Oblique-mode breakdown of the vertical buoyancy layer

Abstract

The O-type transition caused by a pair of small-amplitude oblique waves in a vertical buoyancy layer of a fluid with Prandtl number $0.71$ at a Reynolds number of $200$ is investigated using linear stability analysis and three-dimensional direct numerical simulation. The small-amplitude oblique waves experience linear growth and undergo nonlinear interactions to generate streamwise vortices/streaks, two-dimensional streamwise waves and harmonic oblique waves. The streamwise vortices/streaks and two-dimensional streamwise waves have twice the spanwise or streamwise wavenumber of the original perturbation, respectively. Unlike the O-type transition in isothermal flat-plate incompressible and compressible boundary layers where streaks dominate the transition, in the vertical buoyancy layer, either streaks or two-dimensional streamwise waves can dominate the flow field during the early stages of oblique transition. The growth rates of streaks and two-dimensional waves are dependent on the wavenumber of the initial oblique waves. Streaks dominate the flow for high streamwise wavenumbers, while two-dimensional streamwise waves dominate the flow for low streamwise wavenumbers. Analysis of the turbulent kinetic energy production and the Reynolds stresses reveals that the early stages of the transition differ depending on the wavenumber of the oblique waves. An increase in the initial amplitude of the oblique waves causes a faster transition from laminar flow; however, the growth rates of the streaks and two-dimensional streamwise waves are independent of the initial amplitude. Even though different modes are dominant during the early stages of the O-type transition, the onset of chaotic flow is caused by the breakdown of streak modes.

Large-scale motions in a turbulent natural convection boundary layer immersed in a stably stratified environment

“…Often such NCBLs are modelled using the one-dimensional formulation proposed by Prandtl (1952), referred to as the 'buoyancy layer' from hereon. In line with previous studies (Gill & Davey 1969;McBain, Armfield & Desrayaud 2007;Fedorovich & Shapiro 2009b;Maryada et al 2022), the current study uses the buoyancy layer to model an NCBL immersed in a stably stratified medium.…”

“…This implies that T w and T ∞ are functions of the vertical coordinate x 2 . Here, following (Janssen & Armfield 1996;McBain et al 2007;Zhao et al 2016Zhao et al , 2017Maryada et al 2022), we define x 1 , x 2 and x 3 as the wall-normal, streamwise and spanwise directions, respectively. If T = B, an equilibrium NCBL with a constant boundary layer thickness develops on the heated surface, termed the buoyancy layer (Prandtl 1952).…”

“…It should be noted that the Reynolds number is half the Grashof number (Gr = 2Re = gβ Tδ 3 l /ν 2 ) in the current non-dimensionalisation (Gill & Davey 1969). With the above non-dimensionalisation, the following non-dimensional Navier-Stokes equation with the Oberbeck-Boussinesq approximation for buoyancy and the scalar transport equation are used to solve for the buoyancy layer (Gill & Davey 1969;McBain et al 2007;Maryada et al 2022)…”

“…DNS was performed using an in-house non-staggered finite volume code (Norris 2000;Armfield et al 2003). The code has been used extensively to investigate natural convection flows, and its verification and validation are well documented (Armfield et al 2003;Maryada et al 2022). The spatial terms were discretised using a second-order central difference scheme.…”

“…As the turbulent buoyancy layer is spatially homogeneous in the streamwise and spanwise directions, periodic boundary conditions were imposed in the streamwise and spanwise directions, in line with previous studies (Fedorovich & Shapiro 2009b;Giometto et al 2017;Maryada et al 2022). A no-slip boundary condition for velocity (ũ i = 0) and a constant buoyancy ( θ = 1) boundary condition were used at the heated wall.…”