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1 Portrait of George Batchelor by Rupert Shephard 1984; this portrait hangs in DAMTP, Cambridge, the Department founded under Batchelor's leadership in 1959.
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Turbulence is widely recognized as one of the outstanding problems of the physical sciences, but it still remains only partially understood despite having attracted the sustained efforts of many leading scientists for well over a century. In A Voyage Through Turbulence we are transported through a crucial period of the history of the subject via bi...
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Context 1
... Batchelor , whose portrait (1984) by the artist Rupert Shephard is shown in Figure 8.1, was undoubtedly one of the great figures of fluid dynamics of the twentieth century. His contributions to two major ar- eas of the subject, turbulence and low-Reynolds-number microhydrodynamics, were of seminal quality and have had a lasting impact. ...
Context 2
... on the basis of his work on the elucidation of Kolmogorov's theory, Batchelor was elected in 1947 to a Fellowship at Trinity College, Cambridge, a position that enabled him to devote himself entirely to research over the next four years, during which he published some 15 papers (some with Townsend) on all aspects of turbulent flow. He took his PhD degree in 1948 (see Figure 8.2), and was by October 1949 installed as a lecturer in the Faculty of Mathe- matics, in succession to Leslie Howarth (of von Kármán-Howarth fame) who had left Cambridge to take the Chair of Applied Mathematics at the University of Bristol. ...
Similar publications
Turbulence is widely recognized as one of the outstanding problems of the physical sciences, but it still remains only partially understood despite having attracted the sustained efforts of many leading scientists for well over a century. In A Voyage Through Turbulence we are transported through a crucial period of the history of the subject via bi...
Citations
... The second direction concerns the exploration of small-scale turbulence and its statistical description, assuming the turbulent flow's homogeneity and isotropy. The secondary literature has often emphasized the importance of contributions to the 'semi-empirical' theories of turbulence by Ludwig Prandtl (1875Prandtl ( -1953 and Theodore von Kármán (1881Kármán ( -1963, as well as to the statistical description of a homogeneous and isotropic turbulent flow by Geoffrey Taylor (1886Taylor ( -1975, then Andreï Kolmogorov (1903Kolmogorov ( -1987 (Battimelli 1986;Darrigol 2005;Farge 1992;Eckert 2006;Davidson et al. 2011). In France, the mathematician Joseph Kampé de Fériet (1893-1982 actively participates in these research (Demuro 2018). ...
... In a more general sense, and even if in different forms, the presence of a network of connections between mathematics and experimentation characterizes the statistical theory of turbulence throughout the twentieth century. The contributions of Prandtl, von Kármán, Taylor, and Kolmogorov provide further examples of these interactions (Davidson et al. 2011). These interactions illustrate how the statistical theory of turbulence contributes to the modular evolutive structure of fluid mechanics (Darrigol 2008a, b), driven by empirical challenges and a focus on practical problems in fields such as aeronautics and meteorology. ...
The development of the statistical theory of turbulence mainly take places between 1920 and 1940, in a context where emerging theories in fluid mechanics are striving to provide results closer to experimentation and applicable to practical fluid problems. The secondary literature on the history of fluid mechanics has often emphasized the importance of the contributions of Prandtl, Taylor, and von Kármán to the closure problem of Reynolds equations for a turbulent fluid confined by walls and to the statistical description of an isotropic and homogeneous turbulent flow. During the same period, a new theory of turbulence also surfaces in France. This theory is formulated by a group of researchers led by Philippe Wehrlé (1890–1965), the director of the French National Meteorological Office (Office national météorologique, ONM), and Georges Dedebant (1902–1965), the head of ONM’s Scientific Service. Their objective is to mathematically formalize the turbulence, taking into account the atmospheric turbulence and using a theory of random functions defined from experimental concepts. However, this French theory of turbulence gradually loses international recognition after World War II. After introducing the key figures and the fundamental components of their theory, the article explores various scientific factors why their contribution was increasingly forgotten after the Second World War.
... This review is personal in the sense that it is a result of my own "voyage through turbulence" [16], transitioning from PP to FM research. My journey began in academic PP research (1997)(1998)(1999)(2000)(2001)(2002)(2003)(2004)(2005), where I also had contact with K41 and energy/enstrophy cascades for two-and three-dimensional flows. ...
... The transport barrier (TB) concept was introduced to FM by Prandtl as the laminar/turbulent boundary layer (LBL/TBL) [16,19,20], where the LBL constitutes an edge transport barrier (ETB) using nomenclature from magnetic confinement fusion PP. The boundary layer (BL) is characterised by mean velocity shear and molecular (LBL)/turbulent (TBL) viscosity. ...
... An important difference is that the (traditional) statistical approach considers turbulent flows with a high Re, whereas the dynamical systems analysis is limited to a lower Re. We will focus on the statistical point of view below but will discuss the dynamical systems approach in Section 2. 16. Research on the laminar-turbulent pipe flow transition [31] introduces a third perspective: linear or nonlinear hydrodynamic stability. ...
This review is a first attempt at bringing together various concepts from research on wall- and magnetically-bounded turbulent flows. Brief reviews of both fields are provided: The main similarities identified are coherent (turbulent) structures, flow generation, and transport barriers. Examples are provided and discussed.
... The transport barrier (TB) concept was introduced to fluid mechanics (FM) by Prandtl as the boundary layer (BL) [1][2][3] which constitutes an edge transport barrier (ETB) using nomenclature from magnetic confinement fusion plasma physics (PP). The BL is characterised by mean velocity shear and vorticity generation. ...
... • Increasing core fluctuations for the pipe flow high Reynolds number (Re) transition [12] is similar to controlled confinement transitions in fusion plasmas [13,14] • Travelling wave solutions in pipe flow [15] are reminiscent of the magnetic field structure (islands) in fusion plasmas This paper is personal in the sense that it is a result of my own "voyage through turbulence" [3], which began in academic PP research (1997)(1998)(1999)(2000)(2001)(2002)(2003)(2004)(2005), where I also had contact with K41 and energy/enstrophy cascades for two-and three-dimensional flows. Another major influence was the Kolmogorov-Arnold-Moser (KAM) theorem [16] and the survival or destruction of invariant tori in response to perturbations, which is directly applicable to the magnetic field structure. ...
... where ν kin is the kinematic viscosity. At a certain Re D (∼ 2000), the laminar to turbulent transition takes place [3,24], associated with a steepening of the edge velocity gradient. However, the transition is gradual with Re D ; as it increases, what is observed first are turbulent puffs, which are regions of turbulence separated by laminar regions. ...
This paper is a first attempt at bringing together various concepts from research on wall- and magnetically-bounded turbulent flows. Brief reviews of both fields are provided: The main similarities identified are transport barriers, coherent (turbulent) structures and flow generation. Examples are provided and discussed.
... In incompressible turbulence, the behavior of small inertial scales is commonly described by invoking the physical image of the Richardson cascade [1][2][3]: under the stirring action of velocity gradients, inertial eddies break down into smaller and less energetic ones, that break down into even smaller and less energetic ones, and so on until viscous dissipation occurs. This phenomenological description is intimately linked with two sets of formal results stemming respectively from Kolmogorov's [4,5] and Onsager's [6,7] works. ...
The purpose of this work is to investigate whether a cascading process can be associated with the rotational motions of compressible three-dimensional turbulence. This question is examined through the lens of circulicity, a concept related to the angular momentum carried by large turbulent scales. By deriving a Monin-Yaglom relation for circulicity, we show that an ``effective'' cascade of this quantity exists, provided the flow is stirred with a force having a solenoidal component. This outcome is obtained independently from the expression of the equation of state. To supplement these results, a coarse-graining analysis of the flow is performed. This approach allows to separate the contributions of the transfer and production terms of circulicity and to discuss their respective effects in the inertial range.
... The literature shows that the phenomenon of turbulence has captured the attention of humankind for centuries, see for instance [2]. The discovery of the Euler equations in the mid-18th century and Navier-Stokes equations in the first half of the 19th century are the major scientific and mathematical breakthroughs. ...
... The rest of the paper is divided into three main sections; Sections 2-4. In Section 2 we briefly discuss Fourier transform and its properties, rewrite Equation (2) in Fourier variables, and derive prior estimates. In Section 3 we present and prove our main results whereby we drive the bounds of the spectral energy function (3) and spectral energy bounds. ...
... In fact, (5) implies that the energy of the system (2) in Fourier space is equal to the energy of the system in Cartesian space. To take advantage of (5) we give an equivalent formulation for (2) in Fourier space. This is done in two steps; first we eliminate the pressure term by applying the Leray projector given by (8). ...
The spectral slope of magnetohydrodynamic (MHD) turbulence varies depending on the spectral theory considered; −3/2 is the spectral slope in Kraichnan–Iroshnikov–Dobrowolny (KID) theory, −5/3 in Marsch–Matthaeus–Zhou and Goldreich–Sridhar theories, also called Kolmogorov-like (K-41-like) MHD theory, the combination of the −5/3 and −3/2 scales in Biskamp, and so on. A rigorous mathematical proof to any of these spectral theories is of great scientific interest. Motivated by the 2012 work of A. Biryuk and W. Craig (Physica D 241(2012) 426–438), we establish inertial range bounds for K-41-like phenomenon in MHD turbulent flow through a mathematical rigor; a range of wave numbers in which the spectral slope of MHD turbulence is proportional to −5/3 is established and the upper and lower bounds of this range are explicitly formulated. We also have shown that the Leray weak solution of the standard MHD model is bonded in the Fourier space, the spectral energy of the system is bounded and its average over time decreases in time.
... A transfer of energy is thus required from one scale to the other that is achieved by the energy cascade caused by the nonlinearity of the Navier-Stokes equations (NSEs). The fundamental idea of the energy cascade across scales was first introduced by Richardson [2] and later quantified by Kolmogorov [3]. Under the assumption of scale-similarity, Kolmogorov predicted a power-law behaviour for the energy spectrum E(k) ∝ 2/3 k −5/3 and for the scaling of the moments of velocity's differences across a distance r: |δ r u| p ∝ ( r) p/3 . ...
We investigate numerically the model proposed in Sahoo et al. (2017 Phys. Rev. Lett. 118 , 164501) where a parameter λ is introduced in the Navier–Stokes equations such that the weight of homochiral to heterochiral interactions is varied while preserving all original scaling symmetries and inviscid invariants. Decreasing the value of λ leads to a change in the direction of the energy cascade at a critical value λ c ∼ 0.3 . In this work, we perform numerical simulations at varying λ in the forward energy cascade range and at changing the Reynolds number R e . We show that for a fixed injection rate, as λ → λ c , the kinetic energy diverges with a scaling law E ∝ ( λ − λ c ) − 2 / 3 . The energy spectrum is shown to display a larger bottleneck as λ is decreased. The forward heterochiral flux and the inverse homochiral flux both increase in amplitude as λ c is approached while keeping their difference fixed and equal to the injection rate. As a result, very close to λ c a stationary state is reached where the two opposite fluxes are of much higher amplitude than the mean flux and large fluctuations are observed. Furthermore, we show that intermittency as λ c is approached is reduced. The possibility of obtaining a statistical description of regular Navier–Stokes turbulence as an expansion around this newly found critical point is discussed.
This article is part of the theme issue ‘Scaling the turbulence edifice (part 2)’.
... Literature shows that the phenomenon of turbulence has captured the attention of humankind for centuries, see for instance [15]. The discovery of Euler equations in the mid of the 18th century and Navier-Stokes equations in the first half of the 19th century are the major scientific and mathematical breakthrough developments in terms of having governing rule for fluid flows. ...
... We refer to the review by Verma [47] for the several phenomenological theories on MHD turbulence and the book by Davidson et. al. [15] for the biographies and works of some of the prominent contributors to the area. ...
The spectral slope of Magnetohydrodynamic (MHD) turbulence varies depending on the spectral theory considered; is the spectral slope in Kraichnan-Iroshnikov-Dobrowolny (KID) theory, in Marsch-Matthaeus-Zhou's and Goldreich-Sridhar theories also called Kolmogorov-like (K-41 like) MHD theory, combination of the and scales in Biskamp and so on. A rigorous mathematical proof to any of these spectral theories is of great scientific interest. Motivated by the 2012 work of A. Biryuk and W. Craig [Physica D 241(2012) 426-438], we establish inertial range bounds for K-41 like phenomenon in MHD turbulent flow through a mathematical rigour; a range of wave numbers in which the spectral slope of MHD turbulence is proportional to is established and the upper and lower bounds of this range are explicitly formulated. We also have shown that the Leray weak solution of the standard MHD model is bonded in the Fourier space, the spectral energy of the system is bounded and its average over time decreases in time.
... As mentioned in [16], the local power-law exponent in Equation (50), 0.137, is close to the 1/7 exponent ("the 1/7th law") proposed by von Kármán and Prandtl as a global power-law for the low Reynolds number mean velocity (see Chapter 2.4 in [19] for the historical context). ...
... A characteristic mixing length scale l m (see Chapter 2.5 in [19]) can be found by differentiating the mean velocity log-law: ...
... Other power-law and log-law solutions, inspired by the offset solution [25] and grid-generated turbulence decay (see e.g. Chapter 3.3.1 in [19]), could be: ...
We study the global, i.e. radially averaged, high Reynolds number (asymptotic) scaling of streamwise turbulence intensity squared defined as I^2 = overbar(u^2)/U^2 , where u and U are the fluctuating and mean velocities, respectively (overbar is time averaging). The investigation is based on the mathematical abstraction that the logarithmic region in wall turbulence extends across the entire inner and outer layers. Results are matched to spatially integrated Princeton Superpipe measurements [Hultmark M, Vallikivi M, Bailey SCC and Smits AJ. Logarithmic scaling of turbulence in smooth-and rough-wall pipe flow. J. Fluid Mech. Vol. 728, 376-395 (2013)]. Scaling expressions are derived both for log-law and power-law functions of radius. A transition to asymptotic scaling is found at a friction Reynolds number Re_τ ∼ 11000.
... It has been evidenced long ago that turbulent fluctuations enhance the transport and mixing of particles (Taylor, 1922;Richardson, 1922): Above centimetric scales, small particles suspended in the air diffuse several orders of magnitude faster than prescibed by their molecular diffusivity. As a rough estimate, Lewis Fry Richardson estimates the turbulent diffusion in the atmosphere as 0.2 4/3 with expressed in cm (Davidson et al., 2011;Richardson, 1922). Richardson's formula implies that the turbulent diffusion is already a thousand times more efficient than Brownian diffusion on metric scales, and ten million times more efficient on kilometric scales. ...
This thesis analyses the dynamics of small complex objects immersed in a turbulent environment. Turbulent flows are akin to apparent random fields, that usually display very non-Gaussian and fluctuating statistics, and which are known to enhance the mixing and the transport of the objects that they carry. Here, we focus on the transport of small complex particles, which are characterized by a non-trivial interplay between their mass, their shape and their rheology. Our aim is to gain a physical understanding on how turbulent fluctuations prescribe the dynamics of such complex particles, and lead to various physical phenomena, including preferential concentration, their deformation or catastrophic events such as their fragmentation. Studying such phenomena is relevant for both industrial and sustainability issues. For instance, while volcanic ash has direct impacts for the commercial flight industry, the blooming of diverse types of species such as jellyfishes or phytoplankton has consequences both for the maintenance of power plants and for the thermodynamics of our planet.Our approach relies on a systematic use of massive numerical simulations of the Navier-Stokes equations to generate homogeneous isotropic turbulence at high Reynolds number, and analyze in details the statistics of various types of particles such as inertial spheroids and flexible fibers. For small inertial spheroids, our numerical work shows that the translational and the rotational motion are essentially decoupled. While the translational motion can be described by the motion of a sphere with a suitably defined effective mass, the rotational dynamics displays more intricate features. This reflects in the statistics of the rotation rate and in the concentration properties. Conversely to translational motion, the rotational dynamics is therefore non-universal and depends on the specific shape of the spheroids.For small inertialess fibers, which constitute a paradigmatic example of flexible elongated particles, we find that the dynamics is most of the time closely resembling that of a stiff rod. Yet, in very rare and intermittent episodes, the fibers experience violent buckling events, which correlate to strong local compressions exerted by the local turbulent flow. Besides, detailed statistical investigations reveal that flexibility also produces misalignments of the fibers, e.g. deviations in the statistics of the orientation compared to the dynamics of a completely stiff rod. Our most salient observation is that the coupling between such flexible fibers and the turbulence can be phenomenologically modeled in terms of various activation processes, both for the buckling rate and for the misalignments statistics.We finally investigate scenarios for turbulent fragmentation of brittle fibers. To that end, we implement in our numerics two mechanisms leading to the fibers breaking in smaller pieces, either because of tensile failure or because of flexural failure. We sketch a stochastic description of such violent events that paves the way to better parametrization of turbulent-induced fragmentation of brittle material in industrial codes.
... The description of the laminar-turbulent transition in a pipe flow was first provided by Gotthilf Hagen (1854), but Osborne Reynolds (1879) actually found, for a range of flow velocities, pipe diameters, and viscosities, that the transition occurred roughly always for the same value of a single unifying dimensionless parameter (Davidson et al. 2011). ...
The phenomenon of transport and deposition of particles dispersed on
flows has been studied for decades, with different biases. Understanding
and controlling these phenomena is an open challenge to this day. Different
approaches are used to model these processes and the methods to control
them invariably take into account different approaches. A phenomenological
and numerical approach to the deposition process in two dimensions is
presented in this thesis, highlighting the observables that mostly contribute
to this process. Then, we studied one of the ways to mitigate deposition
rates by applying a magnetic field over the flow.
Initially, we review some important hydrodynamic observables for our
specific modeling issues. Then, we review the basic ideas of the so-called
simplified magnetohydrodynamics (MHDS), a fundamental ingredient of
our discussion on the control, for the purpose of attenuation, of the transport
and deposition processes on flows of electrolytes or liquid metals.
Then, the Boltzmann equation will be discussed for the introduction of the
Lattice Boltzmann Method (LBM), a numerical strategy that will be used to
carry out the simulations to investigate the deposition process. Given its
efficiency and simplicity in implementing boundary conditions and external
volumetric forces, LBM is a great option for this type of study.
We analytically built a model of a static vortex near the wall using the
image method with a vortex and an anti-vortex in the opposite region
respect to the wall. Using this analytical flow model, we introduce a way
of handling particle transport and deposition by pointing to the main
observable candidates to be responsible for the deposition process.
Finally, when using LBM, we analyze how it is possible to attenuate the
deposition process by applying an external magnetic field in the flow. A
reduction in the intensity of the vortical structures is observed, which will
prove to be very relevant in the role that these structures have in launching
the particles towards the walls. This and other effects that are crucial in
determining the deposition rate - an observable of great interest in what
concerns particle deposition - will be attenuated in the presence of the
magnetic field. Our results, therefore, consolidate a mechanical approach to
control deposition in conductive media flows.