A note on the thrust of airfoils

J. Fluid Mech. (2025), vol. 1012, A6, doi:10.1017/jfm.2025.10177 A note on the thrust of airfoils José M. Gordillo Área de Mecánica de Fluidos, Departamento de Ingeniería Aeroespacial y Mecánica de Fluidos, Universidad de Sevilla, Avenida de los Descubrimientos s/n 41092, Sevilla, Spain Corresponding author: José M. Gordillo, [email protected] (Received 4 October 2024; revised 1 April 2025; accepted 22 April 2025) Here, we show that the thrust force of oscillating airfoils calculated within the linearised potential flow approach by means of the vortex impulse theory coincides with the one resulting from the integration of the unsteady pressure distribution around the solid obtained by Garrick (1936) when the vertical component of the wake velocity is calculated self-consistently and the analysis retains the contribution of the flux of horizontal momentum induced by the starting vortex. The limitations of the self-consistent linearised potential flow approach for predicting the thrust force of airfoils oscillating periodically with small amplitudes but large values of the reduced frequency are also discussed, as well as the reasons behind the ability of other results in the literature to approximate measurements better than Garrick’s theory. In fact, for those cases in which the airfoil oscillates periodically, the flux of horizontal momentum induced by the starting vortex is negligible and the vortices in the wake are convected parallel to the free-stream velocity, we have deduced an equation for the mean thrust coefficient which differs from previously published results and is in agreement with experimental and numerical results. In addition, for those cases in which the airfoil is suddenly set into motion, we have also deduced an equation that retains the effect of the starting vortex and correctly quantifies the transient thrust force. Key words: aerodynamics, flow-structure interactions 1. Introduction The quantification of the forces exerted over oscillating airfoils within the potential flow and slender-body limits traces back to the classical works of Wagner (1925), who calculated the unsteady lift force over an airfoil experiencing a sudden change in the angle of attack, of Theodorsen (1935), who considered the analogous case for airfoils performing periodic pitching and heaving motions, of Garrick (1936), who calculated thrust by adding © The Author(s), 2025. Published by Cambridge University Press. This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (https://creativecommons.org/ licenses/by/4.0/), which permits unrestricted re-use, distribution and reproduction, provided the original article is properly cited. 1012 A6-1 https://doi.org/10.1017/jfm.2025.10177 Published online by Cambridge University Press J.M. Gordillo to the suction force at the leading edge of the airfoil the projection in the flight direction of the lift force calculated by Theodorsen, and to the also seminal contribution of von Kármán & Sears (1938), who obtained the same results previously deduced by Wagner (1925)and Theodorsen (1935) but making use of a momentum balance i.e. using the vortex impulse theory. The results of these classical studies, which were originally developed in the field of aeroelasticity, have recently been extended to quantify the unsteady forces experienced by flying or swimming animals at high values of the Reynolds number (Wu 1961; Smits 2019). Experiments, see Mackowski & Williamson (2015) and references therein, as well as the numerical simulations of Young & Lai (2004), reveal that the classical theory due to Garrick (1936) overestimates both thrust and the propulsion efficiency for sufficiently large oscillation frequencies and amplitudes because: (i) the real wake is non-planar (Young & Lai 2004; Godoy-Diana, Aider & Wesfreid 2008; Mackowski & Williamson 2015), a fact which contrasts with the approximations made in the linearised theory, (ii) the viscous drag, which plays an essential role in determining the optimal Strouhal number which maximises the propulsion efficiency (Floryan, Buren & Smits 2018), is neglected in the potential flow approach, (iii) the vortices ejected from the leading edge of the airfoil at large amplitudes of the heaving and pitching motions (Young & Lai 2004, 2007), which tend to reduce thrust, are not captured by the linearised theory and (iv) the three-dimensional effects associated with the finite span of the body (Zurman-Nasution, Ganapathisubramani & Weymouth 2020) were not considered by Garrick in his original contribution. In spite of these drawbacks, a series of recent studies emphasise that the linearised theory due to Garrick is capable of approximating the time-varying value of the thrust force for sufficiently small values of the oscillation amplitudes and reduced frequencies if the effects of static drag are taken into consideration in the modelling (Young & Lai 2004, 2007; Mackowski & Williamson 2015; Saadat et al. 2017;Floryanet al. 2018). Moreover, Floryan et al. (2017) find that Garrick’s result already provides the correct scaling for the thrust force even for large amplitudes of the oscillations. In an attempt to improve the predictions of Garrick’s theory for values of the oscillation frequencies larger than the inverse of the characteristic residence time, a series of very recent contributions (Fernandez-Feria 2016,2017; Alaminos-Quesada & Fernandez-Feria 2020; Sanchez-Laulhe, Fernandez-Feria & Ollero 2023), extends the linearised vortex impulse theory by von Kármán & Sears (1938) with the purpose of calculating the aerodynamic thrust. Newton’s laws dictate that the aerodynamic force calculated using the vortex impulse theory, which results from a momentum balance, must coincide with the one obtained by integrating the pressure distribution around the airfoil (Eldredge 2019) and, hence, the linearised theories by Garrick (1936) and Fernandez-Feria (2016,2017) should provide identical results. However, for values of the reduced frequency of order unity or larger, the predictions by Fernandez-Feria (2016,2017) are in better agreement with the experimental and numerical results reported by Young & Lai (2004,2007)and Mackowski & Williamson (2015) than the ones deduced using Garrick’s theory. Motivated by the better agreement with experimental data, it is explicitly stated in FernandezFeria (2016,2017) and Alaminos-Quesada & Fernandez-Feria (2020) that the vortex impulse formulation of Fernández-Feria corrects the theory due to Garrick, and it is one of the purposes of the present study to find the origin of the differences between the results in Garrick (1936) and in Fernandez-Feria (2016). Indeed, since the predictions in Fernandez-Feria (2016) do not reproduce Garrick’s results, one of the two theories is not self-consistent because, otherwise, the force calculated by the direct integration of the 1012 A6-2 https://doi.org/10.1017/jfm.2025.10177 Published online by Cambridge University Press Journal of Fluid Mechanics pressure distribution around the airfoil would be different from the value obtained through a momentum balance. It will be shown next that the linearised theories due to Garrick (1936)andthe one deduced using the vortex impulse theory, which we develop here by extending the momentum balance in von Kármán & Sears (1938), provide identical results for the aerodynamic force if: (i) the flux of momentum induced by the starting vortex emitted initially from the trailing edge of the airfoil is taken into account and (ii) the vertical velocities of the vortices in the wake are calculated in a self-consistent manner. Indeed, in order to recover Garrick’s result using the vortex impulse formulation, it proves essential that the vertical velocity of the vortices in the wake are calculated self-consistently within the linearised approach, namely, as a result of the vertical velocities induced by the vortex sheet extending along the airfoil and the wake. In contrast, the theory by Fernandez-Feria (2016,2017) does not include the flux of momentum induced by the starting vortex and, in addition, Fernandez-Feria (2016,2017), Alaminos-Quesada & Fernandez-Feria (2020) and Sanchez-Laulhe et al. (2023) do not calculate the vertical velocities of the vortices in the wake in a self-consistent manner but, instead, impose their values: indeed, the assumption in equation (25) in Fernandez-Feria (2016) implies that the vertical velocity of the vortices in the wake is zero. However, we show here that there is no need to impose the value of the vertical velocities of the vortices in the wake because the linearised potential flow theory already permits us to calculate these velocities in a self-consistent manner: in fact, only if this is done does the vortex impulse theory recover the results originally deduced by Garrick, consistently with the fact that the force calculated through a momentum balance must coincide with the value obtained by direct integration of the pressure distribution around the airfoil. One of the main conclusions of this study is that the correct equation for the thrust force within the linearised potential flow approach is the one due to Garrick (1936) or the equation deduced here using the vortex impulse theory in a self-consistent manner, a conclusion that contradicts the assertions in FernandezFeria (2016,2017) and Alaminos-Quesada & Fernandez-Feria (2020). Then, the ability of Fernández-Feria’s results to predict experimental measurements does not mean that Garrick’s theory is incorrect: we show here that the success of Fernández-Feria’s results rests on the fact that the assumption made in Fernandez-Feria (2016,2017) and AlaminosQuesada & Fernandez-Feria (2020) of neglecting the contribution of the starting vortex and of imposing the vertical velocities of the wake vortices to be equal to zero, reflects the realistic nonlinear dynamics of the wake for sufficiently large values of the oscillation frequency. Clearly, these nonlinear effects cannot be accounted for by any self-consistent linear theory. For those cases in which the airfoil oscillates periodically, the flux of horizontal momentum induced by the starting vortex is negligible and the vortices in the wake are convected parallel to the free-stream velocity, we also deduce here an equation for the so-called mean thrust coefficient. This equation differs from previously published results and is in agreement with experimental and numerical results. However, the results in this contribution are not only limited to the study of the thrust force of periodically oscillating airfoils: our results also permit us to calculate the thrust force in transient manoeuvres, like those taking place when an airfoil is impulsively set into motion. In fact, we derive the analytical expression for the thrust force corresponding to the so-called Wagner problem (Wagner 1925) in two different ways, namely, by the direct integration of the pressure distribution around the airfoil and by also using the vortex impulse theory. We validate all the analytical results obtained by means of the numerical code detailed in the Supplementary Material. This contribution is structured as follows: §2is devoted to showing that, within the linearised potential flow approximation, the thrust force calculated by means of the vortex 1012 A6-3 https://doi.org/10.1017/jfm.2025.10177 Published online by Cambridge University Press J.M. Gordillo z e2 ρ, p∞,U∞e1e1xe h(t) α(t) n c + U∞t erɛ 0x Figure 1. Sketch of the canonical flow considered in this study. impulse theory is identical to the classical result due to Garrick once the flux of horizontal momentum induced by the starting vortex is retained in the analysis and the vertical component of the wake velocity is calculated self-consistently. This conclusion will be illustrated by the numerical examples included in §3, where we also establish the limits under which Garrick’s theory can be used to predict experimental measurements. For those cases in which the airfoil oscillates periodically, the flux of horizontal momentum induced by the starting vortex is negligible and the wake vortices are convected parallel to the freestream velocity, we deduce in § 4an analytical equation for the mean thrust coefficient which has been validated using the results of numerical simulations carried out using the vortex-lattice method. The main results are summarised in § 5. 2. Calculation of thrust through a momentum balance The canonical flow to be studied in what follows, which employs the same notation and sign conventions as those used in Bisplinghoff, Ashley & Halfman (1996) except for the fact that, here, the origin of the Cartesian coordinate system is located at the leading edge of the airfoil, is illustrated figure 1: an airfoil of chord cextending along 0 ⩽x⩽c,with xand zdenoting the Cartesian horizontal and vertical coordinates with associated unit vectors e1and e2, forms a time-dependent angle of attack α(t)with an incident uniform stream of density ρand velocity v=U∞e1. The origin of times is set at t=0 and, hence, within the classical linearised potential flow approach, the horizontal position of the starting vortex is x=c+U∞t(Wagner 1925; Theodorsen 1935; von Kármán & Sears 1938;Glauert1983; Ashley & Landahl 1985). Moreover, the vertical position of the point located at a distance x=xe<cfrom the leading edge of the airfoil is za(xe,t)=−h(t) and, hence, for the case of a symmetrical airfoil with zero thickness considered, here, za(x,t)=−h(t)−α(t)(x−xe),(2.1) with the subscripts aand wreferring from now on to quantities corresponding to either the airfoil or the wake. Notice that, in this contribution, positive lift (t)corresponds to a force in the positive z-direction, positive thrust −d(t)is positive in the negative x-direction, whereas positive h(t)corresponds to motion in the negative z-direction and, similarly, positive torque m(t)is in the counterclockwise direction, while positive α(t)gives clockwise rotation. The aerodynamic force f(t)=(t)e2+d(t)e1and torque over such an airfoil, which possesses the two degrees of freedom α(t)and h(t), are calculated for the common case in which the Reynolds number verifies the condition Re =ρU∞c/μ 1, with μindicating the dynamic viscosity; moreover, we will consider that the relative density variations are negligible and that α(t)1, za,w/c1, h/c1, with the vertical positions of the points on the airfoil za(x,t)defined in (2.1)andzw(x,t)referring to the vertical position of the points in the wake. Therefore, under these conditions, 1012 A6-4 https://doi.org/10.1017/jfm.2025.10177 Published online by Cambridge University Press Journal of Fluid Mechanics the thin boundary layer of thickness δsuch that δ/c∝Re−1/21 does not separate and, hence, the classical, linearised potential flow theory summarised in e.g. Ashley & Landahl (1985), is applicable. Indeed, outside the thin boundary layer and the wake, the velocity field is irrotational, namely, v=∇φ,withφ=U∞x+φand φindicating the perturbed velocity potential associated with the perturbed velocity field v=∇φ=ue1+we2verifying the condition |∇φ|/U∞1 except, as will become clear in what follows, at the leading edge, x→0 and, for the case of an airfoil which is suddenly set into motion, at x→c+U∞ti.e. where the starting vortex is located. Then, by virtue of the continuity equation ∇·v=0, the perturbed potential satisfies the Laplace equation ∇2φ=0,(2.2) which must be solved subject to the boundary condition at infinity φ→0andtothe linearised impenetrability condition, which can be expressed as DF Dt=∂F ∂t+U∞e1+∇φ·∇F=0withF=z−za,w(x,t), (2.3) and with za(x,t)given in (2.1). The standard linearisation of (2.3) yields (Ashley & Landahl 1985) w a,w(x,z=0±,t)=∂za,w ∂t+U∞ ∂za,w ∂x,(2.4) with z=0±indicating the upper and lower sides of the airfoil and the wake and w a,w referring to the vertical component of the velocity on the airfoil or at the wake. In view of the linearised impenetrability boundary condition at 0 ⩽x⩽cgiven by (2.1) and (2.4), we seek antisymmetric solutions of the Laplace equation (2.2)intheform of a vortex sheet extending along the airfoil and the wake, whose circulation density is the one satisfying the condition expressed by (2.1)and(2.4). Notice that, making use of the notation φ± =φ(x,z=0±,t),Γ(x,t)=C(U∞e1+∇φ)·d=(φ+−φ−)= 2φ+(x,t)and γ(x,t)=u+ −u− =∂Γ/∂x, in this contribution Γ(x>0,t)=2φ+(x>0,t)=x −∞ γ(x0,t)dx0=x 0 γ(x0,t)dx0(2.5) refers to the clockwise circulation along any closed loop encircling the leading edge of the airfoil and connecting the points (x>0,z=0−)and (x>0,z=0+), whereas γ(x,t)indicates the circulation density. In (2.5) we have taken into account that, since the origin of the vortex sheet is the leading edge of the airfoil, which is located at x=0, γ(x<0,t)=0, and hence the circulation for x<0 is also zero because Γ(x<0,t)= x −∞ γ(x0,t)dx0=0. In the following, Γa,w(x,t)and γa,w(x,t)will indicate the values of the circulation and of the circulation density on the airfoil, which extends along 0⩽x⩽cor at the wake, which extends along x>c. The equation governing the pressure jump at z=0, namely, p(x,z=0,t)= p(x,z=0−,t)−p(x,z=0+,t)=p−(x,t)−p+(x,t),withp=p−p∞indicating the perturbed pressure, can be deduced from the linearised Bernoulli equation particularised at z=0± z=0±:ρ∂φ± ∂t+ρU∞ ∂φ± ∂x+p± =0.(2.6) 1012 A6-5 https://doi.org/10.1017/jfm.2025.10177 Published online by Cambridge University Press J.M. Gordillo Hence, the subtraction of the two equations in (2.6) yields ρ∂Γ ∂t+ρU∞ ∂Γ ∂x=ρ∂ ∂tx 0 γdx0+ρU∞γ=G(x,t)with G(0⩽x⩽c,t)=pa(x,t)(2.7) with pathe pressure jump at the airfoil and, since p=0forx<0andx>c,we conclude that G(x<0,t)=G(x>c,t)=0in(2.7), a fact implying that the material derivatives of both Γand γare zero at z=0forx<0andx>c, namely (Ashley & Landahl 1985), DΓ Dt=∂ ∂xDΓ Dt=0⇒∂Γ ∂t+U∞γ=0,∂γ ∂t+U∞ ∂γ ∂x=0forx<0,x>c. (2.8) Taking into account that Γ=0forx→−∞and also for instants t<0 and that the circulation at the origin of the wake is prescribed by the circulation around the airfoil, namely, Γw(x=c,t)=Γa(x=c,t)=Γe(t)=c 0 γa(x,t)dx,(2.9) with Γe(t)the circulation around the airfoil, we deduce from (2.8)and(2.9)that Γ(x<0,t)=0,Γ(x>c+U∞t,t)=0,Γ w(x=c+U∞(t−t0), t)=c 0 γa(x,t0)dx, and γw(x=c+U∞(t−t0), t)=γw(x=c,t0), (2.10) with γw(x→c,t0)given by (2.8)and(2.10) – see also equations (13)–(27) in Ashley & Landahl (1985) d dtc 0 γa(x,t)dx(t0)+U∞γw(c,t0)=0,(2.11) where γw(x→c,t0)=γw(c,t0)from which we conclude that γw(x0=c+U∞(t−t0), t)=γw(x=c,t0)=− 1 U∞ d dtc 0 γa(x,t)dx(t0) =− 1 U∞ dΓe dt(t0), (2.12) with the circulation around the airfoil Γe(t)defined in (2.9). Equations (2.7)and(2.12) indicate that the unsteady lift force and the torque (t)=c 0 pa(x,t)dxand m(t)=c 0 xpa(x,t)dx,(2.13) as well as the density of circulation along the wake, γw(x,t), can be expressed as a function of γa(x,t). Finally, the density of circulation at the airfoil, γa(x,t), is deduced by imposing that the perturbed vertical velocity induced by the vortex sheet extending along z=0, 0 ⩽x⩽ c+U∞tsatisfies the linearised impenetrability condition given by (2.1)and(2.4), namely (Ashley & Landahl 1985), 1012 A6-6 https://doi.org/10.1017/jfm.2025.10177 Published online by Cambridge University Press Journal of Fluid Mechanics w a(x,z=0±,t)=−dh dt−U∞α(t)−dα dt(x−xe) =1 2πc 0 γa(x0,t) x0−xdx0+1 2πc+U∞t c γw(x0,t) x0−xdx0.(2.14) Introducing the change of variables x0=c+U∞(t−t0)⇒dx0=−U∞dt0,(2.15) and taking into account that the second integral at the right-hand side of (2.14) can be expressed solely in terms of γaby means of (2.12), the equation for γa(x,t)reads w a(x,z=0±,t)=1 2πc 0 γa(x0,t) x0−xdx0−1 2πt 0 dΓe dt0 c+U∞(t−t0)−xdt0 with Γe(t)=c 0 γa(x,t)dx.(2.16) In order to solve the integral equation (2.16) notice first that, since Γ(x→−∞,t)=0 then, by virtue of (2.8), Γ(x<0,t)=0 and hence, φ(z=0±,x<0)=0. Consequently, the local solution of the Laplace equation (2.2) at the leading edge of the airfoil is the one corresponding to the flow around a wedge of angle 2π, namely, φ=U∞cA0(t)r c1/2cos β 2,(2.17) with A0(t)a dimensionless time-dependent constant, r/c1 the radial distance to the leading edge – which is located at z=0, x=0 in the linearised theory – and 0 ⩽β⩽2π indicating the polar angle measured in counterclockwise manner from the horizontal axis. Taking into account: (i) that the Kutta condition ensures that γa(x=c,t)is finite in order to avoid that the flow turns around the trailing edge of the airfoil and (ii) that γa(x/c1,t)is given by ∂φ ∂rβ=0,r c1,t=uz=0+,x c1,t=U∞ 2A0(t)x c−1/2 ⇒γax c1,t=U∞A0(t)x c−1/2 ,(2.18) where use of (2.17) has been made, it can be concluded that the integral equation (2.16) can be solved using Glauert’s method, which relies on expressing the unknown function γa(x,t)as the infinite series (Glauert 1983) γa(x,t) U∞=A0(t)      1−x c x c +∞  n=1 An(t)sin(nθ)=A0(t)1+cos θ sin θ+∞  n=1 An(t)sin(nθ), (2.19) where we have introduced the change of variables x c=1−cos θ 2,(2.20) and, therefore, θ=0atx=0andθ=πat x=c. Notice that the expansion (2.19) implies that γa(x=c,t)=0 and, hence, the density of circulation at x=cdoes not satisfy the 1012 A6-7 https://doi.org/10.1017/jfm.2025.10177 Published online by Cambridge University Press J.M. Gordillo physical condition γa(x=c,t)=γw(x=c,t). However, the continuity of the circulation density at the trailing edge could be enforced following the procedure detailed in Alben (2010) and Eldredge (2019) and references therein by analytically removing the logarithmic singularity at x=cin (2.16). This more elaborate method provides a faster convergence to the solution which, however, can also be found using the more classical procedure followed here by simply retaining a larger number of terms in (2.19), see Alben (2010). Then, the substitution of the expansion (2.19) into the integral equation (2.16) provides us with the values of the time-dependent coefficients Ai(t)as a function of α(t)and h(t), as detailed in the Supplementary Material, see also Wagner (1925), Theodorsen (1935), von Kármán & Sears (1938), Wu (1961) and references therein. Once the values of Ai(t)are known, pa,(t)and m(t)can be determined by means of (2.7)and(2.13), as illustrated, for instance, in the Supplementary Material. Now that (t)is known, the value of the drag force can be obtained by adding to the projection in the flight direction of the lift force the resulting suction force at the leading edge of the airfoil, yielding d(t)=α(t)(t)−ρU2 ∞πcA2 0(t) 4;(2.21) see Garrick (1936), Wu (1961) and references therein, as well as the next subsection. Hence, the thrust force obtained by the direct integration of the pressure distribution around the airfoil is given by TG(t)=−d(t)=−α(t)(t)+ρU2 ∞πcA2 0(t) 4,(2.22) where we have made use of (2.21) and the subscript Gindicates Garrick. Let us point out here that the expression of A0(t)in (2.22) for the case of periodic oscillations of the airfoil was provided by Garrick (1936) using the results of Theodorsen (1935), whereas the corresponding value for arbitrary heaving or pitching motions is deduced elsewhere; see, for instance, the Supplementary Material. 2.1. Forces calculated through a momentum balance So far we have calculated the unsteady lift and thrust forces on the airfoil as a result of the integration of the pressure distribution around the solid. It is now our purpose to calculate the aerodynamic force fthrough a momentum balance using the control volume Ωc(t) limited by a fixed surface Σ∞of dimensionless radius R/c→∞which encircles both the solid and the wake, by the surface Σa,w bounding both the solid and the wake and by Σ, which is a circle of radius →0 centred where the starting vortex is located, namely, at x=c+U∞t. The momentum balance applied at the control volume Ωc(t)defined above yields d dtΩc(t) ρvdω+∂Ωc ρv(v−vc)·ncdσ=∂Ωc (p−p∞)( −nc)dσ, (2.23) with ncthe unit normal pointing outwards the control volume, v=∇φand vcindicating the velocity of the surfaces bounding the control volume, namely, vc=U∞e1at Σand (v−vc)·nc=0atΣa,w. Since there is no relative momentum flux across the surfaces Σa,w, and taking into account that the pressure force at the airfoil is f=Σa(t) (p−p∞)ncdσ, (2.24) 1012 A6-8 https://doi.org/10.1017/jfm.2025.10177 Published online by Cambridge University Press Journal of Fluid Mechanics (2.23) can be written, making use of Gauss’ theorem, as f=−d dtΣaΣw ρφncdσ+Σ ρ∇φ(∇φ−U∞e1)·erdσ+Σ (p−p∞)( −er)dσ, (2.25) where we have taken into account that v=∇φ=∇φ+U∞e1and errefers to the unit vector in polar coordinates, see figure 1.Moreover,in(2.25)wehavemadeuseofthe fact that the integrals evaluated at Σ∞tend to zero because of the Bernoulli equation ρ∂φ/∂t+ρ|∇φ|2/2+p=Kand because Σ∞encircles both the airfoil and the wake and, hence, the circulation around Σ∞is zero, which implies that the perturbed velocity field decays faster than U∞c/Rat infinity, which ensures that both the flux of momentum and the integral of |∇φ|2along Σ∞tend to zero. Then, the aerodynamic force can be written, in the linearised approach, as f=d dtc 0 ρΓandx+d dtc+U∞t c ρΓwndx +Σ ρ∇φ(∇φ−U∞e1)·erdσ+Σ (p−p∞)( −er)dσ, (2.26) with nthe unit vector pointing outwards the side z=0+of Σaand Σw,seefigure 1, and where we have taken into account that the unit normal pointing outwards the side z=0−of Σaand Σwis −n, this being the reason why the integrand in the first two integrals in (2.26)isΓ=φ+ −φ−; in addition, in (2.26)wehavemadeuseofthefact that, by virtue of (2.17) and of the paragraph preceding this equation, where it is shown that Γ(x⩽0,t)=0, the value of the integral extending along a small region near the leading edge tends to zero. Finally, since Γ(x>c+U∞t,t)=0, see (2.10), the leadingorder equation for the perturbed potential at Σ, corresponds to the one characterising the flow around a wedge of angle −π⩽β⩽π, namely, φ=U∞cC sinβ 2r c1/2 ⇒∇φ=C 2U∞sinβ 2r c−1/2er+C 2U∞cosβ 2r c−1/2eβ,(2.27) with er=cos βe1+sin βe2,eβ=−sin βe1+cos βe2,r/cindicating the dimensionless distance to the starting vortex located at x=c+U∞tand Cis a dimensionless constant which does not depend on time because, by virtue of (2.8), the value of the perturbed potential remains constant at Σ; hence, Cis fixed at t=0+, right after the airfoil is set in motion, see Appendix A, where Cis calculated. Notice that the last integral in (2.26) is zero because, by virtue of (2.27), |∇φ|is constant at Σand hence, by virtue of the Bernoulli equation, pis also constant at Σ. Finally, the third integral in (2.26), corresponding to the momentum flux across Σ, is calculated using (2.27), which yields the following expression for the aerodynamic force: f=d dtc 0 ρΓandx+d dtc+U∞t c ρΓwndx−ρU2 ∞πcC2 4e1.(2.28) Taking now into account that α(t)1andh(t)/c1, the linearisation of the normal vector nin equation (2.28) yields n·e2≃1andn·e1=−∂za,w(x,t) ∂x.(2.29) 1012 A6-9 https://doi.org/10.1017/jfm.2025.10177 Published online by Cambridge University Press J.M. Gordillo 0510 τ 15 20 1 2 3 4 tG, VI tG, VI 5 6 7 8 (a)(b) ×10–3 0510 τ 15 20 1 2 3 4 5 6 7 8×10–3 Figure 2. Dimensionless thrust forces tG(τ) (blue line) and tVI(τ) (red line) respectively defined in (3.1) and (3.2), corresponding to the plunging motion prescribed by (3.4), for two different values of the reduced frequency: (a)k=2and(b)k=4. Notice that both results coincide at every instant of time, as expected from the result in (2.52). 05 10 τ 15 20 1 0 –1 –2 2 3 4 tG, FF 5 6 7 8 05 10 τ 15 20 1 0 –1 –2 2 3 4 tG, FF 5 6 7 8 (a)(b) ×10–3 ×10–3 Figure 3. Dimensionless thrust forces tG(τ) (blue line) and tFF(τ ) (black line) respectively defined in (3.1) and (3.6), corresponding to the plunging motion prescribed by (3.4), for two different values of the reduced frequency: (a)k=2and(b)k=4. Notice that the differences between the values of tG(τ) and of tFF(τ) increase with the value of the reduced frequency k. Moreover, the results in the figure show that the mean thrust corresponding to tFF is smaller than the mean thrust corresponding to tG, and also that the differences between the values of the mean thrust become more pronounced for the larger value of the reduced frequency. The results depicted in figures 2 and 3will be discussed in more detail in § 4, where an equation for the mean thrust coefficient is deduced. close to each other, with the tiny differences between the two results attributable to the effect of the numerical discretisation in the evaluation of the integrals in the definition of tVI(τ),see(3.2). Next, figure 3 compares the values of tG(τ) depicted in figure 2 with the values of the dimensionless thrust force defined as tFF(τ) =tVI(τ) −1 ρU2 ∞c×ρU2 ∞cπC 22 −ρc+U∞t c γww wdx,(3.6) 1012 A6-16 https://doi.org/10.1017/jfm.2025.10177 Published online by Cambridge University Press Journal of Fluid Mechanics x/c –0.40123456789 –0.3 –0.2 –0.1 0 0.4 0.3 0.2 0.1 wa,w Figure 4. Vertical velocities induced by the vortex sheet extending along the airfoil and the wake calculated using (2.41). The results depicted in the figure correspond to the plunging motion defined in (3.4)forthree different values of the reduced frequency: k=8(blackcurve),k=2(redcurve)andk=0.5 (blue curve). Notice that the amplitudes of the vertical velocities in the wake for the cases k=2andk=8 are clearly larger than the amplitude of the vertical airfoil velocity, V=5π/180. with tVI(τ) given in (3.2) and which represents the dimensionless thrust force calculated using the vortex impulse theory under the assumptions made in Fernandez-Feria (2016), namely, once the contributions of the starting vortex – which is zero for the particular case of the plunging motion considered in (3.4) – and of the contribution of the vortex thrust force of the vortices in the wake, see (2.40), are set to zero. The results depicted in figure 3 reveal that the values of tG(τ) and of tFF(τ ) are not identical to each other and, in fact, the differences between tG(τ) and tFF(τ) become more visible as the value of the reduced frequency kis increased; notice that the differences between the values of the mean thrust depicted in figures 2 and 3will be discussed in more depth in § 4. Using the results depicted in figure 2 andalsothosein(3.2)and(3.6), we can infer that the values of tG(τ) and of tFF(τ ) would be identical if w w(x>c,t)=0. However, this is not the case, as can be appreciated in figure 4, which shows that, in fact, the amplitudes of the vertical velocities in the wake increase with the value of k, despite the value of Vin (3.4) remaining unchanged. Notice also that figure 3 illustrates one of the main results deduced by Fernández-Feria: the mean thrust predicted in Fernandez-Feria (2016) is similar to the mean thrust predicted by Garrick’s theory for low-to-moderate values of the reduced frequency, but it is smaller than the mean thrust predicted by Garrick (1936) for kO(1). In view of the definitions in (3.1), (3.2)and(3.6) and of the conclusions in §2, the reason for the differences depicted in figure 3 is that the results in Fernandez-Feria (2016) were obtained neglecting the vertical velocities of the wake vortices i.e. assuming that w w(x,t)=0, which implies that the term (2.40) is also zero. However, the values of w w(x,t)calculated within the linearised potential flow theory by means of (2.41)are different from zero, as shown in figure 4 and, hence, the contribution to thrust of the vortex force in (2.40), cannot be neglected. Next, we consider the case in which an oscillating airfoil performs a purely plunging motion with a frequency ωand with a vertical velocity given by w a(0⩽x⩽c,t) U∞=−Vcos (kτ)with V=5π 180 .(3.7) 1012 A6-17 https://doi.org/10.1017/jfm.2025.10177 Published online by Cambridge University Press J.M. Gordillo –6 –4 –2 0 2 4 6 8 1 2 3 4 5 6 7 8 tG, VI ×10 –3 tG ×10 –3 0 0 2 4 6 8 10 12 14 16 18 200.5 1.0 1.5 2.0 ττ (a)(b) Figure 5. (a)Dimensionless thrust forces tG(τ) (blue line) and tVI(τ) (black line) defined, respectively, in (3.1)and(3.2), corresponding to the plunging motion of the airfoil prescribed by (3.7) for a value of the reduced frequency k=2. Notice that both results coincide at every instant of time, as expected from the result expressedby(2.52). The red curve corresponds to the results of (3.9), which does not retain the effect of the starting vortex. (b)After a short transient, the forces corresponding to the plunging motions defined in (3.4)– in blue – and (3.7) – in black – for a value of the reduced frequency k=2convergetothesameresultwithjust a phase shift. It is deduced in Appendix A that C=w a(x=3c/4,t=0+) U∞=−α0+˙ h0 U∞+˙α0 U∞3c 4−xe,(3.8) with quantities with the subscript 0 indicating their corresponding values at t=0+and, hence, the contribution of the starting vortex to the thrust force cannot be neglected if w a(x=3c/4,t=0+)=0. The results depicted in figure 5(a) confirm this is the case: indeed, the blue and black curves representing the values of tG(τ) (blue) and of tVI(τ) (black) defined in (3.1)and(3.2), are very close to each other. However, the red curve, which represents the numerical values of tVIB =tVI(τ) −πC2 4,(3.9) namely, of the thrust force calculated using the vortex impulse theory once the contribution of the starting vortex is set to zero, is well below the black and blue curves. Due to the fact that the value of the thrust force calculated using the vortex impulse theory must be identical to the value obtained by direct integration of the pressure distribution around the airfoil, the result depicted in figure 5(a) illustrates that the contribution of the starting vortex cannot be neglected for those cases in which w a(x=3c/4,t=0+)=0. Moreover, the results in figure 5(b) illustrate that, as expected, the thrust forces of the plunging motions corresponding to (3.4)and(3.7) converge to the same values with just a phase shift. Finally, we analyse the case of an airfoil that is suddenly set into motion, for which w a(0⩽x⩽c,t) U∞=−α0+˙ h0 U∞+˙α0 U∞ (x−xe)H(t), (3.10) with H(t)indicating the Heaviside function. The lift force corresponding to (3.10)was calculated by Wagner (1925) as, see also the Supplementary Material, 1012 A6-18 https://doi.org/10.1017/jfm.2025.10177 Published online by Cambridge University Press Journal of Fluid Mechanics Wagner(τ) =ρU2 ∞cπ⎛ ⎜ ⎜ ⎝− w ax=3c 4,t=0+ U∞⎞ ⎟ ⎟ ⎠ φ(τ), (3.11) with φ(τ) the so-called Wagner function, which is approximated here using the wellknown equation given by Jones (1938) φ(τ)=1−0.165 e−0.0455 τ−0.335 e−0.3τ,(3.12) whereas the corresponding expression for the thrust force is deduced in the Supplementary Material and reads TG,Wagner(τ) =ρU2 ∞cπ w ax=3c 4,t=0+ U∞ φ(τ)α0 +ρU2 ∞cπ⎛ ⎜ ⎜ ⎝ w ax=3c 4,t=0+ U∞ φ(τ)+c 4U∞˙α0⎞ ⎟ ⎟ ⎠ 2 .(3.13) Notice that a special case of (3.13) is given on page 705 of Donovan & Lawrence (1957). Hence, for the case of the so-called Wagner problem, characterised by the linearised impenetrability boundary condition given by (3.10), we find that tG,Wagner(τ) =TG,Wagner(t)+α0Wagner(t) ρU2 ∞c =π⎡ ⎢ ⎢ ⎣ w ax=3c 4,t=0+ U∞ φ(τ)+c˙α0 4U∞⎤ ⎥ ⎥ ⎦ 2 .(3.14) Consistently with the results obtained in § 2notice that, since the Wagner function verifies φ(τ =0)=1/2, see (3.12) and the Supplementary Material for further details, the value of the thrust force given by (3.13)atτ=0, TG,Wagner(τ =0), coincides with the value of TVI(τ =0)calculated using (2.38). The numerical results in figure 6, corresponding to the impulsive motion of the airfoil given by w a(0⩽x⩽c,t) U∞=−α0H(t)with α0=5π 180 ,(3.15) provides further support to (3.14) and also illustrates the relevance of the starting vortex in either of (2.37), (2.38), (2.42) to predict thrust when the airfoil is impulsively set into motion. However, the ability of the prediction in (3.14) to reproduce experimental measurements is limited to those circumstances in which the starting hypotheses of the linearised potential flow theory are not violated. Indeed, notice that the dynamics of the starting vortex generated when the airfoil is suddenly set into motion is described by the selfsimilar solution found by Kaden (1931), see also Pullin (1978). The width of this self-similar region grows in time as ∝(U∞αeq 0c1/2t)2/3,withαeq 0indicating the equivalent angle of attack defined in (A4)ofAppendix A. Hence, it is to be expected that, in 1012 A6-19 https://doi.org/10.1017/jfm.2025.10177 Published online by Cambridge University Press J.M. Gordillo 0 0.002 0.004 0.006 0.008 0.010 0.012 0.014 0.016 0.018 0.020 24 τ 6810 tG, VI Figure 6. (a)Dimensionless thrust forces corresponding to the impulsive motion of the airfoil prescribed by (3.15). Here, tVI(τ) (black line) and tG,Wagner(τ) (blue line) have been calculated using (3.2)and(3.14), respectively. The red curve corresponds to the results of (3.9), which do not retain the effect of the starting vortex. a real experiment, the wake region surrounding the starting vortex experiences large deformations, causing a clear deviation from the linearised approximation, which assumes that the wake is located at z=0. The deviations from the linearised approximation will take place for instants of time tt∗,witht∗estimated from (U∞αeq 0c1/2t∗)2/3∼cand, hence, t∗∼c/(U∞αeq 0)c/U∞if αeq 01. The nonlinear rolling up of the starting vortex described above will certainly modify the thrust force predicted here for large times tt∗; clearly, the validity of our results pertaining to the thrust produced by the starting vortex, which have been deduced within linearised potential flow theory, should be checked against detailed numerical calculations at finite but high Reynolds numbers. However, this additional task exceeds the limits of the present contribution. Similarly, for the cases of airfoils oscillating periodically which are not suddenly set into motion and, hence, the effect of the starting vortex is negligible, the capability of the linearised potential flow results to predict the mean thrust forces is also limited to those cases in which the effect of nonlinearities can be neglected. However, there are a number of experimental conditions that trigger the development of nonlinearities, limiting the applicability of Garrick’s results. First, notice that the linearised approach will only be valid to predict experiments and numerical simulations if H0/c1, with H0denoting the characteristic amplitude of the oscillations, but this necessary condition is not sufficient. Indeed, it has been shown by Ramesh et al. (2014), see also Eldredge (2019) that, in order to prevent flow separation and, hence, the associated ejection of vortices from the leading edge, it is necessary that the value of the adverse pressure gradient at x=0, which can be quantified through the value of A0in the expansion (2.19), is such that A0(t)α∗,with α∗∼O(0.1)indicating a value which can be viewed as a critical angle of attack for flow separation. The value of α∗depends on the type of airfoil and needs to be calculated either experimentally or by means of full numerical simulations. For the case of an oscillating airfoil, the characteristic value of A0can be estimated as the effective angle of attack resulting from the ratio between the characteristic vertical and horizontal velocities, namely, A0∼ωH0/U∞. Hence, in order to prevent flow separation at the leading edge, it is necessary that kH 0/cα∗,withkthe reduced frequency defined in (3.5). In addition, 1012 A6-20 https://doi.org/10.1017/jfm.2025.10177 Published online by Cambridge University Press Journal of Fluid Mechanics even in the case that H0/c1, the self-induced velocities of the vortices ejected from the trailing edge can cause large distortions in the wake for sufficiently large values of the reduced frequency: in fact, the vertical velocities in the wake are appreciably larger than the vertical velocities of the airfoil for large values of the reduced frequency, see figure 4. Indeed, the circulation around the airfoil can be estimated as Γe∼U∞cA0∼cH0ω,with A0estimated above and, moreover, the vertical velocity induced by the vortices ejected during a period of oscillation on the new vortices leaving the trailing edge of the airfoil can be estimated, making use of (2.12), as w 0∼1/U∞dΓe/dt∼cH0ω2/U∞. Hence, in order to prevent large deformations of the wake near the trailing edge of the airfoil, it is necessary that w 0/U∞O(1), namely, that (ω H0/U∞)2(H0/c)−1O(1)or, equivalently, that k2(H0/c)O(1). The emergence of nonlinearities for values of the reduced frequency such that k2(H0/c)O(1)is evidenced, for instance, when the wake is non-planar, as is clearly depicted in the experiments by Godoy-Diana et al. (2008) or in the full numerical simulations by Young & Lai (2004) and Martín-Alcántara et al. (2015). In these references, the vortices in the wake are arranged in such a way that the vertical component of their selfinduced velocities is negligible, as is evidenced by the fact that the wake vortices in these references are convected horizontally. We hypothesise that it is under these circumstances that the predictions in Fernandez-Feria (2016,2017) agree better with experiments than Garrick’s theory. The reason for this is that, when the vortices are convected horizontally, w w≈0 and, hence, the projection of ρ(∇×v)×von the flight direction is negligible, making the contribution of the vortex force term (2.40) also negligible in the real nonlinear flow. Clearly, these nonlinear effects cannot be accounted for by the theory developed by Garrick (1936) or by the self-consistent vortex impulse results presented here, which predict that the vertical velocities of the wake vortices is different from zero, as can be appreciated in figure 4. Then, the capability of the results in Fernandez-Feria (2016,2017) to predict both experiments and numerical results for values of the reduced frequency of order unity or larger, for which the wake experiences large deformations, seems to be justified by the fact that the results in Fernandez-Feria (2016,2017) correspond to a linearised model which takes into account realistic nonlinear effects. The discussion above permits us to conclude that, if the conditions of a particular experiment in which the airfoil oscillates periodically are such that the effects of nonlinearities are negligible and, hence, the starting hypotheses of the linearised potential flow approach are valid, the thrust force should be calculated using the self-consistent results expressed by (2.22)orbyeitherof(2.37), (2.38), (2.42). This explains why the experimental measurements of the mean thrust in Mackowski & Williamson (2015) can be reasonably well predicted by Garrick’s theory, which should be then used for small values of the reduced frequency once viscous effects are taken into consideration, see e.g. Figure 3(a) in Fernandez-Feria (2017). However, when nonlinearities are triggered and the vortices in the wake are arranged in such a way that their vertical self-induced velocities are negligible, namely, w w≈0, the linearised model in Fernandez-Feria (2016, 2017) and Alaminos-Quesada & Fernandez-Feria (2020) should be used instead, once the modifications to the equation for the mean thrust coefficient pointed out in § 4are taken into account, because the contribution to thrust of the vortex force term (2.40) is not calculated self-consistently within the linearised potential flow approach in these contributions but, instead, it is assumed to be zero. The value of the thrust force deduced by Fernandez-Feria (2016,2017), once the modifications in the mean thrust coefficient deduced in §4are taken into account, happens to be a better approximation to the real value 1012 A6-21 https://doi.org/10.1017/jfm.2025.10177 Published online by Cambridge University Press J.M. Gordillo than the one obtained by means of the self-consistent linearised theory because, when the vortices in the wake are arranged in such a way that they are convected horizontally, the projection on the flight direction of ρ(∇×v)×vin the wake is zero, which implies that the contribution to the total vortex force of the vortices in the wake is also zero in the real, nonlinear case. 4. Mean thrust coefficient for the case of airfoils oscillating periodically when the wake vortices are convected downstream at the free-stream velocity For those common cases in which the airfoil starts accelerating smoothly, namely, C=0 in (2.37), and the wake vortices are convected in a direction parallel to the free-stream velocity as a consequence of the development of nonlinearities, which implies that w w=0, the thrust force is (see (2.37)) TVI(t)=−α(t)(t)−ρc 0 γaw adx,(4.1) which is identical to the one used in Fernandez-Feria (2016), as has been discussed in the derivation of (2.38). In this section, our purpose will be to deduce an equation for the thrust force averaged in time when the airfoil oscillates periodically with a frequency ω, the wake vortices are convected with the incident velocity and the contribution of the starting vortex to thrust is set to zero. For that purpose, we first make use of the fact that, since w a(x,t)=−˙ h− U∞α−˙α(x−xe)and c 0 γa(x,t)dx=Γa(x=c,t)=Γe(t), ∂ ∂xΓaw a=γaw a−˙αΓa,(4.2) then −ρc 0 γaw adx=−ρ−˙ h+U∞α+˙α(c−xe)Γe(t)+˙αc 0 Γadx =ρ˙ h+˙α(t)(c−xe)Γe(t)−ρd dtαc 0 Γadx +α(t)ρU∞Γe(t)+ρd dtc 0 Γadx,(4.3) where we have made use of the fact that Γa(x=0,t)=0 and of the identity −˙αc 0 Γadx=−d dtαc 0 Γadx+αd dtc 0 Γadx.(4.4) The substitution of (4.3)into(4.1) yields that TVI(t)=ρ˙ h+˙α(t)(c−xe)Γe(t)−ρd dtαc 0 Γadx,(4.5) where we have made use of the expression for the lift force (t)in (2.32). Consequently, the mean thrust force can be calculated as TVI(t)=ω 2πt+2π/ω t TVI(t)dt=ρω 2πt+2π/ω t˙ h(t)+˙α(t)(c−xe)Γe(t)dt =ρU∞ ω 2πt+2π/ω t wbs(t)Γe(t)dt,(4.6) 1012 A6-22 https://doi.org/10.1017/jfm.2025.10177 Published online by Cambridge University Press Journal of Fluid Mechanics where we have taken into account that, since the thrust is a periodic function of time, t+2π/ω t d dtαc 0 Γadxdt=0,(4.7) and wbs(t)=˙ h(t)+˙α(t)(c−xe) U∞ (4.8) denotes minus the dimensionless velocity of the trailing edge of the airfoil and, consequently, −w U∞x c=3 4,t=−w 3/4(t) U∞=wbs(t)+αU∞−˙αc 4 U∞ .(4.9) The result in (4.6) expresses that, for those cases in which the airfoil oscillates periodically, the contribution of the starting vortex to thrust is negligible and the vortices in the wake are arranged in such a way that they are convected with a velocity parallel to the free-stream velocity, the mean thrust is proportional to the mean of the product of minus the vertical velocity of the trailing edge and the circulation around the airfoil. From now on, since we are considering the case of airfoils oscillating periodically, any generic time-dependent function s(t)will be expressed as the real part of s(t)=s∗eikτ,(4.10) where the constant s∗is a complex number, k=ωc/(2U∞)is the reduced frequency and τ=2tU∞/crefers to the dimensionless time. Using the results in e.g. Ashley & Landahl (1985) or in section V of the Supplementary Material, the analytical expression for the time-periodic circulation around the airfoil, namely, Γe(t)=πU∞c(A0(τ) +A1(τ)/2)/2, is given by the real part of Γe(t)=πU∞c 2 G∗ ik eikτ,(4.11) where we have made use of the notation in (4.10). In (4.11), see Ashley & Landahl (1985) or section V of the Supplementary Material, G∗=−w 3/4 U∞∗ ×2e−ik ∞ 1(1+χ)e−ikχ √χ2−1dχ=−w 3/4 U∞∗ ×2e−ik K0(ik)+K1(ik),(4.12) where Knin (4.12) indicates the modified Bessel function of the second kind of order n. Now, using the results in (4.9), (4.10) and the fact that d/dt=(2U∞/c)d/dτ −w 3/4 U∞∗ =w∗ bs +a0eiφ1−ik 2,w ∗ bs =2ik h0 c+a0eiφ1−xe c, (4.13) with h(t)=h0R(eikτ),α(t)=a0R(eiφeikτ),h0,a0real numbers indicating the amplitudes of the heaving and the pitching motions and φthe phase shift. The substitution 1012 A6-23 https://doi.org/10.1017/jfm.2025.10177 Published online by Cambridge University Press J.M. Gordillo 10–2 10–1 100 k 101 – 0.2 0 0.2 0.4 0.6 0.8 1.0 Figure 7. The figure shows the functions GΓ(k)(blue line) and FΓ(k)(red line) defined in (4.15), as well as the function D(k)(black line) defined in (4.17). of (4.12)and(4.13)into(4.11) yields Γe(t)=πU∞cR⎛ ⎜ ⎜ ⎝w∗ bs +a0eiφ1−ik 2× e−ik (ik) K0(ik)+K1(ik)eikτ⎞ ⎟ ⎟ ⎠ =πU∞cRw∗ bs +a0eiφ1−ik 2×(GΓ(k)−iF Γ(k))eikτ,(4.14) where e−ik (ik) K0(ik)+K1(ik)=GΓ(k)−iF Γ(k). (4.15) The functions FΓ(k)and GΓ(k)in (4.15) are related with the analogous functions F1(k) and G1(k)defined in Fernandez-Feria (2016) in the following way: F1(k)=−π 2FΓ(k), G1(k)=−π 2GΓ(k). (4.16) For our subsequent purposes, it proves convenient to define here the function D(k)=F(k)−GΓ(k)−kFΓ(k) 2(4.17) where F(k)is the real part of the well-known Theodorsen function C(k),definedas C(k)=K1(ik) K0(ik)+K1(ik)=F(k)+iG(k), (4.18) see, e.g. Ashley & Landahl (1985) or section V of the Supplementary Material. The different functions GΓ(k),FΓ(k)and D(k)defined in (4.15)and(4.17) are plotted in figure 7. Now, making use of the fact that: (i) all time-dependent functions are of the form given in (4.10) and (ii) the real part of a complex number s∗is (s∗+s∗c)/2, with the superscript 1012 A6-24 https://doi.org/10.1017/jfm.2025.10177 Published online by Cambridge University Press Journal of Fluid Mechanics caffecting the complex variable s∗indicating the complex conjugate of s∗, the substitution of (4.8), (4.9)and(4.14)into(4.6) yields TVI =ρπU2 ∞c 4 ×w∗ bsw∗c bs (GΓ(k)+iF Γ(k))+a0w∗ bse−iφ1+ik 2(GΓ(k)+iF Γ(k)) +ρπU2 ∞c 4×w∗ bsw∗c bs (GΓ(k)−iF Γ(k))+a0w∗c bs eiφ1−ik 2(GΓ(k)−iF Γ(k)) =2ρπU2 ∞ck 2GΓ(k)×h0 c2 +cos φa0h0 c3 4−a +ρπU2 ∞ckG Γ(k)×k 2a2 0(1−a)1 2−a+a0h0 csin φ −ρπU2 ∞ckF Γ(k)×a0h0 ccos φ+k 2a0h0 csin φ+a2 0 2(1−a),(4.19) and, hence, the expression of the mean thrust coefficient when the contributions to thrust of both the starting vortex and of the vortices in the wake are set to zero, is CT(k)=TVI 1/2ρU2 ∞c=4πk2GΓ(k) ×h0 c2 +a0h0 c3 4−acos φ+a2 0 4(1−a)1 2−a +2πka 0h0 c ×GΓ(k)sin φ−FΓ(k)cos φ−k 2FΓ(k)sin φ−πkF Γ(k)a2 0(1−a), (4.20) where we have made use of the equation xe=(1+a)c/2 relating the position of the pitching axis xewith a. The equation for the mean thrust coefficient given by (4.20), which does not take into account neither the vortex force term (2.40) nor the contribution of the starting vortex, differs from the mean thrust coefficient deduced by Garrick (1936); see also (B4) in Fernandez-Feria (2016), reproduced here for clarity purposes CTG(k)=4πk2F2(k)+G2(k)h0 c2 +πa2 0F2+G21+k21 2−a2 +πa2 0a−1 2F−1 2k2−a+1 2kG −F+πa0h0 c$4kF2+G2sin φ % +πa0h0 c4k21 2−aF2+G2cos φ−2k2(Gsin φ+Fcos φ) +πa0h0 c$2k(Gcos φ−Fsin φ)+k2cos φ%,(4.21) 1012 A6-25 https://doi.org/10.1017/jfm.2025.10177 Published online by Cambridge University Press J.M. Gordillo PULLIN, D.I. 1978 The large-scale structure of unsteady self-similar rolled-up vortex sheets. J. Fluid Mech. 88 (3), 401–430. RAMESH,K.,GOPALARATHNAM,A.,GRANLUND,K.,OL,M.V.&EDWARDS, J.R. 2014 Discrete-vortex method with novel shedding criterion for unsteady aerofoil flows with intermittent leading-edge vortex shedding. J. 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