UNIVERSITY OF CALGARY. Base Region Topology of Turbulent Wake around Finite Wall-Mounted Cylinder with. Application of Low Order Flow Representation

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1 UNIVERSITY OF CALGARY Base Region Topology of Turbulent Wake around Finite Wall-Mounted Cylinder with Application of Low Order Flow Representation by Golriz Boorboor A THESIS SUBMITTED TO THE FACULTY OF GRADUATE STUDIES IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE GRADUATE PROGRAM IN MECHANICAL AND MANUFACTURING ENGINEERING CALGARY, ALBERTA AUGUST, 2014 Golriz Boorboor 2014

2 Abstract Turbulent flow topology around a rectangular cylinder with is characterized by either dipole or quadrupole mean wake depending on the turbulent boundary layer thickness. This study investigates the physical and mathematical mechanisms giving rise to the two distinct types of flow by considering the phase-averaged velocity field. It was found that the vortex stretching and tilting and the time-averaged representation of tilted vortex shedding are the key terms distinguishing between the dipole and quadrupole wake structures. The other objective is to develop a procedure to estimate the higher-rank low order flow based on Extended Proper Orthogonal Decomposition. Within this technique, the global velocity flow can be reconstructed with correlated surface pressure sensors to have a closer look at the mechanisms leading to dipole and quadrupole wakes. It was observed that optimum choice of sensor locations needs analytical considerations to improve the ability of pressure probes in capturing the coherent structures of interest. ii

3 Acknowledgements I wish first to express my gratitude to my Supervisor, Dr. Robert Martinuzzi, for his timely support and guidance through this project. Without his knowledge and assistance this study would not have been possible. I am heartily thankful to my family for their encouragement and unconditional support that enhanced my motivations to set higher targets. Also I am sincerely grateful to my colleagues for their help and foresight and the useful discussions that they provided during this course of work. iii

4 Table of Contents Abstract... ii Acknowledgements... iii Table of Contents... iv List of Tables... vi List of Figures and Illustrations... vii List of Symbols and Abbreviations... xi Chapter 1: Introduction Motivation and methodology Overview of the study... 3 Chapter 2: Analytical Background Formulation of the problem Governing equations Reynolds decomposition and triple decomposition Traditional phase-averaging Proper Orthogonal Decomposition-Extended Proper Orthogonal Decomposition Overview Proper Orthogonal Decomposition (POD) Extended Proper Orthogonal Decomposition (EPOD) Chapter 3: Sources of Data Experimental data Experimental setup Computational Fluid Dynamics (CFD) data Chapter 4: Base Region Vortex Topology of a Wall-Mounted Rectangular Cylinder Introduction to flow topology around truncated cylinders General characterization of the flow Time-averaged transport flow field Fluctuating flow field iv

5 4.5 Instantaneous coherent flow field Summary Chapter 5: Low Order Estimation of Turbulent Unsteady Flows POD representation of flow EPOD-based prediction Optimum location of transducers Summary Chapter 6: Concluding Remarks References v

6 List of Tables Table 5.1: Optimum multi-time delay parameters of EPOD using obstacle pressure sensors Table 5.2: Optimum multi-time delay parameters of EPOD using plate pressure sensors Table 5.3: Optimum multi-time delay parameters of EPOD using plate plus obstacle pressure sensors Table 5.4: Optimum spatial locations of pressure transducers vi

7 List of Figures and Illustrations Figure 2.1: Illustration of POD ability to represent the coherent vortex shedding terms as a function of time in amplitude and phase. The velocity field (around a rectangular cylinder obtained with PIV) is decomposed into steady and unsteady components, where further decomposition is taken to extract the coherent structures associated with vortex shedding modes of velocity Figure 3.1: Wind tunnel schematic and nomenclature Figure 3.2: AR=8, a) the natural and b) the tripped turbulent boundary layers of the on-coming flow Figure 3.3: AR=4, a) the natural and b) the tripped turbulent boundary layers of the on-coming flow Figure 3.4: The boundary layer profiles and root-mean-square stream wise velocity fluctuations for a) natural and b) tripped boundary layers in the case of Figure 3.5: The boundary layer profiles and root-mean-square stream wise velocity fluctuations for a) natural and b) tripped boundary layers in the case of Figure 4.1: a) dipole and b) quadrupole mean flow structures and vectors in the wake of natural and tripped BLs, respectively. Black lines indicate time-averaged vortical structures identified using criterion ( ). Color contours represent the time-averaged stream-wise vorticity Figure 4.2: Induced BLs at the obstacle free-end and base wall. Color contours indicate distribution of time-averaged stream-wise velocity Figure 4.3: Iso-surfaces of time-averaged vortical structures identified using criterion ( ) for a) NBL (dipole wake) b) TBL (quadrupole wake). Dashed lines indicate the edges of the cylinder Figure 4.4: Iso-surfaces of phase-averaged vortical structures ( ) identified using criterion ( ) for a) NBL (half-loop vertical structures) b) TBL (full-loop vertical structures). Dashed lines indicate the edges of the cylinder Figure 4.5: Time-averaged stream-wise vorticity distributed in the wake of a) NBL and b) TBL. Black labels state the core of tip and base vorticities vii

8 Figure 4.6: Time-averaged stream-wise velocity demonstrating the recirculation zone for a) NBL and b) TBL Figure 4.7: Non-dimensional circulation (circulation/ ) of the base vortex Figure 4.8: Non-dimensional net effect of a) vortex stretching and tilting induced by timeaveraged terms, b) time-averaged contribution of fluctuations and c) important terms downstream of the recirculation zone on the base vortex Figure 4.9: Reflection of horseshoe vortex on the cylinder wall. The contour is the time-averaged stream-wise vorticity for a) NBL and b) TBL Figure 4.10: Non-dimensional net effect on the base vortex for TBL Figure 4.11: Non-dimensional a) circulation of the base vortex, b) base vortex area, net effect of c), d) and e) on the base vortex Figure 4.12: Distribution of turbulent kinetic energy within the POD modes for a) NBL and b) TBL Figure 4.13: Power spectral density of the first 6 POD temporal coefficients for a) NBL and b) TBL Figure 4.14: POD spatial modes (x component of velocity) for a) NBL and b) TBL Figure 4.15: POD spatial modes (y component of velocity) for a) NBL and b) TBL Figure 4.16: Phase-averaged stream-wise vorticity ( ) for a) NBL and b) TBL Figure 4.17: Instantaneous span-wise vorticity reconstructed from the first 6 POD modes and streamlines for a) NBL and b) TBL for the same phase of shedding cycle Figure 4.18: Instantaneous span-wise vorticity reconstructed from the first 6 POD modes with streamlines for a) NBL and b) TBL for the same phase of shedding cycle Figure 5.1: POD representation of the velocity wake at. a) Kinetic energy distribution of POD modes, b) PSD function of POD temporal modes and c) POD spatial modes Figure 5.2: POD representation of the velocity wake at. a) Kinetic energy distribution of POD modes, b) PSD function of POD temporal modes and c) POD spatial modes Figure 5.3: POD representation of the velocity wake at. a) Kinetic energy distribution of POD modes, b) PSD function of POD temporal modes and c) POD spatial modes Figure 5.4: POD representation of the velocity wake at. a) Kinetic energy distribution of POD modes, b) PSD function of POD temporal modes and c) POD spatial modes viii

9 Figure 5.5: Demonstration of 90 0 phase shift between the pair modes of first harmonic pair. a) amplitudes of POD mode pair of first harmonic ( and ) representing their temporal evolution. b) phase portrait of and showing the characteristic doughnut pattern Figure 5.6: Demonstration of linear relation between the shift mode amplitude ( ) and sum of the first harmonic amplitudes ( ). Grey line shows the trend line of the scattered amplitude points Figure 5.7: POD representation of the pressure wake at the obstacle lateral walls. a) Kinetic energy distribution of POD modes, b) PSD function of POD temporal modes and c) POD spatial modes Figure 5.8: POD representation of the pressure wake at the mounted plate. a) Kinetic energy distribution of POD modes, b) PSD function of POD temporal modes and c) POD spatial modes Figure 5.9: POD representation of the pressure wake at the obstacle lateral walls and mounted plate. a) Kinetic energy distribution of POD modes, b) PSD function of POD temporal modes and c) POD spatial modes Figure 5.10: Overall residuals of a) u b) v and c) w as functions of multi-time delay parameter for prediction based on pressure information at the obstacle surface Figure 5.11: Overall residuals of a) u b) v and c) w functions of multi-time delay parameter for prediction based on pressure information at the mounted plate Figure 5.12: Overall residuals of a) u b) v and c) w functions of multi-time delay parameter for prediction based on pressure information at the obstacle surface and mounted plate Figure 5.13: Residuals of a) u b) v and c) w for the optimum multi-time delay parameter Figure 5.14: POD representation of the pressure wake at the obstacle lateral walls with application of multi-time delay approach ( ). a) Kinetic energy distribution of POD modes, b) PSD function of POD temporal modes and c) POD spatial modes Figure 5.15: POD representation of the pressure wake at the plate with application of multi-time delay approach ( ). a) Kinetic energy distribution of POD modes, b) PSD function of POD temporal modes and c) POD spatial modes Figure 5.16: EPOD representation of the predicted velocity wake at based on pressure sensors at the obstacle surface. a) Kinetic energy distribution of EPOD modes, b) PSD function of EPOD temporal modes and c) EPOD spatial modes ix

10 Figure 5.17: EPOD representation of the predicted velocity wake at based on pressure sensors at the mounted plate. a) Kinetic energy distribution of EPOD modes, b) PSD function of EPOD temporal modes and c) EPOD spatial modes Figure 5.18: Correlation between the surface pressure data and the POD temporal coefficients of velocity field Figure 5.19: Correlation between the surface pressure data and the POD temporal coefficients of velocity field Figure 5.20: Correlation between the surface pressure data and the POD temporal coefficients of velocity field Figure 5.21: Correlation between the surface pressure data and the POD temporal coefficients of velocity field x

11 List of Symbols and Abbreviations Abbreviation PIV POD EPOD CFD CSs TPA LES NSs RANs CWT SVD AR BL NBL TBL HSV Definition particle image velocimetry proper orthogonal decomposition extended proper orthogonal decomposition computational fluid dynamics coherent structures traditional phase-averaging large eddy simulation Navier Stokes equations Reynolds averaged Navier Stokes equations continuous wavelet transform singular value decomposition aspect ratio boundary layer natural boundary layer tripped boundary layer horseshoe vortex Symbol Definition density obstacle diameter obstacle height Reynold number Strouhal number frequency non-dimensional frequency kinematic viscosity stream-wise coordinate vertical coordinate xi

12 span-wise coordinate stream-wise component of velocity vertical component of velocity span-wise component of velocity time-averaged velocity velocity fluctuations coherent velocity fluctuations incoherent velocity fluctuations pressure vorticity vorticity fluctuations time space phase of oscillatory motion kinetic energy content of the k th POD mode temporal coefficient of the k th POD mode spatial eigenfunction of the k th POD mode boundary layer thickness Kronecker delta number of spatial points number of temporal points multi-time delay parameter time-delay vortex identification criterion circulation iii

13 Chapter 1: Introduction This work focuses on extracting the flow physics characterizing the junction region of the turbulent wake past a finite-length square-cross-section cylinder partially immersed in natural and tripped turbulent boundary layers. In particular the effect of varying boundary layer thickness on the time-averaged structures and large-scale coherent dynamics of the two cases is investigated based on the phase-averaged velocity data collected using Particle Image Velocimetry (PIV) experimental data. The Proper Orthogonal Decomposition (POD) and Extended Proper Orthogonal Decomposition (EPOD) are employed in the case of Computational Fluid Dynamics (CFD) data as a mean to bring about clearer insight into the flow structure and approaches to extend the methodology for future studies. This is achieved by providing a detailed procedure to predict the low order representation of the turbulent velocity field with correlated pressure information distributed on the obstacle surface and mounted plate within the context of remote-sensing based measurements. 1.1 Motivation and methodology Velocity flow fields at moderate to high Reynolds number form a fully turbulent wake due to the growth of instabilities. The flow structure downstream of cylinders is influenced by the separation of the boundary layer on the walls and upstream of the rigid body. The unsteady rollup of these vortices as a result of Kelvin-Helmholtz instabilities gives rise to the so called von Kármán vortices forming a von Kármán vortex street of counter-rotating vortices shed periodically from either sides of the obstacle. In the case of finite wall-mounted cylinders, the turbulent flow field is strongly three dimensional due to the tip and base effects. This behavior is more complicated than for two-dimensional geometries due to the complex interactions between quasi-periodic vortex shedding and the induced structures at the tip and base domains. Within the turbulent flows, fluctuations are driven over a wide range of scales and subsequently smaller-scale terms obscure the behaviour of larger-scale structures. Thus, there is 1

14 a demand for robust flow decomposition approaches to distinguish and extract the important dynamics (coherent structures) of the wake in order to provide a proper tool for analyzing and understanding the complexity of three-dimensional turbulent wakes. In the current work, Traditional Phase-Averaging (TPA) and POD are used for this purpose illustrating the triple decomposition method where the time-dependent fluctuation terms are also decomposed into the deterministic large-scale coherent and small-scale random terms. Therefore, the triple decomposition allows for a further modification to the well-known Reynolds decomposition approach, which provides a double decomposition of the spatio-temporal velocity field into the spatio time-averaged and spatio-temporal unsteady components. In the present study, the time-averaged field and large-scale coherent structures around truncated rectangular cylinders with are investigated using the TPA technique where raw data was obtained by conducting two-component planar PIV measurements. In particular, the main focus of study is on the effect of boundary layer thickness on the base region structures and the interaction with shed vortices within the junction domain. These interactions give rise to two distinct wake topologies causing different distributions of time-averaged stream-wise vorticity (dipole mean structures associated with half-loop vortex shedding and quadrupole mean structures associated with full-loop vortex shedding) for different boundary layer thicknesses. EPOD is employed on the data obtained from CFD for the flow around the finite rectangular cylinder with as another contribution leading to better understanding of the wake. Although the technique of TPA is a useful mean to model the most deterministic vortexshedding dynamics, there are still unresolved terms in representing the velocity fields. These unresolved terms arise as a consequence of not capturing the vortex shedding variations in amplitude from cycle to cycle and the missing dynamics originated from the three dimensionality effects within the tip and base regions. In order to compensate for the limitations with TPA, EPOD can be used to reconstruct the low order representation of the turbulent wake in order to i) account for more complicated coherent interactions, where a dominant frequency does not exist, ii) capture the amplitude variations of the harmonics based on the information within the pressure signals and iii) add additional coherent terms, i.e. higher harmonics, to the resolved dynamics to derive a more representative model. In this approach, the analytical basis allows for making a link between the state spaces of simultaneous pressure measurements and planar velocity fields. Through the reconstruction, i) POD is applied to extract the most energetic 2

15 realizations of the field through a number of pressure sensors. The corresponding modes are calculated in the form of an optimal set of eigenfunctions and increasing levels of approximation are obtained for global state estimation by using more modes considering the optimality of kinetic energy. ii) EPOD is further performed to predict the correlated events of each planar velocity wake with the coherent pressure dynamics. For that purpose, the information of spatiotemporal pressure signals from appropriately placed sensors, where the correlated fluctuation levels are high with good signal-to-noise ratio, is necessary. In order to facilitate the prediction made by EPOD, the optimum locations of pressure transducers should be known. These locations correspond to the highest spatially correlated points with the desirable velocity coherent dynamics to be resolved. This work will also provide a procedure to determine the optimum placement of sensors based on the characteristic flow structures. These quantitative results obtained from post-processing of CFD data are to be extrapolated for the future experimental (stereo PIV) approaches to enhance the capability of remote sensing based predictions to estimate the remote quantity of interest and subsequently to reconstruct the more detailed low order representation of the velocity field in a real situation. Finite-length cylinders in high Reynolds flows are found in a wide variety of engineering applications such as wind patterns behind buildings and terrains, wind turbine blade loading and performance of mixing characteristics in industrial burners. The vast cases where unsteady flow occurs lead to the fundamental study to extract a higher degree of flow physics. 1.2 Overview of the study The general formulation of the problem within the context of classical fluid dynamics and the mathematical description concerning the flow decomposition approaches, TPA based reconstruction of the velocity wake and predicting the turbulent flow field through POD and EPOD techniques accompanied by their analytical literature review are given is Chapter 2. Chapter 3 provides details of data sources from experiments (PIV) and numerical (CFD) simulations. In Chapter 4, the literature review on the flow structures and wake topologies of finite wall-mounted cylinders are provided and then the PIV data analysis and the condition giving rise to different topologies of the time-averaged and time-dependent parts of the flow for different boundary layer thicknesses are studied. Herein, observations are made considering the 3

16 dominant terms driving the transport equation of time-averaged stream-wise vorticity. In Chapter 5, the correlation between the three-component planar velocity fields within the downstream wake of the obstacle with the pressure transducers placed on the plate and obstacle wall is demonstrated based on CFD data obtained from Large Eddy Simulation (LES). This sense of correlation between pressure and velocity dynamics enables the prediction of the velocity wake with the pressure transducers mounted on either the plate, obstacle wall or the combination of plate and obstacle wall. Moreover, the optimum locations where pressure measurements should be performed in order to capture the main physical states in the flow are quantitatively derived. This provides useful information, which is to enhance the further EPOD predictions applied to the experimental approach. Lastly, the main arguments and new findings are summarized while the connection between the application of results obtained from post-processing the CFD data and future PIV measurements is discussed in Chapter 6. 4

17 Chapter 2: Analytical Background This chapter describes the primary equations and techniques employed in Chapter 4 and Chapter 5 to formulate the mathematical framework to extract a physical representation of the significant unsteady dynamics. It starts with introducing the general considerations of the problem and the governing equations within the context of classical fluid dynamics. This is followed by a discussion of the decomposition approaches, underlying the differences between the approaches and corresponding applications in the field of turbulence. The general process of reconstructing the global coherent contributions to the turbulent flow associated with the shedding structures obtained from Traditional Phase-Averaging (TPA) is reviewed. Lastly, Proper Orthogonal Decomposition (POD) is described in detail as an approximation method, which enables the estimation of the lower-rank turbulent field by isolating the most energetic coherent dynamics. The Extended Proper Orthogonal Decomposition (EPOD) technique is presented as a remote sensing based prediction, which provides a rich low order representation of the remote flow dynamics by accounting for detailed flow realizations and additional coherent terms compared with TPA. 2.1 Formulation of the problem Governing equations The equations of fluid motion are generally described with the equations of conservation of laws: i.e. conservation of mass, momentum and energy leading to continuity, momentum and energy transport equations, respectively. The numerical analysis of the flow field taken in this study is under the assumption of incompressibility and Newtonian fluid. By applying the corresponding modifications to the equation of motions, the modified Navier Stokes (NSs) equations are obtained representing a non-linear integral or differential form. The NSs derivation is made for instantaneous realizations and thus represents the Lagrangian frame of reference. 5

18 In order to enable the comparison between numerous terms within the NSs, the nondimensionalized analysis is put forward. The non-dimensional equations are obtained by scaling using characteristic quantities. Here, length is scaled by the cylinder diameter,, velocity by the free-stream velocity,, time by and pressure with as it can be seen from Eq. (2.1a) and Eq. (2.1b). This scaling also allows for the order of magnitude analysis presented in Chapter 4. deonotes the Reynolds number based on the obstacle diameter and is equal to, where is the kinematic viscosity. Here, the velocity is a three-dimensional vector usually expressed in Cartesian components. The other fundamentally important equation is the vorticity transport, which describes the local evolution of the vorticity,, and is defined as the curl of velocity, Vorticity transport is widely used in this work to analyze the rotational structures in terms of vorticities of vortex tubes and sheets within the time-averaged and unsteady parts of the flow. The non-dimensionalized form of the vorticity transport equation reads, Reynolds decomposition and triple decomposition The Reynolds decomposition is a traditional technique of decomposing a time varying function into the time-averaged and time-dependent components. This requires a temporal signal of the function, which has to be expressed on average over the desirable but convergent time domain (i.e. convergent long time series to account for the proper time-averaged realization) and the 6

19 deviation between the instantaneous and the time-averaged fields (fluctuations). This approach was initially proposed by Reynolds (1895), where he described the equation of fluid motions for viscous flows by applying the Reynolds decomposition. Suppose any spatio-temporal function such as or where denotes the three-dimensional spatial coordinate system (Eulerian location) and represents the temporal coordinate. The Reynolds decomposition can be stated as, ( ) ( ) ( ) ( ) ( ) ( ) where the time-averaged field is defined as, ( ) and by definition, ( ) ( ) The time-variant term on the right side of Eq. (2.4) represents the fluctuations of turbulent flow about zero, and thus is averaged out when integrated over the whole time domain. Applying Reynolds decomposition and taking the time-average of the governing equations yield the well-known Reynolds averaged NSs (RANS) equations, which in Einstein notation reads, ( ) The RANS equations represent the average realization of the turbulent fluctuation,, and are equivalent to averaging over the entire spectrum (frequency) domain. These describe well turbulent flows in which the fluctuation energy is uniformly distributed over all nondissipative scales of motion. However, in the case of most turbulent flows (i.e. bluff body flows, separated flows and boundary layers), turbulence is characterized by elevated energy 7

20 concentration at certain scales of energy. Therefore, it is necessary to distinguish the large-scale CSs from the small-scale incoherent components to have an improved tool for flow analysis of important behaviours. For this purpose, Hussain and Reynolds (1970), Hussain (1983), (1986) proposed a triple decomposition, ( ) ( ) ( ) ( ) where represents the larger-scale CSs and denotes the smaller-scale incoherent contributions to the fluctuating field. Coherent terms describe spatially correlated motion of rotational fluid ( ), which are convected and spatially connected volume of fluid motion. Generally, vortices represent an important class of CSs. CSs contributing to spectral concentrations of energy (i.e. statistically significant and non-random) are deterministic in nature (For example, quasi-periodic shed vortices behind the bluff bodies are often the dominant contribution to the energy spectrum), whereas, incoherent fluctuations are typically not spatially correlated and are treated as random. 2.2 Traditional phase-averaging One of the triple decomposition-based techniques for extracting the coherent dynamics from the turbulent velocity field is established through the method of TPA. TPA is executed to reconstruct the low order representation of a typical cycle of the three-dimensional flow by accounting for the most dominant quasi-periodic behaviour of the unsteady part of turbulence raised from vortex shedding instabilities. The dynamics of shed vortices contain a high amount of turbulent kinetic energy compared to the rest of fluctuation components (i.e. in the case of bluff bodies). Therefore, TPA is employed to extract the energetic shed vortices to resolve the most representative part of the fluctuating field. The modified definition of triple decomposition in the case of TPA considers the turbulent flow field to be decomposed into the phase-averaged term and incoherent fluctuations (Hussain and Reynolds, 1970) as shown in Eq. (2.8), ( ) ( ) ( ) 8

21 where represents the phase of oscillatory shed vortices within an averaged shedding cycle. A distinct phase,, can be assigned to each measured time step and ensemble averaging at constant phase of ( ) can be performed to find ( ). The first term on the right side of the Eq. (2.8) (phase-averaged field) represents the time-averaged wake added to the resolved coherent part of the fluctuations as shown in the below equation. ( ) are the remaining components that could not be resolved with TPA, i.e. higher harmonics and stochastic incoherent variations, which also describes the deviation between the instantaneous field and the phaseaveraged field. ( ) ( ) ( ) The data set used in Chapter 4 is obtained from two-component planar PIV measurements and post-processed with TPA to provide a physical representation of the phaseaveraged field. TPA, when implemented with PIV measurements allows for synchronization of numerous planar velocity fields, which is possible through considering a global reference phase. For this purpose, the pressure information containing state-space coherent dynamics obtained from mounted pressure transducers on the obstacle wall or/and plate are acquired simultaneously with each PIV planar measurements (horizontal and vertical velocity planes are captured in order to represent the three-cartesian-components of velocity for the global reconstruction). The oscillatory surface pressure enables the indication of three-dimensional characteristics of the flow due to the high degree of correlation between coherent dynamics of the pressure and the velocity wakes. Moreover, it carries the advantage of being non-intrusive as a remote-sensing approach. The pressure spatio-temporal signal is analyzed to extract the instantaneous phases of vortex shedding dynamics from the total fluctuations by using the Continuous Wavelet transform (CWT). This is done by filtering out the unwanted dynamics by passing the pressure signal through CWT function associated with the centre frequency equal to that of shedding structures and indicating a band-width parameter to account for the minor variations of frequency over different shedding cycles. Then computed phases of the quasi-periodic pressure fluctuations are interpolated onto the planar velocity fields while each shedding cycle is divided into a number of 9

22 bins discretizing the phase of a shedding period and the ensemble averaging of the velocity field at a constant value of phase is obtained over different shedding cycles. The choice for number of bins is based on achieving a compromise between phase resolution and statistical convergence. The TPA representations of transport equations are derived by applying the triple decomposition to the NSs, which returns the equations of phase-averaged field and the unresolved fluctuations shown below with tensor notation (Hussain, 1983), The transport vorticity equations of phase-averaged and unresolved fluctuations read, 10

23 2.3 Proper Orthogonal Decomposition-Extended Proper Orthogonal Decomposition Overview The representation of dominant dynamics can be improved through POD analysis, in which the basic frameworks allow for some modification of turbulent low order modeling compared to TPA. The most significant drawback of TPA is that it only resolves the quasi-periodic von Kármán vortices amongst all the coherent motions while additional energetic coherent terms can be educed and further analyzed to yield the richer representation of deterministic part of the fluctuating field by employing the POD approach. Moreover, the POD estimation of turbulent field illustrates the flow behaviour over the entire time of measurement and hence there is no limitation in describing the motions within only one average cycle as it is in TPA. This improvement with POD adds extra insight to the flow, such as information containing the amplitude modulation of shed vortices over different shedding cycles. POD is one of the most applicable techniques constituting a linear procedure, which extracts the low order representation of the high dimensional data (i.e. large amount of data) associated with non-linear phenomena. In the case of turbulent flow, POD can be employed to calculate a set of orthogonal basis based on an eigenvalue problem within the energy content to represent the most probable realization of the input data. Corresponding modes, therefore, account for the most energetic events within the dynamics of the turbulent flow. The idea of POD was developed by Lumley (1967) as a solution to extract and distinguish the large-scale CSs from the small-scale random fluctuations. The dominant behaviour of a turbulent flow is due to the passage of the most energetic CSs. As CSs describe connected regions of fluid containing high levels of vorticity, and as corollary, the velocity field, which in these regions is spatially correlated, is exploited in POD. The spatial correlation matrix is decomposed to yield the spatial eigenfunctions multiplied by temporal coefficients corresponding to the instantaneous contributions to the flow field. By definition, the POD modes are orthogonal, such that those allow reconstruction of the flow through a simple linear combination of a finite number of POD modes (where the number of modes can be determined based on a criterion of energy optimization). 11

24 An extension of POD technique, EPOD, consists in predicting the coherent part of the turbulent field from the flow behavior observed in a spatially remote (separate) section of the flow field as discussed by Boree (2003), Durgesh and Naughton (2010). In these studies, EPOD is introduced as a mean to predict big eddy structures in real-time using data from a small number of sensors at discrete locations. This remote sensing based prediction is possible through correlating within a domain any physical quantity, for example pressure, to another, for example velocity using suitable basis functions. In the case of velocity data obtained from PIV, the low order modeling via EPOD is proceed through i) predicting the correlated PIV planar fields with a set of unconditional sensors (i.e. pressure transducers) as a lower rank approximation by isolating the CSs from random variations that do not have a deterministic property. And ii) synchronizing the conditional PIV planar measurements for a global volumetric reconstruction by employing the indicated reference of temporal evolution of flow important dynamics obtained from the remote sensors. In this study, the first objective is to develop an effective and computationally economical statistical tool to predict the important events of the three-components planar velocity fields obtained from detailed Computational Fluid Dynamics (CFD) simulations. The second goal is to determine the optimum and highly correlated spatial locations of the obstacle wall and plate with the most important dynamics of the velocity fields to mount pressure transducers on. Corresponding results are to be extrapolated to the experimental approach to enhance the prediction of three-component stereo-piv velocity planes leading to the reconstruction of global field (Chapter 5). Following section describes the underlying mathematical behaviour of POD and EPOD techniques, which are essential to be understood in order to efficiently implement the above goals Proper Orthogonal Decomposition (POD) A definition based on large-scale spatio-temporal coherence requires the analysis of flow patterns in space and time. POD, as an approximation method, identifies and analyzes the spatial and dynamical properties of CSs and constructs low order turbulent representation that exhibit most of the coherent properties of the flow. For this purpose, an approximation of a fluctuating 12

25 flow field (here, as a function of time, t and space, ) is given by a finite-sum in spatial and temporal variable-separated basis (Holmes et al., 1996 & Chatterjee, 2000), ( ) ( ) where approximation of the original flow field is given by, ( ) ( ) ( ) The approximation is made for K number of coherent realizations of the field and one would suggest that the linear combination of temporal coefficients, and spatial eigenfunctions, ( ) results in the exact solution of the field, as K if the orthonormality condition of spatial basis functions (Eq. (2.16)) is taken. This guarantees that they are mutually uncorrelated and that there exists only one ( ) for each choice of ( ) ( ) where denotes the Kronecker delta, { Then, ( ) ( ) To solve for the approximation problem (Eq. (2.14)), we should seek for a set of optimal orthonormal basis functions, solution in a least square sense, ( ) such that it yields the best equivalent minimization 13

26 ( ) ( ( ) ) and state the inner product (also known as cross-correlation) and norm associated with the Hilbert space 1. The original fluctuating field obtained from PIV measurements or numerical simulations consists of data representing the flow information at discrete temporal and spatial points rather than carrying a continuous function 2. Therefore, the so called "snapshot data matrix", Eq. (2.19), is built to arrange the data in a useful format for solving the minimization problem, ( ) and denote the number of discrete points in spatial and temporal domains, respectively. The solution of the minimization problem (Eq. (2.18)) according to the Eckart-Young theorem 3 (Higham, 1989) is then given by the truncated (e.g. rank-k) Singular Value Decomposition (SVD) of the snapshot data matrix,. By definition, SVD representation of the snapshot data matrix reads 4 (Golub and Van Loan, 1990), U and V are unitary spatial and temporal matrices with dimensions of and, respectively, carrying singular vectors of rectangular matrix, and denotes the diagonal matrix including singular values of the matrix. 1 Hilbert ( ) space defines the square-integrable real or complex valued functions equipped with algebra operators (e.g norm and inner product) 2 Since it is the case here, the derivation of POD and EPOD equations is conducted in spatial and temporal discrete domains. 3 This theorem establishes a relationship between the rank-k of the approximant, and the (k + 1)th largest singular value of A. 4 denotes the conjugate transpose of V. 14

27 Now the eigenvalue problem should be executed to find the singular values and vectors of matrix, (Cordiera & Bergmann, 2002). For this reason, the two-point correlation tensor is constructed by calculating or depending on which snapshot or classical POD method is most appropriate. In this study, a snapshot POD (Sirovich, 1987) is chosen for further analysis. By applying the snapshot technique, matrix states the two-point temporal correlation tensor, Eq. (2.21), which simply indicates the correlation between each pair of temporal points while the average operator is considered to be taken over the space domain (rather than demonstrating the two-point spatial correlation tensor associated with the average implemented over the temporal domain in the classical POD approach). This reduces the computational effort while holding the numerical precision, which leads to the view supporting the snapshot POD as a more practical and computationally efficient technique to analyze data obtained from PIV or numerical simulations, where high spatial and moderate temporal resolutions are found. ( ). and return the eigenvectors and eigenvalues of matrix. The relation between snapshot POD and SVD is described through the connection of POD temporal coefficients to the eigenvectors of the equivalent eigenvalue problem. By definition, a basis function representing the coherent realization of the original field should exhibit the largest mean square projection on the observation ( ) (where is any realization of the field), which leads to the following maximization problem (Holmes et al., 1996) equivalent to the maximization problem shown in Eq. (2.18), ( ) ( ) where is evaluated as the spatial average operator specified earlier in the Hilbert space. Now by introducing the two-point temporal correlation tensor,, the eigenvalue problem of Eq. (2.23) is obtained to solve for POD eigenvalues and eigenvectors following Spectral theory (Riesz and Nagy, 1955 & Courant and Hilbert, 1953), which guarantees that the 15

28 maximization problem holds for the largest eigenvalue of the problem, adjoint and non-negative operator of the two-point correlation tensor, under the linear, self which is formulated with respect to the eigenvalue problem represented in Eq. (2.21) and represents the collection of temporal eigenvectors, [ ] It can also be proved that the temporal coefficients, are uncorrelated (orthogonal) and their mean square values represents the POD eigenvalues by using a proper normalization factor, ( ) and correspond to the th and th order of POD modes. In the context of turbulence and if ( ) denotes a velocity field, each eigenvalue indicates the amount of kinetic energy associated with the corresponding POD eigenfunction. POD spatial basis functions are then computed from Eq. (2.24) extended to the projection of onto by using the orthogonality property of the temporal coefficients, ( ) Therefore, is associated with the largest eigenvalue of the problem, and subsequently represents the largest mean square projection on the observation ( ), which satisfies the Eq. (2.22). 16

29 Then by combining the orthogonal temporal coefficients and orthonormal spatial eigenfunctions in a linear manner for rank-k of approximation, one can obtain the low order estimation of the fluctuating field represented in Eq. (2.14) Extended Proper Orthogonal Decomposition (EPOD) EPOD is introduced by Boree (2003) as a tool to analyse correlated events in turbulent flows. The same methodology is used in this study, where EPOD is used as an approach to predict any correlated physical quantity within the domain of with the same or another physical quantity within a domain of, where can be a subdomain of or a separate flow section. Corresponding analysis enables reconstructing the low order representation of the flow field as optimality of kinetic energy is considered. Prediction is based on the definition of CSs dominating the dynamics over the entire flow field such that their deterministic property executes the same coherent events between and domains (i.e. CSs execute same foot print in both pressure and velocity domains). This provides the basis to predict the correlated conditional quantity of interest (i.e. velocity) indicated by ( ) with the coherent characteristics of unconditional measurements (i.e. pressure) indicated by ( ). The first step is applying POD to identify and formulate the most energetic coherent structures within the events of ( ) in, which as an extension of Eq. (2.14) for rank-k approximation reads, ( ) ( ) It should be noted that ( ) indicates the fluctuation components of the original field, ( ) The extended POD modes indicating the spatial and temporal modes of ( obtained as follows, ) are then ( ) ( ) 17

30 ( ) ( ) Then, the low order prediction of ( ) reads, ( ) ( ) By adding the time-averaged realization to the approximated fluctuation terms, one obtains the rank-k predicted representation of the instantaneous flow field, ( ) ( ) ( ) The low order prediction obtained from EPOD analysis carries a number of advantages compared to that of TPA. As it was discussed earlier, a harmonic pair of vortex shedding dominates the dynamics of CSs and therefore is isolated by POD as the most energetic realizations contributing the most to the mean behaviour of the flow field. The amplitude associated with the oscillatory motion of these modes varies in time while the information related to the time-dependent amplitude is lost through TPA but captured by implementing POD and subsequently conserved within EPOD temporal modes (Fig. 2.1). This is due to the fact that, in EPOD, the resolved dynamics are presented for the whole time of the measurement rather than an averaged cycle that is the case in TPA and therefore the variations in amplitude from cycle to cycle can be represented. The other significance is the ability of POD to represent additional coherent terms such as the so called "shift mode" (see section 5.1), "higher harmonics" and modes originating from the complex interactions between vortices (i.e within the junction and free end regions of finite-length cylinders) when higher-rank approximations are considered. 18

31 Figure 2.1: Illustration of POD ability to represent the coherent vortex shedding terms as a function of time in amplitude and phase. The velocity field (around a rectangular cylinder obtained with PIV) is decomposed into steady (in black) and unsteady (in blue) components, where further decomposition is taken to extract the coherent structures associated with vortex shedding modes of velocity (in red). The other objective of this study is to find a quantitative representation of spatial locations for mounting pressure transducers. The idea behind this is to properly locate a number of pressure sensors so that the highest correlation between the coherent dynamics of pressure and velocity fields is captured. For that reason, the most energetic temporal coefficients of the velocity, are obtained though the POD decomposition of the velocity field and the projection onto the pressure data distributed over the domain of interest yields the correlated spatial basis of pressure with the temporal evolution of velocity modes, ( ) ( ) where the corresponding of the maximum of ( ) indicates the highest correlated spatial locations within the pressure domain. 19

32 Multi-time delay measurements One of the analytical improvements to the EPOD technique is possible through conducting the multi-time delay analysis. In order to enhance the prediction, multi-time delay measurement should be replaced with the single-time approach and applied to EPOD calculations. In both approaches, a conditional event (i.e. velocity obtained from PIV measurements) is estimated using the information of an unconditional event (i.e. pressure transducers measurements), while in single-time, measurements of the unconditional event are taken at a single-time and in multitime delay method, the time delays between the events in pressure and velocity domains are considered. Failure of the single-time approach is driven by the phase-lag between the conditional and unconditional events and therefor the time-dependent coefficients cannot be estimated accurately. Let s say if there s a ±90 0 phase difference between the quasi-periodic pressure signals and the time-dependent coefficients of the velocity dynamics, then the crosscorrelation coefficient between the two signals is zero and the EPOD would be unable to estimate the conditional events properly. To avoid this, the phase-lag should be artificially removed by executing the multi-time-delay parameter, into the calculations, which introduces a time-delay at which the optimum cross-correlation is obtained. Consider the case, in which velocity flow field is estimated using the information of a set of pressure data measured simultaneously and hence the snapshot-data matrix introduced in Eq. (2.20) represents pressure information at discrete temporal and spatial points. In the single-time approach, events contained within the pressure field are taken at a single-time leading to the single-time arrangement of the spatio-temporal pressure matrix (Eq. (2.32)), which may fail due to the potential phase-lag between the events of pressure and velocity domains as discussed earlier. 20

33 The modifications established in multi-time delay measurements considers these timedelays for each spatial point by using the information from past and future times extended from the time of measurement indicating the pressure temporal evolution. By applying this method, the highest correlations between the time-varying coefficients of pressure and velocity fields are found and the corresponding value of multi-time delay is used for further calculations, where corresponds to the time delay parameter indicating the optimum time-delay range. The new matrix is constructed by including the information from " " time steps before and after the time of estimation of. Therefore, for each pressure used in estimations, a number of equations are generated equal to the number of time delays included (Durgesh and Naughton, 2010), which in total reads. As is seen in Eq. (2.33), artificial sensor information representing a shift in time is added to the snapshot-data matrix to account for the time-delay and compensate the probable phase-lag between the temporal coefficients of pressure and velocity coherent dynamics. 21

34 Chapter 3: Sources of Data Experimental data obtained from time-resolved Particle Image Velocimetry (PIV) measurements are analyzed in Chapter 4. The main objective of this chapter is to study the time-averaged flow field and dynamics of the large-scale shed vortices within the junction region behind the finitelength square-cross-section cylinder partially immersed in natural and tripped turbulent boundary layers. On the one hand, an cylinder is chosen to be studied through the phase-averaged field and vortex dynamics as the vortices formed at the base are far from the tip vortices and thus their evolution is isolated from any effect approaching from the tip. On the other hand, the most dominant behaviours of the base turbulent flow seem to be correlated with the upstream boundary layer conditions and possibly horseshoe vortex. In order to explore the interaction between the coherent dynamics behind the obstacle and the horseshoe vortices originated at the front, a distinct set of data extending from upstream of the flow are analyzed. This set of data corresponds to cylinder of and are post-processed with Proper Orthogonal Decomposition (POD) to identify the most energetic realizations of the field. The study in Chapter 5 is undertaken on the CFD data for the same prism with ( that is undergoing natural turbulent boundary layer) to approximate the wake of the second experimental set of data (see section ). Numerical analysis is performed using the Large Eddy Simulation to focus on the application of POD and Extended Proper Orthogonal Decomposition (EPOD) techniques to predict the low order representations of velocity coherent fields by the correlated dynamics of obstacle surface and wall pressure. In this case, Computational Fluid Dynamics (CFD) data is a proper candidate to test the earlier defined methodology (see section 2.3) and to be used as a guide for future experimental approaches. The reason is the advantage of CFD, in which the pressure data is spatially distributed within a large number of grids and there is no limitation of taking a few number of pressure transducers as it is in experimental approaches. Hence, the prediction is conducted by employing the complete spatial information of pressure signals and the most optimum location of pressure transducers can be investigated. This chapter outlines the brief description of experimental facility and the setup used in numerical simulations performed by the group members; the first experimental set of data was obtained by Bourgeois (2012), the 22

35 second experimental set of data was obtained by El Hassan (2012) and the CFD simulation was performed by Chen (2012) Experimental data Experimental data is used to study the time-averaged and large-scale coherent structures within the turbulent velocity field in Chapter 4. Measurements are conducted for two obstacles of =4 and 8, respectively, for two on-coming boundary layers of different thicknesses. The first set of data, corresponding to the prism of, looks into the large-scale shed vortices extracted from the TPA approach in addition to the time-averaged structures. This is considered as a tool to investigate the wake structures within the tip and base regions to gain further insights on the dipole and quadrupole flow topologies. Literature review states that the same qualitative wake structures are seen for the obstacles with 5, same boundary and on-coming flow conditions (Okamoto & Sunabashiri 1992; Wang and Zhou 2006 & 2009; Bourgeois et al. 2011). Therefore, the second set of data demonstrates the velocity wake of the prism AR=4 (above ) and focuses on the junction region to study the most dominant fluctuating terms within this section of the flow. These terms seem to originate from the interaction between shed vortices and horseshoe extended legs in a complex manner leading to an existence of a long trailing pair of stream-wise vortices (known as "base vortices"). Therefore, to allow for further investigations, the initial formation region of horseshoe vortex (upstream the obstacle along the base) is covered in the second set of data Experimental setup Measurements were conducted in a suction-type, open-test-section wind tunnel, shown schematically in Fig The inlet flow carrying olive oil particles passes through a set of three 5 Below, the alternate Kármán shed vortices are weakened and in some cases suppressed resulting in a distinct wake topology. 23

36 20 mesh metal grids, a 24 and a 30 mesh metal grid, and one 80 mesh nylon screen before entering the nozzle. The nozzle inlet diameter is 3 m and the inlet-area contraction ratio is 36:1. There is a jet with 0.5 diameter and a flat plate of a sharp leading edge with 1.2 long, 0.8 wide and 0.01 thick placed in the open working section. The rectangular cylinders of ( ) and ( ) machined from aluminum were mounted at from the leading edge of the plate. Mean velocity of, turbulence intensity level of 0.8%, and a Reynolds number of 12,000 state the free-stream flow conditions. The measured Strouhal numbers were for and for with denoting the vortex shedding frequency. The measurements are conducted for two turbulent boundary layers: the naturally developed boundary layer over the flat plate ( ) and a tripped boundary layer generated with a 6.35 rod positioned at the plate leading edge ( ) as shown in Fig. 3.2 and 3.3. Boundary layer profiles of the stream-wise mean velocity and root-mean-square velocity fluctuations are shown in Fig. 3.4 and Fig. 3.5 for both the natural and tripped boundary layers of and measurements, respectively. Figure 3.1: Wind tunnel schematic and nomenclature 24

37 Figure 3.2:, a) the natural and b) the tripped turbulent boundary layers of the on-coming flow Figure 3.3:, a) the natural and b) the tripped turbulent boundary layers of the on-coming flow 25

38 Figure 3.4: The boundary layer profiles and root-mean-square stream-wise velocity fluctuations for a) natural and b) tripped boundary layers in the case of Figure 3.5: The boundary layer profiles and root-mean-square stream-wise velocity fluctuations for a) natural and b) tripped boundary layers in the case of 26

39 Data set 1 The data set-1 measurements were conducted and post-processed to obtain the three-dimensional time-averaged and phase-averaged velocity wakes by Bourgeois (2012). Several horizontal and vertical planes were measured using a LaVision FlowMaster high-framerate PIV system. A Photonics Industries 10mJ Nd:YAG laser system was used to form a 1 - thick laser sheet to illuminate the olive oil particles. Image pairs with a time separation of 50 μs were taken at rates of 500 to 1000 Hz (capturing 4 to 8 data points per shedding cycle) by a HighSpeedStar 5 CMOS camera. Interrogation windows were chosen with 50% overlap corresponding to the spatial resolution of For each plane, a minimum of 5000 image pairs were obtained spanning at least 600 shedding cycles. The estimated uncertainty on individual vector measurements is (Westerweel 2000). For further analysis, TPA was implemented to reconstruct the phase-averaged threedimensional velocity field using the phase of the fluctuating surface pressure difference between the lateral sides of the obstacle (6 pressure transducers were used while positioned symmetrically on the obstacle surface). The sampling rate for the pressure was khz and was synchronized with the PIV measurements. Resultant pressure phase information was interpolated on each PIV vector field to find the phase of the velocity shedding cycle while the shedding period was discretized into twenty equal phase steps, ( ). The detailed derivation of phaseaveraged field and the experimental procedure can be found in Bourgeois (2012). Trilinear interpolation was then used to interpolate the measured phase-averaged data of horizontal and vertical planes onto a three-dimensional Cartesian grid. The redundant x component of velocity data, that was captured from both and planes, was averaged to provide further statistical convergence of the flow statistics Data set 2 The data set-2 measurements were conducted by El Hassan (2012). A Photonics Industries 527 nm Nd:YLF laser light sheet with thickness of 1 was used to illuminate the horizontal plane. Images were acquired at rates of 800Hz, which corresponds to approximately 6 data points per shedding cycle. A HighSpeedStar 5 CMOS camera ( pixels) in double-pulse mode with a pulse separation of 30μs between the images captured the image pairs. Interrogation 27

40 windows of with 50% overlap were executed indicating the spatial resolution of 1000 vector fields were acquired so that statistics were calculated over 900 shedding cycles and the estimated uncertainty on individual vector measurements is (Westerweel 2000) Computational Fluid Dynamics (CFD) data The numerical analysis performed by Chen (2012) simulates the turbulent flow over a finitelength square-cross-section cylinder with conducted at a Reynolds number of and for the naturally developed turbulent boundary layer ( ), where the turbulence intensity reads 0.5%. The flow field was simulated using ANSYS CFX and Large Eddy Simulations (LES) with Smagorinsky sub-grid model. The computational domain consists of 1.4 million hexahedral elements, with spatial resolution of d/30. Validations performed by comparing the CFD results with oil film flow visualizations and PIV experimental data on the same obstacle geometry shows satisfactory agreement. More explanation for CFD setup and validations can be found in Chen (2012). 28

41 Chapter 4: Base Region Vortex Topology of a Wall-Mounted Rectangular Cylinder The present chapter focuses on the base wake topology of a finite-length square-cross-section cylinder with fixed height-to-width ratio partially immersed in two nominally thin turbulent boundary layers of different thicknesses (section 3.1.1). Two distinct wake topologies are observed depending on the boundary layer thickness. For the thinner natural boundary layer (NBL), the flow type is a dipole mean structure associated with the half-loop instantaneous vortex structures while for the thicker tripped boundary layer (TBL), a quadrupole mean structure associated with the full-loop instantaneous vortex structures is seen. A dipole wake describes the existence of a pair of mean stream-wise counter rotating vortices extending from the tip area (the "tip vortices") while for a quadrupole another mean stream-wise pair (the "base vortices") is also present extending along the wall from the junction region. A half-loop structure is attributed to the bending of shed vortices towards the tip region. In a full-loop structure, this bending is also observed along the wall. This work seeks to clarify the physical mechanisms and the influence of BL thickness in the base region. To this end, the dominant terms in the transport equation of base vortex (i.e. the transport equation of the time-averaged stream-wise vorticity) giving rise to these two distinct wake topologies and modal analysis of the flow field are considered. The Particle Image Velocimetry (PIV) data set-1 (section ) is used to investigate the time-averaged flow and large-scale coherent components educed through Traditional Phase- Averaging (TPA) technique. The transport equation of time-averaged stream-wise vorticity is analyzed to identify the driving terms involved in the transport process of base vortices in the flow subject to the TBL. It will be shown that on the one hand, time-averaged vortex tilting and stretching terms dominate the transport of base vorticity inside the recirculation zone. On the other hand, the fluctuation contributions associated with the large-scale shedding coherent dynamics (time-averaged representation of tilted vortex shedding towards the stream-wise direction) become significant downstream of the recirculation zone. As will be shown, the 29

42 scaling of the dominant terms with the upstream BL thickness, δ suggests this parameter as the key role enhancing the base vortices in TBL flow compared to that of NBL. The study is then pursued by employing the PIV data set-2 (section ) and applying Proper Orthogonal Decomposition (POD) analysis to focus on the unsteady field in the near wall region, z/d=0.1. The goal is to characterize the coherent behaviour of the fluctuating field, which in here, cannot be resolved through TPA approach. It will be shown that the shed structures are disturbed as a result of an interaction with the horseshoe vortex (HSV) in the TBL flow. This leads to the view, in which this interaction is responsible for the tilted vortex shedding and hence sustaining the base vortices within the wake far from the obstacle. 4.1 Introduction to flow topology around truncated cylinders The wake flow for two-dimensional cylinders is marked by the alternate shedding of counterrotating span-wise vortices generated by roll-up of the separated shear layers from the opposing side walls (von Kármán vortices). In the case of wall-mounted finite-length cylinders, fully three-dimensional wake structures are formed due to wall-junction and free-end effects. Vortices induced by these effects interact with the large-scale Kármán shed vortices and hence distinguishes these types of wake from two-dimensional flow fields induced downstream of twodimensional geometries. The mean velocity wake for three-dimensional geometries is characterized by either a dipole or quadrupole mean distribution of stream-wise vorticity (Fig. 4.1) depending on the obstacle and δ of the on-coming flow. Bourgeois et al. (2011) and Hosseini et al. (2013) reported the half-loop and full-loop Kármán vortex shedding within their phase-averaged reconstructions associated with dipole and quadrupole mean wakes, respectively. 30

43 Figure 4.1: a) dipole and b) quadrupole mean flow structures and vectors in the wake of natural and tripped BLs, respectively. Black lines indicate time-averaged vortical structures identified using criterion ( ). Color contours represent the time-averaged stream-wise vorticity. The influence of the down-wash flow that originates at the tip area, the up-wash flow from the near wall and the Kármán shed vortices on the generation of tip and base vortices has been investigated by several authors. The down-wash flow is linked to the downward deflection 31

44 of the separated shear layer vorticity at the free-end leading edge, while the up-wash flow is connected to the vorticity induced by the wall BL span-wise gradient. It should be noted that the free-end and wall shear layers generate opposite sign vorticity as the latter develops inside the reverse flow zone ( ) behind the cylinder, while the former is generated at the front side of the obstacle and therefore is isolated from the reverse flow zone effects ( ) (Fig. 4.2). Figure 4.2: Induced BLs at the obstacle free-end and base wall. Color contours indicate distribution of time-averaged stream-wise velocity Etzold and Fiedler (1976), Kawamura and Hiwada (1984), Wang and Zhou (2009), Bourgeois et al. (2011) and Hosseini et al. (2013) have attributed the generation of the tip vortex pair to the Kármán vortex with the axis oblique in the stream-wise direction near the tip. In contrast, Sumner & Heseltine (2004) observed tip vortices below the critical ( ), for which the Kármán shed vortices are shown to be suppressed (Sakamoto and Arie 1983; Okamoto and Sunabashiri 1992; Lee 1997; Heseltine 2003) implying that tip vortices are not the time- 32

45 averaged signature of tilted shed vortices but rather the stream-wise projection of vertical vorticity (induced by the down-wash flow) initiated at the obstacle tip. This is consistent with the view of Etzold and Fiedler (1976), Kawamura and Hiwada (1984), Tanaka and Murata (1999) and Park & Lee (2000), in which, down-wash flow induced by the free-end shear layer is observed to be inherently connected to the tip vortex. Sumner & Heseltine (2004) examined the wake of a circular cylinder with ranging from 3 to 9, where for their experimental condition, the was between 3 and 5. They suggested that the base vortex may represent the time-averaged inclined Kármán shedding with respect to the stream-wise direction supporting the explanation made by Etzold and Fiedler (1976), Wang and Zhou (2009), Bourgeois et al (2011) and Hosseini et al. (2013). This was concluded based on the observation that there was no base vortex pair below, in which shed vortices are annihilated, while it was consistently present for all the cases above associated with the quasi-periodic shedding process. Moreover, Okamoto and Sunabashiri (1992), Tanaka and Murata (1999) and Wang and Zhou (2009) proposed that the down-wash flow adjacent to the free-end and up-wash flow inspired by the velocity gradient of the wall BL interact with the vortex shedding. Subsequently the upper and lower parts of the span-wise vortex roll lean upstream forming a pair of tip and base vortices near the tip and base regions, respectively. Transition between dipole and quadrupole mean flow structures has shown to be a function of cylinder and δ. When falls below a critical value (relatively short cylinders), the von Kármán shedding process is weakened and in some cases (very short cylinders) suppressed. The numerical value of depends on a number of factors such as the BL thickness, on-coming flow turbulence intensity and geometry of obstacle cross-section (Sakamoto & Arie 1983, Kawamura et al. 1984). Sumner et al. (2004) has detected a quadrupole wake for a circular cylinder with above the, while a dipole was seen below. The same observation was made by Wang and Zhou (2009) in the wake of square cylinder with AR varying between 3 and 7. In that study, only is marked as below and the associated flow wake was shown to be a dipole type and free of von Kármán vortex shedding, while quadrupole distribution of vorticity and regular von Kármán shedding were detected for their obstacles with. The absence of base vortices in the wake of obstacles with below was attributed to the predominant free-end downwash flow in short cylinders affecting the 33

46 whole wake, i.e. interacting with and weakening the von Kármán street (Farivar 1981, Okamoto & Sunabashiri 1992, Wang and Zhou 2009), which reattaches and in turn dominates the junction region resulting in the suppression of base vortices (Hussain and Martinuzzi 1996, Krajnovic and Davidson 2002, Wang and Zhou 2009). The dependency of transition on the BL thickness is reported by a number of authors (Park and Lee 2000; Wang and Zhou 2009; Hosseini et al. 2013). In their measurements, the velocity field around cylinders with above (denoting the presence of the shed Kármán structures), is attributed to dipole for a relatively thin BL and quadrupole as BL thickness increases. Wang and Zhou (2009) have also deduced that base vortex is enhanced with increasing boundary layer thickness. Despite the number of studies performed to investigate the near wake flows around the three-dimensional cylinders, the process, by which base vortices are developed and the influence of δ on the vortex generation is not fully understood yet. Hence, the objective of the current study is to gain further appreciation of the mechanism responsible for the formation of base vortex by modifying the BL thickness. For our case of study, the cylinder is greater than the implying the existence of the shedding process and therefore, the only parameter that can distinguish between dipole or quadrupole wake is δ. The present work focuses on the transport mechanism of base vorticity behind the wall-mounted finite-length rectangular cylinder that is partially subject to natural and tripped turbulent BLs. Both BLs are classified as nominally thin boundary layers (BL thickness to obstacle height ratio, ). The study considers the transport equation of time-averaged stream-wise vorticity to investigate the contribution of dominant terms in the transport process of the base vortex. Time-averaged components and coherent dynamics driving the fluctuation contributions to the vorticity equation induced by large-scale shed vortices and extracted from the TPA technique (Bourgeois et al. 2011) are analyzed within the vorticity transport equation. Proper Orthogonal Decomposition (POD) is employed as a method to identify the coherent structures of turbulent flows (Lumley 1967, Holmes et al. 1996, Delville 1999) to recognize the most energetic unsteady modes representing a planar field within the base region and to reconstruct a low order but richer instantaneous wake by adding more coherent components to the vortex shedding dynamics as compared with the TPA method. This also enables exploration of the mechanism through which the upstream BL gradient and HSV interact with the large-scale unsteady terms (i.e. shed vortices) behind the obstacle. 34

47 4.2 General characterization of the flow The time-averaged three-dimensional vortex structures are identified from the criterion (Jeong and Hussain, 1995) and are shown in Fig The time-averaged wakes around the obstacle are marked by dipole and quadrupole structures for NBL and TBL, respectively. While both BLs are considered as nominally thin, different flow structures are observed by only changing the upstream BL thickness from to. The main distinction is the base region topology where the pair of long trailing base vortex exists for the TBL and is absent for the NBL. Whereas the distribution of vortical structures in the vicinity of the tip region indicates the formation of prominent tip vortices for both wake flows. The counterrotating tip vortex pair is seen further from the ground plate (except for a region very close to the free-end) in the TBL compared to the NBL. This is due to the effect of the base vortex in the TBL, which acts to displace the tip vortices in the positive span-wise direction. Figure 4.3: Iso-surfaces of time-averaged vortical structures identified using criterion ( ) for a) NBL (dipole wake) b) TBL (quadrupole wake). Dashed lines indicate the edges of the cylinder. Reprinted with permission from Bourgeois et al. (2012). Copyright 2012, unpublished doctoral dissertation. Fig. 4.4 shows the instantaneous vortex visualizations reconstructed from the TPA technique at of the shedding cycle. The half-loop and full-loop vortical structures are 35

48 clearly illustrated for the NBL and the TBL, respectively. These structures are shed alternately from opposing sides of the obstacle. Again, the main mechanism giving rise to these two different wake topologies is seen to originate from the base region. In the TBL, the shed columnar vortex is bent towards the upstream inside the tip and base regions, while this bending is observed only at the tip for the NBL. These figures then indicate that the dipole and quadrupole mean structures are an artefact of ensemble averaging. Figure 4.4: Iso-surfaces of phase-averaged vortical structures ( ) identified using criterion ( ) for a) NBL (half-loop vertical structures) b) TBL (full-loop vertical structures). Dashed lines indicate the edges of the cylinder. Reprinted with permission from Bourgeois et al. (2012). Copyright 2012, unpublished doctoral dissertation. 36

49 It is worthwhile to study the distribution of the time-averaged stream-wise vorticity (Fig. 4.5) in order to understand the distinct vortex representations of the two flow cases. The timeaveraged vortical structures (Fig. 4.3) are the three-dimensional projection of the time-averaged stream-wise vorticity and its extension into the wake. As it is illustrated in Fig. 4.1, the sense of rotation of tip vortices induces a down-wash flow in the interior wake, while that of base vortices induce an up-wash flow. For the TBL, the pair of base vorticity is always present within the wake as is shown at different stream-wise locations (Fig. 4.5) with comparable size and strength of tip vorticities. The NBL flow exhibits smaller and less concentrated base vorticities, while they start to remarkably descend away from the flow centre line as they are convected further downstream (This is also observed for the case of TBL but in a less significant manner). As base vorticities in NBL descend away from the centre line, a new pair of vorticity has to be induced, with the sign opposite to that of the base vorticity, to satisfy the zero velocity condition at the wake centre-line. These vorticities are labeled in red in Fig. 4.5a. In NBL flow, as can be seen from Fig. 4.3, the pair of base vorticity is not identified by vortex criterion once it passes through the plane. This behaviour can be further studied in Fig. 4.5a at and planes and attributed either to the lower concentration of base vorticity within the region or the canceling effect of the opposing sense of circulations associated with the base vorticity (marked by black labels) and induced vorticity within the inner wake (marked by red labels) as both have the same core strength and size (each inducing same amount of circulation with opposite sign). It should be noted that in both flows, base vorticities become rounder in shape and weaker as they travel down in the stream-wise direction consistent with the observations made by Sumner et al. (2004) and Wang and Zhou (2009). The effect of base and tip vorticities on the recirculation zone is shown in Fig. 4.6 where regions with shorter recirculation length suggest the formation of these vortices. The absence of the base vortex pair in the NBL flow distinguishes the longitudinal extent of the recirculation zone when compared to the TBL case. For both flows, the maximum length of the reverse flow zone reads at for the mid-span ( ), implying the stronger vortex shedding dynamics and weaker influence from the tip (i.e. down-wash flow) and base vortices (i.e. up-wash flow). Typically, the regularly shedding vortices accumulate vorticity within the recirculation region and this leads to a large increase of the recirculation length. 37

50 Figure 4.5: Time-averaged stream-wise vorticity distributed in the wake of a) NBL and b) TBL. Black labels show the core of tip and base vorticities. Figure 4.6: Time-averaged stream-wise velocity demonstrating the recirculation zone for a) NBL and b) TBL 38

51 4.3 Time-averaged transport flow field The time-averaged transport equation of mean stream-wise vorticity is considered to investigate the mechanism causing the different flow structures observed for the two types of flow. For this purpose, individual terms in the vorticity equation are studied to explore the important terms responsible for the formation of the long trailing base vorticities. The transport equation of time-averaged stream-wise vorticity (Eq. (4.2)) is obtained by applying the Reynold s decomposition (Eq. (4.1)) to the transport equation of vorticity (in the stream-wise direction) shown in Eq. (2.3), and then ensemble averaging to yield the transport equation of time-averaged stream-wise vorticity, ( ) ( ) ( ) ( ) The terms on the left hand side represent the convective transport of the time-averaged streamwise vorticity,. The first three terms on the right hand side stand for the mean vortex stretching and tilting arising from the stretching of mean vortex lines due to presence of the velocity gradient,, and rotation of mean vortex lines induced by the velocity gradients, and, respectively. The other product terms associated with describes the molecular diffusion driven by viscous effects. The last four terms denote the fluctuation contribution to the vortex transport. ( ) and originate from the term, in Eq. (2.3) and therefore states the change in vorticity due to fluctuations in strain rates. and are 39

52 the result of in Eq. (2.3) explaining the contribution due to turbulent diffusion. It should be noted that fluctuating terms here represent only the part of the unsteady field induced by the shedding dynamics that are extracted using the TPA method. The reason for neglecting the contributions of the random fluctuation field (i.e. incoherent fluctuations based on TPA definition discussed in section 2.2) is based on the observations made from Fig. 4.4 where formation of the base vortex seems to be explained by the tilted columnar shed vortex at its lower part. Decomposition of into its two components yields, and it can be further deduced that the second term on the right hand side indicates the timeaveraged representation of tilted vortex shedding with respect to the stream-wise direction, where stands for the shed structures and acts to lean the axis of the vortex towards the stream-wise direction. The analysis with regards to the vorticity equation can be simplified by discarding the viscous diffusion terms as the contribution has been found to be negligible compared to the other terms. Also the two vortex tilting terms cancel out parts of one another after being expanded and the remaining terms of vortex stretching and tilting due to time-averaged contributions read, ( ) where and are the remaining portions of the terms and introduced earlier in Eq. (4.2), respectively. Circulation (or strength) of the base vortex, needs to be determined for planes, 40

53 where is the area of base vortex. The area is defined as the closed region occupied by the base vortex for which. As was discussed earlier, Eq. (4.2) formulates the vorticity transport at a spatial point. This formulation, however, is inconvenient in describing the stream-wise evolution of the vortex structure due to the influence of diffusion. For example, in following the evolution of along the centre of the structure, will decrease as a result of diffusion even if were to remain constant. Here, the interest is in defining the so called "net effect" on the entire base vortex structure. It is proposed to characterize this effect as the total contribution of the individual terms of Eq. (4.2) over the entire vortex area,. For instance, Eq. (4.5) illustrates how the net effect of on the base vortex is computed, { } where is evaluated as in Eq. (4.4). Due to the flow field symmetry with respect to the plane, all the calculations are confined to one side of the symmetry plane ( ). Fig. 4.7 represents the non-dimensional circulation of the base vortex. It is noted that in both NBL and TBL flows, the base vortex circulation experiences a gradual change closer to the obstacle while minor change is detected downstream of the recirculation zone for the TBL flow. Fig. 4.8 illustrates the net effect of vortex tilting and stretching and fluctuation contribution terms for both NBL and TBL. By comparing the plots, the dominant terms driving the transport of the base vortex are identified as vortex stretching and the remaining part of the vortex tilting, closer to the obstacle and specifically inside the recirculation zone, where the latter seems to be considerably stronger (it should be recalled that the terms with negative values of net effect contribute to the positive transport of the base vortex as the circulation of the base vortex is negative for the studied part of the region, ). These two terms are initially dominating the base vortex transport for both NBL and TBL flows while they become an order of magnitude larger for the TBL case where the rate of magnitude increases more rapidly for than for. is expected to be significant within the flow where the mean stream-wise vorticity is being 41

54 stretched as the stream-wise velocity is recovered. Also, a portion of this term originates from the reflected horseshoe vortex on the cylinder wall to satisfy the no-slip condition (Fig. 4.9). The horseshoe vortex is initially generated at the front side of the obstacle as the induced vorticity by the upstream BL approaches the adverse pressure gradient at the front face of the bluff-body. The axis of the horseshoe vortex is in the vertical direction after the separation in the upstream region and then by wrapping around the obstacle, the primary axis rotates towards the stream-wise direction forming the extended legs of the horseshoe vortex. In the vicinity of the ground plate, is believed to be notably great as (i.e. the up-wash flow) is formed by the plate BL gradient (also see Fig. 4.2) and is then rotated in the stream-wise direction under the influence of. Figure 4.7: Non-dimensional circulation (circulation/ ) of the base vortex 42

55 Figure 4.8: Non-dimensional net effect of a) vortex stretching and tilting induced by timeaveraged terms, b) time-averaged contribution of fluctuations and c) important terms downstream of the recirculation zone on the base vortex. 43

56 Figure 4.9: Reflection of horseshoe vorticity on the cylinder wall. The contour is the timeaveraged stream-wise vorticity for a) NBL and b) TBL As the flow travels to the far wake in the TBL case, exhibits order of magnitude larger net effect on the base vortex compared to the vortex tilting-stretching and the rest of the terms induced by the unsteady shedding flow field (Fig. 4.8b TBL & 4.8c). By comparing the NBL and TBL flows in Fig. 4.8b, it can be observed that this term does not play a remarkable role among the terms that arise from the fluctuating contributions for the NBL while its significant role is evident in the TBL as it holds the largest net effect on the transport of base vortex (as was discussed before the negative values of net effect has positive contribution to the transport of the base vortex). Fig shows that the dominancy of this term arises from the term rather than or a combination of both. As was discussed earlier, represents the time-averaged foot-print of the tilted vortex shedding to the stream-wise direction. 44

57 Figure 4.10: Non-dimensional net effect on the base vortex for TBL To better appreciate the role of the upstream BL thickness, circulation and area of the base vortex and net effect of the terms driving the transport of base vortex are scaled with the BL thickness for the flows with NBL and TBL (Fig. 4.11) (base vortex circulation, area and net effect of and are scaled with BL thickness while the square of BL thickness is considered for scaling ). It should be recalled that the free-stream velocity and obstacle diameter are kept constant between the two flows while BL thickness changes. The collapse of these data shown in Fig suggests that net effect of the studied terms and circulation of the base vortex are the same between NBL and TBL flows when scaled with BL thickness. This addresses the key effect of upstream BL thickness to enhance the base vortices. This occurs through the process, in which by thickening the BL, the gradient is increased and therefore the product of and its corresponding induced terms with the terms in the vorticity transport equation increase from the order of magnitude point of view. 45

58 Figure 4.11: Non-dimensional a) circulation of the base vortex, b) base vortex area, net effect of c), d) and e) on the base vortex 46

59 4.4 Fluctuating flow field The time-averaged velocity wake allows for identifying the terms responsible for the enhanced base vortices and subsequently the presence of the quadrupole wake rather than the dipole. As one of the most significant terms in the transport of the base vortex was found to be, an applicable technique needs to be put forward to analyze the deterministic behaviour of the unsteady wake. This also allows the determination of how the term, arises in the TBL flow compared to NBL. Accordingly, the study of the fluctuating velocity wake is taken by applying the POD decomposition (section 2.3.2) on the plane ( ) to guarantee that the investigated field is inside the base region and represents the base flow behaviour, which explains the difference between the two types of flow (dipole and quadrupole). The POD temporal coefficients, spatial modes and the kinetic energy contents are then to be compared between the NBL (associated with the dipole mean wake) and TBL (associated with the quadrupole mean wake) as a means to elucidate observations for. Fig shows the distribution of fluctuation kinetic energy of the first 6 POD modes. For the NBL, the first two modes represent nearly half of the total energy, while the energy is almost evenly distributed into the TBL modes. This highlights a more complex dynamic field arises from the competing mechanisms between the modes of TBL flow compared to NBL. Fig. 4.13, 4.14 & 4.15 refer to the power spectral density of the first 6 POD temporal coefficients as a function of non-dimensional frequency ( ) and the spatial modes. It is clearly seen that for the NBL case (Fig. 4.13a, 4.14a & 4.15a), the first 2 POD modes represent the regular alternate vortex shedding with phase shift between them while oscillating at the fundamental shed frequency with the Strouhal number (St) of 0.1. These modes are considered to stand for the most energetic unsteady flow behaviour as constituting 43% of the total kinetic energy. The third and forth modes are assumed to originate from the interactions with induced terms by the upstream BL gradient (as these sort of terms can be detected only for the planes inside the BL region), while the second harmonics of vortex shedding are represented by the fifth and sixth modes with frequency twice shedding frequency. Although, it should be noted that the last 4 modes discussed here have a minor contribution to the energy content of the fluctuating field and therefore are concluded to be negligible for the NBL. 47

60 Figure 4.12: Distribution of turbulent kinetic energy within the POD modes for a) NBL and b) TBL The POD modes associated with the TBL flow depict 2 distinct types of interaction (Fig 4.13b, 4.14b & 4.15b). This was educed as the cross-correlation function was calculated for each pair of the first 6 temporal coefficients (representing the highly correlated modes 1 and 6 and then modes 2, 3, 4, and 5) along with the observations made from the spatial POD modes. The first type seen in modes 1 and 6 (which account for 9% of the kinetic energy) demonstrates the reflection of extended legs of the horseshoe vortex on the obstacle wall followed by convection of its reflected terms (with opposite sign of velocity to that of the horseshoe extended legs) downstream. The time-averaged effect of these modes significantly contributes to the transport of the base vortex through (section 4.3). The strong reflected vorticities on the wall dominate the base wake dynamics preventing the evolution of Kármán vortex shedding explaining the absence of shed vortices in these modes. The second type of behaviour (shown in modes 2, 3, 4 and 5 with 13% of the energy content) correspond to the process where the extended legs of the horseshoe vortex merge with the downstream inner wake and interact with the shed vortices. This mechanism seems to be responsible for the stream-wise tilting of the shed columnar axis (also see Fig. 4.4). This is possible as extended legs of the horseshoe vortex advect the upstream BL gradient into the wake and the resultant product with shed vortices 48

61 then contributes to the time-averaged representation of tilted vortex shedding ( ). The correlation between the shed vortices and horseshoe vortex was also studied by a number of authors but could not be well explored (Eisenlohr and Eckelmann, 1989; Balachandar et al. 2000; Rao and Sumner 2004). Figure 4.13: Power spectral density of the first 6 POD temporal coefficients for a) NBL and b) TBL 49

62 Figure 4.14: POD spatial modes (x component of velocity) for a) NBL and b) TBL 50

63 Figure 4.15: POD spatial modes (y component of velocity) for a) NBL and b) TBL 51

64 4.5 Instantaneous coherent flow field As was discussed earlier in the chapter, one of the main differences between the base regions of dipole and quadrupole wakes is denoted by the increased influence of vortex stretching and tilting caused by contributions of time-averaged terms starting near the back face of the obstacle and persists within the near wake flow in the TBL than in the NBL. The responsible vortex stretching term was shown to act in the form of scaled with. Fig represents the instantaneous stream-wise vorticity reconstructed by TPA adjacent to the base and illustrates how the pronounced horseshoe vortex and its extended legs give rise to the base vortex through (Also see Fig. 4.8 and 4.9) in the TBL. The enhanced horseshoe vortex and subsequently reflected vorticities on the obstacle wall form the dominant timeaveraged field, in which fluctuations and therefore shed structures become less apparent. This is in contrast with the observation made for the NBL, where the fluctuating terms dominate the flow transport and leave no foot-print in the time-averaged field leading to the weaker base vortex pair. This difference between the two wakes becomes even more evident as we move along the span from to. At this plane, the horseshoe vortex and its induced terms are still observed along with the strong base vortices in TBL, while no signature of the horseshoe is seen for the NBL explaining the dominancy of the fluctuating components and weak base vorticities. The above distinction between the two flows arose from the dominance of the timeaveraged field and accounts for the shorter recirculation length in the base region of the TBL flow as the noticeable effect of time-averaged terms obscure the dominancy of shed vortices. This is followed by the entrainment of upstream streamlines (including the extended legs of the horseshoe vortex) into the inner wake closer to the obstacle back face and subsequently interacting with the shed vortices, which will be discussed in further details. The other difference between the base regions of the dipole and quadrupole wakes arises from separate mechanisms modifying the vortex shedding. For the NBL, the periodic shedding of vortices occurred regularly. For the TBL, the shedding seems to be disturbed as a consequence of enhanced vortex stretching and tilting close to the obstacle and due to interaction with the extended legs of horseshoe vortex in the further wake. The latter occurs as the reduced 52

65 recirculation zone causes the saddle point to locate closer to the obstacle and therefore the surrounding flow is sucked into the inner wake at smaller location (Wang and Zhou, 2009). Figure 4.16: Phase-averaged stream-wise vorticity ( ) for a) NBL and b) TBL The interaction between the extended legs of the horseshoe vortex and the shed vortices (section 4.4) seems to be responsible for tilting the shed vortex axis in the stream-wise direction, which is then to be fed into the generation of the base vortex explained through the transport equation and the term, (section 4.3). This can be demonstrated from Fig and 4.18 representing the instantaneous span-wise vorticity reconstructed from the first 6 POD modes discussed earlier in section 4.4. For the purpose of comparison, the same phases of the shedding cycle are taken for both NBL and TBL. As can be seen from the figures, the regular alternate span-wise vorticities (the planar shed terms denote the two-dimensional projection of shedding vortices) are observed to be periodically shed within the whole wake for NBL (Fig. 4.17a & 53

66 4.18a). In contrast the planar span-wise vorticities are confined to a region adjacent to the cylinder and are annihilated as the flow is convected further in the stream-wise direction in the TBL case (Fig. 4.17b & 4.18b) As the distorted shedding process is ascribed to be correlated with the horseshoe system, the streamlines forming upstream representing the initially generated horseshoe and its extended legs travelled down-stream are tracked and marked by letters A, B and C. It is observed that the streamlines representing the convection of the horseshoe system never penetrates into the centre wake for the NBL, while the horseshoe streamlines indeed merge towards the inner wake at about for the TBL, where shed vortices are not detected any more in the wake. This supports the hypothesis that once the extended legs of the horseshoe vortex meet with shed vortices, they act to weaken them through bending their axis in the stream-wise direction by providing the sufficient gradient,. This is concluded only for the TBL flow since the horseshoe vortex is scaled with BL thickness (As it is well-known and addressed by number of authors such as Baker (1980), Ballio et al. (1998) and Simpson (2001) that the HSV is formed by the plate boundary layer vorticity which is induced by the velocity gradients upstream of the obstacle) and therefore the upstream BL gradient (in this case, ) convected by the horseshoe system is an order of magnitude larger than that of the NBL developing a more remarkable product of (it has been already observed that is scaled with the BL thickness confirming the above statement). 54

67 Figure 4.17: Instantaneous span-wise vorticity reconstructed from the first 6 POD modes and streamlines for a) NBL and b) TBL for the same phase of shedding cycle. 55

68 Figure 4.18: Instantaneous span-wise vorticity reconstructed from the first 6 POD modes with streamlines for a) NBL and b) TBL for the same phase of shedding cycle. 56

69 4.6 Summary The chapter focused on the base flow topology around the finite-length square-cross-section cylinder with fixed (greater than ) undergoing two different BL thicknesses (NBL and TBL). The phase-averaged flow was linked to alternately shed Kármán structures with the upper part of the vertical vortex bent upstream near the tip region for NBL and both upper and lower parts bent upstream near the tip and base domains, respectively for TBL. These two distinct phase-averaged flows then gave rise to different time-averaged flow topologies representing a pair of tip vortex for NBL and two pairs of tip and base vortices for TBL. These observations highlighted the different mechanisms within the base region distinguishing the two types of the flow. Therefore, the study was undertaken by identifying the dominant terms responsible for the enhanced and long trailing base vortex in the wake of the TBL compared to that of the NBL. This was done by considering the transport equation of time-averaged stream-wise vorticity within the spatial regions occupied by the base vortex. Moreover, the mathematical explanation of bent shed vortices (, which was found to have a significant net effect on the transport of the base vortex) was extracted and further analyzed to explore the physical process causing this term to contribute to the formation of the base vortex. It was concluded that the dominant terms driving the transport of the base vortex are vortex stretching, and the remaining portion of vortex tilting, (reflection of the horseshoe extended vortices on the obstacle wall and footprint of the up-wash flow induced by the plate gradient, which both in turn are stretched into the stream-wise direction) inside the near wake ( ). Meanwhile, the fluctuation contribution specifically the time-averaged representation of tilted vortex shedding, becomes significant in the far wake ( ). Positive net effect of the latter term on the base vortex and its dominancy further downstream explains the illustration made from the TPA three-dimensional fields (Fig. 4.4), which depicts that the Kármán columnar vortex is indeed connected with the base vortex forming a single structure. POD analysis addressed the distorted vortex shedding as a consequence of interaction with enhanced horseshoe dynamics in the TBL flow compared to the regularly shed vortices 57

70 observed for the NBL. It was deduced from the POD reconstructed fields that the horseshoe streamlines originated at the front side of the cylinder enter the inner wake flow of the TBL. These observations led to the view that the extended legs of the horseshoe vortex interact with the shed vortices acting to tilt the axis of vortex towards the stream-wise direction, which is to generate time-averaged stream-wise vorticity through the term resulting in the long trailing base vortex within the far wake of the TBL case. It was also concluded that it s the stronger effect of the time-averaged wake (i.e. stronger vortex stretching and tilting terms, and ) that provides the means for the horseshoe system to interact with shed vortices in the TBL. Scaling the important terms in the vorticity transport equation with the BL thickness suggested that this parameter is a key factor highly affecting the base wake topology. By increasing the BL thickness in the flow associated with the TBL, the net effect of vortex stretching and tilting due to time-averaged contributions (initial sources of base vortex) becomes stronger leading to the enhanced base vortex and hence the dominant time-averaged wake within the near wake of the base region, which then in turn impairs the shedding process. This results in a shorter stream-wise length of the recirculation zone (compared with the NBL) and the surrounding flow carrying the extended horseshoe vortices moves into the inner wake closer to the obstacle. Then the interaction between the shed vortices evolved within the inner wake and the merged horseshoe system forms the term as the most important source of the base vortex in the far wake. 58

71 Chapter 5: Low Order Estimation of Turbulent Unsteady Flows Coherent structures (CSs) characterizing the deterministic unsteady behaviour of the turbulent flow field are inherently three-dimensional requiring a global representation of the most energetic events. In the case of velocity data acquisition with stereo Particle Image Velocimetry (SPIV), the independently sampled three-component planar velocity fields need to be synchronized in order to perform the global reconstruction of the flow field. The methodology of Extended Proper Orthogonal Decomposition (EPOD) is put forward as a remote-sensing based three-dimensional flow estimation through employing global pressure information accounting for the correlated data with the vortical structures of the velocity wake. The classical statistical reconstruction techniques do not directly account for the dynamic characteristics such that the essential information of the energy distribution amongst CSs is lost. For example, traditional phase-averaging (Hussain and Reynolds (1970)) assumes only one dominant energy-containing harmonic mode while EPOD is an objective estimator of flow features contributing most strongly to the wake dynamics through the correlation between individual coherent modes of motion, which are identified and selected through the energy optimization properties of Proper Orthogonal Decomposition (POD). Therefore, the advantage of EPOD is extracting more useful information from the reference (pressure) sensors, which leads to a more complete dynamical representation. In this chapter, the low order planar velocity wakes are estimated from correlated pressure information distributed on the obstacle plate surfaces providing the global state-space data. The detailed mathematical description of the procedure can be found in section 2.3. Computational Fluid Dynamics (CFD) data (section 3.2) is used for this study as a development tool and guide for experimental implementation since an unlimited number of pressure data (i.e. pressure sensors) are accessible and this enables to examine the most proper locations (i.e. most correlated spatial locations and also where noise plays a lesser role) of transducers to capture the 59

72 CSs of interest. CFD also brings about another benefit of allowing us to illustrate the unbounded distribution of pressure POD spatial modes making the interpretation of modes simpler (i.e. interpretation of spatial distribution of the pressure POD modes would be more convenient by having a non-limited set of data in the spatial domain). This chapter will discuss two aspects: i) The general procedure to estimate the planar velocity fields using simultaneous pressure information collected from surface of the obstacle and plate will be presented. The needs and benefits of applying the multi-time delay approach for remote-sensing application will also be highlighted. ii) The most suitable sensor locations will be determined by correlating the temporal evolution of coherent planar velocity fields with the pressure signals. 5.1 POD representation of flow Remote-sensing based prediction of the global field requires that a certain degree of correlation exists between sensors and the flow dynamics of interest. The passage of CSs results in correlated pressure fluctuations. Recent research (Boree, 2003, Durgesh and Naughton 2010) has demonstrated theoretically the potential to capture the dynamics of these energetic coherent motions by using a small number of pressure sensors. The objective of this study is to provide a procedure for implementing the above statement. This analysis can be used based on the CFD data to improve the experimental measurements. As was discussed earlier, the first step is to guarantee that the local velocity and remote pressure flow fields contain an assured level of common coherent physical events. To that end, the POD modal analysis is taken for the planar velocity fields at (Fig. 5.1), (Fig. 5.2), (Fig. 5.3) and (Fig. 5.4), which are then to be compared with the pressure wake sampled at the surface of the obstacle (Fig. 5.7), plate (Fig. 5.8) and obstacle plus plate (Fig. 5.9). The surface pressure data collected for the obstacle are positioned along the span of the lateral sides of the obstacle to capture the fluctuation energy that is associated with spanwise variations. The energetic content from the POD analysis is ranked by the percentage level of fluctuation kinetic energy (Fig. 5.1a to 5.4a) denoting the energy contents of each POD mode such that the first modes stand for the most energetic events of the flow field. Fig. 5.1b to 5.4b show the Power Spectral Density (PSD) calculated for the POD temporal modes as a function of the non-dimensional frequency, (, where denotes the frequency). 60

73 The POD spatial modes of three components of velocity are shown in Fig. 5.1c to 5.4c representing the projection of the spatio-temporal flow field on the spatial domain. It is seen that the vortex shedding dynamics (modes 1 and 2) and so called "shift mode" (mode 3) represent the first three POD modes standing for the most energetic part of the fluctuating field for velocity at, and. The first two modes are the pair of first harmonics associated with the shed vortices, as it is educed from the PSD functions with their sharp peak at the fundamental shedding Strouhal number (, where denotes the shedding frequency) and the spatial distribution of modes observed in the POD spatial eigenfunctions. The same modes account for the first POD modes of the velocity field at but with different distribution of energy between the modes. At this plane representing the flow field adjacent to the tip region, the highest amount of energy is distributed into the shift mode while the second and third most dominant modes have turned to be the pair of first harmonics. The latter terms also seem to be mixed with some disturbing mechanisms to the shedding dynamics suggested by comparing the PSD function of temporal coefficients and spatial modes of with those of, and. It is necessary to establish a clear insight of what physical flow event each POD mode represents in order to better understand the POD representation of the field and also the EPOD methodology. The most dominant POD modes were observed to be the pair of first harmonics of the shedding cycle with 90 0 phase shift between them. In the other words, the combination of these modes gives the amplitude and phase of shedding. The 90 0 phase shift between these modes can be seen from their spatial modes in Fig. 5.1 to 5.4 and behavior of the temporal coefficients demonstrated in Fig The amplitudes of the pair of first harmonics ( and ) are plotted as a function of time (Fig. 5.5a) and it is observed that the two signals have a 90 0 phase shift between them. That is also deduced from the trend of in terms of resulting in a characteristic doughnut shape (Fig. 5.5b). The First harmonic modes induce an anti-symmetric U and W (and symmetric V) distribution with respect to plane. It represents projection of the three-dimensional vortex shedding on the planar field. The first harmonics of shedding mode are associated with the strouhal number of. The shift mode, which has a very low frequency compared to the shedding frequency, is typically the most energetic mode after the first harmonics. It is identified as the first symmetric U and W (and anti-symmetric V) POD spatial mode. Noack et al. (2003) has defined the shift mode as the difference between the steady 61

74 solution of the Navier Stokes equations (NSs) and time-averaged flow field. This describes the difference between the short time-averaged (with windowing parameter to be ) and the time- Figure 5.1: POD representation of the velocity wake at. a) Kinetic energy distribution of POD modes, b) PSD function of POD temporal modes and c) POD spatial modes 62

75 Figure 5.2: POD representation of the velocity wake at. a) Kinetic energy distribution of POD modes, b) PSD function of POD temporal modes and c) POD spatial modes. 63

76 Figure 5.3: POD representation of the velocity wake at. a) Kinetic energy distribution of POD modes, b) PSD function of POD temporal modes and c) POD spatial modes. 64

77 Figure 5.4: POD representation of the velocity wake at. a) Kinetic energy distribution of POD modes, b) PSD function of POD temporal modes and c) POD spatial modes. 65

78 -averaged of the field, which points out that the shift mode is coupled with the slow-varying amplitude of the first harmonics. This coupling addresses the inefficiency of the Traditional Phase-Averaging (TPA) technique that assumes constant harmonics amplitude. The EPOD tool aims to account for the shift mode in order to better represent a flow field with significant modulation of the harmonic amplitude. The transients of the shift mode can be related to the slowly varying squared amplitude of the first harmonics (Fig. 5.6) when projected onto the POD basis. This behaviour is observed in Fig. 5.6 suggesting a linearity between the sum of squared first harmonics amplitude ( ) and the shift mode amplitude ( ) where the normalized squared amplitude also stands for the kinetic energy of the studied mode (Eq. 2.24); leading to the idea of energy transfer between the shedding harmonics and the shift mode (Bourgeois et al. (2013)). This can be easily observed from Fig. 5.1 to 5.4 where for velocity wake at, the energy carried by first harmonics decreases from 30% to 14% while that of shift mode increases from 3% to 12% as we move along the span to reach. The dominancy of the shift mode and less energetic shedding modes at can be ascribed to the strong down-wash flow and the formation of the tip vortex pair adjacent to the tip region, which as discussed in Chapter 4 acts to weaken the shed vortices. Therefore, one can deduce that at least a portion of the first harmonics energy has been transferred into that of shift mode. 66

79 Figure 5.5: Demonstration of 90 0 phase shift between the pair modes of first harmonic. a) amplitudes of POD mode pair of first harmonic ( and ) representing their temporal evolution. b) phase portrait of and showing the characteristic doughnut pattern Figure 5.6: Demonstration of linear relation between the shift mode amplitude ( ) and sum of the first harmonic amplitudes ( ). Grey line shows the trend line of the scattered amplitude points. 67

80 The POD modal analysis of the surface pressure fluctuations is consistent with the dominancy of the shedding dynamics and shift mode for pressure data distributed on the plate (Fig. 5.8) and the plate plus obstacle (Fig. 5.9). The first harmonics pair is represented by the first two POD modes with the energy content of 48% followed by the shift mode as the third mode standing for 6% of the kinetic energy in both cases. However, the energy distribution between the two modes of shedding first harmonics (difference between the energy of first and second modes of first harmonics) differs between the two sensor arrangements. When only the pressure sensors at the plate are considered, the energy associated with modes one and two are and (Fig. 5.8a) which for the arrangement including the obstacle lateral walls are and (Fig. 5.9a). The difference of the energy distribution has to do with the phase lag introduced due to relative shift of the two domains for the POD integration. This addresses the motivations for using multi-time delay approach to correct this shift while correlating these pressure POD events with those of velocity field. For the pressure field at the obstacle lateral walls shown in (Fig. 5.7), one of the first harmonic modes is seen as the most energetic POD mode constituting nearly 60% of the total kinetic energy and then the shift mode with energy content of 15% signifying the second mode. The next two modes carry only 8% of the energy characterizing a low frequency symmetric U and W (and anti-symmetric V) modes (mode 3) and the other first harmonic mode mixed with perturbing mechanisms (mode 4) evident from its less distinct peaks in the PSD function. The above observations made for pressure fields at the obstacle and plate surface together with the planar velocity fields suggest that the required correlation indeed exists between the velocity and pressure signals as they all recover the main coherent wake dynamics within the POD representation of their flow fields. This satisfies the condition under which EPOD methodology is established to estimate the local velocity wake using the remote pressure sensors. 68

81 Figure 5.7: POD representation of the pressure wake at the obstacle lateral walls. a) Kinetic energy distribution of POD modes, b) PSD function of POD temporal modes and c) POD spatial modes 69

82 Figure 5.8: POD representation of the pressure wake at the mounted plate. a) Kinetic energy distribution of POD modes, b) PSD function of POD temporal modes and c) POD spatial modes 70

83 Figure 5.9: POD representation of the pressure wake at the obstacle lateral walls and mounted plate. a) Kinetic energy distribution of POD modes, b) PSD function of POD temporal modes and c) POD spatial modes 71

84 5.2 EPOD-based prediction In this section, the velocity wakes at,, and will be predicted by implementing the multi-time EPOD technique (section 2.3.2) using the simultaneous and correlated pressure spatio-temporal signals evolved at the either obstacle surface, plate or obstacle surface plus plate. The aim is to explore which pressure field leads to the most representative estimation of CSs of interest indicating the higher degree of correlation between the pressure and different planar velocity fields. Only the first 3 POD modes of the pressure (Fig. 5.7 to 5.9) are taken for velocity predictions as the neglected modes are an order of magnitude less energetic suggesting that no drastic change to the velocity system is expected by their inclusion. In order to enhance the prediction, multi-time delay should be applied to the EPOD analysis to account for the delay between the events of POD and EPOD domains. The residuals are calculated to find for the optimum multi-time delay. Residuals are defined as the normalized discrepancy between the rank-three reconstructed and predicted velocity wakes by POD, (with the first three velocity POD modes) and EPOD, (with the first 3 pressure POD modes), respectively, ( ) ( ) The computed residual is integrated over the two-dimensional spatial domain but still remains as a function of temporal snapshots. Thus to have a better comparison, the mean residual is represented in Fig to 5.12 for varying multi-time delay parameter (from 5 to 60 with increments of 5) at each of the four studied planes predicted with three different pressure fields. The actual time delay ( ) can be found by multiplying the corresponding multi-time delay parameter ( ) by the time step used in CFD measurements ( seconds). 72

85 Figure 5.10: Overall residuals of a) u b) v and c) w as functions of multi-time delay parameter for prediction based on pressure information at the obstacle surface 73

86 Figure 5.11: Overall residuals of a) u b) v and c) w functions of multi-time delay parameter for prediction based on pressure information at the mounted plate 74

87 Figure 5.12: Overall residuals of a) u b) v and c) w functions of multi-time delay parameter for prediction based on pressure information at the obstacle surface and mounted plate 75

88 The optimum multi-time delay parameter leading to the lowest mean residual (Table (5.1) to (5.3)) and therefore, the highest correlation between the POD and EPOD events is then chosen so that a compromise between the three components of velocity is made (overall residual). Table 5.1: Optimum multi-time delay parameters of EPOD using obstacle pressure sensors u v w Overall Table 5.2: Optimum multi-time delay parameters of EPOD using plate pressure sensors u v w Overall

89 Table 5.3: Optimum multi-time delay parameters of EPOD using plate plus obstacle pressure sensors u v w Overall The next step is to determine which of the three pressure fields best predicts the planar velocity wakes. For that purpose, the overall residual of each planar velocity field predicted with different pressure fields using their optimum multi-time delay (found from Table (5.1) to (5.3)) is investigated in Fig It is observed that on the one hand, the pressure information acquired at the plate exhibits the highest correlation with velocity POD modes at, and. The reason is attributed to the distribution of pressure within the stream-wise extent of the plate, which in turn is more capable of expressing the stream-wise variations of fluctuations associated with the stream-wise convection of shed vortices. On the other hand, using pressure signals recorded at the obstacle lateral walls lead to the best prediction of velocity field among all the other imposed pressure fields. This behavior can be explained through analyzing the POD pressure modes of obstacle (Fig. 5.14) and plate (Fig. 5.15) data associated with their optimum multi-time delay parameter ( for obstacle and for plate pressure) used to predict the velocity wake at demonstrated in Fig and 5.17, respectively. The benefit of applying the multi-time delay approach is clearly seen in Fig and 5.15 when compared to the single-time, shown in Fig. 5.7 and 5.8, respectively. By using the multi-time delay technique, extra information accounting for the delay between the pressure and velocity events is recovered and thus the arrangement of pressure data within the two-point 77

90 temporal correlation matrix is changed (Eq. (2.33)) such that a higher correlation with the velocity wake of interest is obtained. As is seen in Fig. 5.14, the pair of first harmonics are the two most energetic POD modes while the shift mode is the third mode for the pressure at the obstacle with a delay of. The common modes captured for the pressure field and velocity wake at suggest the improvement in correlations compared to the case of obstacle pressure POD with zero time delay observed in Fig. 5.7 where not all the common energetic modes between this field and velocity at were captured. Comparison between the pressure POD analysis at the plate surface with (Fig. 5.8) and (Fig. 5.15) also addresses the change in energy content of the three modes. For the predicted velocity wakes, both Fig and 5.17 represent the first harmonics as the most significant energetic modes followed by the shift mode. The main difference between the velocity predictions using obstacle and plate pressure fields is due to the higher level of energy captured for the first mode representing one of the modes of shedding first harmonics (60%) and the shift mode (4%) by obstacle pressure state-space compared to that of plate, which estimates the same energy content of 50% for the first harmonic pair and only less than 1% kinetic energy stands for the shift mode. The higher correlation (the better prediction) between velocity and obstacle pressure compared to the plate pressure seems to arise from the higher amount of kinetic energy captured for the shift mode. This leads to the better prediction as the POD modal analysis of (Fig. 5.4) showed that shift mode indicates the most dominant dynamic of this velocity field. This originates from the corresponding pressure POD eigenfunctions used for velocity predictions where for the obstacle pressure (Fig. 5.14), the shift mode accounts for 12% of the total kinetic energy and only 4% of that is represented by the plate pressure (Fig. 5.15). It is also more expected for sensors on the obstacle (specifically at the obstacle tip) to be capable of capturing the velocity fluctuations at rather than plate as the pressure information read at the plate less contains the global dynamics at the tip due to the fact that distinct flow behavior distinguishes the tip region topology from rest of the wake including the base region adjacent to the plate. 78

91 Figure 5.13: Residuals of a) u b) v and c) w for the optimum multi-time delay parameter 79

92 Figure 5.14: POD representation of the pressure wake at the obstacle lateral walls with application of multi-time delay approach ( ). a) Kinetic energy distribution of POD modes, b) PSD function of POD temporal modes and c) POD spatial modes 80

93 Figure 5.15: POD representation of the pressure wake at the plate with application of multi-time delay approach ( ). a) Kinetic energy distribution of POD modes, b) PSD function of POD temporal modes and c) POD spatial modes 81

94 Figure 5.16: EPOD representation of the predicted velocity wake at based on pressure sensors at the obstacle surface. a) Kinetic energy distribution of EPOD modes, b) PSD function of EPOD temporal modes and c) EPOD spatial modes 82

95 Figure 5.17: EPOD representation of the predicted velocity wake at based on pressure sensors at the mounted plate. a) Kinetic energy distribution of EPOD modes, b) PSD function of EPOD temporal modes and c) EPOD spatial modes 83

96 5.3 Optimum location of transducers The observations made in section 5.2 indicate that remote-sensing based predictions (i.e. EPOD) are indeed biased when it comes to spatial placement of sensors. This leads to the necessity of having a quantified evaluation of optimum locations, where transducers should be positioned to enhance the predictions. The term, optimum, refers to the location exhibiting the highest correlation between the CSs characterizing the remote and local domains. This chapter aims to find the optimum spatial locations at the obstacle surface and mounted plate to read pressure data at for predicting the planar velocity field around the obstacle. This is done by projecting the POD temporal coefficients of velocity field onto the pressure field (Eq. (2.31)), where the detailed mathematical procedure is explained in section Fig to 5.21 demonstrate the spatial correlation between the most energetic temporal evolution of velocity POD modes and the pressure field on the obstacle lateral walls and plate. It is seen that all the three dominant POD modes of velocity (the pair of first harmonics and shift mode) are best captured on the plate, downstream from the obstacle for the nearest plane to the base plate ( ). A 90 phase shift between the maximum and minimum correlation location of mode 1 and mode 2 appears along the direction at which the vortices are being convected. As we move further than the plate, the correlation between the and second mode (which the second mode of first harmonics) and shift mode increases with the pressure signals at the obstacle surface (with points representing the highest correlation located upper along the obstacle span as we move from to for mode three) and decreases with that of plate. But still modes two and three show the same level of correlation with both obstacle and plate at these planes until, where obstacle is more capable of capturing the velocity dynamics of modes two and three (where the area at the very tip of the obstacle shows the highest correlation with mode three) rather than the plate. This behavior is illustrated to be more significant for the third mode. It should be noted that mode one (also the first mode of first harmonics) is consistently higher correlated with the plate even at. 84

97 Figure 5.18: Correlation between the surface pressure data and the POD temporal coefficients of velocity field. Figure 5.19: Correlation between the surface pressure data and the POD temporal coefficients of velocity field. 85

98 Figure 5.20: Correlation between the surface pressure data and the POD temporal coefficients of velocity field. Figure 5.21: Correlation between the surface pressure data and the POD temporal coefficients of velocity field. Recalling that the regions of the highest correlation correspond to the locations where the pressure probes should be mounted, these discrete positions are quantitatively presented in the 86