Experimental studies on transitional separated boundary layers

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(1)UNIVERSIDAD POLITÉCNICA DE MADRID ESCUELA TÉCNICA SUPERIOR DE INGENIEROS AERONÁUTICOS. EXPERIMENTAL STUDIES ON TRANSITIONAL SEPARATED BOUNDARY LAYERS. Tesis Doctoral. José Serna Serrano Ingeniero Aeronáutico. 2013.

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(3) Departamento de Motopropulsión y Termofluidodinámica ESCUELA TÉCNICA SUPERIOR DE INGENIEROS AERONÁUTICOS. EXPERIMENTAL STUDIES ON TRANSITIONAL SEPARATED BOUNDARY LAYERS. Autor José Serna Serrano Ingeniero Aeronáutico. Director Benigno Lázaro Gómez Doctor Ingeniero Aeronáutico. 2013.

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(5) Tribunal nombrado por el Sr. Rector Magfco. de la Universidad Politécnica de Madrid, el día...............de.............................de 20.... Presidente: Vocal: Vocal: Vocal: Secretario: Suplente: Suplente:. Realizado el acto de defensa y lectura de la Tesis el día..........de........................de 20 ... en la E.T.S.I. /Facultad.................................................... Calificación .................................................. EL PRESIDENTE. LOS VOCALES. EL SECRETARIO.

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(7) ABSTRACT Separated transitional boundary layers appear on key aeronautical processes such as the flow around wings or turbomachinery blades. The aim of this thesis is the study of these flows in representative scenarios of technological applications, gaining knowledge about phenomenology and physical processes that occur there and, developing a simple model for scaling them. To achieve this goal, experimental measurements have been carried out in a low speed facility, ensuring the flow homogeneity and a low disturbances level such that unwanted transitional mechanisms are avoided. The studied boundary layers have been developed on a flat plate, by imposing a pressure gradient by means of contoured walls. They generate an initial acceleration region followed by a deceleration zone. The initial region is designed to obtain at the beginning of the deceleration the Blasius profile, characterized by its momentum thickness, and an edge boundary layer velocity, defining the problem characteristic velocity. The deceleration region is designed to obtain a linear evolution of the edge velocity, thereby defining the characteristic length of the problem. Several experimental techniques, both intrusive (hot wire anemometry, total pressure probes) as nonintrusive (PIV and LDV anemometry, high-speed filming), have been used in order to take advantage of each of them and allow cross-validation of the results. Once the boundary layer at the deceleration beginning has been characterized, ensuring the desired integral parameters and level of disturbance, the evolution of the laminar boundary layer up to the point of separation is studied. It has been compared with integral methods, and numerical simulations. In view of the results a new model for this evolution is proposed. Downstream from the separation, the flow near to the wall is configured as a shear layer that encloses low momentum recirculating fluid. The region where the shear layer remains laminar tends to be positioned to compensate the adverse pressure gradient associated with the imposed deceleration. Under these conditions, the momentum thickness remains almost constant. This laminar shear layer region extends up to where transitional phenomena appear, extension that scales with the momentum thickness at separation. These transitional phenomena are of inviscid type, similar to those found in free shear layers. The transitional region analysis begins with a study of the disturbances evolution in the linear growth region and the comparison of experimental results with a numerical model based on Linear Stability Theory for parallel flows and with data from other authors. The results’ coalescence for both the disturbances growth and the excited frequencies is stated. For the transition final stages the vorticity concentration into vortex blobs is found, analogously to what happens in free shear layers. Unlike these, the presence of the wall and the pressure gradient make the large scale structures to move towards the wall and quickly disappear iii.

(8) iv. ABSTRACT. under certain circumstances. In these cases, the recirculating flow is confined into a closed region saying the bubble is closed or the boundary layer reattaches. From the reattachment point, the fluid shows a configuration in the vicinity of the wall traditionally considered as turbulent. It has been observed that existing integral methods for turbulent boundary layers do not fit well to the experimental results, due to these methods being valid only for fully developed turbulent flow. Nevertheless, it has been found that downstream from the reattachment point the velocity profiles are self-similar, and a model has been proposed for the evolution of the integral parameters of the boundary layer in this region. Finally, the phenomenon known as bubble burst is analyzed. It has been checked the validity of existing models in literature and a new one is proposed. This phenomenon is blamed to the inability of the large scale structures formed after the transition to overcome with the adverse pressure gradient, move towards the wall and close the bubble..

(9) ACKNOWLEDGEMENTS The work contained in this thesis has been partially supported by the Spanish Ministry of Education (under project TRA2005-01249 and PhD internship BES-2006-13926), The European Commission through the project “Enviromentally Friendly Aeroengines (VITAL)” under the Sixth Framework Program, and by the enterprise “Industria de Turbo Propulsores S.A. (ITP)” under contracts P020130430 and P060130077. I am indebted to many people who have been essential in the development of this thesis. Firstly, to my advisor, Ph.D. Benigno Lázaro, who provided me the material and scientific resources to design and manufacture the low speed facility and test sections, and the possibility of making use of the most modern experimental flow-diagnostic techniques. He has introduced me into the scientific research, with a scientific rigor difficult to find nowadays. Most of the ideas in this thesis comes from him talks, without his support this work had not been started nor finished. He has also taught me many lessons I am sure I will not forget. I am also grateful to Ezequiel González, I have taken advantage of his deep background in practical issues of experimental fluid mechanics. The software and the tuning of most of the experimental techniques were available through his work and, morally he is co-advisor of the thesis. I consider him as a colleague and friend who has always given good scientific and human advice. Most of the experimental hardware has been specifically manufactured for this thesis. Only two people like Fernando Fernández and José Miguel Velasco have the ability to perform it with the desired high quality, exceeding always the expected results. I have shared hundred of hours with them, and they have always encouraged me. During all these years I have shared pre-doctoral concerns with a lot of friends: Miquel, Manuel, David, Luı́s, Robert, Filippo, Juan Ángel, Marı́a, Elena, ... also my Aero99 friends: Lucas, Juan, Maicol, Elena, Carlos, Javi, Pablo I., Pablo L., Jose, ..., have been a key support. Thank you for the time we have spent together. I am also grateful to my family: my parents and my brother, who have done far more for me than I can express in words. Finally, I am indebted to Inma for her love, unconditional support, and all the time this work has stolen her.. v.

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(11) RESUMEN El estudio de capas lı́mites transicionales con separación es de gran relevancia en distintas aplicaciones tecnológicas. Particularmente, en tecnologı́a aeronáutica, aparecen en procesos claves, tales como el flujo alrededor de alas o álabes de turbomaquinaria. El objetivo de esta tesis es el estudio de estos flujos en situaciones representativas de las aplicaciones tecnológicas, ganando por un lado conocimiento sobre la fenomenologı́a y los procesos fı́sicos que aparecen y, por otra parte, desarrollando un modelo sencillo para el escalado de los mismos. Para conseguir este objetivo se han realizado ensayos en una instalación experimental de baja velocidad especı́ficamente diseñada para asegurar un flujo homogéneo y con bajo nivel de perturbaciones, de modo que se evita el disparo de mecanismos transicionales no deseados. La capa lı́mite bajo estudio se ha desarrollado sobre una placa plana, imponiendo un gradiente de presión a la misma por medio de paredes de geometrı́a especificada. Éstas generan una región inicial de aceleración seguida de una zona de deceleración. La región inicial se diseña para tener en al inicio de la deceleración un perfil de capa lı́mite de Blasius, caracterizado por su espesor de cantidad de movimiento, y una cierta velocidad externa a la capa lı́mite que se considera la velocidad caracterı́stica del problema. La región de deceleración está concebida para que la variación de la velocidad externa a la capa lı́mite sea lineal, definiendo de esta forma una longitud caracterı́stica del problema. Los ensayos se han realizado explotando varias técnicas experimentales, tanto intrusivas (anemometrı́a de hilo caliente, sondas de presión total) como no intrusivas (anemometrı́as láser y PIV, filmación de alta velocidad), de cara a aprovechar las ventajas de cada una de ellas y permitir validación cruzada de resultados entre las mismas. Caracterizada la capa lı́mite al comienzo de la deceleración, y garantizados los parámetros integrales y niveles de perturbación deseados se procede al estudio de la zona de deceleración. Se presenta en la tesis un análisis de la evolución de la capa lı́mite laminar desde el inicio de la misma hasta el punto de separación, comparando con métodos integrales, simulaciones numéricas, y proponiendo un nuevo modelo para esta evolución. Aguas abajo de la separación, el flujo en las proximidades de la pared se configura como una capa de cortadura que encierra una región de fluido recirculatorio de baja cantidad de movimiento. Se ha caracterizado la región en que dicha capa de cortadura permanece laminar, encontrando que se posiciona de modo que compensa el gradiente adverso de presión asociado a la deceleración de la corriente. En estas condiciones, el espesor de cantidad de movimiento permanece prácticamente constante y esta capa de cortadura laminar se extiende hasta que los fenómenos transicionales aparecen. Estos fenómenos son de tipo no viscoso, similares a los que aparecen en una capa de cortadura libre. El análisis de la región transicional comienza con un estudio de la evolución de las vii.

(12) viii. RESUMEN. perturbaciones en la zona de crecimiento lineal de las mismas y la comparación de los resultados experimentales con un modelo numérico y con datos de otros autores. La coalescencia de los resultados tanto para el crecimiento de las perturbaciones como para las frecuencias excitadas queda demostrada. Para los estadios finales de la transición se observa la concentración de la vorticidad en torbellinos, de modo análogo a lo que ocurre en capas de cortadura libres. A diferencia de estas, la presencia de la pared y del gradiente de presión hace que, bajo ciertas condiciones, la gran escala se desplace hacia la pared y desaparezca rápidamente. En este caso el flujo recirculatorio queda confinado en una región cerrada y se habla de cierre de la burbuja o readherencia de la capa lı́mite. A partir del punto de readherencia se tiene una configuración fluida en las proximidades de la pared que tradicionalmente se ha considerado turbulenta. Se ha observado que los métodos integrales existentes para capas lı́mites turbulentas no ajustan bien a las medidas experimentales realizadas, hecho imputable a que no se obtiene en dicha región un flujo turbulento plenamente desarrollado. Se ha encontrado, sin embargo, que pasado el punto de readherencia los perfiles de velocidad próximos a la pared son autosemejantes entre sı́ y se ha propuesto un modelo para la evolución de los parámetros integrales de la capa lı́mite en esta región. Finalmente, el fenómeno conocido como “estallido” de la burbuja se ha analizado. Se ha comprobado la validez de los modelos existentes en la literatura y se propone uno nuevo. Este fenómeno se achaca a la incapacidad de la gran estructura formada tras la transición para vencer el gradiente adverso de presión, desplazarse hacia la pared y cerrar la burbuja..

(13) AGRADECIMIENTOS El trabajo desarrollado para la realización de esta tesis doctoral ha sido parcialmente financiado por el Ministerio de Educación y Ciencia de España a través del proyecto TRA200501249 (Estudios experimentales en capas lı́mites no-estacionarias transicionales) y la ayuda predoctoral de formación de personal investigador (FPI) BES-2006-13926. La Comisión Europea, a través del proyecto “Enviromentally Friendly Aeroengines (VITAL)”, dentro del Sexto Programa Marco. Y por la empresa Industria de Turbo Propulsores S.A. bajo los proyectos P020130430 (Estudios Experimentales en Elementos de Turbomaquinaria. Fase I (TURBEX I)) y P060130077 (Estudios Experimentales en Elementos de Turbomaquinaria. Fase II (TURBEX II)). Esta tesis es el resultado de varios años de trabajo en los que mi vida ha discurrido por caminos diversos. Por cuestiones de espacio, no es este el lugar de mostrar el agradecimiento individual a todas las personas que me han acompañado en esta andadura 1 . Sin embargo, no puedo dejar de dedicar unas palabras a ocho personas fundamentales en este trabajo. A mi director de tesis, D. Benigno Lázaro, le debo estar agradecido por introducirme en el mundo de la investigación cientı́fica, poniendo a mi alcance cuantos medios materiales he necesitado. La mayor parte de la ciencia que hay en esta tesis proviene de sus comentarios. Él ha intentado forjar en mi espı́ritu un rigor en el trabajo que creo difı́cil de encontrar en la sociedad actual. Gracias, Benigno, por todas las lecciones que jamás olvidaré. A D. Ezequiel González, primero mi profesor y con el devenir de los años mi compañero y amigo. Él es el responsable de la mayor parte del sistema de adquisición y de la puesta a punto de los equipos. Sin su ayuda esta tesis no habrı́a llevado ocho años sino ochenta y ocho. Nunca le podré agradecer lo suficiente su ayuda para solventar cualquier problema técnico que haya podido tener, ası́ como los consejos humanos que me ha podido dar. Gracias, Ezequiel, por tu disposicición desinteresada. Esta tesis es una tesis experimental en que las medidas requieren una gran calidad en las instalaciones. Sólo dos grandes profesionales como D. Fernando Fernández y D. José Miguel Velasco, son capaces de fabricar estas con los estándares requeridos (y los medios disponibles). Además de su experiencia y profesionalidad, he tenido la suerte de compartir muchas horas con ellos en las que me han demostrado que son grandes personas. Gracias, Fernando y Jose, por vuestras manos y vuestros consejos. Mi familia, mis padres y mi hermano, siempre me han apoyado en cuantas decisiones 1. Debo dar las gracias por ser los mejores compañeros posibles en estos años a: Manuel, David, Juan Ángel, Miquel, Robert, Filippo, Luı́s, ... que han pasado por una experiencia similar y con los que he compartido muchos cafés y comidas. También a mis amigos Aero99: Lucas, Juan, Maicol, Elena, Carlos, Javi, Pablo I., Pablo L., Jose, ..., porque tuve sed y me disteis de beber, fui huésped y me recogisteis.. ix.

(14) x. AGRADECIMIENTOS. he tomado en la vida. Ellos me han dado todo, empezando por la vida, han escuchado mis problemas y han hecho por mi más de lo que soy capaz de expresar con palabras. Gracias, papá, por ser el espejo en que mirarme; gracias, mamá, por quererme; gracias, Guillermo, por apoyarme incondicionalmente. El trabajo aquı́ expuesto, muy probablemente, no tenga repercusión en la comunidad cientı́fica y es cuestionable que los años y el esfuerzo en él empleados merezcan la pena desde ese punto de vista. Este trabajo tiene sentido por Inma, la mujer de mi vida. Ella ha sufrido los momentos malos a los que me he podido ver abocado. Siempre me ha animado, apoyado y dado su cariño. Este trabajo se ha acabado gracias a ella, para que podamos continuar nuestro proyecto de vida en común. Gracias, Inma..

(15) Contents ABSTRACT. iii. ACKNOWLEDGEMENTS. v. RESUMEN. vii. AGRADECIMIENTOS. ix. Contents. xi. 1 INTRODUCTION 1.1 Boundary layer transition in steady scenarios 1.1.1 Laminar separation bubbles . . . . . . 1.1.2 Transition in separation bubbles . . . 1.2 Technological context. Research motivation . 1.3 Evaluation of the bibliography . . . . . . . . 1.4 Objectives and outline of the thesis . . . . . .. . . . . . .. 1 1 3 5 8 8 10. . . . .. 13 13 15 17 18. . . . . . . . . .. 21 21 21 23 24 24 24 25 25 28. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. 2 PROBLEM DEFINITION 2.1 Boundary layer equations . . . . . . . . . . . . . . . . . . . . . . 2.2 A representative outer boundary layer flow promoting separation 2.3 Technological interest . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Non dimensional parameters characteristic values . . . . . 3 EXPERIMENTAL FACILITIES 3.1 Wind tunnel BV3 . . . . . . . . . . . . . 3.1.1 Contraction . . . . . . . . . . . . 3.1.2 Settling chamber . . . . . . . . . 3.1.3 Diffuser . . . . . . . . . . . . . . 3.1.4 Fan outflow conditioning module 3.1.5 Fan . . . . . . . . . . . . . . . . 3.2 Test sections designs . . . . . . . . . . . 3.2.1 Symmetric test section . . . . . . 3.2.2 Asymmetric test section . . . . . xi. . . . . . . . . .. . . . . . . . . .. . . . . . . . . .. . . . . . . . . .. . . . . . . . . .. . . . . . . . . .. . . . . . . . . .. . . . . . . . . .. . . . . . . . . .. . . . . . . . . .. . . . . . . . . .. . . . . . . . . .. . . . . . . . . .. . . . . . . . . .. . . . . . .. . . . .. . . . . . . . . .. . . . . . .. . . . .. . . . . . . . . .. . . . . . .. . . . .. . . . . . . . . .. . . . . . .. . . . .. . . . . . . . . .. . . . . . .. . . . .. . . . . . . . . .. . . . . . .. . . . .. . . . . . . . . ..

(16) xii. CONTENTS. 4 EXPERIMENTAL INSTRUMENTATION AND MEASUREMENT TECHNIQUES 33 4.1 Laser Doppler Anemometry (LDA) . . . . . . . . . . . . . . . . . . . . . . . . 33 4.1.1 Seeding considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 4.1.2 Postprocessing algorithms . . . . . . . . . . . . . . . . . . . . . . . . . 39 4.1.3 Positioning and optical access . . . . . . . . . . . . . . . . . . . . . . . 40 4.1.4 LDA measurements error sources . . . . . . . . . . . . . . . . . . . . . 41 4.2 Particle Image Velocimetry (PIV) . . . . . . . . . . . . . . . . . . . . . . . . . 42 4.2.1 Experimental Setup and Optical Considerations . . . . . . . . . . . . 44 4.2.2 Data Acquisition and post-processing . . . . . . . . . . . . . . . . . . 46 4.2.3 PIV measurements accuracy . . . . . . . . . . . . . . . . . . . . . . . . 49 4.3 Hot Wire Anemometry (HWA) . . . . . . . . . . . . . . . . . . . . . . . . . . 49 4.3.1 HWA characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 4.3.2 HW error sources estimations . . . . . . . . . . . . . . . . . . . . . . . 52 4.4 Pressure measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 4.4.1 Pressure measurements postprocessing . . . . . . . . . . . . . . . . . . 56 4.4.2 Pressure measurements accuracy . . . . . . . . . . . . . . . . . . . . . 59 4.5 High speed flow visualization . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 4.6 Justification of the selection of the measurement techniques . . . . . . . . . . 62 4.7 Acquisition system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 4.8 Phase measurements technique . . . . . . . . . . . . . . . . . . . . . . . . . . 66 5 PRELIMINARY STUDIES, CONCEPTS VALIDATION AND SELECTION OF FROZEN CONFIGURATION 5.1 Facility flow quality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.1 Seeder effect on flow quality . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Seeding system characterization. Seeding control parameters . . . . . . . . . 5.3 Symmetric test section results . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.1 Upper and lower wall boundary layer control. Suction configuration . 5.3.2 Lateral boundary layers control. Tripping configuration . . . . . . . . 5.3.3 Flow blockage due to the bubble growth . . . . . . . . . . . . . . . . . 5.4 Asymmetric test section preliminary results . . . . . . . . . . . . . . . . . . . 5.4.1 Two-dimensionality of the flow. . . . . . . . . . . . . . . . . . . . . . . 5.4.2 Determination of the characteristic deceleration length . . . . . . . . . 5.4.3 Flow blockage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.4 Implementation of the deceleration wall to measure the lower Reynolds numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 71 71 74 74 80 80 80 83 84 85 85 89 90. 6 PHENOMENOLOGY IN TRANSITIONAL BOUNDARY LAYERS WITH SEPARATION BUBBLES. LAMINAR REGION 95 6.1 Characterization of the boundary layer at the maximum slip velocity station 95 6.2 Evolution of the boundary layer from the maximum velocity station to the separation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98.

(17) CONTENTS. xiii. 6.2.1. 6.3. The relationship between the momentum thickness at the separation and maximum velocity stations . . . . . . . . . . . . . . . . . . . . . . 100 The laminar part of the separated shear layer . . . . . . . . . . . . . . . . . . 105 6.3.1 Experimental data. Verification of boundary layer parameters at the maximum velocity and separation stations . . . . . . . . . . . . . . . . 106 6.3.2 Evolution of the momentum thickness in the laminar shear layer . . . 107 6.3.3 The shear layer angle. Proposals for scaling . . . . . . . . . . . . . . . 108 6.3.4 The shear layer angle. A procedure to determination from measurements113 6.3.5 The laminar shear layer length. Definition . . . . . . . . . . . . . . . . 115 6.3.6 The laminar shear layer length. Experimental results . . . . . . . . . . 118 6.3.7 The shear layer angle. Experimental results . . . . . . . . . . . . . . . 120 6.3.8 Shear layer angles relationships . . . . . . . . . . . . . . . . . . . . . . 123. 7 PHENOMENOLOGY IN TRANSITIONAL BOUNDARY LAYERS WITH SEPARATION BUBBLES. TRANSITIONAL REGION 125 7.1 Linear Stability Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . 125 7.1.1 The implemented stability code . . . . . . . . . . . . . . . . . . . . . . 126 7.1.2 Results for Michalke profiles . . . . . . . . . . . . . . . . . . . . . . . . 127 7.2 The initial stages of the transitional process in a separation bubble . . . . . . 130 7.2.1 Hot wire measurements . . . . . . . . . . . . . . . . . . . . . . . . . . 130 7.2.2 The spectrum evolution in the shear layer . . . . . . . . . . . . . . . . 132 7.2.3 A simplified model for the initial stages of the transition . . . . . . . . 135 7.3 The final stages of the transitional process and the reattachment of the flow . 142 7.3.1 The spectrum evolution . . . . . . . . . . . . . . . . . . . . . . . . . . 142 7.3.2 The reattachment region. Time averaged PIV data . . . . . . . . . . . 143 7.3.3 The reattachment region. The pressure distributions revisited . . . . . 149 7.3.4 The boundary layer evolution at the reattachment region . . . . . . . 149 7.3.5 Qualitative unsteady behavior at reattachment. Flow Visualizations . 155 7.3.6 PIV phase measurements . . . . . . . . . . . . . . . . . . . . . . . . . 160 7.4 The bubble closure and the bursting condition . . . . . . . . . . . . . . . . . 169 7.4.1 Literature review of bursting conditions. . . . . . . . . . . . . . . . . . 169 7.4.2 Analysis of PIV data . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174 8 PHENOMENOLOGY IN TRANSITIONAL BOUNDARY LAYERS WITH SEPARATION BUBBLES. TURBULENT REGION 181 8.1 The pressure distribution on the plate after reattachment . . . . . . . . . . . 181 8.2 The evolution of the boundary layer after reattachment . . . . . . . . . . . . 182 8.2.1 Boundary layer profiles scaling . . . . . . . . . . . . . . . . . . . . . . 182 8.2.2 The inability of integral methods to determine the boundary layer evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188 8.2.3 The survival of big scale structure . . . . . . . . . . . . . . . . . . . . 188 8.3 The separation of the turbulent boundary layer . . . . . . . . . . . . . . . . . 188.

(18) xiv. CONTENTS. 9 CONCLUSIONS 193 9.1 The steady separation bubble model . . . . . . . . . . . . . . . . . . . . . . . 196 9.2 Questions for future research . . . . . . . . . . . . . . . . . . . . . . . . . . . 198 Bibliography. 201. Appendix A. MEASUREMENTS CONSIDERATIONS A.1 The measurement chain and the uncertainty . . . . . . . . . . . A.2 Notions on statistical estimation for finite length samples series A.2.1 Field (intensive) measurements. Unweighted estimators A.2.2 Field (intensive) magnitudes. Weighted estimators . . . A.2.3 Statistical independency of the samples . . . . . . . . . A.2.4 Spectrum estimation . . . . . . . . . . . . . . . . . . . . A.2.5 Derivative magnitudes . . . . . . . . . . . . . . . . . . . A.3 Boundary layer integral thicknesses considerations . . . . . . . A.3.1 Integrals limits . . . . . . . . . . . . . . . . . . . . . . . A.3.2 General error analysis . . . . . . . . . . . . . . . . . . . A.3.3 Error analysis for the momentum thickness . . . . . . . A.4 Measurement sets . . . . . . . . . . . . . . . . . . . . . . . . . . A.4.1 LDV RUN SL1 . . . . . . . . . . . . . . . . . . . . . . A.4.2 LDV RUN SL2 . . . . . . . . . . . . . . . . . . . . . . A.4.3 LDV RUN SB . . . . . . . . . . . . . . . . . . . . . . A.4.4 LDV RUN XIIP . . . . . . . . . . . . . . . . . . . . . A.4.5 LDV RUN X240 . . . . . . . . . . . . . . . . . . . . . A.4.6 LDV RUN X400 . . . . . . . . . . . . . . . . . . . . . A.4.7 PIV RUN1 . . . . . . . . . . . . . . . . . . . . . . . . A.4.8 PIV RUN1 LD615 . . . . . . . . . . . . . . . . . . . A.4.9 PIV RUN2 . . . . . . . . . . . . . . . . . . . . . . . . A.4.10 PIV RUN3 . . . . . . . . . . . . . . . . . . . . . . . . A.4.11 PIV PHM . . . . . . . . . . . . . . . . . . . . . . . . . . A.4.12 PIV PHM ZS . . . . . . . . . . . . . . . . . . . . . . . . A.4.13 HW RUN1 . . . . . . . . . . . . . . . . . . . . . . . . . A.4.14 Visualization . . . . . . . . . . . . . . . . . . . . . . .. 209 209 210 210 211 211 212 213 214 214 214 215 215 215 217 219 219 219 219 219 219 224 224 224 227 227 227. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . ..

(19) Chapter 1. INTRODUCTION The study of transitional boundary layers is of great significance for several technological applications. Particularly, in aeronautical technology, transitional boundary layers appear recursively as key processes in the flow around wings, control surfaces, turbomachinery blades, or wind turbines. Such a variety of flow typologies makes impossible to try to study all the possible phenomenologies related to transitional boundary layers. The purpose of this thesis is the study of transitional boundary layers in scenarios where the transition occurs after local flow separation, which happens when the boundary layer is subjected to an adverse pressure gradient under certain circumstances. As a representative case of this kind of flows, the study will consider flows similar to those developing on the suction side of Low Pressure Turbine Blades (LPT blades from now).. 1.1. Boundary layer transition in steady scenarios. The effort to understand the phenomena related to transitional boundary layers began with Prandtl, who, at the beginning of the twentieth century, developed the theory of the boundary layer [76] and, in the 1920’s, started theoretical studies on the stability of this kind of flows. These studies were complemented almost a decade later with satisfying experimental results on the critical Reynolds number at which the transition occurs [85]. However, until the fifties, a systematic theory to study the stability of laminar boundary layers was not developed. This theory, due to Schlichting [85], is based on analyzing the Orr - Sommerfeld equation. From this theory, the existence of two fundamental types of twodimensional instabilities is derived: non-viscous instabilities, that develop when there is an inflectional point in the velocity profile of the boundary layer, and viscous instabilities, related to the existence and amplification of the so-called two dimensional Tollmien - Schlichting waves (from now TS waves). About the same time, Emmons explained the phenomenon of transition from the formation and growth of turbulent spots following the development and rupture of linear instabilities. This was collected in a physical mathematical formulation for the transition region based on the use of an intermittency function that takes into account the fraction of time, at a given spatial position, when the flow can be considered laminar or turbulent (theory of Emmons’ turbulent spots [28]). This formulation has been the basis for subsequent studies on transition, specially in regard to experimental studies. 1.

(20) 2. CHAPTER 1. INTRODUCTION. In 1956, Klebanoff and Schubauer [86] conducted experiments on a flat-plate without pressure gradient, and by using hot-film sensors determined a correlation for the intermittency function, which was consistent with a Gaussian function. They also observed the mainly triangular, pointed towards the streamwise direction, geometry of the spots and the existence of regions following the spots where the perturbations were attenuated. From those studies, these regions are known as “calming regions”. Following these experiments, Narasimha and Dhawan [21], adjusted the production of turbulent spots to a Dirac delta function (hypothesis of “concentrated laminar breakdown”), suggesting a shape for the intermittency function independent of the way on which the transition developed, and, as in [86], they obtained a proper adjustment of this function by a Gaussian distribution. From these results a relationship between the Reynolds number at the onset of the transition and the Reynolds number based on the length of it was proposed. In addition, a formulation for the determination of the properties of the transitional boundary layer on the intermittency function and the properties of laminar and turbulent boundary layers was developed. These preliminary studies were centered on the phenomenon known as natural transition where all the stages of the transitional process are present. Figure 1.1 shows the stages of the transitional process from a classical point of view, in a zero pressure gradient scenario. The initial stable boundary layer develops and reaches a “critical Reynolds number”, at which it becomes unstable and triggers two dimensional, linear, fluctuations (TS waves), which are convected downstream with approximately a third of the velocity of the main flow. When the amplitude of these waves achieves a large enough value, the growth becomes nonlinear and results in 3D waves, which develop into hairpin vortices. The vortex breakdown occurs in regions of high shear, resulting in three-dimensional fluctuations. This breakdown originates turbulent spots with intense fluctuations. As the turbulent spots propagate, they spread until they coalesce into fully turbulent boundary layer flow. Appart from the natural evolution, two additional transition phenomenologies can be found in the literature: the bypass transition, and the transition taking place in separation bubbles. The bypass transition is characterized by the suppression of several stages of the phenomenon of natural transition, related to the growth of instabilities (see figure 1.1). Thus, after a laminar boundary layer development, the turbulent spots appear suddenly. This phenomenon is explained in [55], where the impossibility of predicting this kind of transition via theoretical models based on the Orr-Sommerfeld equation is argued because no TS waves appear. The bypass of several stages of the transition process is usually related to the presence of a high level of disturbances in the flow or to the existence of a strong adverse pressure gradient. Numerical and experimental studies have been performed to gain a deeper knowledge on the phenomenon. In [55] there is also a model of the bypass transition as a function of the pressure gradient and of the turbulence level of the flow outside and inside the boundary layer, which are the main parameters that affect this kind of transition. A more refined model is found in [66], where the turbulent kinetic energy equation is considered to evaluate the fluctuations of this variable, which, according to the model is at the origin of this transitional process. Detailed DNS simulations have been used to better understand the dynamics of the bypass transition [54]. They show that the origin of the formation of a turbulent spot is a long.

(21) 1.1. BOUNDARY LAYER TRANSITION IN STEADY SCENARIOS. 3. Figure 1.1: Natural boundary layer transition after White [110] (from [98]).. backward jet extending into the upper region of the boundary layer in response to the low frequency eddies present in the freestream turbulence. This zone, with less instantaneous velocity than the mean flow interacts with the fine scale disturbances of the freestream turbulence and avoids the high frequency fluctuations to get into the boundary layer. The small scale motions of the jets originate a kind of Kelvin-Helmholtz instability (from now KH instability) and, when this irregular motion travels to the wall, the turbulent spots appear. The experiments of Matsubara et al. at KTH [30, 64], identified also the backward jets (streaks) with a streamwise length proportional to the displacement thickness of the boundary layer (δ∗ ), and developed a refined model for the spot rate production and the intermittency function that incorporates the effect of the Reynolds number at the beginning of transition Ret , and the freestream turbulence level T u.. 1.1.1. Laminar separation bubbles. The last kind of transitional phenomenology in boundary layers is the transition in a separation bubble that develops from a local laminar flow separation, and which is the object of study of this thesis. A local laminar separation occurs when a laminar boundary layer reaches the separation point before the transition to a turbulent boundary layer happens. For each flow typology, the separation will take place at a certain combination of the flow parameters: Reynolds number, freestream turbulence level, flow configuration, etc. The flow development after the laminar separation point depends strongly on the behavior of the separated laminar shear layer. The classical approach is related to the fact that shear layers are highly unstable, and thus instabilities develop shortly after the separation point.

(22) 4. CHAPTER 1. INTRODUCTION. [50]. The free shear layer instability is originated at the maximum vorticity point of the velocity profile [23], and energy is transferred from the shear flow to the instability waves which are amplified downstream and finally trigger the transition process. Once the flow is turbulent, the momentum transfer across the shear layer is increased, and there is an entrainment of external flow towards the region between the shear layer and the wall. This process leads to the reattachment of the flow and, consequently, generates an enclosed region of low momentum flow known as separation bubble. The classical typology of a separation bubble is shown in Figure 1.2.. Figure 1.2: Time average structure of a short laminar separation bubble. After [50].. Generally, separation bubbles are classified as short or long, but the difference between these two types is controversial and difficult to define for any kind of flow conditions. In early studies, Tani [101] considered that a long bubble causes a global effect in the pressure distribution over an airfoil surface. The difference between the two types was related to the effect of the bubble on the surface pressure distribution: local and limited for a short bubble and appreciable for the long one. Hatman and Wang [45] tried to obtain a more physical description of the transitional bubbles. They distinguished between transitional bubbles (when the transition to turbulence started before the separation point), and laminar bubbles, which, in turn were divided again into short or long if the reverse flow vortex in figure 1.2 was intermittently ejected or not. Also related to this two types of bubbles appears the phenomenon of bursting. Bubble bursting can be seen as either the momentary breakdown (in case of long bubble formation) or definitive breakdown of the turbulent shear layer reattachment process. In traditional models, bubble bursting is interpreted as the inability of the turbulent boundary layer to reattach. More modern models consider that there is an inherent unsteady behavior related to separation bubbles, due to the formation of well defined vortical structures in the rear region of the bubble that are convected downstream..

(23) 1.1. BOUNDARY LAYER TRANSITION IN STEADY SCENARIOS. 5. These structures convey the first stage of the turbulent part of the flow and persist a relatively long distance downstream of the mean reattachment point [68]. The change from an unsteady behavior to a steady one can be the origin of the burst phenomenon according to certain authors [74], but there is no general agreement on this affirmation. The bursting condition was a recurrent concern in early studies on separation bubbles, due to the big impact that it has on the performance of aeronautical profiles. Thus, the first significative studies on this kind of flows, due to Gaster [34], were centered on this phenomenon. Gaster performed pressure and hot wire measurements on a flat surface for different Reynolds numbers and pressure gradients that were continued by Horton [50]. This last author developed a model for separation bubbles based on the scheme in figure 1.3. He divided the flow in a laminar region of constant momentum thickness θ = θsep with a non dimensional length given by: LI 4 · 104 = (1.1) θsep Reθsep . The laminar region was followed by a turbulent shear layer that, at the reattachment point, verifies (see figure 1.3): 1−e ueR cd e II 4H + L ueR (1.2) = cd u eeR = uesep 4H − ΛR with:. cd = 0.0182, H = 1.51, ΛR =. . θ due ue dx. . R. = −0.0082. (1.3). This model fails in representing all the possible cases characterizing separation bubbles, mainly in the location of the bursting point. After Horton, other authors have tried to improve his model. For example, Dunham [27] added the effect of the turbulence level in the freestream. Roberts [82] considered not only the turbulence intensity, but also its length scale. All these simplified models consider that the transition to turbulence is instantaneous, which contradicts the observations obtained early in the study of this kind of flows [34].. 1.1.2. Transition in separation bubbles. The review of Mayle [67] is a good starting point to follow the evolution of transitional models in separation bubbles. Mayle considers that there are two regions in the laminar part of the bubble: an unstable shear layer that extends from the separation point to the point of formation of spots (xt ), and the transition region that covers the rest of the laminar region (from xt to xT ). He obtained correlations for the two lengths using the data of several authors and making a distinction between short and long bubbles. He also assumed that the transition takes place in a zone of constant pressure. Thus, the relationships for zero pressure gradient attached flows could be used. Walker [108] criticized the assumption of constant pressure gradient at the end of the transition, based on studies that showed the transition being completed after the reattachment point. A very important result was reported by Malkiel and Mayle [63], after performing hot wire measurements in the separation bubble developing over the leading edge of a profile subject to low turbulence approaching flow. These researchers found KH waves in the shear layer that triggered the transitional process. The scaling of these waves agreed with the two.

(24) 6. CHAPTER 1. INTRODUCTION. Figure 1.3: Simplified Model of Short Laminar Separation Bubble [50]..

(25) 1.1. BOUNDARY LAYER TRANSITION IN STEADY SCENARIOS. 7. dimensional stability results of Michalke [25], obtained using modified hyperbolic tangent profiles that fitted the mean profiles measured in the separation bubbles. Recently, this scaling has been confirmed by other experiments [23]. Malkiel and Mayle also confirmed the results in [67] because their intermittency measurements fitted quite well with the universal distribution of Narasimha [72]. At present time, most studies agree that, in a low disturbance environment, the transition process in separation bubbles develops from an inviscid KH instability. However, the characteristics and the scaling of this phenomenon are not completely understood. For example, the frequency of the instability, that will lead to the formation and shedding of vortices is not well established. Pauley [74] found that for a laminar bubble, the Strouhal number of the vortex shedding was universal and had a value of St = fu·θe = 0.064. McAuliffe and Yaras found St = 0.011 in [69] and 0.008 < St < 0.013 in [68]. Talan reported 0.010 < St < 0.014 in [100]. Thus, there is a big dispersion of values, in any case being lower than the value found in free shear layers (St ≈ 0.016 [47]). The phenomenon of the inviscid instability has been widely studied by Hatman and Wang [42] [43] [44], that developed a systematic study on a flat plate for different Reynolds numbers and pressure gradients. They found that there could be a transitional regime when the transition started upstream from the separation point via TS waves, or separated flow transition regimes triggered by KH instability at the point of maximum displacement thickness. These authors do not address the physical phenomena related to the transitional mechanisms, and finish his wide study with correlations associated to a Reynolds number based on a length scale that results from correlating the location of the separation point to the imposed pressure gradient. The great potential of DNS techniques has recently permitted to deep in the knowledge on the transitional process. The results of Alam and Sandham [1], in which the flow prior to the separation was excited with periodical disturbances at unstable frequencies, show that full transition to turbulence is characterized by the breakdown of Λ-vortices that appear in the transitional region. These authors also found that transition is due to global instability when the maximum reverse flow velocity in the bubble is bigger than 15 percent of the core flow velocity. In addition, they analyzed the character of the instability associated to the bubble flow: convective in the case of short bubbles and absolute for long bubbles. Other studies, however, have found absolute instabilities in short bubbles as well[97]. Sandberg and Sandham [56] did numerical simulations on a NACA 0012 airfoil in forced and unforced cases, finding that turbulence feedback self-sustained the last case. These authors considered that absolute instability of the two-dimensional vortex shedding is at the origin of this self-sustained mechanism in a manner not predicted by linear stability analysis of the timeaveraged flowfield. Finally, recent DNS results of Simens [93] analyze the scaling of the mean velocity profile with different typical velocities (friction velocity, Zagarola - Smith velocity, ...) and the momentum and energy balances at different stations, trying to match the data with the experimental results of Zang and Hodson [117]. Failure in this pursuit was attributed to the lack of suitable perturbations in the numerical experiment and to the lack of information on the velocity spectrum for the physical one. Global instability analysis of laminar separation bubbles have also been carried out during last years [104] [105], showing the ability of this flow to support self-excited global modes, besides the amplification of incoming perturbations. This is an open research line because.

(26) 8. CHAPTER 1. INTRODUCTION. different behaviors are found (absolute/convective unstable 2D eigenmodes, self-excited 3D stationary global mode) and theories don’t agree with the necessary flow parameters to obtain each flow configuration.. 1.2. Technological context. Research motivation. Transitional separation bubbles are not only a fundamental and complex fluid mechanics problem, which involves laminar, transitional, and turbulent flow, but they appear frequently in key aeronautical applications. This fact is related to the response of laminar boundary layers exposed to adverse pressure gradients. Thus, when the Reynolds number is low enough to avoid flow transition upstream of an adverse pressure gradient region, the scenario is propitious to the generation of a separation bubble. This happens, for example, in the suction side of UAVs’ wing profiles and wind turbines blades, which work at relatively low Reynolds numbers. Another typical scenario for laminar separation bubbles occurs in LPT blades. In modern high bypass ratio (BPR) turbofan engines most of the thrust comes from the fan, which is driven by the LPT. Economic and environmental concerns promote the aeroengine BPR to increase, which results in large fans. Due to efficiency and noise considerations, the fan rotational speed scales inversely with its size. The large power requirement combined with the low rotational speed set the LPT blades to be aerodynamically demanded in a low Reynolds number scenario. Furthermore, the LPT module greatly contributes to the aeroengine weight, price and efficiency. Hodson [49] estimated an improvement of 0.5 % in the specific fuel consumption (SFC) for an increase of 1% in the polytropic efficiency of the LPT, and Wisler [89] increased this improvement up to 0.96% of the SFC. As long as the LPT blades have a large aspect ratio, the profile losses linked to the quasi two dimensional flow established away from the blades’ tips and roots dominate the LPT efficiency. From the different mechanisms contributing to these losses [20], the most important one comes from the boundary layer response over the suction side of the profile [18]. This effect contributes up to 60% of the total profile losses in present day designs. Figure 1.4 shows typical LPT suction side velocity distributions for different profile technologies. It can be seen that the profile suction side is characterized by a rear region with adverse pressure gradient. This increases as the technology advances towards higher lift, more demanded profiles. In addition, the Reynolds number based on the profile chord and characteristic velocity varies from 0.5 · 105 in the final turbine stages of small business jets operating at high altitudes, to 5 · 105 in the first turbine stage of large turbofans at sea level takeoff conditions. All these characteristics are perfect ground for a separation bubble to occur. The flow in LPT blades is further complicated due to the unsteady interaction between the boundary layer and the moving wakes shed from the previous row of relatively moving blades. Under different circumstances this phenomenon can be harmful or beneficial from the profile losses point of view [18, 61].. 1.3. Evaluation of the bibliography. The bibliography related to transitional boundary layers with separation is wide. In section 1.1, the basic references have been given but others can be found in literature. General.

(27) 1.3. EVALUATION OF THE BIBLIOGRAPHY. 9. Figure 1.4: Typical LPT velocity distributions [41]. reviews of the transitional phenomena for wall-bounded flows can be found in the classical book of Schlichting [85], and can be completed with the book of Boiko et al. [6], which includes a complete Chapter on separated flow transition that provides a clear vision of the stability properties of these kind of flows. This book makes continuous references to the review of Michalke et at. [25], a classical work on the use of two-dimensional stability analysis to study the transitional phenomena associated to separated flow. To get a deeper understanding in the techniques of stability analysis the book of Cebeci [15] is a needed reference. The preliminary studies on separation bubbles were due to the group of Queen Mary College, with the works of Gaster [34] and Horton [50]. The model of Horton has been the basis for other models later developed [27], [82] which have improved and completed the flow description with the inclusion of new parameters. A shortcoming in all these models is that they do not include flow physics considerations. They are esentially based on empirical relations which predict specific, restrictive cases. Other authors have developed correlations for the main parameters that define a separation bubble, including the lengths of the different regions: laminar, transitional and complete. A widely used correlation for this flows appear in the review of Mayle [67]. A deep analysis of the global instability processes in separation bubbles can be found in the reviews of Theofilis [104], [105]. The rest of the basic bibliography related to the flow physics in separation bubbles has been presented in Sections 1.1.1 and 1.1.2. For the technological application in LPT scenarios, where transitional separation bubbles appear, the reviews of Hodson [48, 49] and the NASA Minnowbrooks [89, 92] are the reference.

(28) 10. CHAPTER 1. INTRODUCTION. readings.. 1.4. Objectives and outline of the thesis. Despite the big effort done to understand the flow in laminar separation bubbles, there are a lot of questions that remain open or are unclear at present time. Moreover, the interest of the aeronautical community in this phenomenon is rising because of their appearance in many engineering problems, mainly related to high lift designs for wings or turbomachinery blades. The objective of this thesis is to gain light into this kind of flows under steady unexcited scenarios with low disturbances as a first step in a more ambitious project that would try to evaluate the effect of other parameters on the dynamics of the bubbles and develop passive and active control strategies. As an engineering application of the phenomena, physical similarity between the experiments carried out in this thesis and the flow developing in the suction side of LPT blades will be pursued. Specific objectives of the investigation include: 1 Develop a flat plate boundary layer facility with a pressure distribution that leads to a separation bubble on the plate, exhibiting non-dimensional parameters similar to those found in modern LPT blades. 2 Determine the position of the separation point in laminar boundary layers subject to representative LPT pressure gradients and explore the capacities of laminar integral methods and commercial CFD codes to reproduce the separation location and the structure of the flow around this station. 3 Investigate the effect of the pressure gradient on the global parameters of the laminar shear layer: angle between the layer and the wall, shear layer integral thicknesses, and shape factor evolution. 4 Improve the understanding of the transitional process in the shear layer and the parameters characterizing it: length of the laminar part of the shear layer, transitional flow structure, kind, and characteristic frequency of the instability process. 5 Study the complex zone of the flow reattachment: characterize the mean flow properties and the temporal resolved structure, the conditions needed to have a closed bubble, and the conditions that lead to the bursting. 6 Determine the evolution of the “turbulent” boundary / shear layer that develops downstream of the reattachment region. 7 Develop simple models to account for the evolution of laminar separation bubbles in steady scenarios, taking into account the dominant physical phenomena at each zone. To carry out these objectives an experimental approach has been followed. Detailed measurements with pressure probes, hot wire anemometry (HWA), laser doppler anemometry (LDA), particle image velocimetry (PIV), and ultra fast flow visualization have been performed to characterize the different phenomena that control the evolution in laminar separation bubbles..

(29) 1.4. OBJECTIVES AND OUTLINE OF THE THESIS. 11. This thesis has ten chapters and one appendix. This first Chapter is an introduction to the problem studied and a brief literature review, intended to stress the complexity and importance of the problem dealt. The second Chapter presents a formulation of the problem and the parameters to be reproduced at the main experimental facility. The design considerations and features of the experimental facilities where the different experiments have been carried out are presented in Chapter three. Chapter four briefly reviews the experimental techniques, describing the fundamentals of each technique, their specific limitations, requirements, and solutions adopted to comply with them. Chapter five is a summary of the actions done by the author to adequate the experimental facility to the problem under study. From Chapter six to eight the main experimental results are presented. For the sake of clarity, instead of splitting the results by measurement techniques, it has been chosen to introduce the results for the different parts of the flow at study, giving conclusions on the scaling of each part, and showing also numerical computations used to justify or validate certain phenomena. Chapter nine is a summary of the previous results to develop the complete model of flow evolution in the separation bubble, it also shows the conclusions and recommendations for further work. The dissertation is completed with one Appendix, which includes the measurements error analysis and the relevant information about the measurement sets carried out in this work..

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(31) Chapter 2. PROBLEM DEFINITION The objective of this chapter is to define the parameters that govern the problem under study. As has been stated in Section 1.1.1 a laminar separation bubble occurs when a laminar boundary layer separates prior to developing the instabilities which trigger its transition to turbulence, which occurs in the separated shear layer. Therefore, the first part of the problem is related to the development of a laminar boundary layer which separates, the second part to the transition phenomenon and the last one to the evolution of a turbulent shear / boundary layer. To get the boundary layer to separate several techniques can be employed, the use of a backward facing step or the injection of fluid have been strategies followed by certain authors, neverthless, most of the studies impose an adverse pressure gradient. This latter approach is going to be the technique followed in this work. The imposed adverse pressure gradient can have several shapes. In this study, an inviscid linear evolution of the freestream velocity will be selected. This case matches well with the one found in the rear portion of the suction side of LPT blades, highlighting the technological and scientific interest of the obtained results.. 2.1. Boundary layer equations. Laminar and turbulent boundary layers flows are governed by the Navier Stokes equations. In incompressible flows these equations take the form: ∇·u = 0 ∂t u + u · ∇u = −∇p +. (2.1) 1 4u Rec. (2.2). , where the variables are nondimensionalized with the characteristic length (Lc ) and velocity (uc ) of the flow, and the Reynolds number is defined as Rec = ucνLc . For laminar boundary layers the two dimensional assumption can be justified as long as the main flow doesn’t have any component in the spanwise direction z, therefore equations (2.1) and (2.2) can be written as: 13.

(32) 14. CHAPTER 2. PROBLEM DEFINITION. ∂u ∂t ∂v ∂t. ∂u ∂v + ∂x ∂y ∂u ∂u +u +v ∂x ∂y ∂v ∂v +u +v ∂x ∂y. = 0. (2.3) . . ∂2u ∂2u ∂p 1 + 2 + ∂x Rec ∂x2 ∂y 2 ∂p 1 ∂ v ∂2v = − + + ∂y Rec ∂x2 ∂y 2 = −. (2.4) (2.5). The laminar boundary layer theory involves the following assumptions: 1) There is a wall that bounds the flow and imposes a boundary condition of null velocity. 2) The Reynolds number is large enough to confine viscous effects to a thin layer above the wall (thin flow approximation, Lc,y Lc,x ). 3) Outside from this layer, the flow is quasi-unidirectional and steady (v u). Examining the orders of magnitude of the different terms in the equations, it is easy to derive that the flow adjacent to the boundary layer satisfies: ue. ∂ue ∂p =− ∂x ∂x. (2.6). , the characteristic thickness of the boundary layer can be determined by ensuring that the viscous term enters in equation (2.4): u. ∂u 1 ∂2u δl 1 ∼ ⇒ ∼√ 1 ∂x Rec ∂y 2 Lc Rec. (2.7). , with this scaling, taking into account the equation for the external flow (2.6), and considering the steady case, equations (2.3) ,(2.4) and (2.5) are reduced inside the boundary layer to: ∂u ∂v + ∂x ∂y ∂u ∂u u +v ∂x ∂y. = 0. ∂ue 1 ∂2u + ∂x Rec ∂y 2 ∂p 0 = − ∂y = ue. (2.8) (2.9) (2.10). , which are completed with boundary conditions in the x and y coordinates: x = 0 : u = u(0, y). (2.11). y = 0 : u = 0, v = 0. (2.12). y → ∞ : u → ue. (2.13). For turbulent boundary layers it’s common the use of the Reynolds decomposition of each fluid variable in its mean (φ) and fluctuating (φ0 ) values: φ(x, t) = φ(x, t) + φ0 (x, t).

(33) 2.2. A REPRESENTATIVE OUTER BOUNDARY LAYER FLOW PROMOTING SEPARATION15 with: 1 T →∞ T. φ(x, t) = lim. Z. t+T /2. φ(x, t)dt. t−T /2. , where the time average interval T is large enough with respect to the turbulent time scale and small enough with respect to the boundary condition modification time scale. Making use of this decomposition, the Reynolds equations are obtained: ∇·u = 0. 1 ∂t u + u · ∇u = −∇p + 4u − u’ · ∇u’ Rec The new term u’ · ∇u’ represents the divergence of the Reynolds’ stress tensor:   u0 u0 u0 v 0 u0 w 0   u’ · ∇u’ = ∇ ·  u0 v 0 v 0 v 0 v 0 w0  u0 w 0 v 0 w 0 w 0 w 0. (2.14) (2.15). (2.16). . For turbulent boundary layers the hypothesis of instantaneous, 2D flow cannot be justified. However if the domain is considered to be unbounded in the “z” direction, there will be invariance in that direction of the time averaged flow. This implies u0 w0 = v 0 w0 = 0, w = 0, and ∂z φ = 0. Furthermore, assuming stationarity of the main flow and that the thin flow approximation holds for the Reynolds averaged flow, equations (2.14) and (2.15) are reduced to: ∂u ∂v + ∂x ∂y ∂u ∂u u +v ∂x ∂y. = 0. ∂ue ∂ 1 ∂u 0 0 = ue + −uv ∂x ∂y Rec ∂y ∂ 0 = − p + v0 v0 ∂y. (2.17) (2.18) (2.19). completed with the boundary conditions. x = 0 : u = u(0, y). (2.20). y = 0 : u = 0, v = 0, u0 v 0 = v 0 v 0 = 0. (2.21). y → ∞ : u → ue. (2.22). The complete development to obtain the above equations and a deeper analysis of these can be found in [103].. 2.2. A representative outer boundary layer flow promoting separation. e Equations (2.9) and (2.18) include the term ue ∂u ∂x which is the pressure gradient imposed to the outer boundary layer flow. In order to have a representative outer flow which promotes laminar boundary layer separation, a linear deceleration inviscid profile was chosen. This.

(34) 16. CHAPTER 2. PROBLEM DEFINITION 1 "# $. !. =. TRANSITION REGION TO LINEAR PROFILE. "% 0.95. 0.9. 0.85. 0.8. 0.75 -0.1. 0. 0.1. 0.2. &#'&% 0.3 (). x=. Figure 2.1: Simplified model of the deceleration zone under study.. characteristic profile is shown in figure 2.1, where Ld is the characteristic decay length, and u0 is the maximum inviscid velocity, which is placed at x0 . At x0 the boundary layer is considered laminar and a Blasius’ profile is assumed. Using as characteristic length and velocity Ld and u0 respectively, the characteristic Reynolds number in equations (2.17), (2.18) and (2.19) becomes: Rec = ReLd =. u20 due dx linear. L d u0 = ν ν. (2.23). e , with du dx linear being the inviscid velocity gradient in the linear zone. Using the classical scaling for laminar boundary layers: −1. v ∼ δ ∼ ReLd2. (2.24) 1. 1. 2 2 the problem can be formulated in the dilated variables v1 = vReLd , y1 = yReLd , resulting:. ∂u ∂v1 + ∂x ∂y1 ∂u ∂u u + v1 ∂x ∂y1. = 0. x=0 x>0. 1 ∂ue ∂ ∂u 2 + − u0 v 0 ReLd ∂x ∂y1 ∂y1 ∂ ∂p 0 0 +vv ∂y1 ∂y1. = ue. 0 = (. θ01 (. (2.25) . u(y1 ) = uBlasius R∞ = 0 u(1 − u)dy1 = K0. y1 = 0 : u = v = 0 y 1 → ∞ : u → ue = 1 − x. . (2.26) (2.27). (2.28) (2.29).

(35) 2.3. TECHNOLOGICAL INTEREST. 2.3. 17. Technological interest. A representative engineering problem which exhibits the flow previously defined is the scenario found in the suction side of modern LPT blades. Figure 2.2 shows the velocity distribution on a typical LPT blade obtained with the CFD code MISES [26]. The horizontal axis describes the coordinate along the suction side of the profile (s) non dimensionalized with the axial chord (Cx ), whereas the vertical axis represents the velocity non-dimensionalized with the maximum velocity on the profile. Three flow solutions are represented. The black line is the inviscid velocity distribution, whereas the gray ones represent the edge velocities obtained in two viscous cases at different Reynolds numbers. It can be seen that there is a zone where the flow is accelerated and then a deceleration region begins. For the inviscid case, the velocity profile at the deceleration zone can be approximated by an initial nonlinear velocity distribution zone, followed by a linear deceleration. The viscous cases are characterized by the separation of the laminar boundary layer at certain downstream station which is associated to the point where the inviscid and viscous curves disjoin. After flow separation, there is a zone of a weak velocity decay associated to the separated flow segment, where a shear layer develops, followed by an abrupt deceleration to velocity values similar to those found in the inviscid case. This is related to the readherence of the flow that occurs after transitional phenomena and that energizes the shear layer. 1.00. Re 0.90. 0.80. ue=ue*/u0. Inviscid linear diffusion zone 0.70. 0.60. Suction side end 0.50. Maximum Velocity point 0.40 0.00. 0.20. 0.40. 0.60. 0.80 s/Cx. Separation Point. 1.00. 1.20. 1.40. 1.60. Figure 2.2: Characteristic LPT blade suction side velocity distributions.. The later stages in the evolution of the edge velocity for the viscous cases follows an almost linear deceleration and differs from the inviscid case by the boundary conditions imposed at the trailing edge and the growth of the turbulent boundary layer. The description of the boundary layer evolution in the profile segment with outer flow acceleration is simple whenever the turbulence outer flow level is low enough to ensure its laminar character. For these cases, the use of integral methods, as that proposed by Thwaites [85], is justified to obtain the parameters characterizing the boundary layer at the maximum inviscid velocity station. In this case, x0 is the distance from the forward stagnation point.

(36) 18. CHAPTER 2. PROBLEM DEFINITION. to the location of the maximum inviscid velocity station (measured along the suction side of the profile). Application of Thwaites’ method in a curvilinear coordinate system where x runs parallel to the blade surface provides: s √ Z 1 Reθ0 θ0 Re0 = 0.45 =√ u e5 de x (2.30) x0 Re0 0. x u u0 θ 0 u0 x 0 ,x e= , Reθ0 = , and Re0 = . u0 x0 ν ν For a family of blades of currently in-service LPT technology this parameter takes values Re √ θ0 ∼ 12 . The parameter Reθ0 completely describes the boundary layer at x0 , since at this Re0 station the pressure gradient is zero and therefore a Blasius profile develops. The really interesting flow developing region is the suction side zone with adverse pressure gradient. Figure 2.3 shows this part for a typical velocity profile representative of current LPT technology. The approximation done to characterize this zone (gray) is also shown. It consists of a constant velocity region of length x00 − x0 and a linear profile with characteristic decay length Ld . To obtain the virtual origin x00 of a linear deceleration, and the deceleration length Ld , the mean quadratic error between the simplified velocity profile and that provided by a CFD inviscid simulation can be minimized via Lagrange multipliers: " # x eR end (e u−u es )2 de x = I(e x0 , x eend , xe0 0 ) (2.31) , with u0 the maximum inviscid velocity, u e=. x e0. ∂I ∂ xe0 0. = 0 ⇒ xe0 0. , the end of the adjust interval xend can be chosen to be at 95% of the blade chord, in order to avoid the local acceleration effects linked to the trailing edge flow. This choice defines an additional parameter for this kind of flows, known as Backward Surface Diffusion uend BSD = = 1 − xend . u0. 2.3.1. Non dimensional parameters characteristic values. The equations for laminar and turbulent boundary layers presented at 2.1 were non dimensionalized with typical values of the flow. As said, for this problem, the characteristic length and velocity are the deceleration length Lc = Ld and the maximum velocity in the inviscid profile uc = u0 respectively. Besides, the development of the boundary layer in the adverse pressure gradient zone is independent of how is achieved the boundary layer at the beginning of it (parabolic behavior of laminar boundary layer equations), being a Blasius boundary layer profile. Dimensional analysis of boundary layer equations and boundary conditions gives: ! p p p u v ReLd x y ReLd θ0 ReLd x00 , = φ ReLd , , , , H0 , , BSD (2.32) u0 u0 Ld Ld Ld Ld Numerical computations for representative LPT blades geometries were performed to obtain characteristic values for the problem parameters. It is common in specific literature to define the Reynolds number in this kind of geometries with the core flow velocity at the.

(37) 2.3. TECHNOLOGICAL INTEREST. 19. 1.00. INVISCID VELOCITY PROFILE S_END AJUST VIRTUAL ORIGIN OF LINEAR DIFFUSION. 0.95. LINEAR APROXIMATION BSD. v/Vmax. 0.90. Ld/Cx 1. 0.85. 0.80. 0.75 x0/Cx 0.70 0.90. xend/Cx. x0'/Cx. 1.00. 1.10. 1.20. 1.30. 1.40. 1.50. 1.60. s/Cx. Figure 2.3: Simplified model of the deceleration zone in Low Pressure Turbines profiles.. trailing edge of the blade v2 and the suction side length Ss , Re2Ss = Reynolds number ReLd previously defined is related to this one: ReLd = Re2Ss. L d u0 S s v2. v2 Ss ν .. The characteristic. (2.33). Table 2.1 shows characteristic values of the parameters for different modern LPT blades technologies. According to Hodson [48, 49], Re2Ss ranges from about 5 · 104 in the final stage at high altitude in small business jet applications to about 5 · 105 at sea level takeoff in the first stages of large turbofans. Between takeoff and cruise altitude the Reynols number can be divided by 3 or 4. Therefore, the Reynolds number range is high enough to include several schemes of boundary layer behavior. An estimate for High Lift LPT technology, that will be the baseline case for these studies, leads to ReLd ≈ 2.0Re2Ss , thus, the range of interesting Reynolds numbers will be 1 < ReLd · 10−5 < 10. √ θ0 ReLd Ld x0 Technology BSD Ld Ss Ld Conventional 0.23 0.17 2.53 0.014 High Lift 0.30 0.17 1.66 0.033 Ultra High Lift 0.48 0.21 0.99 0.066 High Load 0.24 0.17 1.89 0.030 Table 2.1: Characteristic values of the problem parameters for different LPT technologies. Nomenclature in figures 2.2 and 2.3..

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(39) Chapter 3. EXPERIMENTAL FACILITIES The experimental facilities used in this thesis are presented in this chapter. The experimental work has been done basically in the open-return wind tunnel named BV3 at the Laboratory of Experimental Fluid Dynamics (L.E.F.) of the E.T.S.I. Aeronáuticos, which belongs to the University Politécnica de Madrid. Another facility, designated as BV0, is an open-return wind tunnel conceived to validate concepts and experimental techniques, which for these studies was used to determine the feasibility of the PIV measurement technique to characterize the boundary layer flow. This chapter also includes the design considerations for different concepts of test sections used in the studies.. 3.1. Wind tunnel BV3. The facility BV3 is an open circuit, blow-down wind tunnel that consists of a fan, a diffuser, a settling chamber, a contraction, and a test section. A sketch of the facility is shown in figure 3.1. The design of the different concepts of test sections will be presented in section 3.2. They all share a rectangular inlet cross-section with a h1 = 400mm height and a w = 300mm width. The design range of velocities at the entrance plane of the test section is 1 < v1 < 12.5m/s. The design of this facility was performed as a preliminary effort of the work done for the thesis, and a summary is presented here. A deep description can be found elsewhere [87, 88]. Test sections are described in some detail in Section 3.2. A summary description of the other elements composing the BV3 facility is given next.. 3.1.1. Contraction. The contraction is the element located immediately upstream from the test section and downstream from the settling chamber, where honeycombs and screens are used as preliminary flow conditioning system. Its purpose was to achieve the required mean velocity at the test section entrance and a sufficiently small inlet turbulence intensity, lower than 0.5%. Two constraints were considered to steer its design. First, a limit to the maximum thickness of the boundary layer at the test section entrance was specified. In addition, the contraction should be as short as possible. The selected design implements a rectangular cross-section, 3D contraction. The geometry of each wall consists of a fourth degree polynomial function. 21.

(40) 22. CHAPTER 3. EXPERIMENTAL FACILITIES 3D Contraction. Interface module. Settling chamber. Fan outflow conditioning module Fan. Diffuser. 400. 1400. 450. 1600. 450. 1400. 400. 1200. Test section. 855. 1220. 315. 1100. 900. 300. 5944. All dimensions in mm. Figure 3.1: Elements and main dimensions of the facility BV3.. Figure 3.2 shows a sketch with the dimensions that need to be specified. These are the Ai contraction ratio CR = A = wwei hhei , the contraction length Lc , the position of the inflectional e point xi for the polynomial functions, the inlet length Li and the exit length Le . Inflectional point. Li. Le. hi. xi. Lc. he. we. wi To test section. To settling chamber. Inflectional point. Figure 3.2: Contraction main dimensions.. Both 2D and 3D numerical simulations were performed to define these parameters. On the calculations, a structured mesh was used for the mainflow. The inlet and exit boundaries were placed far enough to avoid perturbation on the boundary conditions, and a Reynolds Stress Model was used coupled to the commercial CFD code FLUENT v6.3 . For the contraction ratio, Mehta [71] suggests a typical value of 10 in small tunnels based on the reduction of the turbulence intensity, the contraction length and the manufacturing cost. For the case under consideration, the contraction ratio was selected to 9 in order to have a 3:1 contraction on fc = Lc and the inflectional each direction. For the nondimensionalized contraction length L hi −he xi point position xei = Lc an iterative, optimization process was performed. Finally, xei = 0.35 fc = 4 were chosen as a compromise solution to obtain balanced values in the exit and L boundary layer thickness, flow uniformity, and capability turbulence level. For the inlet length, the criterion used was to limit the effects of pressure nonuniformity at the contraction.

(41) 3.1. WIND TUNNEL BV3. 23. −pi inlet on the last screen of the settling chamber. A value of pimax ≈ 15 was selected, with ∆ps ∆ps being the pressure drop at the settling chamber, and pimax , pi denoting the maximum and average pressures at the contraction’s inlet plane. From 2D numerical simulations the inlet length was selected to be Li = 0.1hi . The exit length was selected as a compromise to get adequate uniform conditions at the test section entrance and avoiding an undesired growth of the boundary layer thickness. Making use again of numerical simulations Le = he was selected.. 3.1.2. Settling chamber. The settling chamber consists of a serie of honeycombs and screens to straighten the flow and to reduce the turbulence intensity at the inlet of the contraction. Two honeycombs with hexagonal cells were selected. Cell sizes of 6 mm and 3 mm were chosen to get a progressive straightening effect on the flow. The honeycomb length was selected based on other research studies [9, 71]. These recommend values of the order of 6 to 8 cell sizes to obtain an adequate flow straightening effect. The width values were selected as 60 mm and 20 mm respectively. The separation distance between the two honeycombs was selected to allow flow uniformization after the first of them. The existing literature suggests a value of at least 5 cell sizes. Therefore, the two honeycombs were placed 50 mm apart. The distance between the second honeycomb and the first screen was selected as 30 mm. Four screens were used in the settling chamber. This is a compromise number to get the required turbulence reduction without a large increase of the settling chamber length and total pressure loss. The main design parameters of the screens are discussed in [29, 38, 71, 75]. These are listed below: 1 The screen solidity σ, defined as the ratio between projected closed area and total area, should be less than 0.5 to avoid flow instabilities due to the coalescence of jets coming from different cells. 2 The Reynolds number based on the screen wire diameter Red,grid should be greater than 40 in order to have not excessive total pressure loss. 3 The distance between consecutive screens should be greater than 20 mesh sizes to allow isotropic turbulence development. 4 The turbulence reduction along the screen should be such that at the exit of the settling chamber the turbulent intensity is less than 5%, thus, turbulence intensities lower than 0.5% are obtained at the inlet of the test section. The turbulent reduction is estimated as [38]: q u02 dowstream 1 q = (3.1) 1 + ξgrid 02 u upstream being:. ξgrid =. 2∆p −1/3 1 − σ ≈ 6.5Red,grid 2 2 σ ρvgrid. (3.2).

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