Chapter 1
Introduction

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 tutorial in the Flow Stability series on the channel and therefore already has the necessary software installed.

In this tutorial we will compute the direct and adjoint modes of a two-dimensional flow past a cylinder. We will investigate a case in the subcritical regime (Re = 42), below the onset of the Bernard-von Kärmän vortex shedding that is observed when the Reynolds number is above the critical value Re_c ≃ 47; this analysis is important because it allows us to study the sensitivity of the flow, much like that reported by Giannetti and Luchini ( J. Fluid Mech., 2007; 592:177-194). Due to the more complex nature of the flow and the more demanding computational time that is required, only some basic information will be presented in this section, mainly to show the potential of the code for stability analysis.

The files contained in the $NEKTUTORIAL directory are as follows:

• Folder geometry

  • Cylinder.geo - Gmsh file that contains the geometry of the problem

  • Cylinder.msh - Gmsh generated mesh data listing mesh vertices and elements.

• Folder base

  • Cylinder-Base.xml - Nektar++ session file, generated with the $NEK/NekMesh utility, for computing the base flow.

  • Cylinder-Base.fld - Nektar++ field file that contains the base flow, generated using
    Cylinder-Base.xml.

• Folder stability/Direct

  • Cylinder_Direct.xml - Nektar++ session file, generated with $NEK/NekMesh.

  • Cylinder_Direct.bse - Nektar++ field file that contains the base flow.

  • Cylinder_Direct.rst - Nektar++ field file that contains the initial conditions.

• Folder stability/Adjoint

  • Cylinder_Adjoint.xml - Nektar++ session file, generated with $NEK/NekMesh.

  • Cylinder_Adjoint.bse - Nektar++ field file that contains the base flow.

  • Cylinder_Adjoint.rst - Nektar++ field file that contains the initial conditions.

• Folder stability/Receptivity Pre-computed output files for calculating the receptivity.

The mesh is shown in figure 1.1 along with a detailed view around the cylinder. This mesh is made up of 782 quadrilateral elements.

Figure 1.1: Mesh used for the direct stability analysis

Note: It is important to note that stability and transient growth calculations in particular, have a strong dependence on the domain size as reported by Cantwell and Barkley (Physical Review E, 2010; 82); moreover, poor mesh design can lead to incorrect results. Specifically, the mesh must be sufficiently refined around the cylinder in order to capture the separation of the flow and abrupt variations in the size of the elements should be avoided.