Turbulent Problems Research Articles

Bound State Soliton Gas Dynamics Underlying the Spontaneous Modulational Instability.

We investigate the fundamental phenomenon of the spontaneous, noise-induced modulational instability (MI) of a plane wave. The statistical properties of the noise-induced MI, observed previously in numerical simulations and in experiments, have not been explained theoretically. In this Letter, using the inverse scattering transform (IST) formalism, we propose a theoretical model of the asymptotic stage of the noise-induced MI based on N-soliton solutions of the focusing one-dimensional nonlinear Schrödinger equation. Specifically, we use ensembles of N-soliton bound states having a special semiclassical distribution of the IST eigenvalues, together with random phases for norming constants. To verify our model, we employ a recently developed numerical approach to construct an ensemble of N-soliton solutions with a large number of solitons, N∼100. Our investigation reveals a remarkable agreement between spectral (Fourier) and statistical properties of the long-term evolution of the MI and those of the constructed multisoliton, random-phase bound states. Our results can be generalized to a broad class of strongly nonlinear integrable turbulence problems.

Multifidelity design guided by topology optimization

In this study, we present a framework based on the concept of multifidelity design optimization with the purpose of indirectly solving complex—computationally heavy and/or unstable—topology optimization problems. Our primary idea is to divide an original topology optimization problem into two types of subproblems for low-fidelity optimization and high-fidelity evaluation. To realize this idea, artificial design parameters, which we refer to as seeding parameters, are incorporated into the low-fidelity optimization problem for generating various patterns of topology-optimized candidates. The low-fidelity optimization problem is deliberately formulated as an easily solvable one by decreasing the nonlinearity of the original physical phenomena. Notably, selecting valid seeding parameters in the low-fidelity optimization problem is essential for employing the proposed framework. The aim of high-fidelity evaluation is to obtain a satisfactory solution using a high-fidelity analysis model, which considers the nonlinearity of the original physical phenomena, from the data set of the topology-optimized candidates, via the design of experiments. We apply the proposed framework to two case studies of isothermal and thermal turbulent flow problems, and discuss its efficacy as an alternative strategy for solving complex topology optimization problems.

A self-similarity principle for the computation of rare event probability

The probability of rare and extreme events is an important quantity for design purposes. However, computing the probability of rare events can be expensive because only a few events, if any, can be observed. To this end, it is necessary to accelerate the observation of rare events using methods such as the importance splitting technique, which is the main focus here. In this work, it is shown how a genealogical importance splitting technique can be made more efficient if one knows how the rare event occurs in terms of the mean path followed by the observables. Using Monte Carlo simulations, it is shown that one can estimate this path using less rare paths. A self-similarity model is formulated and tested using an a priori and a posteriori analysis. The self-similarity principle is also tested on more complex systems including a turbulent combustion problem with 107 degrees of freedom. While the self-similarity model is shown to not be strictly valid in general, it can still provide a good approximation of the rare mean paths and is a promising route for obtaining the statistics of rare events in chaotic high-dimensional systems.

Viscous dissipation in DG methods for turbulent incompressible flows

AbstractNowadays, (high‐order) DG methods, or hybridised variants thereof, are widely used in the simulation of turbulent incompressible flow problems. For turbulence simulations, and especially in the practically relevant situation of strong under‐resolution, it is important to distinguish between the resolved physical dissipation rate and the contribution of numerical dissipation originating from the underlying method. In this note, a certain ambiguity related to such a decomposition for the viscous effects in a DG‐discretised fluid flow problem, which is due to the discontinuity of the approximate solution, is addressed. A novel but rather natural decomposition into ‘physical’ and ‘numerical’ viscous dissipation is proposed for a class of DG methods. Based on a typical 3D benchmark problem for decaying turbulence, its meaningfulness is confirmed numerically. In order to justify the term ‘dissipation’, both the physical and the numerical contributions for the proposed additive decomposition are provably non‐negative (possibly zero).

IMEX‐ODTLES: A multi‐scale and stochastic approach for highly turbulent flows

AbstractThe stochastic One‐Dimensional Turbulence (ODT) model is used in combination with a Large Eddy Simulation (LES) approach in order to illustrate the potential of the fully coupled model (ODTLES) for highly turbulent flows. In this work, we use a new C++ implementation of the ODTLES code in order to analyze the computational performance in a classical incompressible turbulent channel flow problem. The parallelization potential of the model, as well as its physical and numerical consistency are evaluated and compared to Direct Numerical Simulations (DNSs). The numerical results show that the model is capable of reproducing a representative part of the DNS data at a cheaper computational cost. This advantage can be enhanced in the future by the implementation of a straightforward parallelization approach.

Circulation in High Reynolds Number Isotropic Turbulence is a Bifractal

The turbulence problem at the level of scaling exponents is hard in part because of the multifractal scaling of small scales, which demands that each moment order be treated and understood independently. This conclusion derives from studies of velocity structure functions, energy dissipation, enstrophy density (that is, square of vorticity), etc. However, it is likely that there exist other physically pertinent quantities with uncomplicated structure in the inertial range, potentially resulting in huge simplifications in the turbulence theory. We show that velocity circulation around closed loops is such a quantity. By using a large databases of isotropic turbulence, generated from numerical simulations of the Navier-Stokes equations over a wide range of Reynolds numbers, we show that circulation exhibits a bifractal behavior at the highest Reynolds number considered: space filling for moments up to order $3$ and a mono-fractal with an unchanging dimension of about $2.5$ for higher orders; this change in character roughly at the third-order moment is reminiscent of a "phase transition". We explore the possibility that circulation becomes effectively space filling at much higher Reynolds numbers even though it may technically be regarded as a bifractal. We confirm that the circulation properties depend on only the area of the loop, not its shape; and, for a figure-$8$ loop, the relevant area is the scalar sum of the two segments of the loop.

On the kinematics of scalar iso-surfaces in decaying homogeneous, isotropic turbulence

ABSTRACTScalar iso-surfaces have been used to describe many interface problems in turbulence, such as flame surfaces in turbulent reacting flows and turbulent/non-turbulent interfaces. In this paper we report on direct numerical simulations of the behaviour of iso-surfaces of two differently initialised conserved scalars evolving in isotropic turbulence. The terms in the equation for the iso-surface area density, along with the area density itself, are evaluated for a wide range of iso-surface values using a novel numerical method for evaluating surface integrals. Specifically, we quantify the behaviour of the iso-surface area density production term, related to the strain-rate, and the destruction term, related to iso-surface curvature and molecular diffusion. We find that production of iso-surface area density is not affected by the initial conditions, but it does have a small dependence on the iso-surface value. The destruction term depends not only on the iso-surface value, but also on the initial conditions.

Pair dispersion in inhomogeneous turbulent thermal convection

Due to large scale flow inhomogeneities and the effects of temperature, turbulence small-scale structure in thermal convection is still an active field of investigation, especially considering sophis- ticated Lagrangian statistics. Here we experimentally study Lagrangian pair dispersion (one of the canonical problems of Lagrangian turbulence) in a Rayleigh-B enard convection cell. A sufficiently high temperature difference is imposed on a horizontal layer of fluid to observe a turbulent flow. We perform Lagrangian tracking of sub-millimetric particles on a large measurement volume in- cluding part of the Large Scale Circulation (LSC) revealing some large inhomogeneities. Our study brings to light several new insights regarding our understanding of turbulent thermal convection: (i) by decomposing particle Lagrangian dynamics into the LSC contribution and the turbulent fluc- tuations, we highlight the relative impact of both contributions on pair dispersion; (ii) using the same decomposition, we estimate the Eulerian second-order velocity structure functions from pair statistics and show that after removing the LSC contribution, the remaining statistics recover usual homogeneous and isotropic behaviours which are governed by a local energy dissipation rate to be distinguished from the global dissipation rate classically used to characterise turbulence in thermal convection; and (iii) we revisit the super-diffusive Richardson-Obukhov regime of particle dispersion and propose a refined estimate of the Richardson constant.

Stochastic and deterministic kinetic energy backscatter parameterizations for simulation of the two-dimensional turbulence

AbstractThe problem of modelling 2D isotropic turbulence in a periodic rectangular domain excited by the forcing pattern of prescribed spatial scale is considered. This setting could be viewed as the simplest analogue of the large scale quasi-2D circulation of the ocean and the atmosphere. Since the direct numerical simulation (DNS) of this problem is usually not possible due to the high computational costs we explore several possibilities to construct coarse approximation models and corresponding subgrid closures (deterministic or stochastic). The necessity of subgrid closures is especially important when the forcing scale is close to the cutoff scale of the coarse model that leads to the significant weakening of the inverse energy cascade and large scale component of the system dynamics.The construction of closures is based on thea priorianalysis of the DNS solution and takes into account the form of a spatial approximation scheme used in a particular coarse model. We show that the statistics of a coarse model could be significantly improved provided a proper combination of deterministic and stochastic closures is used. As a result we are able to restore the shape of the energy spectra of the model. In addition the lagged auto correlations of the model solution as well as its sensitivity to external perturbations fit the characteristics of the DNS model much better.

A comparative study of immersed boundary method and interpolated bounce-back scheme for no-slip boundary treatment in the lattice Boltzmann method: Part II, turbulent flows

In the first part of this study [1], we compared the performances of two categories of no-slip boundary treatments, i.e., the interpolated bounce-back schemes and the immersed boundary methods in a series of laminar flow simulations within the lattice Boltzmann method. In this second part, these boundary treatments are further compared in the simulations of turbulent flows with complex geometry to provide a next-level assessment of these schemes. Two non-trivial turbulent flow problems, a fully developed turbulent pipe flow at a low Reynolds number, and a decaying homogeneous isotropic turbulent flow laden with a large number of resolved spherical particles are considered. The major problem of the immersed boundary method revealed by the present study is its incapability in computing the local velocity gradients inside the diffused interface, which can result in significantly underestimated dissipation rate and viscous diffusion locally near the particle surfaces. Otherwise, both categories of the no-slip boundary treatments are able to provide accurate results for most of turbulent statistics in both the carrier and dispersed phases, provided that sufficient grid resolutions are used. The criteria of sufficient grid resolutions for each examined flow are also addressed in this document.

Maximum Entropy Method for Solving the Turbulent Channel Flow Problem.

There are two components in this work that allow for solutions of the turbulent channel flow problem: One is the Galilean-transformed Navier-Stokes equation which gives a theoretical expression for the Reynolds stress (u′v′); and the second the maximum entropy principle which provides the spatial distribution of turbulent kinetic energy. The first concept transforms the momentum balance for a control volume moving at the local mean velocity, breaking the momentum exchange down to its basic components, u′v′, u′2, pressure and viscous forces. The Reynolds stress gradient budget confirms this alternative interpretation of the turbulence momentum balance, as validated with DNS data. The second concept of maximum entropy principle states that turbulent kinetic energy in fully-developed flows will distribute itself until the maximum entropy is attained while conforming to the physical constraints. By equating the maximum entropy state with maximum allowable (viscous) dissipation at a given Reynolds number, along with other constraints, we arrive at function forms (inner and outer) for the turbulent kinetic energy. This allows us to compute the Reynolds stress, then integrate it to obtain the velocity profiles in channel flows. The results agree well with direct numerical simulation (DNS) data at Reτ = 400 and 1000.

Partial regularity of the solutions to a turbulent problem in porous media

A one-equation turbulent model that is being used with success in the applications to model turbulent flows through porous media is studied in this work. We consider the classical Navier–Stokes equations, with feedback forces fields, coupled with the equation for the turbulent kinetic energy (TKE) through the turbulence production term and through the turbulent and the diffusion viscosities. Under suitable growth conditions on the feedback functions involved in the model, we prove the local higher integrability of the gradient solutions to the steady version of this problem.

Direct interaction approximation for non-Markovianized stochastic models in the turbulence problem.

The purpose of this paper is to consider the application of the direct interaction approximation (DIA) [R. H. Kraichnan, J. Math. Phys. 2, 124 (1961); Phys. Fluids 8, 575 (1965)] to the non-Markovianized stochastic models in the turbulence problem. For illustration, a linear damped stochastic oscillator described by a stochastic equation is considered. This process is shown to lead to a functional equation, and construction of solutions of this equation is addressed within the framework of a continued fraction representation. The relation of the DIA solution to the perturbative solution is discussed in the Laplace transform framework, and qualitative differences between the time evolutions of the two solutions are pointed out. The perturbative solution is shown to exhibit an oscillatory behavior indefinitely, in contradiction to the fact that the linear stochastic oscillator in question is damped. The DIA solution, on the contrary, is shown to exhibit a decaying behavior in accord with the latter property. The DIA procedure is applied to the problem of wave propagation in a random medium, which is described by a stochastic differential equation, with the characteristics of the medium represented by stochastic coefficients. The results are compared with those given by the perturbative procedure.

Super-resolution reconstruction of turbulent flows with machine learning

We use machine learning to perform super-resolution analysis of grossly under-resolved turbulent flow field data to reconstruct the high-resolution flow field. Two machine learning models are developed, namely, the convolutional neural network (CNN) and the hybrid downsampled skip-connection/multi-scale (DSC/MS) models. These machine learning models are applied to a two-dimensional cylinder wake as a preliminary test and show remarkable ability to reconstruct laminar flow from low-resolution flow field data. We further assess the performance of these models for two-dimensional homogeneous turbulence. The CNN and DSC/MS models are found to reconstruct turbulent flows from extremely coarse flow field images with remarkable accuracy. For the turbulent flow problem, the machine-leaning-based super-resolution analysis can greatly enhance the spatial resolution with as little as 50 training snapshot data, holding great potential to reveal subgrid-scale physics of complex turbulent flows. With the growing availability of flow field data from high-fidelity simulations and experiments, the present approach motivates the development of effective super-resolution models for a variety of fluid flows.

Breather Turbulence: Exact Spectral and Stochastic Solutions of the Nonlinear Schrödinger Equation

I address the problem of breather turbulence in ocean waves from the point of view of the exact spectral solutions of the nonlinear Schrödinger (NLS) equation using two tools of mathematical physics: (1) the inverse scattering transform (IST) for periodic/quasiperiodic boundary conditions (also referred to as finite gap theory (FGT) in the Russian literature) and (2) quasiperiodic Fourier series, both of which enhance the physical and mathematical understanding of complicated nonlinear phenomena in water waves. The basic approach I refer to is nonlinear Fourier analysis (NLFA). The formulation describes wave motion with spectral components consisting of sine waves, Stokes waves and breather packets that nonlinearly interact pair-wise with one another. This contrasts to the simpler picture of standard Fourier analysis in which one linearly superposes sine waves. Breather trains are coherent wave packets that “breath” up and down during their lifetime “cycle” as they propagate, a phenomenon related to Fermi-Pasta-Ulam (FPU) recurrence. The central wave of a breather, when the packet is at its maximum height of the FPU cycle, is often treated as a kind of rogue wave. Breather turbulence occurs when the number of breathers in a measured time series is large, typically several hundred per hour. Because of the prevalence of rogue waves in breather turbulence, I call this exceptional type of sea state a breather sea or rogue sea. Here I provide theoretical tools for a physical and dynamical understanding of the recent results of Osborne et al. (Ocean Dynamics, 2019, 69, pp. 187–219) in which dense breather turbulence was found in experimental surface wave data in Currituck Sound, North Carolina. Quasiperiodic Fourier series are important in the study of ocean waves because they provide a simpler theoretical interpretation and faster numerical implementation of the NLFA, with respect to the IST, particularly with regard to determination of the breather spectrum and their associated phases that are here treated in the so-called nonlinear random phase approximation. The actual material developed here focuses on results necessary for the analysis and interpretation of shipboard/offshore platform radar scans and for airborne lidar and synthetic aperture radar (SAR) measurements.

Experimental data-based reduced-order model for analysis and prediction of flame transition in gas turbine combustors

In lean premixed combustors, flame stabilisation is an important operational concern that can affect efficiency, robustness and pollutant formation. The focus of this paper is on flame lift-off and re-attachment to the nozzle of a swirl combustor. Using time-resolved experimental measurements, a data-driven approach known as cluster-based reduced-order modelling (CROM) is employed to (1) isolate key flow patterns and their sequence during the flame transitions, and (2) formulate a forecasting model to predict the flame instability. The flow patterns isolated by the CROM methodology confirm some of the experimental conclusions about the flame transition mechanism. In particular, CROM highlights the key role of the precessing vortex core (PVC) in the flame detachment process in an unsupervised manner. For the attachment process, strong flow recirculation far from the nozzle appears to drive the flame upstream, thus initiating re-attachment. Different data-types (velocity field, OH concentration) were processed by the modelling tool, and the predictive capabilities of these different models are also compared. It was found that the swirling velocity possesses the best predictive properties, which gives a supplemental argument for the role of the PVC in causing the flame transition. The model is tested against unseen data and successfully predicts the probability of flame transition (both detachment and attachment) when trained with swirling velocity with minimal user input. The model trained with OH-PLIF data was only successful at predicting the flame attachment, which implies that different physical mechanisms are present for different types of flame transition. Overall, these aspects show the great potential of data-driven methods, particularly probabilistic forecasting techniques, in analysing and predicting large-scale features in complex turbulent combustion problems.

Exact solution of the compressible Euler–Helmholtz equation and the millennium prize problem generalization

SummaryFor the first time the exact strong solution of the three-dimensional (3D) compressible Euler–Helmholtz (EH) vortex equation in unbounded space is obtained. It corresponds to the general solution obtained in Euler variables for the 3D Riemann–Hopf (RH) equation and has a singularity in finite time. The exact closed description for the evolution of all one and multi point moments of the hydrodynamic fields and their spectra is now possible and thus, the mysterious turbulence problem has the exact solution for this case. As an example, for the first time the two-dimensional (2D) energy spectrum in the inertial scale interval is obtained directly from the exact solution of compressible 2D Navier–Stokes (NS) equation. For this purpose, a smooth continuation at all times for these exact solutions to the EH and RH equations is obtained by introducing a super-threshold homogeneous friction or any even extremely low effective viscosity. A new, smooth for all time, analytic solutions to the 1D, 2D and 3D NS equations is obtained. This solution coincides with the above-mentioned modified by viscosity solutions to the EH and RH equations, when also taking into account a linear relation between the pressure and the velocity field divergence, which is well known for the systems far from the equilibrium with relatively large second viscosity. This gives the positive answer to the generalization of the millennium prize problem (www.claymath.org) for the case of the compressible NS equation. Indeed, earlier only a negative answer to the problem of the existence of smooth solutions for any finite time has been a priori considered for the compressible case with the finite initial energy of the smooth velocity field in the unbounded space.

Scale invariance in finite Reynolds number homogeneous isotropic turbulence

The problem of homogeneous isotropic turbulence (HIT) is revisited within the analytical framework of the Navier–Stokes equations, with a view to assessing rigorously the consequences of the scale invariance (an exact property of the Navier–Stokes equations) for any Reynolds number. The analytical development, which is independent of the 1941 (K41) and 1962 (K62) theories of Kolmogorov for HIT for infinitely large Reynolds number, is applied to the transport equations for the second- and third-order moments of the longitudinal velocity increment, $(\unicode[STIX]{x1D6FF}u)$. Once the normalised equations and the constraints required for complying with the scale-invariance property of the equations are presented, results derived from these equations and constraints are discussed and compared with measurements. It is found that the fluid viscosity, $\unicode[STIX]{x1D708}$, and the mean kinetic energy dissipation rate, $\overline{\unicode[STIX]{x1D716}}$ (the overbar denotes spatial and/or temporal averaging), are the only scaling parameters that make the equations scale-invariant. The analysis further leads to expressions for the distributions of the skewness and the flatness factor of $(\unicode[STIX]{x1D6FF}u)$ and shows that these distributions must exhibit plateaus (of different magnitudes) in the dissipative and inertial ranges, as the Taylor microscale Reynolds number $Re_{\unicode[STIX]{x1D706}}$ increases indefinitely. Also, the skewness and flatness factor of the longitudinal velocity derivative become constant as $Re_{\unicode[STIX]{x1D706}}$ increases; this is supported by experimental data. Further, the analysis, backed up by experimental evidence, shows that, beyond the dissipative range, the behaviour of $\overline{(\unicode[STIX]{x1D6FF}u)^{n}}$ with $n=2$, 3 and 4 cannot be represented by a power law of the form $r^{\unicode[STIX]{x1D701}_{n}}$ when the Reynolds number is finite. It is shown that only when $Re_{\unicode[STIX]{x1D706}}\rightarrow \infty$ can an $n$-thirds law (i.e. $\overline{(\unicode[STIX]{x1D6FF}u)^{n}}\sim r^{\unicode[STIX]{x1D701}_{n}}$, with $\unicode[STIX]{x1D701}_{n}=n/3$) emerge, which is consistent with the onset of a scaling range.

Supersonic combustion of hydrogen using an improved strut injection scheme

Numerical investigation of mixing is performed at Mach 2.0 model Scramjet combustor employing parallel strut injection schemes for fuel. In the present investigation, basic strut injector is modified in such a way to produce additional vortices in streamwise direction and improve fuel-air mixing. Air is injected at Mach 2.0 at the combustor inlet and fuel is injected at sonic speed from the blunt end of the strut. The flow field involving high-speed turbulent mixing and heat addition was modeled by three-dimensional Reynolds averaged Navier-Stokes equations. A realizable k-ε model was chosen to close the turbulence problem with the default model constants. Non-premixed combustion of hydrogen and air is modeled using the mixture fraction β-pdf framework. Turbulence-chemistry interactions are handled by a strained flamelet model. Comparisons of numerical results with experimental results have demonstrated the accuracy and applicability of computational grid and a numerical scheme for hot and cold flow solutions. The shock-shear layer interaction present within the combustor increases the local turbulent intensity and has a positive effect on mixing. The mixing efficiency obtained with improved strut injector is compared with the basic strut. Improved strut injection scheme showed a mixing efficiency of >95% with a 45% reduction in length. Further combustion efficiency is calculated in the streamwise direction and plot follows the similar trend as the mixing efficiency. The proposed modification of strut geometry showed improved mixing and combustion performance.

Invited commentary

In laminar flow situations, shear occurs in the zone between the blood vessel wall (velocity = zero) and the central column of flow. In vitro measurement of shear, sometimes difficult, has been achieved with laser Doppler sensors and with magnetic resonance techniques.1Yap C.H. Saikrishnan N. Tamilselvan G. Yoganathan A.P. Experimental technique of measuring dynamic fluid shear stress on the aortic surface of the aortic valve leaflet.J Biomech Eng. 2011; 133: 061007Crossref PubMed Scopus (27) Google Scholar, 2Campagnolo L. Nikolić M. Perchoux J. Lim Y.L. Bertling K. Loubiere K. et al.Flow profile measurement in microchannel using the optical feedback interferometry sensing technique.Microfluidics Nanofluidics. 2013; 14: 113-119Crossref Scopus (54) Google Scholar Casa et al diagrammed the cross section of flow velocity as parabolic for laminar flow and showed how platelets and blood clotting factors can be activated in the shear zone, the region where the velocity curve becomes asymptotic with the wall (Casa Fig 1). The thickness of this zone is related to many parameters, including viscosity. Shear forces are not uniform in the proximal internal carotid artery: under physiologic conditions, the zone of abnormal shear can vary longitudinally as well as along the circumference, as measurements with magnetic resonance techniques have demonstrated.3Markl M. Wegent F. Zech T. Bauer S. Strecker C. Schumacher M. et al.In vivo wall shear stress distribution in the carotid artery: effect of bifurcation geometry, internal carotid artery stenosis, and recanalization therapy.Circ Cardiovasc Imaging. 2010; 3: 647-655Crossref PubMed Scopus (160) Google Scholar The biologic behavior of artery walls in zones of severe stenosis is more difficult to model. Not only is there the problem of turbulence, but measurement accuracy is difficult because of physical limitations of instrumentation. A normal internal carotid artery diameter ranges from 6 to 8 mm, and a significantly stenotic segment is at or less than 1 mm. The axial gate size used to measure carotid flow velocity by duplex ultrasound is typically 1 to 3 mm. The axial resolution for this measurement is on the order of magnitude 0.1 to 0.2 mm, depending on the frequency selected, typically 9 to 12 MHz. Axial resolution is inversely proportional to frequency.4Kremkau F.W. Sonography principles and instruments.9th edition. Elsevier, St. Louis, MO2016: 66Google Scholar Because the formula for shear stress includes 1/(D3), there is potential for any value to have an inherent error between 30% and 70% based on the physics of ultrasound probes, irrespective of operator-related variability. Velocity measurements in a duplex ultrasound scan are sensitive to the beam angle, which is also operator dependent. If shear is to be used as a parameter for risk, it is therefore necessary that the measurement is conducted at the highest probe frequency and is not in an area of severely eccentric plaque, which may distort the measurements. For any vascular laboratory, the overall reproducibility of velocity measurements is not generally ascertained. Understanding that this is the case, reporting standards therefore mandate ranges, not use of precise velocities. Shear stress criteria may need to be formulated in a similar manner. Shear stress is proportional to blood viscosity. For the purposes of clarity, Casa et al did not factor variability of viscosity in their analysis, but it is well established that blood viscosity can vary significantly in any individual over time.5Wells Jr., R.E. Merrill E.W. The variability of blood viscosity.Am J Med. 1961; 31: 505-509Abstract Full Text PDF PubMed Scopus (26) Google Scholar Some literature indicates that viscosity is not really important in terms of carotid-related stroke risk,6Parkhurst K.L. Lin H.F. Devan A.E. Barnes J.N. Tarumi T. Tanaka H. Contribution of blood viscosity in the assessment of flow-mediated dilation and arterial stiffness.Vasc Med. 2012; 17: 231-234Crossref PubMed Scopus (21) Google Scholar but this should be validated in the context of the present study. Therefore, before translating from Casa to clinical utility, as proposed in this paper, it will be necessary for some significant research to occur. In particular, standards for precise vascular laboratory measurement will need to be redefined. The opinions or views expressed in this commentary are those of the author and do not necessarily reflect the opinions or recommendations of the Journal of Vascular Surgery or the Society for Vascular Surgery. Shear rate is a better marker of symptomatic ischemic cerebrovascular events than velocity or diameter in severe carotid artery stenosisJournal of Vascular SurgeryVol. 69Issue 2PreviewThis study was designed to test the hypothesis that the high shear rate of flow in the area of carotid stenosis is associated with the incidence of ischemic symptoms in patients with a high degree of carotid stenosis. Full-Text PDF Open Archive