93 results found
Iyer A, Witherden F, Chernyshenko S, et al., Identifying eigenmodes of averaged small-amplitude perturbations to turbulent channel flow, Journal of Fluid Mechanics, ISSN: 0022-1120
Eigenmodes of averaged small-amplitude perturbations to a turbulent channel flow — which is one of the most fundamental canonical flows — are identified for the first time via an extensive set of high-fidelity GPU-accelerated direct numerical simulations. While the system governing averaged small-amplitude perturbations to turbulent channel flow remains unknown, the fact such eigenmodes can be identified constitutes direct evidence that it is linear. Moreover, while the eigenvalue associated with the slowest-decaying anti-symmetric eigenmode mode is found to be real, the eigenvalue associated with the slowest-decaying symmetric eigenmode mode is found to be complex. This indicates that the unknown linear system governing the evolution of averaged small-amplitude perturbations cannot be self-adjoint, even for the case of a uni-directional flow. In addition to elucidating aspects of the flow physics, the findings provide guidance for development of new unsteady Reynolds-averaged Navier-Stokes turbulence models, and constitute a new and accessible benchmark problem for assessing the performance of existing models,which are used widely throughout industry.
Lakshmi M, Fantuzzi G, Fernández-Caballero J, et al., 2019, Finding extremal periodic orbits with polynomial optimisation, with application to a nine-mode model of shear flow, Publisher: arXiv
Tobasco et al. [Physics Letters A, 382:382-386, 2018] recently suggested thattrajectories of ODE systems which optimise the infinite-time average of acertain observable can be localised using sublevel sets of a function thatarise when bounding such averages using so-called auxiliary functions. In thispaper we demonstrate that this idea is viable and allows for the computation ofextremal unstable periodic orbits (UPOs) for polynomial ODE systems. First, weprove that polynomial optimisation is guaranteed to produce auxiliary functionsthat yield near-sharp bounds on time averages, which is required in order tolocalise the extremal orbit accurately. Second, we show that points inside therelevant sublevel sets can be computed efficiently through direct nonlinearoptimisation. Such points provide good initial conditions for UPO computations.We then combine these methods with a single-shooting Newton-Raphson algorithmto study extremal UPOs for a nine-dimensional model of sinusoidally forcedshear flow. We discover three previously unknown families of UPOs, one of whichsimultaneously minimises the mean energy dissipation rate and maximises themean perturbation energy relative to the laminar state for Reynolds numbersapproximately between 81.24 and 125.
Chernyshenko SI, Zhang C, Butt H, et al., 2019, A large-scale filter for applications of QSQH theory of scale interactions in near-wall turbulence, Fluid Dynamics Research, Vol: 51, ISSN: 0169-5983
An outlook on the recently proposed quasi-steady quasi-homogeneous (QSQH) theory of the effect of large-scale structures on the near-wall turbulence is provided. The paper focuses on the selection of the filter, which defines the large-scale structures. It gives a brief overview of the QSQH theory, discusses the filter needed to distinguish between large and small scales, and the related issues of the accuracy of the QSQH theory, describes the probe needed for using the QSQH theory, and outlines the procedure of extrapolating the characteristics of near-wall turbulence from medium to high Reynolds numbers.
Ghebali S, Chernyshenko SI, Leschziner MA, 2019, Turbulent-drag reduction by oblique wavy wall undulations, ERCOFTAC Series, Pages: 545-551
© Springer Nature Switzerland AG 2019. Reducing the turbulent skin-friction drag over civilian aircraft is a potentially high-reward target, as this drag component accounts for about half of the total drag in cruise conditions. Thus, even modest reductions convert into material savings, resulting in significant cuts in costs. Active-control techniques can be remarkably effective at suppressing turbulence and drag, but pose major engineering challenges in terms of actuation, efficient operation, reliability and maintainability. In contrast, passive techniques based on riblets are easier to implement, but face important durability and maintenance limitations related to the extremely small spacing of the grooves. The alternative passive-control method that is the subject of the present paper was first proposed in Chernyshenko (Drag reduction by a solid wavy wall emulating spanwise oscillations. Part 1. [physics.flu-dyn](arXiv:1304.4638 ), (2013), ). The key characteristic of the method is that it involves wavy surface undulations directed obliquely to the mean flow and having wave lengths two orders of magnitude larger than riblets, and would thus be much more practical to manufacture and maintain.
Ghebali S, Chernyshenko SI, Leschziner M, 2017, Can large-scale oblique undulations on a solid wall reduce the turbulent drag?, Physics of Fluids, Vol: 29, ISSN: 1070-6631
Direct numerical simulations of fully developed turbulent channel flows with wavy walls are undertaken. The wavy walls, skewed with respect to the mean flow direction, are introduced as a means of emulating a Spatial Stokes Layer (SSL) induced by in-plane wall motion. The transverse shear strain above the wavy wall is shown to be similar to that of a SSL, thereby affecting the turbulent flow and leading to a reduction in the turbulent skin-friction drag. However, some important differences with respect to the SSL case are brought to light too. In particular, the phase variations of the turbulent properties are accentuated and, unlike in the SSL case, there is a region of increased turbulence production over a portion of the wall, above the leeward side of the wave, thus giving rise to a local increase in dissipation. The pressure- and friction-drag levels are carefully quantified for various flow configurations, exhibiting a combined maximum overall-drag reduction of about 0.6%. The friction-drag reduction is shown to behave approximately quadratically for small wave slopes and then linearly for higher slopes, whilst the pressure-drag penalty increases quadratically. The transverse shear-strain layer is shown to be approximately Reynolds-number independent when the wave geometry is scaled in wall units.
Ghebali S, Chernyshenko SI, Leschziner MA, Can large-scale oblique undulations on a solid wall reduce the turbulent drag?, Physics of Fluids, Vol: 29, Pages: 105102-105102
Direct numerical simulations of fully-developed turbulent channel flows withwavy walls are undertaken. The wavy walls, skewed with respect to the mean flowdirection, are introduced as a means of emulating a Spatial Stokes Layer (SSL)induced by in-plane wall motion. The transverse shear strain above the wavywall is shown to be similar to that of a SSL, thereby affecting the turbulentflow, and leading to a reduction in the turbulent skin-friction drag. Thepressure- and friction-drag levels are carefully quantified for various flowconfigurations, exhibiting a combined maximum overall-drag reduction of about0.5%. The friction-drag reduction is shown to behave approximatelyquadratically for small wave slopes and then linearly for higher slopes, whilstthe pressure-drag penalty increases quadratically. Unlike in the SSL case,there is a region of increased turbulence production over a portion of thewall, above the leeward side of the wave, thus giving rise to a local increasein dissipation. The transverse shear-strain layer is shown to be approximatelyReynolds-number independent when the wave geometry is scaled in wall units.
Ghebali S, Chernyshenko S, Leschziner M, 2017, Turbulent drag reduction by wavy wall, 10th International Symposium on Turbulence and Shear Flow Phenomena, TSFP 2017
Fully-developed turbulent flow in channels with oblique wavy walls is analysed, from a drag-reduction perspective, by means of Direct Numerical Simulations (DNS). The wavy geometry is chosen to emulate the shear strain produced by a Spatial Stokes Layer (SSL) generated by oscillatory wall motion. As the cost of performing a parametric optimisation is prohibitive, an alternate solution is presented, based on a linear model of a perturbed plane-channel flow, using a turbulent viscosity. Flow properties and levels of drag reduction or increase are reported for various configurations.
Chernyshenko S, 2017, Relationship between the methods of bounding time averages
The problem of finding bounds of time-averaged characteristics of dynamicalsystems, such as for example the bound on the mean energy dissipation rate in aturbulent flow governed by incompressible Navier-Stokes equations, isconsidered. It is shown that both the well-known background flow method ofDoering and Constantin and the direct method proposed by Seis in 2015correspond to the same quadratic storage functional in the framework of theindefinite storage functional method. In particular, a background flow can befound corresponding to the linear functional used in the direct method and viceversa. It is shown that any bound obtained with the background flow method canalso be obtained by the direct method. The reverse is true subject to anadditional constraint. The relative advantages of the three methods arediscussed.
Huang D, Jin B, Lasagna D, et al., 2017, Expensive control of long-time averages using sum of squares and Its application to a laminar wake flow, IEEE Transactions on Control Systems Technology, Vol: 25, Pages: 2073-2086, ISSN: 1558-0865
The paper presents a nonlinear state-feedback con-trol design approach for long-time average cost control, where thecontrol effort is assumed to be expensive. The approach is basedon sum-of-squares and semi-definite programming techniques. Itis applicable to dynamical systems whose right-hand side is apolynomial function in the state variables and the controls. Thekey idea, first described but not implemented in (Chernyshenkoetal.Phil. Trans. R. Soc. A, 372, 2014), is that the difficult problemof optimizing a cost function involving long-time averages isreplaced by an optimization of the upper bound of the sameaverage. As such, controller design requires the simultaneousoptimization of both the control law and a tunable function,similar to a Lyapunov function. The present paper introducesa method resolving the well-known inherent non-convexity ofthis kind of optimization. The method is based on the formalassumption that the control is expensive, from which it followsthat the optimal control is small. The resulting asymptoticoptimization problems are convex. The derivation of all thepolynomial coefficients in the controller is given in terms ofthe solvability conditions of state-dependent linear and bilinearinequalities. The proposed approach is applied to the problemof designing a full-information feedback controller that mitigatesvortex shedding in the wake of a circular cylinder in the laminarregime via rotary oscillations. Control results on a reduced-ordermodel of the actuated wake and in direct numerical simulationare reported.
Chernyshenko SI, Zhang C, Butt H, et al., 2017, Extrapolating statistics of turbulent flows to higher Re using quasi-steady theory of scale interaction in near-wall turbulence, 10th International Symposium on Turbulence and Shear Flow Phenomena, TSFP 2017
A new technique for extrapolating statistical characteristics of near-wall turbulence from medium to higher Re is outlined. Results for extrapolating the velocity two-point correlation from Reτ= 2003 to Reτ= 4179 and for the parameters of an optimized comb probe for detecting the large-scale velocity component required for applying the technique in practice are presented.
Lasagna D, Huang D, Tutty OR, et al., 2016, Sum-of-Squares approach to feedback control of laminar wake flows, Journal of Fluid Mechanics, Vol: 809, Pages: 628-663, ISSN: 1469-7645
A novel nonlinear feedback control design methodology for incompressiblefluid flows aiming at the optimisation of long-time averages of flow quantitiesis presented. It applies to reduced-order finite-dimensional models of fluidflows, expressed as a set of first-order nonlinear ordinary differentialequations with the right-hand side being a polynomial function in the statevariables and in the controls. The key idea, first discussed in Chernyshenko etal. 2014, Philos. T. Roy. Soc. 372(2020), is that the difficulties of treatingand optimising long-time averages of a cost are relaxed by using theupper/lower bounds of such averages as the objective function. In this setting,control design reduces to finding a feedback controller that optimises thebound, subject to a polynomial inequality constraint involving the costfunction, the nonlinear system, the controller itself and a tunable polynomialfunction. A numerically tractable approach to the solution of such optimisationproblems, based on Sum-of-Squares techniques and semidefinite programming, isproposed. To showcase the methodology, the mitigation of the fluctuation kinetic energyin the unsteady wake behind a circular cylinder in the laminar regime atRe=100, via controlled angular motions of the surface, is numericallyinvestigated. A compact reduced-order model that resolves the long-termbehaviour of the fluid flow and the effects of actuation, is derived usingProper Orthogonal Decomposition and Galerkin projection. In a full-informationsetting, feedback controllers are then designed to reduce the long-time averageof the kinetic energy associated with the limit cycle. These controllers arethen implemented in direct numerical simulations of the actuated flow. Controlperformance, energy efficiency, and physical control mechanisms identified areanalysed. Key elements, implications and future work are discussed.
Fantuzzi G, Goluskin D, Huang D, et al., 2016, Bounds for Deterministic and Stochastic Dynamical Systems using Sum-of-Squares Optimization, SIAM Journal on Applied Dynamical Systems, Vol: 15, Pages: 1962-1988, ISSN: 1536-0040
We describe methods for proving upper and lower bounds on infinite-time averages in deterministicdynamical systems and on stationary expectations in stochastic systems. The dynamics and thequantities to be bounded are assumed to be polynomial functions of the state variables. The methodsare computer-assisted, using sum-of-squares polynomials to formulate sufficient conditions that canbe checked by semidefinite programming. In the deterministic case, we seek tight bounds that applyto particular local attractors. An obstacle to proving such bounds is that they do not hold globally;they are generally violated by trajectories starting outside the local basin of attraction. We describetwo closely related ways past this obstacle: one that requires knowing a subset of the basin ofattraction, and another that considers the zero-noise limit of the corresponding stochastic system.The bounding methods are illustrated using the van der Pol oscillator. We bound deterministicaverages on the attracting limit cycle above and below to within 1%, which requires a lower boundthat does not hold for the unstable fixed point at the origin. We obtain similarly tight upper andlower bounds on stochastic expectations for a range of noise amplitudes. Limitations of our methodsfor certain types of deterministic systems are discussed, along with prospects for improvement.
Fantuzzi G, Goluskin D, Huang D, et al., 2016, Bounds for deterministic and stochastic dynamical systems using sum-of-squares optimization, SIAM Journal on Applied Dynamical Systems, Vol: 15, Pages: 1962-1988, ISSN: 1536-0040
We describe methods for proving upper and lower bounds on inﬁnite-time averages in deterministic dynamical systems and on stationary expectations in stochastic systems. The dynamics and the quantities to be bounded are assumed to be polynomial functions of the state variables. The methods are computer-assisted, using sum-of-squares polynomials to formulate suﬃcient conditions that can be checked by semideﬁnite programming. In the deterministic case, we seek tight bounds that apply to particular local attractors. An obstacle to proving such bounds is that they do not hold globally; they are generally violated by trajectories starting outside the local basin of attraction. We describe two closely related ways past this obstacle: one that requires knowing a subset of the basin of attraction, and another that considers the zero-noise limit of the corresponding stochastic system. The bounding methods are illustrated using the van der Pol oscillator. We bound deterministic averages on the attracting limit cycle above and below to within 1%, which requires a lower bound that does not hold for the unstable ﬁxed point at the origin. We obtain similarly tight upper and lower bounds on stochastic expectations for a range of noise amplitudes. Limitations of our methods for certain types of deterministic systems are discussed, along with prospects for improvement.
Zhang C, Chernyshenko SI, 2016, Quasi-steady quasi-homogeneous description of the scale interactions in near-wall turbulence, Physical Review Fluids, Vol: 1, ISSN: 2469-990X
By introducing a notion of an ideal large-scale filter, a formal statement is given of the hypothesis of the quasi-steady quasi-homogeneous nature of the interaction between the large and small scales in the near-wall part of turbulent flows. This made the derivations easier and more rigorous. A method is proposed to find the optimal large-scale filter by multi-objective optimization, with the first objective being a large correlation between large-scale fluctuations near the wall and in the layer at a certain finite distance from the wall, and the second objective being a small correlation between the small scales in the same layers. The filter was demonstrated to give good results. Within the quasi-steady quasi-homogeneous theory expansions for various quantities were found with respect to the amplitude of the large-scale fluctuations. Including the higher-order terms improved the agreement with numerical data. Interestingly, it turns out that the quasi-steady quasi-homogeneous theory implies a dependence of the mean profile log-law constants on the Reynolds number. The main overall result of the present work is the demonstration of the relevance of the quasi-steady quasi-homogeneous theory for near-wall turbulent flows.
Lasagna D, Tutty OR, Chernyshenko S, 2016, Flow regimes in a simplified Taylor-Couette-type flow model, European Journal of Mechanics B - Fluids, Vol: 57, Pages: 176-191, ISSN: 0997-7546
In this paper we introduce a simplified variant of the well-known Taylor–Couette flow. The aim is to develop and investigate a model problem which is as simple as possible while admitting a wide range of behaviour, and which can be used for further study into stability, transition and ultimately control of flow. As opposed to models based on ordinary differential equations, this model is fully specified by a set of partial differential equations that describe the evolution of the three velocity components over two spatial dimensions, in one meridian plane between the two counter-rotating coaxial cylinders. We assume axisymmetric perturbations of the flow in a narrow gap limit of the governing equations and, considering the evolution of the flow in a narrow strip of fluid between the two cylinders, we assume periodic boundary conditions along the radial and axial directions, with special additional symmetry constraints. In the paper, we present linear stability analysis of the first bifurcation, leading to the well known Taylor vortices, and of the secondary bifurcation, which, depending on the type of symmetries imposed on the solution, can lead to wave-like solutions travelling along the axial direction. In addition, we show results of numerical simulations to highlight the wide range of flow structures that emerge, from simple uni-directional flow to chaotic motion, even with the restriction placed on the flow.
Lasagna D, Huang D, Tutty OR, et al., 2016, Controlling Fluid Flows with Positive Polynomials, 35th Chinese Control Conference (CCC), Publisher: IEEE, Pages: 1301-1306, ISSN: 2161-2927
Huang D, Chernyshenko S, 2015, Long-Time Average Cost Control of Polynomial Systems: A Sum-of-Squares-Based Small-Feedback Approach, 8th ASME Annual Dynamic Systems and Control Conference (DSCC 2015), Publisher: AMER SOC MECHANICAL ENGINEERS
This paper provides a proof of concept of the recent novel idea in the area of long-time average cost control. Meanwhile, a new method of overcoming the well-known difficulty of nonconvexity of simultaneous optimization of a control law and an additional tunable function is given. First, a recently-proposed method of obtaining rigorous bounds of long-time average cost is outlined for the uncontrolled system with polynomials of system state on the right-hand side. In this method the polynomial constraints are relaxed to be sum-of-squares and formulated as semi-definite programs. It was proposed to use the upper bound of long-time average cost as the objective function instead of the time-average cost itself in controller design. In the present paper this suggestion is implemented for a particular system and is shown to give good results. Designing the optimal controller by this method requires optimising simultaneously both the control law and a tunable function similar to the Lyapunov function. The new approach proposed and implemented in this paper for overcoming the inherent non-convexity of this optimisation is based on a formal assumption that the amplitude of control is small. By expanding the tunable function and the bound in the small parameter, the long-time average cost is reduced by minimizing the respective bound in each term of the series. The derivation of all the polynomial coefficients in controller is given in terms of the solvability conditions of state-dependent linear and bilinear inequalities. The resultant sum-of-squares problems are solved in sequence, thus avoiding the non-convexity in optimization.
Huang D, Chernyshenko SI, Goulart P, et al., 2015, Sum-of-squares of polynomials approach to nonlinear stability of fluid flows: An example of application, Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences, Vol: 471, ISSN: 0080-4630
With the goal of providing the first example of application of a recently proposed method, thus demonstrating its ability to give results in principle, global stability of a version of the rotating Couette flow is examined. The flow depends on the Reynolds number and a parameter characterising the magnitude of the Coriolis force. By converting the original Navier-Stokes equations to a finite-dimensional uncertain dynamical system using a partial Galerkin expansion, high-degree polynomial Lyapunov functionals were found by sum-of-squares-of-polynomials optimization. It is demonstrated that the proposed method allows obtaining the exact global stability limit for this flow in a range of values of the parameter characterising the Coriolis force. Outside this range a lower bound for the global stability limit was obtained, which is still better than the energy stability limit. In the course of the study several results meaningful in the context of the method used were also obtained. Overall, the results obtained demonstrate the applicability of the recently proposed approach to global stability of the fluid flows. To the best of our knowledge, it is the first case in which global stability of a fluid flow has been proved by a generic method for the value of a Reynolds number greater than that which could be achieved with the energy stability approach.
Huang D, Chernyshenko S, Lasagna D, et al., 2015, Long-time average cost control of polynomial systems: a sum of squares approach, 2015 European Control Conference (ECC), Publisher: IEEE, Pages: 3244-3249
This paper provides a numerically tractable approach for long-time average cost control of nonlinear dynamical systems with polynomials of system state on the right-hand side. First, a recently-proposed method of obtaining rigorous bounds of long-time average cost is outlined for the uncontrolled system, where the polynomial constraints are strengthened to be sum-of-squares and formulated as semi-definite programs. As such, it allows to use any general (polynomial) functions to optimize the bound. Then, a polynomial type state feedback controller design scheme is presented to further suppress the long-time average cost. The derivation of state feedback controller is given in terms of the solvability conditions of state-dependent bilinear matrix inequalities. Finally, the mitigation of oscillatory vortex shedding behind a cylinder is addressed to illustrate the validity of the proposed approach.
Huang D, Chernyshenko S, 2015, Low-order State-feedback Controller Design for Long-time Average Cost Control of Fluid Flow Systems: A Sum-of-squares Approach, 34th Chinese Control Conference (CCC), Publisher: IEEE, Pages: 2479-2484, ISSN: 2161-2927
Huang D, Chernyshenko S, 2015, Long-time Average Cost Control of Stochastic Systems Using Sum of Squares of Polynomials, 34th Chinese Control Conference (CCC), Publisher: IEEE, Pages: 2344-2349, ISSN: 2161-2927
Chernyshenko SI, Goulart P, Huang D, et al., 2014, Polynomial sum of squares in fluid dynamics: a review with a look ahead, PHILOSOPHICAL TRANSACTIONS OF THE ROYAL SOCIETY A-MATHEMATICAL PHYSICAL AND ENGINEERING SCIENCES, Vol: 372, ISSN: 1364-503X
Chernyshenko S, Huang D, Goulart P, et al., 2013, Nonlinear Stability Analysis of Fluid Flow using Sum of Squares of Polynomials, 11th International Conference of Numerical Analysis and Applied Mathematics (ICNAAM), Publisher: AMER INST PHYSICS, Pages: 265-268, ISSN: 0094-243X
Vodop'yanov IS, Nikitin NV, Chernyshenko SI, 2013, Turbulent drag reduction by spanwise oscillations of a ribbed surface, FLUID DYNAMICS, Vol: 48, Pages: 461-470, ISSN: 0015-4628
Blesbois O, Chernyshenko SI, Touber E, et al., 2013, Pattern prediction by linear analysis of turbulent flow with drag reduction by wall oscillation, JOURNAL OF FLUID MECHANICS, Vol: 724, Pages: 607-641, ISSN: 0022-1120
Chernyshenko S, 2013, Drag reduction by a solid wall emulating spanwise oscillations. Part 1
A new idea for turbulent skin-friction reduction is proposed, wherein theshape of the solid wall is designed to create the spanwise pressure gradientacting similarly to the well-known method of drag reduction by in-planespanwise wall motion. Estimates based on the assumption of similarity with dragreduction effect of in-plane wall motion suggest that drag reduction of about2.4% can be achieved in the flow past a wavy wall, with the crests forming anangle of about 38 degrees with the main flow direction, and the wave period inthe main flow direction equal to about 1500 wall units. The required height ofthe wall waves have to be adjusted to achieve the same intensity of thespanwise motion as that created by an in-plane moving wall of the samewavelength and with peak wall velocity equal to 2 wall units. Further researchis being conducted in order to determine this height. Suggestions are made forfurther research on confirming the feasibility of the proposed method and onoptimising the wall shape.
Mathis R, Marusic I, Chernyshenko SI, et al., 2013, Estimating wall-shear-stress fluctuations given an outer region input, Journal of Fluid Mechanics, Vol: 715, Pages: 163-180-163-180
Chernyshenko SI, Marusic I, Mathis R, 2012, Quasi-steady description of modulation effects in wall turbulence
A theoretical description of the phenomenon of modulation of near-wallturbulence by large scale structures is investigated. The description given issimple in that the effect of large-scale structures is limited to aquasi-steady response of the near-wall turbulence to slow large-scalefluctuations of the skin friction. The most natural and compact form ofexpressing this mechanism is given by the usual Reynolds-number-independentrepresentation of the total skin friction and velocity, scaled in wallvariables, where the mean quantities are replaced by large-scalelow-pass-filtered fluctuating components. The theory is rewritten in terms offuctuations via a universal mean velocity and random mean square fluctuationvelocity profiles of the small-scales and then linearised assuming that thelarge-scale fluctuations are small as compared to the mean components. Thisallows us to express the superposition and modulation coefficients of theempirical predictive models of the skin friction and streamwise fluctuatingvelocity given respectively by Marusic et al. (13th Eur. Turb. Conf., 2011) andMathis et al. (J. Fluid Mech. 2011, vol. 681, pp. 537-566). It is found thatthe theoretical quantities agree well with experimentally determinedcoefficients.
Goulart PJ, Chernyshenko S, 2012, Global stability analysis of fluid flows using sum-of-squares, PHYSICA D-NONLINEAR PHENOMENA, Vol: 241, Pages: 692-704, ISSN: 0167-2789
Duque-Daza CA, Baig MF, Lockerby DA, et al., 2012, Modelling turbulent skin-friction control using linearized Navier–Stokes equations, Journal of Fluid Mechanics, Vol: 702, Pages: 403-414-403-414
This data is extracted from the Web of Science and reproduced under a licence from Thomson Reuters. You may not copy or re-distribute this data in whole or in part without the written consent of the Science business of Thomson Reuters.