# Test Case

## Brief description of the study test case

The case considered is the flow past a square cylinder at Re = UD/ν = 22.000, as sketched in Fig.1. The approach flow is uniform (velocity U) and the free stream is confined by two sidewalls (the walls of the water tunnel), the blockage by the cylinder being 7.1 %. In the experiment, the cylinder had of course a finite length (of about 10 D) and was confined by the bottom and top wall of the water tunnel, but here the end effects are not considered and the case of an infinitely long cylinder is taken.

Figure 1: Flow geometry

The principal measured quantities which should be used for an assessment of CFD calculations are as follows:

• Strouhal number St = fD/U, which is the shedding frequency of the vortices f, made dimensionless.
• Mean drag coefficient CD
• RMS of fluctuating drag coefficient, CD', and of fluctuating lift coefficient CL'
• distribution of mean pressure around cylinder faces
• length of mean recirculation region behind cylinder, LR
• distribution of mean velocity UCL along centre line of cylinder
• distribution of kinetic energy of total (periodic and stochastic) fluctuations, ktotal , along centre line of the cylinder.

For a more detailed test of the CFD calculations, the distributions of averaged streamwise and transverse velocity as well as turbulent normal and shear stresses in the vicinity of the wake can be taken either in form of contour plots or profiles at various downstream cross sections. Time-mean averages over all phases are of interest here and for a detailed study of the time-dependent behaviour also phase average values at various phase angles during the period of the vortex shedding.

## Test Case Experiments

The experiments are described in detail in Lyn and Rodi (1994) and Lyn et al. (1995). The cylinder was 40 mm in width and 392 mm long mounted in the rectangular test section of a water channel 392 mm x 560 mm (blockage 7.1 %). The free stream approach velocity U was approximately 0.54 m/s yielding a Reynolds number Re = UD/ν = 22.000. At this Reynolds number the flow is approximately periodic with a Strouhal number St of 0.133 ± 0.004. The approach flow had a turbulence level of about 2 %. The velocity measurements were made above the upper surface and behind the cylinder up to 8 diameters downstream. The measurements were taken with a single component laser Doppler Velocimeter (LDV) just above the upper surface of the cylinder and a two component LDV in the other regions — the locations of the measurement points are given in Fig. 2. Bragg cells were employed to provide an offset frequency necessary to capture the reversal of flow direction in regions of separation. A low-pass filtered pressure signal from a tap on the cylinder side-wall was used to obtain a reference phase for phase-averaging the velocity measurements. 20 phase bins were used for phase-averaging.

Figure 2: Coordinate system and location of measurement points

The quantities measured at the points given in Figure 2 are the following:

• average streamwise flow velocity, U1
• average transverse flow velocity, U2
• average normal stresses,
• average turbulent stress,

The Strouhal number was determined from the fluctuating pressure signal mentioned above as

St = 0.133 ± 0.004 .The drag was not measured directly, but the mean drag coefficient was determined from the velocity distribution in the wake to be CD ≈ 2.1.

The pressure distribution along the cylinder walls has not been measured in these experiments. Here the measurements of other authors performed under similar conditions must be consulted (e.g. Bearman and Obajasu 1982).

The measurement uncertainties are:

U1 and U2 : 5% of approach velocity

: 15% to 25% .

The measured mean velocity along the centre line approaches the free stream velocity rather slowly, see Fig. 3 below. Most calculations yielded a considerably faster recovery of the centre line velocity and hence some doubts were voiced concerning the reliability of the measurements. However, independent measurements of Martinuzzi and Wu (1997) under similar conditions (blockage and turbulence level of the approach flow) yielded very similar results and give credibility to the measurements of Lyn et al. (1995).

The boundary layers on the top and bottom walls where the cylinder was mounted were fairly thin so that the end effects were small. The flow at the mid height of the cylinder where the measurements were taken corresponds closely to the flow past an infinitely long cylinder. The conditions on the boundaries of the flow are well known. They are no-slip conditions on the cylinder walls and on the side walls of the tunnel, but since the boundary layers on the latter were thin these can be considered as frictionless walls. The approach flow was measured 3 cylinder widths upstream and had a turbulence level of about 2% and a centre-line mean velocity deficit of about 5 to 10%. If uniform inflow conditions are to be used in calculations, the inflow boundary should be placed further upstream (5 to 10 D). The intensity of the approach-flow fluctuations is known from the measurements to be about 2% but no information is available from the measurements on the length scale of the turbulence. In RANS calculations this was estimated by assuming a ratio of eddy viscosity to molecular viscosity in the range 10 to 100, and Bosch and Rodi (1998) found that the value of 10 is more suitable for this case.

The measurement data are available both in the ERCOFTAC database Test Case 43 and on the CD of the AGARD-AR-345-report.

## CFD Methods

The study test case was calculated with a large number of RANS and LES methods. In the RANS calculations, 2D unsteady equations were solved thereby attempting to resolve the periodic fluctuating motion and to simulate the superimposed stochastic turbulent motion by a RANS turbulence model so that the calculations were really URANS. A wide variety of turbulence models was used, ranging from the algebraic Baldwin-Lomax model to a Reynolds stress model (RSM).

The most extensive testing was done with variants of the k-ε model and is reported in Rodi (1997) and Bosch and Rodi (1998) -and also reviewed in these papers. Algebraic stress models have also have been applied on this flow (Lakehal et al. 1998). Since the standard k-ε model is known to produce excessive turbulence production in stagnation regions, the Kato-Launder (1993) modification was also tested which suppresses this excessive production. Various near-wall treatments were applied in connection with the different RANS models such as the use of standard wall functions bridging the viscous sublayer but also two-layer approaches in which the basic turbulence model was only used outside the viscous sublayer while this was resolved with a simpler one-equation model. The numerical grids used there were of the order of 100 x 70 when wall functions were applied and 170 x 170 when the two-layer approach was used. In preliminary tests it was found out that these grids yielded accurate numerical solutions. The RANS calculations were obtained with various finite-volume codes, some using staggered and some non-staggered grid arrangements. The details are described in the source papers. It is worth noting however that already in the early calculations it was found out that first order upwind difference schemes cannot be used for vortex shedding calculations as they damp out the periodic shedding motion. In the calculations of Bosch and Rodi (1998), the HLPA (hybrid linear parabolic approximation) method of Zhu (1991) was used, which is low-diffusive and oscillation free. Bosch and Rodi investigated the influence of the location of the inflow boundary and the value of the turbulent length scale specified there. They found that a location of the inflow boundary 4.5 D upstream of the cylinder is too close, as the presence of the cylinder is already felt at this location and the boundary should be shifted further upstream to about 10 D. Also they found a significant influence of the specification of the turbulent length scale at inflow on the calculation results. They determined that a length scale based on the assumption of a ratio of eddy to molecular viscosity of 10 to be more realistic than the previously used value of 100.

In the LES calculations, the 3D, time-dependent Navier-Stokes equations were solved and all motions with scale larger than the mesh size were calculated directly. The effect of the unresolved sub-grid scale motion was in most cases accounted for by a sub-grid scale model but in some cases no such model was used and the dissipative effect of the sub-grid stresses was achieved by numerical damping introduced by a third order upwind scheme for the convection terms. Many calculations were carried out with the Smagorinsky eddy-viscosity model, modifying the length scale near the wall by a van Driest damping function. The popular dynamic approach of Germano et al. (1991) has also been used in a number of cases, mostly with the Smagorinsky model as a base so that the approach then determines the spatial and temporal variation of the Smagorinsky "constant" by making use of the information available from the smallest resolved scales. Some calculations were also carried out with a one-equation model as base in which the velocity scale of the subgrid-scale stresses is calculated from a transport equation for the turbulent subgrid-scale energy. Finally, mixed models combining a scale-similarity model based on a double filtering approach with the Smagorinsky model as well as a filtered structure function model have also been applied. For details of the individual models, the sources must be consulted, but Table 1 lists the subgrid-scale model used in the LES calculations summarised there. The table gives also information on the near-wall treatment and in most calculations no slip conditions have been used at the wall, while in a few calculations wall functions were applied, mainly the Werner-Wengle (1989) approach which assumes a distribution for the instantaneous velocity inside the fist grid cell. A wide variety of numerical solution procedures have been employed, and again the sources must be consulted for details which are however summarised also in the proceedings of the various workshops. The discretization used for convection is again listed for the various calculations in Table 1 and it is important to note here that in LES calculations both spatial and temporal discretization must be at least second order accurate. Usually the numerical grids were stretched in order to achieve a better resolution in the near-wall region with higher gradients; in some cases embedded grids where used to better resolve this region and also semi-structured grids. The schemes are generally explicit with small time steps to resolve the turbulent fluctuations and various time discretization methods were employed —mostly the second order Adams-Bashforth scheme but also the Runge-Kutta, Euler and leap-frog methods. At the outlet, the calculations were all done with the convective conditions. Typical grids used are shown in Table 1. Grid-refinement has generally not been undertaken and it should be noted that LES calculations are not expected to be grid-independent anyway as the subgrid-scale modelling depends an the mesh-size.

The study test case was posed as a test case for an LES workshop held in 1995 at Rottach-Egern, Germany, and nine groups submitted 16 different results that are reported in Rodi et al. (1995) and partly also in a summary paper of Rodi et al. (1997). These calculations were all performed on a computation domain of 4 D in the spanwise direction, extending 4.5 D upstream of the cylinder, 6.5 D on either side of the cylinder (were the tunnel walls were located) and at least 14.5 D downstream of the cylinder. There was a great variety in the results, and hence the same test case was posed again at a workshop held in 1996 in Grenoble, France. The same calculation domain was prescribed and was mostly also used in later calculations. For the Grenoble workshop, seven groups presented 20 results. These are summarized in Voke (1997). Sample results from both workshops are presented and compared in section 6. The full information on both workshops including the results is available in electronic form and will be entered in the Knowledge Base (link will be provided in the final version of this document).

Védy and Voke (2001) studied systematically the influence of resolution in the various directions and to some extent of the size of the computation domain (but not going beyond 4D in the spanwise direction) as well as of the sub-grid scale model.