Chapter 1
Introduction

In this tutorial, we will cover the stability analysis of a two-dimensional channel flow, through both a splitting scheme (the Velocity Correction Scheme) and the direct inversion algorithm (also referred to as the Coupled Linearised Navier-Stokes solver). We will briefly show the application of the stability tools presented to a three-dimensional channel flow test case.

Linear stability analysis is a technique that allows us to determine the asymptotic stability of a flow. By decomposing the velocity and pressure in the Navier-Stokes equations as a summation of a base flow (U,P) and perturbation (u′,p′), such that u = U + ϵu′, p = P + ϵp′, with ϵ « 1, we derive the linearised Navier-Stokes equations,

+ U ⋅∇u′ + u′⋅∇U	= −∇p′ + ∇2u′ + f′,	(1.1)
∇⋅ u′	= 0.	(1.2)

We will consider a parallel base flow through a 2-D channel (known as Poiseuille flow) at Reynolds number Re = 7500. The velocity has the following analytic form:

(1.3)

The domain is Ω = [−π,π] × [−1,1] and it is composed by 48 quadrilateral elements as shown in figure 1.1. The problem has been made non-dimensional using the centreline velocity and the channel half-height.

This mesh was created using the software Gmsh and the first step is to convert it into a suitable input format so that it can be processed by the Nektar++ libraries.

The files in the $NEKTUTORIAL directory are as follows:

• Folder geometry
  • Channel.geo - Gmsh file that contains the geometry of the problem
  • Channel.msh - Gmsh generated mesh data listing mesh vertices and elements.
• Folder base
  • Channel-Base.xml - Nektar++ session file, generated with the $NEK/NekMesh utility, for computing the base flow.
• Folder stability/VCS
  • Channel-VCS.xml - Nektar++ session file, generated with $NEK/NekMesh, for performing the stability analysis.
  • Channel-VCS.rst - Nektar++ field file that contains a set of initial conditions closer to the solution in order to achieve faster convergence.
• Folder stability/Coupled
  • Channel-Coupled.xml - Nektar++ session file, generated with $NEK/NekMesh, for performing the stability analysis.
• Folder stability3D
  • PPF_R10000_3D.xml - Nektar++ session file for performing coupled 3D stability analysis at Re = 10000.
  • PPF_R15000_3D.xml - Nektar++ session file for performing coupled 3D stability analysis at Re = 15000.

1.1 Mesh generation

The first step is to generate a mesh that is readable by Nektar++. The files necessary in this section can be found in $NEKTUTORIAL/geometry/. To achieve this task we use Gmsh in conjunction with the Nektar++ pre-processing utility called $NEK/NekMesh. Specifically, we first generate the mesh in figure 1.1 using Gmsh and successively we convert it into a suitable Nektar++ format using $NEK/NekMesh.