Unification theory of instabilities of visco-diffusive swirling flows

Abstract

A universal theory of linear instabilities in swirling flows, occurring in both natural settings and industrial applications, is formulated. The theory encompasses a wide range of open and confined flows, including spiral isothermal flows and baroclinic flows driven by radial temperature gradients and natural gravity in rotating fluids. By employing short-wavelength local analysis, the theory generalizes previous findings from numerical simulations and linear stability analyses of specific swirling flows, such as spiral Couette flow, spiral Poiseuille flow, and baroclinic Couette flow. A general criterion, extending and unifying existing criteria for instability to both centrifugal and shear-driven perturbations in swirling flows is derived, taking into account viscosity and thermal diffusion, and guiding experimental and numerical investigations in the otherwise inaccessible parameter regimes.

Figures (3)

• Figure 1: Neutral stability curves $(\ref{['a0']})$ parameterized by $k_z$, with their envelope (red, thick) for the non-isothermal BCF. Parameters are $Pr = 5.5$, $\gamma = 0.0004$, $k_r = 1.7\pi$, and $\eta = 0.8$ (selected to facilitate comparison with numerical results from previous studies LepillerYoshikawaGuillermKang2015Kang2023) for the three different $\mu$: (a) Rayleigh unstable, $\mu = 0$; (b) modified Rayleigh line, $\mu = \mu_R \approx 0.63935$ (from $(\ref{['mur']})$); and (c) Rayleigh stable, $\mu = 0.8$. The envelope at $Gr = 0$ in (a) gives $Re_0 \approx 166.8$ from $(\ref{['re0']})$. Green shaded regions are the unions of instability domains for specific $k_z$. Solid oblique straight lines in (c) represent the new unified criterion $(\ref{['elsscf1']})$. All computations for the BCF are performed at the mean geometric radius $r = \sqrt{r_1 r_2}$.
• Figure 2: (a, c) Green shaded area represents the union of individual instability domains defined by the neutral stability curves $(\ref{['a0']})$ in the $(Re_z, Re)$-plane for (a) $k_z=0.3, 0.5, 1, 2, 3, 4$ and (c) $k_z=0.1, 0.3, 0.8, 1.826, 2.8$. The thick red curves show their envelope $(\ref{['masterisc']})$ for the isothermal enclosed SCF (\ref{['escf']}) with $\eta = 0.4$ and $k_r = 3\sqrt{2}$. Panels (a, b) represent $\mu = 0$, and panels (c, d) represent $\mu = 0.5$. The black dashed curves correspond to the terminal $k_z = \frac{\sqrt{2}}{2} k_r = 3$ in (a) and the terminal $k_z = \frac{\sqrt{-2Ro}}{2} k_r \approx 1.826$ in (c). The black solid curves in (a,c) show the boundaries of the individual instability domains corresponding to $k_z$ exceeding the terminal value. The oblique black solid lines in (c) indicate the inviscid LELS criterion (\ref{['lels']}). Vertical solid lines in (c) show $\pm Re_z^{\min}$$(\ref{['rezm']})$. (b,d) Variation of $k_z$ from $0$ to the terminal value according to $(\ref{['kze']})$ as $Re$ increases from $Re_{\infty}$ (dot-dashed line) to (b) $Re_0$ (dotted line) or (d) to infinity. All computations for the SCF are performed at the mean geometric radius $r = \sqrt{r_1 r_2}$.
• Figure 3: For the SCF (\ref{['escf']}) with $k_r = \pi$ and $\eta = 0.5$: (a) The surface of the envelope (\ref{['masterisc']}) in the $(Re_z, Re_2, Re)$-space with a pleat and two folds (cf. the cusp catastrophe model BG1992) and (b) its projection onto the $(Re_z, Re)$-plane. The color scheme is applied solely to enhance visibility of the surface shape and has no physical significance.