2. Formulation of the problem

The flow is as depicted in Figures 1 and 2a. We have a plate $BB'$ (of length $\widetilde{H}$ ) that is normal to a uniform, horizontal stream of speed $U$ ; and this stream is bounded above by a free surface $EF$ . The plate is submerged with the lower edge of the plate at $B'$ at a distance $\widetilde{d}$ below the level of the free surface of the undisturbed stream (i.e. the level of the free surface far upstream). We will refer to $\widetilde{d}$ as the draft of the flow. At some point not known a priori, there is a stagnation point $S$ along the plate $BB'$ and we let $AS$ denote the separation streamline. Note that, since we utilise the open-wake model, the upper and lower streamlines that bound the wake become horizontal (i.e. parallel to the undisturbed stream) at some points downstream. Let $C$ and $C'$ denote these points (not known a priori) at which the flow angle becomes zero along the bounding streamlines of the wake. Throughout this work, $BC$ and $B'C'$ will be referred to as the upper and lower wake boundaries, respectively. The horizontal parts (i.e. where the flow angle is zero) of the bounding streamlines of the wake are denoted by $CD$ and $C'D'$ .

Figure 1. Illustration of the open-wake model for flow past a submerged, finite-length plate that is normal to the oncoming flow bounded above by a free surface.

Figure 2. Complex planes for the open-wake model with a free surface bounding the flow above.

As discussed in the introduction, the pressure within the wake is assumed to be constant and we take the pressure along the wake boundaries to be this same constant, say $p_c$ . Whilst gravity is included on the free surface, we will neglect the effects of gravity on the wake boundaries. This can be justified by noting that a wake contains fluid of the same density as that of the rest of the flow, but it is at (relative) rest and so we have hydrostatic pressure within the wake. The Bernoulli condition tells us that

(1) \begin{equation} p+\rho g y=p_c \end{equation}

within the wake, where $p$ denotes the pressure, $\rho$ is the density and $g$ is the gravitational acceleration. Then, through use of (1) to eliminate the pressure $p$ from the Bernoulli condition, along the wake boundaries we have

(2) \begin{equation} \frac{1}{2}\rho q^2 + p_c = \widetilde{C}, \end{equation}

where $\widetilde{C}$ is a constant and $q$ denotes the speed. Hence, gravity does not appear in this boundary condition and so we can neglect the effects of gravity along the boundaries of the wake.

We will now formulate the overall problem. The flow is as described above and shown in Figure 2a where $z=x+\mathrm{i} y$ with the $x$ -axis set along the level of the undisturbed free surface and the $y$ -axis is set along the vertical plate $BB'$ . Recall that we employ the open-wake model, so the flow angle is zero along $CD$ and $C'D'$ . The effect of gravity is included on the free surface $EF$ that bounds the flow above and we assume constant atmospheric pressure $p=p_a$ along this free surface. On the other hand, gravity is not included along the wake boundaries $BC$ and $B'C'$ ; and the pressure is taken to be constant (say, $p=p_c$ ) along these boundaries. We normalise the velocity such that the uniform, horizontal stream is of unit speed and normalise the distances such that $\pi$ is the normalised magnitude of the net volume flux of the flow above the plate. We denote the characteristic length for the latter normalisation by $L$ and note that the normalised depth of the dividing streamline far upstream is such that it lies along $y=-\pi$ . Then, the Bernoulli condition on the free surface $EF$ is

(3) \begin{equation} \frac{1}{2} q^2 + \frac{y}{F^2} = \frac{1}{2}, \end{equation}

where

(4) \begin{equation} F=\frac{U}{\sqrt{gL}} \end{equation}

is the Froude number. Along the wake boundaries $BC$ and $B'C'$ , we have

(5) \begin{equation} q^2 = K+1, \end{equation}

where

(6) \begin{equation} K=\frac{p_{\infty }-p_c}{\frac{1}{2} \rho U^2} \end{equation}

is the wake underpressure coefficient and $p_{\infty }$ denotes the pressure of the stream in the far-field. The wake underpressure coefficient is a non-dimensional parameter that characterises the flow regime, that is, whether we have a fully or partially established wake. In particular, $K=0$ corresponds to the formation of wake at ambient pressure. Note that, in the present study, the wake is assumed to be fully established with the plate edges $B$ and $B'$ of the plate being the separation points.

We let $f$ denote the complex potential for the flow and $f=\phi + \mathrm{i} \psi$ , where $\phi$ is the velocity potential and $\psi$ is the streamfunction. We set $\psi =0$ along the dividing streamline and $\psi =\pi$ along the free surface (since we have a net volume flux of $\pi$ for the flow between the dividing streamline and the free surface). The velocity potential at the stagnation point is set to be $\phi =1$ . Then, there is a branch cut along the positive $\phi$ -axis for $\phi \gt 1$ . For illustration, the $f$ -plane is as depicted in Figure 2b. We also define an intermediate complex plane by $\zeta =\xi +\mathrm{i} \eta$ via the relation:

(7) \begin{equation} f=\zeta - \log \zeta . \end{equation}

Therefore, the flow region maps to the lower-half of the $\zeta$ -plane. In particular, the free surface $EF$ maps to the negative $\xi$ -axis; and the wake boundaries $BC$ and $B'C'$ , the downstream horizontal approximations $CD$ and $C'D'$ , and the vertical plate $BB'$ all map to the positive $\xi$ -axis. Also note that, by the given mapping (7), $\zeta =1$ corresponds to the stagnation point $S$ (that is situated at some point along the vertical plate in the physical plane). The positions in the $\zeta$ -plane corresponding to the edges of the plate and the points at which the zero flow angle begins along the streamlines bounding the wake are unknown and left to be determined. By $b$ , $b^{\prime }$ , $c$ and $c^{\prime }$ we denote these unknown positions corresponding to $B$ , $B'$ , $C$ and $C'$ in the $\zeta$ -plane, respectively.

Figure 3. $\zeta$ -plane with contour $\widetilde{C}=C_R \cup [R, \zeta +\epsilon ] \cup C_{\epsilon } \cup [\zeta -\epsilon, -R]$ .

Finally, we let $\Omega = \tau - \mathrm{i} \theta$ where $\Omega$ is defined such that $\Omega = \log{({\mathrm{d} f}/{\mathrm{d} z})}$ . Then, we apply the Cauchy integral formula to $\Omega (\zeta )$ for $\zeta \in \mathbb{R}$ with a semi-circular contour $\widetilde{C}=C_R \cup [R, \zeta +\epsilon ] \cup C_{\epsilon } \cup [\zeta -\epsilon, -R]$ in the lower-half of the $\zeta$ -plane, where $C_R$ is the semi-circular arc of radius $R$ centred at the origin, $C_{\epsilon }$ is the semi-circular arc of radius $\epsilon$ centred at $\zeta$ and the intervals $[R,\zeta +\epsilon ]$ and $[\zeta -\epsilon, -R]$ are along the real $\zeta$ -axis. The contour is traversed anticlockwise, and we consider the limit as $R \rightarrow +\infty$ . This contour is depicted in Figure 3. Then, for $\zeta \in \mathbb{R}$ , on the contour, we obtain that

(8) \begin{equation} \tau (\zeta )-\mathrm{i} \theta (\zeta ) =- \frac{1}{\pi \mathrm{i}} \, P.V. \, \int _{-\infty }^{\infty } \frac{ \tau (\sigma )-\mathrm{i} \theta (\sigma )}{\sigma -\zeta } \, \mathrm{d} \sigma, \end{equation}

where $P.V.$ denotes the Cauchy principal value. Here, we have utilised the fact that we have horizontal flow of unit speed in the far-field; hence, $\tau$ and $\theta$ tend towards to zero in the far-field. Taking the real part of (8), we find

(9) \begin{equation} \tau (\zeta )=\frac{1}{\pi } \, P.V. \, \int _{-\infty }^{\infty } \frac{\theta (\sigma )}{\sigma -\zeta } \, \mathrm{d} \sigma . \end{equation}

Then, by using the known flow angles along the vertical plate $BB'$ and the downstream, horizontal parts $CD$ and $C'D'$ of the streamlines bounding the wake, the expression can be rewritten as:

(10) \begin{equation} \tau (\zeta )=\frac{1}{\pi } \bigg [\int _{-\infty }^0 \frac{\theta (\sigma )}{\sigma -\zeta } \, \mathrm{d} \sigma + \int _{c}^b \frac{\theta (\sigma )}{\sigma -\zeta } \, \mathrm{d} \sigma + \int _{b^{\prime }}^{c^{\prime }} \frac{\theta (\sigma )}{\sigma -\zeta } \, \mathrm{d} \sigma \bigg ] + \frac{1}{2} \log \bigg | \frac{(1-\zeta )^2}{(b-\zeta )(b^{\prime }-\zeta )} \bigg |. \end{equation}

Note that one of the integrals in (10) is a Cauchy principal value, depending on the value of $\zeta$ . We recall that the speed of the flow can be expressed as $q=\exp\! (\tau )$ and that the dynamic boundary condition along the wake boundaries is given by (5). It follows that, along $BC$ and $B'C'$ where $\zeta \in \mathbb{R}$ such that $c\lt \zeta \lt b$ or $b^{\prime }\lt \zeta \lt c^{\prime }$ , we have

(11) \begin{equation} \frac{1}{2}\log{(K+1)}=\frac{1}{\pi } \bigg [\int _{-\infty }^0 \frac{\theta (\sigma )}{\sigma -\zeta } \, \mathrm{d} \sigma + \int _{c}^b \frac{\theta (\sigma )}{\sigma -\zeta } \, \mathrm{d} \sigma + \int _{b^{\prime }}^{c^{\prime }} \frac{\theta (\sigma )}{\sigma -\zeta } \, \mathrm{d} \sigma \bigg ] + \frac{1}{2} \log \bigg | \frac{(1-\zeta )^2}{(b-\zeta )(b^{\prime }-\zeta )} \bigg |. \end{equation}

We also look to obtain an integral equation that is valid along the free surface $EF$ . First, we differentiate the Bernoulli condition (3) along $EF$ with respect to $\zeta$ , so we have

(12) \begin{equation} \frac{\mathrm{d} \tau }{\mathrm{d} \zeta } \mathrm{e}^{2\tau } + \frac{1}{F^{2}}\frac{\mathrm{d} y}{\mathrm{d} \zeta }=0. \end{equation}

Note that we can express the derivative that appears in the second term above as:

(13) \begin{equation} \frac{\mathrm{d} y}{\mathrm{d}\zeta }=\mathrm{e}^{-\tau (\zeta )}\Big (1-\frac{1}{\zeta }\Big )\sin{(\theta (\zeta ))}. \end{equation}

Then, using this along with (12) and the fact that $\tau \rightarrow 0$ as $\xi \rightarrow -\infty$ , we find

(14) \begin{equation} \tau (\zeta )=\frac{1}{3}\log{\Bigg (1-\frac{3}{F^2}\int _{-\infty }^{\zeta }\Bigg (1-\frac{1}{\xi }\Bigg )\sin{(\theta (\xi ))} \, \mathrm{d} \xi \Bigg )}. \end{equation}

Finally, eliminating $\tau$ between (10) and (14), then along $EF$ where $\zeta \in \mathbb{R}$ s.t. $-\infty \lt \zeta \lt 0$ , we look to satisfy

(15) \begin{equation} \begin{aligned} \frac{1}{3}\log{\Bigg (1-\frac{3}{F^{2}}\int _{-\infty }^{\zeta }\Bigg (1-\frac{1}{\xi }\Bigg )\sin{\theta (\xi )} \, \mathrm{d} \xi \Bigg )}=\frac{1}{\pi } &\bigg [\int _{-\infty }^0 \frac{\theta (\sigma )}{\sigma -\zeta } \, \mathrm{d} \sigma + \int _{c}^b \frac{\theta (\sigma )}{\sigma -\zeta } \, \mathrm{d} \sigma \\& + \int _{b^{\prime }}^{c^{\prime }} \frac{\theta (\sigma )}{\sigma -\zeta } \, \mathrm{d} \sigma \bigg ] + \frac{1}{2} \log \bigg | \frac{(1-\zeta )^2}{(b-\zeta )(b^{\prime }-\zeta )} \bigg |. \end{aligned} \end{equation}

Note that this is a Nekrasov-type equation [Reference Wehausen, Laitone and Flugge17]. Again, we remark that one of the integrals on the right-hand side of (15) is a Cauchy principal value, depending on the value of $\zeta$ .

Before summarising the problem and discretising to obtain a numerical solution, we first discuss the flow far upstream where, along the free surface, we have $\xi \rightarrow -\infty$ . Considering (10) as $\xi \rightarrow -\infty$ , we have that

(16) \begin{equation} \begin{aligned} \tau (\xi ) &\sim \frac{1}{\xi } \Big (-\frac{1}{\pi }\int _{-\infty }^{\infty } \theta (\sigma )\, \mathrm{d} \sigma \Big ) + \cdots \\ &\sim \frac{1}{\xi } \Bigg [-\frac{1}{\pi }\Bigg (\int _{-\infty }^0 \theta (\xi ) \mathrm{d} \xi + \int _c^b \theta (\xi ) \mathrm{d} \xi + \int _{b^{\prime }}^{c^{\prime }} \theta (\xi ) \mathrm{d} \xi \Bigg ) - 1 + \frac{1}{2}(b+b^{\prime }) \Bigg ] + \cdots \\ &\sim \frac{\beta }{\xi } + \cdots, \end{aligned} \end{equation}

where we let $\beta$ denote the constant coefficient of the $\xi ^{-1}$ term. From this behaviour of $\tau$ far upstream, the Bernoulli condition (3) along the free surface and the fact that $x \sim \xi$ as $\xi \rightarrow -\infty$ (since we have unit horizontal velocity of the undisturbed flow), we have

(17) \begin{equation} \frac{\mathrm{d} y}{\mathrm{d} x} \sim \frac{\beta F^2}{x^2} \text{ as } x \rightarrow -\infty \end{equation}

and hence

(18) \begin{equation} \theta (\xi ) \sim \frac{\beta F^2}{\xi ^2} \text{ as } \xi \rightarrow -\infty, \end{equation}

where $\beta$ is the constant as defined above.

3. Numerical approach

Now, we detail the numerical scheme. We discretise the problem by introducing $N+1$ mesh points along each of the wake boundaries and along the free surface, that is, we take

(19) \begin{equation} \begin{aligned} \xi ^{U}_j - \log \xi ^{U}_j = b-\log b + \frac{j-1}{N}\Big (c-b+\log{\frac{b}{c}}\Big ), \; &\text{ upper wake boundary $BC$}\\ \xi ^{L}_j - \log \xi ^{L}_j = b^{\prime } -\log b^{\prime } + \frac{j-1}{N}\Big (c^{\prime }-b^{\prime }+\log{\frac{b^{\prime }}{c^{\prime }}}\Big ), \; &\text{ lower wake boundary $B'C'$}\\ \xi ^{FS}_j - \log |\xi ^{FS}_j| = X_- + \frac{j-1}{N}(X_+ - X_-), \; &\text{ free surface $EF$} \end{aligned} \end{equation}

where $j=1,2,\ldots, N+1$ and where the finite range $(X_-,X_+)$ is used to truncate the range $({-}\infty, \infty )$ of the velocity potential. Then, the unknowns of the problem are the values of the flow angle $\theta$ at each of these mesh points, that is, we aim to find $\theta _j^{FS}=\theta (\xi ^{FS}_j)$ , $\theta _j^{U}=\theta (\xi ^{U}_j)$ and $\theta _j^{L}=\theta (\xi ^{L}_j)$ for $j=1,2,\ldots, N+1$ . Recall that the positions corresponding to $B$ , $B'$ , $C$ and $C'$ in the $\zeta$ -plane which are $b$ , $b^{\prime }$ , $c$ and $c^{\prime }$ , respectively, are unknowns. We also look to find the constant $\beta$ of the decay of $\tau$ far upstream, as defined through (16). Therefore, overall we have $3N+8$ unknowns.

For the equations in these unknowns, we introduce $N$ collocation points along the free surface and wake boundaries defined by:

(20) \begin{equation} \begin{aligned} \xi ^{U_{\text{inter}}}_j - \log \xi ^{U_{\text{inter}}}_j = b-\log b + \frac{j-\frac{1}{2}}{N}\Big (c-b+\log{\frac{b}{c}}\Big ), \; &\text{ upper wake boundary $BC$}\\ \xi ^{L_{\text{inter}}}_j - \log \xi ^{L_{\text{inter}}}_j = b^{\prime } -\log b^{\prime } + \frac{j-\frac{1}{2}}{N}\Big (c^{\prime }-b^{\prime }+\log{\frac{b^{\prime }}{c^{\prime }}}\Big ), \; &\text{ lower wake boundary $B'C'$},\\ \xi ^{FS_{\text{inter}}}_j - \log |\xi ^{FS_{\text{inter}}}_j| = X_- + \frac{j-\frac{1}{2}}{N}(X_+ - X_-), \; &\text{ free surface $EF$}, \end{aligned} \end{equation}

for $j=1,2,\ldots, N$ . Satisfying the condition (15) at the collocation points along the free surface and satisfying the condition (11) at the collocation points along the upper and lower wake boundaries leads to $3N$ equations. In order to obtain these $3N$ equations, care is required to evaluate the integrals that appear in (11) and (15) since, for a given value of $\zeta$ that corresponds to some point along the free surface $EF$ or the wake boundaries $BC$ and $B'C'$ , one of the integrals in each equation will be a Cauchy principal value. Note that the collocation points (20) are the midpoints between the mesh points (19), so the point to emphasise is that the collocation and mesh points do not coincide. Also, we assume that $\theta (\xi )$ is stepwise linear on $[\xi _j, \xi _{j+1}]$ for $j=1,2,\ldots, N$ (i.e. on each interval of consecutive mesh points (19)). Then, by integrating over each of these intervals (which make up the intervals for integration that appear in (11) and (15)), we can obtain an expression to (analytically) approximate the Cauchy principal value. The approximation here is due to the aforementioned assumption of stepwise linear behaviour of the flow angle. This expression can then be evaluated numerically at each collocation point.

So far, we have $3N$ equations in the $3N+8$ unknowns. We also have several known values of the flow angle $\theta$ . Along the upper wake boundary, we know that

(21) \begin{equation} \theta (b)=\theta ^U_1=\frac{\pi }{2} \; \; \; \text{ and } \; \; \; \theta (c)=\theta ^U_{N+1}=0. \end{equation}

Along the lower wake boundary, we have

(22) \begin{equation} \theta (b^{\prime })=\theta ^L_1=-\frac{\pi }{2} \; \; \; \text{ and } \; \; \; \theta (c^{\prime })=\theta ^L_{N+1}=0. \end{equation}

Along the free surface, we take

(23) \begin{equation} \theta ^{FS}_1=\frac{\beta F^2}{(\xi ^{FS}_1)^2} \; \; \; \text{ and } \; \; \; \theta ^{FS}_{N+1}=0. \end{equation}

The first condition of the above (23) is obtained by evaluating (18) at the furthest mesh point upstream; and by the second condition of (23), we impose zero flow angle at the furthest mesh point downstream along the free surface, which is appropriate for supercritical flow. We also have that the constant $\beta$ is constrained by:

(24) \begin{equation} \beta =-\frac{1}{\pi }\Bigg (\int _{-\infty }^0 \theta (\xi ) \mathrm{d} \xi + \int _c^b \theta (\xi ) \mathrm{d} \xi + \int _{b^{\prime }}^{c^{\prime }} \theta (\xi ) \mathrm{d} \xi \Bigg ) - 1 + \frac{1}{2}(b+b^{\prime }). \end{equation}

This is as derived earlier in (16). Finally, we set the ratio of the (normalised) length $H$ of the vertical plate to the (normalised) draft $d$ by imposing a given value for $H/d$ . In order to impose the value of this parameter, note that expressions to evaluate the values of $H$ and $d$ are required. We have that $y_{C'}=-\pi (1+\beta )$ , where $y_{C'}$ denotes the $y$ -value at $C'$ . This expression can be derived by noting that the depth of the dividing streamline far upstream (i.e. point $A$ in Figure 2a) is $-\pi$ . Also, the $\pi \beta$ contribution to the expression for $y_{C'}$ can be obtained by finding the difference between the $y$ -values far upstream at $A$ and far downstream at $D'$ . This is achieved by integrating around a large arc in the lower-half of the $\zeta$ -plane, taking the limit as the radius of this arc tends to infinity and noting that $\Omega \sim \beta /\zeta$ in the far-field. The draft $d$ of the plate can then be determined by integrating from the point $C'$ on the lower wake boundary to the bottom $B'$ of the plate. Also, the length $H$ of the plate can be calculated by evaluating

(25) \begin{equation} H=y_B-y_{B'}. \end{equation}

Overall, we have $3N+8$ equations in the $3N+8$ unknowns.

To solve, we specify the values of the following parameters: Froude number $F$ , wake underpressure coefficient $K$ and ratio $H/d$ . It is worth noting that, in the current presentation of the method of solution, the problem is posed with the wake underpressure coefficient $K$ as a chosen parameter and the draft is obtained from the solution. This allows for convenient comparison of the results with those found by Tuck and Vanden-Broeck [Reference Tuck and Vanden-Broeck15] in their work on the similar ploughing flows where the bluff body is a semi-infinite, rectangular block (instead of the plate and wake that we investigate in the present study). However, the problem can be reworked so that the draft is prescribed and $K$ is left to be found – this is a more natural way to consider the physical problem of the submerged plate placed at some given distance beneath the level of the undisturbed free surface. This will be considered later in the discussion of results.

Finally, before presenting the results, we introduce Froude numbers based on the draft $d$ of the plate and based on the flow quantities that characterise the channel between the free surface and the upper wake boundary. This will facilitate a quantitative examination of the results. We let $q_{\infty }$ and $h_{\infty }$ denote the non-dimensional speed and width (respectively) of the channel that forms above the wake. Then, we let

(26) \begin{equation} F_d=\frac{F}{\sqrt{d}}, \end{equation}

which is a Froude number defined based on the draft; and we let

(27) \begin{equation} F_h=\frac{F q_{\infty }}{\sqrt{h_{\infty }}}, \end{equation}

which is a Froude number defined based on the downstream channel above the wake. These Froude numbers are as utilised by Tuck and Vanden-Broeck [Reference Tuck and Vanden-Broeck15].