UFR 2-15 Test Case

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

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Test Case Study

Brief Description of the Study Test Case

As previously mentioned, BARC addresses the high Reynolds number, external, unsteady flow over a stationary, sharp-edged smooth rectangular cylinder, and the associated aerodynamic loads [‌2]. The breadth ${(B)}$ to depth ${(D)}$ ratio is set equal to 5. A sketch of the configuration is shown in Fig. 2. The BARC test case gathered new wind tunnel tests in four different facilities [‌5, 7, 54, 56, 57] and computational simulations from six different teams [‌1, 8, 10, 11, 17, 26–28, 46, 71]; the UFR is mainly based on these contributions.

The following common requirements were set for both wind tunnel tests and numerical simulations:

• the depth-based Reynolds number Re_D = UD/ν has to be in the range of 2 × 10⁴ to 6 × 10⁴;
• the incoming flow has to be set parallel to the breadth of the rectangle, i.e. α = 0, α being the angle of attack;
• the maximum intensity of the longitudinal component of the freestream turbulence is set to I_x = 0.01;
• the minimum spanwise length of the cylinder for wind tunnel tests and 3D numerical simulations is set to L/D = 3.

The following additional requirements are specified for wind tunnel tests:

• the maximum acceptable radius of curvature of the edges of wind tunnel models is set to R/D = 0.05;
• the maximum wind tunnel blockage is set to 5%;
• all the points of measurement have to be outside the boundary layers developed at the tunnel walls;
• uniformity of the flow at all measurement points must be checked in the empty tunnel and appropriately documented.

In addition to the main setup described above, sensitivity studies are strongly encouraged. The following additional values of the parameters are suggested for both wind tunnel tests and numerical simulations:

• angles of incidence α = 1^°, 3^°, 6^°;
• Reynolds number Re_D = 10³, 10⁴, 10⁵, 10⁶;
• turbulence intensity I_x = 0.02, 0.05, 0.10.

The flow quantities presented in the following are made dimensionless by using the undisturbed flow field velocity U, the cylinder depth D and the fluid density ρ, unless specified otherwise.

Data can be uploaded to the BARC website by registered participants. Setup information and output data requested for numerical simulations and wind tunnel tests are set in Requests for Computational Simulations [‌3] and Requests for Wind Tunnel Tests [‌4], respectively. To summarize, statistics of pressure over the cylinder central section and at other give sections along the spanwise direction are required both in experiments and in computational contributions. Velocity statistics and time-histories of the aerodynamic loads are required in numerical studies and encouraged in wind-tunnel tests.

Most of the studies adopt incoming flow characteristics in accordance with the range prescribed by the BARC main setup (see above) and/or with the ones suggested for the sensitivity studies [e.g. turbulence intensity and length scale in 56, 57]. The adopted incoming flow features are summarized in Figure 1. As for the freestream turbulence intensity, all the computational studies using Large-Eddy Simulation (LES) or Detached Eddy Simulation (DES) turbulence approaches adopt perfectly smooth incoming flow, mainly because of the difficulties involved in the generation of realistic incoming turbulence features within these approaches. Conversely, perfectly smooth flow conditions cannot be obtained in wind tunnels, where a residual turbulence always exists; on the other hand, grid turbulence generation is a relatively easy and inexpensive task in wind tunnel tests. Hence, the mentioned differences among computational and wind tunnel approach do not allow to compare flowfields obtained exactly in the same conditions, but the complementary features of each approach allow the effects of incoming turbulence to be investigated in a collaborative framework. Figure 1b also shows that another parameter significantly varying among the different contributions is the freestream Reynolds number, even if it keeps the same order of magnitude and most of the values fall in the range specified in the BARC main setup.

Figure 1: Adopted incoming flow features: longitudinal turbulence intensity (a) and Reynolds number (b)

In the following sections, the setup and objectives of both wind tunnel tests and computational simulations are shortly summarized.

Test Case Experiments

The main characteristics of the set-ups of the wind tunnel contributions to BARC are given in table 3. All the experimental contributions are aimed at reproducing as much as possible an unconfined 2D nominal flow, as in the requirements of the BARC benchmark: the ratio between the cylinder spanwise length, L and B ranges from 3 to 10.8, i.e. L/D = 15 – 54, and the blockage is generally lower than 2%, except for one case in Shirato et al. [‌57]. Three of the experimental contributions [‌5, 7, 54] are aimed at obtaining measurements for freestream conditions that are as smooth as possible. A different approach is that of the paper of Shirato et al. (2011), [‌57], in which the characteristics of the spanwise coherence of the aerodynamic action of rectangular cylinders with B/D ratios ranging from 2.2 to 10 is investigated, for different intensity and scale of the incoming turbulence, obtained by means of three grid arrangements placed upstream of the model.

As for reduced scale models, it is worth pointing out that the tests performed by [‌5] are characterised by an ad hoc conceived, aluminum model shaped through a countersink procedure in order to obtain very accurate prismatic shape and sharp edges. The model adopted by [‌7] consists of plastic and wooden parts including interchangeable edge elements with radii of curvature of R/D = 0, 0.01, 0.02 and 0.5 to investigate the influence of edge sharpness on the flow. The model in [‌54] was originally used as a generic H-shaped section in a previous investigation and it has been adapted to obtain a rectangular 5:1 cylinder. All details concerning the model and the experimental arrangement can be found in [‌52]. The model used in [‌56, 57] is made of thin aluminum plate in its B/D = 5 main set-up, while an additional frame made of foamed styrol is inserted between one side surface aluminum plate and the remaining open box-shape model in order to investigate further B/D ratios.

As for measurement techniques and outputs, pressure taps placed on the model surface are used in [‌5, 7, 57] to provide the statistics of the pressure distribution at the body surface. The sampling frequency of pressure measurements was between 350 and 500Hz and in all cases it was found to be adequate to capture the flow fluctuations (we refer to [‌5, 7, 57] for more details). Unsteady aerodynamic loads are measured in [‌54] by means of a high-stiffness piezoelectric balance. Experiments are carried out in [‌54] for different angles of attack and for a wide range of Reynolds numbers (2 × 10⁴ ≤ Re_D ≤ 2 × 10⁶). In the following, only the case at α = 0 will be considered and the values of the forces measured for Re_D = 26400 will be used for comparison, as also done in [‌27, 28], while the effects of Reynolds number will be briefly discussed in the following.

Table 3: Wind tunnel tests: set-up characteristics. L_x is the streamwise integral length scale of oncoming flow turbulence.

${\left.{\text{Source}}\right.}$	${\left.{\text{Re}}_{D}\right.}$	${\left.L/B\right.}$	${\left.{\text{Blockage}}\ (\%)\right.}$	${\left.I_{x}\ \%\right.}$	${\left.L_{x}/D\right.}$
Schewe [‌54]	${\left.26400\right.}$	${\left.10.8\right.}$	${\left.1.83\right.}$	${\left.0.4\right.}$	—
Bronkhorst et al. [‌7]	${\left.50000-100000\right.}$	${\left.4\right.}$	${\left.3.5\right.}$	${\left.0.32-0.56\right.}$	—
Bartoli et al. [‌5]	${\left.20000-80000\right.}$	${\left.7.93\right.}$	${\left.3.75\right.}$	${\left.1.6-2.1\right.}$	${\left.1.27-1.87\right.}$
Shirato et al. [‌57]	—	${\left.3.6\right.}$	—	${\left.10.5,11.5,14\right.}$	${\left.0.92-3.16\right.}$

As suggested in the guidelines of the BARC benchmark, the contributions by [‌7] and [‌5] report calibration studies of the model and of the wind tunnel set-up, and both point out the difficulties associated with obtaining a perfectly symmetric configuration. In particular several causes of asymmetry in the experiment conditions are identified and investigated, the main ones being the disturbances in the incoming flow, misalignment of the model with the incoming flow and inaccuracies in the model geometry. These three causes are separately analyzed in [‌7]. The two research teams use different approaches when aligning the model in the wind tunnel. [‌5] use a trial and error approach, by rotating the model, horizontally placed in the wind tunnel, around its axis, and checking the value of the stagnation pressure coefficient. The tests are then carried out for the angle giving the largest value of the stagnation pressure coefficient, which turns out to be equal to 1. This approach allows compensating for possible flow asymmetries. [‌7], on the other hand, align the model, vertically placed in the wind tunnel, such as to have its faces perpendicular to the tunnel walls (a turntable permits an accuracy of 0.05°). This configuration brings a stagnation pressure coefficient of 1. In spite of the attention paid to the alignment of the model, both studies show a clear asymmetry in the mean and RMS pressure coefficients between the upper and lower faces. Conversely, [‌54], for all the considered Reynolds numbers and α = 0, obtains values of the mean lift coefficient practically equal to zero, which indicate a symmetry of the mean flow. Asymmetries in the mean flow have also been observed in some of the numerical contributions and this issue will be discussed in detail in the Evaluation section.

CFD Methods

The various numerical contributions from 6 teams differ with respect to physical modeling, numerical methods and simulation set-up. We refer to the original papers for a complete description, while an overview is herein given of those aspects which have been the object of extensive sensitivity studies by the contributors.

A first issue is clearly turbulence modeling. The different numerical studies cover a wide range of approaches to turbulence (see Tab. 4), even if the studies based on LES and DES prevail over the ones using Unsteady Reynolds Average (URANS) models, the latter being restricted to the works of Mannini et al. [‌26] and Ribeiro [‌46]. LES simulations represent 51% of the numerical contributions, DES ones 30% and, finally, URANS computations 29%. Nevertheless, a significant number of URANS models have been applied to the test case, thanks to the affordable computational cost of each simulation: 1-equation Spalart–Allmaras model (SA), Linearized Explicit Algebraic Wilcox k – ω model (LEA k – ω), Menter k – ω model (SST k – ω), realizable and RNG k – ε models, Reynolds Stress Model (RSM). Testing the performance of the URANS approach is surely useful in an engineering perspective, as industrial applications often require simple and cheap 2D URANS simulations to investigate a large number of flow parameters and geometry configurations [e.g. in 46]. As for LES, both the classical formulation and the Variational Multi Scale one (VMS-LES)} are tested in conjunction with a number of sub-grid models: Standard and Dynamic Smagorinsky Model (SM and DSM, respectively), Kinetic Energy one-equation model (KET), Wall-Adapting Local Eddy-viscosity (WALE) model. Finally, as for hybrid methods, both classical DES and Improved Delayed Detached Eddy Simulation (IDDES) are employed, where the SA model is adopted in the URANS part of the model. A more detailed description and the precise references for the adopted turbulence approaches and models can be found in the papers cited in Tab 4.

Table 4: Computational studies: turbulence modeling

		${\left.{\text{turbulence model}}\right.}$
${\left.{\text{source}}\right.}$	${\left.{\text{Re}}_{D}\right.}$	${\left.{\text{approach}}\right.}$	${\left.{\text{closures}}\right.}$
Arslan et al. [‌1]	${\left.2.64\times 10^{4}\right.}$	${\left.{\text{LES}}\right.}$	${\left.{\text{SM, DSM, KET}}\right.}$
Bruno et al. [‌8, 10]	${\left.4.0\times 10^{4}\right.}$	${\left.{\text{LES}}\right.}$	${\left.{\text{KET}}\right.}$
Grozescu et al. [‌17, 18]	${\left.2.0\times 10^{4},\ 4.0\times 10^{4}\right.}$	${\left.{\text{VMS-LES}}\right.}$	${\left.{\text{SM, WALE}}\right.}$
Mannini et al. [‌26, 27]	${\left.2.64\times 10^{4},\ 10^{5}\right.}$	${\left.{\text{URANS}}\right.}$	${\left.{\text{SA, LEA}}\ k-\omega \right.}$
Mannini and Schewe [‌28]		${\left.{\text{DES}}\right.}$	${\left.{\text{SA}}\right.}$
Ribeiro [‌46]	${\left.2.64\times 10^{4}\right.}$	${\left.{\text{URANS}}\right.}$	${\left.{\text{RSM, SST}}\ k-\omega \right.}$
			${\left.{\text{real, and RNG}}\ k-\varepsilon \right.}$
Wei and Kareem [‌71]	${\left.10^{5}\right.}$	${\left.{\text{LES}}\right.}$	${\left.{\text{DSM}}\right.}$
		${\left.{\text{IDDES}}\right.}$	${\left.{\text{SA}}\right.}$

As for numerical discretization, commercial codes [e.g. Fluent in 1, 46], open source codes [Openfoam in 10, 71] and proprietary codes [‌17, 28] are used. All codes are based on the finite-volume method, except for the one used by Grosescu et al. [‌17], based on a mixed finite-element/finite-volume discretization. The adopted numerical methods are all second-order accurate in space and time. A crucial issue in LES and hybrid URANS/LES is the introduction of numerical dissipation, which is needed, in particular, to stabilize compressible flow solvers. The numerical viscosity in [‌17, 18] is designed for LES and it has been shown in previous studies to have negligible effects on the quality of the results. The scalar Jameson-Schmidt-Turkel scalar artificial dissipation is used in Mannini et al. [‌27], while a matrix artificial dissipation is employed in Mannini and Schewe [‌28]. Mannini et al. [‌27] and Mannini and Schewe [‌28] investigate the sensitivity of the results to the type and amount of artificial dissipation. No further sensitivity studies to numerics have been carried out.

Figure 2: Computational studies: model and domain geometry

As for the simulation set-up, the geometry of the spatial domain is characterized according to the BARC nomenclature given in Bartoli et al. [‌3] (Fig. 2). The values of the parameters adopted in each study are collected in Tab. 5. Two-dimensional domains are adopted in 30% of the simulations in conjunction with URANS models, except for a single LES simulation performed by Arslan et al. [‌1] and aimed at comparing its accuracy with the 3D LES simulations in the same work. The shape of the domain is mostly prismatic (P) but cylindrical domains (O) are also generated by Mannini and Schewe [‌28] and Ribeiro [‌46], in conjunction with an O-grid in the latter study. In all cases, the spanwise domain size ${{\mathcal {D}}_{z}}$ coincides with the cylinder spanwise length L and periodic boundary conditions are used. On the domain lateral sides, either free-slip [‌1, 10, 11] or far-field [‌17, 18, 28, 46] boundary conditions are imposed.

Table 5: Computational studies: parameters of the computational domain

${\left.{\text{source}}\right.}$	${\left.{\text{dim.}}\right.}$	${\left.{\text{shape}}\right.}$	${\left.{\mathcal {D}}_{x}/B\right.}$	${\left.{\mathcal {D}}_{y}/B\right.}$	${\left.{\mathcal {D}}_{z}/B\right.}$	${\left.\Lambda /B\right.}$	${\left.R/D\right.}$
Arslan et al. [‌1]	${\left.{\text{3D}}\right.}$	${\left.{\text{P}}\right.}$	${\left.23\right.}$	${\left.20.2\right.}$	${\left.1\right.}$	${\left.10\right.}$	${\left.0\right.}$
	${\left.{\text{2D}}\right.}$	${\left.{\text{P}}\right.}$	${\left.23\right.}$	${\left.20.2\right.}$	—	${\left.10\right.}$	${\left.0\right.}$
Bruno et al. [‌8, 10]	${\left.{\text{3D}}\right.}$	${\left.{\text{P}}\right.}$	${\left.41\right.}$	${\left.30.2\right.}$	${\left.1,2,4\right.}$	${\left.15\right.}$	${\left.0\right.}$
Grozescu et al. [‌17, 18]	${\left.{\text{3D}}\right.}$	${\left.{\text{P}}\right.}$	${\left.41\right.}$	${\left.30.2\right.}$	${\left.1\right.}$	${\left.15\right.}$	${\left.0\right.}$
Mannini et al. [‌26, 27]	${\left.{\text{2D}}\right.}$	${\left.{\text{O}}\right.}$	${\left.200\right.}$	${\left.200\right.}$	—	${\left.100\right.}$	${\left.0\right.}$
Mannini and Schewe [‌28]	${\left.{\text{3D}}\right.}$	${\left.{\text{O}}\right.}$	${\left.200\right.}$	${\left.200\right.}$	${\left.1,2\right.}$	${\left.100\right.}$	${\left.0\right.}$
Ribeiro [‌46]	${\left.{\text{2D}}\right.}$	${\left.{\text{O}}\right.}$	${\left.51\right.}$	${\left.50.2\right.}$	—	${\left.25\right.}$	${\left.0\div 0.1\right.}$
Wei and Kareem [‌71]	${\left.{\text{3D}}\right.}$	${\left.{\text{P}}\right.}$	${\left.8\right.}$	${\left.3\right.}$	${\left.0.2,0.4,1\right.}$	${\left.1.5\right.}$	${\left.0\right.}$

No specific sensitivity studies are available in the literature about the domain size for computational simulations of rectangular cylinders, nor has such a study been systematically accomplished during the BARC activity up to now, except for the spanwise dimension ([‌10, 11]). Nevertheless, one may refer to the extensive parametrical studies by [‌44] and [‌45] aimed at assessing the influence of the computational domain size on the simulated flow around an unconfined circular cylinder at low Reynolds numbers. The authors conclude that, in any case, the distance of the cylinder from the inlet boundary Λ_x and from each of the lateral boundaries ${\left.{\mathcal {D}}_{y}/2\right.}$ should be larger than 20D. In spite of the different geometry and Re number, these may be considered as first indications also for the BARC test case. In most cases, the domain dimensions in the x and y directions are about 20 to 50 times the breadth B of the cylinder section, with remarkable exceptions in Mannini and Schewe [‌28] ${({\mathcal {D}}_{x}/B={\mathcal {D}}_{y}/B=200)}$ and in Wei and Kareem [‌71] ${({\mathcal {D}}_{x}/B=8,\ {\mathcal {D}}_{y}/B=3)}$: the former is expected to increase the computational cost but, of course, it avoids spurious effects of the boundary conditions on the simulated flowfield, while the domain size adopted in the latter is suspected to strongly affect the flow. In particular, the blockage in the simulations in Wei and Kareem [‌71] is 6.7%, which is significantly larger than in the other simulations and in experiments and it does not fulfil the requirement of the benchmark of maximum 5% blockage. The ensemble-average value of ${{\mathcal {D}}_{z}/B}$ of the different contributions is close to one, but higher values are adopted in studies addressed to the evaluation of spanwise correlation [‌10, 11, 27], while shorter lengths are adopted by Wei and Kareem [‌71]. As for the cylinder geometry, perfectly sharp edges are adopted in all cases, while Ribeiro [‌46] also varies the radius of curvature of the edges of the cylinder in a broad range.

Most of the grids are hybrid in the x – y plane (i.e. body-fitted, structured in the near wall region and unstructured triangular or quadrilateral elsewhere), and structured along the spanwise direction z. Remarkable exceptions are the fully unstructured grids adopted by Grozescu et al. [‌17] and the structured ones used by Wei and Kareem [‌71] (orthogonal) and Ribeiro [‌46] (non orthogonal).

Table 6: Computational studies: grid resolution. ${\left.\delta _{x}\right.}$ and ${\left.\delta _{z}\right.}$ are the grid spacing in the ${\left.x\right.}$ and ${\left.z\right.}$ directions respectively, while ${\left.n_{w}\right.}$ is the wall normal grid spacing and ${\left.n^{+}\right.}$ is ${\left.n_{w}\right.}$ made dimensionless in wall units. The overline denotes averaging over the grid cells

${\left.{\text{source}}\right.}$	${\left.n_{w}/B\right.}$	${\left.{\overline {n^{+}}}\right.}$	${\left.{\overline {\delta _{x}}}/B\right.}$	${\left.\delta _{z}/B\right.}$
Arslan et al. [‌1]	${\left.2.0\times 10^{-4}\right.}$	${\left.0.5\right.}$	${\left.4.0\times 10^{-3}\right.}$	${\left.0.04\right.}$
Bruno et al. [‌8, 10]	${\left.5.0\times 10^{-4}\right.}$	${\left.1.66\right.}$	${\left.2.0\times 10^{-3}\right.}$	${\left.0.042\div 0.01\right.}$
Grozescu et al. [‌17, 18]	${\left.5.0\times 10^{-4},\ 2.5\times 10^{-4}\right.}$	${\left.\simeq 1\right.}$	${\left.1.0\times 10^{-2},\ 5.0\times 10^{-3}\right.}$	${\left.0.042,\ 0.01\right.}$
Mannini and Schewe [‌28]	${\left.5.0\times 10^{-5}\right.}$	${\left.0.25\right.}$	${\left.1.4\times 10^{-2}\right.}$	${\left.0.0156\right.}$
Ribeiro [‌46]	${\left.4.0\times 10^{-5}\right.}$	${\left.\simeq 1\right.}$	${\left.2.0\times 10^{-3}\right.}$	—
Wei and Kareem [‌71]	${\left.{\text{n.a.}}\right.}$	${\left.0.9-7.44\right.}$	${\left.10^{-2}-3.3\times 10^{-3}\right.}$	${\left.0.04-0.0025\right.}$

Table 6 compares the grid spacing normal to the wall, n_w, and in the x- and z- directions, δ_x and δ_z. Most of the grids adopt 2.0 × 10^-5 ≤ n_w/B ≤ 5.0 × 10^-4 in order to obtain a grid resolution in wall units ${\left.{\overline {n^{+}}}\approx 1\right.}$, and to fully resolve the boundary layer without introducing wall-functions-like approximations. A large scatter is observed in the values of ${\left.{\overline {\delta _{x}}}/B\right.}$: it is worth pointing out that the large value of ${\left.{\overline {\delta _{x}}}/B\right.}$ in Mannini and Schewe [‌28] is justified by the DES model, while the one of the coarsest grid in Grozescu et al. [‌17] has been conceived to test the VMS-LES accuracy in conjunction with very coarse grids. The spanwise grid resolution varies from δ_z/B ≈ 1/24 in coarse grids to δ_z/B ≈ 1/100 in refined grids, with the remarkable exception of an even smaller grid step adopted by Wei and Kareem [‌71].

The overall number of grid cells, n_c varies over 4 orders of magnitude among the studies (Fig. 3a), mainly depending on the approach to turbulence, the domain dimension and size, the grid type. Nevertheless, most of the models adopt an overall number of cells around one million (Fig. 3b). Only Bruno et al. [‌8] provide some quantitative information about the cell skewness in the generated unstructured grid. Sensitivity studies to grid refinement are carried out in [‌10, 11, 17,  18, 26, 46, 71]. Grid independence has been shown only for the URANS simulations in [‌26].

Figure 3: Computational studies: number of cells for each study (a) and its overall distribution (b)

As for the inflow conditions, the numerical contributions cover different values of freestream turbulence intensity and Reynolds number (see Tab. 4 and Fig. 1). Nonetheless, explicit sensitivity studies are carried out only for the Reynolds number in [‌26], [‌17] and [‌18].

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

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