# Statistical Uncertainty of DNS in Geometries without Homogeneous Directions
## 1 Introduction  
The present work represents a synthesis on the statistical uncertainties of the results obtained in a direct numerical simulation (DNS) of flow in a channel and a flow over a confined backward-facing step (BFS). Some of the results of the DNS in BFS were already presented by [PERSON] et al. [1]. To be able to compare results of DNS to experiments and other simulations, the results are presented in a form of averages. As is well known [2], there are two errors attributed to data obtained in a DNS. The more widely known are the errors connected to spatial and temporal resolution. The scale of these errors is usually shown with results obtained with different spatial (temporal) resolutions or with comparisons of smallest turbulent scales to the spatial resolution.  
The second error is accumulated due to finite interval when performing time integration for statistically steady flows. This error drops as a square root of integration time (\(T^{-1/2}\)) and in most simulations, defines the duration of DNS, as well as the computational cost of the DNS. To amend the high computational costs, in simulations with homogeneous directions, these can be utilised to speed-up the reduction of statistical uncertainty by, in addition to temporal integration also performing spatial integration over the homogeneous directions. With such method, the simulation can be one or even two orders of magnitude less demanding. To clarify, according to [PERSON] [3], homogeneous directions are directionsin which statistical quantities do not vary with translation. Similarly, a statistical steady state of a turbulent flow is a state where statistical quantities do not vary with translation of time averaging interval.  
[PERSON] and [PERSON] [4] performed a detailed study of accuracies of different DNS databases of channel flow. The analysis was focused on spatial resolutions, as well as different algorithms. The study showed that higher statistical moments require better spatial resolutions to achieve the same level of accuracy as the first moments of statistics. More recently, [PERSON] et al. [5] performed an analysis of how starting time of averaging and the averaging time affects the convergence of statistical quantities in a DNS. They also explored the effects of different sampling rates. [PERSON] et al. [6] in a recent Large Eddy Simulation (LES) study found that due to large integral time scales, longer averaging times are needed in regions near the walls. As noted by [PERSON] et al. [7], it would be of interest to the turbulent community if more attention would be dedicated to the uncertainties and convergence criteria.  
The DNS results exposed here were presented in the papers [1,8]: the details of the numerical setups can be found there. Only a short description will be given here. A more thorough presentation of all the parameters of the BFS DNS can also be found in [9]. Therefore, we also compared the accuracy of the Nek5000 solver in the channel upstream of the step with the same distribution of elements to the results obtained with a conservative pseudospectral method that was used in the channel simulation. If the reader is interested in a comparison of Reynolds Averaged Navier Stokes (RANS) simulation and LES results, they can be found in [10]. In the present paper, we rather focus on the statistical uncertainties found in the channel flow with two homogeneous directions, and on the BFS flow with no homogeneous directions. In Section 2 we present the numerical setups of channel flow DNS and BFS DNS. In Section 3 we present the results, first for the channel flow and then for the flow over the BFS. The results for the BFS are first presented as profiles with added error bars and later we present results in 49 monitor points. The concluding remarks are in Section 4.