In this section 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 ^{∗}(τ)(τ), with and ^{∗} being the direct and the adjoint operators, respectively.
Also note that the eigenvalue must necessarily be real since ^{∗}(τ)(τ) is self-adjoint in this
case.

The files for this section can be found in the `$NEKTUTORIAL/BackwardStep`

directory.

- 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/MeshConvert`

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/MeshConvert`

, 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 4 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.