# Statistical Uncertainty of DNS in Geometries without Homogeneous Directions
## 3 Results
### A Priori Estimation of the Statistical Uncertainty  
For both cases considered here, one can estimate the order of magnitude of the statistical uncertainty plaguing the averaged velocity prior to the DNS. A coarse estimation of the correlation time \(T_{0}\) can be obtained from a characteristic length \(L\) and velocity scale: \(T_{0}=L/\langle u\rangle\). Then, assuming a given turbulence intensity \(I=\sqrt{k}/\langle u\rangle\), one can compute the relative statistical uncertainty for the averaged velocity corresponding to a given simulation time \(T\):  
\[\frac{2\sigma}{\langle u\rangle}\sim\frac{2}{\langle u\rangle}\sqrt{\frac{k}{ T_{0}-1}}. \tag{4}\]  
For the channel flow case, using wall-units, \(\langle u\rangle\) is around 10, \(L\) is around 180 (in wall units) and the turbulent kinetic energy per unit mass is around 10. This leads to a decorrelation time \(T_{0}\) of around 18. The simulations time is around 54 thousand. The corresponding relative statistical error without spatial averaging is around 1.1 %, which is the order of magnitude observed in the top-left frame of Figure 3.  
For the BFS case, after the step, \(\langle u\rangle\) is around 0.6, \(L\) is around 3.6 and the turbulent kinetic energy per unit mass is around 0.01. This leads to a decorrelation time \(T_{0}\) of around 6.  
Figure 15: Comparison of results with different sampling periods. On the left, the average streamwise velocity component in three points together with the \(2\sigma\) uncertainty is shown. On the right, the uncertainty change with sampling rate for the same three points is shown.  
The simulation time is around 4800. The corresponding relative statistical error is around 1.2 %, which is in the same order of magnitude observed in the top frame of Figure 11.  
Using the correlation time associated with the averaged velocity, one can compute the relative statistical uncertainty for the average temperature from \(I=\sqrt{\langle\theta^{\prime 2}\rangle}/\langle\theta\rangle\). For both passive scalars in the BFS configuration, \(I\) is around 0.1, \(\langle\theta\rangle/\Delta\theta\) is around 1 and 10, \(\langle\theta^{\prime 2}\rangle/(\Delta\theta)^{2}\) is around 0.01 and 1, and the relative statistical uncertainty on the averaged velocity is around 0.7 %, which is in the same order of magnitude observed in the top frame of Figures 13 and 14.  
Indeed, the proposed estimators only provide an order of magnitude for the relative statistical error on the averaged velocity and temperature: the actual correlation time is not constant across the domain and might even change depending on the velocity component considered. However, the authors conjecture that the estimator can be combined with a RANS simulation, from which both \(k\), \(\langle\theta^{\prime 2}\rangle\), \(\langle u\rangle\) and \(\langle\theta\rangle\) can be obtained. The estimator defines the lower limit for uncertainty of RANS simulation due to the turbulent nature of the flow. Although RANS simulation results are burdened with other sources of error, which are usually of greater concern, this might provide an accurate a-priori estimator for the DNS, LES or unsteady RANS simulation and averaging times needed to obtain a given statistical error on the averaged velocity. If the RANS simulation could provide an estimation of the variance of the temperature, one could also build an estimator for the relative statistical error plaguing the averaged temperature. Of course, once a DNS or LES is already running, the estimate of the needed averaging time can be obtained with the arsel toolchain.