Manifestation of the Finite Vortex—Lift, Thrust and Drag 249

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Manifestation of the ﬁnite vortex—lift, thrust and drag

W. Liebe1 & R. Liebe2 1Technical University Berlin, Berlin, Germany. 2Siemens Power Generation, Mülheim a.d.R., Germany.

Abstract

Force production in fluids is effectively achieved by the generation of large, unsteady vortices. The underlying mechanisms are analyzed, new phenomena are described, and the so-called finite vortex model (FVM) is applied to both engineering and nature. First, some fundamentals are given: the characteristic pressure and velocity distributions in ideal vortices; the mathematical idea of the potential vortex (lifting line theory, etc.); the so-called natural vortex, its motion with 2D/3D applications in nature. Second, focus is on the ‘edge flow mechanism’, the finite vortex concept and the ‘Mantelströmung’ (circulation). Inspiration to this physical model came from observation of swimming and flying in nature. Vortex formation and growth are newly discussed, and a mechanism is proposed, which establishes the Kutta condition: The transverse flow around an edge causes flow separation, an initial deflection of the shear layer due to viscous effects; then a strong suction from high local velocities emerges. Apparently the near-field effects of the amplified vortex roll-up dominate. So it is sufficient to consider the dynamics of a finite size vortex, interacting with the structure. The resulting FVM analyses with periodically unsteady flow patterns, force generation and power requirements correlate well with flapping experiments and Navier–Stokes results. In a third part the size of the finite vortices and their detachment frequencies are correlated and quantified. New physical interpretations are given for steady/unsteady as well as high/low Re number flows and boundary layer effects. Several examples use approximations from the conservation of mass and momentum: detachment and repetitive formation of ‘starting vortices’ from an accelerating airfoil (AF); lift force from an inclined, fixed AF (plate) in a steady flow; thrust/lift forces from an oscillating AF (plate) with periodically unsteady vortex formation. Next the FVM is applied to swimming (caudal fin fish propulsion) and flying in nature. Thrust, swimming speed, etc. can be simulated in agreement with pitching AF tests over a wide range of frequencies/amplitudes (shark, dolphin). Thrust from pure flapping wings is interpreted with a ‘gyroscopic effect’ from the vortex cylinder (bird flight). Conclusions concerning the key role of vortex formation/control for thrust/lift and drag in nature as well as in engineering are presented. FVM capabilities and limits are summarized, and future work is suggested.

1 Introduction

Reaction forces from fluid flow around arbitrary bodies in nature, i.e. lift, thrust and drag, are primarily governed by vortices. Moving fins of fish or flapping wings of birds generate and control large, periodically unsteady vortices. The key design element for efficient generation of vortices in nature is the sharp edge, which moves transversely relative to the surrounding fluid. In contrast to this, aircraft engineering employs fixed wings, brought to very high speed. So man needs additional powerful thrust engines and airports (propeller or jet engines, starting/landing tracks). This paper focuses on the following topics: The vortex • fundamentals of vortex motion; the ‘natural vortex’ with examples in nature; • the edge flow and the associated finite vortex model (FVM); • estimates of the finite vortex size (from conservation of mass and momentum).

Force generation in fluids 1. Applications of the FVM in engineering: • lift from a fixed inclined plate in a steady flow; • the trailing edge (TE) of an aircraft wing; generation of lift; • thrust (lift) from a pitching airfoil (AF) with periodically unsteady vortices; • experiments on oscillating AFs and verification of the FVM. 2. Applications of the FVM in nature: • analysis of fish locomotion; swimming parameters, thrust calculation; • analysis of thrust (lift) from flapping wings; bird flight and a gyroscopic effect.

2 Ideal vortices

A vortex in general is whirling or circulating fluid. It is useful but not sufficient to apply the mathematical idealization of the well-known ‘potential vortex’to reality. Real vortices have many surprising features [1]: In the early days Leonardo da Vinci was fascinated by water waves and vortex motion. Later L. Prandtl [2] studied flow separation and vortex roll-up, being inspired by Helmholtz [3] and Thomson’s famous laws for vortex motion in inviscid, ideal fluids. In addition Prandtl [4] extensively studied the behavior of real fluids and vortex motion in water experiments. There are a number of excellent books on aero- and hydrodynamics including vortex formation, motion and decay: the works of Van Dyke [5], Lugt [6], Kuethe [7] show photographs, computer simulations as well as engineering applications in a very clear and balanced fashion. Basically there are two stable configurations of stationary circular motion of fluids, the ‘confined or closed’ vortex and the ‘unconfined or open’ vortex.

2.1 The closed vortex

The simplest mode is a fluid rotation with constant angular velocity ω, shown in Fig. 1. This mode is characterized by a linear rise of the tangential fluid velocity vϕ along the radius r. Here we have no friction or viscous action between neighboring streamlines, though the fluid particles do experience a ‘self-rotation’. It is well known that centrifugal forces and thus the static pressure

Figure 1: The closed or conﬁned vortex: steady motion of liquid in a rotating cylindrical vessel.

Figure 2: The open or unconﬁned vortex: steady state motion of liquid built-up around a rotating cylinder.

2 ps follow an r relationship. The rotating fluid has to be supported at its periphery to remain stationary. Therefore the term ‘closed’ vortex. Figure 1 refers to the popular example of a rotating cylindrical vessel, filled with fluid. After some time a stationary, closed vortex is established.

2.2 The open vortex

The other stable mode of a whirling fluid is resulting from a central ‘excitation’ in a still fluid. This ‘excitation’ may be represented by an immersed, rotating solid cylinder, and Fig. 2 depicts such a rotating, circular cylinder (angular velocity ω, diameter 2rs) being immersed in a still fluid, extending to infinity. Neighboring vortex rings are driven (retarded) at their inner (outer) boundary.

Theoretically this motion is extending to inﬁnity, therefore the term ‘open’ vortex. After build- up and on reaching steady state, we typically obtain a tangential ﬂuid velocity vϕ, which now decreases with increasing radius r. Consequently the static pressure ps increases with r up to the ambient level p∞.

2.3 The potential vortex

The mathematical idea of a flow potential Ø renders possible to determine the 2D distribution of the velocity vector v = (vr, vϕ) in an inviscid fluid. The potential Ø is well known from classical mechanics, where it was introduced in the theory of elasticity and the gravitational potential. In fluid mechanics the potential Ø yields the velocities from eqn (1) [8, 9]: v = grad Ø. (1)

Application to the potential vortex and with the aid of the continuity equation: ∂2Ø/∂r2 + (1/r)∂Ø/∂r + (1/r2)∂2Ø/∂ϕ2 = 0, (2) we obtain the solution for the velocities in the radial and the tangential direction as follows:

vr = 0, vϕ = constant/r = (/2π)(1/r). (3)

As a characteristic result, the potential vortex is described by the throughout constant circulation = 2π(vϕr). characterizes the vortex strength. The briefly outlined potential theory has taken a key role in aircraft engineering since 1910 up to modern times, the so-called panel methods. Also Prandtl’s early lifting line concept (1911) for aircraft wings is based on the theory of potential vortices. In addition, we are used to imposing the condition of smooth flow detachment to the TE. This condition was formulated by Kutta and Joukowski (1904) as a mathematical rule for fixed AFs. To illustrate the basic idea of the lifting line concept, Fig. 3 depicts Prandtl’s lifting line scheme. This powerful tool led to successful aircraft design for over a hundred years. The finite wing (span b) is replaced by a system of vortex carrying lifting lines, each being associated with constant circulations i, contributing to an overall distribution (y) along the wing span. The staggered system of ‘horseshoe-shaped’ vortex lines determines the distribution of wing lift FL(y) as well as the downwash, i.e. the characteristic deflected air velocity vD(y). Figure 3 also indicates the boundaries of the closed vortex sheet: two wing tip vortices (TV) and the so- called system of starting vortices rolling up at the TE at the very beginning of wing motion. These starting vortices (dotted line) lie in the past. They are formed initially during the start of the aircraft acceleration. Roll-up of the starting vortices gradually establishes a ‘smooth-flow detachment condition’ in the very beginning. This condition, however, has to be maintained throughout the entire flight by some physical mechanism. However, to date we do not have physical models to explain these mechanisms, which establish the condition of smooth flow along an aircraft wing. In addition, there is a lack of understanding of local flow around the TE of oscillating AFs, where the Kutta condition is not applicable at all. In this paper as well as in [10, 11] some ideas are presented on mechanisms that are able to continuously establish the ‘Kutta condition’ throughout high velocity cruising flight. The physical explanation of vortex formation and utilization of vortices (or the prevention of parasitic vortices) [12] cannot be achieved with the laws of Thomson; those laws assume the existence of vortices to begin with. Observation plus experiments are needed, as described in the very good review [13]. Recently more findings are utilized in aircraft engineering to prevent

Figure 3: Prandtl’s lifting line theory: The ﬁnite wing is replaced by a staggered system of ‘horse-shoe’ vortex lines, which determine the lift distribution FL(y) along the wing.

premature stall or to reduce the TVs (see Fig. 3), which cause the induced drag of the wing. Nowadays it is increasingly popular to reduce the roll-up of the TVs by ‘winglets’. Some contributions along these lines are given here, and some are found in [12, 14–16] with results on vortex formation (promotion and control), on vortex decay or on means to prevent the generation of parasitic vortices. Potential theory is successfully applicable to large Reynolds numbers—small viscous/large inertial forces—and human fixed wing aircraft design is just relying on high free-stream velocities v∞ with Re > 106–107. Therefore, aircraft design can basically assume inviscid fluids, and consequently the potential flow theory is well applicable. Getting back to the ideal potential vortex with its characteristic flow field (vϕ r) = constant: Here the so-called self-rotation of fluid particles is vanishing, or mathematically rot v = 0. In addition to this, the potential vortex has a typical singularity in its center (r = 0), as well as at flows around sharp edges (velocities are rising to infinity). Finally it should be kept in mind, that potential vortices describe an inviscid fluid with zero mass (ν = 0, ρ = 0), [17].

3 Real vortices

A more realistic open vortex distinguishes two regimes with a different ﬂow structure and velocity/pressure distribution, similar to the Rankine vortex [18], see also Fig. 4:

• An outer ‘O’ regime, extending outwards from r ≥ rs. In this regime the vortex has a potential character. • An inner or central ‘I’ regime, extending between 0 ≤ r ≤ rs. There is no singularity at r = 0. The ﬂuid behaves like a ‘solid cylinder’ of a diameter 2rs. It rotates with a constant angular velocity ω, and rot v = 0, i.e. the ﬂuid particles have a nonzero ‘self-rotation’.

Figure 4: The natural vortex with typical distributions of the tangential velocity vϕ(r) and the static pressure ps(r) in the ‘I’ and ‘O’ regimes.

3.1 The natural vortex

The natural vortex utilizes those two distinctly different regimes [19]; it is schematically shown in Fig. 4. Formation and roll-up of the natural vortex is due to ﬂuid friction [20]. In a natural vortex both viscosity and density are nonzero (real ﬂuids: ν, ρ = const). In both mentioned regimes ‘O’ and ‘I’ the tangential velocity vϕ(r) and static pressure ps(r) have a very different radial distribution, summarized in the following table:

Regime Inner ‘Solid’ Outer ‘Potential’ Range ‘I’ r < rs ‘O’ r ≥ rs

Velocity vϕ(r) Linear Hyperbolic Static pressure ps(r) Parabolic Parabolic ‘Self-rotation’ rot v value Nonzero Zero

The distribution of vϕ(r) and ps(r) can be directly obtained from the general force balance, i.e. the Navier–Stokes equations of motion for viscous ﬂuids with ν, ρ = constant. If we restrict to stationary 2D ﬂow and also add the conservation of mass, two equations emerge for the radial direction [8, 21]: 2 ν 1/r ∂/∂r (r∂vϕ/∂r) − vϕ/r = 0, (4) − 2 = ∂ps/∂r ρvϕ/r 0.

In the regime ‘O’it can be visualized that a circulating ﬂuid particle dm is subjected to a centrifugal = 2 force (outwards, positive r)dFc dm(vϕ/r); dFc being fully balanced by a negative (inwards)

= 2 force dFp dm(vϕ/r) from the characteristic pressure gradient: low (high) pressure in the inner (outer) part of the vortex. This force balance stems from the hyperbolic velocity distribution [eqn (3)], which in turn is established by the retarding friction forces. Each vortex ring in the outer regime r ≥ rs is driven at its inner boundary by the friction force; at its outer boundary the rotation is retarded. The central driver in any natural vortex is the ‘solid body’ regime ‘I’ within r < rs, which excites the ‘O’ regime. As mentioned before, the strength of the surrounding potential vortex is deﬁned by the circulation = 2π(vϕr) = constant. A further detailed analysis yields the following distributions of vϕ(r) and ps(r) [eqn (5)], which are qualitatively shown in Fig. 4:

Natural vortex Inner ‘solid’ Outer ‘potential’ regime ‘I’ ‘O’ Range r < rs r ≥ rs = π = = π Velocities, vϕ, vr vϕ(r) (/2 )(1/rs)(r/rs) ωr vϕ(r) (/2 )(1/r) (5) vr(r) = 0 vr(r) = 0 2 2 Static pressure, ps ps(r) = ps + qs (r/rs) − 1 ps(r) = ps + qs 1 − (rs/r)

Here the following abbreviations are used for the natural vortex at r = rs:

ps(rs) = ps,

vϕ(rs) = vs, (6) = 2 qs (ρ/2)vs ,

= s = 2π(vsrs) = constant. There are many examples in nature for stationary, 3D natural vortices [6, 20], i.e.: • The bath-tub vortex: These types of (discharge) vortices have a nonzero radial inward velocity vr and axial discharge component vz in addition to the swirling tangential vϕ. Streamlines in planes z = constant are logarithmic spirals. Figure 5 gives an example from nature. The inner regime may either be ‘solid’ or ‘hollow’ (when rotation ω = vϕ/r is high). • Vortices in a rotating system: Examples are flow phenomena on planets in general; specifically currents of the sea (gulf stream, etc.) and large scale winds on earth. Due to the superimposed rotation of the earth, the Coriolis force Fcor = 2vϕ is entering the force balance Fc + Fp + Fcor = 0. Different cases are possible: in general high pressure vortices with radial outflow and low pressure vortices with radial inflow. In addition, the rotation may be counterclockwise in the northern hemisphere (clockwise in the southern hemisphere). • Hurricanes and tornados: These well-known vortices are among the most dangerous ones on earth. Hurricanes affect large areas with typical diameters of ca. 500 km. Tornados with higher ratios (Fc/Fcor) develop more concentrated motion (typical diameters ca. 50 m). Figure 5 shows an example for a 3D spiral intake vortex, forming above the inlet of a water turbine (St. Malo, France 22). Its sense of rotation is counterclockwise, due to the action of Coriolis forces on the northern hemisphere. Clearly visible are the large-scale spiral streamlines on the water surface, and also the low pressure vortex core. That core has roughly a diameter of 2rs = 1.5 m, and its suction is seen to accumulate dirt. Figure 6 depicts the typical streamlines of the intake vortex in Fig. 5. It should be noted that the radial flow components in these more complex natural vortices are usually following a relation vr(r, z) = fct(z)/r, which is similar to eqn (3).

Figure 5: Example of a 3D natural vortex: an intake vortex developing above the inlet of a bulb- type hydroturbine in Bretagne/France [22].

Figure 6: Example for a 3D natural vortex: Streamlines with three velocity components vr(r, z), vϕ(r, z), vz(r, z), characterizing the water vortex of Figure 5.

4 The ﬁnite vortex concept

Flow visualizations and many observations in nature lead to another step towards a pragmatic, sufficiently accurate description of the dynamics of natural vortices: the introduction of the finite vortex concept [23]. It has been suggested that an infinitely large vortex be replaced by a finite size vortex with the limited extension r ≤ a only. The idea originates from findings of an oscillating AF in air, simulating caudal fin fish propulsion at medium Reynolds numbers. The underlying basic input came from extensive studies of what is termed the ‘edge flow mechanism’ [12]. In the meantime the FVM has been proven very successful [24].

4.1 The edge vortex

A common mechanism in nature is shown in Figs 7–9: The so-called edge flow mechanism [12, 15]. Basically any normal flow velocity vN hitting a stationary sharp edge generates an abrupt flow separation, because the fluid cannot follow the high local ‘curvature’ of the edge. Then, due to the nonzero fluid viscosity, an unsteady roll-up of a growing edge vortex is initiated; this rotating fluid close to the lee side of the wall then generates a very strong, localized suction. Flow visualizations [6, 25] and also [27] prove that this suction causes a tangential inflow velocity into the edge vortex, thus accelerating the roll-up. Figure 9 shows an early sketch by Prandtl dating back to 1905. He suggests a rather different viewpoint, which is still found in many textbooks: two remote stagnation points marking a separate

Figure 7: Edge flow mechanism: Roll-up of a large starting vortex [25]. In contrast to Fig. 9 experiments of this kind do not show any division of flow/dividing shear layer or a stagnation point at the lee wall. The flow is tangentially following the suction towards the vortex center (see also Fig. 8).

Figure 8: Edge flow mechanism: Sketch of streamlines, redrawn from flow visualization experiments in Fig. 7 [25]. Suction S with tangential inflow; vortex core C. See also test results from [2, 4, 5, 26].

Figure 9: Edge ﬂow mechanism: An early sketch by Prandtl (1905), which is the classical viewpoint of vortex formation [2] till today.

recirculation zone on the luff side. Currently the Prandtl school believes that a separating spiral- shaped discontinuity surface or shear layer controls the vortex formation. It is assumed that there is a closed recirculation zone/no tangential feeding of the vortex. Historically there has been an interesting and very fruitful controversy between the two schools represented by Ahlborn [26] and Prandtl [2]. The subject of this basic controversy was, the role of the boundary layer in the generation and growth of vortices. Even now, after 100 years, this role is still an ongoing point of discussion. The experimental observations and controversies shown in Figs 7–9 shed light on the physical mechanisms involved in the edge flow of real fluids. From the findings we conclude the following sequential steps, which however, are not all in line with the current classical viewpoint [2, 7]

• Due to fluid inertia an abrupt flow separation of vN takes place at the edge. • Fluid viscosity leads to an unsteady roll-up of an edge vortex close to the luff-side wall. • The rotating fluid generates a strong, local suction, which draws ‘near-field fluid’ from the neighborhood into the ‘sink’. • Finally, the increasing suction leads to an almost tangential inflow velocity into the edge vortex, thus accelerating its rotation. From this sequence it can be seen, that all four steps have a local character (‘near-field’ only). This is quite different from the infinitely extending natural vortex, shown in Fig. 4 where the minimum static pressure ( ps − qs) at the ‘sink’ (center r = 0) gradually increases up to the ‘far-field’ pressure level p∞ = ( ps + qs). This pressure level p∞ is reached with r →∞;so the flow phenomena within the edge vortex of Figs 7 and 8 have local character. This is why the edge type vortex can be well represented by a so-called finite vortex [14, 23].

4.2 The ﬁnite vortex model

There are many situations in nature as well as in engineering, where the outlined edge flow mechanism plays a dominating role. In all cases with a moving edge or curved surface we can successfully apply the FVM: The vortex size is limited to the finite radius r ≤ a only, and the finite vortex then is nondimensionalized by the AF chord R. The parameter (2a/R) is found to play a key role in simulating oscillating AF unsteady fluid-structure dynamics [11, 28]. Force generation, power requirements, etc., can be predicted, even for small and large Re numbers of unsteady flows with large frequencies and amplitudes. Early work on unsteady aerodynamics was published by Birnbaum [29], based on classical and potential theory. His analyses make use of the Kutta condition, and so the results are limited to nonzero free-stream velocity, small frequencies and amplitudes. Recent aero- and hydrodynamics oscillating experiments have been successfully analyzed with the FVM, employing an extended version [11] of the original ‘inertial’ concept of 1963 [23]. Recently in 2005 the FVM has even been applied to the high Re number fixed wing cruising flight (see Section 5). Figure 10 is used to describe the essential features of the FVM in more detail. It schematically shows a rectangular plate (angle of attack α, width B, chord R) with the steady flow-through velocity vm at the quarter chord position M. Figure 11 shows recent CFD results [11] for a fixed, inclined plate (streak-lines, ReR = 20 K, turbulent flow), indicating the periodic shedding of large TE vortices. For details of the basic FVM and extensions see [23, 31]. The AF in Fig. 10 may be either visualized as fixed/inclined or as an oscillating AF: (I) Fixed, inclined AF: Constant angle of attack ϕ = α = constant as well as normal velocity component vN ≈ v∞ sin α at the TE point P. The motion of the plate is starting from rest (v∞ = vm = 0); it is rapidly increasing to the full level vm = v∞. Then the plate develops its starting vortex as a finite edge vortex (FEV) with a fast growing radius r > rs around the ‘solid’ core r ≤ rs. We realize that the sharp TE is ‘rounded’ by the FEV. Finally, the FEV reaches a stable radius a, and it has a constant mass mv as well as energy content. For a moment the edge vortex establishes a smooth flow condition (‘Kutta condition’). (II) Oscillating AF: The plate may perform purely pitching, harmonic rotational motions around the leading edge (LE) point O: ϕ(t) = ϕos sin (ωt). Now we have vN(t) = vPRN(t), since vPRN(t) is the time dependent tangential velocity vector vϕ(t) at the finite vortex periphery r = a.Again the FEV is formed.After a short time the diameter 2a, mass mv and the equivalent

Figure 10: Concept of the extended FVM (FVM of 2002): as an example a flat plate is shown— either as a fixed or as an oscillating AF (pitching-down phase) with the flow-through velocity vm, [24].

Figure 11: Same AF as in Fig. 10: CFD LES results for a ﬁxed plate (ﬂow v∞ from left to right, ReR = 20 K). Periodic shedding of large TE vortices, from [30].

moment of inertia θe are built up. The FEV is generated at the moving TE point P via the edge flow mechanism, acting on the normal velocity component vN as before (see also Section 5). In both cases (I) and (II) the FVM is limiting the existence of the outer regime ‘O’ of the natural vortex to a finite radius r ≤ a only. This assumption has five basic advantages (as will be demonstrated in Section 5): (A) We are able to calculate important—now finite—mechanical properties of the vortex, which represents a moving and rotating finite ‘cylinder’: the mass mv, the equivalent (polar) moment of inertia θe (with respect to the axis of vortex rotation), the circulation , the content of total rotational energy ER and of its fraction EP in the outer potential regime. (B) This means that the finite vortex can be utilized as a means to transport translational and rotational momentum. The basic ‘inertial’ version [23] of the FVM has been successfully used to explicitly calculate the vortex–AF interaction (analysis of forces, power, etc.). (C) Approximate physical models can be constructed, serving for quick explicit/numerical analyses, if one wants to avoid extensive Navier–Stokes analyses to begin with. (D) Due to the finiteness (2a/R) the far-field between vortices is neglected. Fluid–solid interaction is sufficiently well approximated by near-field suction (counterrotation, dynamics). (E) Extensions of the original inertial FVM concept [23] were developed [10, 24, 28, 32] as recently described in [11]. The bound vortex is introduced, which generates the circulatory, time-dependent forces F1 F2 (see Fig. 10). It can be shown how to quantify the TE suction, the bound vortex circulation, the interaction between the FEV (at the TE) and the leading edge vortex (LEV). After a fast roll-up due to the local suction S (Fig. 10), the FEV reaches its stable size r = a, and it still stays attached to the TE, as long as vPRN is increasing (during r = a = constant). When vPRN reaches its maximum, the suction decays, and the FEV detaches from the AF [24]. During the attachment phase (Fig. 10: down-pitching phase) the time dependent circulation (t) is built up; it reflects the counterclockwise rotation of the FEV. (t) is given by:

(t) = 2πavPRN(t). (7)

It has been shown in [31], that initiates a counterrotating bound vortex around the AF, which has been termed ‘Mantelströmung’. Any generation of aerodynamic forces (and moments) is due to the interaction of this bound vortex with the instantaneous, resulting incoming flow velocity vector vR(t), [24]. Figure 11 shows numerical large eddy simulation CFD results also for an inclined, fixed plate in a steady, turbulent flow v∞ with ReR = (Rv∞/ν) = 20 K. The counterclockwise rotating vortices shed from the TE are well visible; they confirm the FVM concept depicted in Fig. 10. Calculations along the lines of A) yield explicit expressions for the total rotational energy ER = (EP + ES) with EP from the ‘potential’ regime and ES from the ‘solid’ regime. As an example the variation of (EP/ER) is shown in Fig. 12 with the size of the finite vortex (2a/R) and the inverse size of the vortex core (a/rs). In addition the relative equivalent (polar) moment of inertia fD = (θe/θs)is shown in Fig. 13. The fD compares the θe to a fully ‘solid cylinder’ θs where rs = a, see also [33]. The resulting rotational energy ER and relative moments of inertia fD are plotted in Figs 12 and 13 as a function of vortex geometry. The following table summarizes the expressions found for the finite natural vortex [eqn (8)], with the abbreviations [11]: r ≤ a,(a/rs) = n, vϕ(rs)/vϕ(a) = n.

Figure 12: Properties of a ﬁnite version of the natural vortex in Fig. 4: the total rotational energy content ER and (EP/ER) − EP taken from the potential part only, versus the vortex size (2a/R) and inverse core size (a/rs).

Figure 13: Properties of a ﬁnite version of the natural vortex in Fig. 4: the relative moment of inertia fD = (θe/θs) as a function of the inverse core size (a/rs). fD is for the same moment of momentum of the cylinder mv (liquid = solid). Flapping wings/ﬁns very often show a core size n = 7 yielding fD = 1.980 [34].

2 Content of rotational energy ER = (ρB/4π) [1/4 + ln(n)] 2 Mass of ‘cylinder’ mv = ρπa B 2 (8) Equivalent moment of inertia (the index ‘D’ θe = fD(n)θs, with fD = 2 − (1/n ), and 2 refers to the equivalence: ‘Same moment of θs = (mv/2)a (solid cylinder r = a) momentum’)

4.3 The size of the ﬁnite vortex

Due to fluid inertia it takes a finite time for the FEV to roll-up and reach its stable size (2a/R). ‘Stable’here is defined by ‘no further growth’, i.e. r = a = constant; that means no further suction due to a change in static pressure distribution. This condition is reached in a very short ‘feeding time’ tf . A dimensionless time τf can be approximated by eqn (9), following the results of [15] for the case of a step rise of the angle of attack from zero to α:

τf = (vmtf /R) = (Kτ /4) sin α with Kτ (ReR) (9)

τf is seen to depend on the angle of attack α and a factor Kτ , which essentially is a function of the Reynolds number ReR = (vmR/ν) and the thickness of the TE, involved. These results compare well with findings from Lugt [6], for a similar situation (α = 45◦, inclined elliptic profile, suddenly accelerated to a—rather low—velocity vm (ReR = 200, laminar): he obtained the value τf = 0.20, which corresponds to Kτ = 1.13. In order to estimate the finite vortex size (2a/R), it turned out to be important to distinguish different cases (A), (B) and (C), since the following four independent parameters assume very different ranges: • The first key parameter is the dimensionless (FEV) vortex size (2a/R) itself. • The vortex-detachment frequency or—with the approximation f ≈ 1/tf —the corresponding dimensionless frequency is the second key parameter ( fR/vm). • The velocity level or the Reynolds number ReR = (vmR/ν), i.e. the mean flow through vm. • The velocity ratio (inflow to flapping velocity; for cases (C) only) fratio = (vm/vmax). The following table gives an overview for the relevant cases (A), (B) and (C). The two most important parameters, the dimensionless vortex size (2a/R) and the dimensionless frequency are closely linked to each other. The ranges listed are taken from [11, 28]:

Case FEV size Detachment Velocity level Velocity ratio (2a/R) frequency ReR = (vmR/ν) fratio = ( fR/vm)(vm/vmax)

(A) Fixed AF: Medium size Medium frequency Small ReR NA 2 4 Shedding of 0.10 ...0.30 200 ...10 ReR = 10 ...10 starting vortices (B) Fixed AF: Small size Large frequency Large ReR NA 6 steady cruising 0.03 ...0.06 6000 ...1000. ReR ≥ 10 ﬂight of airplanes (C) Oscillating Large size NA Medium ReR fratio = 3 AF, tests [23] 0.20 ...0.70 ReR = (5 ...50)10 0.21 ...0.22 [23]

Following [15] and recent analyses [11] for the ﬁxed AF cases (A) and (B), simple momentum conservation and eqn (9) yield two expressions for the vortex size and the associated shedding frequency f , with Kτ as a function of ReR (with α as the angle of attack):

(2a/R) = (Kτ /4) sin α, (10) 3 ( fR/vm) = (16cA/π)( cos α/ sin α).

The extended FVM [12] has been successfully applied to a wide range of cases. Correlation of FVM forces, energy partition and circulation with recent experiments and Navier–Stokes ﬁndings is very good [10, 12, 28]. Recent details are found in[11, 33]. For demonstration purposes a few examples are presented in the next section.

5 Applications of the ﬁnite vortex model

The FVM concept is very general; it can be applied to a variety of cases, where ﬁxed or oscillating bodies or sharp-edged AFs are involved. Detachment frequency and size of the vortex are determining the mean force level generated. There is no need to know the static pressure distribution around the bodies to determine the net forces exerted to them. Here usually simple momentum and mass conservation is sufﬁcient to estimate thrust and lift.

5.1 Conservation of mass and momentum

Besides Newton’s basic law of motion, the momentum equation of Euler represents another fundamental relation in structural and fluid mechanics. Quite often this relation is very useful in determining the (reaction) forces acting at the boundaries of any complex mechanical system. For instance in simulating fluid dynamics problems, all types of forces are allowed: viscous (frictional) forces, pressure, inertial as well as gravitational forces. This generality is very practical in many engineering situations. Here we give the relevant equations, describing the mass and momentum balance [8, 21]. Any fluid particle dm, having the velocity v, carries the (translational) momentum, the vector dI = (v dm). Its first derivative in respect to time equals the external force dF (reaction force dFR) exerted on that particle dm:

dF =−(dFR) = dI/dt = d(vdm)/dt. (11)

Spatial integration of all dI = d(ρvdV) around a deﬁned control volume ‘V’ of ﬂuid with its dV as volume elements yields:

F =−FR = d(ρvdV)/dt = ∂(ρv)/∂tdV + (ρv)d(dV)/dt. (12) local global

Generally there are two contributions to F: the local change of the fluid within dV atafixed time t (partial derivative) plus the global, spatial momentum flux through the surface elements dA, making up the surface of the control volume. If we are restricted to fluid dynamics systems with a constant mass content, then (dm/dt) = 0 (stationary flow), and the local, partial derivative term in eqn (12) vanishes. To solve F, we only need the integrands across the in/out cross sections Ain,out

WIT Transactions on State of the Art in Science and Engineering, Vol 3, © 2006 WIT Press www.witpress.com, ISSN 1755-8336 (on-line) Manifestation of the Finite Vortex—Lift, Thrust and Drag 263 around ‘V’. No information is required on details (losses etc.) inside ‘V’. For incompressible ﬂow (ρ = constant) the following vector summation/mass balance is obtained: F =−FR = Iin − Iout = (mv)in − (mv)out = 2 − 2 ρ (VunAin) (VoutAout) , (13) min − mout = 0.

This balance will be used for both: fixed bodies with steady mean flow as well as oscillating bodies with unsteady flow around.

5.2 Flow around ﬁxed bodies

Figure 14 depicts the most important case for aircraft and turbomachinery engineering: a steady free-stream flow velocity v∞ around a fixed, inclined (angle of attack α, width B) AF or plate, 2 generating a lift force FL = CL (ρ/2) v∞ RB (lift coefficient CL). Two viewpoints concerning the ‘smooth flow’ at the TE are indicated: the FVM presents the physical mechanisms, leading to a suction with CL and the lift FL (Fig. 14a). In contrast to this, the current mathematical condition of Kutta–Joukowski shows in Fig. 14b, that a tailored, clockwise rotating, bound circulation is superimposed to the parallel flow field such as to shift the rear stagnation point to the TE for a smooth flow. Figure 14a shows the application of the FVM: The resulting streamlines, schematically the observed edge flow mechanism and a control volume ‘V’ are illustrated: Periodic, high frequency TE vortices are generated in a self-regulating manner [24]. They cause intermittent suction (flow acceleration on the upper side) and a control of the transverse flow component vN (driving the vortex roll-up). This leads to a net deflection of a partial mass flow m along the plate, causing the so-called downwash with its vertical velocity component vD. This deflected flow is equivalent to the above mentioned, bound circulation . Application of the balance equation [eqn (13)] to the control volume ‘V’, indicated in Fig. 14a, allows the vertical reaction force FR = FL (aerodynamic lift) to be determined, if the flow partition (m/mtot) or the partial mass flow m is known, leaving the AF with the geometric sum of the velocities v∞ (horizontal) and the vertical vD = v∞ tan α. Introducing the finite vortex concept allows the quantification m, using the vortex size (2a/R) and the nondimensional vortex detachment frequency ( fR/v∞) (here the detachment cycle time of the counterclockwise vortex is T = 1/f ). From eqn (13) one obtains the following general relations, see also [11]: Momentum flux balance, forces:

I1x = mtotv∞ I1y = 0,

I2x = (mtot − m)v∞ I2y = 0,

I3x = mv∞ I3y =−mv1,

= FRx 0 (zero drag from this simplifying balance), FRy =− Iy,in− Iy, out = 0 − ( − m) = m tan αv∞ = FL (lift force perpendicular to v∞), 2 2 with m = mv/T = (ρπa B)f (one vortex per cycle) and FL = CL(ρ/2)v∞RB.

(a)

(b)

Figure 14: (a) Smooth flow at the TE via periodic vortex detachment (FVM), with vN = 0: deter- mination of the lift coefficient CL from vortex size 2a and detachment frequency f , [11]. (b) Smooth flow at the TE via applying the mathematical Kutta condition: tai- loring a circulation from vN ≡ 0. Lift coefficient CL from according to current practice [7].

The lift coefficient CL is apparently a function of α and a characteristic product ‘vortex size × frequency’, [11] as can be seen from eqn (14): 2 (2a/R) (fR/v∞) = 2CL/(π tan α) = FUNCT (angle of attack α, AF profile, TE thickness). (14) Example: ‘Small inclined plate in a low velocity air flow’. Two unknowns (a, f ) from eqn (14) and an additional condition for (2a/R) – see the extensions given in [10, 11] yield: ◦ Input: α = 15 , CL = 1.20, R = 0.10 m, ReR = 10 K (air: v∞ = 1.58 m/s) Output: (2a/R) = 0.1207 (medium size), a = 6.04 mm, vD = 0.423 m/s, ( fR/v∞) = 196 (medium level), f = 3.09 kHz

(a)

(b)

◦ Figure 15: (a) Flow around a fixed plate, oriented 90 to a free-stream velocity field v∞: tests [25] showing the visualized streamlines, and shed vortices without any large scale recirculation zones. (b) Flow around a fixed plate, oriented 90◦, to a free-stream velocity field v∞: streamlines from a potential flow analysis. Stagnation points properly positioned at A and B. The very large recirculation zones are not realistic.

Figure 14b shows the current viewpoint with the Kutta condition: To satisfy this mathematical condition and to obtain a ‘smooth flow’ at the TE, the tailored, bound, clockwise circulation = π sin α Rv∞ is added to the parallel flow field v∞. This shifts the rear stagnation point exactly to the TE, so that no transverse velocity is resulting (vN = 0): ‘Smooth flow’. Conformal mapping then yields a theoretical value CL = 2π sin α = 2(/Rv∞). The resulting lift is obtained from the well-known formula: FL = (ρB)v∞. (15) Figure 15 indicates a second classical case of flow around a fixed body: a steady, free-stream ◦ flow velocity v∞ against a fixed, long (B >> R) AF/plate, which is oriented 90 to the v∞. Real flow visualization of the streamlines can be seen from Fig. 15a, obtained from tests [25]. Charac- teristic shedding of limited size vortices from the plate edges are visible, without forming large

recirculation zones behind the plate.At medium ReR = 10 K a high drag force FD (drag coefﬁcient CD, [18]) is measured: 2 CD = 2.01 so that FD = CD(ρ/2)v∞ RB. (16) This high force level is due to the strong suction of the high vortex shedding frequency from the experiment shown in Fig. 15a. Early attempts to calculate the CD date far back to Kirchhoff and Newton. The ‘shadow representation’ of Kirchhoff is considering the front stagnation side of the plate only; thus his theory predicts rather low values CD without simulating the strong suction on the rear side (ca. 43% of the measurements) [4]:

CD = 2π/(4 + π) = 0.88. Following the ideas of Section 5.1 and applying the FVM in conjunction with its extensions [11, 24], one can calculate the approximate (2a/R) and ( fR/v∞) from the measured CD. Finally for comparison Fig. 15b gives the result of an inviscid potential flow calculation.To avoid singularities the stagnation points had to be properly positioned: point ‘B’ defining a very large vortex pair/recirculation zone behind the plate. For smooth flow detachment at ‘A’again the vortex circulation A had to be tailored such as to move the stagnation point to ‘A’. However, the resulting extremely large recirculation zones, are not realistic.

5.3 Flow around oscillating bodies

The bodies/AFs should oscillate relative to the still fluid (v∞ = 0) or steady flow (v∞ = 0). Inde- pendent from the level v∞, the oscillation always generates a nonzero ‘mean pumping velocity vm along the AF. The average flow-through velocity at the quarter chord position M is ca. vm = v∞ + vidz/2. Force generation by oscillating edges or appendages has been extensively utilized and perfected by evolution in nature. Compared to fixed bodies, the resulting flow field is much more complex: global static pressures and velocities are highly unsteady and nonlinear. In addition nature operates wings and fins under much more complicated boundary conditions. The following six conditions are completely excluding the use of classical aerodynamics tools [7, 29] to simulate phenomena in nature: 5 • Low to medium Re numbers with Revmax ≤ 10 . There are thick ‘boundary layers’/viscous 6 7 zones. In contrast to airplanes (ReR ≥ 10 to 10 ), friction forces can no longer be neglected. • Vertical take-off and landing capability (with v∞ ≡ 0). Thus no separate ‘engines or airports’, which would be required for horizontal take-off and landing. • Periodically high transverse, unsteady flow velocities vN around the TE; the ‘smooth flow condition’ (Kutta–Joukowski) is periodically established. • AF operation goes deeply into the post-stall regime, taking advantage of the favorable dynamic stall situation. This allows for very high mean CL and CT. ◦ • Oscillation with high frequencies/amplitudes: plunging (so/R) ≥ 0.10; pitching ϕo ≥ 20 . Opti- mum performance in nature is achieved within a rather narrow Strouhal range, which turns out to be St = (2ωso/v∞) = 0.20–0.40 (St definition = inverse reduced frequency). • Highly nonlinear kinematics, vortex roll-up, force generation and large elastic deformations of the shape-adaptive wings/fins. Currently, with the computer power rising, a few specialized Navier–Stokes solvers are capable to analyze swimming and flying in nature [30]. The effort, however, is often still prohibitive. There is the need for effective, physically meaningful models, and here the FVM fills a gap [15, 23].

As an example Fig. 10 has shown a plate, purely pitching around O. The ﬁgure depicts the downstroke, and prescribed harmonic motion is given by:

ϕ = ϕo sin (ωt). (17)

2 Two vortices—each with the ﬁnite mass mv = πa Bρ—are detached during a full cycle of T = 1/f = 2π/ω. Following the purely inertial, basic FVM [23] with centrifugal forces acting on mv the following time-averaged thrust FmT is produced (index m, t = 0, ..., T):

= = π2 2 2 2 3 FmT FRx KT (2a/R) ϕo f R Bρ. (18)

We recognize the important relations: the mean thrust force FmT is proportional to the • square of the vortex size (2a/R)2, • 2 = 2 square of the peak pitching velocity vmax (ϕoωR) , • extension B of the TE, • radius R of the pitching AF, • ﬂuid density ρ.

The dimensionless factor KT turns out to be 4π here. Further extensions of the FVM [10, 12], indicate a more general relation KT (ϕo,2a/R, etc.), depending on the type of modeling. To verify the FVM findings, extensive experiments were conducted back in 1963 [23], to systematically study the generation of fluid flow/thrust forces from a purely pitching AF in a rectangular confinement with an electric drive, producing the harmonic motion equation (17). The corresponding test arrangement is shown in Fig. 16, indicating the air intake/AF with the pitching rear portion R, the rectangular channel Ao = (Bh) and the channel discharge nozzle N,to ◦ ◦ accurately measure the mass flow mN. The six amplitudes ϕo = 12.5 −30.5 are held constant and eight pitching frequencies f = 8−23 Hz are run, to measure the resulting average discharge flow mN through the nozzle N. Figure 16 also shows the mass and momentum conservation between

Figure 16: Pitching AF experiments in air [23]: mean channel ﬂow m = mN (exit velocity vE) and mean reaction force HmT , generated by the frequency f and amplitude ϕo of the AF.

air intake ‘0’ and the two exits ‘2’ and ‘3’ (control volume ‘V’ at the pitching AF). This allows to calculate the time-dependent reaction force H(t), acting at the AF support: 2 2 2 H(t) = movo − m2(t)v2(t) − m3(t)v3(t) = ρ Aovo − Am2vm2 − Am3vm3 , (19) mo = mN = m3(t) + m2(t).

Introduction of the FVM allows to approximately quantify the induced ‘add-on’ velocity vidz and the force H(t) as functions of the vortex size (2a/R), etc. Simplifying to quasi-stationary conditions around the AF, we cite from the evaluations in [12, 28] ﬁve equations for the ﬁve unknowns (index m: averaged over t = 0, ..., T):

HmT = FmT from the FVM and eqn (18),

Am2 = (4/π)BR sin ϕo average jet cross section during T, 2 vidz = (π/4)(2a/R) fR/ sin ϕo two vortices per cycle T = 1/f , inducing the vidz, (20)

vm3 = (mo − mm2)/ρ(Ao − Am2) resulting exit velocity through ‘3’,

mm2 = ρAm2vm2 with vm2 = vo + vidz/2 resulting jet mass ﬂow mm2 exiting through ‘2’.

It should be mentioned that the reaction force HmT is here at ‘standstill’ (zero v∞), i.e. with limited vin or vo due to the mean pumping action vm of the oscillating AF. Superposition of a non-zero free-stream velocity v∞, however, does change the vortex dynamics, i.e. the level of HmT , [33]. Detailed FVM evaluations and validations of the model were reported in [23, 28]. Here only the characteristic linear relationship between the exiting mN or vE and the product (ϕof )isgiven in Fig. 17 (total of ca. 50 test series). The general message is that these and other experiments [34, 35] strongly confirm the validity and capability of the finite vortex concept in a wide range of operation (ϕo, R and f , etc.). Similar to eqn (14) for the fixed AF it should be finally added, that now for the oscillating AF the following nondimensional characteristic product seems to be a function of the overall lift coefficient CL and the amplitude ϕo only [11, 33]: 3 2 2 (2a/R) ( fR/vm) = CL/(π ϕo). (21) A large number of general applications emerged from an extension of the FVM by Liebe [11, 24, 33, 36]. The latest findings over the last 3 years led to practical improvements in the following fields: aircraft as well as turbomachine engineering with hydropumps and turbines, steam turbines and gas turbines. These technical improvements make use of: • new AF/TE geometries towards maximum suction for better aerodynamic performance; • oscillating AF-systems for improved mixing of fluids via unsteady vortices; • new turbulation geometries, generating higher heat transfer coefficients at lower losses; • reduction of cavitation and noise in pumps, turbines fans, etc. In the next two sections the applications of the FVM in nature are described for the swimming of fish and bird flight.

6 Swimming in nature

The swimming of aquatic animals has been studied systematically over the last ca. 150 years [37]. Nowadays ﬁn kinematics of ﬁsh is well documented, but there are still controversial discussions

Figure 17: Measured exiting velocity vE as a function of pitching frequency f and amplitude ϕo from the oscillating AF experiments in air [23].

on swimming performance and on mechanisms for thrust generation. In addition, there are many different modes of locomotion in water: well known ones being the jet action (cuttle fish, jelly fish), the undulation of ‘strip fins’or slender bodies (water snake) and animals, simply utilizing the resistance force from their moving members or appendages (turtles). Solutions for high swimming performance in nature, however, are all based on what is called the ‘caudal fin fish propulsion’ [37–39], relying on the efficient generation and control of large vortices. Figure 18 indicates the principle of this efficient caudal fin fish propulsion [23]: fast transverse motion vN of the tail fin generates two energetic, spinning vortices per beat cycle T. Visualizations of the tail kinematics prove that nature utilizes an efficient combination and phasing of a pure ‘plunging’ of the peduncle, s = so sin(ωt) (via a corresponding body counteraction) and a pure ‘pitching’, ϕ = ϕo sin(ωt +ψ), of the sharp-edged fin itself. This results in a maximum mean transverse velocity vPRN across the TE and in a long attachment time t = ta of the resulting vortices (ta/T > 0.25); this in turn leads to high thrust FT and swimming velocities vs. The mechanisms for thrust generation are along the lines described in Section 4. Fast swimmers (Figs 19–21) in nature are characterized by eight main features [compare eqn (18)]:

• High transverse flow velocities vmax at the TE (Section 5.3). • To get high vN, vmax interestingly the aerobic (red) musculature is strong and concentrated to medial and anterior positions of the body [38]. Those muscles efficiently power the rear portion of the body (plunging amplitude so) and the tail fin (pitching amplitude ϕo).

Figure 18: The principle of the caudal fin fish propulsion [23]: Periodic generation of large vortices ‘c’ at the TE of the elastic caudal fin ‘a’. Sequence t/T = 0 ...0.25. Suction and high velocity streamlines ‘b’ are caused by the mechanism described in Section 4.

(a)

(b)

Figure 19: Swimming performance of the bottlenose dolphin (Tursiops truncatus). (a) Measured frequency f for swimming ﬁsh with different velocities vs at a constant mean amplitude a = 0.53 m, from [40]. (b): Swimming velocity vs as a function of frequency f amplitude a at the TE of the caudal ﬁn. Performance chart from FVM analyses [41].

Figure 20: High performance morphology in nature, from [37]: the bony ﬁsh of the tuna family (Osteichthyes) possess the above eight features for fast cruise swimming.

Figure 21: The optimum result of evolution has not changed over the last ca. 200 million years: the perfect design of the shark [23]. This migratory long-distance, fast swimmer survived the drastic change of climate ca. 265 million years ago, see also [43].

• Minimum undulatory body motion in the front (required as a mechanical counterreaction to the rear motion). Concentration of the driving lateral body motion to the caudal rear. • Optimum hydrodynamics/kinematics of the T E (parameters f , so, ϕo, R and phase difference ψ, etc.); nonharmonic kinematics for maximum vs.

• Long TE extension of the tail fin, i.e. large B. Also ‘sharp’ or thin thickness of the TE. • Slender body for minimum ‘external’ drag CD (against steady swimming velocity vs). • Self-adapting shape of the rear body portion due to a flexible caudal fin region. • Elevated body temperature (countercurrent ‘heat exchangers’) for maximum muscle efficiency [38]. These observations support the opinion of the authors [23, 33] that it is primarily the generation of large caudal fin vortices, not undulatory body motion, that power fast swimmers. The above observations do not support the ‘undulation’ theories such as the models proposed by Lighthill and Hertel et al. [42].

7 Flying in nature

= = 2 Swimming only needs sufficient thrust to overcome drag: FT FD CDA(ρ/2)vs . ‘Active free flight’, however, means the generation of sufficient lift FL and thrust FT, simultaneously, in order to compensate the weight FW and to overcome the drag FD. Intermediate steps towards that goal is ‘passive gliding’, as well as the controlled or wing-beat-assisted, prolonged gliding. Both, evolution [44], as well as human flight [45] went through those steps until performing active flight. Passive gliding utilizes the weight component FT = FW sin ε as continuous thrust to compensate the drag and to reach the flight velocity vF = v∞(ε = gliding angle). The resulting lift FL then compensates the other weight component FW cos ε = FL. Compared to swimming, the higher performance level required for 3D, active, free flight requires higher thrust, good maneuverability as well as vertical take-off and landing capability. Nature achieves this mainly by [compare also eqn (18)]:

1. Design for a higher vN, i.e. instead of fins use wing flapping, (wingbeat = ‘plunging’) around the center line of the body as axis of up/down rotation. Note that in case of fish the fin axis of rotation is perpendicular to the body center line. 2. Design for a longer extension of the TE (birds’ wing span b >> fin extension B of fish). 3. Employing a more complex superposition of the wing ‘plunging’, the chord ‘pitching’(prefer- ably pitching ca. 90◦ ahead of plunging) and the “sculling” of the wings. 4. Use unequal characteristics of both the wing shape and kinematics during the up- and downstroke, respectively. This leads to a higher FL- and FT-level (active down, passive up) when refining towards: • Light wings and body; profile camber. • Blunt, rounded (L E) with maximum suction (interaction of LEV and TEV). • Use of shape-adaptive, flexible wings with a louver-type feather arrangement. Feathers are ‘closed’ during the active downstroke, ‘opened’ during the passive upstroke. • Ability of fast, nonsymmetric, almost arbitrary wing movements for highest maneuverability against gusts, enemies, etc. Some species have special shoulder joints and strong muscles for hovering. • Combination of gliding and active flight to raise efficiency. Compared to this evolution in nature, it is interesting to consider at early human attempts to fly: flapping of cambered, light wings was a breakthrough; this was the dominating, primary concept up to 1900. Figure 22 shows early patents of O. Lilienthal and H.F. Philips towards active and gliding flight machines. There were two other concepts, competing with the successful principleA:

Figure 22: Typical early flying machines as of 1890: The flapping wing concepts were still favored over the fixed wing designs. Patents of O. Lilienthal (1893) and H.F. Philips (1891) for gliding fixed wings as well as flapping gliders, from [45].

(A) Gliders with active, flapping wings (man- or motor-driven) 45% of all patents, (B) Fixed wing gliders with/without thrust generating propellers 26% of all patents, (C) Screw-type propellers (early helicopters) 29% of all patents. O. Lilienthal has successfully built and flown flying machines of the type (A) and (B). From 1890 to 1905 he performed more than 2000 flights, based on bird flight [46]. He analyzed and utilized many features from nature, such as • light-weight, foldable wings and cambered profiles, • optimum profile shapes from experiments (polar diagram), • flapping wing concept (simple or biplane), • lift enhancement by LE and TE flaps, etc. Control was through flaps and shifting of body weight. Later on, in 1903, Orville and Wilbur Wright had their first successful, motor-driven, fixed-wing 50 m flight. From then on human flight ‘departed from nature’. Flying machines of the fixed-wing type were favored for structural and mechanical reasons (early vision of G. Cayley in 1800). Figure 23 depicts the application of the extended FVM to bird flight from [10]. Aerodynamic forces F1 and F2 are generated during the up- and downstrokes. They are originating from the bound vortex (BV) built-up by the driving trailing edge vortex (TEV) during the attachment phase (see also Fig. 10). The TEV/BV vortex pairs are periodically built-up twice during one cycle T, and they are connected by two tip vortices. Those parasitic tip vortices are minimized in nature by ‘forked’ wing tips, similar to winglets in airc engineering. Figure 23 also shows a characteristic ‘ladder wake’ with transverse vortices, formed by the sequence of periodically detaching FEV. Early experiments failed to visualize such transverse vortices behind freely flying birds. Only recent, advanced DPIV techniques [47] and suitable mathematical strategies of data analysis could reveal transverse vortices; also the TEV circulation could be quantified (see [52]). Next a simplified FVM analysis is given for bird flight. It is derived from eqn (18), using the following assumptions (for details see [11]): ◦ • Superposition of wing-beating (‘plunging’) γ (t) = γo cos (ωt)anda90 advancing chord- twisting (‘pitching’) ϕ(t) = ϕo sin (ωt).

(a)

(b)

Figure 23: Application of the extended FVM to ﬂying in nature/oscillating wings. (a) Active attachment phase with interaction between primary FEV and secondary BV around wing. (b) ‘Ladder wake’ formed by periodically detached U/D vortices FEV (vin = v∞), [10].

• Using a constant chord R(y) = R = constant. • Assuming, that the maximum, mean, normal velocity vN across the TE occurs at mid half-wing position y = b/4.

Finally, one obtains a total thrust force FT along the whole span y =−b/2to+b/2 of:

vmax = vN =−b/4ωγo sin (ωtmax) + Rωϕo cos (ωtmax)

with (ωtmax) = arc tan[(b/4R)(γo/ϕo)]. (22)

This equation is now applied to the ﬂapping wing model, shown in Fig. 24, which has been developed for demonstration purposes [48]. The bird mockup can slide along a rail (slope angle

Figure 24: The ﬂapping wing model ‘ANIPROP SM 1’ from [48]: demonstration of thrust generation from pure active ‘plunging’ plus ‘passive pitching’ of the ﬂexible wings.

ε, friction factor λ); the wings have a span b and a mean chord R. The active ‘plunging’ of the wings (amplitude γo) is performed electrically, and the ‘passive pitching’ (amplitude ϕo) results from the wing ﬂexibility. Flapping the model wings with a frequency f causes it to slide upwards. When stopping the electric drive, wings do not move and the mockup slides down again due to its weight (mass m). Using eqn (22) and the following input: ◦ b = 0.80 m γo = 65 m = 4.0kg ◦ R = 0.15 m ϕo = 10 f = 2.0Hz ◦ λ = 0.010 ε = 2.0

The following plausible output is obtained for the model in Fig. 24:

Overall thrust: FT = 0.433 N > friction force FF = 0.392 N, i.e. moving upwards; Downward force: Fdown = 1.37 N > friction force FF = 0.392 N, i.e. moving downwards. Figure 25 depicts a schematic representation of a possible ‘gyroscopic effect’, if considering the finite vortex as a ‘rotating cylinder’ [32]. Finally Figs 26–28 show three birds with remarkable flight performances. The swift as the fastest flyer in nature, reaches maximum velocities of ca. 28 m/s (Fig. 26). Quick estimates for this tiny bird cannot explain such high velocities. One explanation may be an extremely low drag (CD) due to a very strong suction. The ‘Mantelströmung’may act as a ‘lubricating’air film, which strongly reduces the CD. The large, white-headed eagle in Fig. 27 represents a very maneuverable bird, reaching even velocities up to 13 m/s and employing forked wing tips as well as sophisticated high-lift devices such as LE flaps and stall-preventing covering wing feathers. The storm gull is a perfect glider which is also able to hover (Fig. 28).

(a)

(b)

Figure 25: Schematic representation of a gyroscopic effect of the ﬁnite vortex (FV) spinning at the TE of a bird’s wing. (a) Net thrust FT only from pure ﬂapping (plunge). (b) Gyroscopic moment M(x) [32].

8 Results and conclusions

This contribution presents new physical mechanisms of force generation in fluids. The a priori assumption of a smooth flow condition at the TE of anAF is replaced by a more general ‘edge flow mechanism’. In addition the ‘natural vortex’ is introduced, instead of using the classical potential vortex. Specifically, it is shown that ‘steady flow with separation’ really is an unsteady phenomenon, which involves the periodic generation of edge vortices. The resulting dynamic vortex-structure interaction is treated, and it is shown that ‘near-field’ effects are dominating. This is the basis for the finite vortex concept and the resulting FVM, which performs the dynamic simulation with finite size vortices only. The original ‘inertial’FVM of 1963 has been extended in 2002, and many successful applications are reported since then. It is found that four coupled key parameters are affecting the generation of forces in fluids, both in nature and engineering: • vortex size (2a/R), • vortex detachment frequency ( fR/vm), • velocity ratio (vm/vmax), • Reynolds number (vmR/ν).

Figure 26: The fastest ﬂyer in nature, from [49]: the swift, reaching ﬂight velocities up to 28 m/s.

Figure 27: A fast and very maneuverable ﬂyer: the white-headed eagle, reaching ﬂight velocities up to 13 m/s, from [50].

Figure 28: A fast ﬂyer and a perfect glider, which is also able to hover: the storm gull, from [51].

Calculation methods and working equations are derived – partially based on simple momentum conservation. A characteristic product of ‘vortex size ∗ frequency’ is shown to determine the generated force level: thrust (FT), lift (FL) and drag (FD). The resulting numerical results are correlating well with both experiments as well as Navier–Stokes findings. After bracketing the vortex size and interrelated frequency for a wide range of applications, the original FVM equations are applied to four areas of engineering and nature: • The aircraft wing: Periodic formation of starting vortices. The ability to continuously main- tain the Kutta-condition and an ‘average smooth flow’ during cruise flight. Lift prediction from vortex size and detachment frequency, instead of using the mathematical wing circulation. • Pitching AF experiments: Measured mean pumping flow and reaction forces are successfully predicted from FVM simulation. • Swimming in nature: The caudal fin fish propulsion is analyzed in detail. Corresponding equations and ‘swimming charts’ are correlating very well with measurements of fin frequency and amplitude resulting in various swimming velocities. Morphology and body shapes of fast swimmers can be explained from the working equations. • Flying in nature: Simplified superposition of wing flapping (‘plunge’) and twisting (‘pitch’) allows to predict mean thrust and lift forces in birds. Numerical FVM results are presented. In conclusion, high lift and thrust levels as well as good maneuverability in nature is effectively achieved by periodically generating and controlling medium to large vortices along the TE of fins and wings. This flexible principle has been highly perfected in fish, insect and bird evolution. Unsteady aero- and hydrodynamics will play an increasing role in many fields of human engineering. Improvements have already been introduced in: • aircraft and turbomachine design, • process engineering,

• biofluiddynamics, • general understanding and inspiration from nature in many other fields. Finally, additional work is proposed concerning analyses and experiments in the following: • Shedding more light on the role of vortex size, frequency, velocity ratio and Re number. • Clarifying the dynamics and stability of the TE vortex. Dynamic interaction of the TEV and the LEV; possibly further favorable control of that interaction. • Gaining deeper understanding of the ‘Mantelströmung’. Control and maximization of FEV suction towards design improvements for AFs, profiles and TE geometries.

Acknowledgments

This paper represents W. Liebe’s documentation of his development work on physical models to simulate lift and thrust generation in fluids. It covers early ideas concerning the ‘edge flow mechanism’ as well as the justification and formulation of the ‘finite vortex model’ (FVM) from 1963 till today. He also describes recent ideas like the “Mantelströmung” (1999) and the ‘gyroscopic effect’ of the spinning cylinder of an attached finite vortex (2004). W. Liebe was not granted the privilege to complete his contribution to this volume. The paper was finished by his son, R. Liebe, after he passed away in October 2005. The junior author R. Liebe is grateful for his father’s intense discussions and the continuous cooperation over the years. Many stimulating, partially controversial disputes lead to new ideas and extensions of the FVM. A large number of applications in nature as well as engineering have been emerging since 2002. The junior author finally wishes to thank F. Durst [44], F. Fish [37], P. Freymuth [48], W. Send [42] and K. Kawachi [49] for their input and material, contributing to this work.

Nomenclature

a m radius of finite natural vortex A m2 cross-sectional or surface area b m wing span B m extension of airfoil CL – lift coefficient CT – thrust coefficient CD – drag coefficient ER, EP J energy content of one finite natural vortex (total rotational, potential regime only) f Hz frequency of vortex detachment or of AF oscillation fD – ratio of moments of inertia = (θe/θs) fratio – velocity ratio = (vm/vmax) F2 N transverse (Magnus-type or ‘lift’) force on a finite vortex, spinning in a parallel flow of vR F1 N ‘resistance’-type force in line with vR and perpendicular to F2 FL N lift force FT N thrust force

FD N drag force FR N reaction force FC N centrifugal force FCor N Coriolis force Fp N pressure force h m height H N horizontal force component I kg m/s translational momentum of fluid flow Kτ – dimensionless factor for vortex feeding time KL – dimensionless factor for lift generation KT – dimensionless factor for thrust generation L, l m length m kg/s fluid mass flow mv kg mass of finite vortex (cylinder) M Nm gyroscopic moment n – dimensionless, inverse size of vortex core = (a/rs) p, p∞ bar static pressure, free-stream static pressure q bar stagnation pressure R m chord of airfoil (plate), radius (lever) r, rs m radial coordinate, radius of vortex core Re – Reynolds number r, ϕ, z – cylindrical coordinates s, so m plunging oscillation of AF, amplitude of s t, T s time, cycle time T = 1/f = 2π/ω ta, tf s attachment time of FEV, vortex feeding time v, v∞ m/s fluid velocity, free-stream velocity vm m/s mean flow-through velocity at quarter chord point M of AF vD, vE, vs m/s downwash, exit, swimming velocity vN, vmax m/s transverse (normal) velocity across an AF edge, maximum velocity of the TE relative to the fluid vF, vidz m/s flight velocity, induced velocity from vortex detachment at the TE vR m/s overall resulting incidence flow velocity at 1/4 chord point M vPRN m/s transverse (normal) flow component at the TE ‘V’ – fluid control volume x, y, z m Cartesian coordinates α, β, ε rad angle of attack, wake inclination, gliding (slope) γ , γo rad plunging or flapping angle of wing, amplitude of γ ϕ, ϕo rad pitching of the AF chord m2/s circulation 2 θs, θe kg m polar moments of inertia: solid, fluid equivalent ν m2/s kinematic viscosity of fluid ψ rad phase difference Øm2/s flow potential ρ kg/m3 fluid density τf – dimensionless feeding time ω s−1 circular frequency of oscillation (rotation) s−1 circular frequency of the earth’s rotation

References

[1] Liebe, W., Was ist eigentlich ein Wirbel? ZAMM, 65(4), pp. T209–T211, 1985. [2] Prandtl, L., Über Flüssigkeitsbewegung bei sehr kleiner Reibung. Verhandlungen des III Internat. Math. Kongresses, Heidelberg (August 1904), Teubner Verlag, pp. S484–S491, 1905. [3] Helmholtz, H., Über diskontinuierliche Flüssigkeitsbewegungen, Monatsberichte der Königlichen Akademie der Wissenschaften zu Berlin, pp. S215–S228, 1868. [4] Prandtl, L., Oswatitsch et al., Strömungslehre, 7 Auflage, Vieweg Verlag, 1969. [5] Van Dyke, M., An Album of Fluid Motion, Parabolic Press: Stanford, USA, 1982. [6] Lugt, H., Wirbelströmung in Natur und Technik, G. Braun Verlag, 1979. [7] Kuethe, A. et al., Foundations of Aerodynamics, John Wiley & Sons, 4th edn, 1986. [8] Truckenbrodt, E., Fluidmechanik, Band 1, Springer-Verlag, 1980. [9] Platzer, M.F. & Jones, K.D., Concepts of aero- and hydromechanics. Flow Phenomena in Nature, Vol. 1, pp. 5–19, 2006. [10] Liebe, R., Applications of the finite vortex model. Int. Conf. Proc. Design & Nature 2004, Rhodes/Greece, 6/2004. [11] Liebe, R., Unsteady flow mechanisms on airfoils: the extended finite vortex model with applications. Flow Phenomena in Nature, Vol. 1, pp. 283–339, 2006. [12] Liebe, R. & Liebe, W., The finite vortex model for unsteady aerodynamics. Int. Ind. Conf. BIONICS 2004, Hannover, April 2004, Fortschrittsberichte VDI, Reihe 15, Nr. 249, pp. S191–S199, 2004. [13] Elsenaar, A., Vortex formation and flow separation: the beauty and the beast in aerodynamics. The Aeronautics Journal, pp. 615–633, 2000. [14] Liebe, W., Der Auftrieb am Tragflügel: Entstehung und Zusammenbruch, Aerokurier, Heft 12, pp. S1520–S1523, 1979. [15] Liebe, W., Generation and breakdown of aerodynamic lift, Presentation at Hampton, VA, NASA Publication, March 1979. [16] Liebe,W.,Über die Entstehung desAuftriebs am Tragflügel, Instationäre Effekte an schwin- genden Tragflügeln (Sonderdruck), W. Nachtigall, Wiesbaden, June 1980. [17] Liebe, W., Der sogenannte Potentialwirbel. ZAMM 71, 71(5), pp. T545–T546, 1991. [18] Hütte, Die Grundlagen der Ingenieurwissenschaften, 30 Auflage, H. Czichos, Springer- Verlag, 1996. [19] Liebe, W., Der natürliche Wirbel. Forschung im Ingenieurwesen, 51(6), pp. S175–S180, 1985. [20] Liebe, W., Zirkulation und Zähigkeit: Der Rotor im realen fluid. ZAMM, 70(5), pp. T483– T485, 1990. [21] Albring, W., Angewandte Strömungslehre, Akademie Verlag: Berlin, 1978. [22] Liebe, A., Photo Intake vortex at the hydroelectric power plant, St. Malo/France, 2004. [23] Liebe, W., Der Schwanzschlag der Fische. VDI-Zeitschrift, 105(28), pp. S1298–S1302, 1963. [24] Liebe, W., Liebe, R., A general finite vortex model to simulate unsteady aerodynamics in nature. Int. Conf. Proc. Design & Nature 2002, Udine/Italy, 9/2002. [25] Eck, B., Technische Strömungslehre, 7 Auflage, Springer-Verlag, 1966. [26] Ahlborn, F., Die Ablösungstheorie der Grenzschichten und die Wirbelbildung, Jahrbuch der Wissenschaftlichen Gesellschaft für Luftfahrt e.V. (WGL), pp. S171–S188, 1927. [27] Bublitz, P., Geschichte der Entwicklung der Aeroelastik in Deutschland von den Anfängen bis 1945, DFVLR-Mitteilungen 86-25, Köln, 1986.

[28] Liebe, R., Validation of the finite vortex model by analyzing unsteady aerodynamic experiments. Int. Conf. Proc. Design & Nature 2002, Udine/Italy, 9/2002. [29] Birnbaum, W., Das ebene Problem des schlagenden Flügels. ZAMM, 4 , S277–S292, 1924. [30] Durst, F., Numerische Strömungs-Simulationen, Info-Schrift vom LSTM, Universität Erlangen-Nürnberg, Germany, 2004. [31] Liebe, W., Die schwingende Hinterkante als Auftriebshilfe und zur Verminderung des Widerstands am Tragflügel, DGLR-Jahrestagung, Berlin, 1999. [32] Liebe, W., Vortrieb durch Flügelschlag, BIONA-Report 15, 5.Bionik Kongreß in Dessau, 2000. [33] Liebe, R., Swimming performance of an oscillating hydrofoil propulsor. International Journal of Design and Nature (to appear). [34] Wood, C.J. & Kirmani, S.F.A., Visualization of heaving airfoil wakes including the effect of a jet flap. Journal of Fluid Mechanics, 41, pp. 627–640, 1970. [35] Koochesfahani, M.M., Vortical patterns in the wake of an oscillating airfoil. AIAA Journal, 27(9), pp. 1200–1205, 1989. [36] Liebe, W. & Liebe, R., Modell, Berechnung und Anwendung periodisch erzeugter Kanten- wirbel im Turbomaschinenbau, Patent No. DE 102 37 341 A1 (14.8.2002). [37] Fish, F., Review of natural underwater modes of propulsion, Report West Chester Univer- sity, PA USA, 1999. [38] Donley, J.M. & Sepulveda, C.A. et al., Convergent evolution in mechanical design of lamnid sharks and tunas. Nature, 429, pp. 61–65, 2004. [39] Freymuth, P., Vortex visualizations for two-dimensional models of caudal fin fish propulsion. Flow Phenomena in Nature, Vol. 1, WIT: UK, pp. 340–356, 2006. [40] Fish, F., Power output and propulsive efficiency of swimming bottlenose dolphins (Tursops Truncatus). Journal of Experimental Biology, 185, pp. 179–193, 1993. [41] Liebe, W., Konstruktionen der Natur, Gesellschaft f. techn. Biologie und Bionik, RS Nr. 35, Feb. 2002. [42] Hertel, H., Struktur, Form, Bewegung, Krausskopf Verlag, Mainz, 1963. [43] Carwardine, M., Haie, Delius Klasing Verlag: Bielefeld, 2004. [44] Lindhe Norberg, U.M., Evolution of flight in animals. Flow Phenomena in Nature, Vol. 1, WIT: UK, pp. 36–53, 2006. [45] Schwipps, W., 100 Jahre Flugpatent Lilienthal, Luft- und Raumfahrt, 4, pp. S34–S36, 1993. [46] Lilienthal, O., Der Vogelflug als Grundlage der Fliegekunst, Gaertners Verlagsbuchhand- lung: Berlin 1889. [47] Spedding, G.R., Rosen, M.& Hedenström, A., A family of vortex wakes generated ...free flight ...natural range of flight speeds. Journal of Experimental Biology, 206, pp. 2313– 2344, 2003. [48] Send, W., Photo ANIPROP SM 1, Göttingen, 8/2005. [49] Soder, E., Der Mauersegler, NABU-Bundesverband Bonn, 2003. [50] Jones, D., Adler, Könnemann Verlags-GmbH, 1996. [51] Burton, R., Das Leben der Vögel, Kosmos Verlag, Franckh, 1985. [52] Kawachi, K., Unsteady wing characteristics at low Reynolds number. Flow Phenomena in Nature, Vol. 2, pp. 420–434, 2006.