Global linear instability of rotating-cone boundary layers in a quiescent medium

The global linear stability of the family of infinite rotating-cone boundary layers in an otherwise still fluid is investigated using a velocity-vorticity form of the linearized Navier-Stokes equations. The formulation is separable with respect to the azimuthal direction. Thus, disturbance development is simulated for a single azimuthal mode number. Numerical simulations are conducted for an extensive range of cone half-angles ($\psi \in [{20}^{\circ }:{80}^{\circ }]$) and stability parameters (Reynolds number, azimuthal mode number), where conditions are taken to be near those specifications necessary for the onset of absolute instability. A localized impulsive wall forcing is implemented that excites disturbances that form wave packets. This allows the disturbance evolution to be traced in the spatial-temporal plane. When a homogeneous flow approximation is utilised that neglects the spatial variation of the basic state, linear perturbations display characteristics consistent with local stability theory. For disturbances to the genuine spatially dependent inhomogeneous flow, global linear instability characterized by a faster than exponential temporal growth arises for azimuthal mode numbers greater than the conditions for critical absolute instability. Furthermore, a reasonable prediction for the azimuthal mode number needed to bring about a change in global behavior is achieved by coupling solutions of the Ginzburg-Landau equation with local stability properties. Thus, the local-global stability behavior is qualitatively similar to that found in the infinite rotating-disk boundary layer and many other globally unstable flows.

Figure 1

Illustration of the rotating-cone geometry.

Figure 2

Steady velocity flow profiles for four cone half-angles. (a) $F$, (b) $G$, (c) $H$.

Figure 3

Growth rates $\rho$ as a function of the azimuthal mode number $n$ for cone half-angles $\psi =[{20}^{\circ }:10:{90}^{\circ }]$. Cross markers indicate the respective azimuthal mode numbers for critical absolute instability.

Figure 4

Temporal growth rates $g_i$ as functions of time, for disturbances impulsively excited about the location $x_f$ for critical absolute instability, for $n\in [40:10:80]$ and $\psi ={40}^{\circ }$.

Figure 5

Temporal growth rates $g_i$ for disturbances impulsively excited about $x_f=650(\text{Re}=418)$ and for $\psi ={40}^{\circ }$. Development plotted about $x=x_f-25$ (solid lines), $x=x_f$ (dashed), $x=x_f+25$ (chain) and $x=x_f+50$ (dotted), while the solid-crossed lines depict the corresponding solutions in the homogeneous flow about the impulse center. (a) $n=40$, (b) $n=50$, (c) $n=60$, (d) $n=70$.

Figure 6

Spatial-temporal disturbance development of the azimuthal vorticity on the disk surface $|{\omega }_{\theta ,w}|$, for an impulsive center about $x_f=650(\text{Re}=418)$ and for $\psi ={40}^{\circ }$. (a) $n=40$, (b) , (c) $n=60$, (d) $n=70$. Vertical dashed lines in (c) and (d) represent the streamwise location $x_a$ for the onset of absolute instability, associated with the given azimuthal mode numbers.

Figure 7

Spatial-temporal disturbance wave packets and matching temporal growth rates for an impulse centered about $x_f=700$ ($\text{Re}=450$) for $\psi ={40}^{\circ }$. (a) and (c): $n=40$; (b) and (d) $n=60$. Line types for growth rates in (c) and (d) are the same as those given in Fig. 5.

Figure 8

Temporal growth rates $g_i$, for disturbances impulsively excited about the onset of absolute instability $x_f$ for variable $n$. (a) $\psi ={80}^{\circ }$, (b) $\psi ={70}^{\circ }$, (c) $\psi ={60}^{\circ }$, (d) $\psi ={50}^{\circ }$, (e) $\psi ={30}^{\circ }$, (f) $\psi ={20}^{\circ }$.