Dean instability in double-curved channels

Abstract

We study the Dean instability in curved channels using the lattice Boltzmann model for generalized metrics. For this purpose, we first improve and validate the method by measuring the critical Dean number at the transition from laminar to vortex flow for a streamwise curved rectangular channel, obtaining very good agreement with the literature values. Taking advantage of the easy implementation of arbitrary metrics within our model, we study the fluid flow through a doublecurved channel, using ellipsoidal coordinates, and study the transition to vortex flow in dependence of the two perpendicular curvature radii of the channel. We observe not only transitions to 2-cell vortex flow, but also to 4-cell and even 6-cell vortex flow, and we find that the critical Dean number at the transition to 2-cell vortex flow exhibits a minimum when the two curvature radii are approximately equal.

Volumetric Lattice Boltzmann Models in General Curvilinear Coordinates: Theoretical Formulation

“…According to this concept, it is entirely conceivable to formulate LBM models on a cubic Cartesian lattice in non-Euclidean space that corresponds to a general curvilinear mesh in the Euclidean space. The first such attempts were made by Mendosa et al in a series of papers (see [16][17][18]). The significance of their work is that it established this fundamental approach to LBM on curvilinear meshes via a Riemann geometric framework.…”

Lattice Boltzmann model for the simulation of the wave equation in curvilinear coordinates

“…Again, the strategy does not require neither a specific discretization scheme for each problem nor additional interpolation steps. This model has been successfully used to study the Dean's instability in ellipsoidal coordinates [19], the flow through randomly curved media [20] and, more recently, the energy dissipation due to curvature [21].…”

Poiseuille flow in curved spaces

“…In this section, we present a short summary of the method used to simulate Poiseuille flow on Riemann manifolds, equipped with a stationary Riemann metric g. More details and validations of the method can be found in previous publications [24][25][26]. We will use the following notation for partial derivatives throughout the whole paper: …”

“…Here, we have further improved the method in Ref. [26] by improving the treatment of discrete lattice effects, which originate from the forcing term of the LB equation and which lead to spurious source terms in the Navier-Stokes equations (see also Ref. [27]).…”

“…[21][22][23]). Recently, the LB method has been extended to general Riemannian manifolds [24][25][26], allowing us to simulate fluid flow in nearly arbitrary, smooth geometries. Here, we have further improved the method in Ref.…”