Numerical acceleration of aero-engine cavity flow solutions and flutter predictions

Abstract

The research in this thesis concerns aeroelastic flutter computations in turbomachinery labyrinth seals. The presence of annular cavities adjacent to the seal translates numerically into a highly ill-conditioned system of nonlinear equations. As a result, current prediction methods based on the coupling of the nonlinear time-accurate Navier-Stokes equations with structural Finite Element calculations are prohibitively expensive for industrial routine design. To alleviate this issue, the thesis emphasises on numerical acceleration techniques which can be regrouped into two related parts:

The first introduces an uncoupled time-linearized harmonic method for flutter in annular cavity domains. By assuming a periodicity in the flow unsteadiness, the perturbations induced by the seal motion are linearized about the steady-state background flow. The novelty consists in taking advantage of the cyclic symmetry of the vibrating seal nodal diameter and assuming the unsteady flow to be space-periodic in the circumferential direction. In the frequency-domain, the harmonic problem is reduced to two dimensions, therefore allowing the background solution to be axisymmetric, greatly reducing the overall computational cost.

The linear equations defining the unsteady flow are strongly dependant on the solution of the steady-state problem, and so, the unsteady solution accuracy is related to that of the background flow. Unfortunately, obtaining converged steady-state solutions for the compressible Navier-Stokes equations in cavities is a nontrivial task, as the nonlinear set of equations arising from their discretization are exceptionally stiff. This leads to the second aspect of this research which focuses on iterative procedures able to accelerate the convergence rates of the nonlinear steady-state solution in cavities. To that end, a Newton-GMRES method is implemented: the linear problem arising at every Newton artificial time-step is solved using GMRES, a linear multigrid routine for unstructured grids is used to precondition the system. With slight modification, it will also be seen that the linear part of the solver can be recycled for the unsteady problem and that its performance can further be increased by the use of the so-called GMRES with deflated-restart method.

The performance of the implemented iterative methods is assessed for both nonlinear and linearized problems. In some cases, the stiffness of the linearized problem causes the solution procedure to stagnate, this is relieved by the use of deflated GMRES and/or increasing the number of multigrid cycles in the preconditioning iterations. The capability of the linear harmonic method coupled with multigrid-preconditioned GMRES for large parametric studies is demonstrated by obtaining aeroelastic damping criteria across a range of cavity/seal configurations, flow conditions and vibrational mode shapes. This allows the determination of trends in the different mechanisms causing aeroelastic instability in cavity-seal domains.