Inertial waves and mean velocity profiles in a rotating pipe and a circular annulus with axial flow

In this paper we solve the inviscid inertial wave solutions in a circular pipe or annulus rotating constantly about its axis with moderate angular speed. The solutions are constructed by the so-called helical wave functions. We reveal that the mean velocity profiles must satisfy certain conditions to accommodate the inertial waves at the bulk region away from boundary. These conditions require the axial and azimuthal components of the mean velocity to take the shapes of the zeroth and first order Bessel functions of the first kind, respectively. The theory is then verified by data obtained from direct numerical simulations for both rotating pipe and circular annulus, and excellent agreement is found between theory and numerical results. Large scale vortex clusters are found in the bulk region where the mean velocity profiles match the theoretical predictions. The success of the theory in rotating pipe, circular annulus, and streamwise rotating channel suggests that such inertial waves are quite common in wall bounded flow with background rotation.

Figure 1

Velocity profiles of DNS results and the predictions of IW theory in the bulk region for low Reynolds-number cases. (a) ${\overline{u}}_z$ (symbols) compared to Eq. (16) (lines). For clarity the curves are shifted upward from the previous one by $0.5$. (b) ${\overline{u}}_{\theta }$ (symbols) compared to Eq. (15) (lines).

Figure 2

The same curves as in Fig. 1 for high Reynolds-number cases.

Figure 3

The mean velocity profiles in rotating circular annulus. The symbols are numerical results and the curves are given by Eq. (17). The short vertical lines mark the locations $r_0$ of each case. The curves on the left of $r_0$ are computed by $(K_i,A_i)$ and on the right of $r_0$ by $(K_o,A_o)$, respectively. (a) Profiles of mean axial velocity, and (b) profiles of mean azimuthal velocity. In (b) for clarity each curve is shifted upward by $0.2$ from the previous one.

Figure 4

Vortices depicted by $Q$ criterion in the region $1.2<r<1.8$ for case C2. Blue color (dark grey) marks the vortices in region $1.2<r<r_0$ and yellow color (light grey) in region $r_0<r<1.8$, respectively. The main axial flow is from upper-left corner to lower-right corner.