# 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  
Figure 11 shows the average velocity components, together with the \(2\sigma\) uncertainties, which correspond to roughly 95% confidence interval. The top row shows the approximate position of the appropriate points. This is just a guide for the eye, since point designations are not so informative. The streamwise velocity component shows decent convergence. The relative uncertainty in the average \(y\) velocity component seems to be larger, however, this is only because the absolute values of the velocity component are much lower. The  
Figure 8: Temperature fluctuations for both passive scalars with data from monitor points in the plane \(z=0\). Figure 9: Profile of turbulent heat flux components for passive scalar with \(\Pr=0.005\), together with values and uncertainties calculated in monitor points in the plane \(z=0\). absolute uncertainty is actually lower. A similar observation can be made about the \(z\) velocity component. The streamwise velocity component in Figure 11 is mostly positive. Note, however, the points 29 to 31. Since the component changes the sign in these three points, they are located near the stagnation zone. Similarly, the points 20 to 22 lie in a stagnation zone, where the streamwise velocity is close to zero. However, this component has larger absolute value at points 16 to 18 located downstream of the recirculation region and 26 to 28 located in the recirculation region. The \(y\) velocity component at points 16 to 18, 29 to 31, and to a degree in 26 to 28 confirms the structure observed and discussed in [1], where the flow is directed towards the heater at the edges (of \(z\) axis) but is directed away from the heater in the middle (around \(z=0\)). Also of note is that the velocity component immediately downstream of the step, at points 4 to 6 is directed towards the heater while, for example at point 45, it is directed away from the heater. Note that values for streamwise and \(y\) velocity components should be symmetrical over the plane \(z=0\), while the values for the \(z\) velocity component should be anti-symmetrical, with values in the plane \(z=0\) being zero. In fact, such a test can be performed for all the variables shown. Out of 350 such tests that we could perform, 12 tests failed, or around 3.4%. With the above 95% confidence interval, we would expect the failure rate to be around 5%. If such a test is performed with 3\(\sigma\) uncertainty, only one test fails. At monitor point 43, the \(z\) component of turbulent heat flux for passive scalar with \(\Pr=0.1\) is within 4\(\sigma\) of zero; however, outside of 3\(\sigma\) range. For 3\(\sigma\) interval, the confidence interval is around 99.7% and we would expect one test to fail. The \(z\) component of velocity shows the anti-symmetries over the \(z\) axis. However, interestingly, the points in the recirculation region have considerably larger uncertainties at \(z=0\) than at the sides of the domain. This would indicate the flow is fluctuating over this plane, while being more steady at the sides. The only exception seems to be the points 26 to 28 which lie in the recirculation bubble. Figure 12 shows the diagonal components of Reynolds stress tensor. The uncertainties are comparable between different components, however, the asymmetry of turbulence is still apparent (this is apparent just by observing the scale on the left). The asymmetry can also be found in the statistical uncertainties in the recirculation zone. The turbulent kinetic energy seems to be highest through the middle of the domain. However, there are some exceptions. These are at points 26 to 28, located upstream of the reattachment zone, and at points 16 to 18, located downstream of the reattachment zone. This differs from the reattachment zone, located around points 20 to 22. The peak in fluctuations is found around \(x=10\). This is a peculiar result since the flow seems to be more turbulent closer to the lateral walls; however, this is due to the three dimensional structure of the flow.