Autoregressive models (ARMs) can be powerful tools for quantifying uncertainty in the time averages of turbulent flow quantities. This is because ARMs are efficient estimators of the autocorrelation function (ACF) of statistically stationary turbulence processes. In this study, we demonstrate a method for order selection of ARMs that uses the integral timescale of turbulence. A crucial insight into the operating principles of the ARM in terms of the time span covered by the product of model order and spacing between samples is provided, which enables us to develop computationally efficient implementations of ARM-based uncertainty estimators. 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, a method for estimating uncertainties in second-order moments using first-order uncertainties is discussed. These techniques are applied to the time series data of turbulent flow a) through a plane channel and b) over periodic hills. Additionally, we illustrate the potential of ARMs in generating synthetic turbulence time series. Our study presents autoregressive models as intuitive and powerful tools for turbulent flows, paving the way for further applications in the field.

This study introduces vector autoregression (VAR) as a linear procedure that can be used for synthesizing turbulence time series over an entire plane, allowing them to be imposed as an efficient turbulent inflow condition in simulations requiring stationary and cross-correlated turbulence time series. VAR is a statistical tool for modelling and prediction of multivariate time series through capturing linear correlations between multiple time series. A Fourier-based proper orthogonal decomposition (POD) is performed on the two-dimensional (2-D) velocity slices from a precursor simulation of a turbulent boundary layer at a momentum thickness-based Reynolds number, Re-theta=790. A subset of the most energetic structures in space are then extracted, followed by applying a VAR model to their complex time coefficients. It is observed that VAR models constructed using time coefficients of 5 and 30 most energetic POD modes per wavenumber (corresponding to 66% and 97% of turbulent kinetic energy, respectively) are able to make accurate predictions of the evolution of the velocity field at Re-theta=790 for infinite time. Moreover, the 2-D velocity fields from the POD-VAR when used as a turbulent inflow condition, gave a short development distance when compared with 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.

Quantification of uncertainty in results is crucial in both experiments and simulations of turbulence, yet this practice is notably underutilized. This thesis project delves into statistical tools within the framework of uncertainty quantification to systematically quantify uncertainties that occur in the time varying quantities of turbulence. Two main categories of variance estimators for quantifying time averaging uncertainties in turbulent flow time series are examined in detail – the batch-means based methods and autoregressive model-based methods. The batch size is critical to estimation of uncertainty by the batch methods. We discuss reasons for biased estimates and provide guidance on the selection of batch sizes for the non-overlapping, overlapping and batch means-batch correlations estimators, to obtain consistent estimates of uncertainty when dealing with turbulence time samples. The autoregressive model (ARM)-based estimator was found to be more efficient than the batch methods, in terms of computational efficiency and sample requirements. A novel insight into the operating principle of the ARM, enabled fast quantification of uncertainty with few samples and with batch means series. The extension of univariate autoregressive processes to model entire 2D space-time fields of turbulence, through vector autoregression has been discussed and its potential as a turbulent inflow boundary condition has been illustrated. A crucial flow case that questioned the reliability of Computational Fluid Dynamics (CFD), namely flow through Food and Drug Administration benchmark nozzle device was also simulated in this doctoral thesis project, with a well-defined turbulent inflow boundary condition. Novel insights on the flow physics due to geometrical effects were obtained through statistical analysis, anisotropy invariant maps and proper orthogonal decomposition. These insights provide answers to many open questions in this domain. This work provides analyses and methods to increase the reliability of simulations, expanding the scope of CFD to applications where safety and precision are paramount.

Kvantifiering av osäkerhet i resultat är avgörande i både experiment och simuleringar av turbulens, men denna praxis är anmärkningsvärt underutnyttjad. Detta avhandlingsprojekt undersöker statistiska verktyg inom ramen för osäkerhetskvantifiering för att systematiskt kvantifiera osäkerheter som uppstår i de tidsvarierande kvantiteterna av turbulens. Två huvudkategorier av variansskattare för kvantifiering av tidsmedelvärderade osäkerheter i tidsserier av turbulenta flöden undersöktes i detalj;  batch-meansbaserade metoder och autoregressiva modellbaserade metoder. Batchstorleken är avgörande för uppskattning av osäkerheter med de batchbaserade metoderna. Vi diskuterar orsakerna till avvikande  skattningar och ger vägledning kring valet av batchstorlekar för de icke-överlappande, överlappande och batchmedel-batchkorrelationsskattarna, för att få konsekventa skattningar av osäkerheten vid hantering av turbulenstidssampels. Den autoregressiva modellen (ARM)-baserade estimatorn visade sig vara effektivare än batchmetoderna avseende beräkningseffektivitet och samplingskrav. En ny insikt i ARM:s funktionsprincip möjliggjorde snabb kvantifiering av osäkerheter med få stickprov och med batchmedelvärdesserier. utökningen av univariata AR-processer till att modellera hela 2D-rum-tidsfält av turbulens, genom vektorautoregression, har undersökts och dess potential som randvillkor för turbulenta inflöden har illustrerats. Ett avgörande flödesfall som utmanade tillförlitligheten av Computational Fluid Dynamics (CFD), nämligen flöde genom FDA benchmark munstycksanordning, simulerades också i denna avhandling, med ett väldefinierat turbulent inflödesrandvillkor. Nya insikter om flödesfysiken baserad på geometriska effekter erhölls genom statistisk analys, anisotropi-invarianta avbildningar och ortogonala nedbrytningstekniker. Dessa insikter ger svar på många öppna frågor inom denna domän. Detta arbete ökar simuleringarnas tillförlitlighet och utökar omfattningen av CFD till applikationer där säkerhet och precision är av största vikt.

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.

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.

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.

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.

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.  

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.

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.