Magnetohydrodynamic Turbulence at Low Magnetic Reynolds Number

ANRV332-FL40-02 ARI 10 November 2007 15:47

MagnetohydrodynamicTurbulence at LowMagnetic Reynolds NumberBernard Knaepen1 and Rene Moreau2

1Universite Libre de Bruxelles, Boulevard du Triomphe, Campus Plaine CP231,B-1050 Ixelles Belgium; email: bknaepen@ulb.ac.be2Laboratoire EPM, ENSHMG, BP 95, 38402 Saint-Martin d’Heres France;email: r.j.moreau@wanadoo.fr

Annu. Rev. Fluid Mech. 2008. 40:25–45

The Annual Review of Fluid Mechanics is online atfluid.annualreviews.org

This article’s doi:10.1146/annurev.fluid.39.050905.110231

0066-4189/08/0115-0025$20.00

Key Words

magnetohydrodynamics, modeling, simulation

AbstractThis article reviews the main established ideas on the influence ofa magnetic field on turbulence in electrically conducting fluids. Welimit our discussion to the asymptotic range of very small valuesof the magnetic Reynolds number, characterized by the fact thatthe induced magnetic field remains very small in comparison withthe applied magnetic field. We consider three kinds of flows here.The simplest one is freely decaying homogeneous turbulence, whichserves as a test bed to analyze the development of anisotropy resultingfrom the linear damping by the Lorentz force. We then discuss flowsbetween walls perpendicular to the magnetic field and emphasize theinfluence of the Hartmann layers that develop in their vicinity. Wethen review the main features of the possible quasi-two-dimensionalregime that can arise in that context. Finally, we consider magneto-hydrodynamic turbulent shear flows. These are frequent in industrialapplications involving molten metals, such as in metal processing orin the blanket of future nuclear fusion reactors. We pay particularattention to recent attempts to develop specific RANS (Reynolds-averaged Navier-Stokes) models for these flows.

1. INTRODUCTION

Magnetohydrodynamics (MHD) concerns flows of electrically conducting fluids inthe presence of a magnetic field. Those flows obey the coupled Navier-Stokes andMaxwell equations, and one of their key features is the occurrence of induced electriccurrents within the fluid, in turbulent or chaotic regimes as well as in laminar flows.These currents generate ohmic losses, and a new mechanism of dissipation of energy,besides viscous dissipation, is therefore present. This extradissipation, usually namedthe Joule effect, is characterized by the magnetic diffusivity η = 1/μσ of the fluid,where μ stands for magnetic permeability and σ for the electrical conductivity of thefluid.

The typical time scale of this Joule effect is τη = l2/η, where l stands for a typicallength scale of the flow. A crucial nondimensional number is then the magneticReynolds number Rm defined as the ratio between τη and the eddy turnover time ofthe flow τtu = l/u (u stands for a typical velocity scale):

Rm = τη

τtu= ul

η. (1)

In astrophysics, Rm is usually extremely large, which implies that the magneticfield is almost frozen within the flow (this behavior is analogous to the freezingof vorticity in an inviscid fluid). In Earth’s core, Rm is approximately 103 to 104,which is enough to allow for the dynamo effect (see, for instance, Roberts 1967),which self-sustains Earth’s magnetic field. At the laboratory scale, Rm is typically ofthe order of 10−4 or 10−2 for electrically conducting liquids such as molten metals.In that case, the fluctuating magnetic field �b induced by the fluctuating current israpidly smoothed out by the strong magnetic diffusivity. These fluctuations, as wellas those of the electric field, however, can be measured and related to the fluctuatingvelocity field �u. With

−→B0 as the applied magnetic field, the fluctuations �b are of order

Rm−→B0, which means they are negligible in comparison with the applied magnetic field−→

B0. However, their presence has a strong effect on the flow.In this review, we restrict our attention to flows characterized by a magnetic

Reynolds number Rm � 1. This regime covers most cases encountered in indus-trial applications. We can express the currents �j induced by the fluid’s motion in thepresence of the applied magnetic field

−→B0 through the simplified Ohm’s law,

�j = σ (−�∇φ + �u × −→B0), (2)

where φ stands for the local electric potential (without the assumption Rm � 1, onecannot write the electric field �E appearing in Ohm’s law as the pure gradient −�∇φ).The electric potential is related to the velocity via the charge conservation equation

�∇ · �j = 0 ⇒ �φ = −→B0 · �ω, (3)

where �ω = �∇ × �u is the vorticity. The induced electrical currents interact with theapplied magnetic field and give rise to the Lorentz force, which is a source term inthe Navier-Stokes equations and can be expressed as �j × −→

B0. In the limit Rm � 1, we

26 Knaepen · Moreau

can therefore write the MHD equations as

∂tui + ∂ j (u j ui ) = −∂i p/ρ + ν �ui +(

(−�∇φ + �u × −→

)× −→

)i, (4)

�φ = −→B0 · �ω, �∇ · �u = 0, (5)

where p stands for pressure, ρ is the density, and ν is the viscosity.Because little is known about the general case of spatially varying magnetic fields,

we limit ourselves to MHD flows in the presence of a uniform applied magnetic field−→B0. By convention, the magnetic field is oriented in the z or x3 direction, often denotedas x‖. The other two directions, x1 and x2, are equivalent to each other; as is usualin axisymmetric turbulence, they are denoted collectively as x⊥. Here we alreadyencounter one of the most striking characters of MHD turbulence, its anisotropyowing to the privileged direction x‖, which we discuss extensively in the followingsections.

A second important feature of MHD turbulence is the linear Joule damping asso-ciated with Lorentz force. Unlike viscous damping, it does not involve any derivativeof the velocity, which implies that the scales, or wave vectors, which predominantlycontribute to the Joule dissipation, are not concentrated at the small scales, whereviscous dissipation is predominant. This Joule damping is proportional to B2

0 and maybe characterized by the time scale τJ0 = ρ/σ B2

0 , referred to as the Joule time.As in ordinary turbulence, the usual Reynolds number Re is defined as

Re = τν

τtu= ul

ν, (6)

where ν is the kinematic viscosity of the fluid and τν = l2/ν is the time scale of viscousdissipation. Obviously, the flow can only be turbulent for Re 1. In MHD, thestructure and intensity of turbulence can be strongly affected by the applied magneticfield. In an infinite fluid domain, the characteristic nondimensional quantity thatcharacterizes the influence of the magnetic field on the turbulence is the interactionparameter defined as

N = σ B20 l/ρu. (7)

This interaction parameter, built as the ratio of the eddy turnover time and the Jouletime, determines the structure and the decay of turbulence as discussed in Section 2with regard to homogeneous MHD turbulence. In that section, we comment on therecent new insight provided by numerical attempts, either direct numerical simula-tions (DNS) or large-eddy simulations (LES), together with previous experimentalknowledge, to support what we consider to be the basic phenomenology.

In Section 3, we focus on MHD turbulence in domains bounded by walls perpen-dicular to the magnetic field, and we discuss the role and characteristic features ofthe Hartmann layer present along these walls. This Hartmann layer exhibits somespecific properties. Essentially, similar to the Ekman layer in rotating flows, it is notat all a passive layer capable of adjusting to any neighboring core flow while satisfy-ing the no-slip condition at the wall. On the contrary, because of the need to loopthrough the core flow the electric current lines passing in this Hartmann layer, theproperties of that layer have a profound influence on the core flow. In short, they

www.annualreviews.org • MHD Turbulence 27

force the core turbulence to become quasi-two-dimensional (Q2D) in the full sensebecause both the z derivative of the velocity and the uz velocity component becomenegligible. Such a Q2D velocity field does not induce strong electric currents and,as a consequence, is submitted to a weak damping. In other words, in the limit ofa well-established two-dimensionality, the only remaining electromagnetic dampingcomes from the Hartmann layer itself, and according to the key property of that layerwhere viscous friction and the Lorentz force are equal to each other, this damping hasa scaling quite different than that in homogeneous turbulence. Indeed, it is propor-tional to B0 instead of B2

0 and its time scale is τH = (ρ/σν)1/2h/B0 = HaτJ0, whereh is the half-distance between the Hartmann walls and Ha is the Hartmann number.This nondimensional number, defined as

Ha = B0h(σ/ρν)1/2 = h/δ, (8)

with δ = (ρν/σ )1/2/B0, can be very large, which means that the time scale τH maybe much larger than the Joule time scale τJ0. In laboratory experiments with liquidmetals and moderate magnetic fields, it is typically in the range 10 to 103. At largescale, as in the blanket of future fusion reactors, it might reach values of the order of104 or 105. Then the damping of Q2D MHD turbulence may become extremely slow,and the turbulence may persist, whatever the strength of the applied magnetic field.The properties of this kind of turbulence are of course closely related to the classicalproperties of ordinary Q2D turbulence, and the relevant parameter to characterizethe MHD damping is not N but Ha/Re = N/Ha.

In Section 4, we briefly summarize the still ongoing work on the numerical model-ing of MHD turbulence, with RANS (Reynolds-averaged Navier-Stokes) techniques,focusing on the remaining weaknesses of the available models and the current ideasto remove them. In the conclusion, we raise questions that we consider important toimprove the available knowledge and modeling of MHD turbulence and to extendtheir validity to more complex flows.

2. HOMOGENEOUS TURBULENCE

Homogeneous turbulence is the simplest form of turbulence that illustrates howthe applied magnetic field increases the decay rate and forces some anisotropy. It isconvenient to use Fourier transforms, such that �∇ → i �k. Taking the double curl ofthe electrical current (Equation 2), we get �j = −(1/k2)( �B · �k)(�k × �u). This implies thatwe can write the Fourier transform of the Lorentz force as

−→FL(�k) = −σ

0 cos2 θ (�k)�u(�k), (9)

where θ is the angle between the wave vector �k and the magnetic field−→B0 and where

a pure gradient is removed and absorbed in a redefinition of the pressure. Because ofthe minus sign on the right-hand side of Equation 9,

−→FL clearly has a dissipative effect.

This Joule damping acts at all scales because the Lorentz force does not depend onthe length of the wave vector, but it strongly depends on the direction of the wave

vector: The damping rate of a given wave vector is proportional to the square of itscomponent in the direction of the magnetic field.

2.1. Basic Phenomenology

An efficient way to illustrate how the anisotropy develops is to consider an initiallyisotropic turbulence suddenly submitted to the uniform magnetic field. FollowingMoffatt (1967) and Schumann (1976), we assume that the magnetic field can beswitched on instantaneously everywhere in space. Just after the onset of the magneticfield, the system can be described in the framework of rapid distortion theory. TheFourier modes are therefore damped according to Equation 9 and do not significantlyinteract with each other (because Re 1, viscous effects are also neglected). As aconsequence, the evolution of the Fourier transform of the two-point correlationfunction φij(�k) = 〈ui (�k)u∗

j (�k)〉 is governed by

∂tφij(�k, t) = −2σ

0 cos2 θ (�k)φij(�k), (10)

⇒ φij(�k, t) = φ0ij(�k) exp

[−2 cos2 θ (�k)

], (11)

where φ0ij is the initial value of φij. We can obtain a qualitative picture of the evolution

by introducing the so-called Joule cone depicted in Figure 1a. The Fourier modesthat are damped most rapidly are located around θ = 0 (independently from the scaleconsidered), so the region of Fourier space located around the k‖ axis is depletedfrom energy first. As time progresses, energy removal affects modes at increasingangles θ . A cone, centered around the k‖ axis, thus progressively opens and delimits

kykx ky

kz = k|| kz = k||Joule dissipation

Energy-containing modes

Local angularenergy transfers

Figure 1Sketch of the Joule cone in the Fourier space, without (a) and with (b) the inertial energytransfers.

the energy-containing modes from the modes that have lost their energy because ofJoule damping. We can define a semiquantitative measure of the rate of opening ofthe Joule cone as follows. Denoting θ∗

α (t) the angle delimiting the modes that havelost at least the fraction α of their energy, we have

cos θ∗α (t) ∼

( τJ0

)1/2. (12)

During the initial linear damping of turbulence, the rate of opening of the Joule coneis thus of order t−1/2.

In terms of anisotropy, the opening of the Joule cone has two consequences. Thefirst one concerns the partition of energy among the components of the velocity field.Using Equation 11, we can easily show that if the flow is initially isotropic, one has

‖⟩ = ⟨

u2⊥⟩,

∂t⟨u2

‖⟩ = 1

∂t⟨u2

⊥⟩

at t = 0, (13)

where 〈u2‖〉 = 〈u2

z〉 and 〈u2⊥〉 = 1/2(〈u2

x〉 + 〈u2y 〉). Ultimately, the smaller damping of

the parallel component of the velocity comes from the fact that, for an incompressibleflow, the bulk of its energy is located around θ = π/2, where Joule damping vanishes.If one extrapolates the linear decay up to t → ∞, the only contribution from Fourierspace to the energy densities comes from the region θ = π/2 (cos2 θ = 0) so that⟨

u2‖⟩(∞) � 2

⊥⟩(∞). (14)

The parallel component of the velocity thus tends to carry more energy than theperpendicular components, but they are still of the same order of magnitude.

The second form of anisotropy concerns the elongation of structures in the direc-tion of the magnetic field, and it can be pronounced. For instance, one can tentativelydefine the ratio of the length scale in the parallel and perpendicular directions as(

l‖l⊥

d �kk2⊥φi i (�k)

d �kk2‖φi i (�k)

d �kk2(1 − cos2 θ )φi i (�k)2

∫d �kk2(cos2 θ )φi i (�k)

. (15)

As the Joule cone opens up, this ratio increases because the contributions to theintegrals become concentrated in the region θ = π/2. For large times, the behaviorof the above ratio is given by

l‖l⊥

∼ (t/τJ0)1/2. (16)

This indicates that, according to the linear theory, the flow structures elongate withoutany limit in the direction of the magnetic field. Sommeria & Moreau (1982) firstproposed a physical space interpretation of this elongation without assuming anyhomogeneity of the turbulence. Indeed, in physical space, we can write the Lorentzforce (Equation 9) as

−→FL(�x) = −σ

ρ�−1(

−→B0 · �∇)2 �u(�x), (17)

where �−1 denotes the inverse of the Laplacian operator. When eddies are sufficientlyelongated in the direction of

−→B0, one can assume that ∂z � ∂x, ∂y . This implies that

the Lorentz force reduces to

−→FL(�x) = −σ

0�−1⊥

d z2�u(�x). (18)

This equation suggests that for an eddy characterized by an orthogonal length scale l⊥,the action of the Lorentz force reduces to a diffusion in the direction of the magneticfield with the effective diffusivity α = l2

⊥/τJ0. As a consequence (Sommeria & Moreau1982), the length scale l‖ of the eddy in the direction of

−→B0 increases according to

Equation 16.Davidson (1995) provided another elegant explanation of the elongation of flow

structures in the direction of the magnetic field by showing that it is a necessarycondition to satisfy the conservation properties of the Lorentz force. Indeed, thefollowing two integrals

−→FLd V = −

−→B0

�J d V (19)

and ∫V

−→B0 · (�x × �FL)d V = − B2

�∇ ·[−→x⊥

2 �J]

d V (20)

both vanish when the current paths close in the volume considered (this is the case, forinstance, when the flow evolves between insulating boundaries). This implies that theLorentz force conserves both the linear momentum and the component of angularmomentum parallel to the magnetic field. As far as the Lorentz force is concerned,the flow cannot therefore come to rest, requiring the electrical currents to declinesufficiently fast with time. Because the currents are proportional to the gradient ofthe velocity in the direction of

−→B0, i.e., �∇ × �J = σ (

−→B0 · �∇)�u, the flow structures must

therefore elongate along the magnetic field.For turbulent flows, the linear analysis cannot of course be extrapolated for t → ∞.

It is expected, however, that the rapid distortion theory should hold during a durationof the order of the eddy turnover time, whereas triple correlations adapt to the newflow conditions. Depending on the value of the interaction parameter N, severalscenarios are possible. If N 1, the Joule damping time is much smaller than theeddy turnover time, and the turbulence should be almost completely damped beforenonlinear terms can grow large enough to have a sizeable effect on the flow. On thecontrary, when N � 1, the eddy turnover time is much smaller than the Joule time,and the Joule cone does not have time to significantly open up before nonlinear termsbecome important and redistribute efficiently the energy throughout Fourier space.In that case, the flow decay should be similar to what is observed for nonconductingfluids, and no significant buildup of anisotropy can occur. Then, from the point ofview of MHD turbulence, the most complex regime should concern values of theinteraction parameter varying around unity, N ∼ 1.

Alemany et al. (1979) first studied in detail the phenomenology of homogeneousturbulence for N ∼ 1 on the basis of experimental observations. They described decay-ing homogeneous turbulence in mercury in the presence of a homogeneous magneticfield. The nature of this experiment closely resembles Comte-Bellot and Courrsin’s

classical experiment on grid turbulence in wind tunnels. The main contribution ofAlemany et al. (1979) is a description of the energy transfers taking place in MHDturbulence. Contrary to ordinary turbulence, in which these transfers mainly resultfrom a cascade toward small scales, in MHD they have a significant angular compo-nent. Once the Joule cone has sufficiently opened up, the steepest energy gradientsbetween Fourier modes are no longer directed radially but rather orthogonally to thecone’s border. As a consequence, the nonlinearity naturally tries to compensate forthose large gradients by generating angular transfers (see Figure 1b). As the conecontinues to open up, a numerical analysis based on an MHD variant of the EDQNM(eddy-damped quasi-normal Markovian) model (Alemany et al. 1979) suggests that akind of local equilibrium between Joule damping and angular transfers is establishedin the vicinity of the cone’s border. The strongest confirmation of this picture ofangular energy transfers in MHD turbulence comes from its ability to interpret thek−3 inertial range observed in the experiments. Indeed, as the Joule damping timeis scale independent, the equilibrium between Joule damping and energy transferssuggests that the transfer time should also be scale independent. Therefore,

τtransfer = 1/ku(k)

= [k3 E(k)]1/2 ∼ k0 ⇒ E(k) ∼ k−3. (21)

Such a k−3 spectrum is clearly observed in the experiments of Alemany et al. (1979)for interaction parameters N � 3 (k−3 spectra have also been observed in severalexperiments on wall-bounded flows as discussed in Section 3, but these deserve adifferent interpretation). In ordinary turbulence, spectra with a k−3 behavior areusually associated with a direct enstrophy cascade owing to a good two-dimensionality.In Alemany et al.’s (1979) experiments, however, the flow is largely 3D, so 2D effectsin the spectra could only have a marginal influence. The authors also report a powerlaw for the decay of the global kinetic energy, i.e., u2

‖ ∼ t−n with n = 1.7 (only u2‖

could be measured in the experiments). Remarkably, the decay exponent is reportedto be independent of B0 during the power-law decay regime. Although turbulenceis clearly present, the evolution of integral length scales seems compatible with thelinear prediction (Equation 16), thanks to a quite slow variation of l⊥ during thedecay. Indeed, the measurements yield l⊥ ∼ t0.15 and l‖ ∼ t1/2, suggesting that inertiahas only weak influence on the morphological anisotropy.

Because both the length scales and the velocity scales vary during the decay, therelevant values of the interaction parameter and the key time scales are also timedependent. Good estimates of the time scales are τtu(t) = l⊥(t)/u⊥(t) ∼ t and τJ(t) =τJ0(l⊥/ l‖)2 ∼ t so that, whatever the initial value of the interaction parameter, duringthe quasi-steady decay its value must be of the order of unity and stay constant.

2.2. Numerical Investigations

Accurate measurements in MHD flows are challenging and hardly achievable. Most ofthe liquids concerned are either opaque, corrosive, or very hot (such as molten steel).For that reason, numerical simulation seems to be a valuable tool for the study ofMHD turbulence. Several such investigations of homogeneous MHD turbulence

have been performed to date using spectral codes. The first one dates back toSchumann (1976) and concerns freely decaying turbulence. This study largely con-firms the picture of MHD turbulence outlined above and, in particular, the presenceof the angular transfers resulting from anisotropy. Of course, the intensity of turbu-lence present in these numerical simulations is limited by the 323 resolution availableat that time. Forced MHD turbulence was first studied by Hossain (1991) and morerecently by Zikanov & Thess (1998). In both works, the authors used a large-scaleforcing to sustain turbulence and drive the flow to a statistically stationary state. Forthe case of moderate interaction parameter N ∼ 1, Zikanov & Thess (1998) observeda peculiar behavior of the dynamics. They reported that the evolution of the flow con-sists of a succession of Q2D states lasting for several eddy turnover times interruptedby short intermittent bursts of 3D regimes. However, this intermittency might resultfrom the particular type of large-scale forcing used in this attempt, so this observa-tion, which contrasts with the picture of a smooth equilibrium between nonlineartransfers and Joule dissipation, deserves some clarification. Indeed, it questions thestability of the elongated turbulent structures generated by Joule dissipation (Thess& Zikanov 2004).

Another striking feature, mentioned in nearly all the numerical simulations ofhomogeneous turbulence, concerns the partition of energy between the three com-ponents of the velocity field. As stated above, the linear theory of Joule dissipationimplies that the energy carried by the parallel component of the velocity field shouldbe larger than the one carried by the perpendicular components. In forced and decay-ing turbulence, this is observed for a duration of the same order as that of the Jouletime (during which Joule dissipation dominates the dynamics). After that, the decayof the parallel component is stronger than that of the perpendicular components, andthese thus carry more energy (see Figure 2a). This is clearly a nonlinear mechanismtending to restore isotropy and one that has not been elucidated.

Thanks to the higher resolutions that can now be achieved by DNS, the develop-ment of anisotropy during the decay and, namely, its scale independence have recentlyreceived some attention (Vorobev & Zikanov 2007, Vorobev et al. 2005). In our abovediscussion of the global properties of anisotropy, we stress that the level of anisotropyis clearly related to the intensity of the interaction parameter N = τtu/τJ. But, if τtu

is scaled as τtu ∼ (1/k)/[u(k)] as in Equation 21, we can define a scale-dependent in-teraction parameter. It is easy to show that, within the inertial range, N ∼ k−2/3 in theisotropic case (weak magnetic fields) and N ∼ k0 in the case of strong magnetic fieldsin which the spectrum assumes a k−3 slope (21). Therefore, when the intensity of themagnetic field is not strong enough to impose a k−3 spectrum, N should decrease atsmall scales; therefore, the anisotropy should be less pronounced at small scales. Thisargument is, however, contradicted by the numerical simulations of forced turbulenceperformed in Vorobev et al. 2005 and Vorobev & Zikanov 2007. In all cases consid-ered, the level of anisotropy is essentially constant throughout the ranges of scales,despite the fact that the interaction parameter measured at large and small scales issignificantly different (see Figure 2b). These numerical studies seem to indicate thatanisotropy in MHD turbulence is persistent at all scales even when the spectrum doesnot behave as k−3. Whether this conclusion holds for very high Reynolds numbers

0.1 1.00.1

n = 1.6

50 1000

1N0 = 0

Figure 2(a) Evolution of the parallel and perpendicular components of the kinetic energy in decayinghomogeneous magnetohydrodynamic turbulence. E‖ and E⊥ are shown in different colors.The figure includes results from both a nonlinear simulation (solid lines) and a linear simulation(dashed lines). The initial interaction parameter is N = 1, and time is measured in Jouledamping time units. The power law decay rate n of E‖ in the nonlinear case is also shown inthe figure. The curves are obtained from the 5123 simulations of Knaepen & Moin (2004).(b) Scale independence of anisotropy. The curves represent the quantityg(k) = 3 [

∑s (kz2/k2)|u(k)|2]/[

∑s |u(k)|2 for three different global interaction parameters:

N = 0, N = 1, N = 5. To highlight the scale dependence of anisotropy, the sums enteringthe definition of g(k) are restricted to Fourier modes belonging to spherical shells (such as inthe definition of the 3D spectra). Figure 2b taken from Vorobev et al. 2005.

(when small scales are extremely small in comparison with energy-containing scales)is not completely clear because the range of scales that can be explored by numericalsimulation is of course limited by the available numerical resolution.

To alleviate the difficulties inherent to the huge computing cost of DNS, re-searchers have dedicated several recent works to the development of LES in the con-text of MHD turbulence at low magnetic Reynolds numbers. As in hydrodynamics,these studies are particularly relevant to numerous possible industrial applications.The main question to address is how to satisfactorily account for the modification ofFourier space energy transfers in subgrid-scale models. As discussed above, it appearsthat the energy cascade toward small scales is replaced by preferential angular trans-fers. This means that in the traditional LES formulation, in which the subgrid-scalestress tensor models the influence of small scales on large scales, the effective viscosityof the model has to be reduced.

Yoshizawa (1987) was the first to attempt to incorporate this feature in a specificMHD subgrid-scale model in the form of a damping factor to the turbulent viscosity.In terms of the LES filter width �, the expression proposed for the total eddy viscosityis νsgs = νs exp[−(σ/ρ)(Cm�2)B2

0/νs ], where νs gs is the turbulent viscosity appearing inthe definition of the subgrid-scale stress tensor τij = −2νsgs Sij, Sij = (1/2)(∂i u j +∂ j ui ),and the bar denotes the filtering operator. In the limit of vanishing magnetic field B0,

the above model reduces to the traditional Smagorinsky eddy viscosity (Smagorinsky1963), νs = 2Cs �

Sij Sij, which is widely used in hydrodynamic LES; for a veryintense magnetic field, the influence of the model vanishes in agreement with the factthat the turbulent cascade toward small scales becomes less important. The drawbackof the previous model is that besides the Smagorinsky constant, Cs, it contains anextra parameter, Cm.

To avoid having to specify those model parameters, Knaepen & Moin (2004)recently tested the dynamic Smagorinsky model. This model is formally identical tothe original Smagorinsky model except that the constant Cs is replaced by a variable,Cd , that is automatically calibrated during the simulation (Germano et al. 1991).Using this method, the authors demonstrate that, in decaying turbulence, the correctlarge-scale dynamics can be reproduced without introducing an explicit dampingfactor such as that in Shimomura (1991). Indeed, the dynamic procedure can pick avalue for Cd that is compatible with the decrease of the eddy viscosity as the magneticfield is increased. The applicability of the dynamic Smagorinsky model has also beenrecently demonstrated in the case of forced MHD turbulence (Vorobev et al. 2005).

A recent model proposed by Vorobev & Zikanov (2007) explicitly takes into ac-count the anisotropy generated by the magnetic field. It is an extension of the classicalSmagorinsky model in which the constant Cs is replaced by CsG, where, symbolically,G ∼ (l⊥/ l‖)2 (see Equation 15). With this modification, the subgrid viscosity reducesto its hydrodynamic value in the absence of a magnetic field, but it also vanishes in thecase of a strong magnetic field (l⊥/ l‖ → 0). The authors have shown that this model,which is numerically cheaper, behaves as well as the dynamic Smagorinsky model inthe case of forced turbulence but cannot reliably capture the strong transient presentin decaying turbulence.

Finally, a consistent way to build models aiming to reach high Reynolds numbersin MHD turbulence may consist of filtering not only the small scales, but also the wavevectors located within the Joule cone, whose energy has been previously damped. Thisidea is supported by an assessment of the influence of the magnetic field on the numberof relevant degrees of freedom (Potherat & Alboussiere 2003). Of course, such filtersshould be based on an analysis of the inertial transfer mechanisms present around theborder of the Joule cone, whose properties are still far from being well-known.

3. QUASI-TWO-DIMENSIONAL TURBULENCE BETWEENHARTMANN WALLS

We now consider that the fluid domain is limited by two plane walls, which areperpendicular to the applied magnetic field and 2h apart. According to Equation 16,some scales can invade the whole available gap between the Hartmann walls in a timeduration τ2D = τJ0(h/ l⊥)2. In actual experiments, it is possible that l⊥ is large enoughso that τ2D is much smaller than the eddy turnover time τtu = l⊥/u⊥. Then theenergy-containing turbulent structures become column-like before any significantinertial influence can counteract this developing anisotropy.

We first focus on the case of strong magnetic fields for which τ2D is much smallerthan any other time scale. It is then legitimate to distinguish a core flow in which

∂z � ∂x, ∂y and Hartmann layers in which viscosity and the Lorentz force canbalance each other. The Q2D eddies extending between the plane walls have theirends embedded in the Hartmann layers. As a result they suffer some braking that mayeasily be derived from the linear Hartmann layer theory. This behavior is exhibited inSommeria & Moreau (1982) in which the Navier-Stokes equation is integrated fromone Hartmann wall to the other and a model equation for �u⊥ is derived. In the highmagnetic field regime, �u⊥ is almost z-independent in the core flow, and the modelequation reads

∂t �u⊥ + (�u⊥ · �∇)�u⊥ = −1/ρ �∇⊥ p + ν∇2 �u⊥ − �u⊥/τH, (22)

where τH is the Hartmann damping time. The main merit of Equation 22 is thatthe net effect of the Lorentz force is contained in the linear drag −u⊥/τH . In theparticular case of insulating Hartmann walls, Sommeria & Moreau (1982) show thatτH = (ρ/σν)1/2h/B0 = HaτJ0 is Ha times larger than the Joule time scale and thetime τ2D necessary to establish a good two-dimensionality. If the Hartmann wallshave an electrical conductivity σw and a thickness tw [small enough to justify thatthe current density is almost uniform across the wall (see Buehler 1996, Mueller& Buehler 2001)], the expression of this Hartmann damping time becomes τH =(ρ/σν)1/2h/B0[(1 + C Ha)/(1 + Ha)], where C = σwtw/σh. Clearly, if C � 1, whichcorresponds to most practical situations when a liquid metal flows in a metallic duct,τH remains much larger than τJ0 and τ2D. In such conditions, inertia and Hartmannbraking can compete, and the predominant feature is a Q2D turbulence submittedto a linear damping. This regime must depend on only one nondimensional number,the ratio between the two relevant time scales: τH/τtu = Re/Ha = R, R being theReynolds number built on the thickness of the Hartmann layer. Beside this Hartmanndamping, another important role of the Hartmann walls is to completely suppressthe velocity component u‖ (an elementary order of magnitude analysis demonstratesthat u‖ scales as Re/Ha3 at the edge of each Hartmann layer). This is important as itmeans that the turbulence is then Q2D in the full sense because both the derivativein the z direction and the velocity component in that direction are quite small. Ofcourse, the complete picture is not as simple as this because some secondary flowmust enter the eddy at each end and exit the eddy in the core flow, driven by thedisrupted balance between inertia and pressure gradient (in the case of a perfectlycircular eddy, this would just be an Ekman flow within the Hartmann layer).

The existence of such a Q2D regime implies that an inverse energy cascade ex-ists. If τH τtu , the flux of energy toward the large scales, usually denoted ε, isnot significantly affected by the Hartmann damping. The cascade then still obeysthe Kolmogoroff law and exhibits the familiar ε2/3k−5/3 spectral law for scales rang-ing from the largest structures to the forcing mechanism. The existence of thisKolmogoroff-type inverse cascade is demonstrated by Sommeria (1986), Alboussiereet al. (1999), and Messadek & Moreau (2002) (unfortunately, the k−3 part of the spec-trum, related to the enstrophy cascade ranging from the forcing mechanism to thesmall structures, is not observed owing to the lack of resolution). When τH � τtu ,the inverse energy cascade is significantly affected by the Hartmann damping, andthe corresponding range of the energy spectrum then varies also as k−3. Alboussiere

et al. (1999) and Messadek & Moreau (2002) observe this power-law spectra in theMATUR experiment and propose the following interpretation: If a steady law existsin this forced flow, it requires that, at each k in the inertial range, there must be anequilibrium between the local energy transfer, whose time scale is 1/

√k3 E(k), and the

Hartmann damping time, which is k independent. This in turn implies the k−3 spectrallaw, which is also observed in many other experiments, but in more complex condi-tions (Branover 1979, Kljukin & Kolesnikov 1989, Kolesnikov & Tsinober 1974).

The MATUR experiment (Alboussiere et al. 1999, Messadek & Moreau 2002), inwhich turbulence is permanently fed by the instability of a circular free-shear layer,also reveals other noticeable features. Concerning the turbulent velocity fluctuations,the most striking result is certainly their high intensity, which does not decrease whenB0 increases: The ratio between the maximum of the root mean square of either ur

or uθ and the maximum of the mean velocity is of the order of 0.2, typically 10 timeswhat is measured in a turbulence generated by a grid in wind tunnels. As with manyprevious experimental investigations obtained in more complex conditions (Branover1979; Lielausis 1975; Tsinober 1975, 1990), the MATUR experiment also clearlyhighlights the importance of the nondimensional number R = Re/Ha when themagnetic field is large enough to make inertia negligible within the Hartmann layer(R < 300). First, results show that the thickness δ‖ of the free-shear layer scales asδ‖ = 0.1(Re/Ha)n, with an exponent n quite close to 1/2. This makes the thickness ofthe layer larger than its laminar prediction (Hunt 1965) by more than two orders ofmagnitude. Second, the number of large coherent vortices Ns observed follows theempirical law NS = 80(Re/Ha)n, again with n close to 1/2.

We now turn our attention to the regime of intermediate magnetic fields. In thatcase, B0 can be large enough to yield a Q2D turbulent core but too small to preventinertia from affecting the Hartmann layers. The situation is more complex and losesthe elegance of universal mean velocity profiles and the easily interpretable proper-ties summarized above. However, it deserves attention, both because it correspondsto the usual range of parameters found in liquid-metal experiments and because,when Re/Ha becomes of the order of 300, the Hartmann layer becomes unstable(Lingwood & Alboussiere 1999, Krasnov et al. 2004).

Potherat et al. (2000) have made the first theoretical attempt to introduce inertiainto the theory of this layer. The interaction parameter N, which is not relevant in thehigh magnetic field regime, now becomes relevant because the current density withinthe Hartmann layer scales as σ B0U. Technically, Potherat et al. (2000) use a doubleexpansion in terms of the two small parameters 1/N and 1/Ha. At the order 1/N,the major effect is the appearance of the secondary flow already mentioned above,which exits each large vortex in the core and transports some angular momentum.These authors show that its main influence is to enlarge the eddy in its central partand to give it a barrel-like shape. At the order of 1/Ha, which is usually significantlysmaller than 1/N, something similar occurs. According to the current conservationwithin the Hartmann layer (Hunt 1965), some electric current, proportional to thelocal vorticity, exits the layer at its edge and enters the core flow. Potherat et al.(2000) show that at the order of 1/Ha, the electric circuit looks similar to the sec-ondary Ekman flow and also favors the barrel shape, but with different scaling laws.

u2 ≈ O1

Nj ≈ O

Mean flow

Figure 3Sketch of a typicalquasi-two-dimensional eddyin flows bounded byHartmann layers, carryingits O( 1

N ) secondaryrecirculating flow and itsO( 1

Ha ) electric circuit.

The authors also show that, at the same order, the velocity field should assume aparabolic variation in the parallel (wall-normal) direction. Such a z dependence is ob-served in experiments on isolated vortices (Sommeria 1988) and in the simulation ofMHD flows in rectangular ducts with internal obstacles (Mueck et al. 2000) in whichthe Q2D eddies are described as having the shape of cigars. The above picture ofboth the secondary Ekman flow and the electric circuit associated with each vorticalstructure yields an interesting image of the typical Q2D MHD eddy, as illustrated inFigure 3.

We conclude this section on MHD turbulence between Hartmann walls by dis-cussing the case in which the applied magnetic field is quite small so that inertia cannotbe treated as a small disturbance. Many experiments (see Lielausis 1975 for a review)demonstrate the subtle interplay between turbulence and Hartmann damping in thecases in which the Hartmann layer is turbulent. For instance, the relation between thefriction coefficient cf and the strength of the magnetic field is far from monotonic. Onone hand, the magnetic field increases cf through the action of the Hartmann damp-ing; on the other hand, the turbulence suppression mechanism resulting from Jouledissipation has the opposite effect (Brouillette & Lykoudis 1967). To describe thestructure of turbulence inside the Hartmann layer, Alboussiere & Lingwood (2000)advocate that the relevant interaction parameter N is small compared to unity, whichimplies that the turbulence present in the Hartmann layer is of the same nature asthe one encountered in hydrodynamics. In particular, these authors conclude that theturbulent Hartmann layer consists of a viscous layer and a logarithmic layer. Usingthis assumption, Alboussiere & Lingwood (2000) proposed a model for the shearstress based on Prandtl’s mixing length idea and showed that the friction coefficientcomputed from this model compares well with the available experimental data. Thehigh-resolution DNS of a turbulent MHD channel flow performed by Boeck et al.

Figure 4(a) Instantaneous streamwise velocity profile in a plane x = const for a magnetohydrodynamicchannel flow at R = 500 and Ha = 30 (the walls are located at z ± 1). Note that theHartmann layers are clearly turbulent, and the Hartmann number is high enough to produce aquasi-two-dimensional plateau in the core flow. (b) Semilogarithmic plots (in wall units) ofmean velocity profiles for R = 500 and different values of the Hartmann number. Regions I,II, and III correspond to the viscous, logarithmic, and plateau layers, respectively. Note howthe extent of the plateau region increases with increasing Hartmann numbers. Similarbehaviors are observed at different values of R. Figure taken from Boeck et al. 2007.

(2007) largely confirms this picture. These authors further show that, depending onthe strength of the magnetic field, a plateau region can also form in the core of flow,completely in line with the idea of a Q2D core discussed above. To illustrate this,Figure 4 displays some results obtained by Boeck et al. (2007). Finally, we note thatall the above works again indicate that the main control parameter of the turbulencein the Hartmann layer is R = Re/Ha .

4. MODELING OF MAGNETOHYDRODYNAMICTURBULENT SHEAR FLOWS

Most flows encountered in practical applications are much more complex than thosediscussed above. They are present in electromagnetic pumps or flowmeters, in somemetallurgical applications (e.g., the brakes in continuous steel casting), or in theblankets of future fusion reactors. These complex flows still obey the general prin-ciples discussed above, but they are also submitted to other mechanisms and mayexhibit spectacular phenomena. Excluding multiphysics problems (such as solidifica-tion) and time-dependent magnetic fields (skin effect), complexity can arise becauseof nonuniformity, either in the magnetic field (e.g., entry into or exit from a magnet)or in the flow geometry (e.g., change in the duct geometry). Most of these complexflows exhibit an initial length at which they are 3D and their properties are gov-erned by a family of characteristic surfaces (Holroyd & Walker 1978, Kulikovskii1968a,b, Lavrentiev et al. 1990, Walker et al. 1972), which lead to the formation of

high-velocity jets along the walls parallel to the magnetic field. Then the flow is againQ2D, and its dynamics is controlled by the deep coupling between the feeding ofsome Q2D turbulence by the instability of the jets and the turbulent transport ofmomentum, which tends to reduce the mean velocity gradients, often modeled by aneddy viscosity. A good turbulence model for such complex flows should incorporatethis nonlinear coupling, together with both the fast Joule damping, relevant in theinitial 3D zone, and the Hartmann damping, predominant in the downstream Q2Dregion.

Recent attempts aimed at developing RANS techniques for complex MHD flowsare mainly directed toward adapting (k, ε)-type models to include the effect of themagnetic field. At this stage, it appears that none of these efforts is successful inincorporating completely the complex nature of the magnetic damping discussedabove. For instance, Ji & Gardner (1997), Kenjeres & Hanjalic (2000), and Smolentsevet al. (2002) model the effect of the magnetic field through additional damping termsadded to the equations for k and ε. These terms, however, suffer from two defects:(a) They overestimate Joule damping because they act on a time scale τJ0 withouttaking into account the reduction of damping resulting from the growth of anisotropy,and (b) they do not incorporate Hartmann damping in regions of space in which Jouledamping is not relevant. As a result, turbulence is damped quickly after a time of theorder τJ0, whereas experiments (Branover 1979, Lielausis 1975, Messadek & Moreau2002, Tsinober 1975) demonstrate that turbulence persists for much longer times [seealso the recent review by Moreau et al. (2007), which emphasizes the experimentalresults]. Widlund et al. (1998) propose a model, usually named (k, ε, α), that includesa third transport equation for the scalar quantity α ∼ l‖/ l⊥. The advantage of thismodel is that it adjusts the Joule damping time to the actual anisotropy by scalingthe (k, ε) MHD corrections with α. A similar feature is achieved in the interactingparticle representation model and the one-point R-D model in which Joule dissipationis scaled with the dimensionality stress tensor that also provides a measure of l‖/ l⊥(Kassinos & Reynolds 1999). These more elaborate proposals, however, still have thedrawback of completely ignoring the Hartmann damping.

We conclude this section by discussing a recent novelty in the development ofRANS models for MHD flows, suggested by Smolentsev & Moreau (2006, 2007).Focusing on high magnetic fields and the need to model the persisting high level ofQ2D turbulence in actual flows, these authors follow the ideas leading to Equation22 and propose the following equation for the kinetic energy k:

∂tk + u⊥ j ∂ j k = νt(∂ j u⊥i )2 + ∂ j [(ν + νt/σk)∂ j k] − k/τH. (23)

Here, the usual viscous dissipation ε has disappeared because the flux of energy towardsmall scales is zero in the case of strong magnetic fields. For that reason, and becauseonly Hartmann braking is present, we can tentatively refer to this model as a (k, εHa )model. The main advantage of Equation 23 is that the only unknown function neededto close the system is then νt . The only time scale available being τH , Smolentsev &Moreau (2006, 2007) propose to write the eddy viscosity as νt = CνkτH . Theseauthors show that the predicted velocity profiles agree fairly well with the resultsfrom the MATUR experiment with the parameters set to Cν = 0.03 and σk = 1. This

101 102 103 104

≈ 300

(k, Ha)

+ Hartmann effect(k, )

Figure 5Illustration of the threetypical regimes in an (Re,Ha) graph, suggesting themost appropriate RANStechnique in eachsubdomain.

model, however, is not appropriate for flows in which the 3D Joule damping regionis significant.

Figure 5 provides, in an (Re, Ha) diagram, a rough sketch of the domain of expectedvalidity of the different models discussed above.

5. CONCLUDING REMARKS AND PERSPECTIVES

In this review, we emphasize what we consider to be the leading ideas that provide abasic understanding of the specific influence of a magnetic field on MHD turbulence.We have selected the flow conditions that seem to be the simplest, have enoughuniversality, and have governing principles that may be expressed in rather simpleheuristic terms. As a consequence, we disregard many other situations that we considermore complex and for which the published results seem less easy to interpret and lackthe same generality. Therefore, this article does not duplicate previous reviews, such asLielausis (1975), Tsinober (1975), Branover (1979), Tsinober (1990), and Moreau et al.(2007), which present and discuss in more detail a large number of other interestingresults.

Before this decade, the knowledge and phenomenology of MHD turbulence wereessentially based on the available experimental results. However, similar to the trendobserved in hydrodynamic turbulence, numerical simulation has emerged as a leadingtool to get more precise diagnostics and more reliable ideas on the predominantmechanisms. For this reason, this review also discusses the recent attempts to useDNS, LES, or RANS in the context of MHD. We hope and believe that thesetechniques will soon become more adapted to the specific difficulties of MHD andwill therefore yield a much deeper understanding.

The simplest flow configuration discussed in this review consists of homoge-neous turbulence, for which a strong anisotropy can occur under the presence of a

uniform magnetic field. Anisotropy is also encountered in rotating or stratified fluids,in which an external force field with a privileged direction acts on the turbulence.The MHD problem, however, is quite different because the Lorentz force induc-ing the anisotropy provides extra dissipation at all scales, independent of viscosity.From the turbulence viewpoint, the main issue in MHD homogeneous turbulence isthe presence of angular energy transfers, which tend to feed the wave vectors locatedwithin the Joule cone. The properties of these transfers are far from being as clearas those of the radial energy transfers toward viscous dissipation present in hydrody-namic turbulence. Despite some preliminary results suggesting they also have somelocality, further work is needed for this property to be checked and confirmed. Thisis an important point as it might result in a specific LES formulation for MHD, inwhich the filtered wave vectors are those located within the Joule cone and those thathave lost their initial energy.

In all MHD problems, the walls perpendicular to the magnetic field, usually namedthe Hartmann walls, have a tremendous role. We discuss above the possibility of Q2Dturbulence having spectacular properties and the typical shape of the Q2D eddies,illustrated in Figure 3, which seem to have a universal structure. The inverse energycascade can be significantly affected by the Hartmann damping, capable of changingthe spectral law from k−5/3 to k−3. Another striking feature is the level of turbulencethat remains surprisingly high, even for strong magnetic fields. This results from thefact that both Joule damping and viscous effects disappear from the core because oftwo-dimensionality, whereas Hartmann damping and dissipation are localized in theHartmann layers.

However, these asymptotic features, which provide some apparent simplicity, areonly valid when the applied magnetic field is very strong. And a lot remains to beunderstood in the vicinity of the limit Re/Ha ≈ 300, when the Hartmann layeritself becomes unstable. The case of complex turbulent shear flows, which is themost important from the point of view of applications, raises still more questions, inparticular regarding the role of the Shercliff layers located along the walls parallel tothe magnetic field.

Yet there are many other questions, not addressed here, that deserve particularattention. One is the case of a nonuniform magnetic field, whose main property seemsto be the existence of a family of characteristic surfaces along which the streamlinesand the electric current lines must be lying. Recent experiments (Andreev et al. 2006)suggest that turbulence is laminated and suppressed by those surfaces but restartsin the uniform field domain in which it is fed by the instability of the spreading jetsgenerated in the nonuniform field region. Ducts of variable cross section submitted toa uniform field might have similar properties, owing to the analogy between the twoproblems (Mueller & Buehler 2001). Finally, the regimes outside the limit Rm � 1also deserve attention because they raise an interesting question about the transferof a solenoidal vector field (the induced magnetic field �b) by the turbulence. Thisquestion, in a range still far from that of the dynamo action, may also be relevantto large-scale applications such as aluminum reduction cells or the blanket of futurenuclear fusion reactors.

DISCLOSURE STATEMENT

The authors are not aware of any biases that might be perceived as affecting theobjectivity of this review.

ACKNOWLEDGMENTS

The authors are grateful to Boeck et al. (2007) for permission to use the plots con-tained in Figure 4. This review, conducted as part of the award (Modelling andsimulation of turbulent conductive flows in the limit of low magnetic Reynolds num-ber) made under the European Heads of Research Councils and European ScienceFoundation EURYI (European Young Investigator) Awards scheme, was supportedby funds from the Participating Organisations of EURYI and the EC Sixth Frame-work Program. The content of the publication is the sole responsibility of the authorsand it does not necessarily represent the views of the Commission or its services. Thesupport of FRS-FNRS Belgium is also gratefully acknowledged.

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AR332-FM ARI 22 November 2007 21:57

Annual Review ofFluid Mechanics

Volume 40, 2008Contents

Flows of Dense Granular MediaYoël Forterre and Olivier Pouliquen � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 1

Magnetohydrodynamic Turbulence at Low Magnetic ReynoldsNumberBernard Knaepen and René Moreau � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 25

Numerical Simulation of Dense Gas-Solid Fluidized Beds:A Multiscale Modeling StrategyM.A. van der Hoef, M. van Sint Annaland, N.G. Deen, and J.A.M. Kuipers � � � � � � � 47

Tsunami SimulationsGalen R. Gisler � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 71

Sea Ice RheologyDaniel L. Feltham � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 91

Control of Flow Over a Bluff BodyHaecheon Choi, Woo-Pyung Jeon, and Jinsung Kim � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �113

Effects of Wind on PlantsEmmanuel de Langre � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �141

Density Stratification, Turbulence, but How Much Mixing?G.N. Ivey, K.B. Winters, and J.R. Koseff � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �169

Horizontal ConvectionGraham O. Hughes and Ross W. Griffiths � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �185

Some Applications of Magnetic Resonance Imaging in FluidMechanics: Complex Flows and Complex FluidsDaniel Bonn, Stephane Rodts, Maarten Groenink, Salima Rafaï,Noushine Shahidzadeh-Bonn, and Philippe Coussot � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �209

Mechanics and Prediction of Turbulent Drag Reduction withPolymer AdditivesChristopher M. White and M. Godfrey Mungal � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �235

High-Speed Imaging of Drops and BubblesS.T. Thoroddsen, T.G. Etoh, and K. Takehara � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �257

AR332-FM ARI 22 November 2007 21:57

Oceanic Rogue WavesKristian Dysthe, Harald E. Krogstad, and Peter Müller � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �287

Transport and Deposition of Particles in Turbulent and Laminar FlowAbhijit Guha � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �311

Modeling Primary AtomizationMikhael Gorokhovski and Marcus Herrmann � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �343

Blood Flow in End-to-Side AnastomosesFrancis Loth, Paul F. Fischer, and Hisham S. Bassiouny � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �367

Applications of Acoustics and Cavitation to Noninvasive Therapy andDrug DeliveryConstantin C. Coussios and Ronald A. Roy � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �395

Indexes

Subject Index � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �421

Cumulative Index of Contributing Authors, Volumes 1–40 � � � � � � � � � � � � � � � � � � � � � � � � � �431

Cumulative Index of Chapter Titles, Volumes 1–40 � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �439

Errata

An online log of corrections to Annual Review of Fluid Mechanics articles may befound at http://fluid.annualreviews.org/errata.shtml

vi Contents

Magnetohydrodynamic Turbulence at Low Magnetic Reynolds Number

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