Turbulent magnetohydrodynamic flow in a square duct: Comparison of zero and finite magnetic Reynolds number cases

Three-dimensional turbulent magnetohydrodynamic flow in a duct with a square cross section and insulating walls is investigated by direct numerical simulations. The flow evolves in the presence of a uniform vertical magnetic field and is driven by an applied mean pressure gradient. A boundary element technique is applied to treat the magnetic field boundary conditions at the walls consistently. Our primary focus is on the large- and small-scale characteristics of turbulence in the regime of moderate magnetic Reynolds numbers up to $\text{Rm}\sim {10}^2$ and a comparison of the simulations with the quasistatic limit at $\text{Rm}=0$. The present simulations demonstrate that differences to the quasistatic case arise for the accessible magnetic Prandtl number $\text{Pm}\sim {10}^{-2}$ and different Hartmann numbers up to $\text{Ha}=43.5$. Hartmann and Shercliff layers at the duct walls are affected differently when a dynamical coupling to secondary magnetic fields is present. This becomes manifest by the comparison of the mean streamwise velocity profiles as well as the skin friction coefficients. While large-scale properties change only moderately, the impact on small-scale statistics is much stronger as quantified by an analysis of local anisotropy based on velocity derivatives. The small-scale anisotropy is found to increase at moderate $\text{Rm}$. These differences can be attributed to the additional physical phenomena which are present when secondary magnetic fields evolve, such as the expulsion of magnetic flux in the bulk of the duct or the presence of turbulent electromotive forces.

Figure 1

Schematic drawing of the MHD duct flow configuration, with a uniform imposed vertical magnetic field, ${\mathit{B}}_0=B_0{\mathit{e}}_z$. Also sketched are the Hartmann (HL) and Shercliff (SL) layers, which are formed at the field-normal and side walls of the square duct respectively. In dimensionless units, $y\in [-1,1]$ and $z\in [-1,1]$.

Figure 2

Evolution of volume-averaged quantities vs time. Data for runs 1, 2, 3, and 5 are shown. (a) Kinetic energy of fluctuations. (b) Rescaled total enstrophy. (c) Energy of the secondary magnetic field ${\mathit{b}}^s$. The subscript $V$ indicates averaging over the whole duct. Quantities are given in (20, 21, 22). The hydrodynamic Reynolds number is $\text{Re}=14500$ in all cases.

Figure 3

Instantaneous contours of the streamwise velocity $u_x(x_0,y,z,t_0)$ at duct cross section $x_0=\pi$. Left: run 3 at $\text{Rm}=0$. Right: run 5 at $\text{Rm}=400$. The contour range for both panels is indicated by the color bar to the right. Hartmann number $\text{Ha}=43.5$.

Figure 4

Instantaneous magnetic field line plot at $\text{Rm}=400$ and $\text{Ha}=43.5$ for run 5 taken at the same time as in the right panel of Fig. 3. The contour coloring represents the magnitude of the streamwise magnetic field component $b_x$. The mean flow is from the upper left to the lower right.

Figure 5

Structure of the mean magnetic field at $\text{Rm}=400$ and $\text{Ha}=43.5$. Left: Contours of the mean axial magnetic field ${\langle b_x\rangle }_{x,t}$ in the duct cross section. Right: Projected field lines of the mean magnetic field in the cross section, colored by its magnitude ${\langle b_{\perp }\rangle }_{x,t}$.

Figure 6

Profiles of the mean magnetic field components. Left: Mean axial magnetic field ${\langle b_x\rangle }_{x,t}$ along $y=0$. The dashed lines indicate the analytical solution for the corresponding laminar flow [23, 24]. Right: Mean wall-normal magnetic field ${\langle b_z\rangle }_{x,t}$ profiles vs $z$ along lines $y=0$ (solid lines) and versus $y$ along $z=0$ (dashed lines), respectively. The Hartmann number is $\text{Ha}=43.5$ and run 3 is the quasistatic run.

Figure 7

Contours of the mean axial velocity ${\langle u_x\rangle }_{x,t}$ in the cross section at $\text{Ha}=43.5$. (a) $\text{Rm}=0$. (b) $\text{Rm}=400$. (c) Difference ${\langle u_x\rangle }_{x,t}(\text{Rm}=400)-{\langle u_x\rangle }_{x,t}(\text{Rm}=0)$. The contour range spans from ${\langle u_x\rangle }_{x,t}=0$ at the light end to ${\langle u_x\rangle }_{x,t}=1.35$ at the dark end in panels (a) and (b).

Figure 8

Comparison of the profiles of the mean axial velocity ${\langle u_x\rangle }_{x,t}$ along the lines $z=0$ (left panel) and $y=0.5$ (right panel). The Hartmann number is again $\text{Ha}=43.5$ and run 3 is the quasistatic case. Only one half of the cross section is plotted in both panels.

Figure 9

Vectors of the mean secondary flow in the cross section at $\text{Ha}=43.5$. (a) $\text{Rm}=0$. (b) $\text{Rm}=400$. The maximum magnitude of the cross-stream velocity $max\left(\sqrt{{〈u_y〉}_{x,t}^2+{〈u_z〉}_{x,t}^2}\right)\approx 0.019$ for both the cases.

Figure 10

Variation of the individual skin-friction coefficients $C_{f,\mathrm{HL}}$ and $C_{f,\mathrm{SL}}$ with the Hartmann number at $\text{Rm}=0$ and $\text{Rm}=400$.

Figure 11

Comparison of the total turbulent stress components ${\tau }_{11}=-{\langle u_x^{\prime }{u}_x^{\prime }+N{\text{Rm}}^{-1}{b}_x^{\prime }{b}_x^{\prime }\rangle }_{x,t}$ (left panel) and ${\tau }_{12}=-{\langle u_x^{\prime }{u}_y^{\prime }+N{\text{Rm}}^{-1}{b}_x^{\prime }{b}_y^{\prime }\rangle }_{x,t}$ (right panel) along the line $z=0$.

Figure 12

Contours of the turbulent kinetic energy $k_u(y,z)$ in the duct cross section at $\text{Ha}=43.5$. Left: $\text{Rm}=0$. Right: $\text{Rm}=400$. The contour levels for both panels are given by the color bar.

Figure 13

Variation of turbulent kinetic energy production $P_k$ in the spanwise direction $y$ along $z=0$. Run 3 is the quasistatic case.

Figure 14

Contour of the turbulent magnetic energy $k_b(y,z)$ (left) and $x$ component of turbulent electromotive force ${\mathcal{E}}_x(y,z)={\langle u_y^{\prime }{b}_z^{\prime }-u_z^{\prime }{b}_y^{\prime }\rangle }_{x,t}$ (right) in the $y\text{−}z$ cross section for $\text{Rm}=400$ and $\text{Ha}=43.5$ (run 5).

Figure 15

Effect of a moderate $\text{Rm}$ on flow anisotropy. Time-averaged anisotropy coefficients obtained from Eq. (45) are shown along the line $z=0.5$ for the case of $\text{Ha}=43.5$.

Figure 16

Isosurfaces of near-wall vortical structures which are obtained by the $\lambda$ criterion with ${\lambda }_2=-0.75$ for $\text{Rm}=0$ (left) and $\text{Rm}=400$ (right) with the common parameters $\text{Ha}=43.5$ and $\text{Re}=14500$. View is along the $y$ direction (flow is from left to right).

Figure 17

Profiles of the skewness of the wall normal derivative $\partial u_x^{\prime }/\partial z$ along the line $y=-0.75$.

Figure 18

Contours of the difference between $\text{Rm}=400$ and $\text{Rm}=0$ in the skewness of wall normal derivative $\partial u_x^{\prime }/\partial z$, in the duct cross section for $\text{Ha}=43.5$.

Figure 19

Sensitivity of the statistically steady skin friction coefficient $C_f$ vs time $t$. Left: Variation of the grid resolution in the streamwise direction. Right: Variation of grid resolution in the cross section of the duct. The domain size is $4\pi \times 2\times 2$. Parameters are $\text{Rm}=0$, $\text{Re}=14500,$ and $\text{Ha}=43.5$.

Figure 20

Sensitivity to the cross-stream grid resolution of the profiles of: Left: mean axial velocity ${\langle u_x\rangle }_{x,t}$ along $y$ at $z=0$ and along $z$ at $y=0.5$. Right: Reynolds stress component ${\tau }_{12}$. The domain size is $2\pi \times 2\times 2$ and $N_x=512$ for all these cases. Parameters are $\text{Rm}=0$, $\text{Re}=14500,$ and $\text{Ha}=43.5$.