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 threedimensional 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 century2–4 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 matrix-forming 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 difference 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.

Matrix-free time-stepping methods for the solution of TriGlobal instability problems

La inmensa mayoría de los flujos de relevancia ingenieril permanecen sin estudiar en el marco de la teoría de estabilidad global. Esto es debido a dos razones fundamentalmente, las dificultades asociadas con el análisis de los flujos turbulentos y los inmensos recursos computacionales requeridos para obtener la solución del problema de autovalores asociado al análisis de inestabilidad de flujos tridimensionales, también conocido como problema TriGlobal. En esta tesis se aborda el problema asociado con la tridimensionalidad. Se ha desarrollado una metodología general para obtener soluciones de problemas de análisis modal de las inestabilidades lineales globales mediante el acoplamiento de métodos de evolución temporal, desarrollados en este trabajo, con códigos de mecánica de fluidos computacional de segundo orden, utilizados de forma general en la industria. Esta metodología consiste en la resolución del problema de autovalores asociado al análisis de inestabilidad mediante métodos de proyección en subespacios de Krylov, con la particularidad de que dichos subespacios son generados por medio de la integración temporal de un vector inicial usando cualquier código de mecánica de fluidos computacional. Se han elegido tres problemas desafiantes en función de la exigencia de recursos computacionales necesarios y de la complejidad física para la demostración de la presente metodología: (i) el flujo en el interior de una cavidad tridimensional impulsada por una de sus tapas, (ii) el flujo alrededor de un cilindro equipado con aletas helicoidales a lo largo su envergadura y (iii) el flujo a través de una cavidad abierta tridimensinal en ausencia de homogeneidades espaciales. Para la validación de la tecnología se ha obtenido la solución del problema TriGlobal asociado al flujo en la cavidad tridimensional, utilizando el método de evolución temporal desarrollado acoplado con los operadores numéricos de flujo incompresible del código CFD OpenFOAM (código libre). Los resultados obtenidos coinciden plentamente con la literatura. La aplicación de esta metodología al estudio de inestabilidades globales de flujos abiertos tridimensionales ha proporcionado por primera vez, información sobre la transición tridimensional de estos flujos. Además, la metodología ha sido adaptada para resolver problemas adjuntos TriGlobales, permitiendo el control de flujo basado en modificaciones de las inestabilidades globales. Finalmente, se ha demostrado que la cantidad moderada de los recursos computacionales requeridos para la solución del problema de valor propio TriGlobal usando este método numérico, junto a su versatilidad al poder acoplarse a cualquier código aerodinámico, permite la realización de análisis de inestabilidad global y control de flujos complejos de relevancia industrial. Abstract Most flows of engineering relevance still remain unexplored in a global instability theory context for two reasons. First, because of the difficulties associated with the analysis of turbulent flows and, second, for the formidable computational resources required for the solution of the eigenvalue problem associated with the instability analysis of three-dimensional base flows, also known as TriGlobal problem. In this thesis, the problem associated with the three-dimensionality is addressed by means of the development of a general approach to the solution of large-scale global linear instability analysis by coupling a time-stepping approach with second order aerodynamic codes employed in industry. Three challenging flows in the terms of required computational resources and physical complexity have been chosen for demonstration of the present methodology; (i) the flow inside a wall-bounded three-dimensional lid-driven cavity, (ii) the flow past a cylinder fitted with helical strakes and (iii) the flow over a inhomogeneous three-dimensional open cavity. Results in excellent agreement with the literature have been obtained for the three-dimensional lid-driven cavity by using this methodology coupled with the incompressible solver of the open-source toolbox OpenFOAM®, which has served as validation. Moreover, significant physical insight of the instability of three-dimensional open flows has been gained through the application of the present time-stepping methodology to the other two cases. In addition, modifications to the present approach have been proposed in order to perform adjoint instability analysis of three-dimensional base flows and flow control; validation and TriGlobal examples are presented. Finally, it has been demonstrated that the moderate amount of computational resources required for the solution of the TriGlobal eigenvalue problem using this method enables the performance of instability analysis and control of flows of industrial relevance.

Analytical Calculation of Stall-inception and Surge Points for an Axial-flow Compresor Rotor

Recently, a theoretical criterion to calculate the stability of an axial-flow compressor rotor has been presented in the scientific literature. This theoretical criterion was used for determining the locus of the stability line over the rotor map and for predicting the post-stall evolution of the constant-speed line of a rotor. The main objective of this paper is to improve the predictions of such a model. To do that, the paper proposes a different characterization of the characteristic azimuthal length and a calculation of the ratio of specific heats based on a polytropic exponent. Thanks to these new values, the model predicts two bifurcation points in the behaviour of the flow: the inception point of the instability and the surge point. Experimental data from a pure axial compressor are used to validate the model showing that the prediction of the flow coefficient at the surge point has an error inferior to 5%. For the rotor studied, the paper provides a quantitative and qualitative description of the inception of the instability and of the mechanism involved in the instable region of the compressor map. The paper also discusses the role of rotor efficiency in the position of the bifurcations and gives a sensitivity analysis of its position. Finally, it presents a discussion about how the model can explain the different behaviours exhibited by the same rotor when the flow coefficient is reduced

Instability and topology bifurcations on a hemisphere-cylinder at high angle of attack

La configuración de un cilindro acoplado a una semi-esfera, conocida como ’hemispherecylinder’, se considera como un modelo simplificado para numerosas aplicaciones industriales tales como fuselaje de aviones o submarinos. Por tanto, el estudio y entendimiento de los fenómenos fluidos que ocurren alrededor de dicha geometría presenta gran interés. En esta tesis se muestra la investigación del origen y evolución de los, ya conocidos, patrones de flujo (burbuja de separación, vórtices ’horn’ y vórtices ’leeward’) que se dan en esta geometría bajo condiciones de flujo separado. Para ello se han llevado a cabo simulaciones numéricas (DNS) y ensayos experimentales usando la técnica de Particle Image Velocimetry (PIV), para una variedad de números de Reynolds (Re) y ángulos de ataque (AoA). Se ha aplicado sobre los resultados numéricos la teoría de puntos críticos obteniendo, por primera vez para esta geometría, un diagrama de bifurcaciones que clasifica los diferentes regímenes topológicos en función del número de Reynolds y del ángulo de ataque. Se ha llevado a cabo una caracterización completa sobre el origen y la evolución de los patrones estructurales característicos del cuerpo estudiado. Puntos críticos de superficie y líneas de corriente tridimensionales han ayudado a describir el origen y la evolución de las principales estructuras presentes en el flujo hasta alcanzar un estado de estabilidad desde el punto de vista topológico. Este estado se asocia con el patrón de los vórtices ’horn’, definido por una topología característica que se encuentra en un rango de números de Reynolds muy amplio y en regímenes compresibles e incompresibles. Por otro lado, con el objeto de determinar las estructuras presentes en el flujo y sus frecuencias asociadas, se han usado distintas técnicas de análisis: Proper Orthogonal Decomposition (POD), Dynamic Mode Decomposition (DMD) y análisis de Fourier. Dichas técnicas se han aplicado sobre los datos experimentales y numéricos, demostrándose la buena concordancia entre ambos resultados. Finalmente, se ha encontrado en ambos casos, una frecuencia dominante asociada con una inestabilidad de los vórtices ’leeward’. ABSTRACT The hemisphere-cylinder may be considered as a simplified model for several geometries found in industrial applications such as aircrafts’ fuselages or submarines. Understanding the complex flow phenomena that surrounds this particular geometry is therefore of major industrial interest. This thesis presents an investigation of the origin and evolution of the complex flow pattern; i.e. separation bubbles, horn vortices and leeward vortices, around the hemisphere-cylinder under separated flow conditions. To this aim, threedimensional Direct Numerical Simulations (DNS) and experimental tests, using Particle Image Velocimetry (PIV) techniques, have been performed for a variety of Reynolds numbers (Re) and angles of attack (AoA). Critical point theory has been applied to the numerical simulations to provide, for the first time for this geometry, a bifurcation diagram that classifies the different flow topology regimes as a function of the Reynolds number and the angle of attack. A complete characterization about the origin and evolution of the complex structural patterns of this geometry has been put in evidence. Surface critical points and surface and volume streamlines were able to describe the main flow structures and their strong dependence with the flow conditions up to reach the structurally stable state. This state was associated with the pattern of the horn vortices, found on ranges from low to high Reynolds numbers and from incompressible to compressible regimes. In addition, different structural analysis techniques have been employed: Proper Orthogonal Decomposition (POD), Dynamic Mode Decomposition (DMD) and Fourier analysis. These techniques have been applied to the experimental and numerical data to extract flow structure information (i.e. modes and frequencies). Experimental and numerical modes are shown to be in good agreement. A dominant frequency associated with an instability of the leeward vortices has been identified in both, experimental and numerical results.

Molecular Dynamics Simulations of Couette flow

The first steps towards developing a continuum-molecular coupled simulations techniques are presented, for the purpose of computing macroscopic systems of confined fluids. The idea is to compute the interface wall-fluid by Molecular Dynamics simulations, where Lennard-Jones potential (and others) have been employed for the molecular interactions, so the usual non slip boundary condition is not specified. Instead, a shear rate can be imposed at the wall, which allows to obtain the properties of the wall material by means of an iterative method. The remaining fluid region will be computed by a spectral hp method. We present MD simulations of a Couette flow, and the results of the developed boundary conditions from the wall fluid interaction.

BiGlobal and point vortex methods for the instability analysis of wakes

To better understand destruction mechanisms of wake-vortices behind aircraft, the point vortex method for stability (inviscid) used by Crow is here compared with viscous modal global stability analysis of the linearized Navier-Stokes equations acting on a two-dimensional basic flow, i.e. BiGlobal stability analysis. The fact that the BiGlobal method is viscous, and uses a flnite área vortex model, gives rise to results somewhat different from the point vortex model. It adds more parameters to the problem, but is more realistic.

Accurate Parabolic Navier-Stokes solutions of the supersonic flow around and elliptic cone

Flows of relevance to new generation aerospace vehicles exist, which are weakly dependent on the streamwise direction and strongly dependent on the other two spatial directions, such as the flow around the (flattened) nose of the vehicle and the associated elliptic cone model. Exploiting these characteristics, a parabolic integration of the Navier-Stokes equations is more appropriate than solution of the full equations, resulting in the so-called Parabolic Navier-Stokes (PNS). This approach not only is the best candidate, in terms of computational efficiency and accuracy, for the computation of steady base flows with the appointed properties, but also permits performing instability analysis and laminar-turbulent transition studies a-posteriori to the base flow computation. This is to be contrasted with the alternative approach of using order-of-magnitude more expensive spatial Direct Numerical Simulations (DNS) for the description of the transition process. The PNS equations used here have been formulated for an arbitrary coordinate transformation and the spatial discretization is performed using a novel stable high-order finite-difference-based numerical scheme, ensuring the recovery of highly accurate solutions using modest computing resources. For verification purposes, the boundary layer solution around a circular cone at zero angle of attack is compared in the incompressible limit with theoretical profiles. Also, the recovered shock wave angle at supersonic conditions is compared with theoretical predictions in the same circular-base cone geometry. Finally, the entire flow field, including shock position and compressible boundary layer around a 2:1 elliptic cone is recovered at Mach numbers 3 and 4

Instability Analysis of Incompressible Open Cavity Flows

El problema del flujo sobre una cavidad abierta ha sido estudiado en profundidad en la literatura, tanto por el interés académico del problema como por sus aplicaciones prácticas en gran variedad de problemas ingenieriles, como puede ser el alojamiento del tren de aterrizaje de aeronaves, o el depósito de agua de aviones contraincendios. Desde hace muchos a˜nos se estudian los distintos tipos de inestabilidades asociadas a este problema: los modos bidimensionales en la capa de cortadura, y los modos tridimensionales en el torbellino de recirculación principal dentro de la cavidad. En esta tesis se presenta un estudio paramétrico completo del límite incompresible del problema, empleando la herramienta de estabilidad lineal conocida como BiGlobal. Esta aproximación permite contemplar la estabilidad global del flujo, y obtener tanto la forma como las características de los modos propios del problema físico, sean estables o inestables. El estudio realizado permite caracterizar con gran detalle todos los modos relevantes, así como la envolvente de estabilidad en el espacio paramétrico del problema incompresible (Mach nulo, variación de Reynolds, espesor de capa límite incidente, relación altura/profundidad de la cavidad, y longitud característica de la perturbación en la dirección transversal). A la luz de los resultados obtenidos se proponen una serie de relaciones entre los parámetros y características de los modos principales, como por ejemplo entre el Reynolds crítico de un modo, y la longitud característica del mismo. Los resultados numéricos se contrastan con una campaña experimental, siendo la principal conclusión de dicha comparación que los modos lineales están presentes en el flujo real saturado, pero que existen diferencias notables en frecuencia entre las predicciones teóricas y los experimentos. Para intentar determinar la naturaleza de dichas diferencias se realiza una simulación numérica directa tridimensional, y se utiliza un algoritmo de DMD (descomposición dinámica de modos) para describir el proceso de saturación. ABSTRACT The problem of the flow over an open cavity has been studied in depth in the literature, both for being an interesting academical problem and due to the multitude of industrial applications, like the landing gear of aircraft, or the water deposit of firefighter airplanes. The different types of instabilities appearing in this flow studied in the literature are two: the two-dimensional shear layer modes, and the three-dimensional modes that appear in the main recirculating vortex inside the cavity. In this thesis a parametric study in the incompressible limit of the problem is presented, using the linear stability analysis known as BiGlobal. This approximation allows to obtain the global stability behaviour of the flow, and to capture both the morphological features and the characteristics of the eigenmodes of the physical problem, whether they are stable or unstable. The study presented here characterizes with great detail all the relevant eigenmodes, as well as the hypersurface of instability on the parameter space of the incompressible problem (Mach equal to zero, and variation of the Reynolds number, the incoming boundary layer thickness, the length to depth aspect ratio of the cavity and the spanwise length of the perturbation). The results allow to construct parametric relations between the characteristics of the leading eigenmodes and the parameters of the problem, like for example the one existing between the critical Reynolds number and its characteristic length. The numerical results presented here are compared with those of an experimental campaign, with the main conclusion of said comparison being that the linear eigenmode are present in the real saturated flow, albeit with some significant differences in the frequencies of the experiments and those predicted by the theory. To try to determine the nature of those differences a three-dimensional direct numerical simulation, analyzed with Dynamic Mode Decomposition algorithm, was used to describe the process of saturation.

Global Stability Analysis of a Compressible Turbulent Flow around a High-Lift Configuration

The purpose of this work is to analyze a complex high lift configuration for which significant regions of separated flow are present. Current state of the art methods have some diffculty to predict the origin and the progression of this separated flow when increasing the angle of attack. The mechanisms responsible for the maximum lift limit on multi-element wing con?gurations are not clear; this stability analysis could help to understand the physics behind the phenomenon and to find a relation between the flow separation and the instability onset. The methodology presented herein consists in the computation of a steady base flow solution based on a finite volume discretization and a proposal of the solution for a generalized eigenvalue problem corresponding to the perturbed and linearized problem. The eigenvalue problem has been solved with the Arnoldi iterative method, one of the Krylov subspace projection methods. The described methodology was applied to the NACA0012 test case in subsonic and in transonic conditions and, finally, for the first time to the authors knowledge, on an industrial multi-component geometry, such as the A310 airfoil, in order to identify low frequency instabilities related to the separation. One important conclusion is that for all the analyzed geometries, one unstable mode related to flow separation appears for an angle of attack greater than the one correspondent to the maximum lift coe?cient condition. Finally, an adjoint study was carried out in order to evaluate the receptivity and the structural sensitivity of the geometries, giving an indication of the domain region that could be modified resulting in the biggest change of the flowfield.

Heat flow determinations of ODP Hole 111-504B

Hole 504B in the eastern equatorial Pacific has been the focus of five scientific drilling expeditions since it was first drilled in 1979. During these five legs, a series of temperature logs has been obtained over a time span of almost 8 yr, documenting the geothermal and hydrologic state of the oceanic crust in this region. Immediately following reentry at the onset of ODP Leg 111 operations, a high-resolution temperature probe was lowered into the borehole and a precise record of temperature vs. depth in Hole 504B was recorded down to 1300 mbsf. As was observed during previous legs, the temperature gradient in the upper 400 m was reduced, indicating that downhole flow of cool ocean waters through the casing continued, though at a diminished rate. As subhydrostatic pressures in the upper basement have gradually diminished, the volume of flow has decayed from an estimated 6000-7000 L/hr in late 1979 to about 80 L/hr during Leg 111. At depths below 480 mbsf, a predominantly conductive heat transfer environment enabled the temperature gradient log to be analyzed with respect to lithology on both fine and broad scales. Anomalies in the gradient log in the cased section through the sedimentary column were found to correspond to biostratigraphic age markers and/or sharp changes in sediment composition and texture. Broad variations in temperature gradient within the basement correlated with large-scale porosity trends. Conductive heatflow estimates depict a systematic reduction with depth, ranging from approximately 196 mW/m**2 in the sediments to 120 ± 17 mW/m**2 at 1300 mbsf. Possible causes for this observation were examined from several perspectives, but none was suitably convincing. A fluid instability analysis indicated the likely existence of convection cells within the borehole and substantiated the hypothesis of mixing within the borehole postulated from isotopic and chemical studies of borehole waters. However, such mixing of borehole fluids does not provide an adequate explanation for the heatflow variations, and the disparity between surficial and deep values of heat flow remains unresolved.

Capillary jet instability under the influence of gravity

The focus of this paper is on the effect of gravity stretching on disturbed capillary jet instability. Break-up and droplet formation under low flows are simulated using finite difference solution of a one-dimensional approximation of disturbed capillary jet instability chosen from the work by Eggers and Dupont (J. Fluid Mech. 155 (1994) 289). Experiments were conducted using water and aqueous glycerol solutions to compare with simulations. We use a gravity parameter, G, which quantifies gravity stretching by relating flow velocity, orifice size and acceleration and is the reciprocal of the Fronde number. The optimum disturbance frequency Omega(opt) was found to be inversely proportional to G. However, this relationship appears to be complex for the range of G's investigated. At low G, the relationship between Omega(opt) and G appears to be linear but takes on a weakly decaying like trend as G increases. As flows are lowered, the satellite-free regime decreases, although experimental observation found that merging of main and satellite drops sometimes offset this effect to result in monodispersed droplet trains post break-up. Viscosity did not significantly affect the relationship between the disturbance frequency and G, although satellite drops could be seen more clearly close to the upper limit for instability at high G's. It is possible to define regimes of satellite formation under low flows by considering local wavenumbers at the point of instability. (C) 2004 Elsevier Ltd. All rights reserved.

Theoretical investigation of convective instability in inclined and fluid-saturated three-dimensional fault zones.

The convective instability of pore-fluid flow in inclined and fluid-saturated three-dimensional fault zones has been theoretically investigated in this paper. Due to the consideration of the inclined three-dimensional fault zone with any values of the inclined angle, it is impossible to use the conventional linear stability analysis method for deriving the critical condition (i.e., the critical Rayleigh number) which can be used to investigate the convective instability of the pore-fluid flow in an inclined three-dimensional fault zone system. To overcome this mathematical difficulty, a combination of the variable separation method and the integration elimination method has been used to derive the characteristic equation, which depends on the Rayleigh number and the inclined angle of the inclined three-dimensional fault zone. Using this characteristic equation, the critical Rayleigh number of the system can be numerically found as a function of the inclined angle of the three-dimensional fault zone. For a vertically oriented three-dimensional fault zone system, the critical Rayleigh number of the system can be explicitly derived from the characteristic equation. Comparison of the resulting critical Rayleigh number of the system with that previously derived in a vertically oriented three-dimensional fault zone has demonstrated that the characteristic equation of the Rayleigh number is correct and useful for investigating the convective instability of pore-fluid flow in the inclined three-dimensional fault zone system. The related numerical results from this investigation have indicated that: (1) the convective pore-fluid flow may take place in the inclined three-dimensional fault zone; (2) if the height of the fault zone is used as the characteristic length of the system, a decrease in the inclined angle of the inclined fault zone stabilizes the three-dimensional fundamental convective flow in the inclined three-dimensional fault zone system; (3) if the thickness of the stratum is used as the characteristic length of the system, a decrease in the inclined angle of the inclined fault zone destabilizes the three-dimensional fundamental convective flow in the inclined three-dimensional fault zone system; and that (4) the shape of the inclined three-dimensional fault zone may affect the convective instability of pore-fluid flow in the system. (C) 2004 Published by Elsevier B.V.

Flow focusing in microchannels

A volume-of-fluid numerical method is used to predict the dynamics of drop formation in an axi-symmetric microfluidic flow-focusing geometry for a liquid-liquid system. The Reynolds numbers and Weber numbers approximate those of a three-dimensional flow in recently published experiments. We compare the predicted drop formation with the experimental results at various flow rates, and discuss the mechanisms of drop formation in this context. Despite the differences in geometry, we find qualitative correspondence between the numerical and experimental results. Both end-pinching and capillary-wave instability are important for droplet break-up at the higher flow rates.

On the chaotic instability of a nonsliding liquid-filled top with a small spheroidal base via Melnikov-Holmes-Marsden integrals

Chaotic orientations of a top containing a fluid filled cavity are investigated analytically and numerically under small perturbations. The top spins and rolls in nonsliding contact with a rough horizontal plane and the fluid in the ellipsoidal shaped cavity is considered to be ideal and describable by finite degrees of freedom. A Hamiltonian structure is established to facilitate the application of Melnikov-Holmes-Marsden (MHM) integrals. In particular, chaotic motion of the liquid-filled top is identified to be arisen from the transversal intersections between the stable and unstable manifolds of an approximated, disturbed flow of the liquid-filled top via the MHM integrals. The developed analytical criteria are crosschecked with numerical simulations via the 4th Runge-Kutta algorithms with adaptive time steps.

Stochastically driven instability in rotating shear flows

Origin of hydrodynamic turbulence in rotating shear flows is investigated. The particular emphasis is on flows whose angular velocities decrease but specific angular momenta increase with increasing radial coordinate. Such flows are Rayleigh stable, but must be turbulent in order to explain observed data. Such a mismatch between the linear theory and observations/experiments is more severe when any hydromagnetic/magnetohydrodynamic instability and the corresponding turbulence therein is ruled out. The present work explores the effect of stochastic noise on such hydrodynamic flows. We focus on a small section of such a flow which is essentially a plane shear flow supplemented by the Coriolis effect. This also mimics a small section of an astrophysical accretion disk. It is found that such stochastically driven flows exhibit large temporal and spatial correlations of perturbation velocities, and hence large energy dissipations, that presumably generate instability. A range of angular velocity profiles (for the steady flow), starting with the constant angular momentum to that of the constant circular velocity are explored. It is shown that the growth and roughness exponents calculated from the contour (envelope) of the perturbed flows are all identical, revealing a unique universality class for the stochastically forced hydrodynamics of rotating shear flows. This work, to the best of our knowledge, is the first attempt to understand origin of instability and turbulence in the three-dimensional Rayleigh stable rotating shear flows by introducing additive stochastic noise to the underlying linearized governing equations. This has important implications in resolving the turbulence problem in astrophysical hydrodynamic flows such as accretion disks.