Effects of an oscillating magnetic field on homogeneous ferrofluid turbulence

This paper presents the results from direct numerical simulations of homogeneous ferrofluid turbulence with a spatially uniform, applied oscillating magnetic field. Due to the strong coupling that exists between the magnetic field and the ferrofluid, we find that the oscillating field can affect the characteristics of the turbulent flow. The magnetic field does work on the turbulent flow and typically leads to an increased rate of energy loss via two dissipation modes specific to ferrofluids. However, under certain conditions this magnetic work results in injection, or a forcing, of turbulent kinetic energy into the flow. For the cases considered here, there is no mean shear and the mean components of velocity, vorticity, and particle spin rate are all zero. Thus, the effects shown are entirely due to the interactions between the turbulent fluctuations of the ferrofluid and the magnetic field. In addition to the effects of the oscillation frequency, we also investigate the effects of the choice of magnetization equation. The calculations focus on the approximate centerline conditions of the relatively low Reynolds number turbulent ferrofluid pipe flow experiments described previously [K. R. Schumacher et al., Phys. Rev. E 67, 026308 (2003)].

Figure 1

Comparison of the spectrum of the turbulent kinetic energy when using ${32}^3$, ${64}^3$, and ${128}^3$ modes when $H=316\text{Oe}$ and $\Omega =60\text{Hz}$ with the Shliomis (Ref. [8]) magnetization equation. The spectral comparison is shown at the end of 6.19 large eddy turnover times ($T$), where $T=0.00485\text{s}$.

Figure 2

Vortex viscous dissipation due to the antisymmetric part of the stress versus normalized applied magnetic field; $\xi =1.92$ $\left(H=316\text{Oe}\right)$ and $\Omega {\tau }_B=0.02\pi$. Magnetic equations of Shliomis (Ref. [8]), Felderhof and Kroh (Ref. [10]), and Martsenyuk et al. (Ref. [9]). As the magnetic field oscillations in time, $\xi$ goes from $-1.92$ to $+1.92$.

Figure 3

Normalized mean square velocity vs normalized time. The mean square velocity, $⟨u^2⟩$, is normalized by the mean square velocity at $t=0$, $⟨u_0^2⟩$, and the time is normalized by the large eddy turnover time, $T$, where $T=0.00485\text{s}$.

Figure 4

Taylor microscale vs time normalized by the large eddy turnover time.

Figure 5

Mean square spin and mean square vorticity vs time. All panels: dotted line—$\xi =0$, nonoscillating limit; dashed line—$\xi =7.68$, nonoscillating limit; dash-dot line—$\xi =\infty$ nonoscillating limit.

Figure 6

Limit cycles of twice the mean square spin normalized by the mean square vorticity vs nondimensional magnetic field strength. All panels: dotted line—$\xi =0$, nonoscillating limit; dash-dot line—$\xi =7.68$, nonoscillating limit; dashed line—$\xi =\infty$ nonoscillating limit.

Figure 7

Limit cycles of vortex viscous dissipation rate vs $\xi$. Magnetic field amplitude of $\xi =7.68$, and using the Martsenyuk (Ref. [9]) magnetization equation. Oscillation frequencies of (a) $\Omega {\tau }_B=2\pi /100$, (b)$2\pi /10$, and (c)$2\pi$. The labels (A–G) in panel (b) correspond to the spectra in Fig. 9.

Figure 8

Limit cycle of transfer rate of kinetic energy to spin energy vs $\xi$. Magnetic field amplitude of $\xi =7.68$. Oscillation frequencies of (a) $\Omega {\tau }_B=2\pi /100$, (b) $2\pi /10$, and (c) $2\pi$. The labels (A–E) in panel (a) correspond to Figs. 10a, 11a; labels (A–G) in panel (b) correspond to Figs. 10b, 11b; labels (A–E) in panel (c) correspond to the Figs. 10c, 11c.

Figure 9

Vortex viscous dissipation rate spectra vs wavenumber. Magnetic field amplitude of $\xi =7.68$. The labels (A–G) correspond to Fig. 7b.

Figure 10

Spectra plot of the transfer rate of kinetic energy to spin energy vs wavenumber at different times. Magnetic field amplitude of $\xi =7.68$, and using the Martsenyuk (Ref. [9]) magnetization equation. Oscillation frequencies of (a) $\Omega {\tau }_B=2\pi /100$, (b) $2\pi /10$, and (c) $2\pi$. The labels (A–E) in panel (a) correspond to Figs. 8a, 11a; labels (A–G) in panel (b) correspond to Figs. 8b, 11b; labels (A–E) in panel (c) correspond to Figs. 8c, 11c.

Figure 11

Transfer rate of kinetic energy to spin energy vs time. Magnetic field amplitude of $\xi =7.68$, and using the Martsenyuk (Ref. [9]) magnetization equation. Oscillation frequencies of (a) $\Omega {\tau }_B=2\pi /100$, (b) $2\pi /10$, and (c)$2\pi$. The labels (A–E) in panel (a) correspond to Figs. 8a, 10a; labels (A–G) in panel (b correspond to Figs. 8b, 10b; labels (A–E) in panel (c) correspond to Figs. 8c, 10c.

Figure 12

Fractional turbulent kinetic energy (a), and classical viscous dissipation (b), vs normalized time. The oscillation frequencies ($\Omega =60$, 400, and 1000 Hz) and amplitude $\left(\xi =1.92\right)$ correspond with experimental conditions in Schumacher et al. (Ref. [6]). The Shliomis (Ref. [8]) magnetization equation is used.

Figure 13

(a) Fractional turbulent kinetic energy, and (b) classical viscous dissipation rate vs normalized time. Similar magnetic field conditions as Fig. 12, but for flow conditions described in Table . The Shliomis (Ref. [8]) magnetization equation is used.

Figure 14

Fractional turbulent kinetic energy (a), and classical viscous dissipation rate (b), vs normalized time. The simulation using ${64}^3$ Fourier modes is superimposed on the results obtained when using ${128}^3$ modes. Same conditions as the 1000 Hz case in Fig. 13. The Shliomis (Ref. [8]) magnetization equation is used.