# 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  
The uncertainties in monitor points can be overlaid with the profiles that were presented in [1]. The monitor point data cannot be extended to an arbitrary position in space, however, the profile data can be interpolated to the position of the monitor point. In our case, this interpolation was performed in the spectral space. First, we identify the relevant elements in the domain in which the interpolated results are desired. Then, within the elements, which are relevant, the solution in collocation points is transformed into the factors of spectral modes. The appropriate one-dimensional base functions in case of nek5000 are \(P_{n}(x)-P_{n-2}(x)\), where \(P_{n}(x)\) is a Legendre polynomial of order \(n\). Note that this is not connected with the PN/PN-2 formulation, where the polynomials used for pressure are actually of lower order. The range of the polynomials is limited to the interval [\(-\)1,1]. Appropriate translation and scaling has to be performed to obtain the results at the desired location within the element. Since we are using 7 collocation points per direction in each element, the maximal order of base function polynomial is 6. The interpolated data are obtained by evaluating the base function at the desired location within the element and multiplying by appropriate spectral mode factor. To obtain the appropriate line profile, two coordinates of the monitor point are taken as the desired location of the line profile. The positions along the line are the same as the positions of collocation points in this direction. The estimation of statistical uncertainties allows us to put error bars on the results of a DNS. The following figures try to demonstrate the uncertainties within the results in  
Figure 4: Convergence of relative \(2\sigma\) uncertainty of average streamwise component in three monitor points (**a**) and convergence of the decorrelation time expressed as number of time steps (**b**). profiles. Since in the DNS we are usually concerned with other sources of uncertainty it might be unusual to see significant error bars which originate only from statistical sources. Figure 5 shows the profile lines of average velocity components together with values and \(2\sigma\) uncertainties obtained in the relevant monitor points. The profiles are limited to the plane \(z=0\), where most of the monitor points lie. However, due to brevity, only some profiles along with appropriate monitor points in this plane are shown. At \(y<-2\), the heater is present and all velocity values are zero there. For profile \(x=-0.125\), the region below \(y<0\) is located inside the step wall and that is why the velocity values are equal to zero. From the figure it is apparent that the streamwise velocity component is well converged, since the uncertainties are almost invisible. However, for the \(y\) velocity component the uncertainties are relatively significant. Also, location significantly influences the uncertainties, if we compare uncertainties at \(x=10\) and \(x=14\). The \(z\) component of velocity is the least interesting and is not shown in figure. Although there is a weak flow through the plane \(z=0\) immediately after the step (\(x=0.1\)), with the added uncertainties it is clear that it is not statistically significant, as is expected due to the symmetry. The average values fall within \(2\sigma\) uncertainty of zero. Figure 6 shows the profiles of both passive scalars together with the averages and uncertainties calculated in the monitor points. Here, for the locations at \(y<-2\) the fields are not zero, since the thermal field penetrates into the walls and heater. Similar is true for the location at \(x=-0.125\) and values with \(y<0\). The uncertainties in the scalar fields are relatively small, which we attribute to high diffusion due to low Prandtl number. Figure 7 shows the diagonal and non-diagonal components of the Reynolds stress tensor that are expected to be different from zero in the plane \(z=0\) together with the monitor point data.