Transient growth in Taylor-Couette flow of a Bingham fluid

In this paper we investigate linear transient growth of perturbation energy in Taylor-Couette flow of a Bingham fluid. The effects of yield stress on transient growth and the structure of the optimal perturbation are mainly considered for both the wide-gap case and the narrow-gap case. For this purpose we complement the linear stability of this flow subjected to axisymmetric disturbances, presented by Landry et al. [M. P. Landry, I. A. Frigaard, and D. M. Martinez, J. Fluid Mech. 560, 321 (2006)], with the transient growth characteristics of both axisymmetric and nonaxisymmetric perturbations. We obtain the variations of the relative amplitude of optimal perturbation with yield stress, analyze the roles played by the Coriolis force and the additional stress in the evolution of meridional perturbations for the axisymmetric modes, and give the explanations for the possible change of the optimal azimuthal mode (featured by the maximum optimal energy growth $G_{\mathrm{opt}})$ with yield stress. These results might help us in the understanding of the effect of fluid rheology on transient growth mechanism in vortex flows.

Figure 1

Marginal stability curves in the $(B,\mathrm{R}{\mathrm{e}}_1)$ plane for $\eta =0.5$, 0.6, 0.7, 0.75, 0.8, 0.85, 0.9, 0.95, and 0.99 with $\mathrm{R}{\mathrm{e}}_2=1000$.

Figure 2

Constant maximum of optimal growth $G_{\max }$ in the $(B,\mathrm{R}{\mathrm{e}}_1)$ plane for (a) $\eta =0.5$ and (b) $\eta =0.99$. The dashed lines correspond to $G_{\max }(B,\mathrm{R}{\mathrm{e}}_1)=1$.

Figure 3

Amplitudes of the (a) and (c) azimuthal and (b) and (d) radial directions of the optimal perturbation with $\eta =0.5$ at (a) and (b) $\mathrm{R}{\mathrm{e}}_1=1510$ and $k=10.34$ and (c) and (d) $\mathrm{R}{\mathrm{e}}_1=70$ and $k=5.04$.

Figure 4

Amplitudes of the (a) azimuthal and (b) radial direction of the optimal perturbation with $\eta =0.99,\mathrm{R}{\mathrm{e}}_1=340$, and $k=2.82$.

Figure 5

(a) and (c) Meridional velocity fields and (b) and (d) contours of azimuthal velocity of the optimal inputs at (a) and (b) $\eta =0.5,\mathrm{R}{\mathrm{e}}_1=1510$, and $k=10.34$ and (c) and (d) $\eta =0.99,\mathrm{R}{\mathrm{e}}_1=340$, and $k=2.82$.

Figure 6

Time evolution of the energies in the meridional components $E_{rz}$ and in the azimuthal component $E_{\theta }$ of the optimal perturbation for (a) $\eta =0.5,\mathrm{R}{\mathrm{e}}_1=1510$, and $k=10.34$ and (b) $\eta =0.99,\mathrm{R}{\mathrm{e}}_1=340$, and $k=2.82$.

Figure 7

(a) Radial-axial vector plot of the Coriolis force and its Helmholtz decomposition into (b) the rotational part, (c) the vector plot of the additional stress field, and (d) its rotational part, for $\eta =0.5,\mathrm{R}{\mathrm{e}}_1=1510,B=0.25$, and $k=10.34$.

Figure 8

(a) Radial-axial vector plot of the Coriolis force and its Helmholtz decomposition into (b) the rotational part, (c) the vector plot of the additional stress field, and (d) its rotational part, for $\eta =0.5,\mathrm{R}{\mathrm{e}}_1=1510,B=9.1$, and $k=10.34$.

Figure 9

Radial distribution of the energy spectrum function for the wide-gap case for $\eta =0.5,\mathrm{R}{\mathrm{e}}_1=1510,k=10.34$, and (a) $B=0.25$ and (b) $B=9.1$. In the plots of the energy spectrum function (including Figs. 12 and 15), the solid line is for the original optimal perturbation $\mathbf{u}\left(0\right)$, the dashed line is for the superimposed optimal perturbation by the rotational part of the Coriolis force, and the dash-dotted line is for the superimposed optimal perturbation by the rotational part of the additional stress field.

Figure 10

(a) Radial-axial vector plot of the Coriolis force and its Helmholtz decomposition into (b) the rotational part, (c) the vector plot of the additional stress field, and (d) its rotational part, for $\eta =0.99,\mathrm{R}{\mathrm{e}}_1=340,B=0.25$, and $k=2.82$.

Figure 11

(a) Radial-axial vector plot of the Coriolis force and its Helmholtz decomposition into (b) the rotational part, (c) the vector plot of the additional stress field, and (d) its rotational part, for $\eta =0.99,\mathrm{R}{\mathrm{e}}_1=340,B=9.1$, and $k=2.82$.

Figure 12

Radial distribution of the energy spectrum function for the narrow-gap case for $\eta =0.99,\mathrm{R}{\mathrm{e}}_1=340,k=2.82$, and (a) $B=0.25$ and (b) $B=9.1$.

Figure 13

Amplitudes of the (a) and (c) azimuthal and (b) and (d) radial directions of the optimal perturbation for counterrotating cylinders at (a) and (b) $\eta =0.5,\mathrm{R}{\mathrm{e}}_1=320$, and $k=8.34$ and (c) and (d) $\eta =0.99,\mathrm{R}{\mathrm{e}}_1=800$, and $k=3.68$.

Figure 14

Time evolution of the energies in the meridional components $E_{rz}$ and in the azimuthal component $E_{\theta }$ of the optimal perturbation for counterrotating cylinders for (a) $\eta =0.5,\mathrm{R}{\mathrm{e}}_1=320$, and $k=8.34$ and (b) $\eta =0.99,\mathrm{R}{\mathrm{e}}_1=800$, and $k=3.68$.

Figure 15

Radial distribution of the energy spectrum function for counterrotating cylinders for $B=9$ and (a) $\eta =0.5,\mathrm{R}{\mathrm{e}}_1=320$, and $k=8.34$ and (b) $\eta =0.99,\mathrm{R}{\mathrm{e}}_1=800$, and $k=3.68$.

Figure 16

The $G\left(t\right)$ curves with different azimuthal wave numbers for the (a) and (c) $B=0$ and (b) and (d) $B=1$ cases at (a) and (b) $\eta =0.5,\mathrm{R}{\mathrm{e}}_1=320$, and $k=8.34$ and (c) and (d) $\eta =0.99,\mathrm{R}{\mathrm{e}}_1=800,k=3.68$.

Figure 17

Axial vorticity ${\omega }_z$ contours of (a) and (c) the optimal perturbation $\mathbf{u}\left(0\right)$ and (b) and (d) its output $\mathbf{u}\left(t_{\mathrm{opt}}\right)$ at the corresponding optimal time at $\eta =0.5,\mathrm{R}{\mathrm{e}}_1=320$, and $k=8.34$ and (a) and (b) $B=0$ and (c) and (d) $B=1$. Shaded regions and dashed lines denote negative vorticity. Contour levels are (a) [±5.62, ±28.1, 5.62], (b) [±4.64, ±23.2, 4.64], (c) [±6.26, ±31.3, 6.26], and (d) [±4.28, ±21.4,4.28].

Figure 18

Amplitudes of the (a) and (c) azimuthal and (b) and (d) radial directions of the optimal perturbation for the eigenmode $m=3$ at (a) and (b) $\eta =0.5,\mathrm{R}{\mathrm{e}}_1=320$, and $k=8.34$ and (c) and (d) $\eta =0.99,\mathrm{R}{\mathrm{e}}_1=800$, and $k=3.68$.

Figure 19

Time evolution of the inertial term $\mathrm{R}{\mathrm{e}}_1{J}_I$, the viscous dissipative term $J_V$, the yield stress dissipative term $B{J}_Y$, and the energy growth rate $dE/dt$ for the modes $m=3$ and 4 with $\eta =0.5,\mathrm{R}{\mathrm{e}}_1=320$, and $k=8.34$, and $B=0$, 1, 5, and 9. The time of the arrest of energy growth for the modes (i) $m=3$ and (ii) $m=4$ with $B=0$ are indicated.

Figure 20

Time evolution of the inertial term $\mathrm{R}{\mathrm{e}}_1{J}_I$, the viscous dissipative term $J_V$, the yield stress dissipative term $B{J}_Y$, and the energy growth rate $dE/dt$ for the modes $m=0$ and 1 with $\eta =0.99,\mathrm{R}{\mathrm{e}}_1=800,k=3.68$, and $B=0$, 1, 5 and 9.