Flow Induced by a Right Vortex Cylinder

Abstract

The velocity field induced by a right vortex cylinder is derived in details in this chapter. The results of this vortex model provides the basis of many of the analyses presented in this book. The vector potential and velocity field are expressed using the Biot–Savart law. The Biot–Savart law in terms of solid angle is also used. Results for finite, infinite and semi-infinite cylinders are provided. The case of longitudinal and tangential vorticity along the cylinder surface are considered. Cylinders of circular and arbitrary cross sections are investigated. The flow in the full domain or at key locations is presented. Different illustrations of the flow are provided. The derivations steps are provided in details in this chapter. Cylinders of tangential vorticity with arbitrary cross sections are considered in a first section. Different results are derived. It is shown in particular that the velocity induced by an infinite vortex cylinder is constant and equal to the vortex intensity inside the cylinder and equal to zero outside. It is shown that the velocity induced by a finite cylinder of tangential vorticity is linked to the velocity induced by source surfaces using the Neumann-to-Dirichlet map. The vector potential and velocity field induced by a vortex cylinder of circular cross section and tangential vorticity is derived next. Similar developments are used for the cylinder of longitudinal vorticity. A Matlab source code is provided to compute the velocity field induced by a semi-infinite cylinder of tangential vorticity. This code can directly be used to study the induction zone in front of a rotor. Most of the results presented in this section were published in the article titled “Cylindrical vortex wake model: right cylinder” (Branland and Gaunaa, Wind Energy, 524(1), 1–15, 2014, [2]). Results from this chapter are used: in Chaps. 17 and 18 to model a wind turbine (or rotor) in uniform inflow, in Chap. 24 to study the induction zone in front of a wind turbine, in  Chap. 26 to model a wind turbine/rotor under unsteady situations.

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1 Right Cylinder of Tangential Vorticity with Arbitrary Cross Section

A right cylinder of tangential vorticity $\gamma _t$ of arbitrary cross-section extending along the z-axis from $z_1$ and $z_2$ is considered. The cylinder is illustrated in Fig. 36.1. The cylinder volume is delimited by the surfaces $S_1$ at $z=z_1$, $S_2$ at $z=z_2$ and $S_c$ in between. $S_c$ is the only support of vorticity.

Fig. 36.1

Coordinate system used for the right vortex cylinder of tangential vorticity

1.1 Finite Cylinder - General Velocity Field

The velocity field induced by the finite cylinder of tangential vorticity with arbitrary cross-section is obtained by application of the Biot–Savart law (see Eq. 2.224):

$$\begin{aligned} \underline{u}(\underline{x})=\frac{-1}{4\pi } \iint _{S_C} \frac{\left( \underline{x}-\underline{x'}\right) }{||\underline{x}-\underline{x'}||^3}\times \underline{\gamma }_t(\underline{x'}) {\mathrm{d}}S(\underline{x'}) \end{aligned}$$

(36.1)

In the general case this integral needs to be evaluated numerically. The integral is expressed in terms of solid angle in Sect. 36.1.2. Further, the velocity field induced by the cylinder may be determined by the knowledge of the velocity field induced by a source disk of constant intensity $\sigma =\pm \gamma _t$. This result is detailed in Sect. 36.1.4.

1.2 Finite Cylinder - Velocity in Terms of Solid Angle

Introduction The velocity induced by a vortex loop of constant intensity $\gamma _t \mathrm {d}z'$ is given in the book of Saffman [14] (or Eq. B.45) as:

$$\begin{aligned} \mathrm {d}\underline{u}(\underline{x}) = \frac{\gamma _t \mathrm {d}z'}{4\pi } \nabla \varOmega (\underline{x},z') \end{aligned}$$

(36.2)

where $\varOmega (\underline{x},z')$ is the solid angle of the vortex loop centered on $z'$ as seen from the point $\underline{x}$ and where the gradient is taken at $\underline{x}$. Since all the vortex loops of the cylinder have the same shape and radii, the solid angle is a function of $z'-z$, and will be further written: $\varOmega (x,y,z'-z)$. The convention $\varOmega >0$ when $z'>z$ is adopted and the gradient $\partial \varOmega /\partial z>0$ with $z'$ fixed. The total velocity is obtained by integration

$$\begin{aligned} \underline{u}(\underline{x}) = \frac{\gamma _t }{4\pi } \int _{z_1}^{z_2} \nabla \varOmega (x,y,z'-z) \, \mathrm {d}z' \end{aligned}$$

(36.3)

Considering only the axial component, the integrand is $\frac{\partial \varOmega }{\partial z}\, \mathrm {d}z'$. Introducing the change of variable $\zeta =z'-z$:

$$\begin{aligned} u_z(\underline{x})&= \frac{\gamma _t }{4\pi } \int _{z_1}^{z_2} \frac{\partial \varOmega }{\partial z}(x,y,z'-z) \, \mathrm {d}z' = \frac{\gamma _t }{4\pi } \int _{\zeta _1}^{\zeta _2} \frac{\partial \zeta }{\partial z}\frac{\partial \varOmega }{\partial \zeta }(x,y,\zeta ) \, \mathrm {d}\zeta \nonumber \\&= -\frac{\gamma _t }{4\pi } \int _{\zeta _1}^{\zeta _2} \mathrm {d}\varOmega (x,y,\zeta ) \end{aligned}$$

(36.4)

Care should be used when integrating the solid angle since it is discontinuous at $\zeta =0$ when the point of evaluation is within the cylinder cross section. Also the same sign convention should be used for each elementary angle $\mathrm {d}\varOmega$. The dependency with respect to x and y is dropped in the following.

Outside of the cylinder If $\underline{x}$ is not contained within the cylinder, then the solid angle is continuous and:

$$\begin{aligned} u_z(\underline{x}) = \frac{\gamma _t }{4\pi }\left[ \varOmega _z(\zeta _1)-\varOmega _z(\zeta _2)\right] \end{aligned}$$

(36.5)

Above the subscript z has been added to specify that the normal to $S_i$ has been taken as $\underline{e}_z$ for the evaluation of both solid angle. Since the control point is outside of the cylinder, the solid angle of the closed surface $S_1\cup S_c \cup S_2$ is zero (see Sect. B.3.1):

$$\begin{aligned} \varOmega _\text {tot}= \varOmega _{-z}(\zeta _1)+ \varOmega _\text {c}(\zeta _1,\zeta _2)+ \varOmega _{z}(\zeta _2) =0 \end{aligned}$$

(36.6)

where $\varOmega _c$ is the solid angle subtended by the cylinder surface $S_c$. Since $S_1$ is contained in a plane normal to $e_z$, $\varOmega _{-z}(\zeta _1)=-\varOmega _{z}(\zeta _1)$, and thus:

$$\begin{aligned} u_z(\underline{x}) = \frac{\gamma _t }{4\pi } \varOmega _c(x,y,\zeta _1,\zeta _2) \end{aligned}$$

(36.7)

Inside the cylinder The solid angle experiences a discontinuity at $\zeta =0$. The cylinder is split into three cylinders by dividing the original cylinder at $\zeta =\pm \varepsilon$, with $\varepsilon >0$. For the two cylinders such that $|\zeta |>\varepsilon$, the control point is outside of the cylinders and the induced velocity is readily obtained using either Eq. 36.5 or Eq. 36.7. The cylinder delimited by $\zeta =\pm \varepsilon$ is now considered. Within this interval $\mathrm {d}\varOmega$ is discontinuous, but the discontinuity is finite and the integral can be carried on. Since $\mathrm {d}\varOmega$ is odd the integral is 0. The total velocity field is then, using Eq. 36.5:

$$\begin{aligned} u_z(\underline{x})&= \frac{\gamma _t }{4\pi } \lim \limits _{\varepsilon \rightarrow 0^+}\left[ \varOmega _z(\zeta _1)-\varOmega _z(-\varepsilon )+\varOmega _z(\varepsilon )-\varOmega _z(\zeta _2) \right] \end{aligned}$$

(36.8)

$$\begin{aligned}&= \frac{\gamma _t }{4\pi }\left[ \varOmega _z(\zeta _1) -\varOmega _z(\zeta _2) +2\lim \limits _{\varepsilon \rightarrow 0^+}\varOmega _z(\varepsilon ) \right] \end{aligned}$$

(36.9)

where $\varOmega _z(-\varepsilon )=-\varOmega _z(\varepsilon )$ has been used since $\varOmega$ is odd. Given the convention chosen that $\varOmega >0$ for $\zeta >0$, the limit is $2\pi$ and the velocity is:

$$\begin{aligned} u_z(\underline{x}) = \frac{\gamma _t }{4\pi } \left[ \varOmega _z(\zeta _1)-\varOmega _z(\zeta _2)+4\pi \right] \end{aligned}$$

(36.10)

Alternatively, Eq. 36.5 is used with the solid angle of the two cylinder surfaces such that $|\zeta |>\varepsilon$ to give:

$$\begin{aligned} u_z(\underline{x}) = \frac{\gamma _t }{4\pi } \lim \limits _{\varepsilon \rightarrow 0^+}\left[ \varOmega _c(\zeta _1,-\varepsilon ) + \varOmega _c(\varepsilon ,\zeta _2) \right] = \frac{\gamma _t }{4\pi } \varOmega _c(\zeta _1,\zeta _2) \end{aligned}$$

(36.11)

General expression By comparison of Eqs. 36.7 and 36.11 the following expression is obtained for the velocity induced by the cylinder of arbitrary cross section:

$$\begin{aligned} u_z(\underline{x}) = \frac{\gamma _t }{4\pi } \varOmega _c(x,y,\zeta _1,\zeta _2) \end{aligned}$$

(36.12)

where $\varOmega _c$ is the solid angle of the surface of the cylinder. The above formula can also be used for points on the boundary of the cylinder.

1.3 Infinite and Semi-infinite Cylinders of Arbitrary Cross Sections

Infinite cylinder The velocity inside the cylinder is constant, equal to the vortex sheet strength $\gamma _t$, and is 0 outside of the cylinder:

$$\begin{aligned} \underline{u}_{\text {inf. cyl}}&=\begin{Bmatrix}0\\ \gamma _t\end{Bmatrix} \underline{e}_z \end{aligned}$$

(36.13)

where the upper value of the bracket corresponds to points outside of the cylinder and the lower values to points inside the cylinder. The above is directly obtained from the results Sect. 36.1.2 with $\varOmega _z(\zeta _1)=\varOmega _z(\zeta _2)=0$, or by considering that the surface $S_c$ is equivalent to a closed surface for the determination of $\varOmega _c$. The result is found in the book of Batchelor [1, p. 98]. This result can also be proved using the following considerations [7]:

• 1. The vortex cylinder only induces a velocity in the z-direction, i.e. $\underline{u}=u_z$. This is the case because the radial component induced at a point $z=0$ by an elementary vortex at z is canceled by the elementary vortex element at $-z$.

• 2. The velocity is constant with z due to the invariance of the problem in this direction, i.e. $u_z=u_z(r,\theta )$.

• 3. The velocity has a constant value inside and a constant value outside of the cylinder. These results are shown using rectangular circulation contours which sides are parallel to the z and r axes, say delimited by $r_1$ and $r_2$ and of length $\mathrm {d}z$. The circulation along this contour is $\varGamma =[ u_z(r_2,\theta )-u_z(r_1,\theta ) ] \mathrm {d}z$. The circulation is zero as long as the contour does not surround the vortex cylinder.

• 4. The velocity outside of the cylinder is zero since it is constant and since it is zero for infinite radial positions (due to the dependency in $1/r^2$ in the Biot–Savart kernel).

• 5. The velocity inside of the cylinder is $\gamma _t$. This is a consequence of the results 3 and 4. It is shown using a rectangular circulation contour which surrounds part of the vortex cylinder: $\varGamma =-\gamma _t \mathrm {d}z = [ u_z(r_\text {out},\theta ) - u_z(r_\text {in},\theta )]\mathrm {d}z=-u_z(r_\text {in},\theta )$. It also follows that the velocity is not a function of $\theta$.

Semi-infinite cylinder The velocity for $\zeta \rightarrow +\infty$ is the same as the infinite cylinder. The velocity for $\zeta \rightarrow -\infty$ is 0 since the solid angle decreases with the distance. For $\zeta =0$, the velocity is 0 outside of the cylinder and $\gamma _t/2$ inside the cylinder since for these two cases $\varOmega _c=0$ and $\varOmega _c=2\pi$.

1.4 Finite Cylinder of Tangential Vorticity and Link to Source Surfaces

The case to be discussed in this section arise from the application of the Neumann-to-Dirichlet surface map equation presented in Sect. 2.2.5. This application is presented first before discussing the vorticity cylinder with source surfaces.

Application of the surface map It is assumed that the flow consists only of a constant wind $\underline{u}=u_0 \underline{e}_z$. A control volume $D_\text {cyl}$ is defined using a cylindrical control surface such that $\partial D_\text {cyl}=S_1 \cup S_C\cup S_2$ as represented in Fig. 36.2a. This control surface is purely geometrical and does not support any source or vorticity. Indeed, the flow is purely irrotational and divergence free. The surface map equation gives the velocity inside the domain $D_\text {cyl}$ as function of the velocity field on the boundary $\partial D_\text {cyl}$. It is of course known that the velocity inside the domain is $\underline{u}_\text {cyl}=\underline{u}=u_0\underline{e}_z$. The application of the surface map will retrieve this result. Applying Eq. 2.151, the velocity inside of the cylinder as function of the velocity on the boundary $\partial D_\text {cyl}$ is

$$\begin{aligned} \underline{u}_\text {cyl}(\underline{x})&=\int _{\partial D_\text {cyl}} \left[ -\underline{K}(\underline{x}-\underline{x'})\,u_n(\underline{x'})+\underline{K}(\underline{x}-\underline{x'}) \times \underline{u}_\tau \right] \, \mathrm {d}\underline{x'} \end{aligned}$$

(36.14)

where $u_n$ is the component of the velocity field normal to $\partial D_\text {cyl}$ such that $u_n=\underline{u}\cdot \underline{n}$, with $\underline{n}$ pointing towards the interior of the domain, $\underline{u}_\tau =\underline{n}\times \underline{u}$, and $\underline{K}$ is the Biot–Savart kernel defined in Eq. 2.232. The above integral is decomposed into three integral on $S_1$, $S_2$ and $S_C$. On $S_1$, $\underline{n}=\underline{e}_z$ and hence $u_n=\underline{u}\cdot \underline{n}=u_0=\sigma _1$ and $\underline{u}_\tau =\underline{n}\times \underline{u}=\underline{0}$. The notation $\sigma _1$ is introduced since the integral over $u_n$ in Eq. 36.14 has the same expression as an integration over a surface distribution of sources (see Sect. 2.2.5). On $S_2$, $\underline{n}=-\underline{e}_z$ and hence $u_n=-u_0=\sigma _2$ and $\underline{u}_\tau =\underline{0}$. As illustrated in Fig. 36.2b with $u_0>0$, $\sigma _1>0$ is a source surface and $\sigma _2<0$ is a sink surface. On $S_C$, the velocity is everywhere orthogonal to the normal $\underline{n}$ and hence $u_n=0$ and $\underline{u}_\tau =u_0 \underline{\tau }=\underline{\gamma _C}$ where $\underline{\tau }$ is the unit vector $-\underline{e}_z\times \underline{n}$. For a circular cylinder, $\underline{n}=-\underline{e}_r$, and $\underline{\tau }=\underline{e}_\theta$. The velocity inside of the cylinder is hence:

$$\begin{aligned} \underline{u}_\text {cyl}(\underline{x})&=\int _{S_1} -\underline{K}(\underline{x}-\underline{x'})\sigma _1\, \mathrm {d}\underline{x'} +\int _{S_2} -\underline{K}(\underline{x}-\underline{x'})\sigma _2\, \mathrm {d}\underline{x'} +\int _{S_C} \underline{K}(\underline{x}-\underline{x'}) \times \underline{\gamma }\, \mathrm {d}\underline{x'} \end{aligned}$$

(36.15)

$$\begin{aligned}&= \underline{u}_{\sigma =\sigma _1} + \underline{u}_{\sigma =\sigma _2} + \underline{u}_{\gamma =\gamma _C} \end{aligned}$$

(36.16)

where $\underline{u}_{\sigma _1}$ is the velocity field induced by the constant distribution of sources of intensity $\sigma _1$ along the surface $S_1$, and $\underline{u}_{\gamma _C}$ is the velocity field induced by the vortex cylinder of tangential surface vorticity $\underline{\gamma }_C$. The different source and vorticity surfaces are represented in Fig. 36.2b. The velocity inside the cylinder is obviously $\underline{u}_\text {cyl}\equiv \underline{u}$. Outside of the cylinder, the surface map does not have any influence and $\underline{u}_\text {cyl}(\underline{x})\equiv 0$. Since the left hand side of Eq. 36.16 is known, the integral involved in $\underline{u}_{\gamma _C}$ may be determined by the result of the integral involved in $\underline{u}_\sigma$ or vice-versa.

Fig. 36.2

Application of the surface map to a uniform flow. a Uniform flow $\underline{u}$ and cylindrical control volume bounded by $\partial D_\text {cyl}=S_1\cup S_2 \cup S_C$. b Source surfaces and cylinder surface of tangential vorticity resulting from the application of the surface map. The velocity induced by these elements gives the velocity inside the control volume, which is known to be $\underline{u}$

Implications It is noted that $\underline{\gamma }_C$ has the same sign convention as $\underline{\gamma }_t$ introduced as the beginning of this chapter. (see Figs. 36.1 and 36.2b). The results of the previous paragraph are used directly. Since it was seen that $\underline{u}_\tau =u_0 \underline{\tau }\tau =\underline{\gamma _C}$, the value of $u_0$ is given as the norm $\gamma _t$. This results in $\sigma _1=u_0=\gamma _t$, $\sigma _2=-u_0=-\gamma _t$, $\underline{u}=\gamma _t \underline{e}_z$. Hence, the velocity field induced by a finite cylinder of tangential vorticity $\gamma _t$ is:

$$\begin{aligned} \underline{u}_{\gamma _t}&= \begin{Bmatrix}0\\ \gamma _t\end{Bmatrix} \underline{e}_z - \underline{u}_{\sigma =\gamma _t} - \underline{u}_{\sigma =-\gamma _t} \end{aligned}$$

(36.17)

where the value of the upper bracket is to be applied for points outside of the cylinder and the lower value for points inside of the cylinder. This formula is used instead of Eq. 36.1. The results given for an infinite cylinder in Sect. 36.1.3 are obtained directly from Eq. 36.17. The source surfaces are then at infinity and their influence is 0 the velocity reduces to $\gamma _t\underline{e}_z$ inside the cylinder and 0 outside.

In Sect. 36.2.1, a finite cylinder of arbitrary cross section is studied. The velocity field from this cylinder is illustrated in Fig. 36.4. In the figure, it is seen that in the middle of the cylinder, the velocity field is close to constant and directed along the z-direction. Velocity variations and radial flows are present at both extremity of the vortex cylinder but with different direction. It can be pictured that the addition of a source and sink disks at both extremities of the vorticity cylinder will help maintaining a constant velocity parallel to the z axis within the cylinder. The disks will also block the flow outside of the cylinder. Obviously this particular flow is not physical but it is nevertheless a good validation case for the surface map method in case it is implemented numerically (see e.g. Chap. 30 for an example of application to sheared-flows).

2 Right Vortex Cylinder of Tangential Vorticity - Circular Cross Section

The right vortex cylinder with tangential vorticity is not only a classical fluid-dynamic but also magnetostatic problem. The model of a rotor using a semi-infinite vortex cylinder was introduced in 1912 by Joukowski [9]. Joukowski presented analytical results using direct integration of the Biot–Savart equation. The results were not explicitly written in elliptic integral. The author nevertheless mentioned this possibility. In 1926, Müller [12] derived analytical formulae for the magnetic field of a finite solenoid based on the mutual inductance of two circular rings. Foelsch also discussed the topic in 1936 [6]. In 1960, Callaghan and Maslen [4] presented formulae based on the derivation of the vector potential. Their work is applied in fluid-dynamics to obtain the velocity field induced by the tangential vorticity of a right vortex cylinder. In 1974, analytical formulae were also derived by Gibson [8], following a different approach than the previous authors. His results are based on a general Lemma and the use of integration by parts. The current author used direct integration to express the induced velocity field from a finite and semi-infinite cylinder [2]. Details on the derivation are given in the following.

Fig. 36.3

Coordinate system used for the right vortex cylinder of tangential vorticity. A cylinder of finite length represented in this figure

2.1 Finite Vortex Cylinder of Tangential Vorticity

Introduction A finite vortex cylinder of circular cross section is considered. Notations are presented in Fig. 36.3. The cylinder has a radius $r_0$ and extends along the z axis between the coordinates $z_1$ and $z_2$ so that its total length is $L=z_2-z_1$. The vorticity is solely in the tangential direction and reduces to the vortex sheet forming the surface of the cylinder of equation $r=r_0$. No vorticity is assumed to be present in the inlet $z=z_1$ and outlet $z=z_2$ planes of the cylinder. Using the $\delta$-Dirac function and the $\varPi$-gate function, this is formalized as:

$$\begin{aligned} \underline{\omega }(\underline{x})=\gamma _t\,\varPi _{[z_1,z_2]}(z)\,\delta (r-r_0)\, \underline{e}_\theta \end{aligned}$$

(36.18)

The Biot–Savart law writes in terms of vector potential:

$$\begin{aligned} \underline{\psi }(\underline{x})&=\frac{1}{4\pi } \int _V\frac{\underline{\omega }(\underline{x'})}{||\underline{x}-\underline{x'}||} r_0 \mathrm{d}{\theta '}\mathrm{d}{r'}\mathrm{d}{z'} =\frac{\gamma _t}{4\pi } \int _V\frac{\varPi _{[z_1,z_2]}(z')\delta (r'-r_0)\underline{e}_\theta (\theta ')}{||\underline{x}-\underline{x'}||} r_0 \mathrm{d}{\theta '}\mathrm{d}{r'}\mathrm{d}{z'} \nonumber \\&=\frac{\gamma _t r_0}{4\pi } \int _{z_1}^{z_2}\int _{0}^{2 \pi } \frac{\underline{e}_\theta (\theta ')}{\left[ r^2+r_0^2-2 r r_0\cos (\theta '-\theta )+ (z-z')^2 \right] ^\frac{1}{2}} \mathrm{d}{\theta '} \mathrm{d}{z} \end{aligned}$$

(36.19)

In term of velocity $\underline{u}=\nabla \times \underline{\psi }$, the Biot–Savart law writes:

$$\begin{aligned} \underline{u}(\underline{x})&=\frac{-\gamma _t}{4\pi }\int _{z_1}^{z_2} \int _{0}^{2 \pi } \frac{\left( \underline{x}-\underline{x'}\right) \times \underline{e}_\theta (\theta ')}{||\underline{x}-\underline{x'}||^3} r_0\mathrm{d}{\theta '}\mathrm{d}{z'}\nonumber \\&=\frac{\gamma _t r_0}{4\pi } \int _{z_1}^{z_2}\int _{0}^{2 \pi } \frac{(z-z')\underline{e}_r(\theta ')-\left[ r\cos (\theta '-\theta )-r_0\right] \underline{e}_z}{\left[ r^2+r_0^2-2 r r_0\cos (\theta '-\theta )+ (z-z')^2 \right] ^\frac{3}{2}} \mathrm{d}{\theta '} \mathrm{d}{z'} \end{aligned}$$

(36.20)

In these formulae, one recognizes the vortex rings formulae of Chap. 35 integrated over z. As a result of this, it is known that the integration over $\theta$ can be done analytically. Yet, analytical formulae are obtained more easily if the integration over z is done first.

Simplifications (axisymmetry) Similar simplifications to the ones introduced for the vortex rings (see Chap. 35) are applied. From azimuthal symmetry it will be assumed that the control point is located in the plane $x-z$, and thus $\theta =0$. Then, $\underline{e}_r$ may be replaced by $\cos \theta \underline{e}_x$ and $\underline{e}_\theta (\theta ')$ by $\cos \theta '\underline{e}_y$.

Velocity on axis On the ring axis, $r=0$ and $\theta =0$, the radial velocity is 0 by symmetry and the Biot–Savart law Equation 36.20 reduces to:

$$\begin{aligned} u_z(0,0,z)&=\frac{\gamma _t r_0^2}{4\pi } \int _{z_1}^{z_2}\int _{0}^{2 \pi } \frac{1}{\left[ r_0^2+ (z-z')^2 \right] ^\frac{3}{2}} \mathrm{d}{\theta }\mathrm{d}{z'} = - \frac{\gamma _t }{2}\left[ \frac{\zeta }{\sqrt{r_0^2+ \zeta ^2}} \right] _{z-z_1}^{z-z_2} \nonumber \\&=\frac{\gamma _t}{2}\left( \cos \alpha _1-\cos \alpha _2\right) \end{aligned}$$

(36.21)

where $\alpha$ is the half-angle of the cone formed by a point on the axis and a ring on the cylinder as sketched in Fig. 36.3.

Velocity on axis - Using solid angle Equation 36.21 may also be found by integration of the solid angle formula [14] (see Eq. B.45) This approach uses the same results as the one used in the vortex ring section Chap. 35, but is actually simpler since no gradient computation is needed. The cylinder may be seen as a superposition of infinitesimal vortex rings. Considering such a ring at position $z'$, then the solid angle of this ring viewed by the control point on the axis is the one of a cone

$$\begin{aligned} \varOmega = 2\pi \int _{\cos \alpha }^1 \mathrm{d}{\cos \theta } = 2\pi (1-\cos \alpha ) \end{aligned}$$

(36.22)

where $\alpha$ is the cone half angle as illustrated in Fig. 36.3, such that:

$$\begin{aligned} \cos \alpha =\frac{z-z'}{\sqrt{(z-z')^2+r_0^2}} \end{aligned}$$

(36.23)

For an infinitesimal distance $\mathrm {d}z'$, the ring intensity is $\gamma _t \mathrm {d}z'$. The elementary velocity induced by this ring is:

$$\begin{aligned} \underline{\mathrm {d}\underline{u}}(0,0,z)=\frac{-\gamma _t \mathrm {d}z'}{4\pi }\nabla \varOmega =\frac{-\gamma _t \mathrm {d}z'}{4\pi }\frac{\partial \varOmega }{\partial z}\underline{e}_z \end{aligned}$$

(36.24)

By integration along the cylinder axis, introducing the change of variable $\zeta =z-z'$, and eventually using Eq. 36.22:

$$\begin{aligned} u_z(0,0,z)&=\frac{-\gamma _t}{4\pi }\int _{z_1}^{z_2}\frac{\partial \varOmega }{\partial z} \mathrm{d}{z'} =\frac{\gamma _t}{4\pi }\int _{\zeta _1}^{\zeta _2}\frac{\partial \varOmega }{\partial \zeta }\frac{\partial \zeta }{\partial z} \mathrm{d}{\zeta } =\frac{\gamma _t}{4\pi }\left[ \varOmega (\zeta _2)-\varOmega (\zeta _1)\right] \end{aligned}$$

(36.25)

$$\begin{aligned}&=\frac{\gamma _t}{2}\left[ \cos \alpha _1-\cos \alpha _2\right] \end{aligned}$$

(36.26)

which confirms Eq. 36.21.

General equation - Vector potential - Work of Callaghan and Maslen and extension The axisymmetry simplifications mentioned previously are used. Primes are dropped for the integration variable $\theta '$. The vector potential Eq. 36.19 reduces to is tangential component and may be further manipulated as follow [4]:

$$\begin{aligned} \psi _\theta (\underline{x})&=\frac{\gamma _t r_0}{2\pi } \int _{z_1}^{z_2}\int _{0}^{\pi } \frac{\cos \theta }{\left[ r^2+r_0^2+ (z-z')^2-2 r r_0\cos \theta \right] ^\frac{1}{2}} \mathrm{d}{\theta }\mathrm{d}{z'} \qquad \small \text {(even function of }\theta \text {)}\nonumber \\&=-\frac{\gamma _t r_0}{2\pi } \int _{0}^{\pi }\int _{\zeta _1}^{\zeta _2} \frac{\cos \theta }{\left[ r^2+r_0^2+ \zeta ^2-2 r r_0\cos \theta \right] ^\frac{1}{2}} \mathrm{d}{\zeta }\mathrm{d}{\theta } \qquad \small \text {(}\zeta =z-z'\text {)} \end{aligned}$$

(36.27)

$$\begin{aligned}&=-\frac{\gamma _t r_0}{2\pi } \int _{0}^{\pi } \cos \theta \Big [\ln \left( \zeta +\sqrt{\zeta ^2+C(\theta )}\right) \Big ]_{\zeta _1}^{\zeta _2}\mathrm{d}{\theta } \qquad \small \text {(}\zeta \text { integration)} \end{aligned}$$

(36.28)

where for the last step $\frac{\partial }{\partial \zeta } \ln \left( \zeta +\sqrt{\zeta ^2+C(\theta )}\right) =[\zeta +C(\theta )]^{-1/2}$ has been used with $C(\theta )=r^2+r_0^2-2 r r_0\cos \theta$. Equation 36.28 may be rearranged by integrating by part over $\theta$:

$$\begin{aligned} \psi _\theta (\underline{x})&=-\frac{\gamma _t r_0}{2\pi }\left[ \int _{0}^{\pi } \cos \theta \ln \left( \zeta +\sqrt{\zeta ^2+C(\theta )}\right) \mathrm{d}{\theta } \right] _{\zeta _1}^{\zeta _2} \qquad \small \text {(rearranging limits)}\\&=-\frac{\gamma _t r_0}{2\pi }\left[ \left[ \sin \theta \ln \left( \zeta +\sqrt{\zeta ^2+C(\theta )}\right) \right] _0^\pi - \int _0^\pi \frac{r r_0\sin ^2\theta }{\sqrt{\zeta ^2+C(\theta )}\left( \zeta +\sqrt{\zeta ^2+C(\theta )}\right) } \mathrm{d}{\theta } \right] _{\zeta _1}^{\zeta _2} \nonumber \end{aligned}$$

(36.29)

The first term is zero due to the values taken by $\sin \theta$. The second term may be further reduced by multiplying by $\frac{\sqrt{\zeta ^2+C(\theta )}-\zeta }{\sqrt{\zeta ^2+C(\theta )}-\zeta }$, to give:

$$\begin{aligned} \psi _\theta (\underline{x})&=\frac{\gamma _t r_0^2 r}{2\pi }\left[ \int _0^\pi \frac{\sin ^2\theta \left( \sqrt{\zeta ^2+C(\theta )}-\zeta \right) }{\sqrt{\zeta ^2+C(\theta )}\,C(\theta )} \mathrm{d}{\theta } \right] _{\zeta _1}^{\zeta _2}\\&=\frac{\gamma _t r_0^2 r}{2\pi }\left\{ \left[ \int _0^\pi \frac{\sin ^2\theta }{C(\theta )} \mathrm{d}{\theta }\right] _{\zeta _1}^{\zeta _2} -\left[ \int _0^\pi \frac{\zeta \sin ^2\theta }{\sqrt{\zeta ^2+C(\theta )}\,C(\theta )} \mathrm{d}{\theta } \right] _{\zeta _1}^{\zeta _2} \right\} \quad \small \text {(expanding numerator)}\nonumber \end{aligned}$$

(36.30)

$$\begin{aligned}&=-\frac{\gamma _t r_0^2 r}{2\pi }\left[ \int _0^\pi \frac{\zeta \sin ^2\theta }{\sqrt{\zeta ^2+C(\theta )}\,C(\theta )} \mathrm{d}{\theta } \right] _{\zeta _1}^{\zeta _2} \qquad (\text {since }[\text {cst}]_{\zeta _1}^{\zeta _2} =0 ) \end{aligned}$$

(36.31)

Callaghan stops his calculation here, but the expression found for $A_\theta$ may be extended further into elliptic integrals (see Sect. C.4). Introducing the elliptic parameter

$$\begin{aligned} m(\zeta )=\mathrm {k}^2(\zeta )=\frac{4 r r_0}{\zeta ^2+(r+r_0)^2} \end{aligned}$$

(36.32)

with $m_0=m(\zeta =0)$ and using the change of variable $\phi =\theta /2$ in Eq. 36.31:

$$\begin{aligned} \psi _\theta (\underline{x})&=-\frac{\gamma _t r_0^2 r}{2\pi }\left[ \zeta \int _0^\pi \frac{\sin ^2\theta }{\sqrt{\zeta ^2+r^2+r_0^2-2rr_0\cos \theta }(r^2+r_0^2-2rr_0\cos \theta )} \mathrm{d}{\theta } \right] _{\zeta _1}^{\zeta _2} \end{aligned}$$

(36.33)

$$\begin{aligned}&=-\frac{\gamma _t r_0^2 r}{2\pi }\left[ 2\zeta \frac{\sqrt{m}}{2\sqrt{r_0 r}}\frac{m_0}{4r_0r}\int _0^{\pi /2} \frac{\sin ^2(2\phi )}{\sqrt{1-m\cos ^2\phi }(1-m_0\cos ^2\phi )} \mathrm{d}{\phi } \right] _{\zeta _1}^{\zeta _2} \end{aligned}$$

(36.34)

Then, using $\sin ^2(2\phi )=4(\cos ^2\phi -\cos ^4\phi )$

$$\begin{aligned} \psi _\theta (\underline{x})&=-\frac{\gamma _t r_0^2 r}{2\pi }\frac{m_0}{(r_0r)^{3/2}}\left[ \zeta \sqrt{m}\int _0^{\pi /2} \frac{\cos ^2\phi -\cos ^4\phi }{\sqrt{1-m\cos ^2\phi }(1-m_0\cos ^2\phi )} \mathrm{d}{\phi } \right] _{\zeta _1}^{\zeta _2}\nonumber \\&=-\frac{\gamma _t}{2\pi }\sqrt{\frac{r_0}{r}}m_0\Big [\zeta \sqrt{m}(I_1-I_2) \Big ]_{\zeta _1}^{\zeta _2} \end{aligned}$$

(36.35)

the expression is reduced to the sum of two integrals. The first integral is, after the following manipulation of the numerator $\cos ^2\phi =\frac{1}{m_0}(m_0\cos ^2\theta -1+1)$:

$$\begin{aligned} I_1=\int _0^{\pi /2} \frac{\cos ^2\phi }{\sqrt{1-m\cos ^2\phi }(1-m_0\cos ^2\phi )} \mathrm{d}{\phi } = \frac{1}{m_0}\left[ \varPi (m_0,m)-K(m)\right] \end{aligned}$$

(36.36)

For the second integral, the idea is first to transform $\cos ^4$ to make the product $(1-m_0\cos ^2\phi )(1+m_0\cos ^2\phi )$ appear. Usual $-1 +1$ operations are successively used:

$$\begin{aligned} I_2&=\int _0^{\pi /2} \frac{\cos ^4\phi }{\sqrt{1-m\cos ^2\phi }(1-m_0\cos ^2\phi )} \mathrm{d}{\phi }\nonumber \\&=\frac{1}{m_0^2}\int _0^{\pi /2} \frac{m_0^2\cos ^4\phi -1+1}{\sqrt{1-m\cos ^2\phi }(1-m_0\cos ^2\phi )} \mathrm{d}{\phi }\nonumber \\&=\frac{1}{m_0^2}\left[ \int _0^{\pi /2} \frac{-(1+m_0\cos ^2\phi )}{\sqrt{1-m\cos ^2\phi }} \mathrm{d}{\phi } +\varPi (m_0,m)\right] \nonumber \\&=\frac{1}{m_0^2}\left[ -K(m)-\frac{m_0}{m}\int _0^{\pi /2} \frac{m\cos ^2\phi -1+1}{\sqrt{1-m\cos ^2\phi }} \mathrm{d}{\phi } +\varPi (m_0,m)\right] \nonumber \\&=\frac{1}{m_0^2}\left[ -K(m)-\frac{m_0}{m}\left( K(m)-E(m)\right) +\varPi (m_0,m)\right] \end{aligned}$$

(36.37)

Inserting the values of $I_1$ and $I_2$ into Eq. 36.35, gives the following final form for the vector potential:

$$\begin{aligned} \psi _\theta (\underline{x}) =-\frac{\gamma _t}{2\pi }\sqrt{\frac{r_0}{r}}\frac{1}{m_0}\left[ \zeta \sqrt{m} \left( \left( 1-m_0+\frac{m_0}{m}\right) K(m)-\frac{m_0}{m}E(m)+(m_0-1)\varPi (m_0,m)\right) \right] _{\zeta _1}^{\zeta _2} \nonumber \\ \end{aligned}$$

(36.38)

where it is recalled that $m=m(\zeta )$, $m_0=m(0)$ and this equation may be expressed with the variable $\mathrm {k}$ where $m=\mathrm {k}^2$ (see Eq. 36.32).

General equation - Velocity from direct derivation of vector potential The general velocity equations for the finite cylinder may be obtained by derivation of the vector potential as: (see Sect. 2.9.3)

$$\begin{aligned} u_r&=-\frac{\partial \psi _\theta }{\partial z} \end{aligned}$$

(36.39)

$$\begin{aligned} u_z&=\frac{1}{r}\frac{\partial \left( r \psi _\theta \right) }{\partial r}=\frac{1}{\rho }\psi _\theta +\frac{\partial \psi _\theta }{\partial r} \end{aligned}$$

(36.40)

For this purpose, one may take the integrated form with elliptic functions Eq. 36.38 and use the derivatives formulae of the elliptic functions (Sect. C.4.2) to eventually obtain:

$$\begin{aligned} u_r(r,z)&= \frac{\gamma _t}{2 \pi } \sqrt{\frac{r_0}{r}} \left[ \frac{2-\mathrm {k}^2(\zeta )}{\mathrm {k}(\zeta )}K\left( \mathrm {k}^2(\zeta )\right) - \frac{2}{\mathrm {k}(\zeta )}E\left( \mathrm {k}^2(\zeta )\right) \right] _{\zeta _1=z-z_1}^{\zeta _2=z-z_2} \end{aligned}$$

(36.41)

$$\begin{aligned} u_z(r,z)&= -\frac{\gamma _t}{4 \pi \sqrt{r r_0}} \left[ \zeta \mathrm {k}(\zeta ) \left( K\left( \mathrm {k}^2(\zeta )\right) +\frac{r_0-r }{r_0+r}\varPi \left( \mathrm {k}^2(0)|\mathrm {k}^2(\zeta )\right) \right) \right] _{\zeta _1=z-z_1}^{\zeta _2=z-z_2} \end{aligned}$$

(36.42)

with m and $\mathrm {k}$ defined in Eq. 36.32.

The velocity field induced by a finite vortex cylinder is illustrated in Fig. 36.4.

Fig. 36.4

Streamlines and velocity field induced by a finite vortex cylinder illustrated using Line Integral Convolution (LIC) flow visualization. The cylinder is contained within the planes $|z/r_0|\le 2$ and $|r|=r_0$. The vortex intensity is $\gamma _t=-1$

General equation - Velocity from indirect derivation of vector potential The approach chosen by Callaghan [4] consisted in using the integral forms of the vector potential. This approach is less direct but less tedious than having to derive the elliptic integrals as in the previous paragraph.

For the radial component, the integral form Eq. 36.27 is used and combined to derivation properties of integrals,

$$\begin{aligned} \frac{{\mathrm{d}}}{\mathrm {d}z}\int _{z_1}^{z_2} f(z) \mathrm{d}{z} = \big [f(z)]\big ]_{z_1}^{z_2} \end{aligned}$$

(36.43)

gives directly:

$$\begin{aligned} u_r(r,z)=-\frac{{\mathrm{d}} \psi _\theta }{\mathrm {d}z}=-\frac{{\mathrm{d}} \psi _\theta }{\mathrm {d}\zeta }\frac{{\mathrm{d}}\zeta }{\mathrm {d}z}=-\frac{{\mathrm{d}} \psi _\theta }{\mathrm {d}\zeta } =\frac{\gamma _t r_0}{2\pi } \left[ \int _{0}^{\pi } \frac{\cos \theta }{\left[ r^2+r_0^2+ \zeta ^2-2 r r_0\cos \theta \right] ^\frac{1}{2}} \mathrm{d}{\theta } \right] _{\zeta _1}^{\zeta _2} \end{aligned}$$

The above equation may be expressed in the form of elliptic integrals without difficulty. This form was for actually found when establishing the vector potential of a vortex ring: see Eq. 35.23 and the following derivations. Equation 36.41 is readily obtained with this method, and we can further observe that:

$$\begin{aligned} u_r(r,z)=\left[ \psi _{\theta ,\text {Ring}}(r,z')\right] _{z-z_1}^{z-z_2} \end{aligned}$$

(36.44)

For the derivation of the longitudinal component, one may evaluate first $\partial \psi _\theta /\partial r$ using the integral form of Eq. 36.29:

$$\begin{aligned} \frac{\partial \psi _\theta }{\partial r}&=-\frac{\gamma _t r_0}{2\pi }\left[ \int _{0}^{\pi }\cos \theta \frac{\partial }{\partial r} \ln \left( \zeta +\sqrt{\zeta ^2+C(\theta )}\right) \mathrm{d}{\theta } \right] _{\zeta _1}^{\zeta _2} \end{aligned}$$

(36.45)

$$\begin{aligned}&=-\frac{\gamma _t r_0}{2\pi }\left[ \int _{0}^{\pi }\cos \theta \frac{r -r_0\cos \theta }{\sqrt{\zeta ^2+C(\theta )}(\zeta +\sqrt{\zeta ^2+C(\theta )})} \mathrm{d}{\theta } \right] _{\zeta _1}^{\zeta _2} \end{aligned}$$

(36.46)

The steps used from Eqs. 36.30 to 36.31 are applied to the above integral to eventually give:

$$\begin{aligned} \frac{\partial \psi _\theta }{\partial r} =\frac{\gamma _t r_0}{2\pi }\left[ \int _{0}^{\pi }\zeta \cos \theta \frac{r -r_0\cos \theta }{\sqrt{\zeta ^2+C(\theta )}\,C(\theta ))} \mathrm{d}{\theta } \right] _{\zeta _1}^{\zeta _2} \end{aligned}$$

(36.47)

Using Eqs. 36.47 and 36.31, one obtains:

$$\begin{aligned} u_z=\frac{1}{\rho }\psi _\theta +\frac{\partial \psi _\theta }{\partial \underline{r}} =\frac{\gamma _t r_0}{2\pi }\left[ \int _{0}^{\pi }\frac{\zeta (r \cos \theta -r_0) }{\sqrt{\zeta ^2+C(\theta )}\,C(\theta )} \mathrm{d}{\theta } \right] _{\zeta _1}^{\zeta _2} \end{aligned}$$

(36.48)

The same steps as the one used for Eq. 36.33 are applied:

$$\begin{aligned} u_z&=\frac{\gamma _t r_0}{\pi }\left[ \zeta \int _{0}^{\pi /2}\frac{(r \cos (2\phi )-r_0) }{\sqrt{\zeta ^2+C(2\phi )}\,C(2\phi )} \mathrm{d}{\phi } \right] _{\zeta _1}^{\zeta _2} \quad (\phi =\theta /2)\nonumber \\&=\frac{\gamma _t r_0}{\pi }\left[ \zeta \frac{\sqrt{m}}{2\sqrt{rr_0}}\frac{m_0}{4rr_0}\int _{0}^{\pi /2}\frac{2r\cos ^2\phi -(r+r_0)}{\sqrt{1-m\cos ^2\phi }(1-m_0\cos ^2\phi )} \mathrm{d}{\phi } \right] _{\zeta _1}^{\zeta _2} \end{aligned}$$

(36.49)

$$\begin{aligned}&=\frac{\gamma _t r_0}{\pi }\frac{1}{2\sqrt{rr_0}}\frac{m_0}{4rr_0}\Big [\zeta \sqrt{m} \big (2rI_1-(r+r_0)\varPi (m_0,m)\big )\Big ]_{\zeta _1}^{\zeta _2} \end{aligned}$$

(36.50)

$$\begin{aligned}&=\frac{\gamma _t r_0}{\pi }\frac{1}{2\sqrt{rr_0}}\frac{m_0}{4rr_0}\left[ \zeta \sqrt{m}\frac{2r}{m_0}\left( \varPi (m_0,m)-K(m)-\frac{m_0}{2r}(r+r_0)\varPi (m_0,m)\right) \right] _{\zeta _1}^{\zeta _2} \end{aligned}$$

(36.51)

$$\begin{aligned}&=-\frac{\gamma _t}{4\pi }\frac{1}{\sqrt{rr_0}}\left[ \zeta \sqrt{m}\left( K(m)+\left( \frac{m_0}{2r}(r+r_0)-1\right) \varPi (m_0,m)\right) \right] _{\zeta _1}^{\zeta _2} \end{aligned}$$

(36.52)

and Eq. 36.42 is readily obtained.

General equation - Velocity from Biot–Savart law - Indefinite form [2] Instead of using the vector potential, the Biot–Savart law from Eq. 36.20 may be integrated directly. In this perspective, the indefinite integral over z in Eq. 36.20 will be determined below. From the axisymmetry of the flow it may be assumed without loss of generality that the control point lays in the x axis, i.e. $\theta =0$, and that the radial component of the field may be seen as the x-component. We will write $\underline{e}_\rho$ this fake Cartesian representation of the radial component, which is such that $\underline{e}_\theta (\theta ')=\cos \theta '\underline{e}_\rho$. Using these assumptions Eq. 36.20 becomes:

$$\begin{aligned} {\underline{u}}^{z'}(\underline{x}) = \frac{\gamma _t r_0}{4\pi } \int _{0}^{2 \pi }\int _{z'} \frac{(z-z')\cos \theta '{\underline{e}}_\rho -\left[ r\cos (\theta ')-r_0\right] \underline{e}_z}{\left[ r^2+r_0^2-2 r r_0\cos (\theta ')+ (z-z')^2 \right] ^\frac{3}{2}} \mathrm {d}{z'} \mathrm {d}{\theta '} \end{aligned}$$

(36.53)

The notation $C(\theta )=r^2+r_0^2-2 r r_0\cos (\theta )$ is introduced and the change of variable $\zeta =z-z'$ is applied with $\mathrm {d}\zeta =-\mathrm {d}z'$. Noting that the integrand over $\theta '$ is periodic and an even function the integration limit may be reduced to obtain:

$$\begin{aligned} \underline{u}^{\zeta }(\underline{x}) = -\frac{\gamma _t r_0}{2\pi }\int _{0}^{\pi } \int _{\zeta }\frac{\zeta \cos \theta '\underline{e}_\rho -\left[ r\cos (\theta ')-r_0\right] \underline{e}_z}{\left[ C(\theta ')+ \zeta ^2\right] ^\frac{3}{2}} \mathrm {d}{\zeta } \mathrm {d}{\theta '} \end{aligned}$$

(36.54)

The integrals over $\zeta$ are straightforward using the following relations:

$$\begin{aligned} \int \frac{{\mathrm{d}}\zeta }{(C +\zeta ^2)^{\frac{3}{2}}} =\frac{\zeta }{C\sqrt{C+\zeta ^2}},\qquad \int \frac{\zeta \mathrm {d}\zeta }{(C +\zeta ^2)^{\frac{3}{2}}} =\frac{-1}{\sqrt{C+\zeta ^2}} \end{aligned}$$

(36.55)

So that Eq. 36.54 becomes

$$\begin{aligned} \underline{u}^{\zeta }(\underline{x}) = \frac{\gamma _t r_0}{2\pi } \int _{0}^{\pi } \frac{\cos \theta '}{\sqrt{C(\theta ')+\zeta ^2}} \underline{e}_\rho +\frac{\left[ r\cos (\theta ')-r_0\right] }{C(\theta )\sqrt{C(\theta ')+\zeta ^2}} \underline{e}_z \mathrm {d}{\theta '} \end{aligned}$$

(36.56)

The presence of square root of cosine terms evoke elliptic integrals (see Sect. C.4). As a result of this the change of variable $\phi =\theta /2$ is introduced, so that $\cos \theta =\cos (2\phi )=2\cos ^2\phi -1$:

$$\begin{aligned} \underline{u}^{\zeta }(\underline{x}) = \frac{\gamma _t r_0}{\pi } \int _{0}^{\pi /2} \frac{2\cos ^2\phi -1}{\sqrt{C(2\phi )+\zeta ^2}} \underline{e}_\rho +\frac{\left[ 2r\cos ^2\phi -(r_0+r)\right] }{C(2\phi )\sqrt{C(2\phi )+\zeta ^2}} \underline{e}_z \mathrm {d}{\phi } \end{aligned}$$

(36.57)

Developing the expression $C(2\phi )=(r_0+r)^2-2rr_0\cos ^2\phi$, the parameter for the elliptic integral is readily determined and defined as presented in Eq. 36.32. The components of the integral are now treated separately. The radial component is successively reduced by factorizing the elliptical parameter then noting that $2\cos ^2\phi =\frac{2}{m}\left( m\cos ^2\phi - 1 + 1 - \frac{m}{2}\right)$ and using the definition of the complete elliptic integrals of the first and second kind noted K and E respectively. The steps are as follow:

$$\begin{aligned} u^{\zeta }_{r}(\underline{x})&= \frac{\gamma _t r_0}{\pi } \frac{\sqrt{m}}{2\sqrt{rr_0}} \int _{0}^{\pi /2} \frac{2\cos ^2\phi -1}{\sqrt{1-m\cos ^2\phi }} \mathrm {d}\phi \end{aligned}$$

(36.58)

$$\begin{aligned}&= \frac{\gamma _t r_0}{\pi } \frac{\sqrt{m}}{2\sqrt{rr_0}} \left[ -\frac{2}{m}E(m)+\frac{2}{m}\left( 1-\frac{m}{2}\right) K(m) \right] \end{aligned}$$

(36.59)

$$\begin{aligned}&= \frac{\gamma _t }{2\pi } \sqrt{\frac{r_0}{r}}\left[ \frac{2-\mathrm {k}^2}{\mathrm {k}}K(\mathrm {k}^2)-\frac{2}{\mathrm {k}}E(\mathrm {k}^2)\right] \end{aligned}$$

(36.60)

The parameter m or $\mathrm {k}$ is used indifferently, and the dependence of these parameters with respect to $\zeta$ has been dropped to shorten notations. For the longitudinal component, the factorization of the elliptic parameters leads to:

$$\begin{aligned} u_z^{\zeta }(\underline{x})&=\frac{\gamma _t r_0}{\pi }\zeta \frac{\sqrt{m}}{2\sqrt{rr_0}}\frac{m_0}{4rr_0}\int _{0}^{\pi /2}\frac{2r\cos ^2\phi -(r+r_0)}{\sqrt{1-m\cos ^2\phi }(1-m_0\cos ^2\phi )} \mathrm {d}{\phi } \end{aligned}$$

(36.61)

where the notation $m_0=m(0)$ is used. By definition of the elliptic integral of the third kind $\varPi$, the above equation writes:

$$\begin{aligned} u_z^{\zeta }(\underline{x})&=\frac{\gamma _t r_0}{\pi }\frac{1}{2\sqrt{rr_0}}\frac{m_0}{4rr_0}\Big [\zeta \sqrt{m} \big (2rI_1-(r+r_0)\varPi (m_0,m)\big )\Big ] \end{aligned}$$

(36.62)

where $I_1$ has been introduced since it is an integral that needs further development. Using $\cos ^2\phi =\frac{1}{m_0}(m_0\cos ^2\phi -1+1)$ and the definitions of elliptic integrals, the following expression is obtained for $I_1$:

$$\begin{aligned} I_1&=\int _0^{\pi /2}\frac{\cos ^2\phi }{\sqrt{1-m\cos ^2\phi }(1-m_0\cos ^2\phi )} \mathrm {d}\phi = \frac{1}{m_0}\left( \varPi (m_0,m)-K(m)\right) \end{aligned}$$

(36.63)

Inserting Eq. 36.63 in Eq. 36.62 leads to:

$$\begin{aligned} u^{\zeta }_{z}(\underline{x})&=\frac{\gamma _t r_0}{\pi }\frac{1}{2\sqrt{rr_0}}\frac{m_0}{4rr_0}\left[ \zeta \sqrt{m}\frac{2r}{m_0}\left( \varPi (m_0,m)-K(m)-\frac{m_0}{2r}(r+r_0)\varPi (m_0,m)\right) \right] \end{aligned}$$

(36.64)

$$\begin{aligned}&=-\frac{\gamma _t}{4\pi }\frac{1}{\sqrt{rr_0}}\left[ \zeta \sqrt{m}\left( K(m)+\left( \frac{m_0}{2r}(r+r_0)-1\right) \varPi (m_0,m)\right) \right] \end{aligned}$$

(36.65)

$$\begin{aligned}&=-\frac{\gamma _t}{4\pi }\frac{1}{\sqrt{rr_0}}\left[ \zeta \mathrm {k}\left( K(\mathrm {k}^2)+\frac{r_0-r}{r_0+r}\varPi (\mathrm {k}_0^2,\mathrm {k}^2)\right) \right] \end{aligned}$$

(36.66)

General equation - Velocity from Biot–Savart law - Finite cylinder [2] The indefinite integral forms obtained in Eqs. 36.60 and 36.66 may be used on a known interval. Assuming a cylinder extends from $z_1$ to $z_2$ then the induced velocities are simply:

$$\begin{aligned} u_\text {sol,r}(\underline{x})&= \frac{\gamma _t }{2\pi } \sqrt{\frac{r_0}{r}}\left[ \frac{2-\mathrm {k}^2}{\mathrm {k}}K(\mathrm {k}^2)-\frac{2}{\mathrm {k}}E(\mathrm {k}^2)\right] _{\zeta _1=z-z_1}^{\zeta _2=z-z2} \end{aligned}$$

(36.67)

$$\begin{aligned} u_\text {sol,z}(\underline{x})&=-\frac{\gamma _t}{4\pi }\frac{1}{\sqrt{rr_0}}\left[ \zeta \mathrm {k}\left( K(\mathrm {k}^2)+\frac{r_0-r}{r_0+r}\varPi (\mathrm {k}_0^2,\mathrm {k}^2)\right) \right] _{\zeta _1=z-z_1}^{\zeta _2=z-z2} \end{aligned}$$

(36.68)

A regularization of these equations is given in Sect. 36.2.2.

2.2 Semi-infinite Vortex Cylinder of Tangential Vorticity

Results for the semi-infinite cylinder are derived directly from the results of the finite cylinder. Results from this section were published in an article by the author [2].

Introduction and notations An infinite vortex cylinder of circular cross section is considered. Notations are presented in Fig. 36.5. The cylinder has a radius $r_0$ and extends along the z axis between the coordinates $z=0$ and $z=+\infty$. The Biot–Savart law for the vector potential and velocity is given by Eqs. 36.19 and 36.20 using $z_1=0$ and $z_2\rightarrow +\infty$.

Fig. 36.5

Polar coordinate system used for infinite vortex cylinder

Induced velocity field The induced velocity field for the semi-infinite cylinder is obtained with $z_1=0$ and $z_2\rightarrow +\infty$. For the r component, the limit when $z_2$ tends to $+\infty$ is zero. For the z component the limit as $z_2\rightarrow +\infty$ has different value for radii lower or greater than $r_0$. The bracket notation could have been used as in Eq. 39.7, but the same result may be obtained by using an absolute value expression. The results from these calculation leads to:

$$\begin{aligned} u_{r,\text {cyl}}(r,z)&=-\frac{\gamma _t}{2 \pi } \sqrt{\frac{r_0}{r}} \left[ \frac{2-\mathrm {k}^2(z)}{\mathrm {k}(z)}K\left( \mathrm {k}^2(z)\right) - \frac{2}{\mathrm {k}(z)}E\left( \mathrm {k}^2(z)\right) \right] \end{aligned}$$

(36.69)

$$\begin{aligned} u_{z,\text {cyl}}(r,z)&=\frac{\gamma _t}{2}\left[ \frac{r_0-r+|r-r_0|}{2|r-r_0|} +\frac{z \mathrm {k}(z)}{2 \pi \sqrt{r r_0}} \left( K\left( \mathrm {k}^2(z)\right) +\frac{r_0-r }{r_0+r}\varPi \left( \mathrm {k}^2(0)|\mathrm {k}^2(z)\right) \right) \right] \end{aligned}$$

(36.70)

where

$$\begin{aligned} \mathrm {k}^2(z)=m(z)=\frac{4rr_0}{(r_0+r)^2+z^2} \end{aligned}$$

(36.71)

and the dependency in r was omitted to shorten notations. The velocity field induced by a semi-infinite vortex cylinder is illustrated in Fig. 36.6.

Fig. 36.6

Streamlines and velocity field induced by a vortex cylinder illustrated using Line Integral Convolution (LIC) flow visualization. The plane of observation intersects the ring at $z=0$ and $|r|=r_0$. The vortex intensity is $\gamma _t=-1$

Flow near the axis and on the axis The velocity field near the axis is obtained via a Taylor series of the formulae:

$$\begin{aligned} u_{r,t}(r\ll r_0,z)&=-\frac{\gamma _t}{4}\frac{rr_0^2}{\left( r_0^2+z^2\right) ^{3/2}}+O(r^{5/2}),\nonumber \\ u_{z,t}(r\ll r_0,z)&=\frac{\gamma _t}{2}\left[ 1+\frac{z}{\sqrt{r_0^2+z^2}}\right] + O(r). \end{aligned}$$

(36.72)

These equations are evaluated at $r=0$ to get the velocity field on the axis. A Taylor expansion is not necessary to obtain the formulae on the axis though. It is obvious that $u_r=0$ on the axis by symmetry. The component $u_z$ on the axis is derived with the same methods used for the finite cylinder: by direct integration of the Biot–Savart law (Eq. 36.21) or by taking the gradient of the solid angle (Eq. 36.26). The finite cylinder equations are directly applied with $\alpha _2=\pi /2$ and $\cos \alpha _1=z/\sqrt{z^2+r_0^2}$ (Eq. 36.23) and Eq. 36.72 is obtained (see also Stepniewski and Keys [15, p. 155], van Kuik [16]).

Axial induction: analysis of the different terms involved Equation 36.70 consists of the sum of three terms. The contribution of the different terms involved is shown in Fig. 36.7. Absolute values are used in this equation to conveniently write in mathematical form the discontinuity of the first term at $r=r_0$. This first term is the only one that remains when $z=0$. As $z\rightarrow +\infty$, the sum of the two terms involving elliptical integrals tends to $\gamma _t/2$ for $r<r_0$ and tends to 0 for $r>r_0$. Both the elliptic integral of the third kind and the absolute value term have discontinuities at $r=r_0$. The term involving the elliptic integral K is always continuous. When $z<0$, the sum of all three terms is such that the solution is continuous with respect to r and tends to zero as $z\rightarrow -\infty$. On the other hand, for $z>0$, the axial velocity is discontinuous with a jump of value equal to $-\gamma _t$ between the upper and inner part of the vortex cylinder. This jump is half for $z=0$. For all values of z, the axial induction is always highest for $r<r_0$ than for $r\ge r_0$. Axial velocity contours and streamlines are shown in Fig. 36.8.

Fig. 36.7

Contribution of the different terms of Eq. 36.70 to the total axial induction at three different axial positions [2]

Axial induction at particular locations The axial induced velocity is seen to be constant in the rotor plane by inserting $z=0$ in Eq. 36.70 and is also constant in the far-wake by evaluation of the limit of the elliptic integrals. The values at these locations are:

$$\begin{aligned} u_{z,t}(r,0)&=\begin{Bmatrix}0\\ \frac{\gamma _t}{2}\end{Bmatrix} \end{aligned}$$

(36.73)

$$\begin{aligned} u_{z,t}(r,z\rightarrow +\infty )&=\begin{Bmatrix}0\\ \gamma _t\end{Bmatrix} \end{aligned}$$

(36.74)

where the upper value of the bracket corresponds to $r>r_0$ and the lower value to $r<r_0$. The velocity at the vortex sheet is the average of the value on the upper and lower bracket value [8]. This is true for all positive values of z. Noting that the third term of Eq. 36.70 is antisymmetric around $r=r_0$, the axial induction on the cylinder itself reduces to:

$$\begin{aligned} u_{z,t}(r= r_0,z>0)=\frac{\gamma _t}{4}+\frac{\gamma _t}{2}\frac{z \mathrm {k}(z)}{2 \pi \sqrt{r r_0}}K\left( \mathrm {k}^2(z)\right) ), \end{aligned}$$

(36.75)

As previously mentioned, the vortex sheet induces across it a velocity jump in the axial direction of intensity $-\gamma _t$ between the velocities in the outer and the inner part. This jump is exactly half on the leading edge of the cylinder.

Note on the radial induction The radial velocity is an even function of the variable z, i.e. $u_{r,t}(-z)=u_{r,t}(z)$. It is continuous in the entire domain except at the cylinder’s leading edge. In the far wake, the radial velocity tends to zero and the flow is purely in the axial direction. In particular, Eq. 36.69 can be evaluated at the rotor for $r<r_0$. It can also be evaluated on the cylinder itself where it is seen to be non-zero. This means that there is flow going through the cylinder (see discussion by Lewis [10, p. 168] about the leakage through the wall of the semi-infinite cylinder near the free-end side). This is indeed required to satisfy continuity since the cross-sectional area of the cylinder remains constant while the axial velocity varies by a factor 2. The flow thus tends to move the vortex sheet towards a more realistic shape. The converged vortex system, when the vortex sheet is allowed to expand and the convection velocity varies along the wake, has an axial induction that corresponds well with the one from the tangential vorticity cylinder of constant strength [13].

Fig. 36.8

Semi-infinite vortex cylinder velocities. Axial velocity contour plot normalized by $2r_0/\gamma _t$ (left). Streamlines for $\gamma _t=-1$ m/s reveals the strong singularity at the cylinder rim and the radial flow through the cylinder surface (right) [2]

Regularization of the cylinder’s equations A simple regularization of the cylinder’s equation consists in introducing a cut off length $\varepsilon$ in the denominator of the elliptic parameter $\mathrm {k}$, namely:

$$\begin{aligned} m_\varepsilon (\zeta )=\mathrm {k}_\varepsilon ^2(\zeta ) = \frac{4 r r_0}{(r+r_0)^2+\zeta ^2+\varepsilon ^2} \end{aligned}$$

(36.76)

The singularity mainly concerns the radial component of the velocity, so it may be chosen to regularize only this component. The regularized expression of $\mathrm {k}$ can be used for the indefinite form of the radial component (Eq. 2.6.2) but also in the definite form of the radial component of the semi-infinite cylinder (Eq. 36.69). The longitudinal component does not present a singularity, but a discontinuity at $r=r_0$ when $z>=0$. The regularization presented above will remove the discontinuity. For this, the parameter $\mathrm {k}(0)$ should also be replaced by $\mathrm {k}_\varepsilon (0)$ in both the indefinite and definite forms: Eqs. 36.68 and 36.70. For the latter equation, the limit needs to be evaluated again which will affect the first term as follow:

$$\begin{aligned} u_{z,\text {cyl}}(r,z)=\frac{\gamma _t}{2}&\left[ \frac{1}{2}\left( 1+\frac{r_0-r}{(r_0+r)\sqrt{1-\mathrm {k}^2_\varepsilon (0)}}\right) \right. \nonumber \\&+\frac{z \mathrm {k}_\varepsilon (z)}{2 \pi \sqrt{r r_0}} \left( K\left( \mathrm {k}^2_\varepsilon (z)\right) +\frac{r_0-r }{r_0+r}\varPi \left( \mathrm {k}^2_\varepsilon (0)|\mathrm {k}^2_\varepsilon (z)\right) \right) \Bigg ] \end{aligned}$$

(36.77)

Developing the first term, it is clear that when $\varepsilon =0$ the non-regularized form (Eq. 36.70) is retrieved:

$$\begin{aligned} u_{z,\text {cyl}}(r,z)=\frac{\gamma _t}{2}&\left[ \frac{1}{2}\left( 1+\frac{(r_0-r)\sqrt{1+\varepsilon ^2/(r_0+r)^2}}{\sqrt{(r_0-r)^2+\varepsilon ^2}}\right) \right. \nonumber \\&+\frac{z \mathrm {k}_\varepsilon (z)}{2 \pi \sqrt{r r_0}} \left( K\left( \mathrm {k}_\varepsilon (z)^2\right) +\frac{r_0-r }{r_0+r}\varPi \left( \mathrm {k}_\varepsilon (0)^2|\mathrm {k}_\varepsilon (z)^2\right) \right) \Bigg ] \end{aligned}$$

(36.78)

Matlab source code A Matlab  [11] code computing the induced velocity from a semi-infinite vortex cylinder of tangential vorticity is given below. The elliptic integral of the third kind was programmed by N. Troldborg who also used the cylindrical model [3] and kindly accepted the publication of the code here. The implementation of the elliptic $\varPi$ integral is based on the method of Carlson [5] .

3 Vortex Cylinder of Longitudinal Vorticity

A cylinder of circular cross-section is considered. The notations are presented in Fig. 36.9. The cylinder has a radius R and extends along the z-axis between the coordinates $z_1$ and $z_2$. The vorticity is assumed to be solely in the longitudinal direction. It reduces to the vortex sheet forming the surface of the cylinder so that at a point $\underline{x}=(r,\psi ,z)$ the vorticity is: $\underline{\omega }(\underline{x})=\gamma _l\,\varPi _{[z_1,z_2[}(z)\,\delta (r-R)\, \underline{e}_z$. The longitudinal vorticity component of the cylinder will induce a tangential velocity component only. The results presented in this section were published in [2].

Fig. 36.9

Coordinate system and notations for the vortex cylinder of longitudinal vorticity $\underline{\gamma }_l=\gamma _l \underline{e}_z$

3.1 Infinite Cylinder of Longitudinal Vorticity

The tangential velocity induced by an infinite cylinder of longitudinal vorticity is:

$$\begin{aligned} u_{\psi ,l}(r)=\begin{Bmatrix}\frac{\varGamma _\text {tot}}{2\pi r}\\ 0 \end{Bmatrix}=\begin{Bmatrix}\frac{\gamma _l R}{r}\\ 0 \end{Bmatrix} \end{aligned}$$

(36.79)

where the upper value of the bracket corresponds to $r>R$ and the lower value to $r<R$ and where $\varGamma _\text {tot}\underline{e}_z=\underline{\gamma }_l 2 \pi R$ is the total circulation in a plane $z=\text {cst}$. The above result can be obtained by applying the definition of the circulation on contours centered on the z-axis in a plane $z=\text {cst}$ and using the axisymmetry of the flow.

3.2 Finite Cylinder of Longitudinal Vorticity

The Biot–Savart law in term of velocity writes:

$$\begin{aligned} \underline{u}(\varvec{x})&=\frac{\gamma _l R}{4\pi } \int _{z_1}^{z_2}\int _{0}^{2 \pi } \frac{r\underline{e}_\theta (\psi )-R\underline{e}_\theta (\theta ')}{\left[ r^2+R^2+ (z-z')^2-2 r R\cos (\theta '-\psi ) \right] ^\frac{3}{2}} \mathrm {d}{\theta '} \mathrm {d}{z'}. \end{aligned}$$

(36.80)

Using the axisymmetry of the problem, only the tangential component remains:

$$\begin{aligned} u_\theta (\varvec{x})&=\frac{\gamma _l R}{4\pi } \int _{z_1}^{z_2}\int _{0}^{2 \pi } \frac{r-R\cos \theta '}{\left[ r^2+R^2+ (z-z')^2-2 r R\cos \theta ' \right] ^\frac{3}{2}} \mathrm {d}{\theta '} \mathrm {d}{z'} . \end{aligned}$$

(36.81)

The integration over z can be readily done and would lead to an integrated form that would match the one of a semi-infinite filament and only the integral over $\theta$ would need to be computed. Yet, this step may be skipped by using the following analogy. The form taken by Eq. 36.81 directly recalls the one found in the study of the solenoid (see Eq. 36.54). The same integration steps may be followed to eventually lead to the indefinite form:

$$\begin{aligned} u_{\theta }(\underline{x})&=-\frac{\gamma _l}{4\pi }\frac{1}{\sqrt{rR}}\frac{R}{r}\left[ \zeta \mathrm {k}(\zeta )\left( K\left( \mathrm {k}^2(\zeta )\right) -\frac{R-r}{R+r}\varPi \left( \mathrm {k}_0^2,\mathrm {k}^2(\zeta )\right) \right) \right] _{\zeta _1=z-z_1}^{\zeta _2=z-z2}. \end{aligned}$$

(36.82)

where $\mathrm {k}_0=\mathrm {k}(0)$ and the function $\mathrm {k}$ is defined as in Eq. 36.71:

$$\begin{aligned} \mathrm {k}^2(\zeta )=\frac{4 r R}{(r+R)^2+\zeta ^2} \end{aligned}$$

(36.83)

3.3 Semi-infinite Cylinder of Longitudinal Vorticity

The semi-infinite cylinder is assumed to lay between the plane $z_1=0$ and $z_2\rightarrow +\infty$.

Properties The result from the infinite cylinder are applied for planes at infinity (i.e. where $z\rightarrow +\infty$). By geometrical consideration, the induced velocity from the semi-infinite cylinder in the plane $z=0$ is half the values at infinity. Hence using Eq. 36.79:

$$\begin{aligned} u_{\psi ,l}(r,z\rightarrow -\infty )=\begin{Bmatrix}0\\ 0\end{Bmatrix},\quad u_{\psi ,l}(r,0)=\begin{Bmatrix}\frac{\gamma _l R}{2r}\\ 0\end{Bmatrix},\quad u_{\psi ,l}(r,z\rightarrow +\infty )=\begin{Bmatrix}\frac{\gamma _l R}{r}\\ 0 \end{Bmatrix} , \end{aligned}$$

(36.84)

where the upper value of the bracket corresponds to $r>R$ and the lower value to $r<R$. The velocity at the vortex sheet is the average of the value on the upper and lower bracket value. The vortex sheet induces across it a velocity jump in the tangential direction of intensity $\gamma _l$ between the outer and inner part of the cylinder. This jump is half on the leading edge of the cylinder. These results may be verified from the full analytical expression derived in Sect. 36.3.2.

Velocity from the Biot–Savart law Using $z_1=0$ and the limit as $z_2\rightarrow +\infty$ in Eq. 36.82, the tangential velocity component induced by the system of semi-infinite trailed vorticity is obtained as:

$$\begin{aligned} u_{\theta }(r,z)=\frac{\gamma _l}{2}\frac{R}{r}\left[ \frac{r-R+|R-r|}{2|R-r|} +\frac{z \mathrm {k}(z)}{2 \pi \sqrt{r R}} \left( K\left( \mathrm {k}^2(z)\right) -\frac{R-r }{R+r}\varPi \left( \mathrm {k}^2(0),\mathrm {k}^2(z)\right) \right) \right] , \end{aligned}$$

(36.85)

where the function $\mathrm {k}$ is defined in Eq. 36.71. The result is consistent with the one found by Gibson¹ [8] though the procedure leading to it is different. It is also possible to obtain Eq. 36.85 by using a continuous distribution of semi-infinite vortex lines. The induced velocity for a semi-infinite vortex line is given in Eq. 31.42. The obtain result is obviously identical.

The variation of the azimuthal velocity computed with Eq. 36.85 with the axial position is shown in Fig. 36.10. The factor two between the far-wake and the rotor plane velocities is seen on the figure. As expected from Eq. 36.85, the velocity tends to 0 for all regions far from the cylinder edge and is anti-symmetric with respect to z for $r<R$.

Fig. 36.10

Tangential velocity induced by the longitudinal part of the tip vortices. Velocities at the plane $z=0$ are half the ones at “infinity”. On these planes, the velocity is 0 for $r<R$, and equal to $\varGamma _\text {tot}/4\pi r$ and $\varGamma _\text {tot}/2\pi r$ for $r>R$. The velocity tends to 0 for all regions far from the cylinder edge and is anti-symmetric with respect to z for $r<R$ [2]