## Publications

80 results found

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,
et al., 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.

Papadakis G, Vassilicos C, Basbug S,
et al., 2018, Reduced mixing time in stirred vessels by means of irregular impellers, *Physical Review Fluids*, Vol: 3, ISSN: 2469-990X

Previous research has shown that using fractal-like blades instead of regular ones can result in a significant decrease in the power consumption of an unbaffled stirred vessel with a four-bladed radial impeller. In order to fully assess the mixing efficiency of a fractal-like or just irregular impeller with respect to a regular impeller, the mixing time required to homogenize an injected passive scalar was evaluated for both impeller types at Re = 320 and 1600 using direct numerical simulations. It was observed that the irregular impeller can lead to a considerably shorter mixing time. This result was explained by the differences in characteristics of flow and scalar fields generated by the two impellers. We also assess the effect of Re in the transitional regime. Moreover, a simple mathematical model is proposed which can approximate the decay rate of the passive scalar fluctuations integrated over the tank volume.

Papadakis G, Gallis M, Torczynski JR,
et al., 2018, Gas-kinetic simulation of sustained turbulence in minimal Couette flow, *Physical Review Fluids*, Vol: 3, ISSN: 2469-990X

We provide a demonstration that gas-kinetic methods incorporating molecular chaos can simulate the sustained turbulence that occurs in wall-bounded turbulent shear flows. The direct simulation Monte Carlo method, a gas-kinetic molecular method that enforces molecular chaos for gas-molecule collisions, is used to simulate the minimal Couette flow at Re=500. The resulting law of the wall, the average wall shear stress, the average kinetic energy, and the continually regenerating coherent structures all agree closely with corresponding results from direct numerical simulation of the Navier-Stokes equations. These results indicate that molecular chaos for collisions in gas-kinetic methods does not prevent development of molecular-scale long-range correlations required to form hydrodynamic-scale turbulent coherent structures.

Paul I, Papadakis G, Vassilicos JC, 2018, Direct numerical simulation of heat transfer from a cylinder immersed in the production and decay regions of grid-element turbulence, *Journal of Fluid Mechanics*, Vol: 847, Pages: 452-488, ISSN: 0022-1120

The present direct numerical simulation (DNS) study, the first of its kind, explores the effect that the location of a cylinder, immersed in the turbulent wake of a grid-element, has on heat transfer. An insulated single square grid-element is used to generate the turbulent wake upstream of the heated circular cylinder. Due to fine-scale resolution requirements, the simulations are carried out for a low Reynolds number. Three locations downstream of the grid-element, inside the production, peak and decay regions, respectively, are considered. The turbulent flow in the production and peak regions is highly intermittent, non-Gaussian and inhomogeneous, while it is Gaussian, homogeneous and fully turbulent in the decay region. The turbulence intensities at the location of the cylinder in the production and decay regions are almost equal at 11Â %, while the peak location has the highest turbulence intensity of 15Â %. A baseline simulation of heat transfer from the cylinder without oncoming turbulence was also performed. Although the oncoming turbulent intensities are similar, the production region increases the stagnation point heat transfer by 63Â %, while in the decay region it is enhanced by only 28Â %. This difference cannot be explained only by the increased approaching velocity in the production region. The existing correlations for the stagnation point heat transfer coefficient are found invalid for the production and peak locations, while they are satisfied in the decay region. It is established that the flow in the production and peak regions is dominated by shedding events, in which the predominant vorticity component is in the azimuthal direction. This leads to increased heat transfer from the cylinder, even before vorticity is stretched by the accelerating boundary layer. The frequency of oncoming turbulence in production and peak cases also lies close to the range of frequencies that can penetrate the boundary layer developing on the

Thomareis N, Papadakis G, 2018, Resolvent analysis of separated and attached flows around anairfoil at transitional Reynolds number, *Physical Review Fluids*, Vol: 3, ISSN: 2469-990X

We analyze the resolvent operator in three flows around a nominal NACA-0012 airfoil at ReC=50,000and 5∘ angle of attack. In particular, we study two naturally developing flows (around the airfoil with straight and blunt trailing edges) and one tripped flow. The naturally developing flows exhibit laminar separation, transition and turbulent reattachment, while the tripped flow remains attached in the suction side. For all cases, the time-averaged flow fields are computed from separate DNS simulations. The resolvent analysis can identify the areas of maximum receptivity of the flow field as well as the most amplified modes (optimal response). The former was located close to the leading edge, and in the case of naturally developing flows, also in the free-stream. The spatial distribution of optimal forcing was dominated by structures tilted against the mean flow, in agreement with other studies of boundary layer flows. The optimal response of the two naturally developing flows were different. For the NACA-0012 airfoil with straight trailing edge, the response was maximized at the natural frequency of the separating shear layer, and the shape matched closely with the corresponding DMD mode obtained from processing the DNS results. For the blunt airfoil, the receptivity of the separating shear layer was suppressed in the region of natural frequency, while it was maximized at the frequency of vortex shedding from the trailing edge. This is attributed to the lock-in mechanism between the shedding and the separating shear layer observed in the DNS simulations; the lock-in makes the separating shear layer less sensitive to forcing. In the tripped flow, the amplification of the perturbations is significantly diminished, and only by restricting the resolvent analysis to a region close to the suction side can we get a dominant frequency that matches with DNS at that region. The dominant resolvent modes were used to get estimations of the velocity spectra based only on the mean

Paul I, Papadakis G, Vassilicos C, 2018, DNS of heat transfer from a cylinder immersed in the production and decayregions of grid-element turbulence, *Journal of Fluid Mechanics*, ISSN: 0022-1120

The present DNS study, the ﬁrst of its kind, explores the eﬀect that the location of a cylinder, immersed in the turbulent wake of a grid-element, has on heat transfer. An insulated single square grid-element is used to generate the turbulent wake upstream of the heated circular cylinder. Due to ﬁne-scale resolution requirements, the simulations are carried out for a low Reynolds number. Three locations downstream of the gridelement, inside the production, peak and decay regions, respectively are considered. The turbulent ﬂow in the production and peak regions is highly intermittent, non-Gaussian and inhomogeneous, while it is Gaussian, homogeneous, and fully-turbulent in the decay region. The turbulence intensities at the location of the cylinder in the production and decay regions are almost equal at 11%, while the peak location has the highest turbulent intensity of 15%. A baseline simulation of heat transfer from the cylinder without oncoming turbulence was also performed. Although the oncoming turbulent intensities are similar, the production region increases the stagnation point heat transfer by 63%, while in the decay region it is enhanced by only 28%. This diﬀerence cannot be explained only by the increased approaching velocity in the production region. The existing correlations for the stagnation point heat transfer coeﬃcient are found invalid for the production and peak locations, while they are satisﬁed in the decay region. It is established that the ﬂow in the production and peak regions is dominated by shedding events, in which the predominant vorticity component is in the azimuthal direction. This leads to increased heat transfer from the cylinder, even before vorticity is stretched by the accelerating boundary layer. The frequency of oncoming turbulence in production and peak cases also lies close to the range of frequencies that can penetrate the boundary layer developing on the cylinder, and therefore the latter is very responsive to the impinging d

Basbug S, Papadakis G, Vassilicos C, 2018, Reduced power consumption in stirred vessels by means of fractal impellers, *AIChE Journal*, Vol: 64, Pages: 1485-1499, ISSN: 0001-1541

Earlier studies have shown that the power consumption of an unbaffled stirred vessel decreases significantly when the regular blades are replaced by fractal ones. In this paper, the physical explanation for this reduction is investigated using Direct Numerical Simulations at Re = 1600. The gaps around the fractal blade perimeter create jets that penetrate inside the recirculation zone in the wake and break up the trailing vortices into smaller ones. This affects the time‐average recirculation pattern on the suction side. The volume of the separation region is 7% smaller in the wake of the fractal blades. The lower torque of the fractal impeller is equivalent to a decreased transport of angular momentum; this difference stems from the reduced turbulent transport induced by the smaller trailing vortices. The major difference in the turbulent dissipation is seen in the vicinity of trailing vortices, due to fluctuations of velocity gradients at relatively low frequencies.

Papadakis G, Paul I, Vassilicos C, 2018, Evolution of passive scalar statistics in a spatially developingturbulence, *Physical Review Fluids*, Vol: 3, ISSN: 2469-990X

We investigate the evolution of passive scalar statistics in a spatially developing turbulence using Direct Numerical Simulation. Turbulence is generated by a square grid-element, which is heated continuously, and the passive scalar is temperature. The square element is the fundamental building block for both regular and fractal grids. We trace the dominant mechanisms responsible for the dynamical evolution of scalar variance and scalar dissipation along the bar and grid-element centrelines. The scalar-variance is generated predominantly by the action of mean scalar gradient behind the bar and is transported laterally by turbulent fluctuations to the grid-element centreline. The scalar dissipation (proportional to the scalar gradient variance) is produced primarily by the compression of the fluctuating scalar gradient vector by the turbulent strain-rate, while the contribution of mean velocity and scalar fields is negligible. Close to the grid element the scalar spectrum exhibits a well-defined -5/3 power law, even though the basic premises of the Kolmogorov-Obukhov-Corrsin theory are not satisfied (the fluctuating scalar field is highly intermittent, inhomogeneous and anisotropic, and the local Corrsin-microscale-Peclet number is small). At this location, the PDF of scalar gradient production is only slightly skewed towards positive and the fluctuating scalar gradient vector aligns only with the compressive strain-rate eigenvector. The scalar gradient vector is stretched/compressed stronger than the vorticity vector by turbulent strain-rate throughout the grid-element centreline. However, the alignment of the former changes much earlier in space than that of the latter, resulting in scalar dissipation to decay earlier along the grid-element centreline compared to the turbulent kinetic energy dissipation. The universal alignment behaviour of the scalar gradient vector is found far-downstream although the local Reynolds and Peclet numbers (based on the Taylor and Cor

Jones G, Santer M, Debiasi M,
et al., 2017, Control of flow separation around an airfoil at low Reynolds numbers using periodic surface morphing, *Journal of Fluids and Structures*, Vol: 76, Pages: 536-557, ISSN: 0889-9746

The paper investigates experimentally the low Reynolds number flow () around a model that approximates a NACA 4415 airfoil and the control of separation using periodic surface motion. Actuation is implemented by bonding two Macro Fiber Composite patches to the underside of the suction surface. Time-resolved measurements reveal that the peak-to-peak displacement of the surface motion is a function of both the amplitude and frequency of the input voltage signal but the addition of aerodynamic forces does not cause significant changes in the surface behavior. The vibration mode is uniform in the spanwise direction for frequencies below 80 Hz; above this frequency, a secondary vibration mode is observed. The flow around the unactuated airfoil exhibits a large recirculation region as a result of laminar separation without reattachment and consequently produces relatively high drag and low lift forces. Various actuation frequencies were examined. When actuated at , the spectra in the vicinity of the trailing edge and near-wake were found to be dominated by the actuation frequency. Sharp peaks appear in the spectra suggesting the production of Large Coherent Structures at this frequency. The increased momentum entrainment associated with these enabled a significant suppression of the separated region. The result was a simultaneous increase in and decrease in and therefore a large increase in the ratio. In addition, a delay in the onset of stall results in a significant increase in the maximum achievable lift.

Papadakis G, Santer M, Jones G, 2017, Control of low Reynolds number flow around an airfoil using periodic surface morphing: a numerical study, *Journal of Fluids and Structures*, Vol: 76, Pages: 95-115, ISSN: 0889-9746

The principal aim of this paper is to use Direct Numerical Simulations (DNS) to explain the mechanisms that allow periodic surface morphing to improve the aerodynamic performance of an airfoil. The work focuses on a NACA-4415 airfoil at Reynolds number Rec=5×104 and 0° angle of attack. At these flow conditions, the boundary layer separates at x∕c=0.42, remains laminar until x∕c≈0.8, and then transitions to turbulence. Vortices are formed in the separating shear layer at a characteristic Kelvin–Helmholtz frequency of Sts=4.9, which compares well with corresponding experiments. These are then shed into the wake. Turbulent reattachment does not occur because the region of high turbulent kinetic energy (and therefore mixing) is located too far downstream and too far away from the airfoil surface to influence the near-wall flow. The effect of three actuation frequencies is examined by performing the simulations on a computational domain that deforms periodically. It is found that by amplifying the Kelvin–Helmholtz instability mechanism, Large Spanwise Coherent structures are forced to form and retain their coherence for a large part of the actuation cycle. Following their formation, these structures entrain high momentum fluid into the near-wall flow, leading to almost complete elimination of the recirculation zone. The instantaneous and phase averaged characteristics of these structures are analyzed and the vortex coherence is related to the phase of actuation. In order to further clarify the process of reduction in the size of recirculation zone, simulations start from the fully-developed uncontrolled flow and continue for 25 actuation cycles. The results indicate that the modification of airfoil characteristics is a gradual process. As the number of cycles increases and the coherent vortices are repeatedly formed and propagate downstream, they entrain momentum, thereby modifying the near wall region. During this transient period, the separa

Alves Portela, Papadakis G, Vassilicos, 2017, The turbulence cascade in the near wake of a square prism, *Journal of Fluid Mechanics*, Vol: 825, Pages: 315-352, ISSN: 1469-7645

We present a study of the turbulence cascade on the centreline of an inhomogeneous and anisotropic near-field turbulent wake generated by a square prism at a Reynolds number of using the Kármán–Howarth–Monin–Hill equation. This is the fully generalised scale-by-scale energy balance which, unlike the Kármán–Howarth equation, does not require homogeneity or isotropy assumptions. Our data are obtained from a direct numerical simulation and therefore enable us to access all of the processes involved in this energy balance. A significant range of length scales exists where the orientation-averaged nonlinear interscale transfer rate is approximately constant and negative, indicating a forward turbulence cascade on average. This average cascade consists of coexisting forward and inverse cascade behaviours in different scale-space orientations. With increasing distance from the prism but within the near field of the wake, the orientation-averaged nonlinear interscale transfer rate tends to be approximately equal to minus the turbulence dissipation rate even though all of the inhomogeneity-related energy processes in the scale-by-scale energy balance are significant, if not equally important. We also find well-defined near energy spectra in the streamwise direction, in particular at a centreline position where the inverse cascade behaviour occurs for streamwise oriented length scales.

Basbug S, Papadakis G, Vassilicos, 2017, DNS investigation of the dynamical behaviour of trailing vortices in unbaffled stirred vessels at transitional Reynolds numbers, *Physics of Fluids*, Vol: 29, ISSN: 1089-7666

Flow in an unbaffled stirred vessel agitated by a four-bladed radial impeller is investigated by usingdirect numerical simulations atRe= 320 and 1600. We observe fluctuations in the power consumptionwith a peak frequency at ca. three times the impeller rotational speed for both Reynolds numbers. Itis discovered that these fluctuations are associated with a periodic event in the wake of the blades,which involves alternating growth and decay of the upper and lower cores of the trailing vortex pairas well as up-and-down swinging motion of the radial jet. Moreover, the phase relation between thewakes of the different blades is examined in detail. Further studies using fractal-shaped blades showthat the exact blade shape does not have a strong influence on this phenomenon. However, the wakeinteraction between the blades, hence the number of blades, has a direct influence on the unsteadinessof trailing vortices.

Xiao D, Papadakis G, 2017, Nonlinear optimal control of bypass transition in a boundary layer flow, *Physics of Fluids*, Vol: 25, ISSN: 0031-9171

The central aim of the paper is to apply and assess a nonlinear optimal control strategy to suppress bypass transition, due to bimodal interactions [T. A. Zaki and P. A. Durbin, “Mode interaction and the bypass route to transition,” J. Fluid Mech. 531, 85 (2005)] in a zero-pressure-gradient boundary layer. To this end, a Lagrange variational formulation is employed that results in a set of adjoint equations. The optimal wall actuation (blowing and suction from a control slot) is found by solving iteratively the nonlinear Navier-Stokes and the adjoint equations in a forward/backward loop using direct numerical simulation. The optimization is performed in a finite time horizon. Large values of optimization horizon result in the instability of the adjoint equations. The control slot is located exactly in the region of transition. The results show that the control is able to significantly reduce the objective function, which is defined as the spatial and temporal integral of the quadratic deviation from the Blasius profile plus a term that quantifies the control cost. The physical mechanism with which the actuation interacts with the flow field is investigated and analysed in relation to the objective function employed. Examination of the joint probability density function shows that the control velocity is correlated with the streamwise velocity in the near wall region but this correlation is reduced as time elapses. The spanwise averaged velocity is distorted by the control action, resulting in a significant reduction of the skin friction coefficient. Results are presented with and without zero-net mass flow constraint of the actuation velocity. The skin friction coefficient drops below the laminar value if there is no mass constraint; it remains however larger than laminar when this constraint is imposed. Results are also compared with uniform blowing using the same time-average velocity obtained from the nonlinear optimal algorithm.

Paul I, Papadakis G, Vassilicos JC, 2017, Genesis and evolution of velocity gradients in a near-field spatially developing turbulence, *Journal of Fluid Mechanics*, Vol: 815, Pages: 295-332, ISSN: 1469-7645

This paper investigates the dynamics of velocity gradients for a spatially developing flowgenerated by a single square element of a fractal square grid at low inlet Reynolds numberthrough direct numerical simulation. This square grid-element is also the fundamentalblock of a classical grid. The flow along the grid-element centreline is initially irrotationaland becomes turbulent further downstream due to the lateral excursions of vorticalturbulent wakes from the grid-element bars. We study the generation and evolution ofthe symmetric and anti-symmetric parts of the velocity gradient tensor for this spatiallydeveloping flow using the transport equations of mean strain-product and mean enstrophyrespectively. The choice of low inlet Reynolds number allows for fine spatial resolutionand long simulations, both of which are conducive in balancing the budget equations ofthe above quantities. The budget analysis is carried out along the grid-element centrelineand the bar centreline. The former is observed to consist of two subregions: one in theimmediate lee of the grid-element which is dominated by irrotational strain, and onefurther downstream where both strain and vorticity coexist. In the demarcation areabetween these two subregions, where the turbulence is inhomogeneous and developing,the energy spectrum exhibits the best−5/3 power law slope. This is the same locationwhere the experiments at much higher inlet Reynolds number show a well defined−5/3 spectrum over more than a decade of frequencies. Yet, the Q-R diagram remainsundeveloped in the near grid-element region, and both the intermediate and extensivestrain-rate eigenvectors align with the vorticity vector. Along the grid-element centreline,the strain is the first velocity gradient quantity generated by the action of pressureHessian. This strain is then transported downstream by fluctuations and strain self-amplification is activate

Gallis MA, Bitter NP, Koehler TP,
et al., 2017, Molecular-level simulations of turbulence and Its decay, *Physical Review Letters*, Vol: 118, ISSN: 1079-7114

We provide the first demonstration that molecular-level methods based on gas kinetic theory and molecular chaos can simulate turbulence and its decay. The direct simulation Monte Carlo (DSMC) method, a molecular-level technique for simulating gas flows that resolves phenomena from molecular to hydrodynamic (continuum) length scales, is applied to simulate the Taylor-Green vortex flow. The DSMC simulations reproduce the Kolmogorov −5/3 law and agree well with the turbulent kinetic energy and energy dissipation rate obtained from direct numerical simulation of the Navier-Stokes equations using a spectral method. This agreement provides strong evidence that molecular-level methods for gases can be used to investigate turbulent flows quantitatively.

Thomareis N, Papadakis G, 2017, Effect of trailing edge shape on the separated flow characteristics around an airfoil at low Reynolds number: A numerical study, *Physics of Fluids*, Vol: 29, ISSN: 1089-7666

Direct numerical simulations of the flow field around a NACA 0012 airfoil at Reynolds number 50 000 and angle of attack 5° with 3 different trailing edge shapes (straight, blunt, and serrated) have been performed. Both time-averaged flow characteristics and the most dominant flow structures and their frequencies are investigated using the dynamic mode decomposition method. It is shown that for the straight trailing edge airfoil, this method can capture the fundamental as well as the subharmonic of the Kelvin-Helmholtz instability that develops naturally in the separating shear layer. The fundamental frequency matches well with relevant data in the literature. The blunt trailing edge results in periodic vortex shedding, with frequency close to the subharmonic of the natural shear layer frequency. The shedding, resulting from a global instability, has an upstream effect and forces the separating shear layer. Due to forcing, the shear layer frequency locks onto the shedding frequency while the natural frequency (and its subharmonic) is suppressed. The presence of serrations in the trailing edge creates a spanwise pressure gradient, which is responsible for the development of a secondary flow pattern in the spanwise direction. This pattern affects the mean flow in the near wake. It can explain an unexpected observation, namely, that the velocity deficit downstream of a trough is smaller than the deficit after a protrusion. Furthermore, the insertion of serrations attenuates the energy of vortex shedding by de-correlating the spanwise coherence of the vortices. This results in weaker forcing of the separating shear layer, and both the subharmonics of the natural frequency and the shedding frequency appear in the spectra.

Papadakis G, Lu L, Ricco P, 2016, Closed-loop control of boundary layer streaks induced by free-stream turbulence, *Physical Review Fluids*, Vol: 1, ISSN: 2469-990X

The central aim of the paper is to carry out a theoretical and numerical study of active wall transpiration control of streaks generated within an incompressible boundary layer by free-stream turbulence. The disturbance flow model is based on the linearized unsteady boundary-region (LUBR) equations, studied by Leib, Wundrow, and Goldstein [J. Fluid Mech. 380, 169 (1999)], which are the rigorous asymptotic limit of the Navier-Stokes equations for low-frequency and long-streamwise wavelength. The mathematical formulation of the problem directly incorporates the random forcing into the equations in a consistent way. Due to linearity, this forcing is factored out and appears as a multiplicative factor. It is shown that the cost function (integral of kinetic energy in the domain) is properly defined as the expectation of a random quadratic function only after integration in wave number space. This operation naturally introduces the free-stream turbulence spectral tensor into the cost function. The controller gains for each wave number are independent of the spectral tensor and, in that sense, universal. Asymptotic matching of the LUBR equations with the free-stream conditions results in an additional forcing term in the state-space system whose presence necessitates the reformulation of the control problem and the rederivation of its solution. It is proved that the solution can be obtained analytically using an extension of the sweep method used in control theory to obtain the standard Riccati equation. The control signal consists of two components, a feedback part and a feed-forward part (that depends explicitly on the forcing term). Explicit recursive equations that provide these two components are derived. It is shown that the feed-forward part makes a negligible contribution to the control signal. We also derive an explicit expression that a priori (i.e., before solving the control problem) leads to the minimum of the objective cost function (i.e., the fundamental pe

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