## Publications

96 results found

Kantarakias KD, Papadakis G, 2023, Sensitivity-enhanced generalized polynomial chaos for efficient uncertainty quantification, *JOURNAL OF COMPUTATIONAL PHYSICS*, Vol: 491, ISSN: 0021-9991

Mikhaylov K, Rigopoulos S, Papadakis G, 2023, Decomposition of power number in a stirred tank and real time reconstruction of 3D large-scale flow structures from sparse pressure measurements, *CHEMICAL ENGINEERING SCIENCE*, Vol: 279, ISSN: 0009-2509

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Tsagkaridis M, Papadakis G, Jones WP,
et al., 2023, Large eddy simulation of turbulent flame synthesis of silica nanoparticles with an extended population balance model, *Flow, Turbulence and Combustion*, Vol: 111, Pages: 1029-1057, ISSN: 0003-6994

In the present study, a recently proposed extended population balance equation (PBE) model for aggregation and sintering is incorporated into a large eddy simulation-probability density function (LES-PDF) modelling framework to investigate synthesis of silica nanoparticles in a turbulent diffusion flame. The stochastic field method is employed to solve the LES-PBE-PDF equations, characterising the influence of the unresolved sub-grid scale motions and accounting for the interactions between turbulence, chemistry and particle dynamics. The models for gas-phase chemistry and aerosol dynamics are the same as those recently used by the authors to simulate silica synthesis in a laminar flame (Tsagkaridis et al. in Aerosol Sci Technol 57(4):296–317, 2023). Thus, by retaining the same kinetics without any adjustments in parameters, we focus on the modelling issues arising in silica flame synthesis. The LES results are compared with experimental in-situ small-angle X-ray scattering (SAXS) data from the literature. Good agreement is found between numerical predictions and experimental data for temperature. However, the LES model underestimates the SAXS data for the primary particle diameter by a factor of two. Possible reasons for this discrepancy are discussed in view of the previous laminar flame simulations.

Stokes C, Ahmed D, Lind N,
et al., 2023, Aneurysmal growth in type-B aortic dissection: assessing the impact of patient-specific inlet conditions on key haemodynamic indices, *Journal of the Royal Society Interface*, Vol: 20, Pages: 1-16, ISSN: 1742-5662

Type-B aortic dissection is a cardiovascular disease in which a tear develops in the intimal layer of the descending aorta, allowing pressurized blood to delaminate the layers of the vessel wall. In medically managed patients, long-term aneurysmal dilatation of the false lumen (FL) is considered virtually inevitable and is associated with poorer disease outcomes. While the pathophysiological mechanisms driving FL dilatation are not yet understood, haemodynamic factors are believed to play a key role. Computational fluid dynamics (CFD) and 4D-flow MRI (4DMR) analyses have revealed correlations between flow helicity, oscillatory wall shear stress and aneurysmal dilatation of the FL. In this study, we compare CFD simulations using a patient-specific, three-dimensional, three-component inlet velocity profile (4D IVP) extracted from 4DMR data against simulations with flow rate-matched uniform and axial velocity profiles that remain widely used in the absence of 4DMR. We also evaluate the influence of measurement errors in 4DMR data by scaling the 4D IVP to the degree of imaging error detected in prior studies. We observe that oscillatory shear and helicity are highly sensitive to inlet velocity distribution and flow volume throughout the FL and conclude that the choice of IVP may greatly affect the future clinical value of simulations.

Mikhaylov K, Rigopoulos S, Papadakis G, 2023, Three-dimensional characterisation of macro-instabilities in a turbulent stirred tank flow and reconstruction from sparse measurements using machine learning methods, *CHEMICAL ENGINEERING RESEARCH & DESIGN*, Vol: 196, Pages: 276-296, ISSN: 0263-8762

Lu S, Papadakis G, 2023, Flow reconstruction around a surface-mounted prism from sparse velocity and/or scalar measurements using a combination of POD and a data-driven estimator, *Flow, Turbulence and Combustion*, Vol: 110, Pages: 1059-1090, ISSN: 0003-6994

A data-driven algorithm is proposed for flow reconstruction from sparse velocity and/or scalar measurements. The algorithm is applied to the flow around a two-dimensional, wall-mounted, square prism. To reduce the problem dimensionality, snapshots of flow and scalar fields are processed to derive POD modes and their time coefficients. Then a system identification algorithm is employed to build a reduced order, linear, dynamical system for the flow and scalar dynamics. Optimal estimation theory is subsequently applied to derive a Kalman estimator to predict the time coefficients of the POD modes from sparse measurements. Analysis of the flow and scalar spectra demonstrate that the flow field leaves its footprint on the scalar, thus extracting velocity from scalar concentration measurements is meaningful. The results show that remarkably good reconstruction of the flow statistics (Reynolds stresses) and instantaneous flow patterns can be obtained using a very small number of sensors (even a single scalar sensor yields very satisfactory results for the case considered). The Kalman estimator derived at one condition is able to reconstruct with acceptable accuracy the flow fields at two nearby off-design conditions. Further work is needed to assess the performance of the algorithm in more complex, three-dimensional, flows.

Yao H, Papadakis G, 2023, On the role of the laminar/turbulent interface in energy transfer between scales in bypass transition, *JOURNAL OF FLUID MECHANICS*, Vol: 960, ISSN: 0022-1120

Tang HY, Rigopoulos S, Papadakis G, 2023, On the effect of turbulent fluctuations on precipitation: A direct numerical simulation – population balance study, *Chemical Engineering Science*, Vol: 270, Pages: 1-20, ISSN: 0009-2509

The objective of the present paper is to investigate the effect of turbulent fluctuations on precipitationand the implications for modelling. To this end, a coupled direct numerical simulation (DNS) - populationbalance study is conducted on the experiments of Schwarzer et al. (2006) on BaSO4 precipitation in a Tmixer. The unclosed terms in the averaged population balance equation are identified and evaluated viaDNS. A comparison of the average nucleation and growth rates with those computed with the averagevalues shows significant deviations indicative of the importance of fluctuations in precipitation modelling. Furthermore, the correlation between growth and number density is analysed, as well as its contribution to the reactant consumption. The study is performed at Sc ¼ 1 as it is, at present, not possible toresolve the sub-Kolmogorov scales at high Reynolds and Schmidt numbers. However, an attempt toinvestigate the effect of sub-Kolmogorov scales is made by performing a simulation of precipitation ina similar T-mixer at a lower (but still turbulent) Reynolds number and at Sc ¼ 1 and Sc ¼ 10. The findingsindicate the presence of thinner reaction zones at higher Sc, but the effect on the product particle sizedistribution is marginal. An analysis of the results indicates that this is due to a compensation of theeffect of thinner reaction zones by higher reaction rates occurring therein

Kantarakias KD, Papadakis G, 2023, Sensitivity analysis of chaotic systems using a frequency-domain shadowing approach, *Journal of Computational Physics*, Vol: 474, ISSN: 0021-9991

We present a frequency-domain method for computing the sensitivities of time-averaged quantities of chaotic systems with respect to input parameters. Such sensitivities cannot be computed by conventional adjoint analysis tools, because the presence of positive Lyapunov exponents leads to exponential growth of the adjoint variables. The proposed method is based on the well established least-square shadowing (LSS) approach [1], that formulates the evaluation of sensitivities as an optimisation problem, thereby avoiding the exponential growth of the solution. All existing formulations of LSS (and its variants) are in the time domain. In the present paper, we reformulate the LSS method in the frequency (Fourier) space using harmonic balancing. The resulting system is closed using periodicity. The new method is tested on the Kuramoto-Sivashinsky system and the results match with those obtained using the standard time-domain formulation. The storage and computing requirements of the direct solution grow rapidly with the size of the system. To mitigate these requirements, we propose a resolvent-based iterative approach that needs much less storage. Application to the Kuramoto-Sivashinsky system gave accurate results with low computational cost. Truncating the large frequencies with small energy content from the harmonic balancing operator did not affect the accuracy of the computed sensitivities. Further work is needed to assess the performance and scalability of the proposed method.

Bilbao-Ludena JC, Papadakis G, 2023, Structure of vorticity and turbulence fields in a separated flow around a finite wing: Analysis using direct numerical simulation, *PHYSICAL REVIEW FLUIDS*, Vol: 8, ISSN: 2469-990X

Tsagkaridis M, Rigopoulos S, Papadakis G, 2023, Modeling of silica synthesis in a laminar flame by coupling an extended population balance model with computational fluid dynamics, *Aerosol Science and Technology*, Vol: 57, Pages: 296-317, ISSN: 0278-6826

In the present study, we propose a novel extended population balance equation (PBE) model for aggregation and sintering and couple it with computational fluid dynamics (CFD) to investigate synthesis of silica nanoparticles in a laminar diffusion flame. The extended PBE includes finite-rate sintering of primary particles by solving the PBE together with a transport equation for the number concentration of primary particles. In the process simulated, the particles are formed via the oxidation of a vapor precursor, hexamethyldisiloxane (HMDSO), and the aerosol processes include nucleation, condensation, aggregation and sintering. The model is validated with detailed experimental in-situ SAXS data found in the literature and is also compared with a monodisperse and a two-PBE approach. Good agreement is found between the extended one-PBE and two-PBE models, while both of them provide a substantial improvement over the monodisperse one. Furthermore, the coupled CFD-PBE simulation with the extended one-PBE model reduces substantially the computational time as compared with the two-PBE model and requires less than twice the time needed for the monodisperse model. Excellent agreement is found between numerical predictions and experimental data for temperature along the centerline and reasonably good agreement is found between numerical predictions and SAXS data for primary particle diameters. While results for the particle number concentration are in qualitative agreement with the experimental data, the particle formation rate is overpredicted, leading to an overestimation of the number concentration of the primary particles. This is attributed to uncertainties in the experimental data and precursor decomposition kinetics.

Tang HY, Rigopoulos S, Papadakis G, 2022, On the interaction of turbulence with nucleation and growth in reaction crystallisation, *Journal of Fluid Mechanics*, Vol: 944, ISSN: 0022-1120

The objective of this work is to investigate the interaction of turbulence with the nonlinear processes of particle nucleation and growth that occur in reaction crystallisation, also known as precipitation. A validated methodology for coupling the population balance equation with direct numerical simulation of turbulent flows is employed for simulating an experiment conducted by Schwarzer <jats:italic>et al.</jats:italic> (<jats:italic>Chem. Engng Sci.</jats:italic>, vol. 61, no. 1, 2006, pp. 167–181), where barium sulphate nanoparticles are formed by mixing and reaction of barium chloride and sulphate acid in a T-mixer, with the spatial resolution resolved down to the Kolmogorov scale. A unity Schmidt number is assumed, since at present it is not possible to resolve the Batchelor scale for realistic Schmidt numbers (order of 1000 or more). The probability density function, filtered averages and spatial distribution of time and length scales are all examined in order to shed light on the interplay of turbulence and precipitation. Separate Damköhler numbers are defined for nucleation and growth and both are found to be close to unity, indicating that the process is neither mixing nor kinetics controlled. The nucleation length scales are also evaluated and compared with the Kolmogorov scale to show the importance of resolving nucleation bursts. In addition, zones of different rate-determining mechanisms are identified. The ultimate aim of precipitation is to obtain control over the product particle size distribution, and the present study elucidates the synergistic or competing roles of mixing, nucleation and growth on the process outcome and discusses the implications for modelling.

Papadakis G, Yao H, Mollicone JP, 2022, Analysis of interscale energy transfer in a boundary layer undergoing bypass transition, *Journal of Fluid Mechanics*, Vol: 941, ISSN: 0022-1120

The Kármán–Howarth–Monin–Hill equation is employed to study the production and interscale energy transfer in a boundary layer undergoing bypass transition due to free-stream turbulence. The energy flux between different length scales is calculated at several streamwise locations covering the laminar, transitional and turbulent regimes. Maps of scale energy production and flux vectors are visualised on two-dimensional planes and three-dimensional hyper-planes that comprise both physical and separation spaces. In the transitional region, the maps show strong inverse cascade in the streamwise direction near the wall. The energy flux vectors emanate from a region of strong production and transfer energy to larger streamwise scales. To provide deeper insight into the origin of the inverse cascade process, we decompose the energy flux vector into components arising from nonlinear interactions between velocity fluctuations, mean flow inhomogeneity, pressure and viscous effects. The inverse cascade is mainly due to the nonlinear interaction component, and in the earliest stages of transition this component competes with that due to mean flow inhomogeneity. By superposing the instantaneous velocity fields and the energy flux vectors, we relate the inverse cascade process to the growth of turbulent spots. Once the transition process is complete, the maps become very similar to those observed in other fully developed turbulent flows, such as channel flow. Finally we characterise the nonlinear interaction term using probability density functions (PDFs) evaluated at different wall-normal heights. The PDFs are asymmetric and wide-skirted as in homogeneous isotropic turbulence, but are skewed towards positive values reflecting the inverse cascade.

Schlander R, Rigopoulos S, Papadakis G, 2022, Analysis of wall mass transfer in a turbulent pipe flow combining extended POD and FIK identity, *Physical Review Fluids*, Vol: 7, ISSN: 2469-990X

We combine extended proper orthogonal decomposition (EPOD) together with the Fukagata-Iwamoto-Kasagi (FIK) identity to quantify the role of individual coherent structures on the wall mass transfer in a turbulent pipe flow. Direct numerical simulation at a Reynolds number of 5300 (based on bulk velocity) is performed with the passive scalar released at the pipe inlet. The proper orthogonal decomposition (POD) eigenvalues show that the scalar field can be described by a more compact set of modes compared to the velocity field, and that these modes are skewed towards higher azimuthal wave numbers. POD modes for the scalar and EPOD modes for the velocity are visualized in the cross-stream plane to infer the capacity of each mode to transport scalar to and from the wall. A form of the FIK identity is derived for the wall mass transfer coefficient (Sherwood number, Sh) and employed to separate the contributions of the mean and fluctuating velocity and scalar fields. The FIK decomposition shows that the turbulent velocity/scalar correlations account for up to 65.8% of the total Sh. The contribution of each POD and EPOD mode to the Sh number is also computed; it is found that, using azimuthal wave numbers m=1–15 and POD modes n=1–10, it is possible to reconstruct 49% of the turbulent component of Sh, with the velocity modes containing only 31% of the turbulent kinetic energy. Quadrant analysis shows that these modes are related to ejection and sweep events near the wall, with the ejection events dominating.

Tsagkaridis M, Rigopoulos S, Papadakis G, 2022, Analysis of turbulent coagulation in a jet with discretised population balance and DNS, *Journal of Fluid Mechanics*, Vol: 937, Pages: 1-31, ISSN: 0022-1120

The objective of the present study is to investigate turbulence-coagulation interaction via direct numerical simulation (DNS) coupled with the population balance equation (PBE). Coagulation is an important process in several environmental and engineering applications involving turbulent flow, including soot formation, gas-phase synthesis of nanoparticles and atmospheric processes, but its interaction with turbulence is not yet fully understood. Particle dynamics can be described by the PBE, whose Reynolds de-composition leads to unclosed terms involving correlations of number density fluctuations. In this work, we employ a discretisation (sectional) method for the solution of the PBE, which is free of a priori assumptions regarding the particle size distribution (PSD), and couple it with a DNS for the flow field in order to study the behaviour and significance of the unknown correlations. At present, it is not feasible to resolve the Batchelor scales thatresult from diffusion at high Schmidt number, hence a unity Schmidt number is employed. The investigation is conducted on a three-dimensional planar jet laden with monodisperse nanoparticles, and coagulation in the free-molecule regime is considered. The correlations due to turbulent fluctuations of the particle number density are calculated at several points in the domain and found to be positive in most cases, except close to the jet break-up. The transport equation for the moments of the PSD is also studied, and itis found that the correlations make a considerable contribution to the time-averaged coagulation source term, up to 20% on the jet centreline and 40% close to the edges.

Schlander RK, Papadakis G, Rigopoulos S, 2022, RESOLVENT ANALYSIS OF TURBULENT PIPE FLOW LADEN WITH LOW INERTIA PARTICLES

We extend the resolvent framework to turbulent flows laden with low inertia particles. The particle velocities are modelled using the equilibrium Eulerian model, which is valid for Stokes numbers up to 1. We analyse a vertical turbulent pipe flow with a Reynolds number of 5300 based on diameter and bulk velocity, for Froude numbers Fr = 0.4,-0.4 and Stokes numbers St+ = 0-1. A direct numerical simulation (DNS) for a pipe with a length of 7.5 diameters (D) is performed with the particles released uniformly at the pipe inlet. The resolvent formulation can predict some of the physical phenomena observed in inertial particle flows such as localized high concentration due to the vortical centrifuge effect, turbophoresis and gravitational effects. It is shown that the upward flow increases particle concentration in the log layer of the pipe. The downward flow increases concentration near the centre of the pipe: both features have been observed in previous Lagrangian simulations as well as experiments. The main effect of Stokes number is the amplification of smaller streamwise wavelengths, therefore, increasing local scale clustering of particles. The effect of the direction of gravity was also reproduced using a simplified resolvent model which did not require a mean concentration profile as input and simplifies the analysis since no prior simulation or experiment is required for the model to work.

Papadakis G, Rigopoulos S, Mikhaylov K, 2021, Reconstruction of large scale flow structures in a stirred tank from limited sensor data, *AIChE Journal*, Vol: 67, Pages: 1-16, ISSN: 0001-1541

We combine reduced order modelling and system identification to reconstruct the temporal evolution of large scale vortical structures behind the blades of a Rushton impeller. We performed Direct Numerical Simulations at Reynolds number 600 and employed proper orthogonal decomposition (POD) to extract the dominant modes and their temporal coefficients. We then applied the identification algorithm, N4SID, to construct an estimator that captures the relation between the velocity signals at sensor points (input) and the POD coefficients (output). We show that the first pair of modes can be very well reconstructed using the velocity time signal from even a single sensor point. A larger number of points improves accuracy and robustness, and also leads to better reconstruction for the second pair of POD modes. Application of the estimator derived at Re=600 to the flows at Re=500 and 700, shows that it is robust with respect to changes in operating conditions.

Tang HY, Rigopoulos S, Papadakis G, 2020, A methodology for coupling DNS and discretised population balance for modelling turbulent precipitation, *International Journal of Heat and Fluid Flow*, Vol: 86, ISSN: 0142-727X

In this paper, we present a methodology for simulating nanoparticle formation in a turbulent flow by coupling Direct Numerical Simulation (DNS) and population balance modelling. The population balance equation (PBE) is solved via a discretisation method employing a composite grid that provides sufficient detail over the wide range of particle sizes reached during the precipitation process. The coupled DNS/PBE approach captures accurately the strong interaction between the dynamics of turbulent mixing and particle formation processes. It also allows the calculation of the particle size distribution (PSD) of the product and enables an investigation on how it is controlled by turbulent mixing. Finally, it provides the statistics of kinetic processes and their timescales so that further analysis can be performed. The methodology is applied to the simulation of experiments of hydrodynamics and nanoparticle precipitation in a T-mixer (Schwertfirm et al., 2007, Int. J. of Heat and Fluid Flow 28, pp. 1429-1442; Schwarzer et al., 2006, Chem.Eng. Sci. 61, pp. 167-181), and the agreement with the experimental results is very good.

Yao H, Alves Portela F, Papadakis G, 2020, Evolution of conditionally-averaged second order structure functions in a transitional boundary layer, *Physical Review Fluids*, Vol: 5, ISSN: 2469-990X

We consider the bypass transition in a flat plate boundary layer subject to free-stream turbulence and compute the evolution of the second-order structure function of the streamwise velocity, du2(,), from the laminar to the fully turbulent region using DNS. In order to separate the contributions from laminar and turbulent events at the two points used to define du(→x,→r), we apply conditional sampling based on the local instantaneous intermittency, τ (1 for turbulent and 0 for laminar events). Using τ(→x,t), we define two-point intermittencies, γ(TT), γ(LL) and γ(TL) which physically represent the probabilities that both points are in turbulent or laminar patches, or one in turbulent and the other in a laminar patch, respectively. Similarly, we also define the conditionally-averaged structure functions, ⟨du2⟩(TT), ⟨du2⟩(LL) and ⟨du2⟩(TL) and decompose ⟨du2⟩(→x,→r) in terms of these conditional averages. The derived expressions generalise existing decompositions of single-point statistics to two-point statistics. It is found that in the transition region, laminar streaky structures maintain their geometrical characteristics in the physical and scale space well inside the transition region, even after the initial break down to form turbulent spots. Analysis of the ⟨du2⟩(TT) fields reveal that the outer mode is the dominant secondary instability mechanism. Further analysis reveals how turbulence spots penetrate the boundary layer and approach the wall. The peaks of ⟨du2⟩(TT) in scale space appear in larger streamwise separations as transition progresses and this is explained by the strong growth of turbulent spots in this direction. On the other hand, the spanwise separation where the peak occurs remains relatively constant and is determined by the initial inception process. We also analyse the evolution of the two-point intermittency field, γ(TT), at different locations. In particular, we study the growth of the

Papadakis G, Shawki K, 2020, Feedback control of chaotic systems using Multiple Shooting Shadowing andapplication to Kuramoto Sivashinsky equation, *Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences*, Vol: 476, Pages: 1-20, ISSN: 1364-5021

We propose an iterative method to evaluate thefeedback control kernel of a chaotic system directlyfrom the system’s attractor. Such kernels are currentlycomputed using standard linear optimal controltheory, known as Linear Quadratic Regulator (LQR)theory. This is however applicable only to linearsystems, which are obtained by linearising thesystem governing equations around a target state.In the present paper, we employ the PreconditionedMultiple Shooting Shadowing (PMSS) algorithm tocompute the kernel directly from the non-linear dynamics, thereby bypassing the linear approximation.Using the adjoint version of the PMSS algorithm,we show that we can compute the kernel at any point of the domain in a single computation. The algorithm replaces the standard adjoint equation (that is ill-conditioned for chaotic systems) with a well-conditioned adjoint, producing reliable sensitivities which are used to evaluate the feedback matrix elements. We apply the idea to the Kuramoto Sivashinsky equation. We compare the computed kernel with that produced by the standard LQR algorithm and note similarities and differences. Bothkernels are stabilising, have compact support and similar shape. We explain the shape using two-point spatial correlations that capture the streaky structure of the solution of the uncontrolled system.

Alves Portela F, Papadakis G, Vassilicos C, 2020, The role of coherent structures and inhomogeneity in near-field inter-scaleturbulent energy transfers, *Journal of Fluid Mechanics*, Vol: 896, Pages: A16-1-A16-24, ISSN: 0022-1120

We use Direct Numerical Simulation (DNS) data to study inter-scale and inter-space energy exchanges in the near-field of a turbulent wake of a square prism in terms of a Kármán-Howarth-Monin-Hill (KHMH) equation written for a triple decomposition of the velocity field which takes into account the presence of quasi-periodic vortex sheddingcoherent structures. Concentrating attention on the plane of the mean flow and on the geometric centreline, we calculate orientation-averages of every term in the KHMH equation. The near-field considered here ranges between 2 and 8 times the width d of the square prism and is very inhomogeneous and out of equilibrium so that non-stationarityand inhomogeneity contributions to the KHMH balance are dominant. The mean flow produces kinetic energy which feeds the vortex shedding coherent structures. In turn, these coherent structures transfer their energy to the stochastic turbulent fluctuations over all length-scales r from the Taylor length to d and dominate spatial turbulent transport of small-scale two-point stochastic turbulent fluctuations. The orientation averaged non-linear inter-scale transfer rate a which was found to be approximately independent of r by Alves Portela et al. (2017) in the range 6 r 6 0:3d at a distance x1 = 2d from the square prism requires an inter-scale transfer contribution of coherent structures for this approximate constancy. However, the near-constancy of a in the range 6 r 6 d at x1 = 8d which was also found by Alves Portela et al. (2017) is mostlyattributable to stochastic fluctuations. Even so, the proximity of a to the turbulence dissipation rate " in the range 6 r 6 d at x1 = 8d does require inter-scale transfer contributions of the coherent structures. Spatial inhomogeneity also makes a direct and distinct contribution to a, and the constancy of a=" close to 1 would not have been possible without it either in this near-field flow. Finally, the pressure-veloci

Papadakis G, Kantarakias K, 2020, Application of generalized Polynomial Chaos for Quantification of uncertainties of time–averages and their sensitivities in chaotic systems, *Algorithms*, Vol: 13, Pages: 1-16, ISSN: 1999-4893

In this paper, we consider the effect of stochastic uncertainties on non-linear systems with chaotic behavior. More specifically, we quantify the effect of parametric uncertainties to time-averaged quantities and their sensitivities. Sampling methods for Uncertainty Quantification (UQ), such as the Monte–Carlo (MC), are very costly, while traditional methods for sensitivity analysis, such as the adjoint, fail in chaotic systems. In this work, we employ the non-intrusive generalized Polynomial Chaos (gPC) for UQ, coupled with the Multiple-Shooting Shadowing (MSS) algorithm for sensitivity analysis of chaotic systems. It is shown that the gPC, coupled with MSS, is an appropriate method for conducting UQ in chaotic systems and produces results that match well with those from MC and Finite-Differences (FD).

Kantarakias K, Shawki K, Papadakis G, 2020, Uncertainty quantification of sensitivities of time-average quantities in chaotic systems, *Physical Review E*, Vol: 101, ISSN: 2470-0045

We consider time-average quantities of chaotic systems and their sensitivity to system parameters. When the parameters are random variables with a prescribed probability density function, the sensitivities are also random. The central aim of the paper is to study and quantify the uncertainty of the sensitivities; this is useful to know in robust design applications. To this end, we couple the nonintrusive polynomial chaos expansion (PCE) with the multiple shooting shadowing (MSS) method, and apply the coupled method to two standard chaotic systems, the Lorenz system and the Kuramoto-Sivashinsky equation. The method leads to accurate results that match well with Monte Carlo simulations (even for low chaos orders, at least for the two systems examined), but it is costly. However, if we apply the concept of shadowing to the system trajectories evaluated at the quadrature integration points of PCE, then the resulting regularization can lead to significant computational savings. We call the new method shadowed PCE (sPCE).

Shawki K, Papadakis G, 2019, A preconditioned Multiple Shooting Shadowing algorithm for the sensitivity analysis of chaotic systems, *Journal of Computational Physics*, Vol: 398, Pages: 1-19, ISSN: 0021-9991

We propose a preconditioner that can accelerate the rate of convergence of the Multiple Shooting Shadowing (MSS) method [1]. This recently proposed method can be used to compute derivatives of time-averaged objectives (also known as sensitivities) to system parameter(s) for chaotic systems. We propose a block diagonal preconditioner, which is based on a partial singular value decomposition of the MSS constraint matrix. The preconditioner can be computed using matrix-vector products only (i.e. it is matrix-free) and is fully parallelised in the time domain. Two chaotic systems are considered, the Lorenz system and the 1D Kuramoto Sivashinsky equation. Combination of the preconditioner with a regularisation method leads to tight bracketing of the eigenvalues to a narrow range. This combination results in a significant reduction in the number of iterations, and renders the convergence rate almost independent of the number of degrees of freedom of the system, and the length of the trajectory that is used to compute the time-averaged objective. This can potentially allow the method to be used for large chaotic systems (such as turbulent flows) and optimal control applications. The singular value decomposition of the constraint matrix can also be used to quantify the effect of the system condition on the accuracy of the sensitivities. In fact, neglecting the singular modes affected by noise, we recover the correct values of sensitivity that match closely with those obtained with finite differences for the Kuramoto Sivashinsky equation in the light turbulent regime. We notice a similar improvement for the Lorenz system as well.

Guzman Inigo J, Sodar M, Papadakis G, 2019, A data-based, reduced-order, dynamic estimator for reconstruction of non-linear flows exhibiting limit-cycle oscillations, *Physical Review Fluids*, Vol: 4, ISSN: 2469-990X

We apply a data-based, linear dynamic estimator to reconstruct the velocity field from measurements at a single sensor point in the wake of an aerofoil. In particular, we consider a NACA0012aerofoil at Re = 600 and 16◦ angle of attack. Under these conditions, the flow exhibits a vortexshedding limit cycle. A reduced order model (ROM) of the flow field is extracted using proper orthogonal decomposition (POD). Subsequently, a subspace system identification algorithm (N4SID)is applied to extract directly the estimator matrices from the reduced output of the system (thePOD coefficients). We explore systematically the effect of the number of states of the estimator,the sensor location, the type of sensor measurements (one or both velocity components), and thenumber of POD modes to be recovered. When the signal of a single velocity component (in thestream wise or cross stream directions) is measured, the reconstruction of the first two dominantPOD modes strongly depends on the sensor location. We explore this behaviour and provide aphysical explanation based on the non-linear mode interaction and the spatial distribution of themodes. When however, both components are measured, the performance is very robust, and isalmost independent of the sensor location when the optimal number of estimator states is used.Reconstruction of the less energetic modes is more difficult, but still possible. Finally, we assessthe robustness of the estimator at off-design conditions, at Re = 550 and 650.`

Gallis MA, Torczynski JR, Bitter NP, et al., 2019, DSMC simulations of turbulent flows at moderate Reynolds numbers, 31st International Symposium on Rarefied Gas Dynamics (RGD), Publisher: AMER INST PHYSICS, Pages: 1-6, ISSN: 0094-243X

The Direct Simulation Monte Carlo (DSMC) method has been used for more than 50 years to simulate rarefied gases. The advent of modern supercomputers has brought higher-density near-continuum flows within range. This in turn has revived the debate as to whether the Boltzmann equation, which assumes molecular chaos, can be used to simulate continuum flows when they become turbulent. In an effort to settle this debate, two canonical turbulent flows are examined, and the results are compared to available continuum theoretical and numerical results for the Navier-Stokes equations.

Tang H, Papadakis G, Rigopoulos S, 2019, Coupling direct numerical simulations with population balance modelling for predicting turbulent particle precipitation in a T-mixer, 11th International Symposium on Turbulence and Shear Flow Phenomena (TSFP11), Publisher: TSFP

In this study we develop a methodology for predicting the particle size distribution(PSD)inparticulate process, a process used for producing particulate materials,by coupling population balance modelling and direct numerical simulation. Itis employed in investigating the turbulent precipitation of BaSO4in a T-mixer.The high resolution allowed us to capture the dominating mechanisms.Particle formation is most intense in the impingementand the reactantconsumption in each precipitation mechanism depends on the mixing intensity.Different particle formation statesand their characteristics on the PSD in the early stage arethenidentified.Comparisonwith an ideal reactor showsthat the distribution can be controlled by altering the mixing environment.

Papadakis G, Raspaud J, 2019, Wave propagation in stenotic vessels; theoretical analysis and comparison between 3D and 1D fluid–structure-interaction models, *Journal of Fluids and Structures*, Vol: 88, Pages: 352-366, ISSN: 0889-9746

Using analytical expressions for the pressure and velocity waveforms in tapered vessels, we construct a linear 1D model for wave propagation in stenotic vessels in the frequency domain. We demonstrate that using only two parameters to approximate the exact geometry of the constriction (length and degree of stenosis), we can construct a model that can be solved analytically and can approximate with excellent accuracy the response of the original vessel for a wide range of physiologically relevant frequencies. We then proceed to compare the 1D results with full 3D FSI results from the literature for parameters corresponding to an idealized stenotic carotid artery. We find excellent matching with the volume flow rare over the cardiac cycle (less than 1% error). Using results from DNS simulations to parametrize the velocity profile in the stenotic region, we manage to predict also the pressure distribution with small error (a few percentage points). The method proposed in the paper can be used to approximate vessels of arbitrary shape profile and can be extended to cover the whole cardiovascular tree. Recursive expressions make the solution very fast and open the possibility of carrying out sensitivity and uncertainty quantification studies that require thousands (or even millions) of simulations with minimal cost.

Xiao D, Papadakis G, 2019, Nonlinear optimal control of transition due to a pair of vortical perturbations using a receding horizon approach, *Journal of Fluid Mechanics*, Vol: 861, Pages: 524-555, ISSN: 0022-1120

This paper considers the nonlinear optimal control of transition in a boundary layer flow subjected to a pair of free stream vortical perturbations using a receding horizon approach. The optimal control problem is solved using the Lagrange variational technique that results in a set of linearized adjoint equations, which are used to obtain the optimal wall actuation (blowing and suction from a control slot located in the transition region). The receding horizon approach enables the application of control action over a longer time period, and this allows the extraction of time-averaged statistics as well as investigation of the control effect downstream of the control slot. The results show that the controlled flow energy is initially reduced in the streamwise direction and then increased because transition still occurs. The distribution of the optimal control velocity responds to the flow activity above and upstream of the control slot. The control effect propagates downstream of the slot and the flow energy is reduced up to the exit of the computational domain. The mean drag reduction is and in the control region and downstream of the slot, respectively. The control mechanism is investigated by examining the second-order statistics and the two-point correlations. It is found that in the upstream (left) side of the slot, the controller counteracts the near-wall high-speed streaks and reduces the turbulent shear stress; this is akin to opposition control in channel flow, and because the time-average control velocity is positive, it is more similar to blowing-only opposition control. In the downstream (right) side of the slot, the controller reacts to the impingement of turbulent spots that have been produced upstream and inside the boundary layer (top–bottom mechanism). The control velocity is positive and increases in the streamwise direction, and the flow behaviour is similar to that of uniform blowing.

Alves Portela F, Papadakis G, Vassilicos JC, 2018, Turbulence dissipation and the role of coherent structures in the near wake of a square prism, *Physical Review Fluids*, Vol: 3, ISSN: 2469-990X

Between streamwise distances 4d and at least 10d in the planar turbulent wake of a square prism of side length d, the turbulent fluctuating velocities are highly non-Gaussian, the turbulent energy spectrum has a close to −5/3 power law range, and the turbulence dissipation rate obeys the nonequilibrium dissipation scaling if the energy of the coherent structures is not included in the scaling. In this same range of streamwise distances, the coherent structure dissipation rate decays proportionally to the stochastic turbulence dissipation rate and there is a strong tendency of alignment or antialignment between fluctuating velocities and fluctuating vorticities which appears to coincide with the presence of coherent structures.

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