Figure 3.1: u′-component and v′-component of the eigenmode
In the folder $NEKTUTORIAL/stability/Direct there are the files that are required for the
direct stability analysis. Since, the computation would normally take several hours to
converge, we will use a restart file and a Krylov-space of just κ = 4. Therefore, it
will be possible to obtain the eigenvalue and the corresponding eigenmode after 2
iterations.
Cylinder_Direct.rst.The simulation should converge in 6 iterations and the terminal screen should look similar to the one below:
======================================================================= EquationType: UnsteadyNavierStokes Session Name: Cylinder_Direct Spatial Dim.: 2 Max SEM Exp. Order: 7 Expansion Dim.: 2 Projection Type: Continuous Galerkin Advection: explicit Diffusion: explicit Time Step: 0.0008 No. of Steps: 1250 Checkpoints (steps): 1000 Integration Type: IMEXOrder2 ======================================================================= Arnoldi solver type : Arpack Arpack problem type : LM Single Fourier mode : false Beta set to Zero : false Evolution operator : Direct Krylov-space dimension : 4 Number of vectors : 2 Max iterations : 500 Eigenvalue tolerance : 1e-06 ====================================================== Initial Conditions: - Field u: from file Cylinder_Direct.rst - Field v: from file Cylinder_Direct.rst - Field p: from file Cylinder_Direct.rst Writing: "Cylinder_Direct_0.chk" Inital vector : input file Iteration 0, output: 0, ido=1 Writing: "Cylinder_Direct_1.chk" Steps: 1250 Time: 1 CPU Time: 46.5477s Time-integration : 46.5477s Writing: "Cylinder_Direct.fld" Iteration 1, output: 0, ido=1 Writing: "Cylinder_Direct_1.chk" Steps: 1250 Time: 2 CPU Time: 41.7221s Time-integration : 41.7221s Iteration 2, output: 0, ido=1 Writing: "Cylinder_Direct_1.chk" Steps: 1250 Time: 3 CPU Time: 41.8717s Time-integration : 41.8717s Iteration 3, output: 0, ido=1 Writing: "Cylinder_Direct_1.chk" Steps: 1250 Time: 4 CPU Time: 41.9465s Time-integration : 41.9465s Iteration 4, output: 0, ido=1 Writing: "Cylinder_Direct_1.chk" Steps: 1250 Time: 5 CPU Time: 41.987s Time-integration : 41.987s Iteration 5, output: 0, ido=1 Writing: "Cylinder_Direct_1.chk" Steps: 1250 Time: 6 CPU Time: 42.2642s Time-integration : 42.2642s Iteration 6, output: 0, ido=99 Converged in 6 iterations Converged Eigenvalues: 2 Magnitude Angle Growth Frequency EV: 0 0.9792 0.726586 -0.0210196 0.726586 Writing: "Cylinder_Direct_eig_0.fld" EV: 1 0.9792 -0.726586 -0.0210196 -0.726586 Writing: "Cylinder_Direct_eig_1.fld" L 2 error (variable u) : 0.0501837 L inf error (variable u) : 0.0296123 L 2 error (variable v) : 0.0635524 L inf error (variable v) : 0.0355673 L 2 error (variable p) : 0.0344665 L inf error (variable p) : 0.0176009
After the direct stability analysis, it is now interesting to compute the eigenvalues and
eigenvectors of the adjoint operator A* that allows us to evaluate the effects of generic initial
conditions and forcing terms on the asymptotic behaviour of the solution of the
linearised equations. In the folder Cylinder/Stability/Adjoint there is the file
Cylinder_Adjoint.xml that is used for the adjoint analysis.
EvolutionOperator to Adjoint, the Krylov space to 4 and compute
the leading eigenvalue and eigenmode of the adjoint operator, using the restart file
Cylinder_Adjoint.rstThe solution should converge after 4 iterations and the terminal screen should look like this:
======================================================================= EquationType: UnsteadyNavierStokes Session Name: Cylinder_Adjoint Spatial Dim.: 2 Max SEM Exp. Order: 7 Expansion Dim.: 2 Projection Type: Continuous Galerkin Advection: explicit Diffusion: explicit Time Step: 0.001 No. of Steps: 1000 Checkpoints (steps): 1000 Integration Type: IMEXOrder3 ======================================================================= Arnoldi solver type : Arpack Arpack problem type : LM Single Fourier mode : false Beta set to Zero : false Evolution operator : Adjoint Krylov-space dimension : 4 Number of vectors : 2 Max iterations : 500 Eigenvalue tolerance : 0.001 ====================================================== Initial Conditions: Field p not found. - Field u: from file Cylinder_Adjoint.rst - Field v: from file Cylinder_Adjoint.rst - Field p: from file Cylinder_Adjoint.rst Writing: "Cylinder_Adjoint_0.chk" Inital vector : input file Iteration 0, output: 0, ido=1 Steps: 1000 Time: 27 CPU Time: 42.0192s Writing: "Cylinder_Adjoint_1.chk" Time-integration : 42.0192s Writing: "Cylinder_Adjoint.fld" Iteration 1, output: 0, ido=1 Steps: 1000 Time: 28 CPU Time: 37.1084s Writing: "Cylinder_Adjoint_1.chk" Time-integration : 37.1084s Iteration 2, output: 0, ido=1 Steps: 1000 Time: 29 CPU Time: 37.4794s Writing: "Cylinder_Adjoint_1.chk" Time-integration : 37.4794s Iteration 3, output: 0, ido=1 Steps: 1000 Time: 30 CPU Time: 37.3142s Writing: "Cylinder_Adjoint_1.chk" Time-integration : 37.3142s Iteration 4, output: 0, ido=99 Converged in 4 iterations Converged Eigenvalues: 2 Magnitude Angle Growth Frequency EV: 0 0.980493 0.727526 -0.0197 0.727526 Writing: "Cylinder_Adjoint_eig_0.fld" EV: 1 0.980493 -0.727526 -0.0197 -0.727526 Writing: "Cylinder_Adjoint_eig_1.fld" L 2 error (variable u) : 0.434746 L inf error (variable u) : 0.156905 L 2 error (variable v) : 0.698425 L inf error (variable v) : 0.120624 L 2 error (variable p) : 0.216948 L inf error (variable p) : 0.0676028
Note that, in spatially developing flows, the eigenmodes of the direct stability operator tend to be located far downstream while the eigenmodes of the adjoint operator tend to be located upstream and near to the body, as can be seen in figures 3.2 and 3.3. From the profiles of the eigemodes, it can be deducted that the regions with the maximum receptivity for the momentum forcing and mass injection are near the wake of the cylinder, close to the upper and lower sides of the body surface, in accordance with results reported in the literature.
