12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022

LES OF MHD TURBULENT TAYLOR-COUETTE FLOWIN AXIAL MAGNETIC FIELD

Hiromichi Kobayashi
Ryo Sasaki
Department of Physics, Hiyoshi Campus,
Graduate School of Science and Technology,
Keio University
University of Tsukuba
4-1-1 Hiyoshi, Kohoku-ku, Yokohama, 223-8521, Japan 1-1-1 Tennodai, Tsukuba, 305-8573, Japanhkobayas@keio.jp
s2020848@s.tsukuba.ac.jp

Takayasu Fujino
Hidemasa Takana
Faculty of Engineering, Information and Systems,
Institute of Fluid Science,
University of Tsukuba
Tohoku University
1-1-1 Tennodai, Tsukuba, 305-8573, Japan
2-1-1 Katahira, Aoba-ku, Sendai, 980-8577, Japantfujino@kz.tsukuba.ac.jp
takana@tohoku.ac.jp

ABSTRACTal., 2021b). We have already examined two-dimensional MHD
laminar TC flows for aspect ratio 10 and exhibited that the
number of Taylor vortex gradually diminishes from ten to two
as increasing the magnetic field strength (Sasaki et al., 2021c).
In the present study, large eddy simulation (LES) of MHD
turbulent TC flows is carried out to clarify the effect of themagnetic field strength on the flow and electric current fields
in the MHD energy conversion device.

Large eddy simulation (LES) of magnetohydrodynamic(MHD) turbulent Taylor-Couette (TC) flows is conducted toreveal the effect of the Hartmann number proportional to theaxial magnetic field strength on the flow and electric current
fields. The upper-lower walls are insulated and the innerouter sidewalls are set to insulating walls (open condition) or
perfectly conducting walls (short condition). As increasing
the Hartmann number, turbulent fluctuations are suppressed,
and the turbulent flow distributions approach the MHD laminar flow distributions. The orientations of turbulent vortices
change from the azimuthal flow direction to the axial magnetic field direction even in the TC flow. The triplet high-speed
streaky structures are uncovered near the inner wall.

GOVERNING EQUATIONS AND NUMERICALCONDITIONSIn this study, LES is conducted to investigate unsteady
MHD turbulence structures and laminarization phenomena by
the Lorentz force. The low magnetic Reynolds number is assumed. Consequently, induced magnetic fields and the fluctuation of magnetic field are discarded although we can compute the induced magnetic fields as carried out in the previous
study (Kobayashi et al., 2012). Governing equations under the
low magnetic Reynolds number consit of filtered incompressible Navier-Stokes equations with the Lorentz force, a filteredcontinuity equation, filtered Maxwell equations and the generalized Ohm’s law as described below.
(i) Continuity equation:

INTRODUCTION
Taylor-Couette (TC) flow is one of important and fundamental flows (Coles, 1965; Grossmann et al., 2016) and has
been actively studied in various research fields. Direct numerical simulation (DNS) of non-conducting turbulent TC flows
has been conducted, and the fine turbulent vortex structures
and the turbulence statistics of TC flow are clarified (e.g., see
Dong, 2007). The DNS of conducting turbulent TC flows, e.g.,liquid metal flows, is performed with conducting sidewalls under the periodic conditions in the azimuthal and axial directions, and the suppression of the fine turbulent vortices is revealed in the axial magnetic field (Leng et al., 2018). Whenimposing strong magnetic field, it is known that the turbulent
vortex is elongated in the direction of the applied magnetic
field. The aligned vortex structure is the so-called quasi-twodimensional (Q2D) structure and appears in magnetohydrodynamic (MHD) flows (e.g., see Chen et al., 2021).
From the viewpoint of the torque control of wind turbine via the Lorentz force and the recovery of excessive wind
power as electrical output, Takana & Tanida (2017) has proposed a co-axial MHD energy conversion device and demonstrated the performance in experiments. We conducted theone-dimensional theoretical analysis of the flow field (Sasaki
et al., 2021a) and power generation characteristics (Sasaki et∇ · u = 0,(1)(ii) Navier-Stokes equations:11∂u+ (u · ∇)u = − ∇p + ν ∇2 u − ∇ · τ + ( j × B),
∂t
ρ
ρ

(2)

(iii) Maxwell equations:

1∇ × E = 0,

(3)

∇ · j = 0,(4)

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022

Outer ring electrode

rout

ωin

Bz

Rotary shaft electrode
rin

z
L

θ
r

h

I
Load

Figure 1. Computational domain where the inner cylindricalelectrode is rotating and the outer one is at rest.

(iv) Ohm’s law:()
j = σ E +u×B ,(5)

where (¯·) denotes the filtered variable, the velocity u = ui , thepressure p, the subgrid-scale (SGS) stress tensor τ = τi j , theelectric current density j, the electric field E are used. The
magnetic flux density B is set to only the external magnetic
flux density (0, 0, Bz ). The electric potential ϕ is obtained fromthe following Poisson equation:(v) Poisson equation for the electric potential:()∇2 ϕ = ∇ · u × B ,(6)
Figure 2. Mean (upper) and root mean square (lower) profiles of azimuthal velocity for the open condition at aspect ratioof 10. The mean velocity with avarage indicates the averagein the z direction. The mean velocity without average exhibits
the mean velocity at z = L/2.

that is derived from eq.(3) - (5).
The equations are discretized in the r − θ − z cylindricalcoordinate. The magnetic field is imposed in the z (axial) direction. We use the second order Adams-Bashforth method
for the time marching scheme and the second order central
finite difference method for the spatial discretization. The
SMAC scheme is used for the coupling between velocity andpressure. We adopt the coherent structure model (Kobayashi,2005), which is useful subgrid-scale model in the MHD flows(Kobayashi, 2008; Kobayashi et al., 2012).

NUMERICAL RESULTS
Results of Aspect Ratio of 10
First, we show the validation results for TC flow of aspectratio of 10. Figure 2 shows the Mean (upper) and root meansquare (lower) profiles of azimuthal velocity for the open condition at aspect ratio of 10. In the present simulation, upper andlower walls exist. Therefore, we showed two mean velocities:
with average in the z direction and at z = L/2 without average.
The mean and root mean square velocity profiles with average
are good agreement with the DNS results (Dong, 2007).
Let us see the Taylor vortex cells. In Fig. 3, the mean
distributions of stream line and radial velocity for Ha = 0, 50,
70 and 100 of the open condition in the r − z plane at aspectratio of 10 are displayed. As increasing the Hartmann number,
we found that the number of the Taylor vortex or the eddy
current is gradually reduced from ten to two. Note that two
Taylor vortex cells remains due to the existence of the upperand lower walls even for large Hartmann number.
Figure 4 exhibits the mean distributions of the Joule dissipation and the electric current streamline for Ha = 70 and 100of open and short conditions in the r − z plane at aspect ratioof 10. The cells of electric current streamline are comparable
to the Taylor vortex cells for various Hartmann numbers. The
Joule dissipation becomes high at the location of the high ra-

Figure 1 shows the computational domain. The ratio of
the inner radius to the outer radius, namely radius ratio, is
set to 0.5. The aspect ratios of 1 and 10 are adopted for the
cross section in the r − z plane. We use the insulating walls
for the upper and lower walls. We examine two types of the
sidewall condition, (1) open condition: insulating sidewallsand (2) short condition: perfectly conducting sidewalls. The
no-slip boundary condition is utilized on the walls and nopenetration and penetration wall conditions of electric current
are used for open and short conditions, respectively. The computational domain in the azimuthal direction is π /2 and the periodic boundary condition is adopted. The grid points are set to(Nr , Nθ , Nz ) = (32, 64, 128) because LES of aspect ratio of 10with those grid points shows good agreement with the mean
velocity and turbulent intensity profiles of the DNS (Dong,
2007) (not shown here). Reynolds number is set to 8000. The
Hartmann number Ha, the ratio of the Lorentz force to the viscous force, is proportional to the magnetic field strength and isvaried from 0 to 100. The Lorentz force acts against the flow
and suppresses the turbulence.

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12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022
appears on the outer-wall side whereas for that of the open
condition the mean reverse flow never emerges on there.
Figure 6 displays the mean distributions of azimuthal velocity and streamline for Ha = 70 and 100 of open and shortconditions in the r − z plane. Since the upper and lower walls
exist, two Taylor vortices appear for the aspect ratio of 1.
As increasing the Hartmann number, the Taylor vortices arecompressed toward the inner wall owing to the strong Lorentz
force.
Figure 7 shows the mean distributions of the Joule dissipation and the electric current streamline for Ha = 70 and 100of open and short conditions in the r − z plane. Under the open
condition, eddy currents are induced along the Taylor vortices
because all walls are insulated, and electric current cannot penetrate the insulating walls. The electric current concentrates
not only near the inner and upper-lower walls but also on thecentral plane z/h = 0.5, so that the high Joule dissipation (inother words Joule heating) occurs there. However, under the
short condition, the electric current flows near the upper-lower
walls from the inner wall to the outer wall, avoiding the central region where reverse azimuthal velocity appears as shownin Fig. 5(c) and Fig. 6(c)-(d). Moreover, the electric current
flows from the inner wall to itself. The current is the so-called
short-circuit current. The eddy current and short-circuit current cannot be extracted as electrical output.
Figure 8 exhibits vortex structures extracted by the isosurfaces of the second invariant Q = 0.25 of velocity gradient tensor for Ha = 0, 50, 70 and 100 of the open condition.
At Ha = 0, the orientations of the vortices show in the azimuthal direction. At Ha = 50, the vortices are suppressed by
the Lorentz force. For Ha = 70 and 100, the orientations of
the vortices change to the axial direction. The vortices are theso-called quasi-two-dimensional (Q2D) vortices that align tothe direction of the imposed magnetic field (e.g., see Chen etal., 2021). It is found that the Q2D vortices emerge even in the
TC flow.
The contours of instantaneous azimuthal velocity in thevicinity of inner wall for Ha = 0, 50, 70 and 100 of the open
condition are visualized in Fig. 9. As increasing the Hartmann number, the low-speed streaky structures are curbed.
For Ha = 0 and 30, the high and low-speed streaky structures
are advected with the flow velocity. For Ha = 70 and 100,
the high-speed streaky structures have triplet structures andmove slowly although the flow velocities near the inner wall
are higher than those for Ha = 0 and 50 as shown in Fig. 5(a).
The slow velocity may correspond to a group velocity. It will
be investigated in more dial.

Figure 3. Mean distributions of stream line and radial velocity for Ha = 0, 50, 70 and 100 of the open condition in the r − zplane at aspect ratio of 10.

Figure 4. Mean distributions of the Joule dissipation and theelectric current streamline for Ha = 70 and 100 of open andshort conditions in the r − z plane at aspect ratio of 10.

CONCLUSIONS

dial velocity as shown in Fig. 3. This is due to the centrifugal
force. Owing to the open condition, the electric current flows
very close to the wall.

We conducted the LES of MHD turbulent TC flows andinvestigated the effect of the Hartmann number on the velocity
and electric current distributions. When increasing the Hartmann number, turbulent fluctuations are suppressed, and thevelocity and electric current distributions approach the MHD
laminar flow distributions. For high Hartmann numbers, two
Taylor vortices appear, and the eddy currents for the open condition or the short-circuit currents for the short condition are
induced along the TC vortices. The fine turbulent vortices orientating in the azimuthal direction for no magnetic field alignin the axial magnetic field direction, so that we found that the
Q2D structures appear even in TC flow. The triplet high-speed
streaky structures are discovered for high Hartmann numbers
and are transported slowly in the azimuthal (flow) direction.

Results of Aspect Ratio of 1
Let us move on to the results of aspect ratio of 1. The
mean and root mean square profiles of azimuthal velocity foropen and short conditions are shown in Fig. 5. At Ha = 0,mean velocity gradient is sharp and the velocity fluctuation isstrong near the inner and outer walls. This is the typical turbulent TC statistics (Dong, 2007). When increasing the Hartmann number, the velocity fluctuation is suppressed and thenthe mean velocity gradient becomes gentle. For the high Hartmann number of the short condition, the mean reverse flow

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12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022

Figure 5. Mean (left) and root mean square (right) profiles of azimuthal velocity for open (upper) and short (lower) conditions.

Figure 6. Mean distributions of azimuthal velocity and streamline for Ha = 70 and 100 of open and short conditions in the r − z plane.

412th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022

Figure 7. Mean distributions of Joule dissipation and electric current streamline for Ha = 70 and 100 of open and short conditions in
the r − z plane.

Figure 8.

Iso-surfaces of the second invariant Q = 0.25 of velocity gradient tensor for Ha = 0, 50, 70 and 100 of the open condition.

REFERENCES

D ONG , S. 2007 Direct numerical simulation of turbulent
Taylor-Couette flow. J. Fluid Mech. 587, 373–393.
G ROSSMANN , S., L OHSE , D. & S UN , C. 2016
High-Reynolds number Taylor-Couette turbulence.
Annu. Rev. Fluid Mech. 48, 53–80.
KOBAYASHI , H. 2005 The subgrid-scale models based on co-

C HEN , L., P OTHERAT, A., N I , M-J. & M OREAU , R.
2021 Direct numerical simulation of quasi-two-dimensional
MHD turbulent shear flows. J. Fluid Mech. 915, A130.
C OLES , D. 1965 Transition in circular Couette flow. J. Fluid
Mech. 21, 385–425.

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Osaka, Japan, July 19–22, 2022

Figure 9.

Contours of instantaneous azimuthal velocity in the vicinity of inner wall for Ha = 0, 50, 70 and 100 of the open condition.

tron Comm. Jpn. 104, No. 2, pp. 1-9.
S ASAKI , R., F UJINO , T., TAKANA , H. & KOBAYASHI , H.
2021 Theoretical analysis on power generation characteristics for co-axial MHD energy conversion device. IEEJ
Trans. Power Energy 14, No. 10, pp. 642-648 (in Japanese).
S ASAKI , R., F UJINO , T., TAKANA , H. & KOBAYASHI , H.
2021 Quasi-two-dimensional numerical analysis of influence of taylor vortex and eddy current on generation characteristics of co-axial MHD energy conversion device underlaminar flow. IEEJ Trans. Power Energy (in Japanese) (to be
published).
TAKANA , H. & TANIDA , A. 2017 Development and fundamental characteristics of co-axial MHD energy conversion
device. Mech. Eng. Journal 4, No. 1, 16-00500.

herent structures for rotating homogeneous turbulence andturbulent channel flow. Phys. Fluids 17, 045104.
KOBAYASHI , H. 2008 Large eddy simulation of magnetohydrodynamic turbulent duct flows. Phys. Fluids 20, 015102.
KOBAYASHI , H., S HIONOYA , H. & O KUNO , Y. 2012 Turbulent duct flows in a liquid metal magnetohydrodynamic
power generator. J. Fluid Mech. 713, 243–270.
L ENG , X., KOLESNIKOV, Y.B., K RASNOV,D., & L I , B.
2018 Numerical simulation of turbulent Taylor-Couette
flow between conducting cylinders in an axial magnetic
field at low magnetic Reynolds number. Phys. Fluids 30,
015107.
S ASAKI , R., F UJINO , T., TAKANA , H. & KOBAYASHI , H.
2021 Theoretical analysis of annular laminar flows with
MHD interaction driven by rotating co-axial cylinder. Elec-6