Linear stability of magnetohydrodynamic flow in a perfectly conducting rectangular duct
aa r X i v : . [ phy s i c s . f l u - dyn ] J un Under consideration for publication in J. Fluid Mech. Linear stability of magnetohydrodynamic flowin a perfectly conducting rectangular duct
By J ¯A N I S P R I E D E, S V E T L A N A A L E K S A N D R O V Aand S E R G E I M O L O K O V (Received 25 October 2018)
We analyse numerically the linear stability of a liquid metal flow in a rectangular ductwith perfectly electrically conducting walls subject to a uniform transverse magneticfield. A non-standard three dimensional vector stream function/vorticity formulationis used with Chebyshev collocation method to solve the eigenvalue problem for small-amplitude perturbations. A relatively weak magnetic field is found to render the flowlinearly unstable as two weak jets appear close to the centre of the duct at the Hartmannnumber Ha ≈ . . In a sufficiently strong magnetic field, the instability following thejets becomes confined in the layers of characteristic thickness δ ∼ Ha − / located atthe walls parallel to the magnetic field. In this case the instability is determined by δ, which results in both the critical Reynolds and wavenumbers numbers scaling as ∼ δ − . Instability modes can have one of the four different symmetry combinations along andacross the magnetic field. The most unstable is a pair of modes with an even distributionof vorticity along the magnetic field. These two modes represent strongly non-uniformvortices aligned with the magnetic field, which rotate either in the same or oppositesenses across the magnetic field. The former enhance while the latter weaken one anotherprovided that the magnetic field is not too strong or the walls parallel to the field arenot too far apart. In a strong magnetic field, when the vortices at the opposite walls arewell separated by the core flow, the critical Reynolds and wavenumbers for both of theseinstability modes are the same: Re c ≈ Ha / + 8 . × Ha − / and k c ≈ . Ha / . The other pair of modes, which differs from the previous one by an odd distribution ofvorticity along the magnetic field, is more stable with approximately four times highercritical Reynolds number.
1. Introduction
Understanding instabilities in magnetohydrodynamic (MHD) flows in ducts is of greatimportance for liquid metal flows in blankets for fusion reactors (Bühler 2007). Blanketsconsist of rectangular ducts in which the liquid metal flows in a high, transverse magneticfield of between 5 and 10 T. The aim of these devices is to cool plasma chamber and tobreed and to remove tritium. This can be assisted by mixing of the flow by turbulence ifit can be sustained in the presence of a magnetic field. A high magnetic field is known todamp turbulence by means of the Lorentz force. At the same time, the magnetic field canalso affect the base velocity profile in such a way as to create inflection lines (Kakutani1964) and even jets (Hunt 1965) thus making the flow more unstable. These two com-peting effects balancing each other on a certain length scale result in relatively simpleasymptotics for the instability threshold. The most dangerous perturbations are usuallyassociated with the largest length scale on which the magnetic damping becomes compa-rable with the viscous one. This happens in the so-called parallel layers with the relative
J. Priede, S. Aleksandrova and S. Molokov thickness ∼ Ha − / (Braginskii 1960). Linear stability of these layers for a duct withinsulating walls (Shercliff 1953) has been considered in a quasi-two-dimensional approxi-mation by Pothérat (2007), who found the critical Reynolds number Re c ≈ . × Ha / . It is an extremely high, however, typical value for the linear stability of exponential veloc-ity profiles, which have the critical Reynolds number around fifty thousand based on theboundary layer thickness (Drazin & Reid 1981). This high threshold is of little practicalrelevance because the instability for exponential velocity profile is known to be subcriti-cal (Hocking 1975). This is the case also for the stability of Hartmann layer (Lock 1955),which is subcritical too (Lifshits & Shtern 1979; Moresco & Alboussière 2003) with theexperimentally found Reynolds number for the onset of turbulence in straight and an-nular ducts of rectangular cross-sections being respectively around Ha (Murgatroyd1953; Brouillette & Lykoudis 1967) and Ha (Moresco & Alboussière 2004). Marginalturbulent flow states have been observed by Shatrov & Gerbeth (2010) significantly be-low the linear stability threshold in numerical simulations of insulating duct flow subjectto a transverse magnetic field of moderate strength.The stability of MHD flows strongly varies with the electrical conductivity of the ductwalls. For example, Hunt’s flow, which develops in a rectangular duct when the walls per-pendicular to the magnetic field are perfectly conducting while the parallel ones are insu-lating, has a relatively low linear stability threshold Re c ≈ Ha / and ¯ Re c ≈ basedon the maximum and average velocities, respectively (Priede, Aleksandrova & Molokov2010). The low stability of Hunt’s flow is due to two strong jets, which develop in asufficiently strong magnetic field along the insulating walls and attain a velocity ∼ Ha relative to that of the core flow (Hunt 1965). Although the relative velocity of jets reducesas ∼ Ha / /c with the increase of the wall conductance ratio c & Ha − / (Walker 1981),weak jets with the relative velocity O (1) still persist at the parallel walls also in the limitof perfectly conducting duct (Uflyand 1961; Chang & Lundgren 1961). The presence ofjets with inherent inflection points suggests that this flow may also be highly unstablesimilar to Hunt’s flow. It is the aim of the present study to investigate linear stabilityof this flow, which is the last basic MHD duct flow configuration whose linear stabilitymay be not only of theoretical but also of experimental relevance. We first investigatethe case of square duct and find the high-field asymptotics of the instability thresholdwhich are then generalized to arbitrary aspect ratios.The paper is organised as follows. The problem is formulated in § § §
2. Formulation of the problem
Consider a flow of an incompressible viscous electrically conducting liquid with density ρ, kinematic viscosity ν and electrical conductivity σ driven by a constant gradient ofpressure p applied along a straight duct of rectangular cross-section with half-width d and half-height h subject to a transverse homogeneous magnetic field B . The walls ofthe duct are assumed to be perfectly electrically conducting and the field may be appliedacross either the width or the height of the duct.The velocity distribution of the flow is governed by the Navier-Stokes equation ∂ t v + ( v · ∇ ) v = − ρ − ∇ p + ν ∇ v + ρ − f , (2.1) inear stability of MHD flow in a perfectly conducting rectangular duct xyz dh − w ( x,y ) B Figure 1.
The base flow profile in a rectangular duct with perfectly conducting walls subjectto a strong vertical magnetic field for Ha = 100 . where f = j × B is the electromagnetic body force involving the induced electric current j , which is governed by the Ohm’s law for a moving medium j = σ ( E + v × B ) . (2.2)The flow is assumed to be sufficiently slow so that the induced magnetic field is neg-ligible relative to the imposed one, which supposes the magnetic Reynolds number Rm = µ σv d ≪ , where µ is the permeability of vacuum and v is the character-istic velocity of the flow. In addition, we assume that the characteristic time of velocityvariation is much longer than the magnetic diffusion time τ m = µ σd , which allows usto use the quasi-stationary approximation leading to E = − ∇ φ, where φ is the electro-static potential (Roberts 1967). The velocity and current satisfy the mass and chargeconservation ∇ · v = ∇ · j = 0 . Applying the latter to the Ohm’s law (2.2) yields ∇ φ = B · ω , (2.3)where ω = ∇ × v is vorticity. At the duct walls S , the normal ( n ) and tangential ( τ ) velocity components satisfy the impermeability and no-slip boundary conditions v n | s = 0 and v τ | s = 0 . As the walls are perfectly conducting, the tangential electric currentvanishes and Ohm’s law (2.2) yields φ | s = const . We employ the Cartesian coordinates with the origin set at the centre of the duct, x , y and z axes directed along its width, height and length, respectively, as shown in figure1, and the velocity defined as v = ( u, v, w ) . The problem admits a purely rectilinear baseflow with a single velocity component along the duct ¯ v = (0 , , ¯ w ( x, y )) which is shownin figure 1(a) for a strong vertical magnetic field.In the following, all variables are non-dimensionalised by using the maximum velocity ¯ w and the half-width of the duct d as the velocity and length scales, while the time,pressure, magnetic field and electrostatic potential are scaled by d /ν, ρ ¯ w , B = | B | and ¯ w dB, respectively. The dimensionless paramerters defining the problem are the Reynoldsnumber Re = ¯ w d/ν, the Hartmann number Ha = dB p σ/ ( ρν ) and the aspect ratio A = h/d. Note that we use the maximum rather than average velocity as the characteristicscale because the stability of this flow, as shown in the following, is determined by theformer.Linear stability of this flow is analysed using the same method as in our previous study(Priede, Aleksandrova & Molokov 2010). Since the method is based on a non-standardvector stream function formulation, it is briefly outlined in the Appendix.
J. Priede, S. Aleksandrova and S. Molokov -1 -0.5 0 0.5 1 -1-0.5 0 0.5 1 − w − b Ha = 10 Ha = 100 x y ( a ) − w ( x , ) x Ha = 01030100asymptotic: − w (0,0) ~ 0.809 ( b ) Figure 2.
Isolines of the base flow ( y > and electric current lines ( y < for Ha = 10 ( x < and Ha = 100 ( x > shown in the respective quadrants of duct cross-section (a) and the baseflow velocity profiles at y = 0 for Ha = 0 , , , (b) in a vertical magnetic field. The problem was solved by a spectral collocation method on a Chebyshev-Lobatto gridwith even number of points defined by N x + 2 and N y + 2 for the x - and y -directions,where N x,y = 35 · · · were used for various combinations of the control parametersto achieve accuracy of at least three significant figures. Owing to the double reflectionsymmetry of the base flow with respect to x = 0 and y = 0 planes, small-amplitude per-turbations with different parities in x and y decouple from each other. This results in fourmutually independent modes, which we classify as ( o, o ) , ( o, e ) , ( e, o ) , and ( e, e ) accordingto whether the x and y symmetry of ˆ ψ x is odd or even, respectively. Our classificationof modes corresponds to the symmetries I, II, III, and IV used by Tatsumi & Yoshimura(1990) and Uhlmann & Nagata (2006). Thus, four independent problems of differentsymmetries are obtained in one quadrant of the duct cross-section with N x × N y internalcollocation points. The size of matrix for each eigenvalue problem is reduced by a factorof 16 in comparison to the original problem. Further details and the validation of thenumerical method can be found in our previous paper (Priede, Aleksandrova & Molokov2010).
3. Results
Flow in the presence of a vertical magnetic field induces a transverse current in thebulk of the duct which, as seen in the lower part of figure 2(a), almost directly connectsto the perfectly conducting walls parallel to the magnetic field. However, a small partof the current diverts in the corner regions to connect through the Hartmann wallsperpendicular to the magnetic field. This makes the density of the transverse currentand so the resulting electromagnetic force, which opposes the constant driving pressuregradient, slightly lower at the parallel walls than in the core region. As a result, weakjets form along the parallel walls, where the flow becomes slightly faster than in the coreof the duct. As seen in figure 2(a), the formation of jets starts with a velocity minimumappearing in the centre of the duct at Ha ≈ . With the increase of the magnetic field,the velocity in the core becomes almost uniform, while the jets become confined in thinlayers that develop along the walls parallel to the magnetic field and have a thicknessdecreasing as ∼ Ha − / . In a strong vertical magnetic field, asymptotic solution by Hunt(1965) yields the velocity maxima located in the mid-plane of the duct at the distance δ ≈ . A/ Ha ) / (3.1) inear stability of MHD flow in a perfectly conducting rectangular duct . . The latter is seen in figure 2(b) to agree well with our numerical solution.An interesting feature of these jets is that the extra flow rate associated with the velocityover-shoot above the core velocity balances the flow rate deficit at the wall, where thevelocity falls below that of the core (Williams 1962). Thus, the relative contribution ofthese jets to the flow rate is not ∼ Ha − / , as one would expect from simple scalingarguments, but a higher-order small quantity ∼ Ha − / , which is less than the flow ratecorrection due to the Hartmann layers ∼ Ha − . It is confirmed also by our numericalsolution, which yields the best fit of the flow rate for one quarter of a square duct Q ≈ . − . Ha − , (3.2)where the leading-order contribution due to the core flow matches the asymptotic solu-tion. Note that although the correction is ∼ Ha − , its coefficient is not equal to that inthe asymptotic solution by Hunt (1965). The difference is because the asymptotic solu-tion is obtained for a fixed pressure gradient, whereas numerical solution is for a fixedmaximum velocity, which has a O ( Ha − ) higher-order correction. Thus, the maximumvelocity taken as the reference one in this study, results in the same order correction inthe core velocity and, thus, also in the flow rate. Our choice of the maximum velocity asthe reference one is motivated by the following results, which show that the instability inthis flow is associated with the jets at the parallel walls, which seem inherently unstabledue to the inflection points in their velocity profiles. Our results can be rescaled to theaverage velocity using relation (3.2) which becomes sufficiently accurate for Ha & . We start with a square duct, in which the flow without the magnetic field is linearlystable (Tatsumi & Yoshimura 1990). The magnetic field renders the base flow linearlyunstable at Ha & . with respect to a perturbation of symmetry type II. This perturba-tion is characterised by the vorticity component along the magnetic field being an oddfunction in the field direction and an even function spanwise. As shown in the following,the anti-symmetric distribution along the field results in a strong damping when the fieldstrength is increased. The marginal Reynolds number at which the maximum growth ratefor this mode turns zero ( ℜ [ λ ] = 0) is plotted in figure 3(a) against the wavenumber forvarious Hartmann numbers. Besides the marginal Reynolds number, neutrally stable per-turbations are characterised by their oscillation frequency ω = ℑ [ λ ] and the associatedphase velocity − ω/k. It is useful to consider the latter relative to the characteristic baseflow velocity given by Re . This quantity defined as − ω/ ( Re k ) is subsequently referred toas the relative phase velocity and shown in figure 3(b) for mode II. Instability appearsabove the critical Reynolds number Re c , which is defined by the global minimum on theneutral stability curve for the respective Hartmann number. With the increase of Ha , the critical Reynolds number for mode II in figure 3(a) first quickly drops to a minimum Re c ≈ . × at Ha ≈ and then starts to increase. In some ranges of the Hartmannnumber another minimum appears on the neutral stability curve, which may cause thecritical mode to jump from the first to the second minimum as the latter becomes theglobal one. This switchover between global minima shows up as a jump in both the crit-ical wavenumber and frequency, and as a break point in the dependence of the criticalReynolds number on the Hartmann number. Such a jump is noticeable in figure 5 at Ha ≈ , where the instability switches from mode IIa to IIb.However, this jump is of secondary importance because a mode of type I is seen infigure 5(a) to become more unstable than mode II at Ha ≈ . . Although mode Iturns linearly unstable at a slightly higher Hartmann number than mode II, its criticalReynolds number decreases faster with the increase in the Hartmann number than that
J. Priede, S. Aleksandrova and S. Molokov M a r g i n a l R e yno l d s nu m b e r , R e Wavenumber, k IIa IIb Ha = 111215202530100critical points ( a ) R e l a ti v e ph a s e v e l o c it y , − ω / ( R e k ) Wavenumber, k IIa IIb Ha = 111215202530100critical points ( b ) Figure 3.
The marginal Reynolds number (a) and the relative phase velocity (b) versus thewavenumber for neutrally stable modes of type II in a square duct ( A = 1) subject to a verticalmagnetic field at various Hartmann numbers. for mode II. As seen from the neutral stability curves in figure 4(a,b), with the increaseof the Hartmann number, the critical Reynolds number for mode I first quickly dropsto to a minimum Re c ≈ at Ha ≈ and then starts to raise. With the increaseof Ha mode I is quickly approached from above by a mode of type III, which is seenin figure 5(a) to appear at Ha ≈ and become practically indistinguishable from theformer at Ha & . There is also a mode of type IV appearing at Ha ≈ , which inturn approaches mode II in a similar way as mode III approaches mode I. Modes III/IVdiffer from modes I/II by the opposite symmetry across the magnetic field. Namely, formodes III/IV, the vorticity component along the magnetic field is an odd function in thespanwise direction across the magnetic field, whereas it is an even function for modesI/II.The most important feature of the instability seen in figure 5( a,b ) is the criticalReynolds number and the wavenumber for each of two merged pairs of modes increas-ing in strong magnetic field as ∼ Ha / . The relative phase velocity shown in figure5 tends to a constant ∼ . for both pairs of modes. This kind of variation impliesthat in a strong magnetic field the instability is determined by the internal length scale δ ∼ Ha − / , which is the characteristic thickness of the jets developing along the wallsparallel to the magnetic field. The base flow velocity correction of order ∼ Ha − , whichwas discussed above, implies an O ( Ha − / ) correction to the critical Reynolds number.The best fit for modes I/III yields Re c ( Ha ; A = 1) ≈ Ha / + 8 . × Ha − / , (3.3) k c ( Ha ; A = 1) ≈ . Ha / , (3.4)which are seen in figure 5( a,b ) to well approximate numerical results for mode I down to Ha ≈ . Similarly, for modes II/IV, we find Re c ( Ha ; A = 1) ≈ Ha / + 1 . × Ha − / , (3.5) k c ( Ha ; A = 1) ≈ . Ha / , (3.6)where the former is by nearly of a factor of four greater than (3.3).The instability threshold being nearly the same for the modes of opposite spanwisesymmetry in a sufficiently strong magnetic field implies that the perturbations developingin the jets at the opposite walls are effectively separated by the core region of the flow inear stability of MHD flow in a perfectly conducting rectangular duct M a r g i n a l R e yno l d s nu m b e r , R e Wavenumber, k Ha = 20405070100critical points ( a ) R e l a ti v e ph a s e v e l o c it y , − ω / ( R e k ) Wavenumber, k Ha = 20405070100critical points ( b ) M a r g i n a l R e yno l d s nu m b e r , R e Wavenumber, k Ha = 405070100critical points ( c ) R e l a ti v e ph a s e v e l o c it y , − ω / ( R e k ) Wavenumber, k Ha = 405070100critical points ( d ) Figure 4.
The marginal Reynolds number ( a,c ) and the relative phase velocity ( b,d ) versusthe wavenumber for neutrally stable modes of type I ( a,b ) and type III ( c,d ). and, thus, do not affect each other. This is confirmed by the patterns of the criticalperturbations, which are plotted over the duct cross-section in figure 6 for a moderate ( Ha = 15) and a relatively strong ( Ha = 100) magnetic field. As shown in our previouspaper (Priede, Aleksandrova & Molokov 2010), the flow perturbation can be representedby the complex amplitudes of the streamwise ( z ) component of velocity ( ˆ w ) and that ofstream function ( ˆ ψ z ) , whose isolines are plotted in the left ( x < and the right ( x > sides of the cross-section, respectively. The real and imaginary parts of perturbationsplotted at the top and bottom halves of the cross-section show the instant patterns shiftedin time or in the stream-wise direction by a quarter of period or wavelength, respectively.Although the real and imaginary parts of the complex amplitude distributions completelydetermine the evolution of perturbation over the harmonic oscillation cycle, these twoquantities are not uniquely defined. The main ambiguity is due to the free choice of theinitial time instant and the initial stream-wise coordinate. This ambiguity can be partlyeliminated by choosing the phase of the complex velocity perturbation amplitude so that Z S ℜ [ ˆv ] ds = Z S ℑ [ ˆv ] ds, where the integrals are taken over the duct cross-section S. This condition definines thephase up ± π/ , which means swaping the real and imaginary parts. The perturbationamplitude remains defined up to a constant factor which is not important as the linearstability theory predicts only the pattern but not the amplitude of the critical perturba-tions. J. Priede, S. Aleksandrova and S. Molokov C r iti ca l R e yno l d s nu m b e r , R e c Hartmann number, Ha IIa IIb Re c ~ 6.42 × Ha + 8.9 × Ha -1/2 Re c ~ 2.58 × Ha + 1.1 × Ha -1/2 mode IIIIIIIVasymptotic ( a ) C r iti ca l w a v e nu m b e r , k c Hartmann number, Ha IIa IIb k c ~ 0.477 Ha k c ~ 0.419 Ha mode IIIIIIIVasymptotic ( b ) R e l a ti v e ph a s e v e l o c it y , − ω c / ( R e c k c ) Hartmann number, Ha IIa c c ~ 0.911 mode IIIIIIIVasymptotic ( c ) Figure 5.
The critical Reynolds number ( a ), wavenumber ( b ) and relative phase velocity ( c )against the Hartmann number. Positive and negative values of ˆ w are respectively associated with converging and di-verging potential flow component in the cross-section plane. The isolines of ˆ ψ z correspondto the streamlines of the solenoidal flow component in that plane. The critical perturba-tions for modes I and II, which are shown in figures 6( a,b ) and ( d,e ), respectively, differby their vertical symmetry. Namely, the perturbation of ˆ w and ˆ ψ z are respectively evenand odd functions of y for mode I, whereas they are odd and even functions for mode II.Thus, the vortices for mode I rotate in opposite senses in the upper and lower parts ofthe cross-section, whereas for mode II there is one symmetric vortex spanning the wholeheight of the duct. For both of these modes, the pairs of vortices across the vertical mid-plane rotate in the same sense and, thus, represent two parts of a bigger vortex spanningover the whole width of the duct. At Ha = 15 , slightly above the Hartmann numberat which the flow turns linearly unstable, the critical perturbations are seen in figure6( a,d ) to be localised close to the duct centre, where the two velocity maxima discussedabove first appear. In this case, the co-rotating vortices on the opposite sides of the duct,whose symmetric half is shown at x > , are clearly connected by the flow through thevertical mid-plane. However, this is no longer the case for a sufficiently strong magneticfield. As seen in figure in 6( b,e ), at Ha = 100 the critical perturbations for modes I andII are localised at the side walls. Moreover, these perturbations have nearly the samepattern as those for modes III and IV, which are shown in figure 6( c,f ) for the same Ha . It means that in a strong enough magnetic field the perturbations at the oppositewalls are effectively separated by a stagnant core flow and, thus, do not affect each other. inear stability of MHD flow in a perfectly conducting rectangular duct -1 -0.5 0 0.5 1 -1-0.5 0 0.5 1 ψ z w z ℜℑ x y ( a ) -1 -0.5 0 0.5 1 -1-0.5 0 0.5 1 ψ z w z ℜℑ x y ( b ) -1 -0.5 0 0.5 1 -1-0.5 0 0.5 1 ψ z w z ℜℑ x y ( c ) -1 -0.5 0 0.5 1 -1-0.5 0 0.5 1 ψ z w z ℜℑ x y ( d ) -1 -0.5 0 0.5 1 -1-0.5 0 0.5 1 ψ z w z ℜℑ x y ( e ) -1 -0.5 0 0.5 1 -1-0.5 0 0.5 1 ψ z w z ℜℑ x y ( f ) Figure 6.
Amplitude distributions of the real ( y > and imaginary ( y < parts of ˆ w ( x < and ˆ ψ z ( x > of the critical perturbations over one quadrant of duct cross-section for instabilitymodes I ( a,b ), IIa ( d,e ), III( c ) and IV( f ) at Ha = 15 ( a,d ) and Ha = 100 ( b,c,e,f ). This explains the merging of the instability thresholds for the modes with the oppositespanwise symmetries seen in figure 5.In weaker magnetic fields or in narrower ducts across the magnetic field, which will beconsidered later, perturbations at the opposite walls can either enhance or suppress eachother depending on their spanwise symmetry. The first is the case for the perturbationsof type I/II, which, as discussed above, have co-rotating vortices at the opposite wallsconnected by a flow across the vertical mid-plane. These perturbations are more unstablethan those of type II/IV with counter-rotating vortices at the opposite walls, which tendto suppress each other, especially in moderate magnetic fields or in sufficiently narrowducts.Besides the spatial amplitude distributions perturbations can be characterised quan-titatively by their kinetic energy distribution, which can be represented in two differentforms using either the velocity or vorticity/stream function components E ∝ Z S ˆ | v | ds = Z S ℜ [ ˆ ω · ˆ ψ ∗ ] ds, where E is the kinetic energy of perturbation averaged over the wavelength and theasterisk denotes the complex conjugate (Priede, Aleksandrova & Molokov 2010). At themoderate Hartmann number Ha = 15 considered above, most of the kinetic energy, i.e. and for mode I and and for mode II, is carried by the z -componentof velocity and by the y -component of vorticity, respectively. For mode I, next mostenergetic is the x -component of both velocity and vorticity, which contain respectivelyabout and of the energy. For mode II, in this position are the y velocity and the x vorticity components, which contain respectively about and of the energy.The energy distribution for the most unstable modes I and II becomes simpler in astrong magnetic field. For example, at Ha = 100 , of the energy for these modes is0 J. Priede, S. Aleksandrova and S. Molokov ( a ) ( b ) Figure 7.
Isosurfaces of ˆ ψ y (a) and longitudinal velocity ˆ w (b) perturbations over wavelength inone quadrant of the duct cross-section for instability mode I at Ha = 100 in a vertical magneticfield. concentrated in the y -component of vorticity, while the rest is distributed nearly equallybetween the two other vorticity components. This component of vorticity is associatedwith the circulation in the ( x, z ) -planes transverse to the magnetic field. The streamlinesof this circulation are represented by the isolines of ψ y , whose spatial distribution isshown in figure 7( a ). Note that the distribution of ψ y is very close to that of the electricpotential φ because the equations (A 6) and (A 5) governing both quantities are thesame. Moreover, ψ y and φ satisfy the same boundary condition at the wall parallel tothe magnetic field. Thus, both quantities differ only in the vicinity of the wall normal tothe magnetic field, where they have different boundary conditions.The transverse character of circulation for modes I and III is confirmed also by thekinetic energy distribution over the velocity components. Only about of the energy iscarried by the velocity component along the magnetic field, while by the streamwise ( z ) velocity perturbation, whose spatial distribution is shown in 7(b). The relatively lowcontribution of the spanwise ( x ) velocity component, which carries the remaining ofthe energy, is due to the relatively long wavelength of perturbation λ c = 2 π/k c , whichaccording to (3.4) is by approximately a factor of 7 larger than the thickness of the jet(3.1).In a strong magnetic field, the energy distribution in modes II/IV is essentially differ-ent from that in modes I/II considered above. Namely, the latter two have only − of their energy in the spanwise ( x ) velocity component, which implies a circulation con-strained mainly to the ( y, z ) -planes parallel to the side walls. Similar to the previous twomodes, of the energy is carried by the streamwise velocity component. Althoughcirculation mostly occurs in the ( y, z ) -planes, only of the energy is contained thetransverse ( x ) vorticity/stream function component, while are still contained in thevorticity/stream function component along the magnetic field. This scatter of energybetween the vorticity components is due to the confinement of circulation in narrowlayers parallel to the side walls. The confinement causes a strong variation of the veloc-ity perturbation in the spanwise ( x ) direction and, thus, produces a significant vorticitycomponents tangential to the plane of circulation.Finally, we consider the effect of the duct aspect ratio on the instability threshold ofthe two most unstable modes. In order to generalise the above results for square duct toarbitrary aspect ratios it is instructive to start with a horizontal magnetic field, which is inear stability of MHD flow in a perfectly conducting rectangular duct δ defined by (3.1), whichis the characteristic length scale of the instability, varies not only with the Hartmannnumber but also with the aspect ratio A , which defines the size of the duct along themagnetic field. When the magnetic field is directed horizontally along the fixed dimensionof the duct, δ becomes independent of A and varies only with Ha as in the case of squareduct. This simplification is our main motivation for considering first horizontal magneticfield.On changing the magnetic field from vertical to horizontal, modes I/III swap withII/IV, which, thus, become the most unstable ones. For sufficiently large aspect ratios,the critical Reynolds number and the wavenumber are seen in figure 8( a,b ) be the samefor both modes, which agree with the strong field asymptotics (3.3,3.4). As discussedabove, this implies that the instabilities developing in the jets at the parallel walls areeffectively separated by a stagnant core of the flow and, thus, do not affect each other. Itchanges at small aspect ratios, when the walls parallel to the magnetic field are sufficientlyclose to each other. Then the vortices at the opposite walls start to interact, which causesthe thresholds for both modes to diverge. For mode IV, the vortices at the opposite wallscounter-rotate and, thus, tend to suppress each other, which results in the stabilisationof the flow. It is the opposite for mode II, whose instability threshold first drops as theco-rotating vortices at the opposite walls start to enhance each other. However, withfurther reduction of the aspect ratio the critical Reynolds number attains a minimumand then starts to increase following that for mode IV. The raise of Re c for mode II isassociated with the increase of the critical wavenumber. This corresponds to the reductionof wavelength which is required for the vortices to fit between closely spaced parallel walls.Now we turn to vertical magnetic field, which is oriented along the variable height ofthe duct. This slightly more complicated case can be reduced to the previous one bytaking the height of the duct as the length scale and rotating the duct by 90 degrees.This is equivalent to the substitutions A ′ = 1 /A, Re ′ = A Re , Ha ′ = A Ha , and k ′ = Ak, where the parameters with primes correspond to the case of horizontal magnetic fieldconsidered above. Then we can use the result for horizontal magnetic field found aboveaccording to which the critical parameters in strong magnetic field approach those fora square duct given by (3.3,3.4). Substituting the primed parameters into (3.3,3.4) weobtain Re c ( Ha ; A ) = A − Re c ( A Ha ; 1) , (3.7) k c ( Ha ; A ) = A − k c ( A Ha ; 1) , (3.8)which are seen in figure 8(c,d) to fit the numerical results well in the intermediate rangeof aspect ratios, where the thresholds for both most unstable modes I/III merge. In thisrange both the critical Reynolds number and wavenumber for a fixed Ha vary asymptoti-cally with the aspect ratio as ∼ A − / . This variation, which is due to the increase of thejet width (3.1) as ∼ A / , breaks down at both small and large A. In the former limit,asymptotic relations (3.7,3.8) turn inapplicable because the effective Hartmann number A Ha based on the height becomes too small for (3.3,3.4) to be valid. In the latter limit,the jets become so wide that the vortices at the opposite walls start to interact resulting2 J. Priede, S. Aleksandrova and S. Molokov C r iti ca l R e yno l d s nu m b e r , R e c Aspect ratio, A Ha = 2010050mode IIIVasymptotic ( a ) C r iti ca l w a v e nu m b e r , k c Aspect ratio, A Ha = 2050100mode IIIVasymptotic ( b ) C r iti ca l R e yno l d s nu m b e r , R e c Aspect ratio, A Ha = 2050 10 mode IIIIasymptotic ( c ) C r iti ca l w a v e nu m b e r , k c Aspect ratio, A Ha = 205010 mode IIIIasymptotic ( d ) Figure 8.
The critical Reynolds number (a,c) and the wavenumber (b,d) versus the aspect ratiofor modes II/IV in a horizontal magnetic field (a,b) and for modes I/III in a vertical magneticfield (c,d) at various Hartmann numbers. in the divergence of the instability thresholds and the eventual stabilisation describedabove for the case of horizontal magnetic field.
4. Discussion and conclusions
We have presented numerical results concerning linear stability of a liquid metal flowin a rectangular duct with perfectly electrically conducting walls subject to a uniformtransverse magnetic field. It was found that a linearly stable flow in a square duct turnsunstable as a relatively weak magnetic field with the Hartmann number Ha ≈ . isapplied. The instability is due to two weak jets, which first appear at the centre of theduct and then move to the walls parallel to the magnetic field as the field strength isincreased. The instability follows the jets and in a sufficiently strong magnetic field be-comes confined in the layers of characteristic thickness δ ∼ Ha − / located at the parallelwalls. The thickness δ determines the characteristic length scale of the instability, whichresults in both the critical Reynolds and wave numbers scaling as ∼ δ − . Owing to thedouble reflection symmetry of the problem, perturbations with four different symmetrycombinations along and across the magnetic field decouple from each other and, thus,are considered separately. The most unstable is a pair of perturbations with an even dis-tribution of the vorticity along the magnetic field. These two modes represent stronglynon-uniform vortices aligned with the magnetic field, which rotate either in the same oropposite directions across the magnetic field. The former enhance while the latter weaken inear stability of MHD flow in a perfectly conducting rectangular duct Re c ≈ Ha / + O ( Ha − / ) and k c ≈ . Ha / . The other pair of critical perturbations, which differs from the previ-ous one by an odd distribution of vorticity along the magnetic field, is more stable withapproximately four times higher critical Reynolds number.The basic instability characteristics described above resemble those for the Hunt’sflow, which has a similarly increasing, however a significantly lower, critical Reynoldsnumber Re c ∼ Ha / (Priede, Aleksandrova & Molokov 2010). The difference becomessubstantial when the average rather than the maximum velocity is considered. Namely,the critical Reynolds number based on the average velocity for this flow increases in astrong magnetic field in the same way as ¯ Re c ≈ Ha / , whereas it tends to a constant ¯ Re c ≈ for Hunt’s flow. This asymptotically constant Re c is due to a principally differ-ent velocity distribution in Hunt’s flow with the jets of thickness ∼ Ha − / dominatingthe flow rate. Constant ¯ Re c ≈ has been found by Ting et al. (1991) for the linearstability of the flow in a square duct with thin but relatively well-conducting walls ina strong transverse magnetic field. This flow represents an intermediate case in termsof the conductivity of parallel wall between the perfectly conducting one considered inthis study and the insulating one for Hunt’s flow. Although the leading-order velocitydistribution considered by Ting et al. (1991) is nearly the same as that of Hunt’s jet,there is one principal difference between two. Namely, in a duct with thin parallel wallsof a moderate conductance ratio c = σ w d w /σd, where σ w and d w are the electrical con-ductivity and the thickness of wall, satisfying Ha − / ≪ c ≪ Ha / , jets carry a volumeflux of the same order of magnitude as that of the core flow, whereas the contributionof the latter is negligible in Hunt’s flow. In a square duct with thin walls, in which thecore flow in strong magnetic field carries / of the total volume flux (Walker 1981), thejet velocity is by a factor of about four lower than that for Hunt’s flow at the same ¯ Re . It means that jets are more unstable at wall of finite conductivity than in both limitingcases of insulating and perfectly conducting parallel walls.The experiment best matching the model considered in this study has been carriedout by Branover & Gelfgat (1968), who measured a flow of mercury at ¯ Re ≈ × and Ha = 174 in a rectangular copper duct with a relatively high wall conductance ratio,which was and for the Hartmann and the parallel walls, respectively. The magneticfield was applied transversely to the longest side of the duct with the aspect ratio of . . First, the authors found a maximum velocity in jets exceeding the theoretical predictionfor perfectly conducting duct by approximately . At this Hartmann number, we findthat of the excess jet velocity may be due to the finite wall conductivity, when thelatter is included in the numerical solution. Note that the effect of imperfectly conductingwalls is not entirely determined by the relative conductance of the parallel layers c Ha / as it may appear from Hunt (1965) (see also Müller & Bühler 2001, p. 146). Namely, c Ha / ≫ means only that the parallel walls cannot be treated as insulating ( c = 0) .But it does not necessary mean that the walls may be assumed perfectly conducting ( c = ∞ ) . As shown by Walker (1981), the latter approximation requires a much higherwall conductance ratio c Ha − / ≫ . Thus, the effect of imperfectly conducting wallsincreases rather than decreases with the field strength, which results in the jet velocityrelative to that of the core increasing with the field strength as ∼ Ha / /c (Walker 1981).However, the most significant deviation from the laminar flow solution by Hunt (1965)was the jet thickness, which was found by Branover & Gelfgat (1968) to be several times4 J. Priede, S. Aleksandrova and S. Molokov greater than expected. Such a broadening of jets is likely to be due to the turbulencewhich is expected in the experiment at the Reynolds number significantly above thelinear stability threshold ¯ Re c ≈ predicted by our analysis for this setup.In conclusion, note that when the distance of the velocity maximum from the parallelwall (3.1) is taken as the length scale, (3.3) becomes Re δc ≈ . This small criticalReynolds number may be due to the inviscid nature of the instability caused by theinflection point of the base velocity profile. It also implies that the instability may be su-percritical rather than subcritical as for typical shear flows. The supercritical character ofinstability may explain why it appears significantly above the linear stability threshold asrecently reported by Kinet, Knaepen & Molokov (2009). They observed small-amplitudevortices in the jets at the parallel walls for ≤ ¯ Re ≤ at Ha = 200 in the numer-ical simulation of a flow in a rectangular duct with thin walls. These vortices were foundto be subcritically unstable for ≤ ¯ Re ≤ , which may correspond to the largeamplitude instabilities observed in the experiments (Reed & Picologlou 1989; Burr et al. Appendix A. Vector stream function/vorticity formulation
We use the vector stream function ψ , which is introduced to satisfy the incompressiblityconstraint ∇ · v = 0 for the flow perturbation by seeking the velocity distribution in theform v = ∇ × ψ . Since the velocity is invariant upon adding the gradient of arbitraryfunction to ψ , we can impose an additional constraint ∇ · ψ = 0 , (A 1)which is analogous to the Coulomb gauge for the magnetic vector potential A (Jackson1998). Similarly to the incompressiblity constraint for v , this gauge leaves only twoindependent components of ψ . The pressure gradient is eliminated by applying curl to (2.1). This yields two dimen-sionless equations for ψ and ω ∂ t ω = ∇ ω − Re g + Ha h , (A 2) ∇ ψ + ω , (A 3)where g = ∇ × ( v · ∇ ) v , and h = ∇ × f are the curls of the dimensionless convectiveinertial and electromagnetic forces, respectively.The boundary conditions for ψ and ω are obtained as follows. The impermeabilitycondition applied integrally as R s v · ds = H l ψ · dl = 0 to an arbitrary area of wall s encircled by a contour l yields ψ τ | s = 0 . This boundary condition substituted into (A 1)results in ∂ n ψ n | s = 0 . In addition, the no-slip condition applied integrally H l v · dl = R s ω · ds yields ω n | s = 0 . Linear stability of the base flow { ¯ ψ , ¯ ω , ¯ φ } ( x, y ) is analysed with respect to infinitesimal inear stability of MHD flow in a perfectly conducting rectangular duct { ψ , ω , φ } ( r , t ) = { ¯ ψ , ¯ ω , ¯ φ } ( x, y ) + { ˆ ψ , ˆ ω , ˆ φ } ( x, y ) e γt +ß kz , where k is a wavenumber and γ is, in general, a complex growth rate. This expressionsubstituted into (A 2,A 3) results in γ ˆ ω = ∇ k ˆ ω − Re ˆ g + Ha ˆ h , (A 4) ∇ k ˆ ψ + ˆ ω , (A 5) ∇ k ˆ φ − ˆ ω q , (A 6)where ∇ k ≡ ∇ ⊥ +ß k e z ; q and ⊥ respectively denote the components along and transverseto the magnetic field in the ( x, y ) -plane. Because of the solenoidality of ˆ ω , we need onlythe x - and y -components of (A 4), which contain ˆ h ⊥ = − ∂ xy ˆ φ − ∂ q ˆ w, ˆ h q = − ∂ q ˆ φ and ˆ g x = k ˆ v ¯ w + ∂ yy (ˆ v ¯ w ) + ∂ xy (ˆ u ¯ w ) + ß2 k∂ y ( ˆ w ¯ w ) , (A 7) ˆ g y = − k ˆ u ¯ w − ∂ xx (ˆ u ¯ w ) − ∂ xy (ˆ v ¯ w ) − ß2 k∂ x ( ˆ w ¯ w ) , (A 8)where ˆ u = ß k − ( ∂ yy ˆ ψ y − k ˆ ψ y + ∂ xy ˆ ψ x ) , ˆ v = − ß k − ( ∂ xx ˆ ψ x − k ˆ ψ x + ∂ xy ˆ ψ y ) , ˆ w = ∂ x ˆ ψ y − ∂ y ˆ ψ x . The relevant boundary conditions are ˆ φ = ˆ ψ y = ∂ x ˆ ψ x = ∂ x ˆ ψ y − ∂ y ˆ ψ x = ˆ ω x = 0 at x = ± , (A 9) ˆ φ = ˆ ψ x = ∂ y ˆ ψ y = ∂ x ˆ ψ y − ∂ y ˆ ψ x = ˆ ω y = 0 at y = ± A. (A 10) REFERENCES
Braginskii, S. I. (1960) Magnetohydrodynamics of weakly conducting liquids.
Sov. Phys. JETP , 1005–1014.Branover, G. G. & Gelfgat, Yu. M. 1968
Fluid Dynamics , 79.Brouillette, E. C. & Lykoudis, P. S. 1967 Magneto-fluid-mechanic channel flow. I. Experiment. Phys. Fluids , 995–1001.Bühler, L. 2007 Liquid metal magnetohydrodynamics for fusion blankets. In: Molokov, S.,Moreau, R., Moffatt, H.K. (eds.), Magnetohydrodynamics: Historical Evolution and Trends.
Springer, pp. 171–194.Burr, U., Barleon, L., Müller, U. & Tsinober, A. 2000 Turbulent transport of momentum andheat in magnetohydrodynamic rectangular duct flow with strong sidewall jets.
J. FluidMech. , 247–279.Chang, C. C. & Lundgren, T. S. 1961 Duct flow in magnetohydrodynamics.
ZAMP , 100–114.Drazin P.G. & Reid W. H. 1981 Hydrodynamic Stability , Cambridge, § Quart. J.Mech. Appl. Math . , 341–353.Hunt, J. C. R. 1965 Magnetohydrodynamic flow in rectangular ducts. J. Fluid Mech. , 577–590.Jackson, J. D. 1998 Classical Electrodynamics , Wiley, § J. Phys. Soc. Japan , 1041–1057.Kinet, M., Knaepen, B. & Molokov, S. 2009 Instabilities and transition in magnetohydrodynamicflows in ducts with electrically conducting walls. Phys Rev Lett. , 154501.Lifshits, A. M. & Shtern, V. N. 1979 Monoharmonic analysis of the nonlinear stability of Hart-mann flow.
Magnetohydrodynamics , 243–248. J. Priede, S. Aleksandrova and S. Molokov
Lock R. C. 1955 The stability of the flow of an electrically conducting fluid between parallelplanes under a transverse magnetic field.
Proc. Roy. Soc. Lond. (A) , 105–125.Moresco, P. & Alboussière, T. 2003 Weakly nonlinear stability of Hartmann boundary layers,
Eur. J. Mech . B
Fluids , 345–353.Moresco, P. & Alboussière, T. 2004 Experimental study of the instability of the Hartmann layer. J. Fluid Mech. , 167–181.Müller, U. & Bühler, L. 2001
Magnetohydrodynamics in Channels and Containers . Springer.Murgatroyd, W. 1953 Experiments in magnetohydrodynamic channel flow.
Phil. Mag. , 1348–1354.Pothérat, A. 2007 Quasi two-dimensional perturbations in duct flows with a transverse magneticfield. Phys. Fluids , 74104.Priede, J. & Gerbeth, G. 1997 Convective, absolute and global instabilities of thermocapillary–buoyancy convection in extended layers. Phys. Rev . E , 4187–4199.Priede, J., Aleksandrova, S. & Molokov, S. 2010 Linear stability of Hunt’s flow. J. Fluid Mech. , 115–134.Reed, C. B. & Picologlou, B. F., 1989 Side wall flow instabilities in liquid metal MHD flowsunder blanket relevant conditions.
Fusion Tech. , 705–715.Roberts, P. H. 1967 An Introduction to Magnetohydrodynamics , Longmans, § Phys. Fluids , 084101–9.Shercliff, J. A. 1953 Steady motion of conducting fluids in pipes under transverse magnetic fields. Proc. Cambr. Philos. Soc. , 136–144.Tatsumi, T. & Yoshimura, T. 1990 Stability of the laminar flow in a rectangular duct. J. FluidMech. , 437–449.Ting, A. L., Walker, J. S., Moon, T. J., Reed, C. B. & Picologlou, B. F. 1991 Linear stabilityanalysis for high-velocity boundary layers in liquid–metal magnetohydrodynamic flows.
Int.J. Engng. Sci. , 939–948.Uflyand, Ya. S. 1961 Flow stability of a conducting fluid in a rectangular channel in a transversemagnetic field. Soviet Physics: Technical Physics , 1191–1193.Uhlmann, M. & Nagata, M. 2006 Linear stability of flow in an internally heated rectangularduct. J. Fluid Mech. , 387–404.Williams W. E 1962 Magnetohydrodynamic flow in a rectangular tube at high Hartmann num-bers.
J. Fluid Mech. , 262–268.Walker, J. S. 1981 Magnetohydrodynamic flows in rectangular ducts with thin conducting walls. J. de Méchanique20