Effect of roll number on the statistics of turbulent Taylor-Couette flow

A series of direct numerical simulations in large computational domains has been performed in order to probe the spatial feature robustness of the Taylor rolls in turbulent Taylor-Couette flow. The latter is the flow between two coaxial independently rotating cylinders of radius $r_i$ and $r_o$, respectively. Large axial aspect ratios $\mathrm{\Gamma }=7–8$ [with $\mathrm{\Gamma }=L/(r_o-r_i)$, and $L$ the axial length of the domain] and a simulation with $\mathrm{\Gamma }=14$ were used in order to allow the system to select the most unstable wave number and to possibly develop multiple states. The radius ratio was taken as $\eta =r_i/r_o=0.909$, the inner cylinder Reynolds number was fixed to ${\text{Re}}_i=3.4\times {10}^4$, and the outer cylinder was kept stationary, resulting in a frictional Reynolds number of ${\text{Re}}_{\tau }\approx 500$, except for the $\mathrm{\Gamma }=14$ simulation where ${\text{Re}}_i=1.5\times {10}^4$ and ${\text{Re}}_{\tau }\approx 240$. The large-scale rolls were found to remain axially pinned for all simulations. Depending on the initial conditions, stable solutions with different number of rolls $n_r$ and roll wavelength ${\lambda }_z$ were found for $\mathrm{\Gamma }=7$. The effect of ${\lambda }_z$ and $n_r$ on the statistics was quantified. The torque and mean flow statistics were found to be independent of both ${\lambda }_z$ and $n_r$, while the velocity fluctuations and energy spectra showed some box-size dependence. Finally, the axial velocity spectra were found to have a very sharp dropoff for wavelengths larger than ${\lambda }_z$, while for the small wavelengths they collapse.

Figure 1

Instantaneous angular velocity at the midgap for the $\mathrm{\Gamma }=8$ simulation. $\tilde{z}=z/d$ is the axial direction and $\tilde{x}=r_a\theta d$ the rescaled azimuthal direction with $r_a$ the midgap radius. The signature of the Taylor rolls can be seen as a large, stripy pattern, which appears to be fixed in the axial position for the entire azimuthal extent. For comparison, the (smaller) white red line indicates the typical size of a small TC flow computational box of Ref. [7]. The (larger) light gray dashed line indicates the largest box simulated in Ref. [18].

Figure 2

Azimuthally and temporally averaged azimuthal velocity for the G7R2 and G7R3 cases, starting from a random velocity field (left, G7R2) or from a rescaled velocity field with three rolls (right, G7R3 case). From the characteristic signature of the rolls, the different roll wavelength can be appreciated. Both cases have been run for about 100 large eddy turnover times based on the large-eddy turnover time and appear to be stable.

Figure 3

Azimuthally and temporally averaged azimuthal velocity for the G14R7 case.

Figure 4

The blue line shows the temporal evolution of the azimuthally averaged azimuthal velocity at a point in the midgap inside the ejection region of a roll. The orangish line shows the running average of this quantity, and the ochre horizontal line shows the final average. The data used are for only 120 time units, as these are representative of all simulations in this paper.

Figure 5

Mean streamwise velocity profile at the inner cylinder in wall units for the four simulated cases. Symbols as in Table 1. Thin black dashed line indicates $r^{+}=u^{+}$.

Figure 6

Fluctuation velocity profiles at the inner cylinder in wall units for the four simulated cases. Symbols as in Table 1.

Figure 7

Roll-less fluctuation velocity profiles at the inner cylinder in wall units for the four simulated cases. Symbols as in Table 1.

Figure 8

Premultiplied velocity energy spectra $\mathrm{\Phi }k$ in the axial (left) and azimuthal (right) directions for the azimuthal (left) and radial (right) velocities. Symbols as in Table 1.