2.1. Attached flow around a rigid solid body

The complex (dimensional) $z$-plane, $z = x + {\rm i}y$, where i is the imaginary unit, is fixed to the moving frame of the rigid solid body; see figure 1. The origin of this $(x,y)$ frame is centred at the edge whose starting vortex is to be calculated with its $x$-axis parallel to, and directed away from, the solid body surfaces immediately adjacent to this edge. This local Cartesian frame is oriented to form a three-dimensional right-handed coordinate system, whose third (unnamed) basis vector is orthogonal to the $z$-plane and points in the same direction at each edge, e.g. out of the page in figure 1. The exterior of the rigid solid body (white region in figure 1) in the $z$-plane is conformally mapped onto the exterior of a circle which occupies the (dimensionless) $\zeta$-plane (right-hand diagram in figure 1) with its origin at the centre of the circle. The edge under consideration at $z=0$ is mapped onto $\zeta =1$. The $z$-frame is non-dimensionalised,

(2.1)\begin{equation} z = \mathcal{L} \hat{z}, \end{equation}

where $\mathcal {L}$ is the characteristic length scale of the flow field, i.e. the size of the solid body, and $\hat {z}$ is dimensionless. Close to this edge, i.e. $|\hat {z}| \ll 1$, the conformal mapping takes the form

(2.2)\begin{equation} \zeta = 1+ c_{{1}/{2}} \hat{z}^{{1}/{2}} + c_1 \hat{z} + o \! \left(|\hat{z}| \right), \end{equation}

where the $c_{{1}/{2}}$ and $c_1$ are non-zero constants that depend on the body's geometry only. The constant $c_{{1}/{2}}$ is real by construction of the conformal map. The dimensionless velocity potential for the flow generated by the body motion, termed the ‘attached flow’, can be expressed in its Laurent series:

(2.3)\begin{equation} W_a(\zeta,T) = \sum_{n={-}\infty}^{\infty} a_n(T) \zeta^n , \end{equation}

where $T=t/\mathcal {T}$ is dimensionless time and $\mathcal {T}$ is the (convective) timescale for the flow, e.g. the size of the solid body divided by the velocity scale for its motion, and the functions of time, $a_n(T)$, are specified by the particular solid body and its motion.

Substituting (2.2) into (2.3), and expanding for small $\hat {z}$, yields

(2.4)\begin{equation} W_a(\hat{z},T) = 2 \hat{z}^{1/2} f(T) + \hat{z} g(T) + \text{higher order}, \end{equation}

giving the velocity field local to the edge under consideration,

(2.5)\begin{equation} u_a - \text{i} v_a \equiv \frac{{\partial} W_a}{{\partial} \hat{z}} = \hat{z}^{-{1}/{2}} f(T) + g(T) + \text{higher order}, \end{equation}

where $u_a$ and $v_a$ are the corresponding velocity components in the $x$- and $y$-directions, respectively, $f(T)$ and $g(T)$ are functions of time specified by motion of the body (which absorb the coefficients and time-dependent functions in (2.2) and (2.3)), and we have omitted terms in (2.4) that do not depend on $\hat {z}$ because they do not contribute to the velocity field. The form of the attached flow in (2.5) is assumed henceforth, with $f(T)$ and $g(T)$ being the quantities of primary interest in the developed theory. The function, $f(T)$, is associated with the local flow perpendicular to the straight edge, whereas $g(T)$ is parallel to the edge.

As we show in § 3, the two leading-order terms in (2.5) are as specified for several cases of practical interest. Moreover, we are unable to identify an attached flow around a physically realisable solid body of finite extent, whose local flow at one of its edges exhibits higher-order terms in (2.5) that are important. A flow in which the higher-order terms in (2.5) play a dominant role, if it were to exist in reality, may result in vortex dynamics that are not described by the present theory.

2.2. Birkhoff–Rott equation for evolution of a vortex sheet

The vortex sheet generated by (2.5) in the $z$-plane is parameterised by

(2.6)\begin{equation} \hat{z} = \hat{z}_v(\varGamma,T), \end{equation}

where the Lagrangian variable, $\varGamma$, is a dimensionless circulation (scaled by $\mathcal {L}^2/\mathcal {T}$) that gives the total amount of circulation between a point on the vortex sheet and its free end, and the subscript, $v$, refers to the vortex sheet.

The vortex sheet obeys the Birkhoff–Rott evolution equation (Rott Reference Rott1956; Birkhoff Reference Birkhoff1962; Saffman Reference Saffman1995),

(2.7)\begin{equation} \left( \frac{\partial \bar{\hat{z}}_v}{\partial T} \right)_{\varGamma} = u_a - \text{i} v_a + u_v - \text{i} v_v, \end{equation}

where the overbar indicates the complex conjugate, and $u_v - \text {i} v_v$ is the dimensionless velocity field induced by the sheet's interaction with the solid body. This self-induced velocity field guarantees zero total circulation via Kelvin's circulation theorem, and is given by (Milne-Thomson Reference Milne-Thomson1996)

(2.8)\begin{equation} u_v - \text{i} v_v ={-} \left( \frac{\partial \zeta}{\partial \hat{z}} \right)_{\hat{z}_v} \frac{1}{2 {\rm \pi}\text{i}}\, {\int\hskip -1,05em -\,}_{\varGamma_0(T)}^0 \left( \frac{1}{\zeta_v - \zeta_v'} - \frac{1}{\zeta_v -\dfrac{1}{\bar{\zeta'}_v}} \right) \mathrm{d} \varGamma', \end{equation}

where the integral denotes the Cauchy principal value, $\zeta _v \equiv \zeta ( z_v(\varGamma, T))$, $\zeta _v' \equiv \zeta ( z_v(\varGamma ', T))$, and $\varGamma _0(T)$ is the total circulation shed at time, $T$.

From (2.2) and (2.5), it follows that the evolution equation for the vortex sheet is

(2.9)\begin{equation} \left( \frac{\partial \bar{\hat{z}}_v}{\partial T} \right)_{\varGamma} = \hat{z}_v^{-{1}/{2}} f(T) + g(T) - \frac{1}{4 {\rm \pi}\text{i}} \hat{z}_v^{-{1}/{2}} {\int\hskip -1,05em -\,}_{\varGamma_0(T)}^0 \left( \frac{1}{\hat{z}_v^{1/2} - \hat{z}_v^{\prime {1/2}}} - \frac{1}{\hat{z}_v^{1/2} + \bar{\hat{z}}_v^{\prime {1/2}}}\right) \mathrm{d} \varGamma', \end{equation}

where the dash indicates dependence on $\varGamma '$ (as in (2.8)), and we have not included the ${O}(1)$ contributions from the integral in (2.8) because they are always subdominant in the vortex dynamics (as in PS21). The Kutta condition is used to regularise the solution by removing the (unphysical) velocity singularity at the sharp edge, $\hat {z}_v = 0$ (Crighton Reference Crighton1985). This captures the behaviour of a (real) viscous fluid, where viscous diffusion guarantees that the velocity field is always finite. That is, the Kutta condition requires the term of ${O}(\hat {z}_v^{-1/2})$ in (2.9) to vanish at the edge, i.e. as $\hat {z}_v \to 0^+$, which gives the auxiliary condition,

(2.10)\begin{equation} f(T) + \frac{1}{4 {\rm \pi}\text{i}} \int_{\varGamma_0(T)}^0 \left( \frac{1}{\hat{z}_v^{\prime {1/2}}} + \frac{1}{\bar{\hat{z}}_v^{\prime {1/2}}}\right)\, \mathrm{d} \varGamma'=0. \end{equation}

2.3. Self-similar evolution of the vortex sheet generated at an edge

We calculate the evolution of the vortex sheet generated at each edge of the body, immediately following start up, i.e. $T \ll 1$. Motion of the solid body is assumed to obey a power-law dependence on time. It then follows that

(2.11a,b)\begin{equation} f(T) =\text{i}\ f_0 T^k + o\left(T^k\right), \quad g(T) = g_0 T^l + o\left(T^l\right), \end{equation}

where the power laws, $k, l \geqslant 0$, are related to the power laws of the body motion, and $f_0$ and $g_0$ are nonzero (complex) constants. The imaginary unit has been added to the first equation for notational convenience; see below.

The attached flow in (2.5) is devoid of any geometric length scale, enabling the generation of a vortex sheet of similarity form (Pullin Reference Pullin1978),

(2.12a,b)\begin{equation} \hat{z}_v = T^q Z(\lambda) , \quad \lambda = 1- \frac{\varGamma}{\mathcal{J} T^s}, \end{equation}

where the real-valued non-negative power laws, $q$ and $s$, are to be determined, and $\mathcal {J}$ is a shed circulation constant defined by

(2.13)\begin{equation} \varGamma_0(T) = \mathcal{J} T^s. \end{equation}

Equations (2.11a,b) and (2.13) give $0 \leqslant \lambda \leqslant 1$, with $\lambda = 0$ and $1$ coinciding with the sharp edge of the solid body and the vortex sheet's free end, respectively, i.e.

(2.14)\begin{equation} Z(0) = 0. \end{equation}

The Kutta condition in (2.10) becomes

(2.15)\begin{equation} f_0 T^k + \frac{\mathcal{J}}{4 {\rm \pi}} T^{s-{q}/{2}} \int_0^1 \left(\frac{1}{Z^{\prime {1/2}}} + \frac{1}{\bar{Z}^{\prime {1/2}}} \right) \, \mathrm{d} \lambda' = 0, \end{equation}

and it follows that $f_0$ must be real valued. Equation (2.15) furnishes the following relation between the exponents,

(2.16)\begin{equation} s-\frac{q}{2} = k, \end{equation}

and simplifies to give

(2.17)\begin{equation} f_0 + \frac{\mathcal{J}}{4 {\rm \pi}} \int_0^1 \left(\frac{1}{Z^{\prime {1/2}}} + \frac{1}{\bar{Z}^{\prime {1/2}}} \right) \, \mathrm{d} \lambda' = 0. \end{equation}

This allows the governing equation in (2.9) to be recast as

(2.18)\begin{equation} T^{q-1} \left( q \bar{Z} + s (1-\lambda) \frac{\mathrm{d} \bar{Z}}{\mathrm{d} \lambda} \right) = \text{i}\ Z^{-{1/2}} T^{s-q}f_0 \left( 1 + M(Z) \right) + g_0 T^l, \end{equation}

where

(2.19)\begin{equation} M(Z) \equiv \frac{ \displaystyle{\int\hskip -1,05em -\,}_0^1 \left( \dfrac{1}{Z^{1/2}-Z^{\prime {1/2}}} - \dfrac{1}{Z^{1/2}+\bar{Z}^{\prime {1/2}}}\right) \mathrm{d} \lambda' }{ \displaystyle \int_0^1 \left( \dfrac{1}{Z^{\prime {1/2}}} +\dfrac{1}{\bar{Z}^{\prime {1/2}}} \right) \, \mathrm{d} \lambda' } . \end{equation}

Equation (2.18) governs the entire dynamics of the vortex sheet generated at the edge.

The form of (2.18) shows that these dynamics are controlled by the relative strength of the two terms on its right-hand side, i.e. the power laws in time, $s-q$ and $l$. This establishes the existence of three possible vortex types, which we explore in the next section. The analysis generalises the recent work of PS21, who studied the trailing-edge vortex generated by a flat plate undergoing translation and rotation, to an arbitrary body with any number of edges that is executing arbitrary power-law-in-time motion.

2.4. Three possible vortex sheet types

The first term on the right-hand side of (2.18) arises from self-induction of the vortex sheet accounting for the presence of the solid body, the Kutta condition and flow perpendicular to the edge. In contrast, the second term is due to pure translation of the attached flow parallel to the straight edge, cf. (2.5), (2.11a,b) and (2.18). This gives rise to three separate vortex sheet types, depending on whether these terms balance or individually dominate the right-hand side of (2.18).

2.4.1. Type-I vortex sheet

Consider when the first term on the right-hand side of (2.18) dominates the second term, i.e.

(2.20)\begin{equation} l> s-q. \end{equation}

Balancing the left-hand side of (2.18) with its right-hand side, and using (2.16), produces

(2.21a,b)\begin{equation} s = \frac{4 k +1}{3}, \quad q = \frac{2k+2}{3}. \end{equation}

Equation (2.20) then gives the corresponding condition on the power laws for the attached flow,

(2.22)\begin{equation} l >\frac{2k-1}{3}, \end{equation}

with the governing equation (2.18) becoming

(2.23)\begin{equation} \frac{2k+2}{3} \bar{Z} + \frac{4k+1}{3} (1-\lambda) \frac{\mathrm{d} \bar{Z}}{\mathrm{d} \lambda} = \text{i}\ Z^{-{1/2}} f_0 \left( 1+ M(Z) \right), \end{equation}

which is to be solved subject to its initial condition (2.14); this can be performed numerically, as per PS21.

The dimensionless velocity jump across the vortex sheet, at any spatial position parameterised by $\lambda$, is

(2.24)\begin{equation} \gamma = \left.\left\lvert \frac{\mathrm{d} \varGamma}{\mathrm{d} \lambda}\right\rvert \right/ \left\lvert \frac{\mathrm{d} \hat{z}_v}{\mathrm{d} \lambda}\right\rvert = \mathcal{J} T^{({2k-1})/{3}} {\left\lvert \frac{\mathrm{d} Z}{\mathrm{d} \lambda} \right\rvert^{{-}1}}, \end{equation}

which is singular as $T \to 0$, for $0\leqslant k<1/2$, i.e. the velocity jump decays as time evolves. This is in contrast to $k> 1/2$ where the velocity jump is zero initially and grows monotonically, or it is independent of time and finite for $k=1/2$. Note that $k=0$ corresponds to an impulsive start-up flow, with increasing values of $k$ providing a more gradual acceleration at early times. Physically, $0\leqslant k<1/2$ produces a strong concentration of vorticity at the plate edge which may be expected for an impulsive (or near impulsive) start-up flow. The situation resolves for $k \geqslant 1/2$.

2.4.2. Type-II vortex sheet

Next, we consider the complementary case where the second term dominates the first in the right-hand side of (2.18), i.e.

(2.25)\begin{equation} l <\frac{2k-1}{3}. \end{equation}

Balancing the left- and right-hand sides of (2.18), and using (2.16), then gives

(2.26a,b)\begin{equation} s = k+\frac{l+1}{2}, \quad q = 1+l. \end{equation}

The canonical function, $f(T)$, in (2.5) for the attached velocity field is subdominant in this regime, and so the condition of no-penetration on the plate becomes

(2.27)\begin{equation} {\rm Im} \left\{g(T) \right\}=0, \end{equation}

where Im denotes the imaginary component, i.e. $g_0$ in (2.11a,b) is real valued.

The governing equation for $Z$ becomes

(2.28)\begin{equation} (1+l) \bar{Z} + \left(k + \frac{l+1}{2} \right) (1-\lambda) \frac{\mathrm{d} \bar{Z}}{\mathrm{d} \lambda} = g_0. \end{equation}

The corresponding initial condition for $Z$, and the Kutta condition, remain unchanged from (2.14) and (2.17), respectively.

This has exact analytical solution (cf. PS21),

(2.29)\begin{equation} Z =\frac{g_0}{1+l} \left(1-(1-\lambda)^B\right), \end{equation}

where

(2.30)\begin{equation} B = \frac{2(1+l)}{1+l+2k}<1, \end{equation}

Importantly, this type-II vortex sheet, which is a straight line geometrically, cannot be coincident with the solid edge under consideration. This condition imposes the constraint, $g_0>0$, which is illustrated in § 3.

The Kutta condition (2.17) gives

(2.31)\begin{equation} \mathcal{J} ={-}2f_0 g_0^{1/2} \sqrt{\frac{\rm \pi}{1+l}} \frac{\varGamma_c \left(1+ \dfrac{k}{1+l} \right)}{\varGamma_c \left(\dfrac{3}{2}+ \dfrac{k}{1+l} \right)}, \end{equation}

where $\varGamma _c({\cdot })$ is the gamma function (not circulation).

The corresponding velocity jump across the vortex sheet is

(2.32)\begin{equation} {\gamma = T^{k-({1+l})/{2}}\frac{\mathcal{J}}{g_0} \left(k+\frac{1+l}{2} \right) (1-\lambda)^{1-B}}, \end{equation}

which can be expressed in terms of the spatial $X$-coordinate, where $Z=X+ \text {i} Y$, using (2.11a,b), to give

(2.33)\begin{equation} {\gamma = T^{k-({1+l})/{2}} \frac{\mathcal{J}}{g_0} \left(k+\frac{1+l}{2} \right) \left(1- \frac{1+l}{g_0} X \right)^{(1-B)/B}}. \end{equation}

2.4.3. Type-III vortex sheet

The final vortex type occurs when all terms on the right-hand side of (2.18) balance, i.e.

(2.34)\begin{equation} l =\frac{2k-1}{3}, \end{equation}

for which the governing equation (2.18) becomes

(2.35)\begin{equation} \frac{2 k+2}{3} \bar{Z} + \frac{4k+1}{3} (1-\lambda) \frac{\mathrm{d} \bar{Z}}{\mathrm{d} \lambda} = \text{i}\ Z^{-{1/2}} f_0 \left( 1+ M(Z) \right) + g_0, \end{equation}

whose form of solution now depends qualitatively on $f_0$ and $g_0$; this property is absent from type-I and type-II vortex sheets. Type-III vortices resemble type-I or type-II vortices in the cases that $f_0 \gg g_0$ or $g_0 \gg f_0$, respectively.

The velocity jump across the vortex sheet is (cf. (2.24))

(2.36)\begin{equation} \gamma =\mathcal{J} T^{(2k-1)/3} {\left\lvert \frac{\mathrm{d} Z}{\mathrm{d} \lambda} \right\rvert^{{-}1}}, \end{equation}

which is always finite because $(2k-1)/3=l \geqslant 0$, i.e. $k\geqslant 1/2$.

In summary, the three vortex types originally reported by PS21 constitute all possible vortex types generated at a sharp and straight edge of a solid body, provided (2.5) holds. A key implication is that starting vortices in many seemingly different configurations are controlled by the same physical mechanisms. This finding is explored in the sections that follow.

3.1. Trailing-edge vortex of a translating and rotating flat plate

We first demonstrate application of the general theory reported in § 2 to the sudden translation and rotation of a flat plate in an unbounded and quiescent fluid (PS21); see figure 2(a). The required conformal mapping of the flat plate in $z$-space to the unit circle in $\zeta$-space is given in (2.3c) of PS21.

Figure 2.

Starting vortices generated by a flat plate that is translating with velocity, $U(T)$, in a fixed direction in the laboratory frame, with an instantaneous angle of attack, $\alpha (T)$, and an instantaneous angular velocity, $\varOmega (T)$, about a pivot point in the plane of the plate. (a) Trailing-edge vortex as studied by PS21; see § 3.1. (b) Leading- and trailing-edge vortices of the same plate motion; see § 3.2. (c) Leading- and trailing-edge vortices when the plate has an additional translational velocity component, $V(T)$, perpendicular to $U(T)$; see § 3.3.

The plate is of length, $2 a$, and a global $(\tilde {x}, \tilde {y})$-Cartesian frame is referenced to the moving plate, which is aligned along $\tilde {y}=0$ and occupies the region, $-a(1+d)\leqslant \tilde {x} \leqslant a(1-d)$. Note that this global frame differs from the $(x, y)$-Cartesian frame local to each edge; see § 2.1. The rotation pivot position is (i) at the origin of the $(\tilde {x}, \tilde {y})$-frame, and (ii) displaced a (signed) distance of $a d$ from the plate centre. For example, $d = 1/2$ corresponds to a rotation pivot position at a distance of $a/2$ to the right of the plate centre in figure 2, which is the three-quarter-chord position; whereas $d=-1/2$ is the quarter-chord position. The plate moves with the (dimensionless) translational speed, $U(T)= T^m$, and rotates with the (dimensionless) angular velocity $\varOmega (T) = \beta T^p$, where $m, p \geqslant 0$ and $\beta$ is a real-valued constant. The instantaneous angle of attack of the plate is $\alpha (T) = \alpha _0 +\beta T^{1+p}/(1+p)$, where $\alpha _0$ is a real-valued constant. We have used the length scale $\mathcal {L}=a$ and timescale $\mathcal {T}=a/U_0$ for a characteristic translational velocity of $U_0$. In summary, we have

(3.1a{–}c)\begin{equation} U(T)= T^m, \quad \varOmega(T) = \beta T^p , \quad \alpha(T) = \alpha_0 + \frac{\beta}{1+p}T^{1+p}. \end{equation}

The velocity field of the attached flow, local to the trailing edge, is ((2.9) of PS21)

(3.2)\begin{equation} u_a - {\rm i} v_a ={-} \frac{\text{i}}{\sqrt{2}} \, \hat{z}^{-{1/2}} \left(\left[{\dfrac{1}{2}}-d\right] \varOmega +U \sin \alpha \right) + U \cos \alpha + o(1), \end{equation}

which aligns with the assumed form in (2.5), giving

(3.3a,b)\begin{equation} f(T) ={-}\frac{\text{i}}{\sqrt{2}}\left( \left[{\frac{1}{2}}-d\right] \varOmega +U \sin \alpha \right), \quad g(T) = U \cos \alpha. \end{equation}

Note that (2.4) of PS21 separates out a rigid-body-rotation term in the complex potential, $W_a$, and thus differs from the present definition in (2.4). This difference does not affect the overall velocity field, which is the transport quantity of primary interest in the present general theory; see above and (2.9) of PS21.

All results reported by PS21 can be recovered using the general theory in § 2 with (3.3a,b). For example, in the case that $\alpha _0=0$ and $d \neq 1/2$, we find

(3.4a{–}d)\begin{equation} k=p, \quad l=m , \quad f_0 ={-} \frac{\hat{\beta}}{\sqrt{2}}, \quad g_0 =1, \end{equation}

where

(3.5)\begin{equation} \hat{\beta} = \beta \left(\tfrac{1}{2}-d\right). \end{equation}

3.1.1. Vanishing plate rotation and degenerate vortex sheets

We now clarify a feature of this flow reported in § 5.6 of PS21. For non-zero initial angle of attack, $\alpha _0 \neq 0$, this previous study showed that the starting vortex for $\beta \to 0^+$ can differ from identically zero rotation, $\beta =0$ (which was studied by Pullin (Reference Pullin1978)). It was suggested that this difference might be resolved by the inclusion of viscosity. Here, we show that it is a natural consequence of inviscid flow dynamics alone.

Importantly, PS21 took $T\to 0$ before $\beta \to 0^+$, whereas Pullin (Reference Pullin1978) set $\beta = 0$ (no rotation) and then took $T \to 0$. These are complementary, but fundamentally different, limits were reported to produce different results:

1. (i) $(m=1, p < 1, d\neq 1/2)$: Type-I vortices with different growth rates in time, which for non-zero rotation $\beta > 0$, depend on the time power law of angular rotation, $p$; and

2. (ii) $(m=0, p \geqslant 0)$: Identical type-I vortices that are independent of $p$.

This behaviour is now explored through the form of $f(T)$ from the general theory in § 2, and is relevant to understanding the evolution of the starting vortex.

For non-zero initial angle of attack, $\alpha _0 \neq 0$, the vortex dynamics are dominated by $f(T)$; see (2.11a,b), (2.18) and (3.3a,b). From (3.3a,b), it then follows that the vortex dynamics depend on the plate rotation if and only if

(3.6a,b)\begin{equation} {O} \! \left(T^m |\sin \alpha_0| \right) < {O} \! \left( T^p \hat\beta \right) \quad \text{and} \quad d \neq 1/2. \end{equation}

We explore finding (i) in detail.

For $m=1$ and $p<1$, i.e. $m > p$, we observe that (3.6a,b) is satisfied for small time, i.e. the vortex dynamics will depend on plate rotation. Rewriting (3.6a,b) as

(3.7)\begin{equation} T^{m-p} \ll \frac{\hat{\beta}}{|\sin \alpha_0|}, \end{equation}

and choosing a multiplicative factor, $\delta \ll 1$ for this inequality to hold, gives the upper bound in time,

(3.8)\begin{equation} T < T_{max} \equiv \left\vert \frac{\delta \hat{\beta}}{\sin \alpha_0} \right\vert^{{1}/({m-p})}, \end{equation}

for which plate rotation affects the vortex dynamics.

This shows that as $\hat \beta \to 0^+$, a type-I vortex depending on plate rotation arises at $T = 0^+$ which quickly (immediately) morphs into a vortex whose dynamics are independent of plate rotation and coincide with Pullin (Reference Pullin1978). That is, a boundary layer in time exists. This boundary layer is independent of viscosity and can, in principle, be discerned from the present inviscid formulation using the time scale in (3.7).

The above analysis is straightforward to adapt to other settings. Explanation of finding (ii) for the complementary case of $m \leqslant p$ is provided in PS21, is not in contention and, hence, omitted here. The difference in dynamics for $m \geqslant p$ and $m < p$, including the above-described vortex morphing, should be observable in experiment.

3.2. Starting vortex at the leading edge of a translating and rotating flat plate

Next, we examine the starting vortex at the leading edge of the same plate as § 3.1; as per the illustrated left-hand vortex in figure 2(b). This starting vortex configuration has not been reported previously.

By considering a Cartesian frame local to the leading edge of the plate, as shown in figure 2(b), and expanding the attached flow in (2.4) of PS21 about this leading edge, we obtain

(3.9)\begin{equation} u_a - {\rm i}v_a ={-} \frac{\text{i}}{\sqrt{2}} \, \hat{z}^{-{1}/{2}} \left(\left[{\dfrac{1}{2}}+d\right] \varOmega -U \sin \alpha \right) - U \cos \alpha + o(1), \end{equation}

which upon comparison with the canonical form in (2.5), gives

(3.10a,b)\begin{equation} f(T) ={-}\frac{\text{i}}{\sqrt{2}}\left( \left[{\dfrac{1}{2}}+d\right] \varOmega -U \sin \alpha \right), \quad g(T) ={-}U \cos \alpha. \end{equation}

Using the power-law-in-time motion described in § 3.1, these become

(3.11a,b)\begin{equation} f(T) ={-}\frac{\text{i}}{\sqrt{2}}\left[ \tilde{\beta} \, T^p - T^m \sin \alpha(T) \right], \quad g(T) ={-}T^m \cos \alpha(T), \end{equation}

where

(3.12)\begin{equation} \tilde{\beta}\equiv \beta \left(\tfrac{1}{2} +d \right). \end{equation}

The form of $g(T)$ in (3.10a,b) gives $g_0<0$. This establishes that a type-II vortex described in § 2.4.2 cannot exist at the leading edge of the plate; the vortex sheet would lie along the plate itself, which is not allowed.

As an example, we consider the special case of $\alpha _0=0$ and $d \neq - 1/2$, for which the general theory in § 2 yields

(3.13a{–}d)\begin{equation} k=p, \quad l=m , \quad f_0 ={-} \frac{\tilde{\beta}}{\sqrt{2}}, \quad g_0 ={-}1. \end{equation}

Substituting (3.13a{–}d) into (2.23) for type-I vortices, and (2.35) for type III, gives the respective governing equations for these leading-edge vortices. Namely, for type I we obtain

(3.14)\begin{equation} \frac{2p+2}{3} \bar{Z} + \frac{4p+1}{3} (1-\lambda) \frac{\mathrm{d} \bar{Z}}{\mathrm{d} \lambda} ={-}\frac{\tilde{\beta}}{\sqrt{2}} \text{i}\ Z^{-{1/2}} \left( 1+ M(Z) \right), \end{equation}

whereas for type III we have

(3.15)\begin{equation} \frac{2 p+2}{3} \bar{Z} + \frac{4p+1}{3} (1-\lambda) \frac{\mathrm{d} \bar{Z}}{\mathrm{d} \lambda} ={-}\frac{\tilde{\beta}}{\sqrt{2}} \text{i}\ Z^{-{1/2}} \left( 1+ M(Z) \right) -1, \end{equation}

where the function $M(Z)$ is defined in (2.19). This example highlights the utility of the general theory in § 2, which facilitates calculation of the starting vortex at any straight edge of a solid body. All that is required are the constants, $f_0$ and $g_0$, which immediately follow from $f(T)$ and $g(T)$ in the attached flow local to the particular edge.

The special case of vanishing rotation furnishes the following expressions at the leading edge,

(3.16a,b)\begin{equation} f_0 = \frac{\sin \alpha_0}{\sqrt{2}}, \quad g_0 ={-} \cos \alpha_0, \end{equation}

with the corresponding results at the trailing edge given by a sign change in $f_0$ and $g_0$. At both edges, the dominant power law is $k=l=m$. For $0<\alpha _0< {\rm \pi}/2$, the trailing- and leading-edge vortex sheets are both of type I, but have different shapes. For $\alpha _0=0$, there is no leading-edge vortex whereas the trailing-edge vortex sheet is of type II. For normal incidence, i.e. $\alpha _0={\rm \pi} /2$, the vortex sheets are of type I and identical in shape, while being mirror images about the normal direction to the plate surface.

A summary of results for the leading-edge vortex sheets, and their respective regimes of validity, can be calculated in an identical manner to that performed in PS21 for the trailing-edge vortex. These results for the leading-edge vortex of a flat plate are summarised in table 1; see also the complementary table 1 of PS21.

Table 1.

Summary of vortex sheets arising from the leading edge of a flat plate. Results are given as a function of the plate translation and rotation parameters, defined in (3.1a{–}c) and (3.12). Time power laws, $q \geqslant 0$ and $s \geqslant 0$, of their similarity solutions are as defined in (4.1a,b) of PS21, with the local flow near the leading edge of the plate given in (3.9). This table complements table 1 of PS21 that reports results for the trailing-edge vortex. The dominant component of the local attached flow to the edge, that drives the vortex sheet, is indicated in the right-most column.

Comparison of the results in table 1 of PS21 and table 1 establishes that the starting vortices at the trailing and leading edges for $\alpha _0=0$ can differ in type, provided that $d = \pm 1/2$, i.e. rotation occurs at either the quarter- or three-quarter-chord position of the plate. This is because the effect of plate rotation, i.e. $\varOmega (T)$, on the perpendicular component of local flow, $f(T)$, vanishes at one of the plate edges but not the other; the term $\varOmega (T)$ is multiplied by $1/2-d$ and $1/2+d$ for the trailing and leading edges, respectively. This feature is quantified in the phase diagram reported in figure 3 for $d=1/2$. It shows the pairs of vortex types that arise at the leading and trailing edges, as a function of the plate translation and rotation power laws, $m$ and $p$, respectively. Combinations of types I and II, I and III, and II and III are all possible.

Figure 3.

Phase diagram of the vortex sheet pairs that simultaneously arise at the leading edge (LE) and trailing edge (TE) of a flat plate, for $\alpha _0 = 0$ and rotation about the three-quarter-chord position, $d=1/2$. Results are given as a function of the plate translation and rotation power-laws, $m$ and $p$, respectively, defined in (3.1a{–}c). The data are derived from tables 1 of PS21 and table 1; an analogous phase diagram arises for rotation about the quarter-chord position.

This feature is now illustrated using three examples for zero initial angle of attack, ${\alpha _0 = 0}$, for which type-III vortices can arise. Integro-differential equations for the type-III vortex sheets are solved numerically using the method reported in § 5.1.2 of PS21. Results are given for translation and rotation power laws of $m=1$ and $p=2$, respectively, and a dimensionless angular velocity, $\beta = \sqrt {2}$. These results highlight the effect of varying the rotation pivot position, $d$.

Example 3.1 Figure 4(a) gives the starting vortices generated at the edges of a flat plate that is undergoing translation and rotation about its centre, i.e. $d=0$. Type-III vortices are generated at both edges; as per table 1 of PS21 and table 1. Despite the same vortex type, the shape of the vortices at the leading and trailing edges differ. The leading-edge vortex is strongly affected by its proximity to the flat plate, rolling up as it approaches the flat plate, whereas the trailing-edge vortex is swept downstream and, thus, less affected. This difference in behaviour is generated mathematically by the sign change in $g_0$ for the trailing and leading edges, cf. (5.19) of PS21 and (3.15).

Figure 4.

Leading- and trailing-edge vortices (green lines) generated by a flat plate (black lines) that is translating and rotating with time power laws, $m=1$ and $p=2$, respectively. A dimensionless angular velocity of $\beta = \sqrt {2}$ and a convective time of $T=1/2$ are used. (a) Rotation about the plate centre, $d=0$. (b) Rotation about the three-quarter-chord position, $d=1/2$. (c) Rotation about the quarter-chord position, $d=-1/2$. The central region of the flat plate is not shown to accentuate the structure of the starting vortices at its edges; leading edges (LE) and trailing edges (TE) are indicated. The plate chord is $c = 2 a$ and all plots are referenced to this scale as indicated in (a). Type-III vortex sheets computed using the numerical method reported in § 5.1.2 of PS21.

Example 3.2 The configuration used in example 3.1 is now altered to rotate the plate about the three-quarter-chord position, $d = 1/2$. Figure 3 shows that a type-III vortex is generated at the leading edge, whereas a type-II vortex arises at the trailing edge, with the following dominant terms in $f(T)$ and $g(T)$ (see (3.1a{–}c) and (3.3a,b)):

(3.17a,b)\begin{equation} {f(T) ={-}\frac{\text{i}\ T^4 }{3}, \quad g(T) = T,} \end{equation}

giving $f_0=-1/3$, $g_0=1$, $l=1$ and $k=4$. The self-similar solution of the trailing-edge vortex sheet is given by (see (2.11a,b), (2.29) and (2.31)),

(3.18a{–}d)\begin{equation} \hat{z}_v =T^2 Z(\lambda), \quad \lambda = 1- \frac{\varGamma}{\mathcal{J}\,T^5}, \quad Z = \frac{1}{2} \left(1- (1-\lambda)^{2/5} \right), \quad {\mathcal{J}=\frac{16\sqrt{2}}{45}}. \end{equation}

These vortices are shown in figure 4(b) and highlight the difference in dynamics that can occur by simply changing the plate rotation position. Note that the starting vortices are self-similar in time, $T$, and thus no extra information is discerned by changing the time. Changing the dimensionless angular velocity, $\beta$, alters the shape of the type-III leading-edge vortex, whereas the type-II vortex remains unchanged at the trailing edge; as per the nature of these different vortices (data not shown).

3.3. Three independent components of the plate motion

We now address an important and unresolved question: which velocity components of a flat plate independently control its starting vortices?

To achieve this goal, we generalise the problem studied by PS21 to include a translational velocity component, $V(T)$, that acts perpendicular to $U(T)$; see figure 2(c). The velocity field local to the trailing edge becomes

(3.19)\begin{equation} u_a - {\rm i} v_a ={-} \frac{\text{i}}{\sqrt{2}} \, \hat{z}^{-{1/2}} \left(\left[{\frac{1}{2}}-d\right] \varOmega +U \sin \alpha + V \cos \alpha \right) + U \cos \alpha - V \sin \alpha + o(1). \end{equation}

The functions, $f$ and $g$, then take the form,

(3.20a,b)\begin{equation} f(T) ={-}\frac{\text{i}}{\sqrt{2}}\left( \left[{\frac{1}{2}}-d\right] \varOmega +U \sin \alpha + V \cos \alpha \right), \quad g(T) = U \cos \alpha - V \sin \alpha, \end{equation}

which reduce to the canonical forms in (3.3a,b) (with only one translational velocity component) by defining

(3.21a,b)\begin{equation} \hat{U}(T) = \sqrt{U^2 + V^2}, \quad \hat{\alpha}(T) =\alpha+\tan^{{-}1} \left(\frac{V}{U} \right). \end{equation}

This finding proves that there are only two independent components of the plate motion that control the trailing-edge starting vortex. An identical finding holds for the leading-edge vortex by virtue of § 3.2. Consequently, the analysis of PS21 together with the results of § 3.2 encompass the general case, including the starting vortices generated by a plate undergoing pure translational motion. The corollary is that pure translational motion of the plate edge can and does recover the starting vortices reported by PS21, without any plate rotation.

As an example, consider the special case of pure rotation about the plate's trailing edge, i.e. $U=V=0$ and $d=1$. The functions defined in (3.20a,b) become

(3.22a,b)\begin{equation} f(T) = \frac{\text{i}}{2^{3/2}} \varOmega(T), \quad g(T) =0, \end{equation}

where $\varOmega (T) = \beta T^p$, and so a type-I vortex sheet arises. The starting dynamics of this vortex can be identically reproduced by pure translation of the edge, with $\alpha _0=-{\rm \pi} /2$, $U=\beta T^p /2$ and $V=0$, i.e. the plate edge translates normal to its axis.

Together with the above analysis, this establishes that (i) pure translation of an edge or (ii) a combination of translation and rotation of the same edge, are locally equivalent with respect to the starting vortex, and can be mapped onto each other.

This finding pertains only to the flow local to any particular edge. Namely, the (local) starting vortex generated at a single edge of a translating and rotating solid body can always be reproduced by pure translation of the same edge with two velocity components. However, these two velocity components can vary across the different edges of a solid body. This was highlighted in § 3.2, where we showed that the vortices generated at the edges of a flat plate can differ in type, which cannot be reproduced by pure translational motion of the same flat plate. This establishes a fundamental distinction between equivalent local motion of an edge and global motion of the solid body.

3.4. Relationship between Cortelezzi & Leonard (Reference Cortelezzi and Leonard1993) and PS21

Equation (3.20a,b) and the discussion in the previous section now establish that the starting vortex problems studied by Cortelezzi & Leonard (Reference Cortelezzi and Leonard1993) and PS21, while seemingly different, are in fact related. Cortelezzi & Leonard (Reference Cortelezzi and Leonard1993) studied vortex generation by the edge of a semi-infinite plate moving in its plane, with an imposed normal flow. PS21 calculated the trailing-edge vortices of a finite plate undergoing simultaneous translation and rotation.

To make the equivalence between these apparently different studies transparent, we examine the structure of the attached flow local to the plate edge in Cortelezzi & Leonard (Reference Cortelezzi and Leonard1993). The semi-infinite plate studied by Cortelezzi & Leonard (Reference Cortelezzi and Leonard1993) lies along the negative $y$-axis, its Cartesian frame is fixed to the plate with the impinging flow in the $x$-direction; its origin is at the edge. In the notation of Cortelezzi & Leonard (Reference Cortelezzi and Leonard1993), the fluid moves relative to the plate with velocity components, $U_f(T)$ and $-u_p(T)$, in the directions perpendicular and parallel to the plate, respectively. Note that the symbol, $U$, appears in (2) of Cortelezzi & Leonard (Reference Cortelezzi and Leonard1993), which is replaced by $U_f$ here to avoid confusion with the definition in (3.1a{–}c). As per § 2, we orient the $x$-direction parallel to the plate edge: accordingly, we rotate the Cartesian frame used in Cortelezzi & Leonard (Reference Cortelezzi and Leonard1993) by ${\rm \pi} /2$. The velocity field for the attached flow is then (see (2) of Cortelezzi & Leonard Reference Cortelezzi and Leonard1993)

(3.23)\begin{equation} u_a - {\rm i} v_a = \frac{\text{i}}{2} U_f(T) \hat{z}^{-{1/2}} - u_p(T), \end{equation}

with

(3.24a,b)\begin{equation} U_f(T) ={-} \mathcal{V} T^m, \quad u_p(T) ={-} T^n, \end{equation}

where $m$ and $n$ are power laws as per Cortelezzi & Leonard (Reference Cortelezzi and Leonard1993) and $\mathcal {V}$ has been scaled such that it is the ratio of the (dimensional) characteristic velocity perpendicular and parallel to the plate. From (3.24a,b), it follows that the canonical functions, $f$ and $g$, defined in (2.5) take the form

(3.25a,b)\begin{equation} f(T) ={-} \frac{\text{i}}{2} \mathcal{V} T^m, \quad g(T) = T^n, \end{equation}

and, hence, (2.11a,b) gives

(3.26a,b)\begin{equation} f_0 ={-} \tfrac{1}{2} \mathcal{V}, \quad g_0 = 1. \end{equation}

These results together with those provided in (3.3a,b) (of identical form) define the mapping between the starting vortices studied by Cortelezzi & Leonard (Reference Cortelezzi and Leonard1993) and PS21:

(3.27a,b)\begin{equation} (m, p)_{PS} \mapsto (n, m)_{CL}, \quad \hat{\beta}_{PS} \mapsto \tfrac{1}{\sqrt{2}} \mathcal{V}_{CL}, \end{equation}

where the subscripts PS and CL refer to symbols used in PS21 and Cortelezzi & Leonard (Reference Cortelezzi and Leonard1993), respectively.

For example, the critical line for type-III vortices reported by PS21 in the case that $\alpha _0 = 0$ and $d\neq 1/2$, is

(3.28)\begin{equation} m = \tfrac{1}{3} (2 p -1), \end{equation}

whereas the critical line reported by Cortelezzi & Leonard (Reference Cortelezzi and Leonard1993) is

(3.29)\begin{equation} 3 (n+1) - 2 (m+1) = 0. \end{equation}

Equation (3.29) is identical to (3.28) under the mapping defined in (3.27a,b).

The attached flow components, $f(T) \sim U_f(T)$ and $g(T) \sim u_p(T)$, are associated with flow perpendicular and parallel to the straight edge, respectively (Cortelezzi & Leonard Reference Cortelezzi and Leonard1993). A type-I vortex arises when $f(T)$ dominates, whereas a type-II vortex occurs when $g(T)$ dominates; a type-III vortex exists when they are in balance.

3.5. Joukowski aerofoil

Finally, we examine a solid body that differs from the flat plate considered in the previous sections and in the majority of literature reports on the starting vortex. A Joukowski aerofoil is defined in the $z$-plane by

(3.30a,b)\begin{equation} z= \frac{1}{2}\left( \zeta-1+ \frac{{\rm e}^{2\text{i} \gamma}}{\zeta-1+{\rm e}^{\text{i} \gamma}}-{\rm e}^{\text{i} \gamma} \right) {\rm e}^{\text{i} \gamma}, \quad \zeta = 1 - R + R {\rm e}^{\text{i} \theta} , \end{equation}

where $\gamma \in [-{\rm \pi}, {\rm \pi})$ defines the camber of the aerofoil in the $z$-plane, $R$ is the radius of a circle centred at $\zeta = 1-R$ in the (conformally mapped) $\zeta$-plane defining the thickness of the aerofoil, and $\theta \in [-{\rm \pi}, {\rm \pi})$ is the polar angle to the real axis of the $\zeta$-plane that parametrises the circle; see figure 5. The trailing edge of the aerofoil is positioned at $z=0$ and is parallel to the real (horizontal) axis of the $z$-plane; see figure 5. The aerofoil is stationary in the $z$-plane.

Figure 5.

Sample Joukowski aerofoils under the mapping (3.30a,b). Aerofoils in the physical $z$-plane with different thickness parameter, $R=1$ and $1.5$, for increasing camber parameter, $\gamma =n {\rm \pi}/16$, where $n=0,1,2,3$. Corresponding circles in the (conformally mapped) $\zeta$-plane are also illustrated.

The symmetric Joukowski aerofoil, i.e. $\gamma = 0$, has its leading edge at

(3.31)\begin{equation} {z={-}\frac{2R^2}{2R-1},} \end{equation}

giving a dimensionless chord length, $\ell$, of

(3.32)\begin{equation} \ell=\frac{2R^2}{2R-1}. \end{equation}

Equation (3.32) recovers the dimensionless chord length of the flat plate considered in § 3.1, i.e. $\ell = 2$ when $R = 1$ and $\gamma =0$; see figure 5. Thus, results for the symmetric Joukowski aerofoil in this section can be systematically compared to those for the flat plate in § 3.1, as $R$ is varied from unity.

For an imposed uniform flow of (i) velocity, $U(T)$, at fixed angle of attack, $\alpha _0$, to the real axis of the $z$-plane, and (ii) velocity, $V(T)$, perpendicular to the direction of $U(T)$, the complex potential of the attached flow generated by the aerofoil in the $\zeta$-plane is (Batchelor Reference Batchelor1967)

(3.33)\begin{align} W_a = \frac{1}{2} \left( (U- \text{i} V) (\zeta-1+R) \exp({\text{i}(\gamma- \alpha_0)}) + (U+ \text{i} V) \frac{R^2 \exp({\text{i}(\alpha_0- \gamma)})}{\zeta-1+R} \right) , \end{align}

whose corresponding velocity field is $u_a - \text {i} v_a \equiv {\partial } W_a / {\partial } z$. Note that the ‘effective’ angle of attack is $\alpha _0- \gamma$, as required (Batchelor Reference Batchelor1967, chap. 6.7).

We consider the trailing edge of the aerofoil (at $z=0$), and expand its attached velocity field locally to this edge; the characteristic length scale for the flow is taken to be unity, i.e. $\mathcal {L}=1$. This gives

(3.34)\begin{align} u_a - \text{i} v_a &={-}\frac{\text{i}}{\sqrt{2}} \hat{z}^{-{1/2}} \left(U \sin(\alpha_0-\gamma)+ V \cos(\alpha_0 - \gamma) \right)\nonumber\\ &\quad + \frac{U}{2} \left( \exp({-\text{i} \alpha_0}) - \exp({\text{i} (\alpha_0-2\gamma)}) + \frac{2}{R} \exp({\text{i} (\alpha_0-\gamma)}) \right) \nonumber\\ & \quad+ \frac{ \text{i} V}{2} \left(-\exp({-\text{i} \alpha_0})-\exp({\text{i} (\alpha_0-2\gamma)}) + \frac{2}{R} \exp({\text{i} (\alpha_0-\gamma)}) \right) + \text{higher order}, \end{align}

which, as required, reduces to the attached flow past a flat plate when $R=1$ and $\gamma =0$. Equations (2.5) and (3.34) then give

(3.35a)\begin{align} f(T) &={-}\frac{\text{i}}{2^{1/2}}\left( U \sin(\alpha_0-\gamma) + V \cos(\alpha_0-\gamma) \right), \end{align}

(3.35b)\begin{align} g(T) &= \frac{U}{2}\left(\exp({-\text{i} \alpha_0}) - \exp({\text{i} (\alpha_0-2\gamma)}) + \frac{2}{R} \exp({\text{i} (\alpha_0-\gamma)}) \right)\nonumber\\ &\quad + \frac{{\text{i}} V}{2}\left(-\exp({-\text{i} \alpha_0})-\exp({\text{i} (\alpha_0-2\gamma)}) + \frac{2}{R} \exp({\text{i} (\alpha_0-\gamma)}) \right). \end{align}

As before, the imposed flow follows independent power laws in time,

(3.36a,b)\begin{equation} U=\mathcal{V} T^{m}, \quad V = \mathcal{W} T^{n}, \end{equation}

where $\mathcal {V}$ and $\mathcal {W}$ are any real constants and the constant power laws, ${m,n} \geqslant 0$.

When $\alpha _0-\gamma \neq 0, {\rm \pi}/2, {\rm \pi}$, the $f(T)$ term always dominates $g(T)$ in the governing Birkhoff–Rott equation and its associated Kutta condition, i.e. a type-I vortex arises. This case is associated with the dominance of perpendicular flow at the trailing edge. The dominant contribution to $f(T)$, as defined in (2.11a,b), then follows

(3.37a,b)\begin{align} f_0 ={-}\frac{1}{\sqrt{2}} \left\{\!\! \begin{array}{ll} \mathcal{V} \sin(\alpha_0-\gamma) , & {m}<{n} \\ \mathcal{W} \cos(\alpha_0-\gamma), & {m}>{n} \\ \mathcal{V} \sin(\alpha_0-\gamma)+ \mathcal{W} \cos(\alpha_0-\gamma), & {m}={n}\end{array} \right. , \quad k = \left\{\!\! \begin{array}{ll} {m} , & {m}<{n} \\ {n}, & {m}>{n}\\ {m}, & {m}={n}\end{array} \right. . \end{align}

For the special case of $\alpha _0=\gamma$, it is possible for a type-II or type-III vortex sheet to arise because the parallel component of flow at the trailing edge can be dominant. For this case, (3.37a,b) gives

(3.38a,b)\begin{equation} f_0 ={-}\frac{\mathcal{W}}{\sqrt{2}}, \quad k={n}, \end{equation}

whereas (3.35b) reduces to

(3.39)\begin{equation} g(T) = \frac{\mathcal{V} T^{m}}{R} + \frac{{\text{i}} \mathcal{W} T^{n}}{R}\left(1-R \exp(- {\rm i} \alpha_0)\right). \end{equation}

It is evident from the forms of these equations that the trailing-edge vortex exhibits identical types to those of the flat plate with zero initial angle of attack, as the power laws ${m}$ and ${n}$ are varied. The similarity solution for the trailing-edge vortex of the Joukowski aerofoil easily follows from the general formulae in § 2; this is left as an exercise for the reader.

Next, we consider three example vortex sheets generated at the trailing edge of a Joukowski aerofoil.

3.5.1. Case A: symmetric Joukowski aerofoil with type-II vortex sheet

First, consider a symmetric Joukowski aerofoil with initial angle of attack, $\alpha _0=0$, camber parameter, $\gamma = 0$, thickness parameter, $R\geqslant 1$. The power laws in (3.36a,b) are set to ${m=0}$ and $n=2$, i.e. impulsive start-up in the direction parallel to the plate with a smooth (and slower) acceleration in the perpendicular direction. A type-II vortex sheet arises with

(3.40a{–}c)\begin{equation} {k=n=2, \quad l=m=0, \quad g_0 = \frac{ \mathcal{V}}{R},} \end{equation}

and $f_0$ is given by (3.38a,b). The vortex sheet has the self-similar solution,

(3.41a,b)\begin{equation} {\hat{z}_v = T \frac{\mathcal{V}}{R} \left(1-\left(1-\lambda\right)^{{2/5}} \right),} \quad {\mathcal{J} = \frac{16}{15} \mathcal{W} \sqrt{\frac{2 \mathcal{V}}{R}}, } \end{equation}

with a dimensionless length of $\mathcal {V}T/R$, which is inversely proportional to the aerofoil's thickness parameter, $R$. That is, a thicker symmetric Joukowski aerofoil produces a shorter trailing-edge vortex. Equation (2.33) gives the corresponding velocity jump across the vortex sheet,

(3.42)\begin{equation} {\gamma = \frac{8}{3} \sqrt{ \frac{2 R}{\mathcal{V}}} T^{3/2} \left( 1- \frac{R \hat{x}_v}{\mathcal{V} T} \right)^{3/2},} \end{equation}

whereas the total circulation follows from (2.13) and (2.26a,b), and is $\varGamma _0(T) = \mathcal {J}T^{{5}/{2}}$.

3.5.2. Case B: cambered Joukowski aerofoil with type-III vortex sheet

Second, we study an asymmetric (cambered) Joukowski aerofoil with initial angle of attack, $\alpha _0=-3{\rm \pi} /8$, camber parameter, $\gamma = {\rm \pi}/8$, and thickness parameter, $R=1$; see figure 5(a). The power laws in (3.36a,b) are set to $m=2$ and $n=1$, with velocity constants, $\mathcal {V}=-1/\sqrt {2}$ and $\mathcal {W}=1$. A type-III vortex sheet is generated at the trailing edge, with

(3.43a{–}d)\begin{equation} {k=m=2, \quad l=n=1, \quad f_0 ={-}\tfrac{1}{2}, \quad g_0 = 1,} \end{equation}

which is shown in figure 6(a) for $T=1/2$. This vortex sheet is identical to that generated at the trailing edge of the rotating and translating flat plate in figure 4(a).

Figure 6.

Type-III vortex sheet for $T=1/2$ (green) at the trailing edge of (a) a cambered Joukowski aerofoil (case B; § 3.5.2), and (b) a symmetric Joukowski aerofoil (case C; § 3.5.3). The images on the right provide an enlarged-view version of the trailing edge, highlighting the shape of the vortex sheet. The vortex sheets in (a,b) are identical to that at the trailing edge of the flat plate in figure 4(a).

3.5.3. Case C: symmetric Joukowski aerofoil with type-III vortex sheet

Third, we analyse a symmetric Joukowski aerofoil with $\alpha _0=0$, $\gamma = 0$ and $R=11/10$. The power laws in (3.36a,b) are now set to $m=1$ and $n=2$, with velocity constants, $\mathcal {V}=R$ and $\mathcal {W}=-1/\sqrt {2}$. Again, a type-III vortex sheet arises at the trailing edge with

(3.44a–d)\begin{equation} {k=n=2, \quad l=m=1, \quad f_0 ={-}\tfrac{1}{2} , \quad g_0 = 1 }, \end{equation}

which is illustrated in figure 6(b), again for $T=1/2$. The parameters in this third case have been chosen such that the vortex sheet is identical to case B above, despite the different aerofoil geometry. This highlights a principal finding of this study that the vortex sheet is controlled by the flow local to the edge only, which is reflected in identical values for $f_0$ and $g_0$ in (3.43a{–}d) and (3.43a{–}d).

A vortex sheet identical to figure 6(b) is obtained under the mapping, $\mathcal {V} \to - \mathcal {W}$, $\mathcal {W} \to \mathcal {V}$, $m \leftrightarrow n$ and $\alpha _0 \to \alpha _0 + {\rm \pi}/2$, because this produces the same attached flow. This mapping then gives values for $m$ and $n$ that coincide with case B.

4.1. Force generated by a type-II vortex sheet

The aerodynamic force generated by a type-II vortex is now calculated analytically, using (4.5). Substituting (2.29) and (2.31) into (4.5) gives the general formula,

(4.6a,b)\begin{equation} F_x^{(v)}=0, \quad F_y^{(v)} ={-} \frac{2 {\rm \pi}\rho \mathcal{L}^3}{\mathcal{T}^2} K c_{{1}/{2}}\, f_0 g_0 T^{l+k}. \end{equation}

As an example, we consider the trailing-edge vortex of a flat plate with $\alpha _0=0$ and $d\neq 1/2$, for which

(4.7a{–}f)\begin{equation} K=1, \quad c_{{1}/{2}} =\sqrt{2}, \quad k=p, \quad l=m , \quad f_0 ={-} \frac{\hat{\beta}}{\sqrt{2}}, \quad g_0 =1 , \end{equation}

giving the required vortex-induced drag and lift forces,

(4.8a,b)\begin{equation} F_x^{(v)}=0, \quad F_y^{(v)} = \frac{2 {\rm \pi}\rho \mathcal{L}^3}{\mathcal{T}^2} \hat{\beta} T^{m+p} . \end{equation}

This shows that, while a type-II vortex sheet is always parallel to the plate, the lift it generates is linearly proportional to the (rescaled) dimensionless angular velocity, $\hat {\beta }$, in the case that $d \neq 1/2$. It also depends on the sum of the translation and rotation power-laws, $m$ and $p$.

This gives the lift force experienced by the flat plate wing explored in § 3.2.1, for $m = 1$, $p=2$ and $d=-1/2$,

(4.9)\begin{equation} F_y^{(v)} = \frac{2 {\rm \pi}\rho \mathcal{L}^3}{\mathcal{T}^2} \beta T^3 , \end{equation}

since $\hat {\beta } = \beta (1/2-d)$. This cubic dependence of the lift force on time, $T$, and linear dependence on the angular velocity via $\beta$, should be observable in measurement for this experimentally realisable example.

4.2. General formula for type-I vortex sheets at both edges of a flat plate

As a further example of the utility of (4.5), we report general formulae for the lift force generated by type-I vortex sheets, that are shed from the edges of a flat plate. Cases for which $\alpha _0=0$ and $d \neq \pm 1/2$ are considered where identical vortex sheet types arise at both edges; see table 1 of PS21 and table 1. Note that the shape of these vortex sheets in the $z$-plane can only be evaluated numerically; for example, using the numerical method specified in § 5.1.2 of PS21.

Rescaling the Lagrangian coordinates of the vortex sheet and its shed circulation constant according to

(4.10a,b)\begin{equation} Z = | f_0 |^{2/3} \left\{\!\!\begin{array}{ll} \omega, & f_0>0 \\ \bar{\omega}, & f_0<0\end{array} \right. , \quad \mathcal{J} = \operatorname{sgn}(f_0) | f_0 |^{4/3} \tilde{\mathcal{J}}, \end{equation}

substituting (4.7a,b) and the corresponding expressions for the leading edge into (4.5), then gives the lift force experienced by the entire flat plate, arising from both vortex sheets, i.e. at the leading and trailing edges,

(4.11)\begin{align} F_Y^{(v)} &= \frac{ \rho \mathcal{L}^3}{2^{{1/3}} \mathcal{T}^2} \text{sgn}(\beta) | \beta |^{5/3} \tilde{\mathcal{J}} \left(s+{q}/{2}\right) T^{s+{q}/{2}-1} \nonumber\\ &\quad\times \left(\text{sgn} \left(\frac{1}{2}-d \right) \left|\frac{1}{2}-d \right|^{5/3}- \text{sgn} \left(\frac{1}{2}+d \right) \left|\frac{1}{2}+d \right|^{5/3} \right) \int_0^1 \left( {\bar{\omega}}^{'1/2} + {\omega}^{'1/2} \right) \, \mathrm{d} \lambda', \end{align}

where the subscript, $Y$, refers to the $y$-direction at the trailing edge. Note that the lift force vanishes for $d = 0$ because vortices with circulation of identical magnitude, but opposite sign, are generated at the leading and trailing edges.

An equivalent simplified formula for the lift experienced by the plate is not available for type-III vortices. This is because the vortex sheets at each edge exhibit different morphologies. Thus, the lift that each vortex sheet generates must be calculated separately (using (4.5)) and added together.

4.3. Force experienced by a flat plate due to its attached flow

In addition to the force generated by any shed vortex, a solid body experiences an additional force due to its attached flow; this is often referred to as an ‘added apparent mass’. For a flat plate, this additional force is normal to the plate surface and is given by

(4.12a,b)\begin{equation} F_X^{(a)} =0 , \quad F_Y^{(a)} = \frac{{\rm \pi} \rho \mathcal{L}^3}{\mathcal{T}^2} \frac{\mathrm{d}}{\mathrm{d} T} \left( U \sin \alpha - \varOmega \, \mathrm{d} \right), \end{equation}

where we have used (4.1) together with (2.3) of PS21, the superscript, $(a)$, refers to the force arising from the attached flow, with $X$ and $Y$ being the $x$ and $y$-directions at the trailing edge of the plate, respectively.