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∇^{2}u′ |

Equation Type | Dimensions | Projections | Algorithms |

UnsteadyNavierStokes | 2D, Quasi-3D | Continuous | VCS,DS |

`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).

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.

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.

`NEKTAR/solvers/IncNavierStokesSolver/Tests`

. See for example the files
`PPF_R15000_ModifiedArnoldi_Shift.tst`

and `PPF_R15000_3D.xml`

noting that some
parameters are specified in the .tst files.