Reynolds Decomposition Research Articles

Turbulent bulk viscosity

The Reynolds decomposition is used to derive a new set of mean-flow equations for linear viscous fluids, with low compressibility (fluids in the liquid state). Additional shear and normal stresses are related to the velocity turbulent fluctuations in order to interpret the considerable energy dissipation in processes which are accompanied by rapid changes in liquid density, at least within the frame of the local thermodynamic equilibrium (LTE) approximation. A constitutive equation for the normal turbulent stresses, which involves the turbulent bulk viscosity, is given. A closure equation for the turbulent bulk viscosity is suggested. In the field of hydraulic engineering, the model is applied to investigate the classical problem of 1D finite amplitude pressure waves propagation in liquid-filled pipes (water hammer phenomenon). Available experimental pressure data are used to calibrate the model parameters and to demonstrate the capability of the proposed theoretical model in reproducing the experimental trials. The model results are discussed with attention paid to the role played by the turbulent bulk viscosity on the energy dissipation processes.

Effect of the intermittency dynamics on single and multipoint statistics of turbulent boundary layers

The instantaneous velocity field of turbulent flows is traditionally described in a statistical sense as a sum of an ensemble or temporal mean velocity at a specific location and a fluctuating term. While the Reynolds decomposition is well-established for fully turbulent flows, its applicability for flows that show intermittent behaviour is questionable. As intermittency is an integral part of turbulent boundary layer and turbulent jet and wake flows, an analysis of the effect of the intermittency on the statistical quantities is of fundamental interest. In this work, the effect of intermittency on single and multipoint statistics is studied for an experimentally obtained turbulent boundary layer data set at high Reynolds numbers by means of three different decomposition approaches. The analysis clearly shows that the Reynolds decomposition is only valid for single point statistics in the inner fully turbulent region of wall-bounded flows. In the wake region of a turbulent boundary layer, the Reynolds decomposition overestimates the turbulent kinetic energy due to the intermittency of the turbulent/non-turbulent interface. It is shown herein that these artefacts can be reduced or even completely avoided by applying either the boundary layer height or zonal based decomposition approach. In this study, the limitations of the different methods are analysed in detail and the approaches are evaluated comparatively by investigating statistical quantities such as turbulence intensity, anisotropy of fluctuations and characteristic length scales of coherent motions. The results make it possible to better understand and interpret the characteristics of the turbulent statistics of intermittent flows.

Multi-Δt 3D-PTV based on Reynolds decomposition

A novel approach is investigated to extend the range of measurable velocities by 3D-PTV systems. The method is specifically conceived for robotic volumetric PTV measurements, but it has applications for other similar techniques. The multi-Δt method relies upon combining the information from two or more sets of double-frame images with pulse separation of different time duration. Measurements with a short time separation yield a robust particle velocity field estimation with a higher percentage of valid vectors, yet a low measurement precision. Conversely, measurements with longer time separation potentially offer a higher measurement precision but suffer from an increased probability of spurious particle pairing. Reynolds decomposition is used to combine the two (dual-Δt) sets where a predictor for the mean particle displacement and its statistical dispersion is used to pair particle recordings from a longer time separation. For this reason, this method is aimed at the analysis of turbulent flows where the Reynolds decomposition is meaningful (e.g. turbulent flows with steady/quasi-steady boundary conditions). The extent of the search region is selected dynamically, based on the estimate of the velocity fluctuations from the short time separation evaluation. A more advanced variant of the algorithm contemplates the progressive increase of the pulse separation (multi-Δt) until the expected dispersion of data due to turbulent fluctuations eventually exceeds the distance between neighbouring particles.Flow measurements in the near wake of a truncated cylinder obstacle and of an Ahmed body are carried out to examine the performance of the proposed method. Reference data is taken from time-resolved multi-frame analysis based on the Shake-The-Box (STB) algorithm. The two experiments differ for the measurement principle used: the first one is conducted with a tomographic-like system (large aperture), whereas the latter uses coaxial volumetric velocimetry. The rate of correct pairing as well as the velocity dynamic range dependence upon the choice of the time separation are monitored and discussed. The results compare favourably with the STB analysis, indicating that the measurement of the time-average velocity field can be based on dual-Δt 3D-PTV measurements removing the constraint of time-resolved particle motion recording.

Meso‐scale contribution to air–sea turbulent fluxes at GCM scale

AbstractParametrizations of sea surface turbulent fluxes used in general circulation models (GCMs) assume horizontal homogeneity of atmospheric properties at the grid‐cell scale. The present study assesses the contribution of the meso‐scale (i.e., subgrid) to the grid‐scale surface fluxes, for GCM resolution ranging from 20 to 200 km, and thus quantifies the associated GCM surface flux error. A coarse‐graining method allows for an a priori analysis of the subgrid information. It is based on an atmospheric reference dataset produced by the convection‐permitting operational model AROME. The meso‐scale relative contribution to GCM‐scale fluxes exceeding 10% is shown to have large regional patterns, with large values (up to 90%) and high frequency of occurrence (up to 76% of the time). These meso‐scale motions are not necessarily due to convective activity but also occur frequently under dynamical perturbation conditions. Contributions to surface fluxes, at both the GCM scale and the meso‐scale, are disentangled through a Reynolds decomposition. It is found that temperature and humidity meso‐scale heterogeneities do not contribute much to the GCM‐scale fluxes. The subgrid dynamical processes are the main meso‐scale contribution and consist of two parts. The first one represents the wind magnitude and wind direction heterogeneities and corresponds to the so‐called gustiness approach. It is clarified that the contribution of the gustiness wind in the transfer coefficients cannot be neglected. The second part is the contribution of the wind speed subgrid variance, which contributes up to 10% of the GCM‐scale momentum flux.

Flow around an oscillating grid in water and shear-thinning polymer solution at low Reynolds number

The study of turbulence in complex fluids is of great interest in many environmental and industrial applications, in which the interactions between liquid phase rheology, turbulence, and other phenomena such as mixing or heat and mass transfer have to be understood. Oscillating grid stirred tanks have been used for many purposes in research involving turbulence. However, the mechanisms of turbulence production by the oscillating grid itself have never been studied, and oscillating grid turbulence (OGT) remained undescribed in non-Newtonian, shear-thinning, dilute polymer solutions until recently (Lacassagne et al., in Phys Fluids 31(8):083,102, 2019). The aim of this paper is to study the influence of the shear-thinning property of dilute polymer solutions (DPS), such as xanthan gum (XG), on mean flow, oscillatory flows, and turbulence around an oscillating grid. Liquid phase velocity is measured by particle image velocimetry (PIV) in a vertical plane above the central grid bar. Mean, oscillatory and turbulent components of the velocity fields are deduced by triple Hussain–Reynolds decomposition based on grid phase-resolved measurements. Outside of the grid swept region, the amplitude of oscillatory fluctuations quickly become negligible compared to that of turbulent fluctuations, and the triple and classical Reynolds decomposition become equivalent. Oscillatory jets and wakes behind the grid and their interactions are visualized. Turbulent (Reynolds) and oscillatory stresses are used to evidence a modification of oscillatory flow and turbulence intensity repartition in and around the grid swept region. Energy transfer terms between mean, oscillatory and turbulent flows are estimated and used to describe turbulence production in the grid swept region. Energy is injected by the grid into the oscillatory component. In water, it is transferred to turbulence mostly inside the grid swept region. In DPS, oscillatory flow persists outside of the grid swept zone. Energy is transferred not only to turbulence , in the grid swept region, and far from the tank’s walls, but also to the mean flow, leading to an enhancement of the latter. Mean flow production and enhancement mechanisms are explainable by oscillatory jet variable symmetry and intensity, and by time- and space-variable viscosity. Backward transfer from turbulence to oscillatory flow is also evidenced in DPS. Finally, using phased root mean square (rms) values of turbulent velocity fluctuations, it is shown that in water, the decay of turbulence intensity behind an oscillating grid can be related to the decay of fixed grid turbulence for specific grid positions, a result no longer valid in DPS.Graphic abstract

Wake behind a three-dimensional dry transom stern. Part 2. Analysis and modelling of incompressible highly variable density turbulence

We analyse the turbulence characteristics and consider the closure modelling of the air entraining flow in the wake of three-dimensional, rectangular dry transom sterns obtained using high-resolution implicit large eddy simulations (iLES) (Hendricksonet al.,J. Fluid Mech., vol. 875, 2019, pp. 854–883). Our focus is the incompressible highly variable density turbulence (IHVDT) in the near surface mixed-phase region${\mathcal{R}}$behind the stern. We characterize the turbulence statistics in${\mathcal{R}}$and determine it to be highly anisotropic due to quasi-steady wave breaking. Using unconditioned Reynolds decomposition for our analysis, we show that the turbulent mass flux (TMF) is important in IHVDT for the production of turbulent kinetic energy and is as relevant to the mean momentum equations as the Reynolds stresses. We develop a simple, regional explicit algebraic closure model for the TMF based on a functional relationship between the fluxes and tensor flow quantities.A prioritests of the model show mean density gradients and buoyancy effects are the main driving parameters for predicting the turbulent mass flux and the model is capable of capturing the highly localized nature of the TMF in${\mathcal{R}}$.

A critical assessment of turbulence models for 1D core-collapse supernova simulations

Abstract It has recently been proposed that global or local turbulence models can be used to simulate core-collapse supernova explosions in spherical symmetry (1D) more consistently than with traditional approaches for parametrized 1D models. However, a closer analysis of the proposed schemes reveals important consistency problems. Most notably, they systematically violate energy conservation as they do not balance buoyant energy generation with terms that reduce potential energy, thus failing to account for the physical source of energy that buoyant convection feeds on. We also point out other non-trivial consistency requirements for viable turbulence models. The Kuhfuss model from the 1980s proves more consistent than the newly proposed approaches for supernovae, but still cannot account naturally for all the relevant physics for predicting explosion properties. We perform numerical simulations for a $20 \, \mathrm{M}_\odot$ progenitor to further illustrate problems of 1D turbulence models. If the buoyant driving term is formulated in a conservative manner, the explosion energy of ${\sim }2\times 10^{51}\, \mathrm{erg}$ for the corresponding non-conservative turbulence model is reduced to $\lt 10^{48} \, \mathrm{erg}$ even though the shock expands continuously. This demonstrates that the conservation problem cannot be ignored. Although plausible energies can be reached using an energy-conserving model when turbulent viscosity is included, it is doubtful whether the energy budget of the explosion is regulated by the same mechanism as in multidimensional models. We conclude that 1D turbulence models based on a spherical Reynolds decomposition cannot provide a more consistent approach to supernova explosion and remnant properties than other phenomenological approaches before some fundamental problems are addressed.

Calculation of static pressure field of liquid phase in the active zone of cavitation-jet deaerator

In cavitation-jet deaerators, the process of desorption of dissolved gases proceeds when boiling overheated water is directed in the form of a swirling vortex flow into the rarefaction zone. The removal of gases is carried out from the cavitation pocket formed along the flow axis. Such deaerators are characterized by relatively low efficiency. However, their small dimensions and the ability to work without supplying a heating coolant make them useful in cases where effective deaerators of other types cannot be applied by weight and size characteristics or temperature conditions. The expansion of the scope of practical application of cavitation-jet deaerators is hampered by the lack of appropriate mathematical description that would allow solving the problems of their schematic and constructive improvement. One of the main tasks in this case is to calculate the static pressure field of the liquid phase in the deaerator core. There are no ready-made solutions to this problem in the framework of previous studies. Thus, it is relevant from scientific and practical viewpoint to develop a simulation model of active zone of cavitation-jet deaerator, which allows determining the hydrodynamic characteristics of water flows and gas-vapor mixture when the elements, the mode of water supply and drainage evaporation change. Modeling the hydrodynamic situation in the active zone of cavitation-jet deaerator is based on the numerical solution for the Navier-Stokes equations and the continuity equation. These equations are written in the Reynolds decomposition. In this case, the standard k- turbulence model is used to obtain a preliminary solution, which is then refined during the transition to the SST turbulence model. The model is implemented by means of a software package for calculating fluid flows FlowVision. For the first time, a simulation model of the active zone of cavitation-jet deaerator was developed, which made it possible to determine the hydrodynamic characteristics of water flows and gas mixture when the elements design, the mode of water supply and drainage evaporation change. The field of static pressures of liquid phase in the active zone of cavitation-jet deaerator is determined. The results obtained do not contradict the published data describing the operation of cavitation-jet deaerators. In particular, they confirm an increase in the probability of deaerator overflow when gas exhaust is turned on. The results also prove the presence of a significant dependence of the deaeration efficiency on the hydraulic resistance of the evaporation suction tract. Based on the results obtained, a mathematical model of the deaeration process under the considered conditions can be developed.

Global energy fluxes in turbulent channels with flow control

This paper addresses the integral energy fluxes in natural and controlled turbulent channel flows, where active skin-friction drag reduction techniques allow a more efficient use of the available power. We study whether the increased efficiency shows any general trend in how energy is dissipated by the mean velocity field (mean dissipation) and by the fluctuating velocity field (turbulent dissipation). Direct numerical simulations (DNS) of different control strategies are performed at constant power input (CPI), so that at statistical equilibrium, each flow (either uncontrolled or controlled by different means) has the same power input, hence the same global energy flux and, by definition, the same total energy dissipation rate. The simulations reveal that changes in mean and turbulent energy dissipation rates can be of either sign in a successfully controlled flow. A quantitative description of these changes is made possible by a new decomposition of the total dissipation, stemming from an extended Reynolds decomposition, where the mean velocity is split into a laminar component and a deviation from it. Thanks to the analytical expressions of the laminar quantities, exact relationships are derived that link the achieved flow rate increase and all energy fluxes in the flow system with two wall-normal integrals of the Reynolds shear stress and the Reynolds number. The dependence of the energy fluxes on the Reynolds number is elucidated with a simple model in which the control-dependent changes of the Reynolds shear stress are accounted for via a modification of the mean velocity profile. The physical meaning of the energy fluxes stemming from the new decomposition unveils their inter-relations and connection to flow control, so that a clear target for flow control can be identified.

Derivation of a realistic forcing term to reproduce the turbulent characteristics of round jets on the centerline

Turbulence forcing techniques are often required in the numerical simulation of statistically stationary turbulent flows. However, the existing forcing techniques are not based on physics, but rather arbitrary numerical methods that sustain the turbulent kinetic energy. In this work, a forcing technique is devised to reproduce the centerline turbulent characteristics of round jets in a triply periodic box. It is derived from the Navier-Stokes equations by applying a Reynolds decomposition with the mean velocity of the axisymmetric jet. The result is an anisotropic linear forcing term, which is intended to be used in a three-dimensional box to create turbulence. Four direct numerical simulations with different Re_λ have been performed with these forcing terms. The budget of the terms in the kinetic energy equation is very close to the experimental measurement on the centerline. The anisotropy, kinetic energy k, and dissipation rate ɛ of the simulations are also comparable to experimental values. Finally, the kinetic energy spectrum in the axial direction, ϕ(κ_1), is presented. With appropriate normalizations, the spectrum agrees well with the round jet spectrum on its centerline.

Experimental investigation of airflow above waves in a horizontal pipe

We investigate the effect of waves on the airflow in horizontal two-phase pipe flow. Velocity fields in the gaseous phase were acquired by particle image velocimetry (PIV), while interfacial elevation was measured with conductance wave probes. The velocity fields were sampled on a wave-following coordinate system which allows for a decomposition of the velocity field into a mean, wave-coherent and fluctuating component by means of a three-component Reynolds decomposition. Additionally, coherent vortical structures were identified by the swirling strength criterion, and their distribution along the waves is investigated.Results suggest that the interaction between turbulent airflow and propagating waves in a pipe has a number of features reminiscent of wind-wave interaction in open systems. Above waves generated by sufficiently high gas flow rates, there is a distinct region of sheltered airflow, and a lifting of the critical layer on the leeward side of the crest. Streamlines of the phase-averaged flow field show a cat’s eye structure located close to the crest in this region. Above waves of moderate steepness, we observe a shear layer that remains adjacent to the wave surface. Above steeper waves and higher gas flow rate, this layer detaches from the surface just downstream of the crest. Shear layer separation above waves is traditionally linked to the onset of wave breaking, and it is interesting to note that the case where we observe a separated shear layer in the phase-averaged vorticity field is in a regime of amplitude saturation.The swirling strength criterion reveals that vortical structures are shed from the interface and populate the detached shear layer above the trough. Below the detached shear layer, there is a region populated by counter-rotating vortices. The critical height coincides with the border between these two regions.

Dynamically consistent parameterization of mesoscale eddies. Part III: Deterministic approach

This work continues development of dynamically consistent parameterizations for representing mesoscale eddy effects in non-eddy-resolving and eddy-permitting ocean circulation models and focuses on the classical double-gyre problem, in which the main dynamic eddy effects maintain eastward jet extension of the western boundary currents and its adjacent recirculation zones via eddy backscatter mechanism. Despite its fundamental importance, this mechanism remains poorly understood, and in this paper we, first, study it and, then, propose and test its novel parameterization.We start by decomposing the reference eddy-resolving flow solution into the large-scale and eddy components defined by spatial filtering, rather than by the Reynolds decomposition. Next, we find that the eastward jet and its recirculations are robustly present not only in the large-scale flow itself, but also in the rectified time-mean eddies, and in the transient rectified eddy component, which consists of highly anisotropic ribbons of the opposite-sign potential vorticity anomalies straddling the instantaneous eastward jet core and being responsible for its continuous amplification. The transient rectified component is separated from the flow by a novel remapping method. We hypothesize that the above three components of the eastward jet are ultimately driven by the small-scale transient eddy forcing via the eddy backscatter mechanism, rather than by the mean eddy forcing and large-scale nonlinearities. We verify this hypothesis by progressively turning down the backscatter and observing the induced flow anomalies.The backscatter analysis leads us to formulating the key eddy parameterization hypothesis: in an eddy-permitting model at least partially resolved eddy backscatter can be significantly amplified to improve the flow solution. Such amplification is a simple and novel eddy parameterization framework implemented here in terms of local, deterministic flow roughening controlled by single parameter. We test the parameterization skills in an hierarchy of non-eddy-resolving and eddy-permitting modifications of the original model and demonstrate, that indeed it can be highly efficient for restoring the eastward jet extension and its adjacent recirculation zones.The new deterministic parameterization framework not only combines remarkable simplicity with good performance but also is dynamically transparent, therefore, it provides a powerful alternative to the common eddy diffusion and emerging stochastic parameterizations.

Impacts of Mesoscale Eddies on the Vertical Nitrate Flux in the Gulf Stream Region

AbstractThe Gulf Stream (GS) region has intense mesoscale variability that can affect the supply of nutrients to the euphotic zone (Zeu). In this study, a recently developed high‐resolution coupled physical‐biological model is used to conduct a 25‐year simulation in the Northwest Atlantic. The Reynolds decomposition method is applied to quantify the nitrate budget and shows that the mesoscale variability is important to the vertical nitrate supply over the GS region. The decomposition, however, cannot isolate eddy effects from those arising from other mesoscale phenomena. This limitation is addressed by analyzing a large sample of eddies detected and tracked from the 25‐year simulation. The eddy composite structures indicate that positive nitrate anomalies within Zeu exist in both cyclonic eddies (CEs) and anticyclonic eddies (ACEs) over the GS region, and are even more pronounced in the ACEs. Our analysis further indicates that positive nitrate anomalies mostly originate from enhanced vertical advective flux rather than vertical turbulent diffusion. The eddy‐wind interaction‐induced Ekman pumping is very likely the mechanism driving the enhanced vertical motions and vertical nitrate transport within ACEs. This study suggests that the ACEs in GS region may play an important role in modulating the oceanic biogeochemical properties by fueling local biomass production through the persistent supply of nitrate.

Statistical characteristics of falling-film flows: A synergistic approach at the crossroads of direct numerical simulations and experiments

Based on experimental and numerical data from a wide range of flow conditions, relations linking the film-thickness statistics to the bulk-velocity statistics of wavy falling-film flows are developed from a Reynolds decomposition of the local time-varying film thickness and bulk velocity.

Numerical experiments based on a coupled physical–biochemical ocean model to study the Kuroshio-induced nutrient supply on the shelf-slope region off the southwestern coast of Japan

Abstract We developed a triply nested 1/50° ocean model coupled with a NPZD-type lower trophic level ecosystem model and used it to conduct numerical experiments to identify the major processes that supply nutrients on the shelf-slope region north of the Kuroshio. Tosa Bay, an open-type bay facing the Kuroshio, was selected for our experiment. Comparisons of numerical simulations using different grid sizes revealed that a grid size no larger than 1/50° was essential to reproduce a time-independent density structure related to the Kuroshio jet that uplifted nitrate from subsurface waters into the euphotic zone north of the Kuroshio front. The monthly mean budget of nitrate within the euphotic zone on the shelf showed that primary production was nearly balanced by physical advection and the biochemical supply of nitrate via mechanisms such as remineralization of detritus. Eddy advection of nitrate based on Reynolds decomposition, attributable primarily to submesoscale variations, had both positive and negative values within the bay, the indication being that eddy advection functioned regionally to supply or remove nitrate. Lagrangian particle-tracking experiments were performed to examine the major pathways of the nitrate used for primary production in Tosa Bay during the summer, when subsurface maxima of primary production typically appeared. The experiments revealed that when the Kuroshio took a stable nearshore path, nitrate was frequently uplifted around the Kuroshio front and horizontally transported along the front and into the bay via the counterclockwise circulation within the bay; it was sometimes further uplifted onto the shelf.

Analysis of the pressure fields in a swirling annular jet flow

In this paper, we investigate the flow structures and pressure fields of a free annular swirling jet flow undergoing vortex breakdown. The flow field is analyzed by means of time-resolved tomographic particle image velocimetry measurements, which enable the reconstruction of the three-dimensional time-resolved pressure fields using the governing flow equations. Both time-averaged and instantaneous flow structures are discussed, including a characterization of the first- and second-order statistical moments. A Reynolds decomposition of the flow field shows that the time-averaged flow is axisymmetric with regions of high anisotropic Reynolds stresses. Two recirculation zones exist that are surrounded by regions of very intense mixing. Notwithstanding the axisymmetric nature of the time-averaged flow, a non-axisymmetric structure of the instantaneous flow is revealed, comprising a central vortex core which breaks up into a precessing vortex core. The winding sense of this helical structure is opposite to the swirl direction and it is wrapped around the vortex breakdown bubble. It precesses around the central axis of the flow at a frequency corresponding to a Strouhal number of 0.27. The precessing vortex core is associated with a low-pressure region along the central axis of the jet and the maximum pressure fluctuations occur upstream of the vortex breakdown location, where the azimuthal velocity component also reaches peak values as a result of the inward motion of the fluid and the conservation of angular momentum. The POD analysis of the pressure fields suggests that the precessing helical vortex formation is the dominant coherent structure in the instantaneous flow.

Application of Gauss's theorem to quantify localized surface emissions from airborne measurements of wind and trace gases

Abstract. Airborne estimates of greenhouse gas emissions are becoming more prevalent with the advent of rapid commercial development of trace gas instrumentation featuring increased measurement accuracy, precision, and frequency, and the swelling interest in the verification of current emission inventories. Multiple airborne studies have indicated that emission inventories may underestimate some hydrocarbon emission sources in US oil- and gas-producing basins. Consequently, a proper assessment of the accuracy of these airborne methods is crucial to interpreting the meaning of such discrepancies. We present a new method of sampling surface sources of any trace gas for which fast and precise measurements can be made and apply it to methane, ethane, and carbon dioxide on spatial scales of ∼ 1000 m, where consecutive loops are flown around a targeted source region at multiple altitudes. Using Reynolds decomposition for the scalar concentrations, along with Gauss's theorem, we show that the method accurately accounts for the smaller-scale turbulent dispersion of the local plume, which is often ignored in other average mass balance methods. With the help of large eddy simulations (LES) we further show how the circling radius can be optimized for the micrometeorological conditions encountered during any flight. Furthermore, by sampling controlled releases of methane and ethane on the ground we can ascertain that the accuracy of the method, in appropriate meteorological conditions, is often better than 10 %, with limits of detection below 5 kg h−1 for both methane and ethane. Because of the FAA-mandated minimum flight safe altitude of 150 m, placement of the aircraft is critical to preventing a large portion of the emission plume from flowing underneath the lowest aircraft sampling altitude, which is generally the leading source of uncertainty in these measurements. Finally, we show how the accuracy of the method is strongly dependent on the number of sampling loops and/or time spent sampling the source plume.

Invariants of the velocity-gradient tensor in a spatially developing inhomogeneous turbulent flow

Tomographic particle image velocimetry experiments were performed in the near field of the turbulent flow past a square cylinder. A classical Reynolds decomposition was performed on the resulting velocity fields into a time invariant mean flow and a fluctuating velocity field. This fluctuating velocity field was then further decomposed into coherent and residual/stochastic fluctuations. The statistical distributions of the second and third invariants of the velocity-gradient tensor were then computed at various streamwise locations, along the centreline of the flow and within the shear layers. These invariants were calculated from both the Reynolds-decomposed fluctuating velocity fields and the coherent and stochastic fluctuating velocity fields. The range of spatial locations probed incorporates regions of contrasting flow physics, including a mean recirculation region and separated shear layers, both upstream and downstream of the location of peak turbulence intensity along the centreline. These different flow physics are also reflected in the velocity gradients themselves with different topologies, as characterised by the statistical distributions of the constituent enstrophy and strain-rate invariants, for the three different fluctuating velocity fields. Despite these differing flow physics the ubiquitous self-similar ‘tear drop’-shaped joint probability density function between the second and third invariants of the velocity-gradient tensor is observed along the centreline and shear layer when calculated from both the Reynolds decomposed and the stochastic velocity fluctuations. These ‘tear drop’-shaped joint probability density functions are not, however, observed when calculated from the coherent velocity fluctuations. This ‘tear drop’ shape is classically associated with the statistical distribution of the velocity-gradient tensor invariants in fully developed turbulent flows in which there is no coherent dynamics present, and hence spectral peaks at low wavenumbers. The results presented in this manuscript, however, show that such ‘tear drops’ also exist in spatially developing inhomogeneous turbulent flows. This suggests that the ‘tear drop’ shape may not just be a universal feature of fully developed turbulence but of turbulent flows in general.

Vortex identification from local properties of the vorticity field

A number of systematic procedures for the identification of vortices/coherent structures have been developed as a way to address their possible kinematical and dynamical roles in structural formulations of turbulence. It has been broadly acknowledged, however, that vortex detection algorithms, usually based on linear-algebraic properties of the velocity gradient tensor, can be plagued with severe shortcomings and may become, in practical terms, dependent on the choice of subjective threshold parameters in their implementations. In two-dimensions, a large class of standard vortex identification prescriptions turn out to be equivalent to the “swirling strength criterion” (λci-criterion), which is critically revisited in this work. We classify the instances where the accuracy of the λci-criterion is affected by nonlinear superposition effects and propose an alternative vortex detection scheme based on the local curvature properties of the vorticity graph (x,y,ω)—the “vorticity curvature criterion” (λω-criterion)—which improves over the results obtained with the λci-criterion in controlled Monte Carlo tests. A particularly problematic issue, given its importance in wall-bounded flows, is the eventual inadequacy of the λci-criterion for many-vortex configurations in the presence of strong background shear. We show that the λω-criterion is able to cope with these cases as well, if a subtraction of the mean velocity field background is performed, in the spirit of the Reynolds decomposition procedure. A realistic comparative study for vortex identification is then carried out for a direct numerical simulation of a turbulent channel flow, including a three-dimensional extension of the λω-criterion. In contrast to the λci-criterion, the λω-criterion indicates in a consistent way the existence of small scale isotropic turbulent fluctuations in the logarithmic layer, in consonance with long-standing assumptions commonly taken in turbulent boundary layer phenomenology.

On the quantification and visualization of transient periodic instabilities in pulsatile flows

Turbulent-like flows without cycle-to-cycle variations are more frequently being reported in studies of cardiovascular flows. The associated stimuli might be of mechanobiological relevance, but how to quantify them objectively is not obvious. Classical Reynolds decomposition, where the flow is separated into mean and fluctuating velocity components, is not applicable as the phase-average is zero. We therefore expanded on established techniques and present the idea, analogous to Reynolds decomposition, to decompose a flow with transient instabilities into low- versus high frequency components, respectively, to discriminate flow instabilities from the underlying cardiac pulsatility. Transient wall shear stress and velocity signals derived from computational fluid dynamic simulations were transferred to the frequency domain. A high-pass filter was applied to subtract the 99% most-energy-containing frequencies, which gave a cut-off frequency of 25Hz. We introduce here the spectral power index, and compute the fluctuating kinetic energy, based on the high-pass filtered velocity components, both being frequency-based operators. The efficacy was evaluated in an aneurysm model for multiple flow rates demonstrating transition to turbulent-like flows. The frequency-based operators were found to better correlate with the qualitatively observed flow instabilities compared to conventional descriptors, like time-averaged wall shear stress or oscillatory shear index. We demonstrate how the high frequencies beyond the physiological range could be analyzed and/or transferred back to the time domain for quantification and visualization purposes. We have introduced general frequency-based operators, easily extendable to other cardiovascular territories based on a posteriori heuristic filtering that allows for separation, isolation, and quantification of cycle-invariant turbulent-like flows.