# Statistical Uncertainty of DNS in Geometries without Homogeneous Directions
## 3 Results
### Uncertainties in Flow over Backward-Facing Step (BFS)
#### 3.2.2 Uncertainties in Monitor Points
That  
Figure 10: Profile of turbulent heat flux components for passive scalar with \(\Pr=0.1\) toghether with values and uncertainties calculated in monitor points through the plane \(z=0\). is, the reattachment does not have a shape of a straight line but is prolonged towards the outflow closer to the lateral walls. Figure 11: \(x\)-axis: number of point according to Figure 2. Top: Mean \(x\)-direction velocity. Middle: \(y\)-direction velocity. Bottom: \(z\)-direction velocity. Red dots represent average values with \(\pm\) 2\(\sigma\) range (left scale), blue dots represent (2\(\sigma\)) uncertainty (right scale). Figure 13 shows the average value, as well as the square of RMS fluctuations for the passive scalar with \(\mathrm{Pr}=0.005\). The average values have some expected properties: they are higher near the heater and downstream of the step. The drop in temperature through the points 45 to 49 may be surprising; however, fluid from the recirculation zone is heating  
Figure 12: \(x\)-axis: number of point according to Figure 2. Top: Fluctuations of \(x\)-direction velocity. Middle: Fluctuations of \(y\)-direction velocity. Bottom: \(z\)-direction velocity. Red dots represent root-mean-square values with \(\pm\) 2\(\sigma\) range (left scale), blue dots represent (2\(\sigma\)) uncertainty (right scale). fluid at points 32 to 38 and 39 to 44. According to the structure of the flow, described in [1] the cold fluid is flowing towards the heater at the sides (\(z=\pm 1.8\)). More interesting are the fluctuations of the passive scalar. Immediately one can notice the relatively high uncertainties in fluctuations in points located close to the heater. Unfortunately, no monitor point was placed in the heater to see the behaviour of uncertainties of thermal fluctuations in the solid. Consistently, thermal fluctuations in monitor points located inside or near the recirculation zone show higher statistical uncertainties, compared to thermal fluctuations in monitor points outside of recirculation zone. Figure 14 shows the similar graph for the passive scalar with \(\Pr=0.1\). Note that the averaging time here is shorter so we expect slightly higher relative uncertainties. This is the case in the figure, however, we can observe the same trend of uncertainties of thermal fluctuations being higher at the points near the heater. Nevertheless, there are subtle differences, which would indicate that the influence of solid walls in connection with the Prandtl number is stronger than the proximity to the stagnation zones. These differences can be seen in first 6 monitor points 13 to 15, as well as 29 to 31. However, since the lower Prandtl number means wider thermal layer a wider influence of stagnation zones could be observed. Figure 13: \(x\)-axis: number of point according to Figure 2. Top: Mean temperature for \(\Pr=0.005\). Bottom: \(\langle\theta^{2}\rangle/\Delta\theta^{2}\) for \(\Pr=0.005\). Red dots represent root-mean-square values with \(\pm\) 2\(\sigma\) range (left scale), blue dots represent (2\(\sigma\)) uncertainty (right scale). The higher uncertainties in thermal fluctuations, as well as in average values of passive scalars, were traced back to an order of magnitude higher decorrelation times. By Equation (3) this increases the statistical uncertainties. The higher decorrelation time is estimated since the temperature oscillations have longer periods, although the amplitudes of the fluctuations do not seem to be vastly different. The uncertainties of Reynolds stresses and turbulent heat fluxes can be found in the Appendix A.  
Graphs in Figures 11-14 were produced by taking into account monitor data at every time step. An interesting observation can be made if we skip some of the time steps in the calculation of averages. Our time step length was approximately 0.1 of the time step calculated through the Courant-Friedrichs-Lewy condition (CFL time step). We found that the average results do not change by much if values from every hundredth time step (or every tenth CFL time step) are taken.