Articulada Induzidos por Escoamento/ Sina Mirzaei Sefat. – Rio de ..... Com base em vários ensaios exaustivamente realizados, foram possíveis definir os limites de .... autorrotação, mas mostra o papel da energia inicial no limite de transição entre eles. Em ...... In fact, the different flow regimes of the fully and partially.
TATALAMENTO E AUTORROTAÇÃO DE PLACA PLANA VERTICAL ARTICULADA INDUZIDOS POR CORRENTE UNIFORME
Sina Mirzaei Sefat
Tese de Doutorado apresentada ao Programa de Pós-graduação
em
Engenharia
Oceânica,
COPPE, da Universidade Federal do Rio de Janeiro, como parte dos requisitos necessários à obtenção do título de Doutor em Engenharia Oceânica. Orientador: Antonio Carlos Fernandes
Rio de Janeiro Novembro 2011
TATALAMENTO E AUTORROTAÇÃO DE PLACA PLANA VERTICAL ARTICULADA INDUZIDOS POR CORRENTE UNIFORME
Sina Mirzaei Sefat
TESE SUBMETIDA AO CORPO DOCENTE DO INSTITUTO ALBERTO LUIZ COIMBRA DE PÓS-GRADUAÇÃO E PESQUISA DE ENGENHARIA (COPPE) DA UNIVERSIDADE FEDERAL DO RIO DE JANEIRO COMO PARTE DOS REQUISITOS NECESSÁRIOS PARA A OBTENÇÃO DO GRAU DE DOUTOR EM CIÊNCIAS EM ENGENHARIA OCEÂNICA.
Examinada por:
Prof. Antonio Carlos Fernandes, Ph.D.
Prof. Sergio Hamilton Sphaier, D.Sc.
Prof. Juan Bautista Villa Wanderley Ph.D.
Prof. Celso Pupo Pesce, D.Sc.
Dr. Ricardo Franciss D.Sc.
RIO DE JANEIRO, RJ – BRASIL NOVEMBRO DE 2011
Mirzaei Sefat, Sina Tatalamento e Autorrotação de Placa Plana Vertical Articulada Induzidos por Escoamento/ Sina Mirzaei Sefat. – Rio de Janeiro: UFRJ/COPPE, 2011. XV, 117 p.: il.; 29,7 cm. Orientador: Antonio Carlos Fernandes Tese (doutorado) – UFRJ/ COPPE/ Programa de Engenharia Oceânica, 2011. Referências Bibliográficas: p. 113-117. 1. Autorrotação. 2. Tatalamento. 3. Manifold. I. Fernandes, Antonio Carlos. II. Universidade Federal do Rio de Janeiro, COPPE, Programa de Engenharia Oceânica. III. Título.
iii
To my parents, Mohammad and Zahra, and my sisters Azade and Atefe.
iv
Agradecimentos
Thanks God for all the blessings that he brings into my life.
I am in debit to my adviser Prof. Antonio Carlos Fernandes for his guidance and support during the research, also all of his favors and supports from the first day that I came to Brazil up to now.
I would like to thank Fabio Moreira Coelho, Luís Victor Cascão, Mohammad Mehdi Armandei and Amanda Silva de Albuquerque for their technical assistance in experimental tests and for many discussions and suggestions.
I like to acknowledge Agência Nacional de Petróleo – ANP, for all of their financial supports during my PhD period.
I also would like to thanks from all staffs of LOC/COPPE/UFRJ (Laboratório de Ondas e Correntes da COPPE, da Universidade Federal do Rio de Janeiro): Ivan Bragança Marinho Falcão, Luiz Antônio Ferreira, Anderson Araújo do Santos and Werner de Barros.
Finally, I would like to send my especial thanks to my top friends because of their friendship and all of their supports in my life: Saman Karimi (University of Illinois, USA), Meisam Zadhosseini (Derak Co., Iran), Arash Roshanzamir (ISOIO Co., Iran), Reza Firoozkoohi (Cesos – NTNU, Norway) and Arash Abbasnia (Amirkabir University of Technology, Iran).
v
Resumo da Tese apresentada à COPPE/UFRJ como parte dos requisitos necessários para a obtenção do grua de Doutor em Ciências (D. Sc.)
TATALAMENTO E AUTORROTAÇÃO DE PLACA PLANA VERTICAL ARTICULADA INDUZIDOS POR CORRENTE UNIFORME
Sina Mirzaei Sefat Novembro/2011
Orientador: Antonio Carlos Fernandes
Programa: Engenharia Oceânica
O objetivo deste estudo é entender o problema de oscilação que pode ocorrer com o lançamento livre de dispositivos oceânico (offshore) na água durante sua a instalação no fundo do mar. Este estudo aborda a rotação de uma placa plana vertical articulada torno de seu eixo vertical excitada apenas por um escoamento de água horizontalmente incidente. No caso de placa fixa, as cargas agindo sobre ela foram medidas para os ângulos de ataque de 0˚ a 90˚, para diferentes números de Reynolds. Para o caso dinâmico, uma análise dimensional mostra que o movimento de rotação induzida é governado essencialmente pelo número de Reynolds, pelo momento adimensional de inércia. Um modelo quase-permanente desenvolvido pela tese é capaz de reproduzir tanto o tatalamento (fluttering) quanto a autorotação obtidas no LOC. Confirmou-se a existência da frequência natural de tatalamento (fluttering) pleno que é linearmente proporcional à velocidade do escoamento incidente e inversamente proporcional à largura da placa.
vi
Abstract of Thesis presented to COPPE/UFRJ as a partial fulfillment of the requirements for the degree of Doctor of Science (D. Sc.)
FLUTTERING AND AUTOROTATION OF A HINGED VERTICAL FLAT PLAT INDUCED By UNIFORM CURRENT
Sina Mirzaei Sefat
November/2011
Advisor: Antonio Carlos Fernandes
Department: Ocean Engineering
The objective of this study is to understand the oscillation motions that may occur when an offshore device is released in the water during the installation in seabed. This study addresses the rotation of a hinged flat plate allowed to rotate about a vertical axis under the influence of a uniform current. In the static case, the loads and moments acting on the flat plate were measured for angles of attack ߠ ൌ0˚െ90˚and for different Reynolds numbers. For the dynamic case, the dimensional analysis proves that the motion in flow induced rotation is governed essentially by dimensionless moment of inertia and Reynolds number. A quasi-steady model is suggested to model the fluttering motion as obtained in LOC’s flume. It is confirmed that the natural and response frequencies of fluttering motion are linearly proportional to the incoming flow velocity and inversely proportional to the flat plate width.
vii
Sumário Lista de Figuras
x
Lista de Tabelas
xv
Capítulo 1: Introdução
1
Capítulo 2: Experimentos
4
Capítulo 3: Análise Teórica da Placa Estática
6
Capítulo 4: Modelagem Teórica Quase-Permanente de Rotação Induzida de Placa Plana
8
Capítulo 5: Análise de Estabilidade da Rotação Induzida pelo Escoamento
9
Capítulo 6: Conclusões e Sugestões para Trabalhos Futuros
10
Apêndice 1: Introduction
12
Apêndice 2: Experiments in LOC’s Current Flume
19
2.1 Experimental Setup
19
2.2 Fixed Plate
23
2.2.1 Hydrodynamic Forces and Moment on a Fixed Flat Plate
23
2.2.2 Hydrodynamic Loads and Moment Results
24
2.2.2.1 Lift Moment versus Munk Moment
27
2.2.3 Strouhal Number (St)
32
2.3 Flow Induced Rotation
33
2.3.1 Dimensional Analysis
33
2.3.2 Equivalent Harmonic Angle of Fluttering
36
2.3.3 Response Frequency of Autorotation
38
2.3.4 Natural Frequency of Fluttering
39
2.3.5 Phase diagram of Fluttering and autorotation
41
Apêndice 3: Theoretical Analysis of Static Plates 3.1 Introduction
44
3.2 Kirchhoff’s Model
46
3.3 Roshko’s Model
48
viii
44
3.4 Wu’s Model
50
3.5 Kutta’s Model for Moment Coefficient
58
Apêndice 4: Theoretical Modeling of Flow Induced Fluttering and Autorotation
61
4.1 Fluttering
61
4.1.1 Natural Frequency of Fluttering Motion 4.2 Autorotation (Tumbling)
64 68
4.2.1 Equation of Motion of Autorotation
71
4.2.1.1 Wu’s Streamline Theory Moment coefficient model
72
4.2.1.2 Kutta’s Moment Coefficient Model
73
4.2.1.3 LOC Moment Coefficient Model
75
Apêndice 5: Stability Analysis of Flow Induced Rotation
77
5.1 Fixed Points
77
5.2 Conservative System and Phase Diagrams
79
5.2.1 Kutta’s Moment Coefficient Model
81
5.2.2 Wu Streamline Theory Moment Coefficient Model
82
5.2.3 LOC Moment Coefficient Model
83
5.3 Experimental Phase Diagram
84
Apêndice 6: Conclusions
87
Apêndice A: Computational Fluid Analysis
90
A.1 Turbulence Modeling
93
A.2 ANSYS CFX Modeling
96
Apêndice B: Free Streamline Theory
100
B.1 Kirchhoff’s Model
100
B.2 Roshko’s Model
106
B.3 Wu’s Model
110
References
113
ix
Lista de Figuras Fig. 1-1 Schematic of flow induced rotation of a flat plate hinged at the center and is free to rotate in uniform current
15
Fig. 1-2 Pendulum Installation Method (PIM) of a manifold, (i) first phase: almost a free fall in water since the installation cable is slack, (ii) second phase: is characterized by the curvilinear motion with the installation cable becoming taut, (iii) is just the smooth vertical trajectory down to the bottom
15
Fig. 1-3 During several model testing by Fernandes et al. [11,12], Oscillatory behavior of a manifold during the first phase of the PIM (Pendulous Installation Method); (a) (manifold), (b)
first model of a complex object
second model showing clearly the fluttering
behavior
16
Fig. 1-4 The vertically hinged flat plate in the current flume of LOC/COPPE/UFRJ (Laboratory of Waves and Current of COPPE, Federal University of Rio de Janeiro)
16
Fig. 2-1 (a) Schematic illustration of reference axis, (b) Picture of the experimental setup in the LOC/COPPE/UFRJ (Laboratory of Waves and Current of COPPE, Federal University of Rio de Janeiro) Fig. 2-2 Assembly details of the torque sensor and load cell
19 20
Fig. 2-3 Data Acquisition System for: (a) the load cell and (b) the torque sensor
21
Fig. 2-4 Diagram of forces and center of pressure setup around the flat plate; note that for
0˚ the plate is transversal to the flow while for 24
90˚ it is aligned to the flow Fig. 2-5 Drag coefficient obtained by experiments at LOC and calculated with ANSYS CFX
25 x
Fig. 2-6 Lift coefficient obtained by experiments at LOC and calculated with ANSYS CFX
25
Fig. 2-7 Comparison of normal coefficient of experimental results at LOC and Hoerner [15] and also numerical results
26
Fig. 2-8 Position of the center of pressure of numerical modeling compared with experiments at LOC and two other experimental results of Flachsbart [14] and Tachikawa [37] and linear segments as 27
proposed by (2-6-a) and (2-6-b) Fig 2-9 Schematic of the interaction of uniform current with a 2D slender
28
body Fig. 2-10 Moment coefficient obtained numerical modeling and experiments at LOC and also the Munk moment coefficient given by (2-15), and moment coefficient segments, proposed in Table 2-2
30
Fig. 2-11 Strouhal numbers of experimental results at LOC experimental and the numerical simulation results for different block coefficients
32
Fig. 2-12 (a) Time history of autorotation motion for I*=0.25 and for different Reynolds numbers, (b) Time history of autorotation motion for Re=99245 and for different dimensionless moment of 36
inertia Fig. 2-13 Equivalent harmonic angle of fluttering motion versus current velocity for flat plates with different dimensionless moment of
37
inertia Fig. 2-14 The experimental results of response frequency of autorotation motion as a function of different velocities and for different plates
39
Fig. 2-15 The experimental results of natural frequency as a function of different velocities and for different plates for fluttering motion
40
Fig. 2-16 Natural frequency of flow induced fluttering versus vortex 41
shedding frequency Fig. 2-17 Phase diagram to classify different motions of flow induced rotation based on Reynolds number and dimensionless moment of inertia, obtained by experiments at LOC. Each point means a xi
different test; preliminary phase diagrams will be discussed later in 42
Apêndice 5
45
Fig. 3-1 Schematic of free streamline theory, for flow past a flat plate Fig. 3-2 The z-plane of the actual flow based on Kirchhoff’s theory and other
46
conformal mappings planes [14]. Fig. 3-3 Flow assumption of Roshko’s modeling [32,33]
48
Fig. 3-4 The flow in the physical space (z-plan, where z = x+ iy)
51
Fig. 3-5 Comparison of normal coefficient of streamline theory with σ
1.25, and experimental results at LOC and Hoerner [15], 53
and also ANSYS CFX numerical results Fig. 3-6 Location of stagnation point and center of pressure of flat plate [43];
54
µ is position of stagnation point [43] Fig. 3-7 Comparison of position of the center of pressure with streamline theory (Eq. (3-21)) and ANSYS CFX and experimental results of
55
LOC, Flachsbart [14] and Tachikawa [38] Fig. 3-8 The comparison of the linear segmented results of position of the center of pressure found by streamline theory and numerical
56
ANSYS CFX results Fig. 3-9 Comparison of moment coefficient of streamline theory (Eq. 3-23) with σ
1.0 and also σ
1.25 and numerical results of ANSYS 57
CFX and experiments at LOC Fig. 3-10 Comparison of the moment coefficients proposed by experimental
60
results of LOC, streamline theory, Munk and Kutta Fig. 4-1 (a) Time series of fluttering motion of the hinged flat plate about the vertical axis submitted to a uniform current (Re=1.26 105), (b) The
62
respective frequency domain response Fig. 4-2 Undamped oscillation when the difference of the natural and exciting force frequency is small, and it results beating motion, and
2 are
1
the first and second waves frequency
Fig. 4-3 Comparison between experimental results from LOC and Eq. (4-5), I*=0.0012, CB=0.142; Segment.1: CM1 = -0.1452, Segment.2: CM1 = xii
63
65
-0.16015, Segment.3: CM1 = -0.18791 Fig. 4-4 Comparison between experimental results from LOC and Eq. (4-5), I*=0.0038, CB=0.214; Segment.1: CM1 = -0.1452, Segment.2: CM1 =
66
-0.16015, Segment.3: CM1 = -0.18791 Fig. 4-5 Comparison between experimental results from LOC and Eq. (4-5), I*=0.0047, CB=0.214; Segment.1: CM1 = -0.1452, Segment.2: CM1 =
66
-0.16015, Segment.3: CM1 = -0.18791 Fig. 4-6 Comparison between experimental results from LOC and Eq. (4-5), I*=0.0378, CB=0.428; Segment.1: CM1 = -0.1452, Segment.2: CM1 = -0.16015, Segment.3: CM1 = -0.18791
67
Fig. 4-7 The experimental setup for autorotation motion
69
Fig. 4-8 The phase diagram as a function of angle and angular velocity for 70
four different Reynolds numbers, I*=0.1508 Fig. 4-9 The autorotation motion of plate with I*=0.15 and Re=9.95 x 104
72
Fig. 4-10 The autorotation motion of plate with I*=0.25 and Re=9.95 x 104
73
Fig. 4-11 The autorotation motion of plate with I*=0.1522 and Re=9.95 x 104
74
Fig. 4-12 The autorotation motion of plate with I*=0.25 and Re=9.95 x 104
74
Fig. 4-13 The autorotation motion of plate with I*=0.1508 and Re=9.95 x 104
75
Fig. 4-14 The autorotation motion of plate with I*=0.25 and Re=9.95 x 104
76
Fig. 5-1 The phase diagram of Kutta’s moment coefficient model
81
Fig. 5-2 The phase diagram of Streamline theory moment coefficient model
82
Fig. 5-3 The phase diagram of LOC adjusted moment coefficient model
83
Fig. 5-4 Typical phase diagram of the small fluttering state, obtained by experiments at LOC, Re= 1.65 x 104, I* = 0.0047 and
= π/2
84
Fig. 5-5 Typical phase diagram of fluttering motion, obtained by experiments at LOC, Re= 5.85 x 104, I* = 0.0047 and
85
= π/2
Fig. 5-6 Typical phase diagram of autorotation motion, obtained by experiments at LOC, Re= 8.65 x 104, I* = 0.15 and θ =
π/2; note
that the minimum is close to zero (around 0.005 rad/s)
85
Fig. A-1 The velocity in RANS equations represent as time average and 91
fluctuation velocity terms xiii
Fig. A-2 Mesh generation model
97
Fig. A-3 CFX pre model
98
Fig. A-4 Time history of drag and lift coefficients for different angles of
99
attack Fig. B-1 Plans under the assumption of Kirchhoff’s modeling [20]
102
Fig. B-2 Position of the center of pressure base on Kirchhoff model (Eq. (B105
30)) Fig. B-3 The hodograph of Roshko and Wu modeling
106
Fig. B-4 Plans under the assumption of Roshko’s modeling [33,34]
106
Fig. B-5 Plans under the assumption of Wu’s modeling [43]
110
Fig. B-6 Lift coefficients as function of angle of attack for different wake 112
under-pressure coefficients [43] Fig. B-7 Drag coefficients as function of angle of attack for different under-
112
pressure wake coefficients [43]
xiv
Lista de Tabelas Table 2-1 The experiment setup properties
22
Table 2-2 List of moment coefficient segments
31
Table 2-3 List of dimensions of the variables
34
Table 3-1 Drag coefficient values for different values of k [33]
49
Table 3-2 The moment coefficients proposed by experiments, streamline 59
theory, Munk and Kutta Table 4-1 Linear segments of moment coefficient
64
Table 4-2 The root mean square of angle of rotation for different Reynolds 71
numbers Table 4-3 Equations of motion based on moment coefficient proposed by
75
experimental results of LOC Table 5-1 Fixed points for differential equation models of motion
77
Table 5-2 Stability analysis of the linearized equations in the neighborhood 78
of the fixed pints Table 5-3 The conservative energy systems for different differential equation
80
models Table 5-4 The value of energy level for separation trajectory of fluttering to
84
autorotation Table A-1 RANS equation terms
92
Table A-2 Mesh properties
96
Table A-3 Properties of fluid domain and boundary condition for k-epsilon 97
modeling in ANSYS CFX
xv
Capítulo 1
Introdução A interação de um corpo em movimento com corrente é um dos primeiros problemas a ter recebido atenção. Entretanto, a maioria das pesquisas se concentra nos movimentos de translação, pelo menos em Engenharia Naval e Oceânica. Apenas alguns têm realizado investigações sobre o movimento de rotação. Consequentemente, alguns aspectos importantes da rotação induzida por escoamento ainda não são bem compreendidos. Este estudo aborda a rotação de uma placa plana vertical articulada em torno de seu eixo vertical, excitada
apenas por um escoamento de água horizontalmente
incidente. O objetivo deste estudo é entender o problema de oscilação (que pode ser excessiva) que pode ocorrer com o lançamento livre de dispositivos oceânicos (offshore) na água, durante sua a instalação no fundo do mar. O comportamento da placa plana é estudado experimentalmente e teoricamente e o estudo evidencia a ocorrência de tatalamento (fluttering)1 e de autorrotação conforme fatores como a intensidade da corrente, momento de inércia da placa, condições iniciais, entre outros, como explicado na tese. A tese está organizada do seguinte modo: ela tem seis capítulos e a cada capítulo corresponde um apêndice de mesmo número onde o assunto é apresentado pelo autor 1
O Dicionário Michaellis Português‐Inglês (Edição 2000‐2009) assim se refere ao verbo Tatalar: sm bras whir as a wing stroke vint (...) 2. to whir, to flutter 1
que é mais fluente em inglês e onde, portanto, pode expressar-se melhor. A tese tem também mais dois apêndices, o A e B, onde aspectos teóricos são ainda mais aprofundados. Estes últimos apêndices são citados pelos capítulos e apêndices numéricos. Assim sendo, o Capítulo 2 descreve a configuração experimental para medições no canal de correntes do LOC/COPPE/UFRJ (Laboratório de Ondas e Correntes do Programa de Engenharia Naval e Oceânica da COPPE, Universidade Federal do Rio de Janeiro). No caso de placa fixa (sem movimento), as cargas agindo sobre ela foram medidas para os ângulos de ataque de 0˚ a 90˚, para diferentes números de Reynolds. Para o caso dinâmico (placa livre para oscilar apenas em rotação em torno de eixo vertical), uma análise dimensional mostra que o movimento de rotação induzida é governado essencialmente pelo número de Reynolds, pelo momento adimensional de inércia, pela largura da placa e pelas condições iniciais. O Capítulo 3 é alocado para explicar uma teoria que permite a modelagem analítica pela Teoria da Linha de Corrente (Streamline Theory). Esta teoria fornece resultados que podem ser usados para calcular as cargas (forças e momentos) agindo na placa plana. Além disso, a tese desenvolveu uma modelagem numérica usando o sistema ANSYS CFX. O Capítulo apresenta comparação dos resultados das três metodologias citadas: experimentos, ANSYS CFX e Teoria da Linha de Corrente (Streamline Theory). No Capítulo 4, o trabalho sugere um modelo quasi-permanente analítico tanto para o tatalamento (fluttering), como para a autorrotação. Novamente, seguem comparações com resultados dos experimentos no canal de correntes do LOC. Neste capítulo, identifica-se e estuda-se a freqüência natural de vibração do movimento de tatalamento (fluttering) bem como a frequência de autorrotação.
2
No Capítulo 5, o trabalho realiza uma análise de estabilidade através da confecção de diagrama de fase. Aborda o comportamento da rotação induzida pelo fluxo identificando a natureza da transição do tatalamento (fluttering) para a autorrotação. Evidencia-se também a dependência das condições iniciais. Finalmente, o Capítulo 6 fecha a tese, apresentando Conclusões e Sugestões de Trabalhos para o Futuro. Ver mais detalhes no Apêndice 1.
3
Capítulo 2
Experimentos Este capítulo apresenta os experimentos realizados, começando com a configuração experimental para medições no LOC/COPPE/UFRJ. No caso da placa fixa, as forças e momentos agindo sobre a placa plana foram medidos para os ângulos de ataque θ=0˚90˚ e para diferentes números de Reynolds. Os resultados estáticos são úteis para se compor as propriedades importantes para o caso dinâmico nos movimentos de tatalamento (fluttering) e no de autorrotação induzidos apenas pelo fluxo. Da mesma forma, os resultados para o número de Strouhal, isto é, identificando a freqüência de geração de vórtices, são também apresentados. Trata-se de mais uma contribuição da tese, uma vez que o número de Strouhal é bem conhecido apenas para corpos cilíndricos. Por outro lado, para o caso dinâmico, a análise dimensional mostra que o movimento de rotação induzida é governado essencialmente por adimensional do momento de inércia, número de Reynolds, largura da placa e condições iniciais. Com base em vários ensaios exaustivamente realizados, foram possíveis definir os limites de bifurcação de estabilidade quando o tatalamento (fluttering) muda para a autorrotação. Para tal, um diagrama de fase foi preparado baseado nos experimentos. Observaram-se três tipos de estados: o estado de pequeno tatalamento (flutterig), o
4
tatalamento (fluttering) pleno e o estado de autorrotação. A dependência é clara do número de Reynolds, momento de inércia adimensional e condições iniciais. Por outro lado, os experimentos e a análise dimensional confirmaram que a freqüência natural é linearmente proporcional à velocidade de entrada e inversamente proporcional à largura da placa. Também com base na análise dimensional e experimentos, foi confirmado que o ângulo de rotação equivalente harmônico de rotação é aproximadamente constante em diferentes velocidades. Ver mais detalhes no Apêndice 2.
5
Capítulo 3
Análise Teórica da Placa Estática Neste Capítulo, a tese sistematizou modelagem teórica para o caos da placa estática submetida a escoamento uniforme incidindo por vários ângulos de ataque. Usou a chamada Teoria da Linha de Corrente (Streamline Theory). Esta teoria basicamente modela, com bons resultados, o escoamento a jusante. O princípio básico da Teoria da Linha de Corrente (Streamline Theory) assume que o fluxo bidimensional permanente e espacialmente constante à jusante para cada ângulo de ataque, caracterizando uma situação plenamente desenvolvida. As bordas da placa são assumidas cantos vivos e considera-se o escoamento precocemente separado. Mostra-se que a dependência do número de Reynolds é insignificante. Além disso, neste capítulo, a espessura da placa plana é assumida nula. Assumir que a jusante a distribuição da pressão é constante significa que a velocidade perto dos dois pontos de separação é quase a mesma. Vórtices se formam alternadamente em cada lado. Esta formação de vórtices ocorre à jusante de qualquer corpo rumbudos e a freqüência de desprendimento de vórtices é uma característica da forma do corpo. São três as diferentes Teorias da Linha de Corrente (Streamline Theories) descritas neste capítulo (Kirchoff, Roshko e Wu), além de uma teoria heurística (Kutta). Para cada uma, o coeficiente adimensional de momento foi calculado para diferentes
6
ângulos de ataque, e a aderência com resultados experimentais é notável para a Teoria de Wu. Isto permite a utilizacão dos sempre úteis resultados analíticos. Ver mais detalhes no Apêndice 3.
7
Capítulo 4
Modelagem Teórica Quase-Permanente de Rotação Induzida de Placa Plana O capítulo apresenta uma aproximação quasi-permanente com o objetivo da obtenção de mais conhecimentos sobre a natureza do fenômeno de tatalamento (fluttering), bem como para a autorrotação da placa plana induzida pelo escoamento. Mostra-se ser possível modelar a dinâmica da placa através de uma equação diferencial ordinária com contribuições de momento de inércia adicional e coeficientes de momento obtidos dos capítulos anteriores. Baseado nesta modelagem quasi-permanente do movimento, confirma-se que para o caso de tatalamento (fluttering), a freqüência natural do movimento de oscilação é linearmente proporcional à velocidade do escoamento de entrada e inversamente proporcional à largura da placa. A modelagem matemática mostra que a freqüência natural é quase igual ao resultado experimental da frequência de desprendimento de vórtices o que leva a um movimento com características de batimento no caso de tatalamento (fluttering). Por outro lado, o método de Runge-Kutta foi usado para estimar as trajetórias de movimento de autorrotação com um modelo quasi-permanente semelhante. Estas estimativas teóricas têm boa aderência com as trajetórias experimentalmente obtidas nos ensaios do LOC. Ver mais detalhes no Apêndice 4. 8
Capítulo 5
Análise de Estabilidade da Rotação Induzida pelo Escoamento Este capítulo desenvolve uma análise de estabilidade sobre a rotação induzida pelo escoamento uniforme e aprofunda-se na bifurcação entre o fenômeno de tatalamento (fluttering) para autorrotação. Primeiramente, os pontos fixos para os diferentes modelos são obtidos. Cada ponto é classificado por meio da equação linearizada. Encontraram-se apenas centros e pontos de sela. Em seguida, os diagramas de fase são montados em função da velocidade angular e ângulo de rotação. Os níveis de energia iniciais são identificados e mostram-se essenciais para entender a bifurcação citada. Mostram-se os resultados para os modelos citados da Teoria da linha de corrente (Streamline Theory), Kutta e um advindo dos experimentos, chamado de LOC. Como era de se esperar, o diagrama de fase com base no modelo do LOC se porta melhor quando comparado com os experimentos. Na verdade, neste modelo de aproximação do coeficiente de momento é mais próximo da realidade em todos os ângulos.
9
Capítulo 6
Conclusões e Sugestões para Trabalhos Futuros A tese apresentou o estudo experimental combinado com sistematização teórica sobre a hidrodinâmica instável de uma placa plana vertical articulada num eixo fixo quando submetida apenas a um escoamento uniforme. Identificou-se a existência de três tipos de comportamento: o pequeno tatalamento (fluttering) , o tatalamento (fluttering) pleno e a autorrotação. As cargas hidrodinâmicas estáticas foram obtidas experimentalmente no canal de correntes do LOC. Houve também o processamento numérico através do ANSYS CFX. Finalmente um método analítico, com base na Teoria da Linha de Corrente (Streamline Theory), corroborou os resultados anteriores. Estes resultados estáticos são evidentemente úteis para a obtenção de resultados dinâmicos de rotação induzida pelo escoamento. Um modelo quase-permanente desenvolvido pela tese é capaz de reproduzir tanto o tatalamento (fluttering) quanto a autorotação obtidos no LOC. Para que este modelo tivesse mais consistência, a tese recorreu com sucesso à uma análise dimensional. Confirmou-se a existência da frequência natural de tatalamento (fluttering) pleno que é linearmente proporcional à velocidade do escoamento incidente e inversamente proporcional à largura da placa. Como era de se esperar, devido à mesma dimensão, o mesmo tipo de dependência se aplica à rotação no fenômeno de autorrotação.
10
Finalmente, o diagrama de fase da análise de estabilidade corrobora a existência dos três estados de pequeno tatalamento (fluttering) , o tatalamento (fluttering) pleno e a autorrotação, mas mostra o papel da energia inicial no limite de transição entre eles. Em particular, explicitou-se a dependência das condições iniciais. Com estes resultados, nem que seja qualitativamente, espera-se contiribur para a melhoria do Método de Lançamento Pendular de Manifold (PIM - Pendoulous Installation Method) que foi a motivação inicial da presente investigação. No futuro, sugere-se continuar as investigações, sempre usando uma abordagem combinando teoria e experimentos, como a utilizada na tese. Como o objetivo é diminuir o tatalamento (fluttering) e a autorrotação, sugere-se estudar corpos com fairings e bordo de ataque, conforme já iniciado por (Fernandes et al -- OMAE2012). Finalmente, sugere-se estudar corpos com espessura.
11
Apêndice 1
Introduction The interaction of a moving body with current is one of the earliest hydrodynamic problems to received attention. However, most of the researches concentrate on translational motions, and just few investigations have been performed on rotational motion. Hence, some important aspects of the flow induced rotation are still not well understood, although this phenomenon has gained interest in various areas such as morphology of plants and animals, meteorology and aeroballistics, aerospace engineering, windborne debris occurring in windstorms [2,17,39] and the falling object [1,4,13,23,26,37]. This explains the choice of this thesis about studying the behavior of a hinged flat plate allowed to rotate about a vertical axis under the influence of a uniform current (see Fig. 1-1). This is indeed a fundamental problem with only one degree of freedom that, as shown next, requires a vast knowledge of several hydrodynamic issues as lift, drag, vortex shedding, self excitation, separation, etc. The fluttering and autorotation are two different phenomena which may occur in flow induced rotation of a plate about a fixed axis, which is free to rotate in the current. In the autorotation, the plate rotates continuously around a vertical axis and it never damps out. On the other hand, the fluttering motion is another unexpected periodic oscillation of the plate around a stable position which the plate is normal to the flow. For a free falling object, the fluttering motion is called the same and the body oscillates 12
either periodically or chaotically from side to side as it descends alternating gliding at low angle of attack and fast rotational motion [26]. But the autorotation motion for the falling object is called tumbling which is characterized by the end-over-end rotational motion of body. Maxwell [23] was the first who tried to explain tumbling motion of a flat-plate. Later, several experimental researches under both free-fall and fixed-axis conditions were made in order to classify and quantify the types of these rotational motions. The theoretical and experimental results before 1979, such as Dupleich [4], Smith [34], and etc, have been evaluated and summarized by Iversen [18]. According Iversen [18], the motion of a freely falling and also fixed axis rectangular plate is governed by the Reynolds number (Re=Ub/v), the aspect ratio (c/b), thickness ratio (t/b) and the non-dimensional moment of inertia of the flat plate (I*=I/ ρf cb4), where b, c, t and I are width, span, thickness and moment of inertia of plate respectively. Also U, ρf and v are the velocity, density and kinematic viscosity of fluid, respectively. Iversen shows that, the transition from fluttering to tumbling should be completely governed by these parameters. Tachikawa [39] firstly, presented a two-dimensional equation of motion to explain the trajectories of free falling of square and rectangular plates in a uniform flow. Secondly, he obtained by experiments, the aerodynamic coefficients of autorotation in different square and rectangular flat plates, which are required for solving the equations of motion. The results show that the trajectories of a flat plate released into a flow from rest is closely related to the initial mode of motion and is distributed over a wide range. Lugt [21,22] argued that, for a viscous fluid, at least qualitatively, a fifth-order polynomial for the damping terms in the pendulum equation is necessary to simulate the
13
self-excited oscillation (fluttering) and autorotation for bodies with axis fixed in a parallel flow. Field et al. [13] prepared a bifurcation diagram based on experimental data, showing the dynamical behavior of falling disks as a function of the two parameters: the dimensionless moment of inertia and Reynolds number. According to this bifurcation diagram, depending on the dimensionless moment of inertia, the motion could be fluttering for smaller dimensionless moment of inertia, or tumbling for larger dimensionless moment of inertia. Based on these measurements, the transition from fluttering to tumbling is nearly independent of Reynolds number and it appears at I* 0.04. Tanabe et al. [37] gave a phenomenological model for free fall of a paper, assuming zero thickness and incompressible ideal fluid. They discovered five different patterns of: periodic rotation, chaotic rotation, chaotic fluttering, periodic fluttering and simple perpendicular fall. Mittal et al. [25] show a computational study on flow induced motion of a hinged plate pinned at its center. Their focus is on the effect of Reynolds number and plate thickness ratio and non-dimensional moment of inertia on vortex induced rotation of plates. Based on their numerical results, they suggest that the flutter and tumble frequencies of large aspect-ratio plates are governed by the Von-Kármán vortex shedding process. Anderen et al. [1] investigated the dynamics of freely falling plates experimentally, numerically and by quasi-steady modeling, at Reynolds number of 103, which describes the motion of freely falling rigid plates based on detailed measurement of the plate trajectories and from this to assess the instantaneous fluid forces.
14
Fig. 1-1 Schematic of flow induced rotation of a flat plate hinged at the center and is free to rotate in uniform current.
In the present work, controlling the oscillatory behavior of manifolds in the pendulous installation method (PIM) [8,10,11,12,35] is the first motivation to study the flow induced rotation. Fig 1-2 describes the PIM and Fig 1-3 shows some selected pictures with fluttering from other experimental tests [29]. Though, the study on flow induced rotation of a vertical hinged flat plate under the influence of uniform current is a 1-DOF problem, but it aims at the understanding of the fluttering problem of falling objects in air or water. The results could be generalized for some other phenomena which are 3-DOF, like as PIM method, falling objects, windborne debris, etc.
Fig. 1-2 Pendulum Installation Method (PIM) of a manifold, (i) first phase: almost a free fall in water since the installation cable is slack, (ii) second phase: is characterized by the curvilinear motion with the installation cable becoming taut, (iii) is just the smooth vertical trajectory down to the bottom.
15
(a) (b) Fig. 1-3 During several model testing by Fernandes et al. [11,12], Oscillatory behavior of a manifold during the first phase of the PIM (Pendulous Installation Method); (a) first model of a complex object (manifold), (b) second model showing clearly the fluttering behavior.
In this work, the flow induced rotation of a fixed axis flat plate is studied experimentally and theoretically and both the fluttering and the autorotation have been clearly identified. The experiments were conducted in a flume (22m x 1.4m x 0.5m) at the LOC/COPPE/UFRJ (Laboratory of Waves and Current of COPPE, Federal University of Rio de Janeiro), see Fig. 1-4.
Fig. 1-4 The vertically hinged flat plate in the current flume of LOC/COPPE/UFRJ (Laboratory of Waves and Current of COPPE, Federal University of Rio de Janeiro).
16
Apêndice
2
describes
the
experimental
setup
for
measurements
at
LOC/COPPE/UFRJ. In the static case, the loads and moments acting on the flat plate were measured for angles of attack θ
0˚ 90˚and for different Reynolds numbers. On
the other hand, for the dynamic case, the dimensional analysis proves that the motion in flow induced rotation motion is governed essentially by dimensionless moment of inertia and Reynolds number. Certain combinations define the stability boundaries between fluttering and autorotation. Hence in this apêndice, a bifurcation diagram prepared from the experiments to classify different states, observed small fluttering, fluttering and autorotation based on different Reynolds number and dimensionless moment of inertia. For the analytical modeling, the so-called Streamline theory has been used to approximate the wake far downstream and Apêndice 3 is allocated to explain this theory. A comparison of the results among the three methodologies of experiments, ANSYS CFX numerical modeling and Streamline theory is discussed. In Apêndice 4, a quasi-steady model is suggested to model the trajectory of autorotation motion via analytical modeling. The comparison of analytical modeling with experiments at water flume of LOC was made. It should be mentioned that most of the previous researches on flow induced rotation, conducted experimentally and rarely numerically. In this apêndice, the natural frequency of fluttering motion has also been suggested. In Apêndice 5, a stability analysis is performed on flow induced rotation phenomenon to gain insight into the nature of transition from fluttering to autorotation. At first, the fixed points for different models of motion are obtained and each point analyzed using the linearized equation. Secondly, the phase diagrams as a function of angular velocity and angle of rotation have been presented for different dynamic models
17
from Apêndice 3. Finally, phase diagrams obtained by experiments at LOC for small fluttering state, fluttering and autorotation are presented. Apêndice 6 shows the conclusions and suggestions of future works.
18
Apêndice 2 Experiments in LOC’s Current Flume 2.1 Experimental Setup The experiments were conducted in a current flume with 22 m in length, 1.4 m in width and 0.5 m in depth at LOC/COPPE/UFRJ (Laboratory of Waves and Current of COPPE, Federal University of Rio de Janeiro), shown in Fig. 2-1. The experimental set up was assembled to measure forces, moment and the center of pressure for different angles of flow incidence on the fixed flat plate and for different Reynolds number. Subsequently, free rotation of the flat plate was also assessed.
(a) (b) Fig. 2-1 (a) Schematic illustration of reference axis, (b) Picture of the experimental setup in the LOC/COPPE/UFRJ (Laboratory of Waves and Current of COPPE, Federal University of Rio de Janeiro). 19
The experimental apparatus was mounted in accordance with the details shown in Fig. 2-2. A load cell and a torque sensor were positioned at the top of the plate. These sensors were calibrated already positioned in the apparatus.
Fig. 2-2 Assembly details of the torque sensor and load cell.
The vortex shedding frequencies for the static flat plate were obtained through spectral analysis of the signal acquired from an Acoustic Doppler Velocimeter (ADV) which was positioned at a point in the wake of the plate. The velocity of the current flowing in the channel was obtained from a turbine type flow-meter. The rotation was measured by two different systems. The first one is a Qualisys system and the other is potentiometer. The Qualisys system supplies a range of hardware and software products for motion capture and analysis of movement data. The key components of the system are the cameras and the Qualisys Track Manager (QTM) software to measure the rotation angle and also frequency.
20
The potentiometer placed on the top of the flat plate was powered by a voltage source. The output voltage of the potentiometer was obtained with the same acquisition system used for the load cells. The oscillation frequency was obtained by spectral analysis of the rotation signal acquired for each velocity of incident current. The data acquisition from the load cell was performed using a system (NI - 9172 module in conjunction with the universal NI Daq - 9219) which has A/D converters and customized connections for the strain-gauges bridge complement required for the cell. The digital signal was recorded by software built in LabView 8.2. The acquisition of data from the torque sensor was performed similarly, but using an analog signal conditioner that amplifies and filters the signal before being digitalized. This configuration was required due to the sensor full scale compared to the low magnitude of the torque signal for some angles of flow incidence. Fig. 2-3 (a) and (b) present the mentioned data acquisition systems.
(a) (b) Fig. 2-3 Data Acquisition System for: (a) the load cell and (b) the torque sensor. 21
Initially an aluminum plate with width of 0.6 m, draft of 0.5 m, and thickness of 5 mm was tested. This means that the blockage coefficient (CB=plate width/flume width) is 0.428. In this block coefficient, some wall blockage effect was observed. To decrease this effect a similar aluminum plate 0.3 m in width was tested (CB=0.214) under the same experimental conditions. Finally, to observe the effect of the inertia variation in the behavior of the oscillating flat plate, two PVC plates with width 0.3 m (CB=0.214) and 0.2 m (CB=0.142) were tested. Their thicknesses were 5 and 8 mm, respectively. The water depth was kept constant at 0.5m. The characteristics of the flat plates tested are summarized in Table 2-1: Plate Properties Width Materials
0.2, 0.3 and 0.6 m Aluminum, PVC
Fluid Parameters Fluid used Velocity range
Water 0.05 – 0.4 m/s
Table 2-1 The experiment setup properties.
Based on tests in wind channels, by previous researchers [13,17] and also the experiments in LOC, it is proved that the autorotation (tumbling) occurs at large dimensionless moment of inertia. In the water flume case, because of large density of water in comparison with the air, increasing of the dimensionless moment of inertia is not straight forward. For this purpose a unique apparatus was constructed to model the autorotation which is discussed in Apêndice 4.
22
2.2 Fixed Plate 2.2.1 Hydrodynamic Forces and Moment on a Fixed Flat Plate The loads acting on a flat plate immersed in a flow stream may be decomposed in normal pressures and tangential stresses on its surface. When integrated, these pressures and stresses give rise to the resultant forces and moment. The component of the load which is in the same direction of the flow is the drag and the lift force is perpendicular to the flow direction. There is also a binary given by the moment of the force about an axis. If this axis is taken at the center of pressure, by definition, the binary is zero. The dimensionless lift CL, drag CD, normal CN and moment CM coefficients are defined as below.
CL
L 0.5 f U 2 S
(2-1)
CD
D 0.5 f U 2 S
(2-2)
CN
N 0.5 f U 2 S
(2-3)
CM
Mz 0.5 f U 2 Sb
(2-4)
where L is the lift, D is the drag, N is the normal force, MZ is the moment about z axis (the flow is in x,y plane),
is the fluid density, U is the upstream flow velocity and
S is a reference surface. Here S=bh, where b is the flat plate width and
is the depth of
the water in the flume. The distance from the centre of pressure to the centre of the plate is c. The relationship between the moment coefficient, the normal force coefficient (CN) and the position of the centre of pressure is:
C M (c / b ) C N
(2-5)
23
The dimensions of the flat plate and the diagram of drag and lift forces setup around the flat plate, as well as the angle of attack are shown in Fig. 2-4.
Fig. 2-4 Diagram of forces and center of pressure setup around the flat plate; note that for transversal to the flow while for 90˚ it is aligned to the flow.
0˚ the plate is
2.2.2 Hydrodynamic Loads and Moment Results The loads and moment coefficients were measured as a function of angle of attack by experiments at LOC. Similar measurements were carried out with a CFD simulation using the ANSYS CFX code (See Apêndice A). Figs 2-5 and 2-6 show respectively the drag and lift coefficients as a function of angle of attack for both the experiment results and the numerical simulation with ANSYS CFX. The Reynolds Number was kept around 1.17 x 105. In both figures, the numerical simulation of lift and drag coefficients show good agreement with experimental results.
24
2.1
CFX - CB=0.06 CFX - CB=0.214 LOC - CB=0.214
1.8
CD
1.5 1.2 0.9 0.6 0.3 0 0
40 50 60 70 80 90 θ[deg.] Fig. 2-5 Drag coefficient obtained by experiments at LOC and calculated with ANSYS CFX.
1.75
10
20
30
CFX - CB=0.06 CFX - CB=0.214 LOC - CB=0.214
1.5
CL=2πα
1.25
CL
1 0.75 0.5 0.25 0 0
10
20
30
40 50 60 70 80 90 θ[deg.] Fig. 2-6 Lift coefficient obtained by experiments at LOC and calculated with ANSYS CFX.
A classical result of potential theory for flat plate is that for a small angle of attack
2
, [20,42] (Note that here
25
⁄2
). Fig. 2-6 also shows the
comparison of this result with experimental and numerical results. It is clear that, for the cases with smaller blockage effect the results match well with the theoretical small incident angle limit. Flachsbart [14] measured the normal coefficient of a two dimensional flat plate. These data are reported by Hoerner [15]. Fig. 2-7 shows the comparison of normal coefficient of numerical simulation with ANSYS CFX and the experimental results in LOC and Hoerner [15] results. 2.25
LOC
2
CFX - CB 0.06 HOERNER
1.75
CN
1.5 1.25 1 0.75 0.5 0.25 0 0
10
20
30
40 50 θ [deg.]
60
70
80
90
Fig. 2-7 Comparison of normal coefficient of experimental results at LOC and Hoerner [15] and also numerical results.
Fig. 2-8 shows a good agreement between the position of the center of pressure found by numerical modeling and experiments at LOC and two other experimental results of Flachsbart [14] and Tachikawa [40]. Considering these results, two linear regions may be devised as show in Fig. 2-8. These two linear segmented models adjusted to fit the position of the center of pressure are given by Eqs (2-6-a) and (2-6-b).
26
0.35
LOC CFX - CB=0.06 TACHIKAWA FLACHSBART assumed
Center of pressure (c/b)
0.3 0.25
CP = 0.381θ - 0.2745
0.2 0.15 0.1 0.05
CP = 0.1012θ
0 0
10
20
30
40
50
60
70
80
90
θ[deg.] Fig. 2-8 Position of the center of pressure of numerical modeling compared with experiments at LOC and two other experimental results of Flachsbart [14] and Tachikawa [40] and linear segments as proposed by (2-6-a) and (2-6-b).
c b 0.1012
for angle of attack 55
(2-6-a)
c b 0.381 0.2745
for angle of attack 55 90
(2-6-b)
2.2.2.1 Lift Moment versus Munk Moment Based on the D'alembert's paradox the net force on a body which is immersed in an inviscid flow is zero, but not necessarily a zero moment and any shape other than a sphere generates a moment. This moment may be called Munk moment in the elongated body context [41]. The Munk moment arises from the asymmetric location of the stagnation points, where the pressure is highest. The Munk moment usually arises for the case of a long slender body inclined at angle unbounded fluid (see Fig. 2-9).
27
to the direction of motion in an
Fig 2-9 Schematic of the interaction of uniform current with a 2D slender body.
Resolving the free stream velocity U into components parallel and normal to the axis of the body, the cross flow velocity (v) and the added mass ( mdx ) gives us an elemental momentum ( m vdx ) and if this changes with time, Newton’s second law gives us the relation
dF
d mdx v dt
(2-7)
Or
dF d dm dv m v v m dx dt dt dt
(2-8)
Munk omitted the second term because he was not considering accelerated flow. By considering v U sin and dx dt U cos , (2-8) may be written as (2-9)
dF dm dm dx 1 dm 2 v v U sin 2 dx dt dx dt 2 dx
(2-9)
The added mass is [27]
m y 2 S
(2-10)
where
S y2
(2-11)
By considering (2-10) and (2-11), the result is
28
dF 1 dS U 2 sin 2 dx 2 dx
(2-12)
The integral over the entire length of the body is [27]
F
1 U 2 sin 2 S B 2
(2-13)
where SB is the base area, if the body is truncated. The Munk moment about the vertical axis of the body can be obtained in a manner similar to that used to obtain the lateral force. For steady motion, we obtain the following expression [27]
1 1 M m U 2 xdm sin 2 myy mxx U 2 sin 2 2 b 2
(2-14)
where mxx is added mass along the body x-axis (forward), and myy is along the body lateral y-axis, and also should be noted that
⁄2
. This moment may be
approximated for small angles of incidence. According to the above statements and extending them, for the 2D case, the Munk moment coefficient for a flat plate can be express as
C M ( ) sin 2 4
(2-15)
On the other hand, based on the potential theory for a flat plate in small angles of attack, the lift coefficient is [42]:
CL 2 sin
(2-16)
The location of the center of pressure, for small α values is located at one forth of plate width (b/4) from the nose. Hence, the center of pressure could be approximated as [30,31]: (c / b) (1/4) cos
(2-17)
Therefore the lift moment coefficient can be express as 29
1 C M 2 sin ( ) cos ( ) sin cos ( ) sin 2 4 2 4
(2-18)
which it is exactly matched by the formulation of the moment coefficient of Munk moment for flat plates (Eq. 2-15). The comparison of the moment coefficient calculated by numerical modeling and experiments at LOC and also the Munk moment coefficient given by (2-15) is presented in Fig. 2-10. This comparison shows that the range of its validity is (75 θ[deg.]
0 -0.1
90).
0
10
20
30
40
50
60
70
80
90
CM = -0.1452θ
-0.2 -0.3
CM = -(π/4)sin2θ
CM
-0.4 -0.5 -0.6 -0.7 -0.8 -0.9
LOC CFX - CB=0.06 Munk
Fig. 2-10 Moment coefficient obtained numerical modeling and experiments at LOC and also the Munk moment coefficient given by (2-15), and moment coefficient segments, proposed in Table 2-2.
Table 2-2 shows a model adjusted to fit the available experimental and numerical data for moment coefficient. For angles of attack less than 30˚ (
30˚), the normal
coefficient is almost constant; hence it is obvious that by multiplying with the linear term of center of pressure, the resultant moment coefficient is linear too. But the normal coefficient in angles of attack between 30˚ to 75˚ (30˚
75˚), is
not constant and could be expressed as a polynomial function. So it expected by 30
multiplying the normal coefficient with linear term of center pressure, the resultant moment coefficient also be a higher order polynomial function. But because of small values of the nonlinear terms, a surprising linear behavior for moment coefficient shows up. This is shown in Fig 2-10 and Table 2-2. So indeed we have a Hydrodynamic Torsional Spring for angles of attack less than 75˚. Moment Coefficient C M 0.1452
Angle
0 75
C M 4 sin 2 75 90 Table 2-2 List of moment coefficient segments
The reason for such a surprisingly high range of validity of the linear behavior may be understood that the pressure distribution in the wake part of the flat plate is almost constant (this will be addressed in Apêndice 3). It means that this region do not contribute to the moment. In fact the main contribution to the moment comes from the front part of the plate in a flow that may receive a potential theory treatment that leads to the linear behavior, provided a singularity in the corners are taking off as will show in Apêndice 3 by the Streamline theory.
31
2.2.3 Strouhal Number (St) The Strouhal number is a dimensionless number characterizing the vortex shedding frequency and is given by (2-19) St
f sb U
(2-19)
where fs is the frequency of vortex shedding, b is the flat plate width and U is the velocity of the fluid. The experimental results of Strouhal number for the different flat plates in terms of Reynolds number and also the numerical simulation results are presented in Fig. 2-11. All results are consistent except the flat plat with CB=0.428 which it is due to the large blockage effect. Fig. 2-11 suggests that the Strouhal number without the blockage effect for the flat plate is in the range of 0.19 to 0.235. 0.35 0.3
Strouhal Number
0.25 0.2 0.15 0.1
LOC - CB=0.142 LOC - CB=0.214
0.05
LOC - CB=0.428 CFX - CB=0.06
0 0
50000
100000
150000
200000
250000
Reynolds Number
Fig. 2-11 Strouhal numbers of experimental results at LOC experimental and the numerical simulation results for different block coefficients.
32
2.3 Flow Induced Rotation With the understanding of the static behavior in one hand, the work went on releasing the flat plate to rotate. Three different behaviors could be devised experimentally: small fluttering state, fluttering and autorotation. Several tests were performed at the LOC to investigate these motions of plates with different dimensionless moment of inertia and different ranges of velocity. An important observation during the tests was that the autorotation motion just occur under two conditions. The first is initial condition: the plate must be released with an angle less then stall angle (-15˚
15˚), then the motion is affected by the large lift
force. The other condition is that the dimensionless moment of inertia should be high enough to store sufficient angular momentum to overcome the hydrodynamic torsional spring. The result will be shown next, but before this a dimensional analysis will be performed.
2.3.1 Dimensional Analysis In order to improve the understanding of flow induced rotation of flat plates, a simple dimensional analysis is performed. The flow induced rotation motion of a flat plate may be characterized by eight dimensional parameters: b the width of the plate, the thickness of the plate,
the density of the plate,
the density of the fluid, I the moment of inertia
of the plate, A66 added moment of inertia, v the kinematic viscosity of the fluid, and U the uniform flow velocity. The scheme that has been used for reducing these dimensional variables into dimensionless groups is the Buckingham-pi theorem, which has six steps [5]:
33
1. List and number the variables involved in the problem. As mentioned before, the number of variables is 9 (n=9) as listed below:
R f b , t , s , f , I , A66 , ,U
(2-20)
2. List the dimensions of each variable according to {MLT} or {FLT}. The list is given in Table 2-3.
R
b
t
s
f
I
A66
U
M
0
0
0
1
1
1
1
0
0
L
0
1
1
-3
-3
2
2
2
1
T
0
0
0
0
0
0
0
-1
-1
Table 2-3 List of dimensions of the variables.
3. Find . In this study the j is equal to 3, thus the number of dimensionless numbers will be 6 (n j=6). 4. Select j scaling parameters which do not form a pi product. Hereby we picked up the uniform flow velocity, plate width, and fluid density. 5. Algebraically find the exponents which make the product dimensionless. In our case the results are:
t s A66 I , , , 5 5 b f f b f b U b
R f ,
(2-21)
6. Write the final dimensionless function. From the five parameters of step 5, we form five non-dimensional numbers, the thickness ratio, density,
⁄
t/b, the dimensionless
, the dimensionless added moment of inertia, Iˆ I / A66 , and
Reynolds number, Re=U.b/v
34
R f , * , Iˆ,
1 Re
(2-22)
Therefore the flow induced rotation motion of a flat plate is governed by the Reynolds number, thickness ratio, dimensionless density and dimensionless moment of inertia of the flat plate. It should be noted that the dimensionless moment of inertia can be rewritten as a function of dimensionless density and thickness ratio. Actually the motion in flow induced rotation motion and in particular, the transition from fluttering to autorotation, should be completely governed by dimensionless moment of inertia and Reynolds number. It should be mentioned that for the comparison purpose, instead of Iˆ , in this research, we will use I * 128 Iˆ as used by Iversen [18] and Field et al. [13]. Fig. 2-12(a) and 2-12(b) show the time history of autorotation motion for different Reynolds number and dimensionless moment of inertia. 30
Re=99245 Re=78010 Re=66800 Re=58360 Re=47715 Re=37575
25
θR [rad.]
20 15 10 5
(a)
0 0
4
8
12 Time [sec.]
35
16
20
30
I*=0.25 I*=0.1522
25
I*=0.0805
θR [rad.]
20 15 10 5 (b)
0 0
4
8
12 16 20 Time [sec.] Fig. 2-12 (a) Time history of autorotation motion for I*=0.25 and for different Reynolds numbers, (b) Time history of autorotation motion for Re=99245 and for different dimensionless moment of inertia.
Fig 2-12(a) and 2-12(b) illustrate the proven dimensional analysis results. The increase in Reynolds number and moment of inertia results in an increase of flow induced motion.
2.3.2 Equivalent Harmonic Angle of Fluttering The equivalent harmonic angle of rotation of the fluttering motion may be defined as Eq. (2-23) N
eq 2
2 i
(2-23)
i
N
where eq is the equivalent harmonic angle of fluttering, N is the number of data that has been registered, i is the angle of plate in iteration i. The dimensional analysis for equivalent harmonic angle leads to (2-24) 36
eq f ( * , )
(2-24)
which for a constant thickness and specific mass, it could be written as (2-25)
eq const
(2-25)
Hence, based on the dimensional analysis, we expect that the equivalent harmonic angle of rotation for fluttering motion of a hinged plate be approximately constant in different velocities. Fig. 2-13 presents the equivalent harmonic of rotation versus current velocity for different flat plate models.
Eq. Harmonic Angle [deg.]
90
LOC - I*=0.0012 LOC - I*=0.0047
75
LOC - I*=0.0378
60 45 30 15 0 0
0.05
0.1
0.15 0.2 0.25 0.3 0.35 Velocity [m/sec.] Fig. 2-13 Equivalent harmonic angle of fluttering motion versus current velocity for flat plates with different dimensionless moment of inertia.
As shown in Fig. 2-13, we can consider the equivalent harmonic angle of rotation approximately constant in different velocities. This average value increases from 21˚ to 28˚ and 34˚ for plates with dimensionless moment of inertia of 0.0012, 0.0047 and 0.0.387 respectively, which confirms the proven dimensional analysis results. The
37
increase in moment of inertia results in an increase of equivalent harmonic angle of rotation of fluttering.
2.3.3 Rotational Frequency of Autorotation Similar dimensional analysis for the rotational frequency of flow induced autorotation of a flat plate leads to [5]:
R
1 f , * , I * , U b Re
(2-26)
For bodies with sharp corners like the flat plate, the dependence of Reynolds number is negligible because of early separation. On the other hand, the dimensionless added inertia can be rewritten as a function of dimensionless density and thickness ratio. Therefore, (2-26) may be written as (2-27)
R
U b
f ( * , )
(2-27)
Hence for a constant thickness and specific mass, the dimensionless rotational frequency is constant. This statement is confirmed in Fig. 2-14.
38
0.3 0.25
LOC - I*=0.25 LOC - I*=0.1522 LOC - I*=0.0805
fr .(b/U)
0.2 0.15 0.1 0.05 0 30000
50000
70000
90000
110000
Reynolds Number Fig. 2-14 The experimental results of rotational frequency of autorotation motion as a function of different velocities and for different plates.
2.3.4 Natural Frequency of Fluttering The dimensional analysis for the natural frequency of flow induced fluttering of a flat plate leads to [5]:
n
U b
f ( *, )
(2-28)
Hence for constant thickness and specific mass, (2-28) may be written as (2-29)
n
U b
const
(2-29)
In other words, the natural frequency for fluttering motion of a hinged flat plate is linearly proportional to the incoming velocity U and inversely proportional to the plate width b. This statement is confirmed in Fig. 2-15. The comparison of experimental results between natural frequency as a function of different velocities and for different plates is shown in Fig. 2-15. 39
0.4 0.35 0.3
fn .(b/U)
0.25 0.2 0.15 0.1
LOC - I*=0.0012
0.05
LOC - I*=0.0047 LOC - I*=0.0378
0 0
50000
100000
150000
200000
250000
Reynolds Number Fig. 2-15 The experimental results of natural frequency as a function of different velocities and for different plates for fluttering motion.
On the other hand, Fig. 2-16 indicates a linear relationship between the natural frequency (fn) of the plate and the vortex shedding frequency (fS) measured by experiments at LOC. Fig. 2-16 shows very well that the shedding frequency and the natural frequency of flow induced rotation are almost identical. In other words, it is clear that the oscillation of the plate is a result of the vortex shedding, which is confirmed by experimental observations. As the vortices are generated the flat plate rotates following the shedding. The proximity between fn and fs justifies the beating type of response.
40
0.4
LOC - I*=0.0012
0.35
LOC - I*=0.0047
0.3
LOC - I*=0.0378
fn [Hz.]
0.25 0.2 0.15 0.1 0.05 0 0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
fs [Hz.] Fig. 2-16 Natural frequency of flow induced fluttering versus vortex shedding frequency.
2.3.5 Bifurcation diagram of Fluttering and autorotation As mentioned in Apêndice 1, some investigations were performed on the dynamical behavior of falling objects and also plates that are free to rotate about a fixed axis, as function of different parameters as like Reynolds number, dimensionless moment etc, to classify the fluttering and autorotation (tumbling) motions [13,25,37]. Mittal et al. [25] studied numerically the flow induced motion of a hinged plate, pinned at its center. They kept fixed the non-dimensional moment of inertia and focused on the effect of plate thickness ratio for different Reynolds number on vortex induced rotation of plates. Based on their numerical results, they suggest that the flutter and tumble frequency of large aspect-ratio plates is governed by the Karman vortex shedding process. Field et al. [13] prepared a bifurcation diagram base on experimental data, showing the dynamical behavior of falling disks as a function of the dimensionless moment of 41
inertia and Reynolds number. Based on those measurements, the transition from fluttering to tumbling is nearly independent of Reynolds number and it appears at I* 0.04. Based on the experiments at LOC, the bifurcation diagram of Fig. 2-17 is presented as function of different Reynolds numbers and dimensionless moment of inertia to classify the small fluttering state, fluttering and autorotation motions. It should be mentioned here that the small fluttering state is characterized by small oscillations in the range of
10˚
10˚. Actually, it means that the small fluttering state is fluttering
motion with equivalent harmonic angle less than 10˚. 0.25 Fluttering
Small Fluttering State
Autorotation Small Fluttering State
0.2
0.15
I*
Autorotation
0.1
Fluttering 0.05
0 0
15000
30000
45000
60000
75000
90000
105000
120000
Reynolds Number Fig. 2-17 Bifurcation diagram to classify different motions of flow induced rotation based on Reynolds number and dimensionless moment of inertia, obtained by experiments at LOC. Each point means a different test; preliminary phase diagrams will be discussed later in Apêndice 5.
42
As shown in Fig. 2-17, at low Reynolds numbers (approximately less than 2500), the plate stays almost at the same initial orientation (small fluttering state). At higher Reynolds numbers, depending on the dimensionless moment of inertia, the motion could be fluttering for smaller dimensionless moment of inertia, or autorotation (tumbling) for large dimensionless moment of inertia. Note that the transition from fluttering to autorotation is nearly independent of Reynolds number and it appears at I*=0.05 to 0.06, which is close to the falling disc case reported by Field et al. [13] (I* 0.04). Therefore, based on the bifurcation diagram of Fig. 2-17, in order to have autorotation motion, it is necessary that the dimensionless moment of inertia be larger than 0.05 or 0.06 and also it plate releases in a proper initial condition (angular position or velocity).
43
Apêndice 3 Theoretical Analysis of Static Plates 3.1 Introduction The target of the Apêndice 3 is to develop an analytical investigation on the loads and moment acting on a flat plate for different angles of attack. A flat plate has a simple shape and actually is the simplest airfoil with no thickness. For small incidence angles, several results were recovered for inviscid and irrotational flow such as the classical results of vortex sheet approach [42]. But for large incidence angles because of the presence of the wake behind the flat plate, it is not possible to assume that the flow is inviscid and viscous terms play an important role on the loads and moment acting on the flat plate. For this purpose, the Streamline theory has been applied to calculate the loads and moments acting on the flat plate under the interaction of a uniform current. This method was first introduced by Helmholts and extended by Kirchhoff and others. The main objective of this theory is to find the free streamlines to define the wake. Outside the wake, the flow is assumed potential. The aim is to compute the resultant pressure in the plate. According to the experimental observations, for bodies with sharp corners like the flat plate, the flow generally separates from certain points on the obstacle resulting in wake formation and the dependence of flow on Reynolds number is negligible because of the early separation. Downstream of the separation points, the distribution of pressure is remarkably constant for
44
almost any form of bluff body. This means that the velocity close to the two separation points is almost the same. The shear layers do not continue far downstream as assumed and roll up to form vortices, alternately on each side. This vortex formation occurs behind all bluff bodies, at a frequency which is a characteristic of each body shape (Strouhal number effect). As the pressure on the free streamline tends to remain constant for some distance downstream of the separation point, the flow past a bluff body is considered in two parts (see Fig. 3-1). Near the body, it may be described by the free-streamline theory. This part of the wake will be called the near-wake, or the free-streamline range. The pressure in this region will be called the wake under-pressure. Further downstream beyond the near-wake region, the shear layer gradually becomes larger and broader and become unstable and does not continue smoothly far downstream. The vortices mix and diffuse rapidly and are eventually dissipated in the wake. In such a range, the shape of the free streamline cannot be determined definitely. This part of the wake will be called the far-wake, or the mixing range. Along the far-wake, the mean pressure increases gradually from the wake under-pressure and finally recovers the main stream pressure far downstream.
Fig. 3-1 Schematic of free streamline theory, for flow past a flat plate.
45
3.2 Kirchhoff’s Model Kirchhoff studied the flow past a flat plate normal to the stream, which it is based on an assumption which results in a considerable loss of reality. By this assumption, the velocity on the free streamline at separation is equal to the free-stream velocity U. In this case, the pressure at the separation points and behind the separation points is equal to the free-stream pressure. This is not in agreement with experience, which shows that the pressure is actually always lower than the free-stream value. Fig. 3-2 shows the z-plane of the actual flow based on Kirchhoff’s theory and also other conformal mappings planes. As usual the solution consists of mapping z onto the plane of the complex potential hodograph or -plane, where
. This is accomplished through the complex ⁄
is the complex velocity.
Fig. 3-2 The z-plane of the actual flow based on Kirchhoff’s theory and other conformal mappings planes [14].
As mentioned before, the essential assumption of Kirchhoff is that everywhere on the free streamlines
and
, the velocity is equal to the free-stream value U. Then, in the hodograph
plan the free streamline is simply the circle | |
U (=1 after normalization). Secondly, the trace
of the plate SS in the hodograph plan is known a priori and is of simple form, so that the mapping to the w-plane is easily accomplished. In practice, it is convenient to use ζ instead of v, where
1/v = (1/q)
= dz/dw. Thus ζ gives the true flow direction and the reciprocal of the 46
velocity magnitude at the corresponding point in the physical plane. Once the mappings are known, the solution is given completely by z dw z w
q2
1
2
(3-1)
q 2 w
(3-2)
where q2 determines the pressure, since Bernoulli’s equation may be used to evaluate the pressure coefficient
CP 1 q 2
(3-3)
In particular, at the separation point and all along the free streamline CP = CPS = 0, since qs = 1. The pressure coefficient in the wake and on the back side of the plate is also zero. The normal, drag and lift then are simply due to the excess of pressure on the front. Applying the definitions, their values in the Kirchhoff theory are (see Appendix B)
CN
2 sin 4 sin
(3-4)
CD
2 sin 2 4 sin
(3-5)
CL
sin 2 4 sin
(3-6)
Which for the cases that plate is normal to current, CD = 0.88. The difference between the computed drag according Kirchhoff and the actual experience is seriously considerable. Experiments showed that the drag is very larger, being of the order CD
2. Hence, it would be
necessary to modify the theory. There are various wake-flow models, such as the Riabouchinsky model [20], the re-entrant jet model [43], Roshko [32,33,34] and Wu [43,44,45].
47
3.3 Roshko’s Model
As shown in Section 3-2, the difference between the drag coefficient based on the Kirchhoff’s model and the experiments is considerable. This difference is due to suction on the back of the plate. Corresponding to this, the velocity on the free streamlines at separation points are higher than the free stream velocity value. Kirchhoff’s model is modified by Roshko only by assuming that the velocity at separation point is qs = kU and to be at this value along the free streamline until the later becomes parallel to the free stream [32,33]. Fig. 3-3 shows the schematic of Roshko’s model. (To see the mapping planes from z- to the w-plan and the mapping formulations see Appendix B).
Fig. 3-3 Flow assumption of Roshko’s modeling [32,33].
At infinity, the flow must have returned to point I. The velocity at separation point has been normalized to qs = 1, which leads to the velocity at infinity to be U = 1/k. At B and free streamlines become parallel, separated by the distance
the
, which may be found by equation
(3-7) [32,33]: d k
k2 1 k 2 1
(3-7)
To find the drag coefficient, the average pressure on the front of the plate is: 48
C PF
1 d
k 2 1 2 k 2 k 2 1 tan 1 2 k k 1
(3-8)
On the back, the pressure is constant, so the average pressure there is simply (3-9)
C PS 1 k 2
Finally, the drag coefficient is (3-10)
C D C PF C PS
Table 3-1 shows the drag coefficient values for different values of k. Note that when k 1.5, the drag coefficient based on Roshko’s modeling has a good agreement with experiments (CD
2). ′
/
1.00
0.880
0.880
∞
∞
1.05
0.868
0.970
9.460
59.400
1.10
0.855
1.065
5.073
15.900
1.15
0.843
1.166
3.615
7.600
1.20
0.831
1.271
2.888
4.610
1.30
0.806
1.496
2.168
2.390
1.40
0.782
1.742
1.814
1.570
1.50
0.758
2.008
1.606
1.170
1.60
0.735
2.295
1.471
0.936
1.80
0.690
2.930
1.308
0.692
2.00
0.650 3.650 1.217 0.659 Table 3-1 Drag coefficient values for different values of k [33].
Note that the Roshko’s hypothesis is for the Wu is discussed, which applies for
0.
49
/
0 case. Next a more complete theory by
3.4 Wu’s Model
According to Wu’s theory, the hodograph and mapping is the same as in the Roshko’s model. But in this model, instead of k, an artifice is introduced to admit the under-pressure coefficient as a free parameter, to account for the essential feature of a very complicated process of viscous dissipation in the wake. It also allows the replacement of the actual wake flow of a real fluid by a simplified model within the framework of an equivalent potential flow. The dimensionless parameter is usually called the wake under-pressure coefficient [43]. This parameter characterizes the wake flow. In fact, the different flow regimes of the fully and partially developed flows can also be indicated by different ranges of .
P p 0.5 U 2
(3-11)
where P denotes the pressure of the undisturbed free stream, U is its relative velocity, is the fluid density, p is pressure in the near-wake region. According to the definition of and comparison by the definition of k in Roshko’s theory, the result is:
k 2 1
(3-12)
The wake flow will be called fully developed, if the region of the constant-pressure nearwake extends beyond the trailing edge of the plate. It will be called partially developed, if the near-wake region terminates in front of the trailing edge. For brevity these two flow regimes will also be called the full wake flow and the partial wake flow. Fig. 3-4 shows the flow in the physical space (z-plan, where z = x+ iy).
50
Fig. 3-4 The flow in the physical space (z-plan, where z = x+ iy).
Hereby, the velocity at separation is normalized to q = 1, and remains at this value along the free streamlines until the latter reach points
and
where
0. Downstream of these two
points, the free streamlines are parallel to the free stream on
and
along which q
decreases from unity back to the free stream value U. In order to the under-pressure coefficient of the flow be with q = 1 on
and
, U takes the value (1+ )1/2, as can be shown by
applying Bernoulli’s equation
p
1 1 q2 P U 2 2 2
(3-13)
The plate at each angle of attack is physically a different bluff body. Hence, the pressure in the wake part of the plate is different in each angle of attack. Hence, each angle of attack leads to a different under wake coefficient ( ). According to (3-4), from Kirchhoff’s classical result, by considering that normal coefficient of flow past a flat plate (
CN
, the
0) is rewritten as below
2 cos 4 cos
(3-14)
51
⁄2
In the fully wake flow past a flat plate, the local pressure’s force is everywhere normal to the plate, and there is no singular force at the leading edge. Therefore, CL and CD should satisfy the condition
CD CL tan
; For all
(3-15)
and 0 and σ
Based on above the conclusions, when
0, the drag and lift coefficients of
Roshko’s model could be reduce to [43]:
CD
2 1 0.05 2 4
(3-16)
CL 0
(3-17)
In the special cases of (i)
0 or (ii)
close to zero and
0, the Eqs (3-15) to (3-17)
are obviously satisfied. As noted before, each angle of attack must be assumed a different bluff body, which has a different pressure on the downstream the separation points. But in the general case, because of the complicated manner in which the dependence on
and
appears, the
results of streamline theory show that for small angles of , less than 45 degree, the values of and
approach respectively the asymptotes [43]
CL ( , ) (1 ) CL (0, )
(3-18)
CD ( , ) (1 ) CD (0, )
(3-19)
C N ( , ) (1 )
2 cos 4 cos
(3-20)
This argument is supported by experimental and numerical evidences. Fig. 3-5 shows the comparison of normal coefficient for the streamline theory with
1.25 and experimental
results at LOC and Hoerner [15], and also the numerical results with ANSYS CFX.
52
2.25 2 1.75 1.5 CN
1.25 1 0.75 0.5
Streamline Theory - σ = 1.25 CFX - CB=0.06 LOC Horner
0.25 0 0
10
20
30
40 50 60 70 80 90 θ [deg.] Fig. 3-5 Comparison of normal coefficient of streamline theory with 1.25, and experimental results at LOC and Hoerner [15], and also ANSYS CFX numerical results.
Fig. 3-5 shows that the streamline theory results agree very well with numerical and experimental results, especially when
is less than 55 degrees. This is about the range of interest
for modeling the fluttering motion. Fig. 3-6 shows the location of stagnation point and also the center of pressure for different range of . As expected, because of constant pressure distribution on the downstream side of separated side of the plate, the location of center of pressure is independent of all practical range of . The constant pressure distribution in the downstream side of plate leads to the center of pressure of this side of the plate to be always in the center of plate. Hence, the overall center of pressure of plate is only depending on the pressure distribution on the upstream side of plate. This was mentioned before in Apêndice 2, during the discussion about the hydrodynamic torsional spring effect.
53
0.6
Center of Pressure for σ=0 Center of Pressure for σ=0.2 Stagnation point for all σ
0.5
(µ/b), (c/b)
0.4 0.3 0.2 0.1 0 0
10
20
30
40 50 60 70 80 90 θ [deg.] Fig. 3-6 Location of stagnation point and center of pressure of flat plate [43]; is the position of stagnation point [43].
Based on the above discussion, the Kirchhoff’s classical formulation for the center of pressure in term of the width is (see Appendix B) c
3 cos b 4 4 sin
(3-21)
Fig. 3-7 shows a good agreement of (3-21) with numerical results of ANSYS CFX and experimental results of LOC, and also Flachsbart [14] and Tachikawa [40] for
54
< 55˚.
0.35
LOC CFX - CB=0.06
Center of pressure (c/b)
0.3
TACHIKAWA FLACHSBART
0.25
Kirchhoff's Classical Model
0.2 0.15 0.1 0.05 0 0
10
20
30
40
50
60
70
80
90
θ[deg.] Fig. 3-7 Comparison of position of the center of pressure with streamline theory (Eq. (3-21)) and ANSYS CFX and experimental results of LOC, Flachsbart [14] and Tachikawa [38].
The result suggests that the position of the center of pressure has a linear relation with the angle of attack. Hence, Eq. (3-21) could be expanded in Taylor’s series around
0˚ (
π/2).
The linear term is given by:
c b 3
1 4 4
(3-22)
Fig. 3-8 shows the comparison of the linear segmented results for the position of the center of pressure, found by streamline theory, and the numerical ANSYS CFX results.
55
Center of pressure (c/b)
0.15
0.12
(c/b) = 0.105θ
0.09 (c/b) = 0.1012θ 0.06
0.03 Streamline Theory ANSYS CFX - CB = 0.06
0 0
10
20
30
40 50 θ[deg.]
60
70
80
90
Fig. 3-8 The comparison of the linear segmented results of position of the center of pressure found by streamline theory and numerical ANSYS CFX results.
The moment coefficient could be obtained by considering Eq. (3-21) for center of pressure and Eq. (3-20) for normal coefficient. Hence, the moment coefficient formulation is:
3 sin 2 CM (1 ) 4 4 cos 2
(3-23)
As noted before, the center of pressure position is independent of , and also the normal coefficient formulation of streamline theory has a good accuracy with numerical and experimental results with
1.25. Fig. 3-9 shows the comparison of the moment coefficient
from the streamline theory (Eq. 3-23) with
1.0 and also
from LOC and numerical modeling of ANSYS CFX.
56
1.25 and experimental results
θ [ang.]
0 0
10
20
30
40
50
60
70
80
90
-0.05 -0.1
CM
-0.15 -0.2
-0.25 -0.3 -0.35 -0.4
LOC CFX - CB=0.06 Streamline Theory - σ=1.0 Streamline Theory - σ=1.25
Fig. 3-9 Comparison of moment coefficient of streamline theory (Eq. 3-23) with numerical results of ANSYS CFX and experiments at LOC.
1.0 and also
1.25 and
According to Fig. 3-9 the linear piecewise behavior coming from the streamline theory is in a good agreement with numerical and experimental results, for angles of attack smaller than 55 degree. This means that the streamline theory works very well in this range. Perhaps, the more important point of Fig. 3-9 is that the linear behavior that is observed in the moment coefficient curve has a technical support in this large range. Based on Fig. 3-9 and Eq. (3-23), the moment coefficient could be approximated by Taylor’s series for σ
1.0 and also σ
1.25 as below:
CM
13 0.16015 5 4 2
; for σ
1.0
(3-24)
CM
3 0.18791 4 2
; for
1.25
(3-25)
It should be noted that if σ
0.95, then the slope of the linear piecewise streamline theory
exactly match the hydrodynamic torsional spring, which CM = 0.1452. 57
3.5 Kutta’s Model for the Moment Coefficient
Now trying to get heuristic results for all
(0˚-90˚), this work presents another approach named
here Kutta’s model as explained below. For a plate, in the case of perpendicular angle of attack (
0˚), the drag coefficient as
mentioned in Apêndice 2, for Re > 104 is 2 ( C D. 2 ). If the fluid flow is parallel to the plate, the drag coefficient can be expressed as a function of the Reynolds number as given in [42] 0.455 1700 C D.II 2 2.58 log(Re ) Re
(3-26)
Between the two extremes ( CD. and C D. II ) the coefficient could be approximated by the analytical function [31]
C D C D. sin 2
(3-27)
This approximation is qualitatively acceptable with experimental and numerical results. Thus, the drag force can be approximated using an analytical function as [31]
D C D ( ) U 2 A C D. sin 2 U 2 A 2 2
(3-28)
The lifting force can be expressed in a similar way as [31] L C L ( )
2
U 2 A C L. max sin cos
2
U 2A
(3-29)
where CL.max represents the maximum value of the lift coefficient of the plate, which is based on Fig. (2-5) and (2-6). The value of CL.max is 1, and C D. 2C L. max . As mentioned in Apêndice 2, the location of the center of pressure, for small values is located at one forth of plate width from the stagnation point, which it has been validated by analytical calculations and also by several experiments. It is also clear that for symmetric profiles
58
like a flat plate, the moment has to be vanished while
0, so in this case, the center of pressure
should at origin. Hence, we have chosen [31]
(c / b) (1/4) cos
(3-30)
Using these approximations, for drag and lift forces and also the center of pressure, the moment represents as: l cos L cos D sin 4 b cos C L. max sin cos 2 2 sin 3 U 2 A 4 2 b cos sin C L. max cos 2 2 sin 2 U 2 A 4 2 b cos sin C L. max 1 sin 2 U 2 A 4 2
M 0 k ( ) N
(3-31)
In this text, this procedure of calculation of moment coefficient will be called here as the Kutta’s moment coefficient. Table 3-2 and also Fig. 3- 10 demonstrate the moment coefficients proposed by experimental results of LOC, streamline theory, Munk and Kutta. Experimental results of LOC CM 0.1452
0 75
C M 4 sin 2
75 90
Munk (Lift) C M 4 sin 2
All angles
Streamline Theory CM 0.1452 0.0047
All angles
Kutta
All angles 1 C L. max sin 2 2 cos 2 8 Table 3-2 The moment coefficients proposed by experiments, streamline theory, Munk and Kutta. CM
59
θ[deg.]
0 -0.1
0
10
20
30
40
50
60
70
80
90
-0.2 -0.3
CM
-0.4 -0.5 -0.6 -0.7 -0.8 -0.9 -1
LOC Streamline Theory - σ=1.0 Munk
CFX - CB=0.06 Kutta
Fig. 3-10 Comparison of the moment coefficients proposed by experimental results of LOC, streamline theory, Munk and Kutta.
It can be concluded that, the Munk (Lift) moment is accurate for calculation of the moment in incidence angle of 75
90 where the stall does not happen. The Streamline
theory formulation for larger angle (
< 75˚), is a very good approximation for moment
coefficient of a flat plate. Anyway, once the full range should be considered Kutta’s model could be used.
60
Apêndice 4 Theoretical Modeling of Flow Induced Fluttering and Autorotation In Apêndice 4, to gain further insight into the nature of the flow induced rotation phenomenon, a quasi-steady approximation is applied to model the dynamics of a flat plate free to rotate under interaction of a uniform flow. An ordinary differential equations with fluid moment contributions from added mass and moment coefficients will be presented. The fluid moment is expressed in terms of the kinematic variables of the static plate in different angles of attack. The trajectories of autorotation motion are modeled analytically and compared with free rotation experiments carried out in LOC’s flume.
4.1 Fluttering One of the observations of this research is to devise a quasi-steady modeling to match with the fluttering experimental results. As shown on Fig. 2-17, according to the tests in the water LOC’s flume and also experiments at wind channels by previous researchers [13,25], exists a maximum dimensionless moment of inertia (I*) below which, only fluttering occurs for any initial condition. Several experiments for fluttering motion of plates with different dimensionless moment of inertia and different ranges of velocity have been performed at LOC. Fig 4-1
61
exemplifies one fluttering time trace as recorded at LOC (The initial angular position is =0). As shown in Fig. 4-1(a), the time series resembles a beating behavior. However, it is important to notice that there are several low frequency responses and not only one, as confirmed by the frequency response in Fig. 4-1(b). 80
(a)
Rotation [deg.]
60 40 20 0 -20 -40 -60 -80 0
50
100
150
200
250
Time [s.]
Energy Spectrum [deg2 s]
16000
(b)
12000
8000
4000
0 0
0.15
0.3
0.45
0.6
Frequency [Hz.] Fig. 4-1 (a) Time series of fluttering motion of the hinged flat plate about the vertical axis submitted to a uniform current (Re=1.26 105), (b) The respective frequency domain response.
62
It should be mentioned that the pure beating motions occur when the natural frequency is close to the excitation frequency (see Fig. 4-2), and it is interesting to remind that as shown in Fig. 2-16, the fluttering of the plate is a result of the vortex shedding and the natural frequency of fluttering and vortex shedding frequency are almost identical.
Fig. 4-2 Undamped oscillation when the difference of the natural and exciting force frequency is small, and it results beating motion, 1 and 2 are the first and second waves frequency.
The experiments for fluttering motion in LOC show that the spectra for all Reynolds numbers have some other spikes (is shown in Fig. 4-1-a). The height of other spikes increase gradually till the main natural frequency spike which is comparably larger than other spikes. The existence of these spikes causes a type of chaotic beating which is different from a pure beating motion.
63
4.1.1 Natural Frequency of Fluttering Motion In fluttering, the plate oscillates periodically around a stable position with the plate normal to the free stream. The fluttering angular motion range is always larger than the stall angle ( 75˚
75˚). Hence, based on the above conclusions, a simple linear
model for the flow induced fluttering of a flat plate in a uniform current is suggested. It has only one degree of freedom and applying the angular moment theorem [28], the following Eq. (4-1) is suggested for the undamped case
( I A66 )
2 d 2 f CM SbU 2 dt 2
(4-1)
By using the linear segments of moment coefficient presented in Table 4-1 and dividing (4-1) by the added moment of inertia, the governing equation is given by
1 Iˆ ddt K 2
2
*
0
(4-2)
where
K *
64 C M 1 U 2 b2
(4-3)
By considering Eq. (4-3), the natural frequency for small angles of the flow induced rotation of a hinged flat plate is given by (4-4):
n U /b
8
CM 1 (1 A66* )
(4-4)
where CM1 may be obtained from Table 4-1 from the moment coefficient proposed by experimental results of LOC and by the streamline theory. Segment I: Proposed by LOC Experiments CM1
0.1452
Segment II: Streamline Theory with 1 13 0.16015 5 4 2
Table 4-1 Linear segments of moment coefficient.
64
Segment III: Streamline Theory with 1.25 3 0.18791 4 2
As predicted by (2-29) and also Eq. (4-4),
is linearly proportional to U and
inversely proportional to b. The comparison between experimental and mathematical results of natural frequency as a function of different velocities and for different plates is shown in Figs 4-3 to 4-6. Eq. (4-4) is very interesting since shows the role of the coefficient CM1 which is presented in Table 4-1. That is the restoring moment part of natural frequency. 2.4 2
ωn.(b/U)
1.6 1.2 0.8
LOC - I*=0.0012 - CB=0.142 Mathematical model by Segment.1
0.4
Mathematical model by Segment.2 Mathematical model by Segment.3
0 35000
45000
55000
65000
75000
Reynolds Number Fig. 4-3 Comparison between experimental results from LOC and Eq. (4-4), I*=0.0012, CB=0.142; Segment.1: CM1 = -0.1452, Segment.2: CM1 = -0.16015, Segment.3: CM1 = -0.18791.
65
2.4 2
ωn.(b/U)
1.6 1.2 0.8 LOC - I*=0.0038 - CB=0.214 Mathematical model by Segment.1
0.4
Mathematical model by Segment.2 Mathematical model by Segment.3 0 55000
70000
85000
100000
115000
Reynolds Number Fig. 4-4 Comparison between experimental results from LOC and Eq. (4-4), I*=0.0038, CB=0.214; Segment.1: CM1 = -0.1452, Segment.2: CM1 = -0.16015, Segment.3: CM1 = -0.18791. 2
ωn.(b/U)
1.6
1.2
0.8 LOC - I*=0.0047 - CB=0.214 Mathematical mode by Segment.1
0.4
Mathematical mode by Segment.2 Mathematical mode by Segment.3 0 50000
60000
70000 80000 90000 100000 110000 Reynolds Number Fig. 4-5 Comparison between experimental results from LOC and Eq. (4-4), I*=0.0047, CB=0.214; Segment.1: CM1 = -0.1452, Segment.2: CM1 = -0.16015, Segment.3: CM1 = -0.18791.
66
2.4 2
ωn.(b/U)
1.6 1.2 0.8 LOC - I*=0.0378 - CB=0.428 Mathematical model by Segment.1
0.4
Mathematical model by Segment.2 Mathematical model by Segment.3 0 50000
90000
130000
170000
210000
Reynolds Number Fig. 4-6 Comparison between experimental results from LOC and Eq. (4-4), I*=0.0378, CB=0.428; Segment.1: CM1 = -0.1452, Segment.2: CM1 = -0.16015, Segment.3: CM1 = -0.18791.
The results show that because of the blockage effect for the flat plate with I*=0.0378 and CB=0.428, the mathematical modeling of all different linear segments of moment coefficient, and experimental results are not close to each other. Actually the blockage effect results in an increase of local velocity. But for the other plates, the natural frequency calculated with the mathematical model based on experimental linear segment of moment coefficient and also mathematical model based on streamline theory linear segment of moment coefficient with σ
1, is almost equal to experimental results of vortex shedding frequency, what
leads to fluttering with beating characteristics.
67
4.2 Autorotation (Tumbling)
As mentioned in Apêndice 2, based on several trajectory tests for fluttering and autorotation motions, an important observation that show up is that there are different trajectories and rotational behavior, depending on the initial angle of attack at release. In practice, it was observed that when the plate was released at an angle less than stall angle (approximately 15˚
15˚) and if the dimensionless moment of inertia is
large enough, the autorotation will occur. On the other hand, if the plate is released from rest, at an angular position in which the flow was already stalled (approximately 75˚
75˚), the plate starts to fluttering around a stable position, normal to the
current. Therefore, the requirements for autorotation are a large enough moment of inertia to overcome hydrodynamic torsional spring and a proper initial condition (angular position or velocity). This is clearly showed in the bifurcation diagram of Fig. 2-17. In the water flume case, because of the large density of water in comparison to air, reaching a certain dimensionless moment of inertia the occurrence of the autorotation (tumbling) is difficult. For this purpose, as shown in Fig. 4-7, in order to increase the moment of inertia of the plate, a bar with length of one meter was installed on the top of plate. Also two masses with 1 Kg were added to system, which it could be placed in different levels of the bar to provide different moment of inertia.
68
Fig. 4-7 The experimental setup for autorotation motion.
Fig. 4-8 shows the time history of autorotation motion for four different Reynolds numbers. It should be noted that because of the symmetric shape of the flat plate, we can confined the angle of the rotation between π/2 to -π/2. The figures presented as examples which are registered for 10 seconds and as observed by increasing the Reynolds number, the energy level increases and the plate rotates faster. It is interesting to see in Fig 4-8 by increasing the Reynolds number, the periodic shape of the graphs gets more harmonic.
69
Angle [rad.]
π/2
0
5
-π/2
10 t
Re=44940
Angle [rad.]
π/2
0
5
-π/2
10 t
Re=58475
Angle [rad.]
π/2
0
5
10
t
-π/2 Re=71715
Angle [rad.]
π/2
5
0
10
t
-π/2 Re=99445
Fig. 4-8 Time history of autorotation motion for four different Reynolds numbers, I*=0.1508.
Table 4-2 presents the root mean square of the angle of the rotation for different Reynolds numbers, which also shows the effect of the increasing of the Reynolds
70
number. The root mean square for a harmonic function is
⁄√2, and for a linear
periodic function is 2⁄3 , where is amplitude of the periodic motion. Root Mean Square of Angle of Rotation X π 44940 0.8677 58475 0.843 71715 0.809 99445 0.785 Table 4-2 The root mean square of angle of rotation for different Reynolds numbers. Reynolds Number
As we see, in higher Reynolds numbers, the value of the root mean square is closer to the root mean square of harmonic sinusoidal motion, and in lower Reynolds number, it is closer to root mean square value of linear periodic functions.
4.2.1 Equation of Motion of Autorotation
A simple linear model for the flow induced rotation of a flat plate in a uniform current which has only one degree of freedom is suggested by applying the angular moment theorem [28]
1 ( I A66 ) f CM SbU 2 2
(4-5)
By incorporating dimensionless variables Iˆ I A66 and t * t (U b) , the equation of motion become
1 Iˆ dtd 64 C 2
*2
(4-6)
M
This equation can be solved numerically using Runge-Kutta fourth order method. The normal force coefficient and centre of pressure positions obtained for static plates at various angles of attack, described in Apêndices 2 and 3, can be used to obtain moment coefficients in the above equation. The numerical calculations were made to compare with the experimental trajectories for autorotation motion recorded at LOC.
71
4.2.1.1 Wu’s Streamline theory moment coefficient model
The equation of motion based on the streamline theory moment coefficient model is
1 Iˆ dtd 64 32 2
*2
sin 2 0 4 cos 2
(4-7)
Figs. 4-9 and 4-10 show an agreement of the autorotation trajectories of experiments at LOC and the numerical simulation based on (4-7). The Reynolds number and initial angular positions were kept around 9.95 x 104 and 0 0 respectively and
initial angular velocity is zero. 7
LOC Experiments - I*=0.15 Quasi-Steady Modeling - I*=0.15
Angle of Rotation [rad.]
6 5 4 3 2 1 0 0
0.5
1
1.5
2 2.5 3 3.5 4 Time [sec.] Fig. 4-9 The autorotation motion of plate with I*=0.15 and Re=9.95 x 104.
72
7 LOC Experiments - I*=0.25 Quasi-Steady Modeling - I*=0.25
Angle of Rotation [Rad.]
6 5 4 3 2 1 0 0
0.5
1
1.5 2 2.5 3 3.5 Time [sec.] Fig. 4-10 The autorotation motion of plate with I*=0.25 and Re=9.95 x 104.
The numerical calculations of Eq. (4-7) confirmed that the autorotation just occur with proper initial conditions and also when the moment of inertia is large enough.
4.2.1.2 Kutta’s moment coefficient model
According to the Kutta moment coefficient model, the equation of motion is represented as (4-8)
1 Iˆ dtd 64 18 C 2
*2
2 L. max sin 2 (2 cos ) 0
(4-8)
Figs 4-11 and 4-12 show the comparison of the autorotation trajectories of the experiments at LOC and the numerical simulation based on Eq. (4-8). The Reynolds Number and initial angular position the same as the other model were kept around 9.95 x 104 and 0 0 respectively.
Figures show that the numerical modeling simulation of autorotation motion is also in a good agreement with experimental results.
73
8 LOC Experiments - I*=0.1522
Angle of Rotation [rad.]
7
Quasi-Steady Modeling - I*=0.1522
6 5 4 3 2 1 0 0
0.5
1
1.5 2 Time [sec.]
2.5
3
3.5
Fig. 4-11 The autorotation motion of plate with I*=0.1522 and Re=9.95 x 104. 8 LOC Experiments - I*=0.25
Angle of Rotation [Rad.]
7
Quasi-Steady Modeling - I*=0.25
6 5 4 3 2 1 0 0
0.5
1
1.5 2 2.5 3 3.5 Time [sec.] Fig. 4-12 The autorotation motion of plate with I*=0.25 and Re=9.95 x 104.
74
4.2.1.3 LOC moment coefficient model
Finally with the moment coefficient proposed from the experimental results of LOC, the obtained equations of motion are shown at Table 4-3: Equation of Motion
1 Iˆ dtd 64 32 2
*2
Angle
sin 2
4 cos 2
0
1 Iˆ dtd 64 4 sin 2 0 2
*2
0 75
75 90
Table 4-3 Equations of motion based on moment coefficient proposed by experimental results of LOC.
Figs. 4-13 and 4-14 show the comparison of the autorotation trajectories recorded by experiments at LOC and the numerical simulation based on the equations of motion of Table 4-3. The Reynolds Number was kept around 9.95 x 104. The initial angular position in all Figures is 0 90 ( 0 0 ). 4
LOC Experiments - I*=0.1508
Angle of Rotatin [rad.]
3.5
Quasi-Steady Modeling - I*=0.1508
3 2.5 2 1.5 1 0.5 0 0
0.4
0.8 1.2 Time [sec.]
1.6
2
Fig. 4-13 The autorotation motion of plate with I*=0.1508 and Re=9.95 x 104.
75
4 LOC Experiments - I*=0.25
Angle of Rotation [rad.]
3.5
Quasi-Steady Modeling - I*=0.25
3 2.5 2 1.5 1 0.5 0 0
0.4
0.8 1.2 1.6 2 Time [sec.] Fig. 4-14 The autorotation motion of plate with I*=0.25 and Re=9.95 x 104.
As we expected the numerical modeling simulation of autorotation motion based on LOC moment coefficient model has a better agreement with experiments, in comparison to the other two models, because in the LOC moment coefficient model, the effect of lift moment in small angels has been considered.
76
Apêndice 5 Stability Analysis of Flow Induced Rotation To gain more insight into the nature of transition between fluttering and autorotation, the work considered different phenomenological models of the flow induced rotation of a flat plate hinged at center on different equations of motion that discussed in Apêndice 4. The aim of Apêndice 5 is to do a stability analysis on different dynamic models of flow induced rotation phenomenon.
5.1 Fixed Points The fixed points for differential equation models of motion are presented in Table 5-1: LOC moment coefficient model
1 Iˆ dtd 96 2
*2
sin 2 0 4 cos 2
1 Iˆ dtd 16sin 2 0 2
*2
Fixed points
0 75
( , ) ( n 2 , 0), n 0, 1, 2, ..
75 90
( , ) ( n 2 , 0), n 0, 1, 2, ..
Wu Streamline Theory moment coefficient model ( =1)
1 Iˆ dtd 96 2
*2
sin 2 0 4 cos 2
( , ) ( n 2 , 0), n 0, 1, 2, ..
Kutta’s moment coefficient model
1 Iˆ dtd 8 C 2
*2
L. max sin 2
( , ) ( n 2 , 0), n 0, 1, 2, ..
(2 cos2 ) 0
Table 5-1 Fixed points for differential equation models of motion.
77
For linearization of equation of motions of Table 5-1, to obtain a system of ODEs, by setting
,
and also considering the Taylor’s series. The linearized
equations in the neighborhood of the fixed points can be obtained as in Table 5-2: Linearized Equations of Motion Streamline theory moment coefficient model
Fixed Points ⁄ , , n=0, 2, 4, .. : , where 1/ 4 . hence, we
For fixed points: 192 / 1 have:
0
1 0
́
Where =0, det 4q 4 192 / 1 0 and ∆ express that, the critical points 2, 4,...are all Centers.
y1 y2 y 2 k * y1
⁄ , , n= , where
For fixed points: 192 / 1 we have:
0 ́
192 / 1 0, so we can nπ⁄2 , 0 , n=0,
, , .. : 1/ 4
.
hence,
1 0
Where =0, det 192 / 1 0 and ∆ 4q 4 192 / 1 0, so we can express that, the critical points nπ⁄2 , 0 , n= 1, 3,...are all Saddle points.
Munk (or Lift) theory moment coefficient model
⁄ , , n= , , .. : For fixed points: 16/ 1 . Hence, we have: 0 1 ́ 0 =0, det 16/ 1 0 Where and ∆ 4q 64/ 1 0, so we can express that, the critical points nπ⁄2 , 0 , n= 1, 3,...are all Saddle points.
y1 y2 y 2 k * y1
Kutta’s moment coefficient model
y1 y2 y 2 k * y1
⁄ , , n=0, 2, 4, .. : For fixed points: 16/ 1 . Hence, we have: 0 1 ́ 0 =0, det 16/ 1 0 Where and ∆ 4q 64/ 1 0, so we can express that, the critical points nπ⁄2 , 0 , n=0, 2, 4,...are all Centers.
⁄ , / 1
For fixed points: 16/ · .
, n=0, 2, 4, .. : . Hence, we have: 0 1 ́ 0 =0, det 16/ · Where / 1 0 and ∆ 4q 64/ · . / 1 0, so we can express that, the critical . points nπ⁄2 , 0 , n=0, 2, 4,...are all Centers. ⁄ , , n= , , .. : / 1 . Hence, we have: 0 1 ́ 0 =0, det 16/ · Where / 1 0 and ∆ 4q 64/ · . / 1 0, so we can express that, the critical . points nπ⁄2 , 0 , n= 1, 3,...are all Saddle points. For fixed points: 16/ ·
.
Table 5-2 Stability analysis of the linearized equations in the neighborhood of the fixed pints. 78
5.2 Conservative System and Phase Diagrams The Newton’s second law (
) is the source of many important second-order
systems. We can show that energy is conserved, as follow. Let potential energy, defined by
mx
⁄
denote the
. Then
dV 0 dx
(5-1)
By multiplying both sides by , we have
mx x
dV d 1 x 0 mx 2 V ( x) 0 dx dt 2
Hence, for a given solution
(5-2)
, the total energy is constant as a function of time:
1 E mx 2 V ( x) 2
(5-3)
The energy is often called a conserved quantity and these systems called conservative systems. The conservative energy systems for different differential equation models of flow induced rotation motion are presented in Table 5-3. Figs. 5-1 to 5-3 show trajectories for various values of E and for different equations of motions of Kutta, Wu streamline theory (
1) and LOC adjusted moment
coefficient model. These graphs continue periodically with period
to the left and right.
We can see some of the trajectories are ellipse-like and closed (fluttering motion) and some other are wavy (autorotation motion).
79
Kutta’s moment coefficient model
d 2 8 1 Iˆ *2 CL. max sin 2 (2 cos2 ) 0 dt
d 1 2 16 CL. max 2 1 4 cos cos 0 * 2 4 dt (1 Iˆ) 1 4 1 2 8 CL. max 2 cos cos E ˆ 4 2 (1 I )
Streamline theory moment coefficient model d 2 96 sin 2 0 1 Iˆ *2 dt 4 cos 2
192 4 d 1 2 ln(4 cos ) 0 2 * 2 ˆ dt (1 I ) (4 cos )
1 2 96 4 ln(4 cos ) E 2 ˆ 2 ( 4 cos ) (1 I ) Munk (Lift) theory moment coefficient model
1 Iˆ dtd 16 sin 2 0 2
*2
d 1 2 8 cos 2 0 * 2 ˆ dt (1 I ) 1 2 8 cos 2 E 2 (1 Iˆ) Table 5-3 The conservative energy systems for different differential equation models.
80
5.2.1 Kutta’s Moment Coefficient Model Fig. 5-1 is the phase diagram that we get from Kutta’s moment coefficient model. Based on Table 5-2, the direction of motion changes at the point: y2 0 then:
8 CL. max 2 1 4 cos cos E ˆ 4 (1 I ) If
(5-4)
y1 2 , then C E , where C (6 ) (CL.max /(1 Iˆ)). Hence, if
C E C , then plates reverse its direction for a y1 2 , and these values of E with E C the plate oscillates. This corresponds to the closed trajectories in the figure (fluttering). However, if E C then y2 0 is impossible and the plate makes a whirly motion that appears as wavy trajectories in the phase diagram (autorotation). Finally, the value E C corresponds to the two “separating trajectories” in Fig. 5-1 connecting the saddle points. 12
Fluttering Autorotation
E>C 6
E=C
0 -6
-3
0
3
EC
3.5
E=C
0 -5
0
5
EC
Autorotation
3.5
E=C
0 ‐5
0
5
E
