11.4 Session file configuration: Linear stability analysis

Stability analyses of incompressible flow involves solving the linearised Navier-Stokes equations

+ L(U,u′) = -∇p + ν∇2u′,

where L is a linear operator, its adjoint form, or both. The evolution of the linearised Navier-Stokes operator, which evolves a solution from an initial state to a future time t, can be written as

The adjoint evolution operator is denoted as A^*. This section details the additional configuration options, in addition to the standard configuration options described earlier, relating to performing this task.

11.4.1 Solver Info

• Eqtype: sets the type of equation to solve. For linear stability analysis this must be set to

  Equation Type	Dimensions	Projections	Algorithms
  UnsteadyNavierStokes	2D, Quasi-3D	Continuous	VCS,DS

• EvolutionOperator: sets the choice of the evolution operator:
  • Nonlinear (standard non-linear Navier-Stokes equations).
  • Direct (A – linearised Navier-Stokes equations).
  • Adjoint (A^* – adjoint Navier-Stokes equations).
  • TransientGrowth (A^*A – transient growth evolution operator).
• Driver: specifies the type of problem to be solved:
  • Standard (normal 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, HomModesZ sets number of modes).
  • SingleMode (BiGlobal Stability Analysis: full-complex mode. Overrides HomModesZ to 1.).
  • HalfMode (BiGlobal Stability Analysis: half-complex mode u.Re v.Re w.Im p.Re).
  Note: For visualization of Homogeneous results with FieldConvert you can use --output-points-hom-z to set output number of modes to a desired value. To process results obtained with HalfMode you can convert to SingleMode using FieldConvert module halfmodetofourier.

11.4.2 Parameters

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

• kdim: sets the dimension of the Krylov subspace κ. Can be used with: ModifiedArnoldi and Arpack. Default value: 16.
• evtol: sets the tolerance of the iterative eigenvalue algorithm. Can be used with: ModifiedArnoldi and Arpack. Default value: 1 × 10^-6.
• nvec: sets the number of converged eigenvalues sought. Can be used with: ModifiedArnoldi and Arpack. Default value: 2.
• nits: sets the maximum number of Arnoldi iterations to attempt. Can be used with: ModifiedArnoldi and Arpack. Default value: 500.
• realShift: provide a real shift to the direct solver eigenvalue problem by the specified value to improve convergence. Can be used with: Arpack only.
• imagShift: provide an imaginary shift to the direct solver eigenvalue problem by the specified value to improve convergence. Can be used with: Arpack only.
• LZ: sets the length in the spanswise direction L_z. Can be used with Homogeneous set to 1D. Default value: 1.
• HomModesZ: sets the number of planes in the homogeneous directions. Can be used with 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.

11.4.3 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.