-v- TABLE OF CONTENTS PAGE Chapter Introduction 1 II The Formulation of Lagerstrom-Chester Theory 4 III The Solution of the Lagerstrom-Chester Paradox l0 IV A New Slip Condition 25 Forma[r]

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different terms in the turbulent kinetic energy balance equation. The computations were conducted at 5 degrees angle of attack at the low **Reynolds** number case and at 15 degrees for the higher **Reynolds** number case. The freestream Mach number for the low and higher **Reynolds** **numbers** cases are M = 0.4, and 0.2, respectively. The simulation results at the low **Reynolds** number showed a long laminar separation bubble in the forward part of the airfoil and an attached thick shear layer in the aft part of the airfoil. For this case, the computations of the drag coefficient revealed that the form drag contributed about 76% of the total drag and the friction drag contributed only 24% of the drag at an angle of attack of 5 degrees. It was observed that the flow near the reattachment consists of an outer region that resembles a turbulent mixing layer. It was also noted that the outer shear layer that originated from the separation bubble persisted up to the trailing edge at this low **Reynolds** number. The developing wall boundary layer along the wall recovers slowly toward an equilibrium boundary layer. It was also observed that turbulent intensities and **Reynolds** shear stresses peak near the reattachment region and in the middle part of the shear layer. The budget of the terms in the turbulent kinetic energy equation showed that the roles of the turbulent diffusion, advection, and dissipation terms change as the boundary layer evolves over the airfoil. In the separation bubble region, the dissipation is **small** compared to the diffusion and advection terms. Dissipation becomes larger than the diffusion and advection terms downstream of the reattachment point

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the Eulerian hydrodynamic conservation laws implemented through an implicit large eddy simulation (ILES) method- ology (Passot et al., 1988; Blaisdell et al., 1993). This is because it is commonly accepted that an ILES formulation of the Euler equations provides a good estimation for the Navier–Stokes equations in the limit of infinite **Reynolds** **numbers** (Bos and Bertoglio, 2006; Zhou et al., 2014; Sytine et al., 2000). However, two conditions must be enforced in order to satisfy the aforementioned assumption. Firstly, vor- ticity must be introduced via either boundary and/or initial conditions since the Euler equations are incapable of gener- ating vorticity from irrotational **flows**. Secondly, an artificial viscosity must be incorporated into the simulation mecha- nism to mimic the preservation of dissipative behavior of the Navier–Stokes equations in the inviscid limit (Moura et al., 2017). The ILES mechanism is a suitable approach for artifi- cial dissipation through the use of numerical truncation errors and is our simulation algorithm of choice for the high-fidelity numerical experiments in this investigation.

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The results in this paper have indicated how the **flows** generated during one stroke and rotation can increase lift production in the subsequent stroke. However, it would probably be inappropriate to view these interactions solely as a means of maximizing flight performance. More impressive than the absolute augmentation in lift due to the kinematics of the preceding stroke is the sensitivity of aerodynamic performance to subtle changes in those kinematics. **Small** changes in either the duration or angle of attack of the downstroke, as well as in the angular velocity of stroke reversal, could produce substantial changes in lift and drag production during the successive half-stroke. Such sensitivity both necessitates the use of a sophisticated control feedback system (Wehrhan, 1987; Hengstenberg, 1991; Egelhaaf, 1991) and allows for remarkable aerial maneuvers (Wagner, 1986). A recent behavioral analysis has shown that Drosophila actively controls the timing of ventral flip rotations along with changes in stroke amplitude during visually induced turns (Dickinson et al. 1993). Since changes in the timing of rotation can be kinematically similar to changes in the axis of wing rotation (Ellington, 1984c), such behavior could generate large and rapid changes in flight control forces.

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This example is the Mach 10.01 viscous flow of Nitrogen around a sphere of radius 0.1524 m. The free stream temperature and pressure are 200 K and 0.0468 Pa, respectively. The **Reynolds** number based on the radius of the sphere is 26.99, which is **small** enough to assume a laminar flow. The computational grid, shown in Figure 19 (left), consists of 2,995,100 nodes, 16,477,103 tetrahedra and 380,800 prisms and the surface of the sphere is discretized with 234,952 triangles. The near wall region contains 40 layers of prisms and it is approximately 0.683 radii thick. The initial CFL is 10 -5 , exponentially increasing to 10 in 3,000 iterations. The stabilization scheme is

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Despite the potential relationships among vortex ring formation, impulse and stroke ratio during pulsed jetting at intermediate Re, little is known about jet **flows** produced by squid hatchlings. Specifically, do paralarvae produce vortex rings at these scales? How fast is water expelled relative to swimming velocities and what stroke ratios are observed? How efficient is the jet in these **small** worlds? In the present study we seek to answer these questions by using digital particle image velocimetry (DPIV) to directly measure bulk ring properties (e.g. circulation, impulse, kinetic energy) and other jet features (e.g. L/D, jet velocity, vorticity structure) in free- swimming paralarval Doryteuthis pealeii [formerly Loligo pealeii (see Vecchione et al., 2005)]. Although some of the questions above were addressed briefly in a previously published overview paper on squid jetting throughout ontogeny (see Bartol et al., 2008), we present a more detailed and expansive data set in this paper.

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ity profile with an infiection point. In the Hall and Smith (1991) studies the boundary layer was restricted to the interactive classical P ran d tl type. However, in the studies presented here a more generalised boundary layer albeit with some restrictions is considered where this boundary layer could be of the triple-deck, the classical P ran d tl type or possibly a related form. The condition of the boundary layer with this particular infiection point profile represents a neutral point for the corresponding 2-D Rayleigh equa tion which possesses a non-trivial solution. It is at this neutral point th a t the 3-D non-linear interaction is deemed to begin. At the point of interac tion a critical layer is initiated and the vortex fiow develops d o w n stream ^ The 2-D infiection point condition is eroded as the three-dim ensionality of the vortex flow gradually develops downstream b ut the neutrality of the Rayleigh wave is preserved because the wave num ber remains real. The three-dim ensionality of the vortex evolves as the cross-flow derivative of the streamwise vortex component becomes non-zero. However, in this study we are concerned with the behaviour of the fiow in the neighbourhood of the initiation of the interaction. Therefore, although in C hapter 1 we see the erosion of the inflection point condition, we are sufficiently far enough up stream for this not to have a great effect on the solution obtained. In C hapter 2 however, where the region under consideration is much nearer to the neighbourhood of the initiation of the critical layer, the a:-scale is suf ficiently **small** th a t the inflection point condition and also the neutrality of the Rayleigh wave is preserved. We have m ade some assum ptions about the form of the wave such th a t for convenience we have defined the transverse component of the wave to consist of a single harm onic at the initial station. Therefore, the ^-dependency of the wave is taken as cos ^ qz where z is a

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Over the last three decades, most of the simulations for designing aircraft have been reduced to second order finite volume methods, mainly with RANS (**Reynolds** Averaged Navier-Stokes) models for turbulence. The usual trick for having an accurate solution with a RANS simulation consists in performing a sequence of computations on more and more refined meshes. However, the RANS approach can be inefficient for simulating some problems that are intrisically time dependent. Indeed, RANS model is a time averaged model, and thus stationary. For time dependent simulations, the most accurate method is called DNS (Direct Numerical Simulation) and consists in meshing the domain at the turbulent scale, without any turbulence model. However, due to the required mesh fineness, DNS is often considered for low **Reynolds** **numbers** only (since the number of cells increases as Re 9/4 ) and/or for academic configurations, for which efficient finite difference methods can be used. An intermediate solution, between DNS and RANS approach consists in applying a spatial filter to the Navier-Stokes equations. The resulting model is time dependent, but shall be closed for taking into account **small** scales. This method is called the Large Eddy Simulation (LES) approach. For both DNS and LES, the accuracy yielded by adaptive mesh refinement is harder to implement because the problem is time dependent. It would therefore imply theoretically performing a mesh adaptation at each time step, which is very costly. It also raises issues regarding dynamic load balancing as far as parallel computing is concerned.

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However, both theoretical frameworks used for analyzing detached vortices on airfoils, effective camber increase and detached vortex lift, were developed to explain steady-state conditions. The **small** vortices bound in wing corrugations are stable and produce a constant increase in effective camber (Rees, 1975; Newman et al. 1977). In the case of delta-wing aircraft, the large leading edge vortex is stable because of the presence of constant axial flow through the vortex center. Thus, the corrugations of insect wings and the axial flow on delta-wing aircraft are analogous in that they stabilize vortices on the lifting surface. Wu et al. (1992) have recently reviewed a number of additional mechanisms in airfoil design that enhance lift production through the stabilization of detached vortices. In contrast, the **flows** on the two-dimensional wing model started from rest are unsteady, and the performance of the wing eventually decays as the leading edge vortex is shed. However, lift augmentation by detached vortex lift may still be crucial for insect flight for two reasons. First, by analogy with delta wings, the leading edge vortex on actual insect wings might be stabilized by the presence of axial flow. Just such a mechanism was seen by Maxworthy (1979) in his three-dimensional modelling studies of the fling motion. Second, and perhaps more important, is the previous calculation that owing to the short translation distance only about one quarter cycle of von Karman shedding is encountered by the wing during each half-stroke. The limited translation distance of the wing stroke enables the fly to reap the benefits of increased lift from the temporary presence of a leading edge vortex, even though the flow is not stable over many chord lengths of translation. This strategy of exploiting the benefits of unstable leading edge bubbles over a short wing stroke was previously proposed by Ellington (1984b) in his discussion of delayed stall.

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Various hybrid methods that couple continuum with molecular solvers seamlessly within the same simulation have been proposed. These can be broadly categorised into either domain decomposition (DD) (O’Connell & Thompson 1995; Hadjiconstantinou & Patera 1997; Nie et al. 2004) or heterogeneous multiscale methods (HMM) (Ren & E 2005; Yasuda & Yamamoto 2008; Kessler, Oran & Kaplan 2010; Asproulis, Kalweit & Drikakis 2012; Borg, Lockerby & Reese 2013a). DD methods are applicable to configurations where the domain can be split distinctly into molecular and continuum-fluid regions, with overlapping regions placed at the molecular–continuum interfaces where two-way coupling based on fluxes or state properties occurs. Typically a Schwarz alternating method is used to converge the solutions in the disparate regions iteratively (Hadjiconstantinou & Patera 1997). A review of DD methods for dense fluids can be found in Mohamed & Mohamad (2010). As an alternative, the HMM is more effectively applied to rheological **flows**, or problems in which a constitutive relation and boundary information do not exist. The HMM places a continuum-fluid solver grid over the whole domain, and microscopic simulations dispersed at the nodes of the computational grid provide the missing information.

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In Eulerian schemes based on a fixed mesh there are two very popular methods for the capturing of the phase interface, namely the volume of fluid (VOF) method [85] and the level set (LS) method [142, 115, 112]. A combination of VOF and LS proposed in [141, 154] has shown to produce a robust method for **flows** with complex geometries and interface deformations in the setting of incompressible fluids. The disadvantage of Eulerian methods applied to multiphase flow problems is a significant amount of numerical dissipation, which requires proper interface reconstruction techniques to avoid excessive smearing of the phase boundary and to restore a sharp interface. However, this may become rather cumbersome in complex configurations. The level set method also needs a periodic re- initialization to restore the signed distance function property of the level set function, which requires the additional solution of a Hamilton-Jacobi equation. Furthermore, VOF has difficulties in simulating highly **compressible** multi-phase **flows**, hence most of the applications of the VOF method are restricted to the simulation of incompressible fluids. While the VOF method is perfectly conservative, the level-set approach is not. Due to the piecewise linear interface reconstruction (PLIC) used in the VOF context and due to the signed distance function property of the level-set method, both approaches are called sharp interface methods.

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molecular simulation. This is discussed in more detail in Borg, Lockerby & Reese (2013c), Patronis et al. (2013) and Patronis & Lockerby (2014). The internal-flow multiscale method (IMM) of Borg et al. (2013c) was developed instead for these kind of geometries, and for isothermal, incompressible, steady **flows**. The method consists of iteratively solving a one-dimensional (1D) steady continuity equation; MD simulations are applied at regularly spaced nodes along the streamwise direction of the domain, with heights chosen to match the local geometry of the full channel (see figure 1). Measurements of mass flow rates are extracted from these MD subdomains, which are then used to evaluate macro continuity errors. Based on these errors the equivalent pressure gradients (i.e. body forces) applied to each MD simulation are adjusted. A converged solution is obtained after only a few (typically two or three) iterations. As the fluid and wall conditions are defined from a microscopic (intermolecular) perspective in the micro elements, non-continuum models for slip, density layering, and non-local viscosity and stress are not required in the macro description. The flow rate is the sole parameter measured in the micro solutions, and provides an adequate and sufficient micro-to-macro coupling. However, several problems with this simple approach have become evident, in particular, the fluid compressibility is not incorporated and the method is only applicable to steady **flows**.

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is especially true for high speed ﬂows at high **Reynolds** number where the correct resolution of thin boundary layer requires very ﬁne meshes. Let’s note also the hy- brid pressure-density based method [21], which is based on the AUSM+up ﬂuxes, but still uses a pressure cor- rection loop similar to PISO algorithm and the maximal time step is therefore also limited.

Computational analysis is carried out to solve a flow field in two-dimensional NACA 4415 Aerofoil to analyze flow characteristics, and the effect of increase **Reynolds** number.Aerofoil is generated by using an online aerofoil generator from which the co-ordinates were imported to create the geometry of that aerofoil. After obtaining the co-ordinates, they were imported to Ansys 14.0 Workbench for creating the desired geometry. The 2D view of the aerofoil was shown in Fig.2.1. Standard k−ε model is used to predict the flow field. A UN steady

In terms of operation and effectiveness, bubble-based phenomena will not be considered because bubble air, in **small** quantities, can be assimilated by the fluid and does result in significant performance losses. Besides, bub- ble behavior is very complicated to simulate using software packages and its analysis depends largely on observ- er subjectivity. Thus, type 5 vortex is discarded. As for superficial vortices, they have no impact on pump suction, which means type 1 and 2 vortices will not be considered. This all means that there are only three different types of vortices potentially impacting submergence: 3, 4 & 6. The first two do not imply air entry and so they are dis- carded because of their very morphological nature. Hence, submergence is conditioned by type 6 vortex. Critical submergence is determined when the type 6 vortex vertex reaches the upper part of the suction structure. In that moment, a massive entry of air occurs, as represented in Figure 3.

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A recently developed second-moment **Reynolds** stress model from DLR was applied to two challenging high-lift **flows**: (1) transonic flow over the ONERA M6 wing, and (2) subsonic flow over the DLR-F11 wing-body configuration from the second AIAA High Lift Prediction Workshop. The **Reynolds** stress model computational results were contrasted with those obtained from other simpler turbulence models. Multiple codes, grids and turbulence models were used to compute transonic, high angle-of-attack solutions for the ONERA M6 wing. At most span stations, the range of results showed only relatively **small** variations from each other, with little difference between RSM and other models. However, at the 90% span station, the range of results showed larger variation in the prediction of the shock location and shock-induced separation. The one- and two-equation model predictions showed slightly larger areas of shock-induced separation than the RSM predictions, but all models were fairly close to the limited experimental surface pressure data, making it difficult to draw firm conclusions regarding model fidelity. A previously reported significant overprediction of shock-induced separation by the SA and SST models was not repeated in the current study. However, there is some indication of a multi-valued solution with TAU using SA-neg on Grid 1. This possibility remains an open question.

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A study has been carried out to compare steady jet and synthetic jet heat transfer distributions at low **Reynolds** **numbers**. Both jets issued from a 5mm diameter orifice plate with air for the steady jet being supplied by a compressor via a plenum chamber. Tests were conducted for **Reynolds** **numbers** ranging from 1000 to 4000, and for non-dimensional surface to jet exit spacings (H/D) from 1 to 6. Dimensionless stroke length (L o /D) for the synthetic jet was held constant at 8. A

A closer look at performance of finite aspect ratio, three- dimensional flapping foils is warranted at higher **Reynolds** **numbers**, in order to design vehicles to swim with optimal kinematics and maximizing thrust while optimizing efficiency. In an effort to understand this trade-off, tests were performed using a three-dimensional, linearly tapered hydrofoil forced to move in roll and pitching motions, and the measured force and hydrodynamic efficiency data are presented, discussed and compared with data reported from previous tests (Flores, 2003; McLetchie, 2004; Polidoro, 2003; Read, 2000).

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This article attempts to highlight characteristics of the aerodynamic forcing on a rigid circular cylinder experiencing dry galloping vibrations. Observations from a series of wind tunnel tests are studied comparatively with the literature on rain-wind cable vibrations and on flow past inclined lifting bodies such as missiles, for drawing similarities. Un- steadiness and spatial variation of the flow, both previously undermined, are significant during the large cylinder motions recorded. Thus, they are here suspected to play a role in triggering unstable behaviour. Instabilities were restricted to specific ranges of cable- wind angles and **Reynolds** **numbers**. The transitional features identified refute the view of simple bursting separation bubbles that rhythmically produce lift and suggest that there is a multitude of paths for energetically feeding dry galloping. Finally explanations are provided and a mechanism incorporating unstable features is proposed for future modelling.

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Undoubtedly signiﬁcant insight into the ﬂuid dynamics of a disk has been achieved through these numerous experimental and numerical simulations. Nevertheless, there are still some aspects that are not fully understood and merit further investigation. With regard to the ﬂow conﬁgurations, the steady ﬂow over a ﬁxed circular disk or a freely falling/rising circular disk in a stationary ﬂuid were considered in the previous studies. In both of these cases, the ﬂow relative to the disk is only in one direction, and the wake ﬂow is always behind the disk. By contrast, the oscillatory ﬂow over a circular cylinder or a circular cylinder oscillating in a quiescent ﬂuid has been extensively studied (see, e.g., Bearman 1984; Williamson & Govardhan 2008; An et al. 2011). Unfortunately, to the best of our knowledge, similar studies have not yet been performed for the circular disk. For a disk oscillating in a ﬂuid at rest, the disk moves back and forth, accelerates and decelerates. Some new and interesting phenomena are expected to be revealed. This is the motivation of the present study, and four questions are to be answered: (i) what is the wake of the oscillating disk, (ii) how many types of ﬂow regimes exist, (iii) what is the dependence of the threshold on the oscillating amplitude and **Reynolds** number, and (iv) what is the connection between the wake instabilities of a ﬁxed disk and those of an oscillating disk.

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