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

+ L(U,u′) = -∇p + ν∇^{2}u′, |

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

u(t) = A(t)u(0). |

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.

`Eqtype`

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

.

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_start`

: sets the start number of temporal slices for Floquet stability analysis. Default value: 0.`N_skip`

: sets the number skip of temporal slices for Floquet stability analysis. Default value: 1.`N_slices`

: sets the number of temporal slices for Floquet stability analysis. Files sequence`N_start, N_start + N_skip, ..., N_start + N_skip * (N_slices-1)`

will be loaded.`BaseFlow_interporder`

: sets the interpolation order of temporal slices for Floquet stability analysis. If`BaseFlow_interporder`

< 2, the baseflow is taken as periodic and trigonometric functions are used for interpolation. It should be noted that the file`N_start + N_skip * N_slices`

is at t =`period`

and should not be loaded for the periodic case. If`BaseFlow_interporder`

>= 2, the flow is taken as aperiodic and Lagrange interpolation is used. In this case, the file`N_start + N_skip * (N_slices-1)`

is at t =`period`

and should be loaded. Default value: 0.`period`

: sets the time span (if`BaseFlow_interporder`

>= 2) or period (if`BaseFlow_interporder`

< 2) of the base flow. Default value: 1.

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.