The three-dimensional structure of the flow behind a heaving and pitching finite-span wing is investigated using dye flow visualization at a Reynolds number of 164. Phase-locked image sequences, which are obtained from two orthogonal views, are combined to create a set of composite images that give an overall sense of the three-dimensional structure of the flow. A model of the vortex system behind the wing is constructed from the image sequences. Variations of the Strouhal number, pitch amplitude and heave/pitch phase angle are qualitatively shown to affect the structure of the wake.

A visual study of the flow structures behind a heaving and pitching finite-span wing

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Foundations Of Potential Theory

Vorticity transport is analysed within the leading-edge vortex generated on a rectangular flat plate of aspect ratio 4 undergoing a starting rotation motion in a quiescent fluid. Two analyses are conducted on the inboard half of the blade to better understand the vorticity transport mechanisms responsible for maintaining the quasi-equilibrium state of the leading-edge vortex. An initial global analysis between the and spanwise positions suggests that, although spanwise velocity is significant, spanwise convection of vorticity is insufficient to balance the flux of vorticity from the leading-edge shear layer. Subsequent detailed analyses of vorticity transport in planar control volumes at the and spanwise positions verify this conclusion and demonstrate that vorticity annihilation due to interaction between the leading-edge vortex and the opposite-sign layer on the plate surface is an important, often dominant, mechanism for regulation of leading-edge-vortex circulation. Thus, it provides an important condition for maintenance of an attached leading-edge vortex on the inboard portion of the blade.

Vorticity transport in the leading-edge vortex on a rotating blade

Experiments on leading-edge vortex (LEV) growth and detachment from a plunging profile have been conducted in a free-surface water tunnel Direct-force and velocity-field measurements have been performed at a Reynolds number of Re = 10,000, a reduced frequency of k = 025, and a Strouhal number of St = 016, for three varying leading-edge geometries The leading-edge shape is shown to influence the shear layer feeding the LEV, and thus to some extent the development of the LEV and associated flow topology This effect in turn influences the arrival time of the rear (LEV) stagnation point at the trailing edge, which, once breached, constitutes a detachment of the LEV It is found that despite minor phase changes in LEV detachment through leading-edge shape, the position of the trailing edge (chord length) should be chosen as the characteristic length scale for the vortex separation process

Characteristic length scales for vortex detachment on plunging profiles with varying leading-edge geometry

1 Complex Numbers Sums and Products Basic Algebraic Properties Further Properties Vectors and Moduli Complex Conjugates Exponential Form Products and Powers in Exponential Form Arguments of Products and Quotients Roots of Complex Numbers Examples Regions in the Complex Plane 2 Analytic Functions Functions of a Complex Variable Mappings Mappings by the Exponential Function Limits Theorems on Limits Limits Involving the Point at Infinity Continuity Derivatives Differentiation Formulas Cauchy-Riemann Equations Sufficient Conditions for Differentiability Polar Coordinates Analytic Functions Examples Harmonic Functions Uniquely Determined Analytic Functions Reflection Principle 3 Elementary Functions The Exponential Function The Logarithmic Function Branches and Derivatives of Logarithms Some Identities Involving Logarithms Complex Exponents Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions 4 Integrals Derivatives of Functions w(t) Definite Integrals of Functions w(t) Contours Contour Integrals Some Examples Examples with Branch Cuts Upper Bounds for Moduli of Contour Integrals Antiderivatives Proof of the Theorem Cauchy-Goursat Theorem Proof of the Theorem Simply Connected Domains Multiply Connected Domains Cauchy Integral Formula An Extension of the Cauchy Integral Formula Some Consequences of the Extension Liouville's Theorem and the Fundamental Theorem of Algebra Maximum Modulus Principle 5 Series Convergence of Sequences Convergence of Series Taylor Series Proof of Taylor's Theorem Examples Laurent Series Proof of Laurent's Theorem Examples Absolute and Uniform Convergence of Power Series Continuity of Sums of Power Series Integration and Differentiation of Power Series Uniqueness of Series Representations Multiplication and Division of Power Series 6 Residues and Poles Isolated Singular Points Residues Cauchy's Residue Theorem Residue at Infinity The Three Types of Isolated Singular Points Residues at Poles Examples Zeros of Analytic Functions Zeros and Poles Behavior of Functions Near Isolated Singular Points 7 Applications of Residues Evaluation of Improper Integrals Example Improper Integrals from Fourier Analysis Jordan's Lemma Indented Paths An Indentation Around a Branch Point Integration Along a Branch Cut Definite Integrals Involving Sines and Cosines Argument Principle Rouche's Theorem Inverse Laplace Transforms Examples 8 Mapping by Elementary Functions Linear Transformations The Transformation w = 1/z Mappings by 1/z Linear Fractional Transformations An Implicit Form Mappings of the Upper Half Plane The Transformation w = sin z Mappings by z2 and Branches of z1/2 Square Roots of Polynomials Riemann Surfaces Surfaces for Related Functions 9 Conformal Mapping Preservation of Angles Scale Factors Local Inverses Harmonic Conjugates Transformations of Harmonic Functions Transformations of Boundary Conditions 10 Applications of Conformal Mapping Steady Temperatures Steady Temperatures in a Half Plane A Related Problem Temperatures in a Quadrant Electrostatic Potential Potential in a Cylindrical Space Two-Dimensional Fluid Flow The Stream Function Flows Around a Corner and Around a Cylinder 11 The Schwarz-Christoffel Transformation Mapping the Real Axis onto a Polygon Schwarz-Christoffel Transformation Triangles and Rectangles Degenerate Polygons Fluid Flow in a Channel Through a Slit Flow in a Channel with an Offset Electrostatic Potential about an Edge of a Conducting Plate 12 Integral Formulas of the Poisson Type Poisson Integral Formula Dirichlet Problem for a Disk Related Boundary Value Problems Schwarz Integral Formula Dirichlet Problem for a Half Plane Neumann Problems Appendixes Bibliography Table of Transformations of Regions Index

Complex variables and applications

We employ experiments to study aspect ratio ( ) effects on the vortex structure, circulation and lift force for flat-plate wings rotating from rest at 45° angle of attack, which represents a simplified hovering-wing half-stroke. We use the time-varying, volumetric data of Carr et al. (Exp. Fluids, vol. 54, 2013, pp. 1–26), reconstructed from phase-locked, phase-averaged stereoscopic digital particle image velocimetry (S-DPIV), and an volumetric data set matching the span-based Reynolds number ( ) of . For and of ( – ), we directly measure the lift force. The total leading-edge-region circulation for and 4 compares best overall using a span-based normalization and for matching rotation angles. The total circulation increases across the span to the tip region, and is larger for . After the startup, the total circulation for each has a similar slope and a slow growth. The first leading-edge vortex (LEV) and the tip vortex (TV) for move past the trailing edge, followed by substantial breakdown. For the outboard, aft-tilted LEV merges with the TV and resides over the tip, although breakdown also occurs. Where the LEV is ‘stable’ inboard, its circulation saturates for and the growth slows for . Aft LEV tilting reduces the spanwise LEV circulation for each . Both positive and negative axial flow are found in the first LEV for and 4, with the positive component being somewhat larger. This yields a generally positive (outboard) average vorticity flux. The average lift coefficient is essentially constant with from 1 to 4 during the slow growth phase, although the large-time behaviour shows a slight decrease in lift coefficient with increasing . The S-DPIV data are used to obtain the lift impulse and the spanwise and streamwise components contributing to the lift coefficient. The spanwise contribution is similar for and 4, due to similar trailing-edge vortex interactions, LEV saturation behaviour and total circulation slopes. However, for the streamwise contribution is much larger, because of the stronger, coherent TV and aft-tilted LEV, which will create a relatively lower-pressure region over the tip.

Aspect-ratio effects on rotating wings: circulation and forces

We investigate experimentally the unsteady, three-dimensional vortex formation of low-aspect-ratio, trapezoidal flat-plate fins undergoing rotation from rest at a 90° angle of attack and Reynolds numbers of O(103). The objectives are to characterize the unsteady three-dimensional vortex structure, examine vortex saturation, and understand the effects of the root-to-tip flow for different velocity programs. The experiments are conducted in a water tank facility, and the diagnostic tools are dye flow visualization and digital particle image velocimetry. The dye visualizations show that the low-aspect-ratio plate produces symmetric ring-like vortices comprised mainly of tip-edge vorticity. They also indicate the presence of the root-to-tip velocity. For large rotational amplitudes, the primary ring-like vortex sheds and a secondary ring-like vortex is generated while the plate is still in motion, indicating saturation of the leading vortex. The time-varying vortex circulation in the flow symmetry plane provides quantitative evidence of vortex saturation. The phenomenon of saturation is observed for several plate velocity programs. The temporal development of the vortex circulation is often complex, which prevents an objective determination of an exact saturation time. This is the result of an interaction between the developing vortex and the root-to-tip flow, which breaks apart the vortex. However, it is possible to define a range of time during which the vortex reaches saturation. A formation-parameter definition is investigated and is found to reasonably predict the state corresponding to the pinch-off of the initial tip vortex across the velocity programs tested. This event is the lower bound on the saturation time range.

Vortex formation and saturation for low-aspect-ratio rotating flat-plate fins

High-incidence lift generation via flow reattachment is studied. Different reattachment mechanisms are distinguished, with dynamic manoeuvres and tip vortex downwash being separate mechanisms. We focus on the latter mechanism, which is strictly available to finite wings, and isolate it by considering steadily translating wings. The tip vortex downwash provides a smoother merging of the flow at the trailing edge, thus assisting in establishing a Kutta condition there. This decreases the strength/amount of vorticity shed from the trailing edge, and in turn maintains an effective bound circulation resulting in continued lift generation at high angles of attack. Just below the static lift-stall angle of attack, strong vorticity is shed at the trailing edge indicating an increasingly intermittent reattachment/detachment of the instantaneous flow at mid-span. Above this incidence, the trailing-edge shear layer increases in strength/size representing a negative contribution to the lift and leads to stall. Lastly, we show that the mean-flow topology is equivalent to a vortex pair regardless of the particular physical flow configuration.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/C21AE509A6471CA0AB5525B454E40903/S0022112016008491a.pdf/div-class-title-on-the-mechanism-of-high-incidence-lift-generation-for-steadily-translating-low-aspect-ratio-wings-div.pdf

On the mechanism of high-incidence lift generation for steadily translating low-aspect-ratio wings

We investigate experimentally the unsteady, three-dimensional vortex formation of lowaspect-ratio flat plates undergoing rotation from rest at fixed angles of attack and low Reynolds number (order 10 3 ). Two configurations are investigated: a trapezoidal plate at 90° angle of attack that is a simplified model of a fish-fin starting flow, and rectangular plates at 45° angle of attack that are very simplified models of a hovering-wing half-stroke. The objectives are to characterize the time-varying, three-dimensional vortex structure for various velocity profiles and geometries, to understand the effect of the significant root-to-tip flow on the vortex formation, and to investigate whether vortex saturation (“formation number”) effects are present. The experiments are performed in a water tank facility, and the diagnostic tools are dye flow visualization and digital particle image velocimetry (DPIV). For the fin-like starting configuration, the flow is dominated by the tip vortex and the overall flow structure is a symmetrical ring-like vortex. At high rotational amplitudes and speeds this vortex sheds while the plate is still in motion, indicating that it has saturated (achieved maximum circulation). The time-varying vortex circulation calculation, obtained from DPIV measurements in the symmetry plane, also shows evidence of saturation effects for some cases. For the majority of the velocity profiles and amplitudes tested, the behavior of the circulation with time is complex, making an objective determination of the saturation time difficult. This complexity is directly related to flow phenomena such as the strong rootto-tip velocity induced by the tip and side-edge vortices, and a Kelvin-Helmholtz-like instability inherent in the separated shear layers at this Reynolds number. For the wing-like starting case, rectangular wings of aspect ratio (AR) 2 and 4 are compared. Dye flow visualization reveals a strong spanwise (root-to-tip) flow over the entire wing and an attached helical leading-edge vortex (LEV) early in the motion, consistent with prior research. We observe LEV bursting for both AR’s, however for the AR = 2 wing the bursting occurs relatively later. For the AR = 2 case the flow consists of an overall, three-dimensional vortex loop made up of a connection among the LEV, the tip vortex (TV), and the trailingedge vortex (TEV), similar to previous studies. For the AR = 4 wing the flow is less coherent and exhibits massive separation near the tip late in the motion. For this flow the KelvinHelmoltz-like instability is also observed in the separated shear layers, and is relatively larger with distance toward the tip. This instability most likely contributes to the breakdown of the LEV structure, which has been found previously.

http://enu.kz/repository/2011/AIAA-2011-396.pdf

Vortex Formation and Saturation for Low-Aspect-Ratio Rotating Flat Plates at Low Reynolds Number

The use of flow field information to compute the fluid dynamic force on a body is investigated with specific application to experimental volumetric measurements The calculation method used avoids the explicit evaluation of the pressure on the boundaries It is shown that errors in the data introduce an artificial dependence of the calculations on the position origin, and also that these errors are amplified by the position vector A statistical description of the calculation variation associated with origin dependence is presented A method is developed that objectively determines an origin which reduces the effect of the amplified error The method utilises mathematical identities which relate the measurements to the main sources of error in a physically meaningful way, and is also found to be effective for changes of the external and internal boundaries of the fluid

Adam DeVoria

Papers

Aspect-ratio effects on rotating wings: circulation and forces

Vortex formation and saturation for low-aspect-ratio rotating flat-plate fins

On the mechanism of high-incidence lift generation for steadily translating low-aspect-ratio wings

Vortex Formation and Saturation for Low-Aspect-Ratio Rotating Flat Plates at Low Reynolds Number

On calculating forces from the flow field with application to experimental volume data