- Email: [email protected]
Cylinder wake control by magnetic fields in liquid metal flows
Experimental Thermal and Fluid Science 16 (1998) 92±99
Cylinder wake control by magnetic ®elds in liquid metal ¯ows G. Mutschke a b
a,* ,
V. Shatrov b, G. Gerbeth
a
Research Center Rossendorf Inc., P.O. Box 510119, D-01314 Dresden, Germany Institute of Theoretical and Applied Mechanics, Novosibirsk, Russian Federation
Received 14 October 1996; received in revised form 14 February 1997; accepted 3 March 1997
Abstract In this paper we are concerned with the control of wake instabilities in the ¯ow of an electrically conducting ¯uid around a circular cylinder by means of external magnetic ®elds. Besides the Reynolds number (Re) a second parameter N appears describing the strength of the magnetic body force. This oers, depending on the direction of the magnetic ®eld, a large variety of ¯ow con®gurations and therefore dierent transition regimes. We perform a numerical simulation of the unsteady two-dimensional ¯ow and characterize the dierent ¯ow regimes. Strong magnetic ®elds are capable to stabilize the 2-D ¯ow and to suppress the shedding of vortices. We present curves of neutral 2-D stability in the (Re, N)-parameter plane separating steady and periodic ¯ow regimes. We further perform a linear 3-D stability analysis of the 2-D ¯ow being either steady or periodic and show how the magnetic ®eld in¯uences the 3-D instabilities. We pay special attention to the case when the magnetic ®eld is aligned with the oncoming ¯ow. Here we ®nd 3-D instability in parameter regions above the 2-D stability curve in the (Re, N)-plane where the ¯ow is 2-D stable (steady). This ®rstly con®rms a general result of Hunt (J.C.R. Hunt, Proc. Roy. Soc. A 293 (1996) 342) obtained from a stability analysis of parallel ¯ows and shows that the magnetic ®eld in¯uences 2-D and 3-D instabilities in a dierent way. Ó 1998 Elsevier Science Inc. All rights reserved. Keywords: Cylinder wake; Flow control; Instabilities; MHD; Wake transition
1. Introduction The ¯ow around a circular cylinder is one of the most investigated and best understood blu body wake problems. The ®rst instability leading to the von Karman vortex street arises at Re 49 [2], recently the nature of the secondary instability towards a spanwise-periodic 3-D ¯ow at Re 188 was understood [3,4]. In this paper we present numerical results on the control of cylinder wake instabilities in the ¯ow of an electrically conducting ¯uid due to externally applied magnetic ®elds. The motivation for trying to control those instabilities by external magnetic ®elds is on one hand a recently growing interest in using external body forces for turbulence control [5,6]. On the other hand, there is still a basic interest in such magnetohydrodynamic (MHD) ¯ows. Besides the usual Reynolds number Re an additional parameter N B20 =v1 appears describing the
*
Corresponding author. Tel.: +49 351 260-2373; fax: +49 351 2602007; e-mail: [email protected] 0894-1777/98/$19.00 Ó 1998 Elsevier Science Inc. All rights reserved. PII: S 0 8 9 4 - 1 7 7 7 ( 9 7 ) 1 0 0 0 7 - 3
strength of the magnetic body force. Depending on the direction of the magnetic ®eld, a large variety of ¯ow con®gurations and therefore also transition scenario exist. Little is known about instabilities and transition scenario in such MHD wake ¯ows except the general belief that magnetic ®elds have a damping in¯uence. Already in 1963 Tsinober et al. [7] reported a downstream shift of the separation point at the cylinder surface due to a magnetic ®eld, thereby stabilizing the ¯ow. Early experiments (for a summary see Tsinober p [8]) found a growing mean drag proportional to N as the magnetic ®eld is increased. Recent experiments from Lahjomri et al. [9± 11] show a curve of neutral stability in the (Re, N) parameter plane separating Karman-like periodic ¯ows from steady ones. However, all known experimental results about 2-D stability are restricted to relatively high Re because of serious measuring problems in liquid metal ¯ows. Additionally, to our knowledge, nothing yet is known about 3-D transitions in those wake ¯ows that are doubtless hard to be detected experimentally because of strong aspect ratio in¯uences [12]. Therefore, there is good reason to investigate the onset of these 2-D and also 3-D instabilities in a numerical experiment from
G. Mutschke et al. / Experimental Thermal and Fluid Science 16 (1998) 92±99
that we might also get more insight into the corresponding hydrodynamic problem. A ®nal motivation came from an early analytical work of Hunt in the frame of a local parallel ¯ow analysis. He found for the particular case of a magnetic ®eld being aligned with the ¯ow that 3-D instability exists in parameter regions where the ¯ow is 2-D stable [1]. His conclusion is based on the interesting point that the Squire-theorem [13] is no longer valid in the MHD case. We therefore expect an interesting interplay between 2D and 3-D instabilities in the MHD cylinder wake, in particular for an aligned magnetic ®eld. The paper is organized as follows. In Section 2 we de®ne the MHD ¯ow problem and discuss the governing equations. In Section 3 we brie¯y describe the applied numerical method. Section 4 discusses our main results on 2-D ¯ows and on 2-D and 3-D instabilities. In conclusion we give a short summary and an outlook to future work.
We consider the ¯ow of an electrically conducting, incompressible viscous liquid around an insulating circular cylinder in an external uniform magnetic ®eld B0 (see Fig. 1). We have in mind a laboratory-scale experiment with molten metal where for the magnetic Reynolds number Rm lrv1 d 1 holds. All ¯uid parameters (density q, kinematic viscosity m g=q, electric conductivity r, permeability l) are assumed to be constant. Scaling the length by the cylinder radius a d=2, the velocity by v1 , the pressure by qv21 =2, the time by a=v1 , the magnetic ®eld by B0 , the electric ®eld by B0 v1 , and the electric current density by rv1 B0 , respectively, yields the following non-dimensional governing equations in inductionless approximation (see [14]): @v 2 N v rv ÿ rp Dv f m ; @t Re 2 f m j B0 ;
1
r v 0;
2
j E v B0 ;
3
E ÿrU:
The Navier±Stokes equation (1) contains a Lorentz force term acting on the ¯uid due to induced currents j described by Ohms law (4). Using the conservation of electric charge (3) and Ohms law (4), the electric potential U can be obtained from DU r v B0 : 5 The two characteristic parameters are: Re dv1 =m Reynolds number, N rB20 d=qv1 interaction parameter (N is related to the Hartmann number Ha as N Ha2 =Re); additionally one has to distinguish three fundamental directions of the magnetic ®eld. In the following we will refer to them as cases: (a) B0 ex , aligned ®eld, (b) B0 ey , transverse ®eld, (c) B0 ez , parallel ®eld, where ex ; ey ; ez denote the unit vectors in the x; y; z directions, respectively. 2.1. 2-D ¯ow and 2-D instabilities
2. Problem de®nition
r j 0;
93
4
At the ®rst stage, we are concerned with the 2-D ¯ow in the (x,y)-plane perpendicular to the cylinder axis, i.e. T velocity ®elds v0 u; v; 0 not depending on z. In cases (a) and (b) it can be easily shown that the electric ®eld vector E consists only of a constant z-component. Its value determines the current density at in®nity but has no in¯uence on either the ¯ow or its stability. Choosing E to vanish, the Navier±Stokes equation (1) with a Lorentz force term (a) f m ÿvey , aligned ®eld, (b) f m ÿuex , transverse ®eld, completely describes the 2-D ¯ow and its 2-D stability. From here one already might suppose that case (b) has a stronger impact on the 2-D ¯ow than case (a) because the force scales with the streamwise velocity component in case (b) whereas only the transverse component determines the force in case (a). In case (c) the electric ®eld lies in the (x,y)-plane depending on both coordinates, but the curl of the body force f m vanishes. This means, that the 2-D ¯ow and its 2-D stability are not aected by the magnetic ®eld at all. However, this holds for in®nitely extended systems and is not in contradiction to experimental results of Papailiou [15] who investigated the ¯ow at the surface of liquid metals, probably further in¯uenced by electrical boundary conditions. 2.2. Linear 3-D instabilities We perform a linear 3-D-stability analysis by assuming small 3-D disturbances v0 to a basic 2-D ¯ow v0 (steady or periodic)
Fig. 1. Sketch of the geometry.
v v0 v0 ; jv0 j jv0 j 6 and linearize the Navier±Stokes equation (1) with respect to v0 . The electric potential is only due to the 3D perturbation U U0 since, according to the above arguments, electric ®elds from the 2-D ¯ow do not
94
G. Mutschke et al. / Experimental Thermal and Fluid Science 16 (1998) 92±99
have to be taken into account. Therefore, Eq. (5) modi®es to 7 DU0 r v0 B0 : Assuming periodicity in the cylinder spanwise direction z characterized by a wave number k, each Fourier mode of the 3-D disturbances can be investigated separately, and the problem reduces essentially to a 2-D one v0 v00r cos kz; v00h cos kz; v00z sin kzT ; U0 U00 sin kz; where standard polar coordinates are used.
8 9
3. Numerical method The 2-D ¯ow is determined using an established streamfunction vorticity ®nite dierence method (v0 ÿr W0 ez , x0z DW0 ) similar to [16], details are described in [17]. For the hydrodynamic cylinder ¯ow it is known that at Re 49 the steady 2-D ¯ow becomes absolutely unstable starting to shed alternating vortices known as the von K arm an vortex street. Switching on a magnetic ®eld of certain strength N at ®xed Reynolds number then for t ! 1 either damps out the vortex shedding yielding a steady ¯ow or leads to a modi®ed vortex street of a still periodic wake ¯ow. The linear 3-D problem was treated in a velocity±vorticity ®nite dierence formulation similar to [18]. The total accuracy is second order, whereby all elliptic equations including the Poisson equation for the electric potential were solved in fourth order. At some initial time t 0 a 3-D vorticity disturbance k h a0 sin2 ; 4 sin hT 10 r3 2 2r of amplitude a0 10ÿ2 was added to a developed 2-D ¯ow being either steady or periodic. Marching in time by simultaneously integrating the 2-D and the linearized 3-D equations, after some transient time the 3-D perturbation enters the expected linear regime where it will either increase or decay exponentially in time. To decide about the global 3-D stability of the ¯ow we monitor in time the total 3-D energy w00r ; w00h ; w00z T 0; a0
Z Z 2p=k Z E3D t
dx dy dz
v02 2
Fig. 2. Typical behavior of the logarithm of the global 3-D energy E3D in time (Re 200, N 0:4, k 0:8, aligned magnetic ®eld). After a transient time the ¯ow enters a linear regime, and the TGR can be detected.
h-direction, with up to 256 ´ 256 points. Boundary conditions are v0 0, v0 0 and @U0 =@r 0 at r 1, hydrodynamic potential ¯ow for v0 , v0 0 and U0 0 at rmax , and out¯ow conditions for v0 , v0 and U0 in an out¯ow angle of 70 downstream at rmax . To validate our numerical approach, we performed intensive tests. The 2-D calculations without magnetic ®eld were in good agreement to results of Fornberg [19] with respect to recirculation length, vorticity at the cylinder surface and drag coecients; also the Strouhal frequency and the vortex street are nicely reproduced. The main error on the TGR in the 3-D calculations was due to the ®nite spatial grid spacing, therefore dense grids had to be applied. Fig. 3 shows a Richardson extrapolation for the TGR with up to 256 ´ 256 points reducing the error to less than 5%. Further, we compared our results for the purely hydrody-
11
integrated over the computational domain. In the following, the slope of the logarithm of this quantity in the linear regime is called the temporal growth rate (TGR), and positive (negative) values of TGR correspond to absolute 3-D instability (stability) of the ¯ow. In general, our global linear stability analysis has the advantage that it detects only absolute instabilities and excludes the convective ones. Fig. 2 shows an example of the temporal growth rate obtained by this method at Re 200, N 0.4, k 0.8 for an aligned magnetic ®eld. For both 2-D and 3-D simulations we used the same exponential polar grid, ri eci , rmax 50, equidistant in
Fig. 3. Richardson extrapolation for the temporal growth rate of E3D at Re 200, N 0:4, k 0:8, aligned magnetic ®eld. NP denotes the number of grid points in both radial and azimuthal directions.
G. Mutschke et al. / Experimental Thermal and Fluid Science 16 (1998) 92±99
95
namic problem at Re 200 with data published recently by Henderson [20] and found an excellent agreement (see Fig. 8). 4. Flow and instability control by magnetic ®elds 4.1. 2-D ¯ow properties To enlighten the action of the magnetic body force on the 2-D ¯ow, we show in Figs. 4 and 5 isolines of the 2D streamfunction W0 of the steady ¯ow at Re 100 for dierent interaction parameter N and aligned and transverse ®elds, respectively. Of course, some of the shown ¯ows are not really steady because of 2-D instabilities (see Section 4.2), this was arti®cially forced by applying a symmetry condition. In both ®gures it is to be seen that an increasing magnetic ®eld acts on the wake by decreasing the length of the recirculation bubble and lowering the separation angle. However, for stronger magnetic ®elds separation is not ®nally suppressed in case (a), rather some asymptotic state seems to be reached with widened and straightened streamlines caus-
Fig. 5. Isolines of the 2-D streamfunction of a steady 2-D solution at Re 100 for a transverse ®eld at dierent N .
ing even upstream a region of very slow ¯ow called ``upstream wake'' [8]. In case (b) the body force acts stronger than in case (a) in terms of recirculation length at the same value of N, further it leads to a complete suppression of the separation for N > 0:8 [17]. For more details of the 2-D ¯ow we refer to [17]. We only want to mention that the vorticity in the boundary layer in general increases dueptothe action of the body force and therefore leads to a N -like increase of the total drag (mainly due to the pressure part) with increasing N. However, as can be seen in the next section, increasing N may lead to a steady ¯ow with a smaller total drag than the time-averaged drag of the corresponding unsteady ¯ow without magnetic ®eld at the same Reynolds number. For example, at Re 200 holds CDaveraged N 0 1:3, whereas CDsteady N 0:3 0:9. 4.2. 2-D stability
Fig. 4. Isolines of the 2-D streamfunction of a steady 2-D solution at Re 100 for an aligned ®eld at dierent N .
For the hydrodynamic cylinder ¯ow it is known that at Re 49 the steady 2-D ¯ow becomes absolutely unstable (undergoing a Hopf bifurcation [2]) starting to shed alternating vortices known as the von Karman vortex street. From the two ®gures above one might already
96
G. Mutschke et al. / Experimental Thermal and Fluid Science 16 (1998) 92±99
For the hydrodynamic cylinder ¯ow today it seems to be clear that at a Reynolds number of about 188.5 the periodic 2-D ¯ow becomes unstable to spanwise periodic 3-D disturbances with a wavelength of about 4 diameters. Two recent papers show excellent agreement between numerical results from a linear stability
investigation [4] and experiments [3]. However, the evolution of these instabilities at higher Re > 220 is not yet completely understood. With increasing Re the most unstable mode changes signi®cantly, further there is large scatter in both experimental and numerical results (for a summary see [22,12]). Therefore we restrict our discussion for the MHD ¯ow mainly to the case of Re 200. We now focus on MHD case (a) with aligned magnetic ®eld. Fig. 7 shows its neutral 2-D stability curve in the (Re, N) parameter plane. Above the curve, any 2-D disturbance will be damped out in time towards a steady ¯ow, whereas below the curve the magnetic ®eld is too weak and can only modify the persisting vortex street. Hunt [1] performed an analytical work on the relation between 2-D and 3-D instabilities, but limited to a local analysis of parallel ¯ows. He found, that for an aligned magnetic ®eld the Squire-theorem is no longer valid. Transferring his results to the present case results in a tangent to our 2-D stability curve beginning at the origin (see Fig. 7). The parallel ¯ow analysis would now distinguish between three dierent regions in the parameter space: Region I: Above the tangent and above the 2-D stability curve before their point of contact the ¯ow is stable to both 2-D and 3-D disturbances, i.e. steady and 2D Region II: Below the 2-D stability curve the ¯ow is primarily 2-D unstable, 3-D instability will occur at higher Re Region III: Between 2-D stability curve and tangent beyond their point of contact the ¯ow is 2-D stable but 3-D unstable! Of course, the MHD cylinder wake is neither a parallel ¯ow nor leads local instability always to global instability. Therefore, the results of Hunt cannot be applied straightforwardly. However, there is a ®rst qualitative hint that similar eects to the violation of the Squiretheorem for parallel MHD ¯ows might exist for the MHD cylinder wake. First results of our linear 3-D stability analysis are shown in Figs. 8±11. Fig. 8 shows curves of temporal growth rates for dierent N at
Fig. 7. Neutral 2-D stability curve for an aligned magnetic ®eld with a tangent from the parallel ¯ow analysis of Hunt.
Fig. 8. Temporal growth rates of the global 3-D energy at Re 200, aligned ®eld.
Fig. 6. Neutral 2-D stability curve for aligned and transverse magnetic ®eld.
expect, that the magnetic ®eld in case (a) and (b) is capable to suppress the shedding of vortices. Fig. 6 shows the curves of neutral 2-D stability in the (Re, N) plane for cases (a) and (b) respectively, separating steady ¯ows above the curve from periodic ones below it. As already expected from the discussion of the 2-D ¯ow above, for (b) weaker magnetic ®elds than for (a) can stabilize the ¯ow. For a comparison with experimental results of Josserand and Weier [10,11,21] at larger Re > 1000 we refer to [21]. The Strouhal frequency of the vortex street modi®ed by a magnetic ®eld decreases only slightly with increasing N until the ¯ow becomes steady at the critical value of N. 4.3. 3-D stability
G. Mutschke et al. / Experimental Thermal and Fluid Science 16 (1998) 92±99
97
Fig. 9. Snapshot of W0 , x0 and e3D (from top to bottom) at Re 200, N 0, k 0.5 (aligned ®eld) in the linear regime.
Fig. 11. Snapshot of W0 , x0 and e3D (from top to bottom) at Re 200, N 0.4, k 0.6 (aligned ®eld) in the linear regime.
Fig. 10. Snapshot of W0 , x0 and e3D (from top to bottom) at Re 200, N 0.05, k 0.7 (aligned ®eld) in the linear regime.
Re 200. Compared to the hydrodynamic case at N 0, a weak magnetic ®eld of N 0.05 seems to decrease the level of 3-D instability. But the ¯ow is still 3-D unstable, the most unstable wave number is slightly decreased. A widening of the curve leads to a band of unstable wave numbers of approximately the same width as for N 0 although shifted to smaller wave numbers. The larger value of N 0:4 shows a broader range of unstable wave numbers than at N 0. The maximum TGR value is only slightly lower than for the hydrodynamic case, i.e. a slight damping in¯uence seems to exist. A further increase of the magnetic ®eld up to N 1 leads to a complete suppression of 3-D instability for Nc3D > 0:75. Here it is interesting to note that for N 0:4 the ¯ow is 3-D unstable, whereas the corresponding 2-D ¯ow is already steady, as can be seen from Fig. 7. The critical value of neutral 2-D stability at Re 200 is about Nc2D 0:27, so a 3-D instability above the 2-D threshold is clearly detected. To characterize the 3-D instabilities, in Figs. 9±11 we plotted isolines of the 2-D streamfunction W0 , the 2-D vorticity x0z and the local 3-D energy density e3D v02 =2 at a time arbitrarily chosen inside the linear regime, for dierent values of N. Both unsteady 2-D ¯ows at N 0 and N 0:05 look similar, the maxima of the energy density e3D are closely arranged with the 2-D vortices shed from the cylinder. For N 0:4 the e3D maxima are closely coupled to the steady 2-D recirculation eddies and seem to be stretched along the magnetic ®eld lines. Fig. 12 shows the 3-D energy density on
98
G. Mutschke et al. / Experimental Thermal and Fluid Science 16 (1998) 92±99
Fig. 12. Logarithm of the 3-D energy density versus radius (downstream distance, at y 0) at Re 200, k 0.8, aligned ®eld.
the downstream axis. For N 0:05 the local variation of e3D resembles the vortex street oscillations of the 2-D ¯ow, whereas for N 0:4 the steady 2-D ¯ow results in a smooth shape of e3D . In both cases, the absolute maximum of e3D is located at about three diameters downstream, with larger distances the 3-D energy density decreases exponentially. First results for a larger value of Re 220 show for the same Interaction parameter N always larger values of the TGR compared to Re 200. This would therefore result in a larger critical value Nc3D and could lead to a rather steep 3-D stability line in the (Re, N) plane. Finally, we would like to summarize ®rst results for cases (b) and (c) at Re 200. Here, already for weak magnetic ®elds a strong damping action can be found, and 3-D instability seems to exist only far below the critical value of neutral 2-D instability. 5. Conclusions and outlook We were numerically investigating the in¯uence of magnetic ®elds on the relation between 2-D and 3-D instabilities in the MHD cylinder ¯ow. The most interesting case turned out to be an aligned magnetic ®eld. By monitoring in time the total energy of the 3-D disturbances we detected global 3-D instability in a parameter region where the ¯ow is 2-D stable, i.e. steady. This is qualitatively in line with the result of Hunt, notwithstanding the fact that his work was restricted to a local parallel ¯ow stability analysis. 3-D instability of a 2-D stable ¯ow is unique for the cylinder problem. It is further interesting to observe that even this standard hydrodynamic problem of the ¯ow around a cylinder represents one example, where the in¯uence of an applied magnetic ®eld on instabilities is non-monotonic and not purely damping but strongly depends on the character of the instability. It would be highly desirable to verify these results experimentally. Recent experiments [21] show a slight ampli®cation of low frequency ¯uctuations due to an aligned magnetic
®eld and support the widening of the unstable wave number region found here. In the following we will continue this work to larger Re and stronger magnetic ®elds; for the aligned ®eld case we will check the existence of a 3-D stability line. We further continue to investigate the other two directions of the magnetic ®eld in order to get a better understanding of the reasons for the dierent behavior in all three cases. For an increasing transverse ®eld it is known that not only the vortex street is damped but also the steady recirculation eddies of the 2-D ¯ow vanish [17]. The damping of 3-D instabilities we found at Re 200 is the strongest here of all three cases considered. Therefore, one might suppose that the arising of 3-D instability is connected to boundary layer separation and vortex shedding ``of a certain strength'' in the near wake. Nomenclature All quantities listed without units are nondimensional. B0 vector of the external magnetic ®eld CD drag coecient E electric ®eld vector E3D total energy of the 3-D disturbance N Interaction parameter Re Reynolds number Rm magnetic Reynolds number TGR temporal growth rate of 3-D disturbances a0 amplitude of the 3-D vorticity disturbance d cylinder diameter ex ,ey ,ez unit vector in x; y; z-direction, respectively fm Lorentz force vector j electric current density vector k wave number in z-direction r radius coordinate (standard polar system) t time v velocity vector v1 free-stream velocity Greek symbols h U W0 m l q r x
angle coordinate (standard polar system) electric potential 2-D streamfunction kinematic viscosity (m2 /s) permeability (Vs/Am) density (kg/m3 ) electric conductivity (1/Xm) vorticity vector
Subscripts C
critical value for neutral stability
Superscripts 2D, 3D with respect to 2-D or 3-D, respectively 0 quantity of the 2-D ¯ow
G. Mutschke et al. / Experimental Thermal and Fluid Science 16 (1998) 92±99 0
T
quantity of the linearized 3-D ¯ow transposed representation of a vector
Operator symbols r
Nabla vector operator inner product cross product
Acknowledgements Financial support under grant No. Ge 682/3-4 and grants for several visits of V. Shatrov at Research Center Rossendorf from ``Deutsche Forschungsgemeinschaft'' are gratefully acknowledged. The work is supported by the European INTAS program under grant No. 941504. We gratefully acknowledge use and support of the computer facilities at HLRZ Supercomputing Center Juelich.
References [1] J.C.R. Hunt, On the stability of parallel ¯ows with parallel magnetic ®elds, Proc. Roy. Soc. A 293 (1996) 342±358. [2] C.P. Jackson, A ®nite-element study of the onset of vortex shedding in ¯ow past variously shaped bodies, J. Fluid Mech. 182 (1987) 23±45. [3] C.H.K. Williamson, Mode A secondary instability in wake transition, Phys. Fluids 8 (1996) 1680±1682. [4] R.D. Henderson, D. Barkley, Secondary instability in the wake of a circular cylinder, Phys. Fluids 8 (6) (1996) 1683±1685. [5] G.L. Brown, Vorticity in the Electromagnetic Control of Turbulence, APS-DFD Meeting, Atlanta, Nov. 1994; D.M. Nosenchuck, Electromagnetic Turbulent Boundary Layer Control, APS-DFD Meeting, Atlanta, Nov. 1994. [6] J.C.S. Meng, Experimental Studies of a Seawater Superconducting Electromagnetic Thruster: A Continuing Quest for Higher Magnetohydrodynamic Propulsive Eciency, Proceedings of the Second International Conference on Energy Transfer in MHD Flows, Aussois (France), 1994, pp. 491±500.
99
[7] A.B. Tsinober, A.H. Shtern, E.V. Shcherbinin, On the separation of the magneto-hydrodynamic boundary layer, Latvijas PSR Zinatnu Akad. Vestis 12 (1963) 49±56. [8] A.B. Tsinober, MHD ¯ow around obstacles (in Russian), Zinatne Publ., Riga (Latvia), 1970. [9] J. Lahjomri, P. Caperan, A. Alemany, The cylinder wake in a magnetic ®eld aligned with the velocity J, Fluid Mech. 253 (1993) 421±448. [10] J. Josserand, P. Marty, A. Alemany, G. Gerbeth, MHD ¯ow around a circular cylinder in an aligned magnetic ®eld, AIAA Report on Progress in Astronautics and Aeronautics 148 (1992) 551±565. [11] J. Josserand, P. Marty, A. Alemany, Pressure and drag measurements on a cylinder in a liquid metal ¯ow with an aligned magnetic ®eld, Fluid Dynam. Res. 11 (1993) 107±117. [12] C.H.K. Williamson, Vortex dynamics in the cylinder wake, Annu. Rev. Fluid Mech. 28 (1996) 447. [13] H.B. Squire, On the stability for three-dimensional disturbances of viscous ¯uid ¯ow between parallel walls, Proc. Roy. Soc. A 142 (1933) 621. [14] R. Moreau, Magnetohydrodynamics, Kluwer Academic, Dordrecht, 1990. [15] D.D. Papailiou, Magneto-¯uid-mechanic turbulent vortex streets, Progr. Astron. Aeron. 100 (1992) 152±173. [16] T.P. Loc, Numerical analysis of unsteady secondary vortices generated by an impulsively started circular cylinder, J. Fluid Mech. 100 (1980) 111±128. [17] V. Shatrov, G. Mutschke, G. Gerbeth, Numerical simulation of the two-dimensional MHD-¯ow around a circular cylinder, Proceedings of the Second International Conference on Energy Transfer in MHD Flows, Aussois (France), 1994 pp. 745±756, also in Magnetohydrodynamics 33 (1997) 3±13. [18] P.J. Roache, Computational Fluid Dynamics, Hermosa Publishers, Albuquerque, 1976. [19] B. Fornberg, A numerical study of steady viscous ¯ow past a circular cylinder, J. Fluid Mech. 98 (1980) 819±855. [20] D. Barkley, R.D. Henderson, Three-dimensional Floquet stability analysis of the wake of a circular cylinder, J. Fluid Mech. 322 (1996) 215. [21] T. Weier, G. Mutschke, G. Gerbeth, A. Alemany, A. Pilaud, On the stability of the MHD ¯ow around a circular cylinder in an aligned magnetic ®eld, Proceedings of the Second International Conference on Energy Transfer in MHD Flows, Aussois (France), 1994, pp. 561±570. [22] H.-Q. Zhang, U. Fey, B.R. Noack, On the transition of the cylinder wake, Phys. Fluids 7 (1995) 779±795.
Cylinder wake control by magnetic ®elds in liquid metal ¯ows G. Mutschke a b
a,* ,
V. Shatrov b, G. Gerbeth
a
Research Center Rossendorf Inc., P.O. Box 510119, D-01314 Dresden, Germany Institute of Theoretical and Applied Mechanics, Novosibirsk, Russian Federation
Received 14 October 1996; received in revised form 14 February 1997; accepted 3 March 1997
Abstract In this paper we are concerned with the control of wake instabilities in the ¯ow of an electrically conducting ¯uid around a circular cylinder by means of external magnetic ®elds. Besides the Reynolds number (Re) a second parameter N appears describing the strength of the magnetic body force. This oers, depending on the direction of the magnetic ®eld, a large variety of ¯ow con®gurations and therefore dierent transition regimes. We perform a numerical simulation of the unsteady two-dimensional ¯ow and characterize the dierent ¯ow regimes. Strong magnetic ®elds are capable to stabilize the 2-D ¯ow and to suppress the shedding of vortices. We present curves of neutral 2-D stability in the (Re, N)-parameter plane separating steady and periodic ¯ow regimes. We further perform a linear 3-D stability analysis of the 2-D ¯ow being either steady or periodic and show how the magnetic ®eld in¯uences the 3-D instabilities. We pay special attention to the case when the magnetic ®eld is aligned with the oncoming ¯ow. Here we ®nd 3-D instability in parameter regions above the 2-D stability curve in the (Re, N)-plane where the ¯ow is 2-D stable (steady). This ®rstly con®rms a general result of Hunt (J.C.R. Hunt, Proc. Roy. Soc. A 293 (1996) 342) obtained from a stability analysis of parallel ¯ows and shows that the magnetic ®eld in¯uences 2-D and 3-D instabilities in a dierent way. Ó 1998 Elsevier Science Inc. All rights reserved. Keywords: Cylinder wake; Flow control; Instabilities; MHD; Wake transition
1. Introduction The ¯ow around a circular cylinder is one of the most investigated and best understood blu body wake problems. The ®rst instability leading to the von Karman vortex street arises at Re 49 [2], recently the nature of the secondary instability towards a spanwise-periodic 3-D ¯ow at Re 188 was understood [3,4]. In this paper we present numerical results on the control of cylinder wake instabilities in the ¯ow of an electrically conducting ¯uid due to externally applied magnetic ®elds. The motivation for trying to control those instabilities by external magnetic ®elds is on one hand a recently growing interest in using external body forces for turbulence control [5,6]. On the other hand, there is still a basic interest in such magnetohydrodynamic (MHD) ¯ows. Besides the usual Reynolds number Re an additional parameter N B20 =v1 appears describing the
*
Corresponding author. Tel.: +49 351 260-2373; fax: +49 351 2602007; e-mail: [email protected] 0894-1777/98/$19.00 Ó 1998 Elsevier Science Inc. All rights reserved. PII: S 0 8 9 4 - 1 7 7 7 ( 9 7 ) 1 0 0 0 7 - 3
strength of the magnetic body force. Depending on the direction of the magnetic ®eld, a large variety of ¯ow con®gurations and therefore also transition scenario exist. Little is known about instabilities and transition scenario in such MHD wake ¯ows except the general belief that magnetic ®elds have a damping in¯uence. Already in 1963 Tsinober et al. [7] reported a downstream shift of the separation point at the cylinder surface due to a magnetic ®eld, thereby stabilizing the ¯ow. Early experiments (for a summary see Tsinober p [8]) found a growing mean drag proportional to N as the magnetic ®eld is increased. Recent experiments from Lahjomri et al. [9± 11] show a curve of neutral stability in the (Re, N) parameter plane separating Karman-like periodic ¯ows from steady ones. However, all known experimental results about 2-D stability are restricted to relatively high Re because of serious measuring problems in liquid metal ¯ows. Additionally, to our knowledge, nothing yet is known about 3-D transitions in those wake ¯ows that are doubtless hard to be detected experimentally because of strong aspect ratio in¯uences [12]. Therefore, there is good reason to investigate the onset of these 2-D and also 3-D instabilities in a numerical experiment from
G. Mutschke et al. / Experimental Thermal and Fluid Science 16 (1998) 92±99
that we might also get more insight into the corresponding hydrodynamic problem. A ®nal motivation came from an early analytical work of Hunt in the frame of a local parallel ¯ow analysis. He found for the particular case of a magnetic ®eld being aligned with the ¯ow that 3-D instability exists in parameter regions where the ¯ow is 2-D stable [1]. His conclusion is based on the interesting point that the Squire-theorem [13] is no longer valid in the MHD case. We therefore expect an interesting interplay between 2D and 3-D instabilities in the MHD cylinder wake, in particular for an aligned magnetic ®eld. The paper is organized as follows. In Section 2 we de®ne the MHD ¯ow problem and discuss the governing equations. In Section 3 we brie¯y describe the applied numerical method. Section 4 discusses our main results on 2-D ¯ows and on 2-D and 3-D instabilities. In conclusion we give a short summary and an outlook to future work.
We consider the ¯ow of an electrically conducting, incompressible viscous liquid around an insulating circular cylinder in an external uniform magnetic ®eld B0 (see Fig. 1). We have in mind a laboratory-scale experiment with molten metal where for the magnetic Reynolds number Rm lrv1 d 1 holds. All ¯uid parameters (density q, kinematic viscosity m g=q, electric conductivity r, permeability l) are assumed to be constant. Scaling the length by the cylinder radius a d=2, the velocity by v1 , the pressure by qv21 =2, the time by a=v1 , the magnetic ®eld by B0 , the electric ®eld by B0 v1 , and the electric current density by rv1 B0 , respectively, yields the following non-dimensional governing equations in inductionless approximation (see [14]): @v 2 N v rv ÿ rp Dv f m ; @t Re 2 f m j B0 ;
1
r v 0;
2
j E v B0 ;
3
E ÿrU:
The Navier±Stokes equation (1) contains a Lorentz force term acting on the ¯uid due to induced currents j described by Ohms law (4). Using the conservation of electric charge (3) and Ohms law (4), the electric potential U can be obtained from DU r v B0 : 5 The two characteristic parameters are: Re dv1 =m Reynolds number, N rB20 d=qv1 interaction parameter (N is related to the Hartmann number Ha as N Ha2 =Re); additionally one has to distinguish three fundamental directions of the magnetic ®eld. In the following we will refer to them as cases: (a) B0 ex , aligned ®eld, (b) B0 ey , transverse ®eld, (c) B0 ez , parallel ®eld, where ex ; ey ; ez denote the unit vectors in the x; y; z directions, respectively. 2.1. 2-D ¯ow and 2-D instabilities
2. Problem de®nition
r j 0;
93
4
At the ®rst stage, we are concerned with the 2-D ¯ow in the (x,y)-plane perpendicular to the cylinder axis, i.e. T velocity ®elds v0 u; v; 0 not depending on z. In cases (a) and (b) it can be easily shown that the electric ®eld vector E consists only of a constant z-component. Its value determines the current density at in®nity but has no in¯uence on either the ¯ow or its stability. Choosing E to vanish, the Navier±Stokes equation (1) with a Lorentz force term (a) f m ÿvey , aligned ®eld, (b) f m ÿuex , transverse ®eld, completely describes the 2-D ¯ow and its 2-D stability. From here one already might suppose that case (b) has a stronger impact on the 2-D ¯ow than case (a) because the force scales with the streamwise velocity component in case (b) whereas only the transverse component determines the force in case (a). In case (c) the electric ®eld lies in the (x,y)-plane depending on both coordinates, but the curl of the body force f m vanishes. This means, that the 2-D ¯ow and its 2-D stability are not aected by the magnetic ®eld at all. However, this holds for in®nitely extended systems and is not in contradiction to experimental results of Papailiou [15] who investigated the ¯ow at the surface of liquid metals, probably further in¯uenced by electrical boundary conditions. 2.2. Linear 3-D instabilities We perform a linear 3-D-stability analysis by assuming small 3-D disturbances v0 to a basic 2-D ¯ow v0 (steady or periodic)
Fig. 1. Sketch of the geometry.
v v0 v0 ; jv0 j jv0 j 6 and linearize the Navier±Stokes equation (1) with respect to v0 . The electric potential is only due to the 3D perturbation U U0 since, according to the above arguments, electric ®elds from the 2-D ¯ow do not
94
G. Mutschke et al. / Experimental Thermal and Fluid Science 16 (1998) 92±99
have to be taken into account. Therefore, Eq. (5) modi®es to 7 DU0 r v0 B0 : Assuming periodicity in the cylinder spanwise direction z characterized by a wave number k, each Fourier mode of the 3-D disturbances can be investigated separately, and the problem reduces essentially to a 2-D one v0 v00r cos kz; v00h cos kz; v00z sin kzT ; U0 U00 sin kz; where standard polar coordinates are used.
8 9
3. Numerical method The 2-D ¯ow is determined using an established streamfunction vorticity ®nite dierence method (v0 ÿr W0 ez , x0z DW0 ) similar to [16], details are described in [17]. For the hydrodynamic cylinder ¯ow it is known that at Re 49 the steady 2-D ¯ow becomes absolutely unstable starting to shed alternating vortices known as the von K arm an vortex street. Switching on a magnetic ®eld of certain strength N at ®xed Reynolds number then for t ! 1 either damps out the vortex shedding yielding a steady ¯ow or leads to a modi®ed vortex street of a still periodic wake ¯ow. The linear 3-D problem was treated in a velocity±vorticity ®nite dierence formulation similar to [18]. The total accuracy is second order, whereby all elliptic equations including the Poisson equation for the electric potential were solved in fourth order. At some initial time t 0 a 3-D vorticity disturbance k h a0 sin2 ; 4 sin hT 10 r3 2 2r of amplitude a0 10ÿ2 was added to a developed 2-D ¯ow being either steady or periodic. Marching in time by simultaneously integrating the 2-D and the linearized 3-D equations, after some transient time the 3-D perturbation enters the expected linear regime where it will either increase or decay exponentially in time. To decide about the global 3-D stability of the ¯ow we monitor in time the total 3-D energy w00r ; w00h ; w00z T 0; a0
Z Z 2p=k Z E3D t
dx dy dz
v02 2
Fig. 2. Typical behavior of the logarithm of the global 3-D energy E3D in time (Re 200, N 0:4, k 0:8, aligned magnetic ®eld). After a transient time the ¯ow enters a linear regime, and the TGR can be detected.
h-direction, with up to 256 ´ 256 points. Boundary conditions are v0 0, v0 0 and @U0 =@r 0 at r 1, hydrodynamic potential ¯ow for v0 , v0 0 and U0 0 at rmax , and out¯ow conditions for v0 , v0 and U0 in an out¯ow angle of 70 downstream at rmax . To validate our numerical approach, we performed intensive tests. The 2-D calculations without magnetic ®eld were in good agreement to results of Fornberg [19] with respect to recirculation length, vorticity at the cylinder surface and drag coecients; also the Strouhal frequency and the vortex street are nicely reproduced. The main error on the TGR in the 3-D calculations was due to the ®nite spatial grid spacing, therefore dense grids had to be applied. Fig. 3 shows a Richardson extrapolation for the TGR with up to 256 ´ 256 points reducing the error to less than 5%. Further, we compared our results for the purely hydrody-
11
integrated over the computational domain. In the following, the slope of the logarithm of this quantity in the linear regime is called the temporal growth rate (TGR), and positive (negative) values of TGR correspond to absolute 3-D instability (stability) of the ¯ow. In general, our global linear stability analysis has the advantage that it detects only absolute instabilities and excludes the convective ones. Fig. 2 shows an example of the temporal growth rate obtained by this method at Re 200, N 0.4, k 0.8 for an aligned magnetic ®eld. For both 2-D and 3-D simulations we used the same exponential polar grid, ri eci , rmax 50, equidistant in
Fig. 3. Richardson extrapolation for the temporal growth rate of E3D at Re 200, N 0:4, k 0:8, aligned magnetic ®eld. NP denotes the number of grid points in both radial and azimuthal directions.
G. Mutschke et al. / Experimental Thermal and Fluid Science 16 (1998) 92±99
95
namic problem at Re 200 with data published recently by Henderson [20] and found an excellent agreement (see Fig. 8). 4. Flow and instability control by magnetic ®elds 4.1. 2-D ¯ow properties To enlighten the action of the magnetic body force on the 2-D ¯ow, we show in Figs. 4 and 5 isolines of the 2D streamfunction W0 of the steady ¯ow at Re 100 for dierent interaction parameter N and aligned and transverse ®elds, respectively. Of course, some of the shown ¯ows are not really steady because of 2-D instabilities (see Section 4.2), this was arti®cially forced by applying a symmetry condition. In both ®gures it is to be seen that an increasing magnetic ®eld acts on the wake by decreasing the length of the recirculation bubble and lowering the separation angle. However, for stronger magnetic ®elds separation is not ®nally suppressed in case (a), rather some asymptotic state seems to be reached with widened and straightened streamlines caus-
Fig. 5. Isolines of the 2-D streamfunction of a steady 2-D solution at Re 100 for a transverse ®eld at dierent N .
ing even upstream a region of very slow ¯ow called ``upstream wake'' [8]. In case (b) the body force acts stronger than in case (a) in terms of recirculation length at the same value of N, further it leads to a complete suppression of the separation for N > 0:8 [17]. For more details of the 2-D ¯ow we refer to [17]. We only want to mention that the vorticity in the boundary layer in general increases dueptothe action of the body force and therefore leads to a N -like increase of the total drag (mainly due to the pressure part) with increasing N. However, as can be seen in the next section, increasing N may lead to a steady ¯ow with a smaller total drag than the time-averaged drag of the corresponding unsteady ¯ow without magnetic ®eld at the same Reynolds number. For example, at Re 200 holds CDaveraged N 0 1:3, whereas CDsteady N 0:3 0:9. 4.2. 2-D stability
Fig. 4. Isolines of the 2-D streamfunction of a steady 2-D solution at Re 100 for an aligned ®eld at dierent N .
For the hydrodynamic cylinder ¯ow it is known that at Re 49 the steady 2-D ¯ow becomes absolutely unstable (undergoing a Hopf bifurcation [2]) starting to shed alternating vortices known as the von Karman vortex street. From the two ®gures above one might already
96
G. Mutschke et al. / Experimental Thermal and Fluid Science 16 (1998) 92±99
For the hydrodynamic cylinder ¯ow today it seems to be clear that at a Reynolds number of about 188.5 the periodic 2-D ¯ow becomes unstable to spanwise periodic 3-D disturbances with a wavelength of about 4 diameters. Two recent papers show excellent agreement between numerical results from a linear stability
investigation [4] and experiments [3]. However, the evolution of these instabilities at higher Re > 220 is not yet completely understood. With increasing Re the most unstable mode changes signi®cantly, further there is large scatter in both experimental and numerical results (for a summary see [22,12]). Therefore we restrict our discussion for the MHD ¯ow mainly to the case of Re 200. We now focus on MHD case (a) with aligned magnetic ®eld. Fig. 7 shows its neutral 2-D stability curve in the (Re, N) parameter plane. Above the curve, any 2-D disturbance will be damped out in time towards a steady ¯ow, whereas below the curve the magnetic ®eld is too weak and can only modify the persisting vortex street. Hunt [1] performed an analytical work on the relation between 2-D and 3-D instabilities, but limited to a local analysis of parallel ¯ows. He found, that for an aligned magnetic ®eld the Squire-theorem is no longer valid. Transferring his results to the present case results in a tangent to our 2-D stability curve beginning at the origin (see Fig. 7). The parallel ¯ow analysis would now distinguish between three dierent regions in the parameter space: Region I: Above the tangent and above the 2-D stability curve before their point of contact the ¯ow is stable to both 2-D and 3-D disturbances, i.e. steady and 2D Region II: Below the 2-D stability curve the ¯ow is primarily 2-D unstable, 3-D instability will occur at higher Re Region III: Between 2-D stability curve and tangent beyond their point of contact the ¯ow is 2-D stable but 3-D unstable! Of course, the MHD cylinder wake is neither a parallel ¯ow nor leads local instability always to global instability. Therefore, the results of Hunt cannot be applied straightforwardly. However, there is a ®rst qualitative hint that similar eects to the violation of the Squiretheorem for parallel MHD ¯ows might exist for the MHD cylinder wake. First results of our linear 3-D stability analysis are shown in Figs. 8±11. Fig. 8 shows curves of temporal growth rates for dierent N at
Fig. 7. Neutral 2-D stability curve for an aligned magnetic ®eld with a tangent from the parallel ¯ow analysis of Hunt.
Fig. 8. Temporal growth rates of the global 3-D energy at Re 200, aligned ®eld.
Fig. 6. Neutral 2-D stability curve for aligned and transverse magnetic ®eld.
expect, that the magnetic ®eld in case (a) and (b) is capable to suppress the shedding of vortices. Fig. 6 shows the curves of neutral 2-D stability in the (Re, N) plane for cases (a) and (b) respectively, separating steady ¯ows above the curve from periodic ones below it. As already expected from the discussion of the 2-D ¯ow above, for (b) weaker magnetic ®elds than for (a) can stabilize the ¯ow. For a comparison with experimental results of Josserand and Weier [10,11,21] at larger Re > 1000 we refer to [21]. The Strouhal frequency of the vortex street modi®ed by a magnetic ®eld decreases only slightly with increasing N until the ¯ow becomes steady at the critical value of N. 4.3. 3-D stability
G. Mutschke et al. / Experimental Thermal and Fluid Science 16 (1998) 92±99
97
Fig. 9. Snapshot of W0 , x0 and e3D (from top to bottom) at Re 200, N 0, k 0.5 (aligned ®eld) in the linear regime.
Fig. 11. Snapshot of W0 , x0 and e3D (from top to bottom) at Re 200, N 0.4, k 0.6 (aligned ®eld) in the linear regime.
Fig. 10. Snapshot of W0 , x0 and e3D (from top to bottom) at Re 200, N 0.05, k 0.7 (aligned ®eld) in the linear regime.
Re 200. Compared to the hydrodynamic case at N 0, a weak magnetic ®eld of N 0.05 seems to decrease the level of 3-D instability. But the ¯ow is still 3-D unstable, the most unstable wave number is slightly decreased. A widening of the curve leads to a band of unstable wave numbers of approximately the same width as for N 0 although shifted to smaller wave numbers. The larger value of N 0:4 shows a broader range of unstable wave numbers than at N 0. The maximum TGR value is only slightly lower than for the hydrodynamic case, i.e. a slight damping in¯uence seems to exist. A further increase of the magnetic ®eld up to N 1 leads to a complete suppression of 3-D instability for Nc3D > 0:75. Here it is interesting to note that for N 0:4 the ¯ow is 3-D unstable, whereas the corresponding 2-D ¯ow is already steady, as can be seen from Fig. 7. The critical value of neutral 2-D stability at Re 200 is about Nc2D 0:27, so a 3-D instability above the 2-D threshold is clearly detected. To characterize the 3-D instabilities, in Figs. 9±11 we plotted isolines of the 2-D streamfunction W0 , the 2-D vorticity x0z and the local 3-D energy density e3D v02 =2 at a time arbitrarily chosen inside the linear regime, for dierent values of N. Both unsteady 2-D ¯ows at N 0 and N 0:05 look similar, the maxima of the energy density e3D are closely arranged with the 2-D vortices shed from the cylinder. For N 0:4 the e3D maxima are closely coupled to the steady 2-D recirculation eddies and seem to be stretched along the magnetic ®eld lines. Fig. 12 shows the 3-D energy density on
98
G. Mutschke et al. / Experimental Thermal and Fluid Science 16 (1998) 92±99
Fig. 12. Logarithm of the 3-D energy density versus radius (downstream distance, at y 0) at Re 200, k 0.8, aligned ®eld.
the downstream axis. For N 0:05 the local variation of e3D resembles the vortex street oscillations of the 2-D ¯ow, whereas for N 0:4 the steady 2-D ¯ow results in a smooth shape of e3D . In both cases, the absolute maximum of e3D is located at about three diameters downstream, with larger distances the 3-D energy density decreases exponentially. First results for a larger value of Re 220 show for the same Interaction parameter N always larger values of the TGR compared to Re 200. This would therefore result in a larger critical value Nc3D and could lead to a rather steep 3-D stability line in the (Re, N) plane. Finally, we would like to summarize ®rst results for cases (b) and (c) at Re 200. Here, already for weak magnetic ®elds a strong damping action can be found, and 3-D instability seems to exist only far below the critical value of neutral 2-D instability. 5. Conclusions and outlook We were numerically investigating the in¯uence of magnetic ®elds on the relation between 2-D and 3-D instabilities in the MHD cylinder ¯ow. The most interesting case turned out to be an aligned magnetic ®eld. By monitoring in time the total energy of the 3-D disturbances we detected global 3-D instability in a parameter region where the ¯ow is 2-D stable, i.e. steady. This is qualitatively in line with the result of Hunt, notwithstanding the fact that his work was restricted to a local parallel ¯ow stability analysis. 3-D instability of a 2-D stable ¯ow is unique for the cylinder problem. It is further interesting to observe that even this standard hydrodynamic problem of the ¯ow around a cylinder represents one example, where the in¯uence of an applied magnetic ®eld on instabilities is non-monotonic and not purely damping but strongly depends on the character of the instability. It would be highly desirable to verify these results experimentally. Recent experiments [21] show a slight ampli®cation of low frequency ¯uctuations due to an aligned magnetic
®eld and support the widening of the unstable wave number region found here. In the following we will continue this work to larger Re and stronger magnetic ®elds; for the aligned ®eld case we will check the existence of a 3-D stability line. We further continue to investigate the other two directions of the magnetic ®eld in order to get a better understanding of the reasons for the dierent behavior in all three cases. For an increasing transverse ®eld it is known that not only the vortex street is damped but also the steady recirculation eddies of the 2-D ¯ow vanish [17]. The damping of 3-D instabilities we found at Re 200 is the strongest here of all three cases considered. Therefore, one might suppose that the arising of 3-D instability is connected to boundary layer separation and vortex shedding ``of a certain strength'' in the near wake. Nomenclature All quantities listed without units are nondimensional. B0 vector of the external magnetic ®eld CD drag coecient E electric ®eld vector E3D total energy of the 3-D disturbance N Interaction parameter Re Reynolds number Rm magnetic Reynolds number TGR temporal growth rate of 3-D disturbances a0 amplitude of the 3-D vorticity disturbance d cylinder diameter ex ,ey ,ez unit vector in x; y; z-direction, respectively fm Lorentz force vector j electric current density vector k wave number in z-direction r radius coordinate (standard polar system) t time v velocity vector v1 free-stream velocity Greek symbols h U W0 m l q r x
angle coordinate (standard polar system) electric potential 2-D streamfunction kinematic viscosity (m2 /s) permeability (Vs/Am) density (kg/m3 ) electric conductivity (1/Xm) vorticity vector
Subscripts C
critical value for neutral stability
Superscripts 2D, 3D with respect to 2-D or 3-D, respectively 0 quantity of the 2-D ¯ow
G. Mutschke et al. / Experimental Thermal and Fluid Science 16 (1998) 92±99 0
T
quantity of the linearized 3-D ¯ow transposed representation of a vector
Operator symbols r
Nabla vector operator inner product cross product
Acknowledgements Financial support under grant No. Ge 682/3-4 and grants for several visits of V. Shatrov at Research Center Rossendorf from ``Deutsche Forschungsgemeinschaft'' are gratefully acknowledged. The work is supported by the European INTAS program under grant No. 941504. We gratefully acknowledge use and support of the computer facilities at HLRZ Supercomputing Center Juelich.
References [1] J.C.R. Hunt, On the stability of parallel ¯ows with parallel magnetic ®elds, Proc. Roy. Soc. A 293 (1996) 342±358. [2] C.P. Jackson, A ®nite-element study of the onset of vortex shedding in ¯ow past variously shaped bodies, J. Fluid Mech. 182 (1987) 23±45. [3] C.H.K. Williamson, Mode A secondary instability in wake transition, Phys. Fluids 8 (1996) 1680±1682. [4] R.D. Henderson, D. Barkley, Secondary instability in the wake of a circular cylinder, Phys. Fluids 8 (6) (1996) 1683±1685. [5] G.L. Brown, Vorticity in the Electromagnetic Control of Turbulence, APS-DFD Meeting, Atlanta, Nov. 1994; D.M. Nosenchuck, Electromagnetic Turbulent Boundary Layer Control, APS-DFD Meeting, Atlanta, Nov. 1994. [6] J.C.S. Meng, Experimental Studies of a Seawater Superconducting Electromagnetic Thruster: A Continuing Quest for Higher Magnetohydrodynamic Propulsive Eciency, Proceedings of the Second International Conference on Energy Transfer in MHD Flows, Aussois (France), 1994, pp. 491±500.
99
[7] A.B. Tsinober, A.H. Shtern, E.V. Shcherbinin, On the separation of the magneto-hydrodynamic boundary layer, Latvijas PSR Zinatnu Akad. Vestis 12 (1963) 49±56. [8] A.B. Tsinober, MHD ¯ow around obstacles (in Russian), Zinatne Publ., Riga (Latvia), 1970. [9] J. Lahjomri, P. Caperan, A. Alemany, The cylinder wake in a magnetic ®eld aligned with the velocity J, Fluid Mech. 253 (1993) 421±448. [10] J. Josserand, P. Marty, A. Alemany, G. Gerbeth, MHD ¯ow around a circular cylinder in an aligned magnetic ®eld, AIAA Report on Progress in Astronautics and Aeronautics 148 (1992) 551±565. [11] J. Josserand, P. Marty, A. Alemany, Pressure and drag measurements on a cylinder in a liquid metal ¯ow with an aligned magnetic ®eld, Fluid Dynam. Res. 11 (1993) 107±117. [12] C.H.K. Williamson, Vortex dynamics in the cylinder wake, Annu. Rev. Fluid Mech. 28 (1996) 447. [13] H.B. Squire, On the stability for three-dimensional disturbances of viscous ¯uid ¯ow between parallel walls, Proc. Roy. Soc. A 142 (1933) 621. [14] R. Moreau, Magnetohydrodynamics, Kluwer Academic, Dordrecht, 1990. [15] D.D. Papailiou, Magneto-¯uid-mechanic turbulent vortex streets, Progr. Astron. Aeron. 100 (1992) 152±173. [16] T.P. Loc, Numerical analysis of unsteady secondary vortices generated by an impulsively started circular cylinder, J. Fluid Mech. 100 (1980) 111±128. [17] V. Shatrov, G. Mutschke, G. Gerbeth, Numerical simulation of the two-dimensional MHD-¯ow around a circular cylinder, Proceedings of the Second International Conference on Energy Transfer in MHD Flows, Aussois (France), 1994 pp. 745±756, also in Magnetohydrodynamics 33 (1997) 3±13. [18] P.J. Roache, Computational Fluid Dynamics, Hermosa Publishers, Albuquerque, 1976. [19] B. Fornberg, A numerical study of steady viscous ¯ow past a circular cylinder, J. Fluid Mech. 98 (1980) 819±855. [20] D. Barkley, R.D. Henderson, Three-dimensional Floquet stability analysis of the wake of a circular cylinder, J. Fluid Mech. 322 (1996) 215. [21] T. Weier, G. Mutschke, G. Gerbeth, A. Alemany, A. Pilaud, On the stability of the MHD ¯ow around a circular cylinder in an aligned magnetic ®eld, Proceedings of the Second International Conference on Energy Transfer in MHD Flows, Aussois (France), 1994, pp. 561±570. [22] H.-Q. Zhang, U. Fey, B.R. Noack, On the transition of the cylinder wake, Phys. Fluids 7 (1995) 779±795.