### 1 Two-dimensional Channel flow (optional)

Note: For speed you may wish to go to the next section since all mesh input files have been provided and return to this section when time permits.

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) ∇⋅ u′ = 0. (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:

 (3)

The domain is Ω = [−π,π] × [−1, 1] and it is composed by 48 quadrilateral elements as shown in figure 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 for this section can be found in the `\$NEKTUTORIAL/Channel` directory.

• 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/MeshConvert` utility, for computing the base flow.
• Folder `Stability/VCS`
• `Channel-VCS.xml` - Nektar++ session file, generated with `\$NEK/MeshConvert`, 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/MeshConvert`, for performing the stability analysis.