The complex unsteady flow within a fluid-filled annulus and its transition to turbulence
Sophie Calabretto, Department of Engineering Science
The spin-up from rest of a fluid enclosed by a container can yield some interesting and beautiful phenomena.
Initially, the container and the fluid are both stationary. A sufficiently long time after the container is set spinning, the fluid rotates with the container as if it were a solid body.
Between these two simple, well-understood, equilibrium states a complicated process occurs involving the transient formation of boundary layers (thin layers of moving fluid) and turbulence (an unsteady, disorganised, three-dimensional state of flow).
The aim of this project is to investigate the unsteady dynamics resulting from these rotating flows. The transition between slightly disturbed laminar flow and fully turbulent flow is of particular interest. Turbulent transient flows appear in a broad range of fields, including the study and optimisation of pipe networks, and the analysis of abnormalities in the cardiovascular system.
Figure 1 (left): Snapshots of the azimuthal velocity component (velocity in the direction of rotation), showing the development of an unsteady boundary layer and secondary phenomena.
Figure 2 (right): Close-up of the secondary non-linear Görtler instabilities seen on the inner wall of an
annulus with a rectangular aspect ratio.
The flow of fluid in a rotating annulus is used as a paradigm for studying the complex dynamics of these transient flows. As with all real fluid motion, the flow in this problem is governed by the full, three-dimensional, Navier-Stokes equations. Analytic solutions are known only for a small number of particular cases, and so computational schemes are used to calculate approximations to the pressure and velocity fields. In this project, the rotationally symmetric Navier-Stokes equations are numerically solved by exploiting the capabilities of semtex (http://users.monash.edu.au/~bbun/semtex.html) a quadrilateral spectral element direct numerical simulation (DNS) code that is ideal for solving problems in cylindrical coordinate systems.
Solving the rotationally symmetric Navier-Stokes equations
Semtex is a family of spectral element simulation codes, written in C++,developed by Prof. Hugh Blackburn of Monash University, Australia. Semtex uses parametrically mapped quadrilateral elements, the classic Gauss-Lobatto-Legendre nodal shape function basis, and continuous Galerkin projection to solve the conservation of momentum (Navier-Stokes) equations and the conservation of mass (continuity) equation that govern the flow of an incompressible fluid. In order to obtain a solution that is converged both spatially and temporally, a very fine computational mesh and a small time step are used when unning the DNS code. Using multiple threads on a single node, may require 20 GB of RAM and CPU times of up to a week, while producing datasets of up to 80 GB.
The next step will be to use the Pan cluster to run fully three-dimensional simulations of the flow (rather than looking at the rotationally symmetric cross-section). Pan will be crucial for this task, as the code will be run in parallel (using MPI to distribute jobs over a number of processors). Distributing this huge problem over many nodes makes the simulation feasible – drastically reducing the computation time required, while still allowing highly-resolved, converged solutions. Without access to the NeSI Pan cluster, these three-dimensional simulations would not be practicable.
