Efficient numerical methods for global linear instability. Analysis and modeling of non-orthogonal swept attachment-line boundary layer flow

The aim of this thesis is to study the mechanisms of instability that occur in swept wings when the angle of attack increases. For this, a simplified model for the a simplified model for the non-orthogonal swept leading edge boundary layer has been used as well as different numerical techniques in order to solve the linear stability problem that describes the behavior of perturbations superposed upon this base flow. Two different approaches, matrix-free and matrix forming methods, have been validated using direct numerical simulations with spectral resolution. In this way, flow instability in the non-orthogonal swept attachment-line boundary layer is addressed in a linear analysis framework via the solution of the pertinent global (Bi-Global) PDE-based eigenvalue problem. Subsequently, a simple extension of the extended G¨ortler-H¨ammerlin ODEbased polynomial model proposed by Theofilis, Fedorov, Obrist & Dallmann (2003) for orthogonal flow, which includes previous models as particular cases and recovers global instability analysis results, is presented for non-orthogonal flow. Direct numerical simulations have been used to verify the stability results and unravel the limits of validity of the basic flow model analyzed. The effect of the angle of attack, AoA, on the critical conditions of the non-orthogonal problem has been documented; an increase of the angle of attack, from AoA = 0 (orthogonal flow) up to values close to _/2 which make the assumptions under which the basic flow is derived questionable, is found to systematically destabilize the flow. The critical conditions of non-orthogonal flows at 0 _ AoA _ _/2 are shown to be recoverable from those of orthogonal flow, via a simple analytical transformation involving AoA. These results can help to understand the mechanisms of destabilization that occurs in the attachment line of wings at finite angles of attack. Studies taking into account variations of the pressure field in the basic flow or the extension to compressible flows are issues that remain open. El objetivo de esta tesis es estudiar los mecanismos de la inestabilidad que se producen en ciertos dispositivos aerodinámicos cuando se aumenta el ángulo de ataque. Para ello se ha utilizado un modelo simplificado del flujo de base, así como diferentes técnicas numéricas, con el fin de resolver el problema de estabilidad lineal asociado que describe el comportamiento de las perturbaciones. Estos métodos; sin y con formación de matriz, se han validado utilizando simulaciones numéricas directas con resolución espectral. De esta manera, la inestabilidad del flujo de capa límite laminar oblicuo entorno a la línea de estancamiento se aborda en un marco de análisis lineal por medio del método Bi-Global de resolución del problema de valores propios en derivadas parciales. Posteriormente se propone una extensión simple para el flujo no-ortogonal del modelo polinomial de ecuaciones diferenciales ordinarias, G¨ortler-H¨ammerlin extendido, propuesto por Theofilis et al. (2003) para el flujo ortogonal, que incluye los modelos previos como casos particulares y recupera los resultados del analisis global de estabilidad lineal. Se han realizado simulaciones directas con el fin de verificar los resultados del análisis de estabilidad así como para investigar los límites de validez del modelo de flujo base utilizado. En este trabajo se ha documentado el efecto del ángulo de ataque AoA en las condiciones críticas del problema no ortogonal obteniendo que el incremento del ángulo de ataque, de AoA = 0 (flujo ortogonal) hasta valores próximos a _/2, en el cual las hipótesis sobre las que se basa el flujo base dejan de ser válidas, tiende sistemáticamente a desestabilizar el flujo. Las condiciones críticas del caso no ortogonal 0 _ AoA _ _/2 pueden recuperarse a partir del caso ortogonal mediante el uso de una transformación analítica simple que implica el ángulo de ataque AoA. Estos resultados pueden ayudar a comprender los mecanismos de desestabilización que se producen en el borde de ataque de las alas de los aviones a ángulos de ataque finitos. Como tareas pendientes quedaría realizar estudios que tengan en cuenta variaciones del campo de presión en el flujo base así como la extensión de éste al caso de flujos compresibles.

Low-dimensional modeling of streaks in a wedge flow boundary layer

This paper is concerned with the low dimensional structure of optimal streaks in a wedge flow boundary layer, which have been recently shown to consist of a unique (up to a constant factor) three-dimensional streamwise evolving mode, known as the most unstable streaky mode. Optimal streaks exhibit a still unexplored/unexploited approximate self-similarity (not associated with the boundary layer self-similarity), namely the streamwise velocity re-scaled with their maximum remains almost independent of both the spanwise wavenumber and the streamwise coordinate; the remaining two velocity components instead do not satisfy this property. The approximate self-similar behavior is analyzed here and exploited to further simplify the description of optimal streaks. In particular, it is shown that streaks can be approximately described in terms of the streamwise evolution of the scalar amplitudes of just three one-dimensional modes, providing the wall normal profiles of the streamwise velocity and two combinations of the cross flow velocity components; the scalar amplitudes obey a singular system of three ordinary differential equations (involving only two degrees of freedom), which approximates well the streamwise evolution of the general streaks.

Low-Dimensional Modelling for optimal streaks in the blasius boundary layer

This paper is concerned with the low dimensional structure of optimal streaks in the Blasius boundary layer. Optimal streaks are well known to exhibit an approximate self-similarity, namely the streamwise velocity re-scaled with their maximum remains almost independent of both the spanwise wavenumber and the streamwise coordinate. However, the reason of this self-similar behavior is still unexplained as well as unexploited. After revisiting the structure of the streaks near the leading edge singularity, two additional approximately self-similar relations involving the velocity components and their wall normal derivatives are identified. Based on these properties, we derive a low dimensional model with two degrees of freedom. The comparison with the results obtained from the linearized boundary layer equations shows that this model is consistent and provide good approximations.

Linear global instability of non-orthogonal incompressible swept attachment-line boundary layer flow

Instability of the orthogonal swept attachment line boundary layer has received attention by local1, 2 and global3–5 analysis methods over several decades, owing to the significance of this model to transition to turbulence on the surface of swept wings. However, substantially less attention has been paid to the problem of laminar flow instability in the non-orthogonal swept attachment-line boundary layer; only a local analysis framework has been employed to-date.6 The present contribution addresses this issue from a linear global (BiGlobal) instability analysis point of view in the incompressible regime. Direct numerical simulations have also been performed in order to verify the analysis results and unravel the limits of validity of the Dorrepaal basic flow7 model analyzed. Cross-validated results document the effect of the angle _ on the critical conditions identified by Hall et al.1 and show linear destabilization of the flow with decreasing AoA, up to a limit at which the assumptions of the Dorrepaal model become questionable. Finally, a simple extension of the extended G¨ortler-H¨ammerlin ODE-based polynomial model proposed by Theofilis et al.4 is presented for the non-orthogonal flow. In this model, the symmetries of the three-dimensional disturbances are broken by the non-orthogonal flow conditions. Temporal and spatial one-dimensional linear eigenvalue codes were developed, obtaining consistent results with BiGlobal stability analysis and DNS. Beyond the computational advantages presented by the ODE-based model, it allows us to understand the functional dependence of the three-dimensional disturbances in the non-orthogonal case as well as their connections with the disturbances of the orthogonal stability problem.

Tranversal motion and flow structure of fully nonlinear streaks in a laminar boundary layer

Typical streak computations present in the literature correspond to linear streaks or to small amplitude nonlinear streaks computed using DNS or nonlinear PSE. We use the Reduced Navier-Stokes (RNS) equations to compute the streamwise evolution of fully non-linear streaks with high amplitude in a laminar flat plate boundary layer. The RNS formulation provides Reynolds number independent solutions that are asymptotically exact in the limit $Re \gg 1$, it requires much less computational effort than DNS, and it does not have the consistency and convergence problems of the PSE. We present various streak computations to show that the flow configuration changes substantially when the amplitude of the streaks grows and the nonlinear effects come into play. The transversal motion (in the wall normal-streamwise plane) becomes more important and strongly distorts the streamwise velocity profiles, that end up being quite different from those of the linear case. We analyze in detail the resulting flow patterns for the nonlinearly saturated streaks and compare them with available experimental results.

Linear instability of orthogonal compressible leading-edge boundary layer flow

Instability analysis of compressible orthogonal swept leading-edge boundary layer flow was performed in the context of BiGlobal linear theory. 1, 2 An algorithm was developed exploiting the sparsity characteristics of the matrix discretizing the PDE-based eigenvalue problem. This allowed use of the MUMPS sparse linear algebra package 3 to obtain a direct solution of the linear systems associated with the Arnoldi iteration. The developed algorithm was then applied to efficiently analyze the effect of compressibility on the stability of the swept leading-edge boundary layer and obtain neutral curves of this flow as a function of the Mach number in the range 0 ≤ Ma ≤ 1. The present numerical results fully confirmed the asymptotic theory results of Theofilis et al. 4 Up to the maximum Mach number value studied, it was found that an increase of this parameter reduces the critical Reynolds number and the range of the unstable spanwise wavenumbers.

Computation of nonlinear streaky structures in boundary layer flow

En esta tesis se integran numéricamente las ecuaciones reducidas de Navier Stokes (RNS), que describen el flujo en una capa límite tridimensional que presenta también una escala característica espacial corta en el sentido transversal. La formulación RNS se usa para el cálculo de “streaks” no lineales de amplitud finita, y los resultados conseguidos coinciden con los existentes en la literatura, obtenidos típicamente utilizando simulación numérica directa (DNS) o nonlinear parabolized stability equations (PSE). El cálculo de los “streaks” integrando las RNS es mucho menos costoso que usando DNS, y no presenta los problemas de estabilidad que aparecen en la formulación PSE cuando la amplitud del “streak” deja de ser pequeña. El código de integración RNS se utiliza también para el cálculo de los “streaks” que aparecen de manera natural en el borde de ataque de una placa plana en ausencia de perturbaciones en la corriente uniforme exterior. Los resultados existentes hasta ahora calculaban estos “streaks” únicamente en el límite lineal (amplitud pequeña), y en esta tesis se lleva a cabo el cálculo de los mismos en el régimen completamente no lineal (amplitud finita). En la segunda parte de la tesis se generaliza el código RNS para incluir la posibilidad de tener una placa no plana, con curvatura en el sentido transversal que varía lentamente en el sentido de la corriente. Esto se consigue aplicando un cambio de coordenadas, que transforma el dominio físico en uno rectangular. La formulación RNS se integra también expresada en las correspondientes coordenadas curvilíneas. Este código generalizado RNS se utiliza finalmente para estudiar el flujo de capa límite sobre una placa con surcos que varían lentamente en el sentido de la corriente, y es usado para simular el flujo sobre surcos que crecen en tal sentido. Abstract In this thesis, the reduced Navier Stokes (RNS) equations are numerically integrated. This formulation describes the flow in a three-dimensional boundary layer that also presents a short characteristic space scale in the spanwise direction. RNS equations are used to calculate nonlinear finite amplitude “streaks”, and the results agree with those reported in the literature, typically obtained using direct numerical simulation (DNS) or nonlinear parabolized stability equations (PSE). “Streaks” simulations through the RNS integration are much cheaper than using DNS, and avoid stability problems that appear in the PSE when the amplitude of the “streak” is not small. The RNS integration code is also used to calculate the “streaks” that naturally emerge at the leading edge of a flat plate boundary layer in the absence of any free stream perturbations. Up to now, the existing results for these “streaks” have been only calculated in the linear limit (small amplitude), and in this thesis their calculation is carried out in the fully nonlinear regime (finite amplitude). In the second part of the thesis, the RNS code is generalized to include the possibility of having a non-flat plate, curved in the spanwise direction and slowly varying in the streamwise direction. This is achieved by applying a change of coordinates, which transforms the physical domain into a rectangular one. The RNS formulation expressed in the corresponding curvilinear coordinates is also numerically integrated. This generalized RNS code is finally used to study the boundary layer flow over a plate with grooves which vary slowly in the streamwise direction; and this code is used to simulate the flow over grooves that grow in the streamwise direction.

Nonlinear description of transversal motion in a laminar boundary layer with streaks

The nonlinear streamwise growth of a spanwise periodic array of steady streaks in a flat plate boundary layer is numerically computed using the well known Reduced Navier-Stokes formulation. It is found that the flow configuration changes substantially when the amplitude of the streaks grows and the nonlinear effects come into play. The transversal motion (in the wall normal-spanwise plane), which is normally not considered, becomes non-negligible in the nonlinear regime, and it strongly distorts the streamwise velocity profiles, which end up being quite different from those predicted by the linear theory. We analyze in detail the resulting flow patterns for the nonlinearly saturated streaks, and compare them with available experimental results.

Nonlinear description of transversal motion in a laminar boundary layer with streaks

The nonlinear streamwise growth of a spanwise periodic array of steady streaks in a flat plate boundary layer is numerically computed using the well known Reduced Navier- Stokes formulation. It is found that the flow configuration changes substantially when the amplitude of the streaks grows and the nonlinear effects come into play. The transversal motion (in the wall normal-spanwise plane), which is normally not considered, becomes non-negligible in the nonlinear regime, and it strongly distorts the streamwise velocity profiles, which end up being quite different from those predicted by the linear theory. We analyze in detail the resulting flow patterns for the nonlinearly saturated streaks, and compare them with available experimental results.

Comparison of turbulence models for the computational fluid dynamics simulation of wind turbine wakes in the atmospheric boundary layer

An elliptic computational fluid dynamics wake model based on the actuator disk concept is used to simulate a wind turbine, approximated by a disk upon which a distribution of forces, defined as axial momentum sources, is applied on an incoming non-uniform shear flow. The rotor is supposed to be uniformly loaded with the exerted forces estimated as a function of the incident wind speed, thrust coefficient and rotor diameter. The model is assessed in terms of wind speed deficit and added turbulence intensity for different turbulence models and is validated from experimental measurements of the Sexbierum wind turbine experiment.

Parameterization of the atmospheric boundary layer for offshore wind resource assessment with a limited length-scale k-ε model

The structure of the atmospheric boundary layer (ABL) is modelled with the limited- length-scale k-ε model of Apsley and Castro. Contrary to the standard k-ε model, the limited-length-scale k-ε model imposes a maximum mixing length which is derived from the boundary layer height, for neutral and unstable atmospheric situations, or by Monin-Obukhov length when the atmosphere is stably stratified. The model is first verified reproducing the famous Leipzig wind profile. Then the performance of the model is tested with measurements from FINO-1 platform using sonic anemometers to derive the appropriate maximum mixing length.

CFD modelling of the interaction between the Surface Boundary Layer and rotor wake. Comparison of results obtained with different turbulence models and mesh strategies

A simplified CFD wake model based on the actuator disk concept is used to simulate the wind turbine, represented by a disk upon which a distribution of forces, defined as axial momentum sources, are applied on the incoming non-uniform flow. The rotor is supposed to be uniformly loaded, with the exerted forces function of the incident wind speed, the thrust coefficient and the rotor diameter. The model is tested under different parameterizations of turbulence models and validated through experimental measurements downwind of a wind turbine in terms of wind speed deficit and turbulence intensity.

Laminar-turbulent transition induced by a discrete roughness element in a supersonic boundary layer

The linear instability and breakdown to turbulence induced by an isolated roughness element in a boundary layer at Mach 2:5, over an isothermal flat plate with laminar adiabatic wall temperature, have been analysed by means of direct numerical simulations, aided by spatial BiGlobal and three-dimensional parabolized (PSE-3D) stability analyses. It is important to understand transition in this flow regime since the process can be slower than in incompressible flow and is crucial to prediction of local heat loads on next-generation flight vehicles. The results show that the roughness element, with a height of the order of the boundary layer displacement thickness, generates a highly unstable wake, which is composed of a low-velocity streak surrounded by a three-dimensional high-shear layer and is able to sustain the rapid growth of a number of instability modes. The most unstable of these modes are associated with varicose or sinuous deformations of the low-velocity streak; they are a consequence of the instability developing in the three-dimensional shear layer as a whole (the varicose mode) or in the lateral shear layers (the sinuous mode). The most unstable wake mode is of the varicose type and grows on average 17% faster tan the most unstable sinuous mode and 30 times faster than the most unstable boundary layer mode occurring in the absence of a roughness element. Due to the high growthrates registered in the presence of the roughness element, an amplification factor of N D 9 is reached within 50 roughness heights from the roughness trailing edge. The independently performed Navier–Stokes, spatial BiGlobal and PSE-3D stability results are in excellent agreement with each other, validating the use of simplified theories for roughness-induced transition involving wake instabilities. Following the linear stages of the laminar–turbulent transition process, the roll-up of the three-dimensional shear layer leads to the formation of a wedge of turbulence, which spreads laterally at a rate similar to that observed in the case of compressible turbulent spots for the same Mach number.

Computation of nonlinear streaky structures in boundary layer flow

In this work, the Reduced Navier Stokes (RNS) are numerically integrated, and used to calculate nonlinear finite amplitude streaks. These structures are interesting since they can have a stabilizing effect and delay the transition to the turbulent regime. RNS formulation is also used to compute the family of nonlinear intrinsic streaks that emerge from the leading edge in absence of any external perturbation. Finally, this formulation is generalized to include the possibility of having a curved bottom wall

Design of a Two-Phase buck converter with fourth-order output filter for envelope amplifiers of limited bandwidth

The use of techniques such as envelope tracking (ET) and envelope elimination and restoration (EER) can improve the efficiency of radio frequency power amplifiers (RFPA). In both cases, high-bandwidth DC/DC converters called envelope amplifiers (EA) are used to modulate the supply voltage of the RFPA. This paper addresses the analysis and design of a modified two-phase Buck converter optimized to operate as EA. The effects of multiphase operation on the tracking capabilities are analyzed. The use of a fourth-order output filter is proposed to increase the attenuation of the harmonics generated by the PWM operation, thus allowing a reduction of the ratio between the switching frequency and the converter bandwidth. The design of the output filter is addressed considering envelope tracking accuracy and distortion caused by the side bands arising from the nonlinear modulation process. Finally, the proposed analysis and design methods are supported by simulation results, as well as demonstrated by experiments obtained using two 100-W, 10-MHz, two-phase Buck EAs capable of accurately tracking a 1.5-MHz bandwidth OFDM signal.