Chapter 1Introduction

This tutorial further explores the use of the spectral/hp element framework Nektar++ to perform 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.

This tutorial assumes the reader has already completed the previous tutorials in the Flow Stability series on the channel and cylinder and therefore already has the necessary software installed.

Task: 1.1 Prepare for the tutorial. Make sure that you have:

• Downloaded the tutorial files: flow-stability-bfs.tar.gz
Unpack it using `unzip flow-stability-bfs.tar.gz` to produce a directory `flow-stability-bfs` with subdirectories called `tutorial` and `complete` We will refer to the `tutorial` directory as `\$NEKTUTORIAL`.

In this tutorial we will perform a transient growth analysis of the flow over a backward-facing step. This is an important case which allows us to understand the effects of separation due to abrupt changes of geometry in an open flow. The transient growth analysis consists of computing the maximum energy growth, G(τ), attainable over all possible initial conditions u′(0) for a specified time horizon τ. It can be demonstrated that it is equivalent to calculating the largest eigenvalue of A(τ)A(τ), with A and A being the direct and the adjoint operators, respectively. Also note that the eigenvalue must necessarily be real since A(τ)A(τ) is self-adjoint in this case.

• Folder `geometry`
• `bfs.geo` - Gmsh file that contains the geometry of the problem
• `bfs.msh` - Gmsh generated mesh data listing mesh vertices and elements.
• Folder `base`
• `bfs-Base.xml` - Nektar++ session file, generated with the `\$NEK/NekMesh` utility, for computing the base flow.
• `bfs-Base.fld` - Nektar++ field file that contains the base flow, generated using
`bfs-Base.xml`.
• Folder `stability`
• `bfs_tg.xml` - Nektar++ session file, generated with `\$NEK/NekMesh`, for performing the transient growth analysis.
• `bfs_tg.bse` - Nektar++ field file that contains the base flow. It is the same as the `.fld` file present in the folder `Base`.

Figure 2.1 shows the mesh, along with a detailed view of the step edge, that we will use for the computation. The geometry is non-dimensionalised by the step height. The domain has an inflow length of 10 upstream of the step edge and a downstream channel of length 50. The mesh consist of N = 430 elements. Note that in this case the mesh is composed of both triangular and quadrilateral elements. A refined triangular unstructured mesh is used near the step to capture the separation effects, whereas the inflow/outflow channels have a structure similar to the previous example. Therefore in the `EXPANSION` section of the `bfs-Base.xml` file, two composites (`C[0]` and `C[1]`) are present. For this example, we will use the modal basis with 7th-order polynomials.

We will perform simulations at Re = 500, since it is well-known that for this value the flow presents a strong convective instability.