Chapter 1
Introduction

The aim of this tutorial is to introduce the user to the spectral/hp element framework Nektar++ and its use for global stability computations. Information on how to install the libraries, solvers, and utilities on your own computer is available on the webpage www.nektar.info.

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.