F. Gómez

Four Decades of Studying Global Linear Instability: Progress and Challenges

Global linear instability theory is concerned with the temporal or spatial development of small-amplitude perturbations superposed upon laminar steady or time-periodic three-dimensional flows, which are inhomogeneous in two (and periodic in one) or all three spatial directions. After a brief exposition of the theory, some recent advances are reported. First, results are presented on the implementation of a Jacobian-free Newton-Krylov time-stepping method into a standard finite-volume aerodynamic code to obtain global linear instability results in flows of industrial interest. Second, connections are sought between established and more-modern approaches for structure identification in flows, such as proper orthogonal decomposition and Koopman modes analysis (dynamic mode decomposition), and the possibility to connect solutions of the eigenvalue problem obtained by matrix formation or time-stepping with those delivered by dynamic mode decomposition, residual algorithm, and proper orthogonal decomposition analysis is highlighted in the laminar regime; turbulent and three-dimensional flows are identified as open areas for future research. Finally, a new stable very-high-order finite-difference method is implemented for the spatial discretization of the operators describing the spatial biglobal eigenvalue problem, parabolized stability equation three-dimensional analysis, and the triglobal eigenvalue problem; it is shown that, combined with sparse matrix treatment, all these problems may now be solved on standard desktop computers.

A reduced-order model of three-dimensional unsteady flow in a cavity based on the resolvent operator

A novel reduced-order model for nonlinear flows is presented. The model arises from a resolvent decomposition in which the nonlinear advection terms of the Navier-Stokes equation are considered as the input to a linear system in Fourier space. Results show that Taylor-Gortler-like vortices can be represented from a low-order resolvent decomposition of a nonlinear lid-driven cavity flow. The present approach provides an approximation of the fluctuating velocity given the time-mean and the time history of a single velocity probe.

On the origin of frequency sparsity in direct numerical simulations of turbulent pipe flow

The possibility of creating reduced-order models for canonical wall-bounded turbulent flows based on exploiting energy sparsity in frequency domain, as proposed by Bourguignon et al. [Phys. Fluids26, 015109 (2014)], is examined. The present letter explains the origins of energetically sparse dominant frequencies and provides fundamental information for the design of such reduced-order models. The resolvent decomposition of a pipe flow is employed to consider the influence of finite domain length on the flow dynamics, which acts as a restriction on the possible wavespeeds in the flow. A forcing-to-fluctuation gain analysis in the frequency domain reveals that large sparse peaks in amplification occur when one of the possible wavespeeds matches the local wavespeed via the critical layer mechanism. A link between amplification and energy is provided through the similar characteristics exhibited by the most energetically relevant flow structures, arising from a dynamic mode decomposition of direct numerical simulation data, and the resolvent modes associated with the most amplified sparse frequencies. These results support the feasibility of reduced-order models based on the selection of the most amplified modes emerging from the resolvent model, leading to a novel computationally efficient method of representing turbulent flows.

Estimation of unsteady aerodynamic forces using pointwise velocity data

A novel method to estimate unsteady aerodynamic force coefficients from pointwise velocity measurements is presented. The methodology is based on a resolvent-based reduced-order model which requires the mean flow to obtain physical flow structures and pointwise measurement to calibrate their amplitudes. A computationally-affordable time-stepping methodology to obtain resolvent modes in non-trivial flow domains is introduced and compared to previous existing matrix-free and matrix-forming strategies. The technique is applied to the unsteady flow around an inclined square cylinder at low Reynolds number. The potential of the methodology is demonstrated through good agreement between the fluctuating pressure distribution on the cylinder and the temporal evolution of the unsteady lift and drag coefficients predicted by the model and those computed by direct numerical simulation.

On the Coupling of Direct Numerical Simulation and Resolvent Analysis

The present contribution explores the relationship between response and forcing via amplification mechanisms in the Navier–Stokes equations applied to a turbulent pipe flow. A novel numerical method coupling direct numerical simulation with the resolvent model [J. Fluid Mech. 658, 336-382 (2010)] is developed in order to reveal the exact distribution of the nonlinear forcing terms, originally unknown in the model. The obtained results highlight the major role of the nonlinear terms in the energy spectra.

On the role of global flow instability analysis in closed loop flow control

Control of linear flow instabilities has been demonstrated to be an effective theoretical flow control methodology, capable of modifying transitional flows on canonical geometries such as the plane channel and the flat-plate boundary layer. Extending the well-developed theoretical flow control techniques to flows over or through complex geometries requires addressing the issue of efficient capturing of the leading members of the global eigenspectrum pertinent to such flows. The present contribution describes state-of-the-art modal global instability analysis methodologies recently developed in our group, based on matrix formation and time-stepping, respectively. The relative performance of these algorithms is assessed on the recovery of BiGlobal and TriGlobal eigenspectra in the spanwise periodic and the cubic lid-driven cavity, respectively; the adjoint eigenspectrum in the latter flow is recovered for the first time. For three-dimensional flows without any homogeneous spatial direction, the time-stepping methodology was found to outperform the matrix-forming approach and permit recovering the leading TriGlobal eigenmodes in an three-dimensional open cavity of aspect ratio L : D : W = 5 : 1 : 1; theoretical flow control of this configuration is underway.

Streamwise-varying steady transpiration control in turbulent pipe flow

The effect of streamwise-varying steady transpiration on turbulent pipe flow is examined using direct numerical simulation at fixed friction Reynolds number Re_τ=314. The streamwise momentum equation reveals three physical mechanisms caused by transpiration acting in the flow: modification of Reynolds shear stress, steady streaming and generation of non-zero mean streamwise gradients. The influence of these mechanisms has been examined by means of a parameter sweep involving transpiration amplitude and wavelength. The observed trends have permitted identification of wall transpiration configurations able to reduce or increase the overall flow rate −36.1% and 19.3%, respectively. Energetics associated with these modifications are presented. A novel resolvent formulation has been developed to investigate the dynamics of pipe flows with a constant cross-section but with time-mean spatial periodicity induced by changes in boundary conditions. This formulation, based on a triple decomposition, paves the way for understanding turbulence in such flows using only the mean velocity profile. Resolvent analysis based on the time-mean flow and dynamic mode decomposition based on simulation data snapshots have both been used to obtain a description of the reorganization of the flow structures caused by the transpiration. We show that the pipe flows dynamics are dominated by a critical-layer mechanism and the waviness induced in the flow structures plays a role on the streamwise momentum balance by generating additional terms.

Structural analysis on a hemisphere-cylinder at moderate Reynolds number and high angle of attack

Three-dimensional DNS has been performed on a hemisphere-cylinder at Reynolds number Re= 1000 and angle of attack AoA= 20◦ in order to analyze flow structures and wake frequencies. PIV experiments have also been carried out over the same geometry and flow conditions to validate the numerical results. Critical point theory has been applied in order to determine the topology patterns over the surface of the body. Critical points and separation lines on the body surface show the presence of three different flow patterns: separation bubble, ”horn vortices” and ”leeward vortices”. Both, ”horn vortices” and ”leeward vortices” are found to be asymmetric and unsteady. The frequency related to ”leeward vortices” oscillations has been identified both experimentally and numerically. Two more dominant frequencies, related to two different wake shedding modes have been found. On the other hand, POD has been performed and the four most energetic POD modes is found to be composed of a mixture of these three frequencies. They are modes on the wake shedding. Finally, DMD modes associated with these three frequencies are found on the symmetry plane close to the nose area. These modes represent different shear layer instabilities. Flow separation was found to be intrinsically linked with the observed shear-layer instability.

Global Linear Instability at the Dawn of its 4th Decade: A List of Challenges (A Practical Guide on how to Contain the Euphoria and Avoid the Oversell)

Global linear instability theory is concerned with the temporal or spatial development of small-amplitude perturbations superposed upon laminar steady or time-periodic three-dimensional flows, which are inhomogeneous in two (and periodic in one) or all three spatial directions. 1 The theory addresses flows developing in complex geometries, in which the parallel or weakly nonparallel basic flow approximation invoked by classic linear stability theory does not hold. As such, global linear theory is called to fill the gap in research into stability and transition in flows over or through complex geometries. Historically, global linear instability has been (and still is) concerned with solution of multi-dimensional eigenvalue problems; the maturing of non-modal linear instability ideas in simple parallel flows during the last decade of last century 2‐4 has given rise to investigation of transient growth scenarios in an ever increasing variety of complex flows. After a brief exposition of the theory, connections are sought with established approaches for structure identification in flows, such as the proper orthogonal decomposition and topology theory in the laminar regime and the open areas for future research, mainly concerning turbulent and three-dimensional flows, are highlighted. Recent results obtained in our group are reported in both the time-stepping and the matrixforming approaches to global linear theory. In the first context, progress has been made in implementing a Jacobian-Free Newton Krylov method into a standard finite-volume aerodynamic code, such that global linear instability results may now be obtained in compressible flows of aeronautical interest. In the second context a new stable very high-order finite dierence method is implemented for the spatial discretization of the operators describing the spatial BiGlobal EVP, PSE-3D and the TriGlobal EVP; combined with sparse matrix treatment, all these problems may now be solved on standard desktop computers.

Effect of the Trailing Edge Geometry on the Unsteadiness of the Flow Around a Stalled NACA 0015 Airfoil

The effect on the flow unsteadiness on a NACA 0015 airfoil at Reynolds number \(Re = 200\) and Angle of Attack \(AoA = 18^{\circ }\) is investigated numerically. Four different geometries based on the NACA 0015 airfoil and with trailing-edge modifications are compared. Long-time integration of the incompressible two-dimensional Navier-Stokes equations shows that the recovered flow field is steady independently of the trailing-edge geometry.