## Abstract

The horseshoe vortex is given in Oseen flow as a constant spanwise distribution of lift Oseenlets. From this, the vortex line is represented in steady, incompressible Oseen flow. The velocity near to the vortex line is determined, as well as near to and far from the far field wake. The velocity field in the transverse plane near to the vortex line is shown to approximate to the two-dimensional Lamb–Oseen vortex, and the velocity field in the streamwise direction is generated by the bound vortex line of the horseshoe vortex giving a streamwise decay much faster than that of the Batchelor vortex. The far field wake description is shown to be consistent with laminar wake theory.

## 1. Introduction

Consider the steady, trailing vortex line and wake that emanate from a body such as a wing in uniform flow at large Reynolds number. In inviscid flow, Lanchester (1907, 1908) and Prandtl (Prandtl 1936; Glauert 1959) develop a model for flow past a wing based upon the horseshoe vortex. However, in the downstream wake Goldstein (1960, p. 131–134) argues that this model is inappropriate because vorticity, without the action of fluid viscosity, does not diffuse. Instead of taking the double limiting operation *μ*→0, and then *t*→∞ on the Navier–Stokes equations in that order, Goldstein argues that the order *t*→∞ and then *μ*→0 is more appropriate. (Here, *t* is time and *μ* is the viscosity coefficient.) Batchelor (1956) gives a similar argument, and states that the effect of viscosity, although small, determines the appropriate leading order solution. To obtain the far field representation of the trailing vortex line (Batchelor 1964), Batchelor retains the viscous component but linearizes the velocity to the uniform stream, yielding the Oseen equations. He then determines the asymptotic form near to the vortex line of the velocity components in the transverse plane to the uniform stream. However, he demonstrates that this solution gives a pressure imbalance that can only be resolved by the inclusion of a streamwise velocity component. Batchelor proposes this velocity component is symmetric about the axis of the vortex line, and then shows it has a streamwise decay of order log *x*_{1}/*x*_{1}, where *x*_{1} is the streamwise Cartesian coordinate. We propose an alternative possible solution by considering a streamwise velocity component with different symmetry properties but yet still rectifying the pressure imbalance flagged by Batchelor. The decay of this streamwise velocity is only , and arises from the description of the horseshoe vortex in Oseen flow.

In the present paper, the Oseen representation is considered for steady, incompressible flow to model the trailing vortex line and wake. First, the horseshoe vortex representation in Oseen flow is given by a constant distribution of spanwise lift Oseenlets. This is inferred from the result that the potential part of the lift Oseenlet is an infinitesimal inviscid horseshoe vortex (Chadwick 2005). By using Stokes's theorem, the representation of a vortex line in Oseen flow is then obtained. This representation appears to be novel and, although much has been written on Oseen flow especially at low Reynolds number, the application and interpretation of Oseen flow within the context of the Lanchester–Prandtl model for aircraft (based on the horseshoe vortex) appears to be new. (General reviews of the different models currently used in vortex dynamics are given in Fukumoto & Moffatt (2000), Jacquin (2002) and Delbende *et al*. (2004).) The vortex line in steady, incompressible Oseen flow is then shown to approximate to the Lamb–Oseen vortex representation near the vortex line. The streamwise velocity component is generated by the bound vortex line over the span of the horseshoe vortex. This velocity component does not give the same streamwise decay as that given by Batchelor. Even so, this velocity field still satisfies the Oseen pressure equation since it has been generated by a spanwise distribution of lift Oseenlets. Furthermore, since streamlines separate from the generating body (for example, wing tip) at the boundary of the vortex core, there must also be a potential outflow which can only be produced by a streamwise drag Oseenlet in the Oseen flow formulation. This gives an additional streamwise velocity component which decays as 1/*x*_{1} and so is the leading order streamwise velocity term. Finally, the far field drag and lift Oseenlets are considered near to and far from the far-field wake, and the resulting flow is shown to reduce to that given by laminar wake theory (Landau & Liftshitz 1959, pp. 74–75).

## 2. Statement of the problem

The time independent Navier–Stokes equations describing an incompressible fluid are given by (Lamb 1932, p. 577)(2.1)in the Cartesian coordinate system (*x*_{1}, *x*_{2}, *x*_{3},), with *i* and *j* integers such that 1≤*i*, *j*≥3; a repeated suffix within a particular term implies a summation over the suffix values, for example . and *p*^{†} are the Navier–Stokes velocity and pressure, respectively. *ρ* and *μ* are the fluid density and the dynamical coefficient of viscosity, respectively, and are both assumed to be constant. denotes the gradient operator and the Laplacian operator.

Assuming uniform flow past a streamlined body, we expect that the disturbance to the uniform stream is small. Applying the small disturbance approximation (Katz & Plotkin 2001, p. 77) but on the Navier–Stokes equations, we linearize the fluid velocity to a uniform stream *U* by assuming that (Chadwick 2002)(2.2)where ‘*O*’ means ‘of the order of’, . if *i*=*j*, and zero otherwise. Also, we have positioned the uniform stream along the *x*_{1} axis. *u*_{i} and *p* are the Oseen velocity and pressure, respectively. This yields the Oseen equations (Oseen 1927, pp. 30–38)(2.3)(2.4)We shall consider these equations rather than the potential flow equations; For manoeuvring problems the Reynolds number is generally large and so the coefficient in front of the viscous term of equation (2.3) in dimensionless variables is small. Assuming the viscous term is negligible yields the potential flow equations. However, we shall find that the viscous term as well as the potential flow term is singular within the shed vortex wake behind the body. This means that the viscous term cannot be ignored even though the coefficient in front of it may be very small.

The following three fundamental solutions (*u*_{i}^{(m)}, *p*_{i}^{(m)}), 1≤*m*≤3, to the Oseen equations each yield a unit force in the *m*-direction of acting at the origin (Oseen 1927, p. 34). They are represented by the potentials *φ* and *Χ* such that(2.5)where(2.6)andConsequently,The velocity field of the wake behind the streamlined body is then defined as(2.7)such that the induced force is(2.8)where there is a summation over the integer *j*. Hence, we assume for a repeated integer within a term of the type *a*_{i}*b*^{(i)} then this implies that . *V*_{B} is the volume of the body and d*V*_{y} is an element of this volume. *F*_{j} is the total force on the body due to the action of the fluid flow around the body. This field is matched to a near field, since on the body surface the Oseen approximation breaks down. For a thin wing, this representation can be simplified to (Chadwick 2005)(2.9)where *A*_{B} is the area over which the wing resides. For a wing with high aspect ratio, this can be further simplified to (Chadwick 2005)(2.10)where the wing span is oriented in the *x*_{3} direction. In particular, the lift on the wing is(2.11)Consider a constant spanwise distribution of lift Oseenlets, and let us define this as a horseshoe vortex. Then the velocity induced by the horseshoe vortex is given by(2.12)In §3, we shall reformulate this to give(2.13)where , , , , , ; is the unit vector in the stream direction, and so for a trailing vortex we expect . The integral path *C*_{h} is along the horseshoe contour made of the three straight lines: ; and . Hence, over a general contour *C* we define the velocity field of the vortex line as(2.14)If we let *k*→∞, then the standard definitions of the horseshoe vortex and vortex line in potential flow are recovered. In §3*d*, we then determine the flow close to the shed vortex line.

Finally, the far field wake simplifies to(2.15)From this, in §4 we determine the flow in the far field wake close to the wake line and far from the wake line.

## 3. The horseshoe vortex

From equation (2.12), we define a horseshoe vortex as a line of constant strength lift Oseenlets over a span *s* such that *L* is the total lift generated by the distribution. The lift Oseenlet is given in terms of a potential velocity term *ϕ* and a wake potential term *Χ*. We investigate the velocity induced by each term in turn. Hence, we consider(3.1)

### (a) Contribution from φ

We first investigate the velocity induced by the potential velocity term *φ*, which is given by(3.2)where and , and where the identity(3.3)has been used; . This identity is used and proved in Chadwick (2005).

Since the Laplacian of 1/*R* is zero for *R*>0 , then we can rewrite equation (3.2) as(3.4)where for , (2, 3, 1) or (3, 1, 2), for (*i*, *j*, *q*)=(1, 3, 2), (2, 1, 3) or (3, 2, 1), and otherwise. As *l*≠2, we can apply Stokes's theorem to get an integral over the horseshoe vortex contour *C*_{k} defined along the integral path consisting of the three straight lines: ; and (see figure 1).

Applying Stokes's theorem and letting we get(3.5)where the tangent vector in the anticlockwise sense, as represented in figure 1, is given by . This is the result for the potential part of the horseshoe vortex given by equation (2.13).

### (b) Contribution from Χ

Now consider the contribution from the wake potential *Χ*. This is given by(3.6)where we have let . However, *Χ* satisfies , and so(3.7)Applying Stokes's theorem and letting we get(3.8)

### (c) The vortex line

Combining equations (3.5) and (3.8) gives(3.9)which is the result equation (2.13).

Consider the arm of the trailing vortex line positioned from (0, 0, 0) to (∞, 0, 0). From equation (2.14) this is given by(3.10)

### (d) The velocity near to the vortex line

We can use this result to find the velocity near a vortex line. For small *R*−*x*_{1}, *x*_{1}>0, we have *R*−*x*_{1}∼*r*^{2}/2*x*_{1}, where , and so the velocity potential part of the vortex line is such that(3.11)where is the unit vector direction of the *θ* cylindrical polar coordinate such that , . Hence the potential associated with this velocity is , near to the vortex line, which is the inviscid result. Similarly, the velocity near the vortex line is given by(3.12)which in the plane of constant *x*_{1} gives the core radius , and yields the Lamb–Oseen vortex (Delbende *et al*. 2004)(3.13)Batchelor (1964) demonstrates that the solution given by equation (3.12) yields a pressure deficit. However, from the horseshoe vortex description equation (3.9), this deficit is implicitly accounted for by the inclusion of the axial velocity term generated by the bound vortex line(3.14)which then means that the Bernoulli equation is satisfied to first order. To first order this is which is also the Oseen pressure equation. Seeking an asymptotic form of equation (3.14), gives(3.15)as *s*→∞, where the function erf is the error function (Abramowitz & Stegun 1965). This solution (3.15) is antisymmetric in *x*_{2}. This is to be expected for a solution related to the generation of lift in the *x*_{2} direction. This is also to be expected as the Oseen solution is a linearization of the velocity, and so can be represented by a distribution of Oseenlet solutions generated by the potentials *φ*, *Χ* and *Χ**. Hence, given the symmetry of the velocity in the transverse plane, it is possible to infer the symmetry in the streamwise direction. This suggests that, for the streamwise velocity, any functions of *x*_{2} and *x*_{3} are odd. In contrast, Batchelor seeks a term symmetric in *x*_{2} and *x*_{3} to satisfy the pressure imbalance given by the Bernouilli equation. This yields an entirely different solution, which to leading order is(3.16)for some constant *A*. If Batchelor had sought an antisymmetric solution in *x*_{2} instead, then the solution (3.15) rather than (3.16) would have been obtained. We suggest that for flow past a body that can be represented by the horseshoe vortex model, then equation (3.14) is the appropriate streamwise component to balance pressure and equation (3.15) is the appropriate near vortex asymptotic approximation.

Furthermore, as streamlines separate from the generating body (for example, wing tip) at the boundary of the vortex core, there must also be a potential outflow which can only be produced by a streamwise drag Oseenlet in the Oseen flow formulation. This gives an additional streamwise velocity component, which reduces to(3.17)near to the vortex line, where *D* is the drag. Assuming *x*_{1} is again to first order constant this reduces to the Batchelor vortex representation in the plane of constant *x*_{1}, or *q*-vortex (Delbende *et al*. 2004) such that(3.18)for some constant Δ*U*. Taking a further approximation to equation (3.12), gives(3.19)Hence, in the eye of the vortex line, the fluid velocity is finite and is the uniform stream velocity to first order.

## 4. The far field wake

Consider the force , then the far field velocity is given by(4.1)

### (a) Flow near to wake line

Close to the wake line,(4.2)and so(4.3)which is the far field laminar wake solution of Landau & Liftshitz (1959, pp. 74–75) near to the wake line. This solution is also the three-dimensional extension of the two-dimensional result from boundary layer theory for flow past a flat plate (Katz & Plotkin 2001, p. 472; Schlichting 1979, p. 177). Hence, in the far field wake, boundary layer flow and Oseen flow give the same solution. Furthermore, we see a relation for the near field flows of the vortex line and the lift Oseenlet: The lift Oseenlet flow is the same as the flow obtained in the limit as the span of the horseshoe vortex tends to zero: In other words, and in the near field. This relation between the lift Oseenlet and horseshoe vortex for the potential velocity is given in Chadwick (2005).

### (b) Flow far from wake line

Far from the wake line, the vortex wake potential *Χ* is negligible. So *Χ*^{(2)}∼0, *Χ**∼0 and(4.4)In spherical polars , , , then(4.5)Similarly, *Χ*^{(1)}∼0, and(4.6)which is the far field laminar wake solution of Landau and Lifshitz (Landau & Liftshitz 1959, p. 76) away from the wake line.

## 5. Discussion

A description of the horseshoe vortex in Oseen flow is given by a spanwise distribution of lift Oseenlets. Applying Stokes's theorem, the far field flow near to the trailing vortex line is then obtained such that the flow in the transverse plane to the stream direction reduces to the two-dimensional Lamb–Oseen vortex, and the streamwise vorticity is generated by the bound vortex line of the horseshoe vortex and decays as . This is in contrast to Batchelor's streamwise solution which decays as log *x*_{1}/*x*_{2}. Also, it is shown that the solution presented here satisfies symmetries expected from the Oseen velocity potentials and a solution giving lift in the *x*_{2} direction.

In addition to the vortex solution, it is argued that a streamwise drag Oseenlet centred on the vortex line is also possible. This gives an additional streamwise velocity component which decays as 1/*x*_{1} and so is the leading order streamwise velocity term. Although this is a different decay rate, the form of the solution of the Batchelor vortex in the plane of constant *x*_{1}, or *q*-vortex, used in stability analysis (Jacquin 2002; Olendraru & Sellier 2002; Delbende *et al*. 2004), remains unaltered.

From the description of the horseshoe vortex in Oseen flow, a description of the vortex line over a general contour is given.

Finally, a description of the far field wake by the drag and lift Oseenlet is presented near to and far from the wake. This description is consistent with that presented for laminar flow theory (Landau & Liftshitz 1959) and far field boundary layer theory (Schlichting 1979).

Furture work is to adopt this approach for the time evolution of the vortex line, ring and wake by using the time dependent Oseenlets. In particular this leads to an investigation into whether it is possible to obtain complete solutions, up to the Oseen approximation, without having to apply involved matching procedures (Fukumoto & Moffatt 2000).

## Footnotes

- Received June 27, 2005.
- Accepted October 12, 2005.

- © 2005 The Royal Society