In fluid dynamics, Non-overlapping Batch Means (NOBM) estimators have been a conventional choice for quantifying uncertainty in time averages of turbulent flow quantities, due to their speed and simplicity. These estimators aim to transform the autocorrelated time series of a turbulent flow variable to a series of uncorrelated batch means. Estimators derived from NOBM, such as the Overlapping Batch Means (OBM) estimator and the recent Batch Means and Batch Correlations (BMBC) of Russo and Luchini, also fall under the category of batch-based variance estimation. A parameter fundamental to the reliable quantification of uncertainty for these three estimators is the batch size. The questions of how to determine this batch size intrinsic to each estimator, how many batches to have, and if they can be computed using physical timescales, have not been clarified in extant literature. Our work examined the potential of the three batch means estimators, namely NOBM, BMBC and OBM in their ability to quantify time-averaging uncertainties accurately. Additionally, we explored how the batch size and number of batches affected the uncertainty quantification. We demonstrate through applications to turbulent channel flow data and autoregressive synthetic turbulence series, the criteria on which the batch size for each estimator depends on, by inferring their relations to the autocorrelation of the time series. Our analysis allowed us to establish guidelines for selecting optimal batch sizes intrinsic to each estimator, ensuring unbiased estimates of uncertainty. With the guidelines and discussions presented in this study, these estimators can be confidently employed for estimating uncertainty in the time averages of statistically stationary turbulent flow quantities. 

Autoregressive models (ARMs) can be powerful tools for quantifying uncertainty in the time averages of turbulent flow time series data.  This is because ARMs are efficient estimators for the autocorrelation function of statistically stationary turbulence processes. Since the variance of an estimated time average is a function of the autocovariance of the time series, ARMs can be used to formulate more robust estimators for variance of the sample mean. In this work, we discuss the construction of such a variance estimator for computing time-averaging uncertainties in turbulent flow variables. We highlight the differences of the ARM-estimated autocorrelation with the sample-estimated autocorrelation function. Rather than relying on complex order selection criteria, we demonstrate a method that leverages the integral timescale of turbulence to obtain the order of the autoregressive model. A crucial insight about the operating principle of the ARM in terms of the temporal duration covered by the product of model order and spacing between samples, enabled us to develop computationally efficient implementations of the ARM variance estimator. This approach facilitates the quantification of uncertainty in downsampled time series and on a series of autocorrelated batch means, with minimal loss of accuracy. Furthermore, the method for estimating uncertainties in second order moments, using first order uncertainties is also derived. Through these techniques, we applied the ARM-based uncertainty estimator to quantify the uncertainty in the time averaged quantities for turbulent flow a) through a plane channel and b) over periodic hills, respectively. Additionally, we also illustrate the potential of the ARM in generating synthetic turbulence signals, emulating a reference turbulence time series in statistics until the second order.  Our study presents autoregressive models as intuitive yet powerful tools for turbulent flow time series data, paving the way for new applications in the field.  

An automatic method is proposed for the removal of the initialization bias that is intrinsic to the output of any statistically stationary simulation. The general techniques based on optimization approaches such as Beyhaghi et al. [1] following the Marginal Standard Error Rules (MSER) method of White et al. [16] were observed to be highly sensitive to the fluctuations in a time series and resulted in frequent overprediction of the length of the initial truncation. As fluctuations are an innate part of turbulence data, these techniques performed poorly on turbulence quantities, meaning that the local minima was often wrongly interpreted as the minimum variance in the time series and resulted in different transient point predictions for any increments to the sample size. This limitation was overcome by considering the finite difference of the slope of the variance computed in the optimization algorithm. The start of the zero slope region was considered as the initial transient truncation point. This modification to the standard approach eliminated the sensitivity of the scheme, and led to consistent estimates of the transient truncation point, provided that the finite difference time interval was chosen large enough to cover the fluctuations in the time series. Therefore, the step size for the finite difference slope was computed using both visual inspection of the time series and trial and error. We propose the Augmented Dickey-Fuller test as an automatic and reliable method to determine the truncation point, from which the time series is considered stationary and without an initialization bias.

The FDA benchmark nozzle was introduced as an idealized geometry for assessing the performance of Computational Fluid Dynamics (CFD) in blood carrying medical flow devices. While many CFD simulations have been performed on the flow through this nozzle in its \emph{sudden expansion} orientation, there are significant variations in the reported jet-breakdown location and turbulence statistics at the `transitional’ and `turbulent’ Reynolds numbers. Experimental data also contain large uncertainties in viscous stresses and centreline velocities. Previous studies’ inconsistencies and the notable absence of a well-defined turbulent inflow at the FDA-specified Reynolds numbers have led to questionability of the results, making it difficult to establish a reliable reference case for validation. In this work, we performed the direct numerical simulation of the flow through the FDA nozzle at a throat Reynolds number of 7500. This is the first simulation with a well-defined turbulent inflow, facilitating validation of turbulence statistics in the inlet pipe. Our analysis diverges from traditional studies, focusing on the flow features entering the sudden expansion rather than within it. Flow analysis revealed the growth of axial turbulence intensities along the annular region of the throat section. Anisotropy invariant maps across cross-sections, indicated a strong two-component turbulence in the nozzle, which subsequently transitioned to a one-component turbulence in the throat. A proper orthogonal decomposition (POD) of the velocity field upstream of the sudden expansion, identified low wavenumber rod-like vortices in the throat region as the most energetic modes. These structures were characterized by alternate low and high velocities and covered the annulus of the throat. In contrast, flow analysis, statistics, and POD of the flow with a parabolic inlet confirmed that numerical noise led to the breakdown of the jet in the sudden expansion section. Reduced order models of velocity fields with POD modes from the turbulent inflow simulation were used to investigate the sensitivity of the jet breakdown in the sudden expansion section. The results illustrated a downstream shift in the position of the jet breakdown when more modes entered the sudden expansion. The findings in this research advance our understanding of the impact of inflow condition on the ensuing flow dynamics and enhance the efficacy of CFD in simulating and analyzing such flows.

In this paper, we present a general description of the flow structures inside a two-dimensional Food and Drug Administration (FDA) nozzle. To this aim, we have performed numerical simulations using the numerical code Nek5000. The topology patters of the solution obtained, identify four different flow regimes when the flow is steady, where the symmetry of the flow breaks down. An additional case has been studied at higher Reynolds number, when the flow is unsteady, finding a vortex street distributed along the expansion pipe of the geometry. Linear stability analysis identifies the evolution of two steady and two unsteady modes. The results obtained have been connected with the changes in the topology of the flow. Finally, higher-order dynamic mode decomposition has been applied to identify the main flow structures in the unsteady flow inside the FDA nozzle. The highest-amplitude dynamic mode decomposition (DMD) modes identified by the method model the vortex street in the expansion of the geometry.

This study introduces vector autoregression (VAR) as a linear procedure that can be used for synthetizing turbulence time series over an entire plane, allowing them to be imposed as efficient turbulent inflow conditions in simulations requiring stationary and cross-correlated turbulence time series. A VAR model is applied to the complex time coefficients derived from a Fourier-based proper orthogonal decomposition (POD) of the velocity fields of the precursor simulation of a turbulent boundary layer at a momentum thickness based Reynolds number, Re_theta=790. VAR is a statistical tool for modelling and prediction of multivariate time series through capturing linear correlations between multiple time series. By performing POD, firstly a subset of the most energetic structures in space are extracted, and then a VAR model is fitted to their time coefficients. It is observed that VAR models constructed using time coefficients of 5 and 30 most energetic POD modes per wave number (corresponding to >40% and >90% of turbulent kinetic energy across all wave numbers, respectively), are able to make accurate predictions of the evolution of the velocity field at Re_theta=790 for infinite time. Moreover, the two-dimensional velocity fields from the low-order POD-VAR are used as a turbulent inflow condition and compared against other common inflow methods. Since the VAR model can produce an infinite number of velocity planes in time, this enables reaching statistical stationarity without having to run an extremely long precursor simulation or applying ad-hoc methods such as periodic time series. 

The efficiency of ensemble averaging over standard time averaging for large scale simulations of turbulence has been investigated recently in the context of high performance parallel computing. The comparisons between the ensemble mean and the time averaging are usually reported in terms of parallel efficiency metrics. A common limitation of these studies is the failure to have large number of realizations that are statistically significant for the ensemble mean, due to high computational cost. As a result, there is a lack of clarity regarding the conclusions drawn from these studies. We have proposed the autoregressive model as a way to generate ensembles of signals that replicate statistical properties of turbulent flow time series, including an initial transient duration. This approach facilitated quick analysis of ensemble averages. Our results showed that the number of time series utilized in the ensemble plays a crucial role in the convergence of the ensemble mean. The autoregressive model is fit to a reference time series from a turbulent flow simulation to determine the parameters of the model. These parameters are then used to generate new “synthetic turbulent” series of any desired length. The error in the ensemble mean and variance of the ensemble were analysed through increasing number of realizations and samples per realization. Both factors were observed to influence the ensemble mean convergence. The present study offers insights for researchers seeking to determine whether ensemble averaging or time averaging is more suitable for their specific problem and simulation time. The source code used to conduct this study in a parallel environment has also been provided.

A framework is introduced for accurate estimation of time-average uncertainties in various types of turbulence statistics. A thorough set of guidelines is provided to adjust the different hyperparameters for estimating uncertainty in sample mean estimators (SMEs). For high-order turbulence statistics, a novel approach is proposed which avoids any linearization and preserves all relevant temporal and spatial correlations and cross-covariances between SMEs. This approach is able to accurately estimate uncertainties in any arbitrary statistical moment. The usability of the approach is demonstrated by applying it to data from direct numerical simulation (DNS) of the turbulent flow over a periodic hill and through a straight circular pipe.