Low cost 3D global instability analysis and flow sensitivity based on dynamic mode decomposition and high‐order numerical tools

Abstract

SUMMARYWe explore the recently developed snapshot-based dynamic mode decomposition (DMD) technique, a matrix-free Arnoldi type method, to predict 3D linear global flow instabilities. We apply the DMD technique to flows confined in an L-shaped cavity and compare the resulting modes to their counterparts issued from classic, matrix forming, linear instability analysis (i.e. BiGlobal approach) and direct numerical simulations. Results show that the DMD technique, which uses snapshots generated by a 3D non-linear incompressible discontinuous Galerkin Navier-Stokes solver, provides very similar results to classical linear instability analysis techniques. In addition, we compare DMD results issued from non-linear and linearised NavierStokes solvers, showing that linearisation is not necessary (i.e. base flow not required) to obtain linear modes, as long as the analysis is restricted to the exponential growth regime, that is, flow regime governed by the linearised Navier-Stokes equations, and showing the potential of this type of analysis based on snapshots to general purpose CFD codes, without need of modifications. Finally, this work shows that the DMD technique can provide three-dimensional direct and adjoint modes through snapshots provided by the linearised and adjoint linearised Navier-Stokes equations advanced in time. Subsequently, these modes are used to provide structural sensitivity maps and sensitivity to base flow modification information for 3D flows and complex geometries, at an affordable computational cost. The information provided by the sensitivity study is used to modify the L-shaped geometry and control the most unstable 3D mode.

A Reduced Order Model to Predict Transient Flows around Straight Bladed Vertical Axis Wind Turbines

“…The solver has been widely validated for a variety of flows, including bluff body flows, airfoil and blade aerodynamics under static and rotating conditions [15][16][17], global instability analysis [20,21] and turbulent regimes [19]. The advantages of high order over low order methods to compute turbine flows have been discussed in [22].…”

Sensitivity Analysis to Control the Far-Wake Unsteadiness Behind Turbines

“…Equation (3) may be advanced in time to simulate the growth or decay of perturbation uponq [8,44].Alternatively, it is possible to solve the above perturbation equation in the frequency domain by introducing the ansatz, q =qe σt , which leads to an eigenvalue problem…”

“…The shape of this mode suggests that the wake may be divided in two regions: a middle-distance-wake, x/D < 10 and a very-far-wake , x/D > 10, as observed in the previous section: vorticity curves in Figure 6 and the shape of the direct mode in Figure 8a. A spatial stability analysis performed using spatial-DMD [8,[52][53][54]] reveals a spatial growth of 0.4 for the middle-distance-wake and of 0.9 for the very-far-wake showing that the convective instability of the wake is enhanced aft 10 turbine diameters. The two distinct wake regions will lead to varying sensitivities (see next section).…”

Discrete linear stability and sensitivity analysis of fluid flows

“…Contrary to other authors, [64,40,34], the compressible NS are used in this study, therefore, in order to reproduce previous results and for validation purposes, the Mach number Ma is set to 0.2, being a compromise between numerical accuracy and compressible effects. …”

“…Equation (2.6) may be advanced in time to simulate the growth or decay of perturbation uponq [11,34]. The perturbations are solved in the frequency domain by characterising them as normal modes, q ′ =qe σt , whereq is an eigenmode of the system.…”