# Statistical Uncertainty of DNS in Geometries without Homogeneous Directions
## 3 Results
### Uncertainties in Flow over Backward-Facing Step (BFS)  
Here we present the results in all the 49 monitor points that were used during the DNS of BFS. The data from the first 31 monitor points were averaged over 12 million time steps for the velocity, pressure and the passive scalar with \(\Pr=0.005\). Samples from monitor points 32 to 49 were averaged only over about 10 million time steps. The data for the second passive scalar, with \(\Pr=0.1\) were averaged over 7 million time steps. These discrepancies are because of late addition of monitor points 32 to 49 to the simulation, and late addition of the second passive scalar, which also had to reach a statistical steady state. Note that the averaging intervals differ in beginning only, but have identical end of time integration. The detailed integration span for uncertainty estimation is given in Table 1.  
Nevertheless, the profile data were obtained by integrating for the whole duration of 12 million time steps for velocity, pressure and passive scalar with \(\Pr=0.005\). Therefore the uncertainty estimation for monitor points 32-49 is a conservative estimate for uncertainties of the profiles. The profiles involving passive scalar with \(\Pr=0.1\) were obtained from the same time span as uncertainty data.  
The flow through time in the BFS case was approximately 60 dimensionless time units. With a time step of \(4\times 10^{-4}\) dimensionless time units, this is equal to approximately 150 thousand time steps. Thus, our simulation spanned more than about 80 flow-through times.  
The statistical steady state for the flow field and the passive scalar with \(\Pr=0.005\) was determined with the help of volumetric averaging in the area downstream the step. No particular tools were used to determine this. Since the second scalar with \(\Pr=0.1\) was  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Description** & **Points** & \
\begin{tabular}{c} **Monitor Points/Millions** \\ **of Time Steps** \\ \end{tabular} &
\begin{tabular}{c} **Profile Data/Millions** \\ **of Time Steps** \\ \end{tabular} \\ \hline Velocity, pressure, passive scalar with \(\Pr=0.005\) & 1–31 & 12 & 12 \\ Velocity, pressure, passive scalar with \(\Pr=0.005\) & 32–49 & 10 & 12 \\ Passive scalar \(\Pr=0.1\) & 1–49 & 7 & 7 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Number of time steps used for integration.  
Figure 3: Various turbulence statistics (solid lines, left scale) in channel flow with uncertainties (symbols; right scale).  
added later in the simulation, the statistical steady state was determined separately. For the determination of the turbulent heat flux, the average velocity is needed. The shortened time integration interval was used to determine the average velocity that is used in the calculation of the turbulent heat flux for the passive scalar with \(\Pr=0.1\).  
This is also confirmed in the simulation and can be observed in Figure 4, where the convergence of relative \(2\sigma\) uncertainty for the streamwise velocity component with number of steps is shown. The figure was produced by calculating the average and the uncertainty approximately every \(5\times 10^{4}\) time steps. However, not every time sample was used in the calculation. The data set was down sampled to include every tenth time step. As we show later, this does not have a significant effect on the estimation of uncertainty. The figure also shows convergence of the decorrelation time \(T_{0}\), or rather the number of time steps \(N_{0}\) associated with it.