Instabilities in buoyant flows under localized heating

Abstract

We study, from the numerical point of view, instabilities developed in a fluid layer with a free surface in a cylindrical container which is nonhomogeneously heated from below. In particular, we consider the case in which the applied heat is localized around the origin. An axisymmetric basic state appears as soon as a nonzero horizontal temperature gradient is imposed. The basic state may bifurcate to different solutions depending on vertical and lateral temperature gradients and on the shape of the heating function. We find different kinds of instabilities: extended patterns growing on the whole domain, which include those known as targets, and spiral waves. Spirals are present even for infinite Prandtl number. Localized structures both at the origin and at the outer part of the cylinder may appear either as Hopf or stationary bifurcations. An overview of the developed instabilities as functions of the dimensionless parameters is presented in this article. © 2007 American Institute of Physics. ͓DOI: 10.1063/1.2714295͔ Spiral and target wave instabilities in Rayleigh-Bérnard convection were experimentally found in the early 1990s and several theoretical mechanisms have been proposed to justify their appearance. In this article we study the instabilities generated in the flow by a nonuniform heating function. In this setup spiral and target waves are the linearly growing modes of nontrivial axisymmetric basic states. We find that localized structures also may appear. These patterns grow at different positions in the domain depending on the basic flow from which they bifurcate. This article reports a systematical numerical study of these transitions.

Biot number constraints on the thermal regime of acidic magmas in the subvolcanic crust: an integrated approach using numerical modelling and petrology

“…. , M (Navarro et al 2007;Pla et al 2009). Therefore, a linear system AX ¼ B is derived at each iteration step of the Newton method, where X is a vector of P ¼ 4 × N × M unknowns and A is a matrix of order P × P.…”

“…We utilized the Chebyshev collocation method to solve the basic state, such that the partial derivatives to time of v and temperature u in equations (1-3) are 0 (Foster 1969;Canuto et al 1988;Navarro et al 2007;Pla et al 2009), as well as a Newton-like iterative method to treat the non-linearity (Navarro et al 2007). This applies until a difference in the two successive steps is less than 10 27 (Pla & Herrero 2011).…”

Bifurcation phenomena in a convection problem with temperature dependent viscosity at low aspect ratio

“…(a) Critical instability curves R(m, Γ ) for a fluid layer with constant viscosity (µ = 0); (b) critical instability curves R(m, Γ ) for a fluid layer with temperature dependent viscosity (µ = 0.0862).Stationary equations and boundary conditions are solved by an iterative Newton-like method, similarly to the procedure used to compute basic states in[33]. In this method an approximate solution v s x , v s z , P s , θ s is required at the beginning (step s = 0).This choice is the key to success of the iterative method, and in fact all branch continuation techniques depend on finding the most appropriate one[34].…”

Secondary whirls in thermoconvective vortices developed in a cylindrical annulus locally heated from below

“…The numerical method is described in detail and tested in Refs. [14,16]. L = 33 and N = 21 are considered in our computations.…”