10.4 Stability analysis Session file configuration

The type of equation which is to be solved is specified through the EqType option in the session file. This can be set to any of the following:

Equation to solve

+ L(U,u′) = −∇p + ν∇2u′

10.4.1 Solver Info

• Eqtype: sets the type of equation to solve, according to the table above.
• TimeIntegrationMethod: the following types of time integration methods have been tested with each solver:

  	Explicit	Diagonally Implicit	IMEX	Implicit
  UnsteadyNavierStokes	X		X

• Projection: the Galerkin projection used may be either
  • Continuous: for a C0-continuous Galerkin (CG) projection;
  • Discontinuous: for a discontinous Galerkin (DG) projection.
• EvolutionOperator:
  • Nonlinear (non-linear Navier-Stokes equations).
  • Direct (linearised Navier-Stokes equations).
  • Adjoint (adjoint Navier-Stokes equations).
  • TransientGrowth ((transient growth evolution operator).
• Driver: specifies the type of problem to be solved:
  • Standard (time integration of the equations)
  • ModifiedArnoldi (computations of the leading eigenvalues and eigenmodes using modified Arnoldi method)
  • Arpack (computations of eigenvalues/eigenmodes using Implicitly Restarted Arnoldi Method (ARPACK) ).
• ArpackProblemType: types of eigenvalues to be computed (for Driver Arpack only)
  • LargestMag (eigenvalues with largest magnitude).
  • SmallestMag (eigenvalues with smallest magnitude).
  • LargestReal (eigenvalues with largest real part).
  • SmallestReal (eigenvalues with smallest real part).
  • LargestImag (eigenvalues with largest imaginary part).
  • SmallestIma (eigenvalues with smallest imaginary part ).
• Homogeneous: specifies the Fourier expansion in a third direction (optional)
  • 1D (Fourier spectral method in z-direction).
• ModeType: this specifies the type of the quasi-3D problem to be solved.
  • MultipleMode (stability analysis with multiple modes).
  • SingleMode (BiGlobal Stability Analysis: full-complex mode).
  • HalfMode (BiGlobal Stability Analysis: half-complex mode u.Re v.Re w.Im p.Re).

10.4.2 Parameters

The following parameters can be specified in the PARAMETERS section of the session file:

• Re: sets the Reynolds number
• Kinvis: sets the kinematic viscosity ν.
• kdim: sets the dimension of the Krylov subspace κ. Can be used in: ModifiedArnoldi and Arpack. Default value: 16.
• evtol: sets the tolerance of the eigenvalues. Can be used in:
• imagShift: provide an imaginary shift to the direct sovler eigenvalue problem by the specified value. lttModifiedArnoldi and Arpack. Default value: 10⁻⁶.
• nits: sets the maximum number of iterations. Can be used in: ModifiedArnoldi and Arpack. Default value: 500.
• LZ: sets the length in the spanswise direction L_z. Can be used in Homogeneous set to 1D. Default value: 1.
• HomModesZ: sets the number of planes in the homogeneous directions. Can be used in Homogeneous set to 1D and ModeType set to MultipleModes.
• N_slices: sets the number of temporal slices for Floquet stability analysis.
• period: sets the periodicity of the base flow.
• realShift: provide a real shift to the direct sovler eigenvalue problem by the specified value.

10.4.3 Functions

When using the direct solver for stability analysis it is necessary to specify a Forcing function “StabilityCoupledLNS” in the form:

This is required since we need to tell the solver to use the existing field as a forcing function to the direct matrix inverse as part of the Arnoldi iteration.