# Turbulent Flow over Confined Backward-Facing Step: PIV vs. DNS
## 3 Computational Set-Up of Direct Numerical Simulation  
The DNS code Nek5000 [31] was used to solve dimensionless Navier-Stokes equations for incompressible fluid with internal energy equation. Details of the simulation, with a focus on heat transfer, are described in [2]. Since the passive scalar heat transfer equations do not affect the velocity field, they are irrelevant for the present work. The code Nek5000 is based on the spectral element method. Similar to the finite element method, the domain was divided into elements. Within each element, a spectral method was used to search for the solution. In this way, the method can be applied to irregularly shaped geometries, which are usually simulated with the finite element method. However, the spectral element method retains the quick convergence rate and the high accuracy of the spectral method. In addition, the elements serve as the unit for parallelization. The computational domain in the DNS shown in Figure 2 is shorter than the one shown in Figure 1. A four-fold shorter inlet of 12 dimensionless units was applied and half of that inlet length was used to produce the fully developed turbulent inflow with the cyclic boundary condition. The inlet boundary condition ensured a constant mass flow rate through the domain. The outlet section in the computation was shorter too (22 dimensionless units); however, it was long enough to eliminate the influence of the outflow boundary condition on the recirculation vortex behind the step. Figure 2: **Top**: DNS spectral element mesh. **Bottom**: reference mesh dimension divided by the local Kolmogorov scale in the z = 0 plane. The computational domain was divided into approximately 153,000 elements, shown in the top drawing of Figure 2. The sizes of the elements varied in the wall normal directions and were smaller near the wall. However, the elements were of the same length in the streamwise x direction. Within each element, seven collocation points were used in each direction, bringing the number of points within each element to 343. Distances between the collocation points in the domain were in the following intervals: \(\Delta\)x E [0.0212, 0.0586], \(\Delta\)y E [0.0037, 0.0269] and \(\Delta\)z E [0.0040, 0.0493]. These points define the basic cells but are not shown in Figure 2. The points on the boundaries between the elements coincided, so the total number of unique points within the domain was approximately 31 million. The spatial resolution of the simulation is shown in the bottom drawing of Figure 2, where the ratio of the mesh cell and the local Kolmogorov scale, a posteriori obtained from the DNS results, is shown. Such a resolution is typical for DNS in 3D geometries. The second-order accurate time integration is a combination of an explicit Adams-Bashfort scheme for convective and source terms and a semi-implicit backward differentiation for diffusion terms. The time step of 0.0004 dimensionless time units corresponds to around 10% of the CFL (Courant-Friedrichs-Lewy condition). Our tests have shown that the time step must be significantly lower than the stability limit of 0.5 CFL in order to ensure time step-independent results. Since the domain was three-dimensional with walls surrounding the BFS, there were no homogeneous directions. This meant that no spatial averaging can be employed to ease the averaging procedure, except for the symmetry in the spanwise (z) direction, and DNS required long time-averaging intervals. The presented results are shown with averaging over about 4900 dimensionless time units or approximately 12 million time steps (3 million Intel Xeon-E5 CPU hours). The averaging time corresponds to roughly 80 flow-through times of the DNS domain. This is an order of magnitude more than in comparable studies, which had periodic boundary conditions in the spanwise direction and compensate the shorter averaging times with spatial averaging over the spanwise direction.