UFR 2-15 Evaluation

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

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Bulk parameters

The main flow bulk parameters obtained in the different wind tunnel and numerical studies are reported in Tables 7 and 8: t - avg(CD) and t - avg(CL) are the time- and spanwise-averaged drag and lift coefficients per unit length, respectively; t - std(CL) is the standard deviation of the time variation of the lift coefficient; StD = fsD/U is the Strouhal number, where the shedding frequency fs 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 [‌5354] 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.

UFR2-15 figure04.png
Figure 4: Computational results: histograms of the bulk parameters; (a) over 36 realizations, (b) over 36 realizations, (c) over 30 realizations, (d) over 25 realizations.

The values of t - avg(CD) 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 t - avg(CD) equal to 1.029 at ReD = 26400, while the measured values vary very little (0.94 ≤ t - avg(CD) ≤ 1.05) for 2 × 104 ≤ ReD ≤ 2 × 106. 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 t - avg(CD) ∈ [0.96, 1.04], while the URANS and hybrid simulations give t - avg(CD) ∈ [0.965, 1.295]. 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 StD 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 t - std(CL) 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, [‌1011] and [‌1718] point out a decrease of t - std(CL) with increasing grid resolution, [‌1718] a decrease with increasing Reynolds number and [‌101127] a decrease with increasing the spanwise extent of the computational domain. Finally, turbulence modeling has also a significant impact on the predictions of t - std(CL): 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 t - std(CL) of [‌71] are significantly larger than those of the DES in [‌27]. Finally, [‌1718] 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 t - avg(CL) 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 t - avg(CL) 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
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Bartoli et al. [‌5]
Schewe [‌5354]
Nakamura and Mizota [‌33]
Nakamura and Yoshimura [‌34]
Nakamura and Nakashima [‌35]
Nakamura et al. [‌36]
Okajima [‌41]
Okajima [1983][1]
Parker and Welsh [‌43]
Stokes and Welsh [‌60]
Knisely [‌22]
Matsumoto [2005][1]
Ricciardelli and Marra [‌47]
  1. 1.0 1.1 Reported in Mannini et al. [‌27]

Table 8: Bulk parameters: numerical results
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Arslan et al. [‌1] 2D LES
Arslan et al. [‌1] 3D LES
Mannini et al. [‌27]
Mannini et al. [‌26]
Mannini and Schewe [‌28]
Ribeiro [‌46]
Grozescu et al. [‌18]
Grozescu et al. [‌17]
Bruno et al. [‌10][1]
Bruno et al. [‌8]
Wei and Kareem [‌71]
ensemble average[2]
standard deviation[2]
Shimada and Ishihara [‌55]
  1. Also in Grozescu et al. [‌17]
  2. 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. 5. Figure 5(a) collects the wind tunnel measurements, while Figure 5(b) collects the computational results. The abscissa denotes the distance from the cylinder leading edge measured along the cylinder side (see also Fig. 2). For the sake of completeness Figure 5(a) (and Fig. 8(a) in the following) also includes the data obtained in high turbulent incoming flows by Le et al. [‌23], even if the experimental setup significantly differs from the BARC main one. The data of Galli (2005) and Matsumoto (2005) are taken from Mannini et al. [‌27]. 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 ≤ s/D ≤ 6, are very close to each other and in good agreement with the experimental data available in the literature (also reported in Fig. 5(a)), with the only exception of the RANS computation with the RSM model in Ribeiro [‌46]. As already mentioned, this leads to very similar predictions of the time averaged drag.

UFR2-15 figure05.png
Figure 5: Side-averaged, spanwise-averaged and time-averaged distributions: wind tunnel (a) and computational (b) results.

Conversely, a significant spread of both wind tunnel and computational pressure values is observed in Fig. 5 on the lateral side of the cylinder. Figure 6 summarizes the ensemble statistics of the Cp distributions on the lateral cylinder side; in particular, the range of the wind tunnel and numerical Cp values is reported for 24 locations over the cylinder lateral side, together with the median, the 25-th and the 75-th percentile values p25 and p75, respectively, whiskers and outliers, if any. Points are drawn as outliers if they are larger than p75 + w(p75 - p25) or smaller than p25 - w(p75 - p25), being w = 1.5 the maximum whisker length. It is worth pointing out that the subset of wind tunnel pressure data over which statistics are evaluated does not include the realizations obtained in high turbulent incoming flows [‌23], because such measurements do not belong to the same ensemble of smooth conditions.

The results of Bartoli et al. [‌5] are rather away from the remaining ones, which conversely well agree with each other. As for the numerical results, although most of the distributions are contained in a narrower range than the wind tunnel one (Figure 6), the shape of the mean pressure distribution significantly varies among the different simulations. The spread in the predictions of the mean pressure distribution is strictly linked with the differences in the mean flow topology obtained in the various simulations (see Fig. 7). Therefore, before analyzing in more detail the behaviour of the mean pressure coefficient on the cylinder lateral size, let us briefly describe the main features of the mean flow topology. The mean flow on the cylinder lateral side is characterized by a main recirculation zone (or main vortex as in [‌8]), whose size and shape significantly vary in the different contributions. To give a more precise quantification of this variability, Table 9 shows the x coordinate at which the mean recirculation zone ends (mean reattachment location xr) and the coordinates of the centre of the main recirculation zone. Only a subset of the contributions to BARC, for which these data were made available, is considered; as previously for the bulk coefficients, detailed values are not given in Table 9 for the sake of brevity (we refer to the cited papers), and only the range of the results obtained in all the simulations carried out in each single contribution is reported, together with the ensemble average and the standard deviation over all the available data. The value of xr deduced by Matsumoto et al. [‌29] from wind tunnel pressure measurements is also shown for reference {both in Table 9 and in Fig. 7 (red line). The data in Tab. 9 confirm a significant variability of the length and of the x position of the centre of the mean vortex, while the normal distance from the cylinder of its centre remains almost constant. This leads to very different shapes and curvature of the mean streamlines at the edge of this main recirculation zone. Note how both the 2D simulation with no turbulence model by Tamura et al. [‌61] and the 2D LES by Arsan et al. [‌1] significantly underestimate the size of the main recirculation region. This again confirms that these kinds of 2D simulations, as it could have been anticipated, do not give reliable results. In most of the numerical contributions to BARC, a smaller recirculation is also visible very close to the lateral wall and immediately downstream of the upstream corner, which was already detected in [‌8] (see in particular Fig. 11 of [‌8]}, in which the mean flow structures are sketched). Again, its dimensions and shape vary significantly among the different simulations; in fact, in some cases it is hardly visible.

UFR2-15 figure06.png
Figure 6: Statistics of the side-averaged, spanwise-averaged and time-averaged distributions: wind tunnel (a) and computational (b) results. The data of the 2D LES in [‌1] and those in [‌71] have not been included.

The distribution of the standard deviation of the time variation of the Cp pressure coefficient, averaged also spanwise and between the upper and lower half perimeters, is plotted in Fig. 8. As an overall remark, a large spread among the different experimental and numerical predictions of this quantity is evident. In this case the variability is larger for the numerical results than for the wind tunnel measurements. More particularly, on the upstream face of the cylinder the Cp standard deviation is very low in all the numerical simulations, while in most of the experiments is roughly around 0.05. This may be explained by the difference in the free-stream conditions, which are smooth in most of the numerical simulations, while turbulent fluctuations are present in the experiments, although the turbulence intensity is kept low. This is confirmed also by the fact that the data of [‌23], obtained for high levels of oncoming flow turbulence and reported in Fig. 8, show very large values of the Cp standard deviation in the upstream part of the cylinder side. As for its mean value, the largest variability Cp standard deviation is observed on the cylinder lateral side. In all cases there is a peak located slightly upstream of the reattachment of the main mean recirculation vortex, in the zone where the mean Cp increases. On average, the peak is located more downstream and is more intense in numerical simulations than in experiments. By comparing the distributions of the standard deviation of Cp obtained on the cylinder lateral surface in DES and LES simulations to those given by URANS models, quite surprisingly the differences in the intensity and location of the main peak are rather small; therefore, it seems that turbulence modeling has an effect on the dynamics of the flow over the lateral cylinder sides which is comparable to that of other sources of uncertainties present in the simulations and experiments. The main difference is that in DES and LES simulations, consistently with the wind tunnel measurements, the value of the standard deviation of Cp has an unique peak along the cylinder side, while in the RANS ones a minimum is also found at a distance of approximately 2D from the upstream corner. The reasons for this behaviour are not clear at this stage.

UFR2-15 figure07.png
Figure 7: Time and spanwise-averaged streamlines

Table 9: Mean reattachment location and coordinates of the centre of the mean recirculation zone
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Arslan et al. [‌1] 2D LES
Arslan et al. [‌1] 3D LES
Mannini et al. [‌27][1]
Mannini et al. [‌25]
Grozescu et al. [‌17]
Grozescu et al. [‌18]
Bruno et al. [‌8] [2] [2]
Ensemble average
Standard deviation
Matsumoto et al. [‌29]
  1. Reported in Arslan et al. [‌1]
  2. 2.0 2.1 In percent of the ensemble-average value

UFR2-15 figure08.png
Figure 8: Side-averaged, spanwise-averaged distributions of the standard deviation in time of the : wind tunnel (a) and computational (b) results

Sensitivity to modelling and simulation parameters

Let us analyze herein in more detail the sensitivity of the numerical predictions of the pressure distribution and of the mean flow on the cylinder lateral surface to different modelling and simulation parameters, in order to possibly identify some trends or explanations of the result dispersion.

Let us consider first the mean pressure distribution, which has been previously shown to be linked to the mean flow topology. The approach to turbulence, viz. URANS, LES or hybrid RANS/LES, is expected to play a key role. However, none of these approaches seem to reduce the result dispersion, i.e. results significantly deviating from the ensemble average are found for LES, URANS and hybrid simulations.

For URANS, as expected, the largest sensitivity is to turbulence model; in particular, [‌46] obtains a mean Cp distribution very close to the ensemble average for the realizable k - ε and the RSM models (the latter, however, giving a wrong prediction of the base pressure), while the distributions obtained with the RNG k - ε and the SST k - ω models largely overestimate the pressure increase in the downwind zone of the cylinder side. Finally, the URANS simulation of [‌28], which uses the LEA k - ω model, provides a mean Cp distribution close to the ensemble average.

For LES and hybrid URANS/LES approaches, the analysis is more complicated, since different parameters seem to have a comparable impact. As a first remark, it is pointed out once again that the 2D LES in [‌1] yields results significantly different from the 3D ones, and, in particular, a main recirculation zone much shorter than in 3D simulations. The effects of SGS modeling in LES have been investigated in Grozescu et al. [‌1718] and in [‌1]. Both these studies show that the SGS model has a limited impact, in spite of the quite coarse grid resolution. As for the hybridization strategy in hybrid simulations, the mean Cp distributions obtained in the IDDES simulations of Wei and Kareem [‌71] deviate much more from the ensemble average than those of the DES simulations in [‌28]; this is probably due to the large difference in the domain size adopted in the studies. The sensitivity to grid refinement has been investigated in LES simulations by Grozescu et al. [‌1718], Bruno et al. [‌1011] and Wei and Kareem [‌71]; in Grozescu et al. [‌1718] and Wei and Kareem [‌71] the refinement is carried out in all directions, while Bruno et al. [‌1011] focus on refinement in the spanwise direction. All these studies agree on the qualitative behaviour, i.e. grid refinement leads to a decrease of the main vortex length and to an upstream displacement of its centre with the previously highlighted consequences on the pressure distribution. However, from a quantitative viewpoint, the effect of grid refinement is more limited in Grozescu et al. [‌1718] than in Bruno et al. [‌1011] and Wei and Kareem [‌71]; this is rather surprising since the grid resolution in Grozescu et al. [‌1718] is significantly coarser than in the other studies and it might be due to the use of unstructured grids, which allows more local refinements, and of the VMS-LES approach, which has been shown in previous studies to give more accurate results of classical LES on coarse grids. In particular, Bruno et al. [‌1011] carried out a systematic investigation of the effects of the grid resolution in the spanwise direction for a spanwise domain size L/D = 5, by considering δz/D = 0.21, 0.1, 0.05. The results show a striking impact of the spanwise grid resolution on the flow features on the cylinder side and on the related quantities; however, for the finest considered grid a very short length of the main recirculation zone is obtained, close to that of the 2D LES in [‌1], and, consequently the mean pressure distribution becomes quite far from the ensemble average of the numerical predictions. The reasons of this behaviour are not clear at the moment and this point definitely deserves further investigation. The sensitivity to the numerical method, and, in particular, to the numerical dissipation is investigated only in the DES simulations of Mannini and Schewe [‌28], in which it is shown that the length of the main recirculation zone becomes smaller with decreasing numerical dissipation. Note, however, that the sensitivity is likely to depend on the specific used numerical scheme (see also the discussion in CFD methods). The effect of the Reynolds number has finally been investigated in Grozescu et al. [‌1718], in which it is shown that the main recirculation region becomes shorter as the Reynolds number is increased; however, the impact is definitely limited compared to the overall dispersion of the results.

Finally, the analysis of the distribution of pressure standard deviation is even more complex. In general, since the peak location is related to the length of the mean recirculation region, reference can be made to the discussion above. As for the intensity of the peak and of the pressure fluctuations in general, differences in the results obtained by different turbulence approaches, which capture different amounts of turbulence fluctuations, can be expected; for instance, larger fluctuations are expected in LES simulations, especially those associated with very fine grid resolutions, than in URANS. The impact of the approach to turbulence has been previously discussed, pointing out that the observed behaviours do not completely fulfill the expectations and are not fully understood at present.

Symmetry of the mean flow

Pressure coefficient distributions previously commented result from the averaging between the upper and lower half perimeters, so that possible asymmetries with respect to the axis can not be pointed out. In fact, the pressure measurements in wind tunnel tests [‌57] and some computational simulations [‌1117] have systematically, independently and explicitly discussed unexpected emerging differences in the statistics of pressure fields over the upper and lower lateral surfaces. Significant differences between the upper and lower surfaces (up to Δ(t - avg(Cp)) ≈ 0.1 and Δ(t - std(Cp)) ≈ 0.05 are obtained in wind tunnel measurements and in some computational simulations.

The causes of the occurrence of single asymmetric flow realizations are at this stage unclear. Systematic errors due to insufficient sampling window in time and related lack of convergence in evaluating the pressure statistical moment are avoided by convergence assessment, as described at the beginning of the evaluation section. In the preliminary wind tunnel studies by [‌7] and [‌5] different possible causes of the non-symmetry were identified, the main being the disturbances in the incoming flow (e.g. by the upstream pitot-static tube), misalignment of the model with the incoming flow (e.g. misalignment of the model and/or asymmetry in the flow) and inaccuracies in the model geometry (e.g. degree of sharpness of the four edges, pressure taps disturbances and or quality of the rectangular shape of the model section). These three causes are separately and systematically addressed in [‌7], while  [‌5] focus on the last ones. Based on the performed investigations, [‌7] conjecture that the asymmetry results from an inaccuracy in model shape, while misalignment of the relative flow is tentatively suspected by [‌5]. In spite of a number of trial re-alignments and model flippings, neither study yields a perfect symmetry of the time-averaged pressure field along the side surfaces.

The computational models are not affected by the wind tunnel inaccuracies cited above. In particular, the mathematical model accounts for perfectly rectangular cylinder section with perfectly sharp edges, while the laminar incoming flow and the other boundary conditions are fully symmetric. Hence, asymmetric flow can be triggered only by numerical issues. Local asymmetries exist in the spatial grid generated by [‌11] outside the grid boundary layer at the wall, while the mesh generated by [‌17] is fully symmetric with respect to the plane. The procedures adopted for the numerical discretization and numerical solution of the governing equations (e.g. multigrid partition and parallelisation) can represent further potential causes of an asymmetric solution. The computational simulations performed by  [‌17] provide smaller differences than the ones pointed out by  [‌11]. The systematic grid refinement in the spanwise direction performed in  [‌11] suggests that the asymmetric time and spanwise averaged pressure field can be simulated only by using a dense grid, while negligible differences take place with the coarsest mesh. The authors conjecture that small coherent flow structures in the spanwise direction past the trailing edges are responsible for the whole asymmetric flow field.

In general, the scrutinized studies show that the BARC flow has an high sensitivity to a number of factors which may trigger a significant asymmetry in the time-averaged flow field. It is worth recalling that some analogous well known examples of time-averaged asymmetric flows around nominally symmetric setups exist in bluff body aerodynamics, e.g. the circular cylinder in its critical regime [‌651], or the side-by-side arrangement of two rectangular cylinders [B/D = 1.28, S/D = 0.5, where S is the gap between cylinders, 30]. Another example, which is more similar to the flow setup discussed in this paper, is provided by [‌39], where the flow at ReD = UD/ν = 103 around rectangular cylinders with different breadth to depth ratios 3 ≤ B/D ≤ 9 is simulated by solving the 2D Navier-Stokes equations without turbulence models. The simulated time-averaged flow is symmetric for all the considered B/D ratios (included the one characterized by B/D = 5, except for B/D = 4. It is worth noting that this asymmetry is not revealed in previous wind tunnel tests performed by the same authors 36] under the same incoming flow nominal conditions.

The spanwise correlation of pressure and forces were also provided in one experimental study [‌56] and in two numerical contributions [‌1127] to BARC. They are not discussed here for the sake of brevity.

Contributed by: Luca Bruno, Maria Vittoria Salvetti — Politecnico di Torino, Università di Pisa

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