## Abstract

Consider uniform, steady potential and incompressible flow past a fixed thin wing inclined at a small angle to the flow. An investigation is conducted into the physical interpretation and consequences of the revision by Chadwick (Chadwick 2005 *Proc. R. Soc. A* **461**, 1–18) of the Lanchester–Prandtl lifting wing theory in Euler flow. In the present paper, the lift is evaluated from the pressure distribution over the top and bottom surfaces together with a contribution across the trailing edge of the wing. It is shown that this contribution across the trailing edge has previously been erroneously omitted in the standard approach but confirms and provides a physical explanation for the discrepancy in the lift calculation found by Chadwick. This results in a reduction of the lift by a half, but this reduction in lift from the additional calculation is not the right answer, and instead arises from a mathematical discrepancy with the physically observed lift. The discrepancy is due to the pressure becoming singular at the trailing edge in the Euler model. The physical explanation is that in real flow the pressure is regularized by the action of viscosity and so is not singular at the trailing edge. So this lift force at the trailing edge is present in the Euler model but not in a real flow. In a real flow, the viscous effects prevent the pressure becoming singular and so there is no lift force, and consequently no large torque, concentrated at the trailing edge. That the lift force at the trailing edge has been ignored in the Lanchester–Prandtl theory in Euler flow has led to fortuitous agreement with the experimental results on real flows. This shows that the Euler model does not properly predict forces for this problem in which there are singularities (vorticity) within the flow field. We propose a revision to the Euler model by allowing a counterbalancing singular viscous velocity term to reside on the trailing vortex sheet, which is derived from the lift oseenlet. This viscous term ensures that the pressure and velocity are not singular in the flow field. The consequences for the flow due to the inclusion of this term for extending triple-deck and similar asymptotic theories to the case for flow past wings rather than aerofoils are discussed, as well as for the (ideal) high Reynolds number limit and for slender body lift.

## 1. Introduction

This paper investigates the shortcomings in the Lanchester–Prandtl theory developed in Euler flow to model lift on a wing. There have been theoretical concerns (Goldstein 1960, pp. 131–134) with the Lanchester–Prandtl model, but these have been muted due to its enormous success at accurately predicting aircraft lift. One concern is the representation of the trailing line vortex, which is the building block of the vortex sheet model. At the centre of the line vortex, the Cauchy stress principle, required in the derivation of the fluid equations, breaks down. This manifests itself in the generation of infinite kinetic energy around its centre. This leads to Goldstein's principal concern with the model (which is also part of free-streamline theory) which is the lack of viscosity to diffuse these singularities within the fluid. Saffman (1992) describes how viscosity can be introduced to remove these singularities, and in the case of the rolled vortex sheet from a wing tip refers to the Batchelor vortex (Batchelor 1964) giving a solution in an Oseen flow field. In the case of a flow field generated by a horseshoe vortex, a solution with a viscous core is also provided by Chadwick (2006). There has also been research on the importance of the viscous component in the time evolution of the vortex ring (Fukumoto & Moffatt 2000), and work on the evolution of the vortex sheet by again recognizing the importance of a viscous flow field (Abid & Verga 2002; Krasny *et al*. 2003). In a physically realistic model the vortex sheet is not completely inviscid, and the effects of viscosity are important. Thwaites (1960, pp. 292–293, figs. VIII.1–VIII.3) considers a thin disc held in a uniform stream; initially, the disc is perpendicular to the flow and a viscous wake region emanates encompassed by vortex layers. As the inclination of the disc is reduced to a small angle, the vortex wake still contains a small but viscous region and is not purely inviscid. An extension to wing theory would give the model described in figure 1, where the contained viscous wake region originates from the boundary layer that detaches from the body and is transported downstream (Wu 1972).

Chadwick (2005) investigates the Lanchester–Prandtl model in Euler flow describing the flow and forces on a wing. In particular, he represents the vortex sheet that emanates from the trailing edge of the wing by a discrete summation of horseshoe vortices, proposed by Lanchester (1907, 1908) and Prandtl (Glauert 1959). The lift contribution for each horseshoe vortex is then determined from its circulation, and the total lift is the summation of all the contributions from each horseshoe vortex. In calculating the lift from the circulation of a horseshoe vortex, standard theory assumes that the flow is locally two dimensional (Batchelor 1967, p. 585, fig. 7.8.4). However, when modelling a three-dimensional flow such that a vortex sheet emanates from the trailing edge, Chadwick (2005) demonstrates that this assumption breaks down and shows that the lift contribution from a single horseshoe vortex is half that of the generally accepted result. Since an isolated horseshoe vortex is considered, it is unclear what the physical interpretation of this result is in terms of the physical flow variables of pressure and velocity past a wing.

In the present paper, we aim to provide a physical interpretation of this result by considering the pressure distribution in the flow field around a thin wing inclined at a small angle to the flow. First, we represent the vortex sheet by an integral distribution of singularities over the wing and vortex sheet. In particular, one can represent the sheet by an integral distribution of infinitesimal horseshoe vortices (Prandtl 1936; Thwaites 1960, p. 301). We determine the lift from the pressure distribution over the top and bottom surfaces of the wing as in the standard approach (Thwaites 1960, p. 301). However, the lift contribution from the pressure distribution across the trailing edge itself is not included in the standard approach. We shall include this additional evaluation across the trailing edge itself and show that it reduces the lift evaluation by exactly a half, and so is consistent with and confirms the results from Chadwick (2005). The implication of this result is that the Euler model is fundamentally flawed if applied to this lifting problem, and Chadwick (2005) provides a revision by considering a lift oseenlet. In the present paper, we investigate the consequences of including this viscous term in the model described by figure 1. We find that now the pressure is not singular and so there is no (unphysical) lift at the trailing edge. The consequences for triple-deck theory and for ideal flow are also discussed.

## 2. Potential flow

### (a) Equations of motion

We start with the time-independent incompressible Navier–Stokes equations (Lamb 1932, p. 577) with the viscosity set to zero, and also the continuity equation for incompressible flow(2.1) and *p*^{†} are the velocity and pressure, respectively, in suffix notation for the Cartesian coordinate system (*x*_{1}, *x*_{2}, *x*_{3}). *ρ* is the fluid density and is assumed to be constant.

Assuming the slip body boundary condition,(2.2)on the body surface *S*_{B}, where *n*_{i} is the outward-pointing normal.

Finally, we assume that away from the vortex wake region, the velocity can be represented by a potential velocity *ϕ*^{†} such that . Then,(2.3)where *p*_{0} is a constant. These equations given in (2.3) are the Bernoulli equation and Laplace equation, respectively. We assume that the pressure tends to zero in the far field, *p*^{†}→0, and the velocity tends to a uniform stream predominantly in the *x*_{1} direction (∂*ϕ*^{†}/∂*x*_{i})→*Uδ*_{i1}+*Vδ*_{i2} as ; the velocities are such that *U*≫*V*, and *δ*_{ij} is defined such that *δ*_{ij}=1 for *i*=*j* and *δ*_{ij}=0 for *i*≠*j*. So *p*_{0}=1/2*ρ*(*U*^{2}+*V*^{2}).

### (b) Wing representation

Consider the flow around a thin wing by a distribution of bound and free vortices over an area *A* within the thin wing. For simplicity, let us restrict our attention to an area *A* that lies on the rectangular area 0≤*x*_{1}≤*X*_{1}, *x*_{2}=0, 0≤*x*_{3}≤*X*_{3} (figure 2).

The perturbation velocity given by *u*_{i}=∂*ϕ*/∂*x*_{i} is(2.4)where is the velocity induced by a distribution of source terms necessary to give the thin wing finite non-zero thickness and *u*_{i} is the velocity associated with lift. The slip (or impermeability) condition is then given as(2.5)However, for a thin wing, the source terms providing outflow are of lower order of magnitude than the cross-flow terms providing lift. The lifting solution is therefore given by a velocity *u*_{i} antisymmetric in the cross-flow direction *x*_{2} such that the boundary condition (Katz & Plotkin 2001, p. 332)(2.6)holds. The complementary boundary condition is satisfied by the lower order source potential. However, that this velocity is of lower order and therefore neglected does not then imply that the wing has zero thickness. One could solve for if so inclined, but the solution does not then affect the solution for *u*_{i} determined by satisfying the condition (2.6). To completely satisfy the impermeability condition would require determining all the lower order terms. However, our aim is to evaluate the lift force, and in §2*c* it shall be shown that the complete solution is not necessary for the lift force calculation; the lift force is linear, in the sense that it is represented by a summation of the independent lifting forces from individual horseshoe vortices in the model. There is no nonlinear contribution from the interaction between one horseshoe vortex on another, or on a source, or on any lower order term. Thus, to evaluate the lift force it is necessary to determine only the leading order term which is obtained by satisfying the slip/impermeability boundary condition to leading order given by (2.6). In the two-dimensional equivalent of thin aerofoil theory, Batchelor (1967, p. 470) determines how accurate the approximation is compared with the exact model for the particular case of Joukowski aerofoils mapped to the circular cylinder. He found that the order was small and related to the thinness parameter, and a similar result can be expected for flow past wings.

The velocity *u*_{i} is given by Thwaites (1960, p. 301, fig. VIII. 8, eqns (20) and (21)), who considers an integral distribution of infinitesimal horseshoe vortices. This representation appears to have first been given by Prandtl (1936). The infinitesimal horseshoe vortex potential *ϕ*^{infinitesimal} is derived in Thwaites (1960, p. 391, eqn (76)) as(2.7)where and *Q* is the infinitesimal horseshoe vortex strength. Rearranging terms by noting that and , gives(2.8)Hence, the integral distribution given in Thwaites can be represented in the form (Chadwick 2005)(2.9)where , *x*_{11}=*x*_{1}−*y*_{1}. Thwaites describes the function *l*_{13} as the load function (Thwaites 1960, p. 301), and it is related to the circulation strength of the vortex *Γ* such that *l*_{13}(*y*_{1}, *y*_{3})=∂*l*_{3}(*y*_{1}, *y*_{3})/∂*y*_{1} and *l*_{3}(*y*_{1}, *y*_{3})=*ρUΓ*(*y*_{1}, *y*_{3}) (Chadwick 2005). Hence, *l*_{13} is a singularity strength distribution over the area *A*. The vortex strength in the *x*_{1} direction is given by *γ*_{1}=−*l*_{33}/*ρU* and in the *x*_{3} direction is given by *γ*_{3}=*l*_{13}/*ρU* where *l*_{33}=∂*l*_{3}/∂*y*_{3}. In appendix A, this velocity representation (2.9) is shown to satisfy the standard dynamic and kinematic boundary conditions, as well as the Kutta condition for wakes. The shed vortex wake is also a singular sheet that we represent by an equivalent singularity distribution *l*_{13}(*y*_{1}, *y*_{3}) (figure 3).

Let us also define the quantity *L* by(2.10)(Hence, for a single horseshoe vortex of strength *Γ*, *L*=*ρUΓs*, where *s* is the span of the horseshoe vortex.)

### (c) Force integral representation

The force is represented by an integral distribution of the surface pressure. However, since the pressure is undefined on the vortex sheet and this sheet intersects the wing surface at the trailing edge, the integral cannot be determined at the trailing edge. Therefore, a contour *C*_{δ} is considered (which is defined later) which lies on the wing near to and enclosing the trailing edge, a distance *δ* away from the vortex sheet. The force is then given as an integral over the body surface *S*_{B} except for the surface on the body *S*_{δ} (which is the surface enclosed by the contour *C*_{δ}), in the limit as *δ*→0. Finally, the force integral is evaluated over a symmetric surface *S*_{symm} in order to simplify the calculation. Thus, before proceeding with the force representation, the surface *S*_{symm} is defined and from this the contour *C*_{δ} is defined.

Consider the surface *S*_{symm} defined as the surface a distance *ϵ* away from the rectangular area *A*. (*A* is defined as lying in the region *ϵ*≤*x*_{1}≤*X*_{1}−*ϵ*, *x*_{2}=0, *ϵ*≤*x*_{3}≤*X*_{3}−*ϵ*.) Then *S*_{symm} consists of the top and bottom surfaces *S*_{tb}, the side surfaces *S*_{side}, the trailing edge surface *S*_{te} and the front edge surface *S*_{fe}.

The top and bottom surfaces *S*_{tb} consist of the top surface *S*_{t} parameterized by (*p*, *ϵ*, *q*) where *ϵ*≤*p*≤*X*_{1}−*ϵ*, *ϵ*≤*q*≤*X*_{3}−*ϵ* and the bottom surface *S*_{b} parameterized by (*p*, −*ϵ*, *q*) where *ϵ*≤*p*≤*X*_{1}−*ϵ*, *ϵ*≤*q*≤*X*_{3}−*ϵ*.

The side surfaces *S*_{side} consist of the left side surface parameterized by (*p*, *ϵ* cos *q*, *ϵ* sin *q*) where *ϵ*≤*p*≤*X*_{1}−*ϵ*, −*π*≤*q*≤0 and the right side surface parameterized by (*p*, *ϵ* cos q, *X*_{3}−*ϵ*+*ϵ* sin *q*) where *ϵ*≤*p*≤*X*_{1}−*ϵ*, 0≤*q*≤*π*.

The trailing edge surface *S*_{te} consists of the half cylindrical surface parameterized by (*X*_{1}−*ϵ*+*ϵ* cos *p*, *ϵ* sin *p*, *q*) where −*π*/2≤*p*≤*π*/2, *ϵ*≤*q*≤*X*_{3}−*ϵ*, the quarter sphere parameterized by (*X*_{1}−*ϵ*+*ϵ* sin *p* sin *q*, *ϵ* cos *p*, *ϵ* sin *p* cos *q*) where 0≤*p*≤*π*, *π*/2≤*q*≤*π* and the quarter sphere parameterized by (*X*_{1}−*ϵ*+*ϵ* sin *p* sin *q*, *ϵ* cos *p*, *X*_{3}−*ϵ*+*ϵ* sin *p* cos *q*) where 0≤*p*≤*π*, 0≤*q*≤*π*/2.

The front edge surface *S*_{fe} consists of the half cylindrical surface parameterized by (*ϵ* cos *p*, *ϵ* sin *p*, *q*) where *π*/2≤*p*≤3*π*/2, *ϵ*≤*q*≤*X*_{3}−*ϵ*, the quarter sphere parameterized by (*ϵ* sin *p* sin *q*, *ϵ* cos *p*, *ϵ* sin *p* cos *q*) where 0≤*p*≤*π*, *π*≤*q*≤3*π*/2 and the quarter sphere parameterized by (*ϵ* sin *p* sin *q*, *ϵ* cos *p*, *X*_{3}−*ϵ*+*ϵ* sin *p* cos *q*) where 0≤*p*≤*π*, 3*π*/2≤*q*≤2*π* (figure 4).

Then, let the contour *C*_{ϵ} be where the plane *x*_{1}=*X*_{1}−*ϵ* intersects the surface *S*_{symm} (figures 5 and 6). Similarly, let the contour *C*_{δ} be where the plane intersects the surface *S*_{symm}, where *δ*≫*ϵ* (see figures 5 and 6). Hence, to first order, the contour *C*_{δ} is a distance *δ* away from the vortex sheet.

Then the force is represented by a surface integral of the normal pressure over the body surface (Lamb 1932) such that(2.11)where *f*_{i} is the force on the body due to the fluid; d*s* is an element of surface and *S*_{B}−*S*_{δ} is part of the body surface not lying on *S*_{δ}. (Standard calculations essentially calculate the force integral over the body surface *S*_{B}−*S*_{ϵ}, but the contribution to the force integral across the trailing edge is not then considered.)

Since on the body surface, then(2.12)where *S* is a general surface enclosing the body except for the constraint that *S*_{δ} must lie on *S*. The following result has been used: that the integrand of the volume integral enclosing the two surfaces *S*−*S*_{δ} and *S*_{B}−*S*_{δ} is identically zero from Green's integral theorem. The integral (2.12) can be further simplified by substituting in for the fluid velocity given by (2.4). It is then noted that all lower order terms including lower order source terms do not contribute to the force integral: consider the integral surface given in (2.12) such that the surface is spherical with radius *R* and *R*→∞. Note that the source velocity is of order 1/(*R*^{2}), the integral surface is of order *R*^{2} and the integrand includes a normal direction component. This means that in the limit, the force contribution due to sources or any lower order terms is zero. The second component of force, *f*_{2}, is therefore given by(2.13)Consider the integral over the surface such that *S*=*S*_{symm} which is symmetric about *x*_{2}=0. All the terms in the integrand that are antisymmetric in *x*_{2} give zero integral contribution. Also, the term in the integrand −*ρUVn*_{1} gives zero contribution over the closed surface. Thus, from the form of *u*_{i} given in (2.9), the equation for *f*_{2} reduces to(2.14)

This gives the well-known aerodynamics result that the lift integral is linear, in the sense that the lift can be determined by summing up contributions from individual horseshoe vortices. There is no nonlinear lifting contribution from the interaction between one horseshoe vortex on another, or one horseshoe vortex on a source, or any other lower order term.

## 3. Lift force over top and bottom surfaces of the wing

The force integral (2.14) is over a surface within the fluid *S*_{symm} and this is not the same as the wing surface, as shown in figure 7. Thus, the circular arc around the trailing edge lies within the fluid and not on the body surface. This means that lower order source terms are not located on this arc. The thin wing theory assumes a representation by an integral distribution of infinitesimal horseshoe vortex singularities over the area *A* defined in §2*c*. Thus, the surface *S*_{symm} can be brought arbitrarily close to within a distance *ϵ*>0 of the area *A*.

A near-field approximation *ϕ*^{nf} is first calculated for *ϕ*. The inner integral of (2.9) is given in slender body theory (Chadwick 2002) and the near-field approximation discussed extensively in Tuck (1992). (Note that the standard form of the slender body integral, , can be rewritten as .) Applying slender body analysis (Thwaites 1960, IX.11), a near-field approximation *ϕ*^{nf} for *ϕ* such that *ϕ*=*ϕ*^{nf}(1+*O*(*ϵ*^{2} ln *ϵ*)) is given as(3.1)where . (We note here that the Kutta condition states that *l*_{13}=0 and not *l*_{3}=0 at the trailing edge. Indeed, the quantity *l*_{3} is related to the circulation *Γ* and so is expected to be non-zero at the trailing edge for a non-zero total lift on the wing (Thwaites 1960, pp. 300–302, eqns (20)–(24)).) The force integral in the two-direction over the top and bottom surfaces, which we shall denote by , is over the top surface *S*_{t} and bottom surface *S*_{b}. From (3.1), it is clear from symmetry arguments that the contribution from the bottom surface is the same as from the top surface.

Therefore, substituting (3.1) into (2.14) gives(3.2)The force over the top and bottom surfaces of the wing is therefore *L*. However, there is an additional contribution to the lift force across the trailing edge of the wing.

## 4. Lift force across the trailing edge of the wing

It is noted that *u*_{1} is not singular at the trailing edge, and so there is no contribution to the integral from the term *u*_{1}*n*_{1} across the trailing edge in the limit as *δ*→0. Thus, from (2.14), the force in the two-direction across the trailing edge of the wing is(4.1)where the inner limit is taken first. Also, *A*_{te} is the projection of the surface *S*_{te} onto the plane of constant *x*_{1}, similarly with *A*_{δ}, and d*A* is an element of area in the *x*_{1} plane. Hence, we have assumed that *l*_{3} is a slowly varying function in *x*_{1}. Applying the divergence theorem, the area integral is changed to a contour integral such that(4.2)where *C*_{ϵ} and *C*_{δ} are given in figures 5 and 6, and ∮ is an integration around a closed contour in the anticlockwise sense.

Substituting for *ϕ*^{nf} from (3.1) and changing the order of integration give(4.3)where the contour *C*_{δ} can now be replaced by the circular contour of radius *δ* at *y*_{3}=*x*_{3} (figure 8).

### (a) Contribution from contour

The contribution to (4.3) from the contour , denoted by , is(4.4)where , *x*_{2}=*r*_{3} cos *θ*_{3} and *x*_{3}−*y*_{3}=*r*_{3} sin *θ*_{3}.

### (b) Contribution from contour C_{ϵ}

Let the contribution to the force integral over the contour C_{ϵ} be denoted by . Owing to symmetry, the contribution from the upper half contour *x*_{2}≥0 is equal to the contribution from the lower half contour *x*_{2}≤0. Also, the contribution from the semicircular contours is of lower order for small *ϵ*. Therefore,(4.5)

Therefore, at the trailing edge there is a jump in the lift force of value .

## 5. Total lift force on the wing

Hence, the total lift force on the wing is, , half of that expected by considering the pressure distribution over the top and bottom surfaces of the wing only.

The force can also be calculated from a far-field integral enclosing the body: consider a closed surface enclosing a volume of fluid with the integrand given in (2.14). From Green's integral theorem, this is identically zero, and so the lift force can be represented by an integral over a surface enclosing the shed vortex wake and over a far-field surface enclosing the body (figure 9). It shall be shown next that the first of these integrals is zero, and then the force shall be calculated from a far-field integral enclosing the body.

### (a) The force integral over a surface enclosing the shed vortex wake

First, we determine the form of the potential in the near field of the shed vortex wake, . The potential is given from (2.9) as(5.1)Applying the Taylor series expansion gives(5.2)Substituting this expression for into (2.9) gives(5.3)which gives the same result as (3.1) with *x*_{1}=*X*_{1}. From Thwaites (1960, IX.11), it is seen that *ϕ*=*ϕ*^{nf}(1+*O*(*δ*^{2} ln *δ*)) for a point in the fluid a distance *δ* away. In the limit, this lower order term does not contribute to the force calculation and so is ignored. Representing the flow near the wake in the form(5.4)gives, for *y*_{1}>*X*_{1},(5.5)This is consistent with the standard aerodynamic horseshoe vortex description of the shed vortex wake that vortex lines are parallel and have constant strength.

The force integral over the trailing vortex wake can now be determined. To first order, from (2.14), substituting in for *ϕ* the value from (5.4) gives(5.6)However, using the result (5.5), along the wake and so .

### (b) The force integral over a spherical surface radius *R*→∞ enclosing the body

In the far field, from (2.9), applying the Taylor series expansion about a point (0, 0, 0) centred on the body gives(5.7)where , and *n* and *m* are positive integers. The expansion is convergent except on the vortex wake *x*_{2}=0, 0≤*x*_{3}≤*X*_{3}, and also the coefficient *a*_{00}=*L*/(4*πρU*). Consider the surface *S*_{R} such that , and the surface which lies on the surface *S*_{R} such that where 0≤*s*≤*X*_{3}. Therefore, the points on the surface are a distance at most *δ* away from the trailing vortex sheet. Then from (2.14), the force over the far-field surface enclosing the body is given by(5.8)On substituting the expansion for *ϕ* from (5.7) into (5.8), only the first term in the expansion, *a*_{00}=*L*/(4*πρU*), is non-zero in the limit as *R*→∞, and gives(5.9)Hence, the total lift force on the wing is .

## 6. An alternative model derived from a distribution of lift oseenlets

Consider an intermediate region outside the near field close to the body, such that the Oseen flow is valid (figure 10). This assumes that the fluid velocity is to first order the uniform stream velocity *U*, and linearizing the Navier–Stokes equations in this way yields the Oseen equations (Chadwick 1998). The representation of the vortex line and infinitesimal horseshoe vortex in Oseen flow has been given by Chadwick (2006). From this, by comparison with the representation of a thin wing given in §2*b*, the intermediate region description by a distribution of infinitesimal horseshoe vortices over an area *A* within the wing becomes(6.1)where is the lift oseenlet such that (Oseen 1927)(6.2)where(6.3)*k*=*ρU*/2*μ*. In the high Reynolds number ideal flow limit, the viscous terms then reside on the vortex sheet only, and off the vortex sheet the velocity is given by *u*^{Oseen}=∇*ϕ*^{Oseen} such that(6.4)which is the same as the potential expression (2.9). Hence, the potential in Oseen flow *ϕ*^{Oseen} is seen to be a continuation for the inviscid flow potential *ϕ* into the near field, in the ideal flow high Reynolds number limit. Therefore, off the vortex sheet the two velocity descriptions, inviscid flow and Oseen flow in the high Reynolds number limit, are identical. Thus, the slip boundary condition together with the Kutta condition will provide a unique solution for the potential *ϕ*^{Oseen}=*ϕ*. However, the crucial difference between the two velocity descriptions is the presence of the viscous term that exists even in the high Reynolds number limit, but collapses onto the vortex sheet. This term is crucial as it regularizes the pressure and so ensures that there is no unphysical lift force contribution at the trailing edge. We note that in the far field, the singular sheet rolls up and is directed along a curved trajectory rather than along the *x*_{1}-axis, and so a closer approximation could consider curvilinear coordinates such that the oseenlet singularity in the potential lies along this curve. However, in the present paper we are concerned with the lift force calculation, for which the assumption that the vortex sheet lies along a flat plane is made. If, instead, one is interested in the velocity representation in the far-field wake, then wake roll-up would have to be taken into account.

## 7. Discussion

Euler codes that evaluate the lift force from the pressure distribution do so by evaluating the forces over the top and bottom surfaces only. This calculation fortuitously omits the contribution across the trailing edge itself. This contribution can be evaluated using the integral representation given by Thwaites, who in turn refers to Prandtl. This representation of the vortex sheet is based upon an integral distribution of infinitesimal horseshoe vortices, which for the particular case of high-aspect ratio wings, as Chadwick (2005) shows, reduces to Prandtl's lifting line. In appendix A, it is further demonstrated that this representation satisfies the dynamic and kinematic wake boundary conditions, and in doing so satisfies the Kutta condition at the trailing edge. Although not used in the development of standard numerical Euler codes, this formulation is amenable to analysis and so is used here. The result is an additional contribution from across the trailing edge, which reduces the lift by a half. By properly evaluating the lift in this way, the lift evaluation does not now agree well with experiment, being a half less than expected. Given this large discrepancy with experiment, it is clear that the Lanchester–Prandtl representation does not model the lifting force in a realistic way. This lack of realism has been identified with the removal of viscous effects in the Euler model, leading to unrealistic singularities in the pressure and velocity and as a consequence an unrealistic lift contribution at the trailing edge. A revision is proposed to the Lanchester–Prandtl model in Euler flow by including the viscous contribution through the use of oseenlets. By using a viscous model, it is evident that there is no contribution across the trailing edge itself, which is consistent with experimental predictions for real flows and which is why this omission in the Euler codes has been so fortuitous. Even in the high Reynolds ideal flow limit, the inclusion of viscous effects still resides as delta functions over the wake vortex sheet. This demonstrates that the high Reynolds ideal flow limit of the Navier–Stokes equations is not Euler flow for most lifting practical problems, which has consequences for free-streamline theory. This outcome was predicted by Goldstein (1960, pp. 131–134). Goldstein states that retaining the viscous term in the limiting process rather than setting it to zero in the Euler equations will lead to a different ideal flow (free streamline) limit, and the correct limiting procedure should include the retention of the viscous term. For ideal flow (high Reynolds number limit) the width of the viscous wake vanishes, and so outside of the trailing vortex sheet the velocity is given by the (inviscid) potential velocity only. Since the viscous singularity term is represented by delta functions distributed over the vortex sheet surface, the boundary conditions are satisfied by the (inviscid) potential only, and so are satisfied by applying the Euler slip (impermeability) boundary condition together with the Kutta condition. Thus, the solution for velocity and pressure fields away from the trailing vortex sheet is identical to that given by the Euler model. The crucial difference with the standard Euler solution is the presence of the singular viscous term that has collapsed onto the vortex sheet, and is essential for ensuring a correct evaluation of the lift in agreement with experiment.

This difference will particularly have effects on triple-deck theory for wings, and also slender body lifting problems, as discussed below.

For slender bodies with circular cross-section, the additional viscous lifting force has been determined empirically (Allen & Perkins 1951). Instead, by considering lift oseenlets (Chadwick 2002) it is possible to include the viscous contribution within the theoretical model itself producing a lifting force in agreement with experiment (Fishwick 2006).

In triple-deck theory (Stewartson 1969; Messiter 1970), the outer flow is matched to the boundary layer through intermediate ‘decks’. In this theory, the outer laminar wake region is represented by the detached boundary layer, (Katz & Plotkin 2001, p. 473) and is identical to the form predicted from Oseen flow (Chadwick 2006). For lifting flow past a wing, the dominant outer viscous term of the lift oseenlet must originate from a near-field term from within the boundary layer itself. Extending triple-deck theory for flow past wings rather than aerofoils requires this term to be considered, and future work will investigate the form of the near field that generates this term, and so develop a triple-deck theory for this problem.

## Footnotes

- Received November 28, 2006.
- Accepted April 2, 2007.

- © 2007 The Royal Society