# Statistical Uncertainty of DNS in Geometries without Homogeneous Directions
## 3 Results
### Uncertainties in Flow over Backward-Facing Step (BFS)
#### 3.2.1 Uncertainties in Profiles of BFS Results
Some discrepancy between profile and monitor point data can be observed. This is due to data from these particular monitor point was only averaged over around 10 million time steps, while the profile was averaged over 12 million time steps. This is the case for profile of \(\left<w^{\prime 2}\right>\) at \(x=10\) and monitor point 47 at \(y=-0.2\). The averages seem adequately converged and the anisotropy of turbulence immediately after the step is clearly observable (compare profile of fluctuations at \(x=0.1\), where fluctuations in \(x\) direction are almost twice the fluctuations in \(y\) direction). However, this turbulent anisotropy persists towards the outflow (profile and monitor points at \(x=10\) and \(x=14\)). The data from monitor points support the claim that the anisotropy is statistically significant. Of course, the anisotropy of turbulence can also be observed near the walls. Figure 5: Selected profiles of average velocity components through the midplane (\(z=0\)) at positions where monitor points are present together with the \(2\sigma\) uncertainty in these points. The only non-diagonal component of the Reynolds stress tensor that is non-zero through the plane \(z=0\) is the \(\langle u^{\prime}v^{\prime}\rangle\). At monitor point 48 with \(x=14\) and \(y=-0.2\) a similar discrepancy can be observed between the profile and the monitor data due to difference in the integration time. Figure 8 shows the fluctuations of temperature through the middle of the BFS domain. The points located in the stagnation zones and close to the heater exhibit rather high uncertainties in the fluctuations. This is especially apparent in the case of lower Prandtl number. However, our analysis showed that the significant increase of the uncertainties in these regions is more connected to the length of decorrelation time, than to amplitude of oscillations. If we observe the history of temperature fluctuations in various points, it is apparent that at different monitor points the temperature oscillates at different timescales or frequencies. The smoother temperature signals have higher decorrelation times and thus higher uncertainties. Figure 6: Passive scalar profiles together with monitor point data uncertainties at specific positions in the plane \(z=0\). Figure 7: Profile of non-zero components of the Reynolds stress tensor at certain positions in the plane \(z=0\). Figure 9 shows the profile of turbulent heat flux components for the passive scalar with \(\Pr=0.005\) through the plane \(z=0\). The \(x\) and \(y\) components of turbulent flux are increasing towards the outflow. The third component is zero within the statistical uncertainty and is not shown. Figure 10 shows the profile of turbulent heat flux for passive scalar with \(\Pr=0.1\) together with values and uncertainties calculated in the monitor points in the plane \(z=0\). The \(x\) and \(y\) components of turbulent heat flux for both Prandtl numbers show that higher uncertainties are found where the average values are higher (unlike with temperature fluctuations above). However, a general observation is that the relative uncertainties in the recirculation zone are a bit higher than outside of it. This is visible with comparison of turbulent heat flux results with similar values, such as \(\langle\
u^{\prime}\theta^{\prime}\rangle\) component for \(\Pr=0.005\) at points 5 (\(x=0.1\), \(y=-1.5\), cyan) and 35 (\(x=2\), \(y=0.8\), blue) or 41 (\(x=4\), \(y=0.8\), red), or component \(\langle\
u^{\prime}\theta^{\prime}\rangle\) with \(\Pr=0.1\) at points 5 (\(x=0.1\), \(y=-1.5\), cyan) and 43 (\(x=10\), \(y=0.8\), black) or 44 (\(x=14\), \(y=0.8\), magenta).