Flow induced by a rotating cone: Base flow and convective stability analysis

The steady flow over a rotating cone is investigated theoretically and numerically in order to improve the traditional von Kármán solution by proposing a self-similar correction which is an explicit function of the cone angle. The effect of the correction on the linear stability analysis of the rotating-cone flow is successively investigated through a weakly divergent approach. Both the base flow correction and the results of the stability analysis are validated against dedicated numerical simulations. As for the base flow, the comparison shows a clear improvement obtained by using the proposed correction in comparison with the classical von Kármán solution. As for the stability properties of the flow, the com- parison against the reference simulations shows a good agreement among all the approaches for large azimuthal wave numbers, but a better performance is obtained with the weakly divergent approach for lower wave numbers. The latter approach provides a lower critical Reynolds number than that predicted by parallel theory and, most importantly, changes the interplay between modes I and II with respect to what predicted by the parallel stability calculations. Finally, it is observed that the proposed correction of base flow has a slight effect on the stability analysis of the considered cases, but it may have important effects for low cone angles. Thus, while the classical Kármán solution is appropriate for large cone angles, the proposed correction is recommended for future stability analyses of slender cones.

Figure 1

Schematic representation of the rotating cone flow and the used reference frame.

Figure 2

Self-similar profiles of the first-order velocity correction: (solid line) $U_1$, (dashed line) $V_1$, (dotted line) $W_1$.

Figure 3

Perspective sketch of the computational domain used to compute the base flow.

Figure 4

Profiles of the meridian and wall-normal velocity components for (a), (b) $X=10$, (c), (d) $X=50$, and (e), (f) $X=100$ for $\psi =\pi /3$. Dashed line: von Kármán solution. Solid black line: corrected profile according to the theory described in Sec. 2b. Solid red line: results from the DNS.

Figure 5

Profiles of the meridian and wall-normal velocity components for (a), (b) $X=10$, (c), (d) $X=50$, and (e), (f) $X=100$ for $\psi =\pi /6$. Dashed line: von Kármán solution. Solid black line: corrected profile according to the theory described in Sec. 2b. Solid red line: results from the DNS.

Figure 6

Growth rate for $\psi =\pi /3$ and $n=24$ from LNS (circles), parallel-stability analysis (black dashed line), and weakly divergent approach (black solid line).

Figure 7

Profiles of the eigenfunctions for $\psi =\pi /3$ and $n=24$ at $X=350$. See Fig. 6 for the used list of symbols.

Figure 8

Real part of the eigenfunctions for $\psi =\pi /3$ and $n=24$. (a), (d) $\widehat{u}$, (b), (e) $\widehat{v}$, (c), (f) $\widehat{w}$. Left: LNS. Right: weakly divergent approach.

Figure 9

Imaginary part of the eigenfunctions for $\psi =\pi /3$ and $n=24$. (a), (d) $\widehat{u}$, (b), (e) $\widehat{v}$, (c), (f) $\widehat{w}$. Left: LNS. Right: weakly divergent approach.

Figure 10

Growth rate for $\psi =\pi /3$ and $n=17$ from LNS (circles). Dashed line: parallel-stability analysis, solid line: weakly divergent approach. Black lines are associated to mode I, and red lines to mode II.

Figure 11

Growth rate for $\psi =\pi /3$ and $n=17$ from the LNS projected into mode I (black circles) and mode II (red pluses). Dashed line: parallel-stability analysis, solid line: weakly divergent approach. Black lines are associated to mode I, and red lines to mode II.

Figure 12

Profiles of the eigenfunctions for $\psi =\pi /3$ and $n=17$ at $X=340$. See Fig, 10 for the used list of symbols.

Figure 13

Profiles of the eigenfunctions for $\psi =\pi /3$ and $n=17$ at $X=390$. See Fig. 10 for the used list of symbols.

Figure 14

Real part of the eigenfunctions for $\psi =\pi /3$ and $n=17$. (a), (d) $\widehat{u}$, (b), (e) $\widehat{v}$, (c), (f) $\widehat{w}$. Left: LNS results. Right: weakly divergent approach.

Figure 15

Real part of the eigenfunctions for $\psi =\pi /3$ and $n=17$. (a), (d) $\widehat{u}$, (b), (e) $\widehat{v}$, (c), (f) $\widehat{w}$. Left: LNS results projected into mode I. Right: weakly divergent approach.

Figure 16

Growth rate for different wave numbers for (a) $\psi =\pi /2$, (b) $\psi =\pi /3$, and (c) $\psi =\pi /6$. Dashed black line: parallel-stability analysis (mode I), solid black line: weakly divergent approach (mode I), dashed red line: parallel-stability analysis (mode I and mode II), solid red line: weakly divergent approach (mode I and mode II). The growth-rate contours start from 0 with steps of 0.015. The two dotted lines in panel (b) indicate the wave numbers studies in Secs. 6a and 6b, $n=17$ and $n=24$, respectively.

Figure 17

Growth rate for different wave numbers for (a) $\psi =\pi /3$ and (b) $\psi =\pi /6$ for mode I only without base-flow correction (black) and with base-flow correction (red). Dashed lines: parallel-stability analysis, solid line: weakly divergent approach. The growth-rate contours start from 0 with steps of 0.015.

Figure 18

Profiles of the velocity obtained extracting the eigenfunction for $\psi =\pi /3$ and $n=24$ from LNS at $X=350$: grid convergence tests. See Table 3 for details of the different cases.

Figure 19

Profiles of the velocity obtained extracting the eigenfunction for $\psi =\pi /3$ and $n=17$ from LNS at $X=350$: grid convergence tests. See Table 4 for details on the different cases.