left.0.115\right.}$
Nakamura et al. [‌36]	—	—	—	${\left.0.118\right.}$
Okajima [‌41]	—	—	—	${\left.0.115\right.}$
Okajima [1983]^[1]	—	—	—	${\left.0.105\right.}$
Parker and Welsh [‌43]	—	—	—	${\left.0.105\right.}$
Stokes and Welsh [‌60]	—	—	—	${\left.0.105\right.}$
Knisely [‌22]	—	—	—	${\left.0.106\right.}$
Matsumoto [2005]^[1]	—	—	—	${\left.0.132\right.}$
Ricciardelli and Marra [‌47]	—	—	—	${\left.0.116\right.}$

↑ ^1.0 ^1.1 Reported in Mannini et al. [‌27]

**Table 8: Bulk parameters: numerical results**
${\left.\right.}$	${\left.t-avg(C_{D})\right.}$	${\left.t-avg(C_{L})\right.}$	${\left.t-std(C_{L})\right.}$	${\left.St_{D}\right.}$
Arslan et al. [‌1] 2D LES	${\left.1.39\right.}$	—	${\left.0.82\right.}$	${\left.0.16\right.}$
Arslan et al. [‌1] 3D LES	${\left.0.984{\text{ -- }}1.04\right.}$	—	${\left.0.59{\text{ -- }}0.84\right.}$	${\left.0.107{\text{ -- }}0.113\right.}$
Mannini et al. [‌27]	${\left.0.968{\text{ -- }}1.071\right.}$	${\left.0.0032{\text{ -- }}0.047\right.}$	${\left.0.42{\text{ -- }}1.075\right.}$	${\left.0.094{\text{ -- }}0.102\right.}$
Mannini et al. [‌26]	${\left.1.015{\text{ -- }}1.172\right.}$	—	${\left.0.108{\text{ -- }}1.12\right.}$	${\left.0.087{\text{ -- }}0.105\right.}$
Mannini and Schewe [‌28]	${\left.0.965{\text{ -- }}1.016\right.}$	${\left.-0.087{\text{ -- }}0.085\right.}$	${\left.0.173{\text{ -- }}0.553\right.}$	${\left.0.087{\text{ -- }}0.119\right.}$
Ribeiro [‌46]	${\left.1.17\right.}$	—	${\left.0.9\right.}$	${\left.0.073\right.}$
Grozescu et al. [‌18]	${\left.0.97{\text{ -- }}0.98\right.}$	${\left.-0.097{\text{ -- }}0.0043\right.}$	${\left.0.52{\text{ -- }}0.65\right.}$	${\left.0.107{\text{ -- }}0.11\right.}$
Grozescu et al. [‌17]	${\left.0.96\right.}$	${\left.0.0022\right.}$	${\left.0.35\right.}$	${\left.0.122\right.}$
Bruno et al. [‌10]^[1]	${\left.0.96{\text{ -- }}1.03\right.}$	${\left.-0.315{\text{ -- }}-0.0024\right.}$	${\left.0.2{\text{ -- }}0.73\right.}$	${\left.0.112{\text{ -- }}0.122\right.}$
Bruno et al. [‌8]	${\left.1.03\right.}$	—	${\left.0.73\right.}$	${\left.0.112\right.}$
Wei and Kareem [‌71]	${\left.1.165{\text{ -- }}1.305\right.}$	${\left.-0.33{\text{ -- }}0.42\right.}$	${\left.0.495{\text{ -- }}1.465\right.}$	—
ensemble average^[2]	${\left.1.002\right.}$	${\left.-0.04\right.}$	${\left.0.54\right.}$	${\left.0.107\right.}$
standard deviation^[2]	${\left.0.047\right.}$	${\left.0.10\right.}$	${\left.0.25\right.}$	${\left.0.011\right.}$
Shimada and Ishihara [‌55]	${\left.0.975\right.}$	—	${\left.0.03{\text{ -- }}0.12\right.}$	${\left.0.103{\text{ -- }}0.119\right.}$

↑ Also in Grozescu et al. [‌17]
↑ ^2.0 ^2.1 The data of the 2D LES in [‌1] and those in [‌71] have not been included in the computation of the ensemble average and of the standard deviation

Main flow features and statistics

The distribution of the pressure coefficient $Cp$, averaged in time ($t-avg$ in the following), in the spanwise direction ($z-avg$ in the following) and between the upper and lower half perimeters ($side-avg$ in the following), is plotted in Fig. \ref{fig:cp_mean}. Figure \ref{fig:cp_mean}(a) collects {the} wind tunnel measurements, while Figure \ref{fig:cp_mean}(b) the computational results. The abscissa $s$ denotes the distance from the cylinder leading edge measured along the cylinder side (see also Fig. \ref{fig:domain_cfd}). For the sake of completeness Figure \ref{fig:cp_mean}(a) (and Fig. \ref{fig:cp_std}(a) in the following) also includes the data obtained in high turbulent incoming flows by \citet{le2009}, even if the experimental setup significantly differs from the BARC main one. The data of Galli(2005) and Matsumoto(2005) are taken from \citet{mannini2011}. As a first remark, the mean pressure values given by the different wind tunnel and computational contributions to BARC on the rear side of the cylinder, {$5.5 \le s/D \le 6$}, are very close to each other and in good agreement with the experimental data available in the literature (also reported in Fig. \ref{fig:cp_mean}(a)), with the only exception of the RANS computation with the RSM model in \citet{ribeiro2011}. As already mentioned, this leads to very similar predictions of the time averaged drag.