# Statistical Uncertainty of DNS in Geometries without Homogeneous Directions
## 2 Numerical Model
### Backward Facing Step Flow  
The details of the numerical set-up of the BFS case are described by [PERSON] et al. [1]. A sketch of the domain is shown in Figure 1. The flow enters at left into the narrow duct. Twelve dimensionless units downstream of the inflow, the duct expands into a wider rectangular duct. The outflow is located 22 dimensionless units downstream of the step. All the boundaries of the domain, except for the inflow and outflow, are no-slip velocity boundary conditions. The no-slip velocity boundary condition is also applied at the lateral sides. The thermal domain is a bit larger since it includes two solid walls: the step wall (cyan) and the heater (red). The thickness of the walls is 0.25 dimensionless units. The heater contains an internal volumetric heat source. The outer parts of the domain are thermally insulated, while at the intersection of fluid and solid domain a conjugate heat transfer boundary condition is used.  
The standard incompressible Navier-Stokes system of equations, without gravity effects, is solved, however, here a constant mass influx was used as a boundary condition. The appropriate Reynolds number based on bulk velocity at the inflow and hydraulic diameter at the inflow was set to 7089. The calculated average friction Reynolds number in the channel upstream was around 207. In addition to the Navier-Stokes system of equations, the energy equations for two passive scalars with Prandtl numbers of 0.005 and 0.1 are also solved for. Buoyancy is neglected, as are thermally dependant material properties of fluid and walls.  
The equations are solved using the spectral element method implemented in the open source code nek5000, which is developed at the Argonne National Laboratory. The elements are smaller at the walls and larger in the middle of the channels. In the \(x\) direction the elements are of constant 0.25 length units. In the \(y\) direction the element sizes are varying from 0.044 to 0.115, while in the \(z\) direction the sizes vary from 0.047 to 0.210. Within each element there are seven collocation points in each direction. The calculations are performed with dealiasing with 11 collocation points in each direction. The PN/PN formulation is used, that is, the mesh for pressure variable is the same as the mesh for velocity. Second order time marching scheme (EXT2) is used, however, the time step is chosen so that the CFL number is oscillating around 0.1.  
Since there are no homogeneous directions in the step geometry, the results are presented in a different manner. 49 positions were selected in which the values of velocity, pressure and the two passive scalars were exported at each time step. These 49 points are not computational points but, rather, the values there are obtained through spectral interpolation.  
Figure 2 shows the distribution of monitor points in which we can calculate the statistical uncertainty associated with time averaging of DNS results. Their precise position is given in the Table A1 in the Appendix A. The position of each monitor point was chosen during the set-up of the DNS and during the initial runs. The positions were selected where during these preliminary runs, interesting features of the flow or temperature fields, such as maxima of components of turbulence fluctuations, averages, etc., were observed. Some points were also chosen as a means to determine the quality of the DNS.  
Figure 1: Sketch of backward-facing step (BFS) domain used in the direct numerical simulation (DNS). Flow flows from the narrow end towards the wide end. The blue and red coloured blocks are solid walls. Except for the inflow and outflow, all other surfaces are no-slip velocity boundary conditions.  
Figure 2: Distribution of points as seen from positive \(z\) axis.