2. Formulation

The 2-D problem is to find the depth $h$ and velocity components $u$ and $v$ in the $x$ and $y$ directions using the following depth-averaged equations:

(2.1)$$\begin{gather} \frac{\partial h}{\partial t}+\frac{\partial({uh})}{\partial x}+\frac{\partial({vh})}{\partial y}=0, \end{gather}$$

(2.2)$$\begin{gather}\frac{\partial{(uh)}}{\partial t}+\frac{\partial( {u^2 h}+\frac{1}{2}g'h^2)}{\partial x}+\frac{\partial({uvh})}{\partial y}= g'hS_o-\frac{c_f}{2}u\sqrt{u^2+v^2}, \end{gather}$$

(2.3)$$\begin{gather}\frac{\partial {(vh)}}{\partial t}+\frac{\partial({vuh})}{\partial x}+\frac{\partial( {v^2 h}+\frac{1}{2}g'h^2)}{\partial y}={-}\frac{c_f}{2}v\sqrt{u^2+v^2}, \end{gather}$$

where $g$ is the gravitational acceleration, $c_f$ is the quadratic friction coefficient and $S_o=\tan \theta$ is the channel slope. The components of the gravity acceleration are $g'=g \cos \theta$ and $g \sin \theta =g'S_o$ in the $z$ and $x$ directions, respectively. The $x$-axis is downward at an angle $\theta$ relative to the horizontal and the $z$-axis is upward perpendicular to the inclined $x$–$y$ plane. The water depth $h$ is measured along the $z$-axis.

The initial condition is an undisturbed flow of uniform depth $H$ with a longitudinal velocity $U$ downward parallel to the channel bed in the $x$ direction. The balance between gravity and bed friction in the undisturbed flow yields the dimensionless relation

(2.4)\begin{equation} S_o = \tfrac{1}{2} {c_f} {Fr}^2, \end{equation}

where $Fr =U/\sqrt {g'H}$ is the undisturbed-flow Froude number.

3. Roll-wave packet and its front runner

A roll-wave packet, such as the one shown in figure 1(b), is a spatial instability developed from a local disturbance. To reproduce this instability, we introduced a slight modification to the uniform flow, with the depth $h$ and unit discharge $q_x =uh$ specified at the inlet as follows:

(3.1a)$$\begin{gather} h(x=0,y,t)=\begin{cases} H\left[1+\epsilon\sin\left(\dfrac{2{\rm \pi} t}{T}\right)\right] & {\rm for} \ 0< t\leq\dfrac{T}{2}\\ H & {\rm for} \ t>\dfrac{T}{2} \end{cases}, \end{gather}$$

(3.1b)$$\begin{gather}q_x (x=0, y, t) = {Fr} \sqrt{g' [h(x=0,y,t)]^3},\quad q_y (x=0, y, t ) = 0. \end{gather}$$

The depth of the inlet disturbance $h(x=0,y, t)$ is the upper half of a sinusoid of period $T$; the amplitude of the sinusoidal disturbance is $\epsilon$. The inlet $q_x(x=0,y, t)$ is selected to keep the Froude number of the disturbance equal to the Froude number of the undisturbed flow.

This inlet perturbation defined by (3.1) is a ‘type-c disturbance of constant $Fr$’. The six inlet disturbances examined by Yu & Chu (Reference Yu and Chu2022) were type-a, type-b and type-c of constant $Fr$ or constant $q$. Type-a is a periodic disturbance of constant amplitude. Type-b is one full sinusoid from $t = 0$ to $T$, while type-c is one half of a sinusoid from $t = 0$ to $T/2$. All six inlet disturbances produced the same front runner according to Yu & Chu (Reference Yu and Chu2022). This study considers only the roll waves produced by ‘type-c disturbance of constant $Fr$’.

Figure 1(b) shows an example of a roll-wave packet developed from the type-c disturbance of constant $Fr$ with $\epsilon =0.2$ and $T=0.94$ s on an undisturbed flow with a Froude number $Fr = 3.71$. The flow is convectively unstable as both the front runner and the trailing edges of the roll-wave packet move downstream from the source (Criminale, Jackson & Joslin Reference Criminale, Jackson and Joslin2003). The front-runner amplitude $\hat {h}_{FR}/H$ increases with time as the packet advances and disperses in space. The spatial extent of the wave packet $S_o\lambda _p/H$ and the number of steep wavefronts within the wave packet $N_p$ increase with the dimensionless time $S_otU/H$. In this example, $N_p= 3$, 5, 7 and 9, $S_o\lambda _p/H= 52.75$, 137.9, 232.9 and 333.2 and $\hat {h}_{FR}/H=2.79$, 3.62, 3.95 and 4.07, at times $S_otU/H=162.95$, 293.31, 436.71 and 586.63, respectively.

We carried out a series of simulations of the roll waves for undisturbed-flow Froude number $Fr = 3.71$, 4.63 and 5.60. Figure 2 shows all simulation results for (a) the amplitude $\hat {h}_{FR}/H$, (b) the celerity of the front runner ${c}_{FR}/U$ and (c) the spatial extent $S_o \lambda _p /H$ of the wave packet, and their dependence on the front-runner position $S_o x_{FR}/H$ and the Froude number $Fr$. The amplitude and celerity of the front runner approach the long-wavelength limit of the Dressler (Reference Dressler1949) solution. The green dashed lines in the figure indicate the asymptotic solutions.

Figure 2. The dependence of (a) the amplitude $\hat {h}_{FR}/H$ and (b) the celerity of the front runner ${c}_{FR}/U$ and (c) the spatial extent of roll-wave packet $S_o \lambda _p/H$ on Froude number $Fr = U/\sqrt {g'H}$ and front-runner distance of advance $S_o x_{FR}/H$. (d) The three tables on the right-hand side list the conditions of four simulation cases for verifying the effects of the perturbation amplitude $\epsilon$ and the friction coefficient $c_f$. The lines and the $\circ$, $\square$ and $\triangle$ symbols denote case 1, case 2, case 3 and case 4, respectively. The black, red and blue colours of the lines and symbols refer to $Fr = 3.71$, 4.63 and 5.60, respectively. The green dashed lines in (a) and (b) represent the long-wavelength limit of Dressler's solution.

The tables embedded in figure 2(d) show the simulation conditions for different undisturbed-flow Froude numbers. Simulation cases 1, 2, 3 and 4 were conducted to examine the effect of bed friction $c_f$ and the inlet perturbation amplitude $\epsilon$.

For the reference case 1, the friction coefficient and channel slope are $(c_f, S_o)=(0.00728, 0.0501)$, (0.00786, 0.0843) and (0.00760, 0.119) for $Fr = 3.71$, 4.63 and 5.60, respectively. These are the same conditions as in the laboratory experiment by Brock (Reference Brock1967) on a smooth channel bed. For a given $c_f$ and $Fr$, Brock (Reference Brock1967) selected the channel slope $S_o$ to satisfy (2.4).

In simulation case 2, we doubled the friction and channel slope $(c_f, S_o)$. Remarkably, the front runner amplitude $\hat {h}_{FR}/H$ and the extent of the wave packet $S_o \lambda _p /H$, marked by the open circles in figure 2, are nearly identical to the results for reference case 1 denoted by the solid lines.

The amplitude of the inlet disturbance for both cases 1 and 2 is $\epsilon = 0.2$. For cases 3 and 4, the inlet perturbations are $\epsilon =0.1$ and 0.05. The change in perturbation amplitude by a factor of 4 has only a negligibly small effect on the roll-wave packet and its front runner, as shown by the triangular and square symbols in the figure.

Therefore, we conclude that the nonlinear development of the roll-wave packet depends mainly on two dominant parameters: the undisturbed-flow Froude number $Fr$ and the front-runner distance of advance $S_o x_{FR}/H$.

4. Numerical method and adaptive mesh

Figure 3 sketches the 2-D simulation problem to determine the impact force of the front runner on a circular prism of diameter $D$. The top panels of the figure show the depth profiles on the centre plane, while the bottom panels show the depth-contour maps. Panels (a) and (b) refer to two different times, one before and the other after the impact of the front runner against the structure.

Figure 3. The depth profiles on the centre $x$–$z$ plane (top) and the depth-contour maps on the inclined $x$–$y$ plane (bottom) (a) before the arrival of the front runner at time $S_o (t-t_{arr}) U/H = -4.02$ and (b) after the impact when the front runner just past the circular prism positioned at a distance $S_ox_b/H=251.2$ ($x_b/D=66.7$) from the inlet at time $S_o (t-t_{arr}) U/H= 1.59$ where the arrival time $S_o t_{arr} U/H=166.32$. The simulation results in this example are for an undisturbed-flow Froude number of ${Fr}=3.71$. The green stars mark the prism's position $(x_b, y_b)$. The yellow lines are iso-depths of $h/H=2$, 4, 6, 8.

We used the Gerris Flow Solver by Popinet (Reference Popinet2003, Reference Popinet2011) on an adaptive quadtree mesh that dynamically refines and coarsens depending on the flow feature. The computational efficiency of the quadtree adaptive mesh for 2-D shallow-water flow simulation has been reported in several publications. Liang et al. (Reference Liang, Zang, Borthwick and Taylor2007) and An & Yu (Reference An and Yu2012) simulated shock reflection by a circular cylinder. They found approximately 80 % saving of the CPU time compared with using uniform mesh. Popinet (Reference Popinet2015) simulated Tohoku tsunami and he found 97 % saving of CPU time.

Figure 4(a) shows the mesh on an $x$–$y$ plane for the entire length of the channel at time $S_o(t-t_{arr})U/H=-2.68$ for the undisturbed-flow Froude number $Fr = 3.71$. Panels (b) and (d) of the figure show the close-up views of the mesh at time $S_o(t-t_{arr})U/H=-2.68$ and $+$2.68, before and after the arrival of the front runner, respectively. Panel (c) shows the adaptive mesh at time $S_o(t-t_{arr})U/H=0$ when the impact force reaches its peak.

Figure 4. The adaptive quadtree mesh used in the simulation for ${{Fr}=3.71}$ and $x_b/D=66.7$ ($S_o x_b/H= 251.2$). (a) Global view of the mesh. Panels (b) and (d) are the close-up views of the mesh before and after the impact by the front runner and panel (c) is the close-up view at the instant when the impact force reached its peak.

The shock-capturing scheme for spatial discretization is a generalized MINMOD limiter (GML) which uses the limiter function

(4.1)\begin{equation} \psi = {\rm minmod}\left(\beta\frac{q_i-q_{i-1}}{\varDelta},\frac{q_{i+1}-q_{i-1}}{2\varDelta}, \beta\frac{q_{i+1}-q_{i}}{\varDelta}\right) \end{equation}

to reconstruct the cell-interface flux with a free parameter varied from $\beta =1$ to 2 (Sweby Reference Sweby1984; Nessyahu & Tadmor Reference Nessyahu and Tadmor1990; Kurganov & Tadmor Reference Kurganov and Tadmor2002). In the limiter function, $q_i$ is the cell-centred value of the unknown variable and $\varDelta$ is the size of the $i$th cell. The GML reduces to the dissipative classical MINMOD limiter when $\beta =1$. It becomes the least dissipative SUPERBEE limiter if $\beta =2$. We used $\beta =1.50$, balancing dissipation and computational stability. The implementation of the GML for an adaptive mesh is due to Popinet (Reference Popinet2011).

The temporal discretization was a second-order predictor–corrector scheme for the hyperbolic part of the equations and a third-order total variation diminishing Runge–Kutta scheme for the source term (Gottlieb & Shu Reference Gottlieb and Shu1998). The second-order accurate Cartesian cut-cell method of An & Yu (Reference An and Yu2012) and Causon, Ingram & Mingham (Reference Causon, Ingram and Mingham2001) defines the boundary of the circular prism.

The impact on the obstacle is a nonlinear problem of wave scattering. The integration of the hydrostatic pressure force per unit length $f_{hs}=\frac {1}{2} \rho g' h^2$ along the closed path $s$ following the boundary of the obstacle gives the wave force $F_w$ as follows:

(4.2)\begin{equation} F_w ={-} \oint_s f_{hs} \,\boldsymbol{i} \boldsymbol{\cdot} \boldsymbol{n} \, {\rm d}s, \end{equation}

where $\boldsymbol {{i}}$ is the unit vector in the direction of the undisturbed flow and $\boldsymbol {{n}}$ is the outward normal unit vector perpendicular to the obstacle's solid boundary. We ignored the viscous drag due to the turbulent boundary layer around the obstacle in calculating the ‘wave’ force by (4.2).

The wave force $F_w$, adimensionalized by the product of the frontal projection area $HD$ and the stagnation pressure $\frac {1}{2} \rho U^2$, is the wave force coefficient

(4.3)\begin{equation} C_{HU^2} = \frac{F_w}{ \frac{1}{2} \rho U^2 H D }. \end{equation}

Before the arrival of the front runner, the wave force coefficient has an undisturbed value $\bar {C}_{HU^2}$. The force coefficient reaches its peak value of $\hat {C}_{HU^2}$ at the arrival of the front runner. We define the impact-duration coefficient

(4.4)\begin{equation} C_{\mathcal{T}}=\frac{ S_o{\mathcal{T}} U}{H} = \frac{\displaystyle\int_{t_{b}^*}^{t_{i}^*} [C_{HU^2}-\bar{C}_{HU^2}]\, {\rm d}t^*}{\hat{C}_{HU^2}-\bar{C}_{HU^2}}, \end{equation}

where ${\mathcal {T}}$ is the duration of the front-runner impact. The integration in (4.4) is over a period of time from $t_{b}^* = S_o t_{b} U/H$ to $t_{i}^* = S_o t_{i} U/H$ when $C_{HU^2}$ is greater than the undisturbed value $\bar {C}_{HU^2}$. The dimensionless parameter $S_o{\mathcal {T}}U/H$ reflects the impact duration of the front runner against the obstacle. The pink-coloured rectangle in figure 5(a) defines the peak force coefficient $\hat {C}_{HU^2}$ and impact-duration coefficient $S_o{\mathcal {T}}U/H$. It is equal to the area below the solid line (representing the time series) and above the dotted line (indicating the undisturbed-flow value).

Figure 5. (a) The wave force coefficient $C_{HU^2}$, (b) shock stand-off distance $x_s/D$ and (c) centre-plane run-up height $R/D$ as a function of the dimensionless time $S_o(t-t_{arr})U/H$. The red dashed line denotes the analytical solution obtained by Forbes & Schwartz (Reference Forbes and Schwartz1981).

The total number of computation cells in the adaptive mesh varies with time. But there are a maximum mesh size $\varDelta _{max}$ and a minimum size $\varDelta _{min}$. In our numerical simulation for the impact force over a computational domain of $L_x \times L_y=162\ {\rm m} \times 3\ {\rm m}$, the numbers of computational cells are $L_x/\varDelta _{min}=27\,648$, $L_y/\varDelta _{min} =512$, $L_x/\varDelta _{max}=1728$ and $L_y/\varDelta _{max}= 32$. We conducted a mesh refinement study using the method of Stern et al. (Reference Stern, Wilson, Coleman and Paterson2001) to determine the accuracy of the simulation. The fractional error of the peak force coefficient is

(4.5)\begin{equation} FE(\%) = \frac{|\hat{C}_{HU^2}-[\hat{C}_{HU^2}]_{\varDelta \rightarrow 0}|}{[\hat{C}_{HU^2}]_{\varDelta \rightarrow 0}} \times 100. \end{equation}

The exact value $[\hat {C}_{HU^2}]_{\varDelta \rightarrow 0}$ was estimated by extrapolation. The detailed calculation for a simulation with $Fr = 3.71$ and $S_ox_b/H=1005$ in Appendix B gives the fractional error $FE(\%)=1.175$%.

5. Impact on a circular prism for $Fr = 3.71$

We analyse the impact on a circular prism at a distance $S_ox_b/H = 251.2$ for an undisturbed-flow Froude number $Fr = 3.71$. In this example, the inlet perturbation has a period $S_oTU/H= 6.08$ and an amplitude of $\epsilon = 0.2$. The smooth channel bed has a friction coefficient $c_f= 0.00728$. The dimensionless diameter of the circular prism is $D/H= 75$. Figure 5 shows the temporal variation of (a) the wave force coefficient $C_{HU^2}$, (b) the shock stand-off distance $x_s/D$ and (c) the centre-plane run-up height $R/D$. The impact force peaks when the front runner arrives at time $S_ot_{arr}U/H= 166.32$. The front runner has an amplitude $\hat {h}_{FR}/H= 2.94$, a velocity of $\hat {u}_{FR}/U = 1.44$ and a celerity of ${c}_{FR}/U = 1.64$ before its impact on the circular prism.

The nonlinear dynamics is the impingement of the front runner on the cylinder, the rapid rise of the impact force to a first peak, then a second peak and eventually back to the undisturbed flow. These stages for $Fr = 3.71$ are identified in figure 5 by the labels a, b, c, d, e, f, g, h, i and j for their occurrences at dimensionless times $t^*=S_o (t-t_{arr}) U/H = -2.70$, $-$0.75, $-$0.41, 0.00, 0.44, 0.65, 1.66, 4.37, 8.43 and 32.7, respectively. Figure 6(a–i) shows the corresponding depth-contour maps of the bow shock waves and the front runners on the $x$–$y$ plane.

Figure 6. Panels (a–i) show the depth $h/H$ contours of the bow shock waves on the $x$–$y$ plane at times $t^*=S_o (t-t_{arr}) U/H = -2.70$, $-$0.75, $-$0.41, 0.00, 0.44, 0.65, 1.66, 4.37 and 8.43, respectively. The corresponding wave force coefficient $C_{HU^2}$, stand-off distance $x_s/D$ and run-up height $R/D$ are shown in figure 5.

The stages represented in figure 6 are: (a) before the arrival of the front runner, (b) the beginning of the front-runner interaction with the bow shock wave, (c) the rise of the run-up to its maximum height, (d) the maximum compression of the stand-off distance and the rise of the impact force to its peak, (e–i) the relaxation of the bow shock wave with the reduction of the impact force and (j) the arrival of the second waves in the packet. We provide further description of these stages in the following subsections.

6. Dependence on Froude number and obstacle's location

We conducted simulations of the impact on a circular prism to find the impact coefficients on two dominant influencing parameters. Figure 7 shows the dependence of (a) the peak wave force coefficient $\hat {C}_{HU^2}$, and (b) the impact-duration coefficient ${ S_o{\mathcal {T}} U}/{H}$ on the undisturbed-flow Froude number $Fr$ and the location of the obstacle $S_ox_b/H$. The colours of the lines and symbols black, red and blue in the figure define the Froude numbers $Fr = 3.71$, 4.63 and 5.60, respectively. Four series of simulations designated as cases 1, 2, 5 and 6 are conducted. The tables embedded in the figure list the condition of the simulations.

Figure 7. Dependence of (a) the peak wave force coefficients $\hat {C}_{HU^2}$ and (b) the impact-duration coefficients $S_o{\mathcal {T}}U/H$ for the circular prism on two parameters: $Fr$ and $S_ox_b/H$. The black, red and blue colours refer to $Fr = 3.71$, 4.63 and 5.60, respectively. The green dashed lines in (a) and (b) represent the asymptotic limits. (c) The tables list the conditions of the simulations for the effects of the perturbation period $S_oTU/H$ and the friction coefficient $c_f$.

7. Shapes and orientations of the structures

We used the same numerical scheme to calculate the wave impact force on structures of different shapes and orientations. The obstacles examined are (a) square prism (SP), (b) circular prism (CP), (c) rotated-square prism (RP) and (d) triangular prism (TP). The base of the TP is an equilateral triangle. The depth-contour maps of $h/H$ in figure 8 show the patterns of the scattering waves around these obstacles on the $x$-$y$ plane at different times.

Figure 8. Details of the contour maps of $h/H$ on the $x$–$y$ plane for the wave impact on (a) square prism, (b) circular prism, (c) rotated-square prism and (d) triangular prism located at $x_b/D=266.67$ ($S_ox_b/H=1004.7$), at times $S_o(t-t_{arr})tU/H = -2.68$, 0, $+$2.68. The simulations were conducted for the dimensionless width of the obstacles $D/H =75$ and the undisturbed Froude number ${Fr} = 3.71$.

The stand-off distance $x_s$ of the bow shock wave and the run-up height $R$ depend on the obstacles’ shape. We found a stand-off distance particularly small for structures with a sharp and pointy front, such as the RP and TP. The small stand-off distance leads to a small run-up height and a small pile-up of water in front of the obstacle. Therefore, the wave impact force on the TP can be significantly smaller compared with the impact on a SP with the same dimensionless width $D/H$.

Figure 9(a–c) shows the dependence of the peak wave force coefficient $\hat {C}_{HU^2}$ on the two parameters $Fr$ and $S_o x_b/H$ for the obstacles of the four different cross-sections. All things being equal, the highest impact force is on the SP. The CP is the second, and the RP is the third in the wave impact force ranking. Finally, the least impact force is on the TP.

Figure 9. The dependence of the peak wave force coefficients $\hat {C}_{HU^2}$ (a–c) and the impact-duration coefficient $S_o{\mathcal {T}} U/H$ (d–f), on $Fr$ and $S_o x_b/H$ for obstacles of different shapes and orientations. The symbols $\square$, $\circ$, $\diamond$ and $\triangleleft$ refer to the results obtained for the SP, CP, RP and TP, respectively.

Depending on the obstacle's shapes and orientations, the wave force coefficient can be as high as $\hat {C}_{HU^2} \simeq 46.2$ for the front runner with ${Fr} = 3.71$ impacting on SP and as low as $\hat {C}_{HU^2} \simeq 8.12$ for the impact on TP located at $S_ox_b/H = 1000$. At the same location $S_ox_b/H = 1000$, the force coefficients on SP and TP are $\hat {C}_{HU^2} \simeq 84.0$ and 12.1 for ${Fr} = 4.63$, and $\hat {C}_{HU^2} \simeq 145$ and 15.6 for ${Fr} = 5.60$, respectively. The peak wave force coefficients $\hat {C}_{HU^2}$ of all obstacles at $S_ox_b/H = 1000$ are more than an order of magnitude greater than the textbook values of the drag coefficients of turbulent flow on obstacles.

Figure 9(d–f) shows impact-duration coefficients $C_{\mathcal {T}}={S_o{\mathcal {T}} U}/{H}$. The longest duration coefficient is associated with the impact on TP, while the SP has the shortest impact duration. Table 1 summarizes the numerical values for the peak wave force coefficient $\hat {C}_{HU^2}$ and the impact-duration coefficient $C_{\mathcal {T}}$ and the product of the two coefficients $\hat {C}_{HU^2} C_{\mathcal {T}}$ for the obstacles SP, CP, RP and TP located at $S_ox_b/H=1000$.

Table 1. The peak wave force coefficient $\hat {C}_{HU^2}$, impact-duration coefficient $C_{\mathcal {T}}$ and the product of $\hat {C}_{HU^2}$ and $C_{\mathcal {T}}$ for SP, CP, RP and TP located at $S_ox_b/H=1000$.

7.1. Direction cosine of the oblique surface

The orientation of the obstacle surface is provided by the direction cosine, $\cos \phi = - \boldsymbol {{i}\,{\cdot }\,{n}}$, defined by the unit vector $\boldsymbol {{i}}$ in the direction of the incoming waves and the outward unit vector $\boldsymbol {{n}}$ perpendicular to the oblique surface. As shown in figure 10, the direction cosines are (a) $-\boldsymbol {i} \boldsymbol {\cdot } \boldsymbol {n}=\cos 45^\circ$ and (b) $-\boldsymbol {i} \boldsymbol {\cdot } \boldsymbol {n}=\cos 60^\circ$ for the oblique surfaces of the RP and the TP, respectively.

Figure 10. The incident waves and reflected waves on the oblique surfaces of (a) the RP and (b) TP. The direction cosines are (a) $-\boldsymbol {i} \boldsymbol {\cdot } \boldsymbol {n}=\cos 45^\circ$ and (b) $-\boldsymbol {i} \boldsymbol {\cdot } \boldsymbol {n}=\cos 60^\circ$ for the oblique surfaces of the RP and the TP, respectively.

We propose the following heuristic interpretation for impact reduction. The pressure force per unit length $f_{hs}$ in (4.2) is assumed proportional to the normal component of the incident and reflected waves. If the incident wave momentum flux is $\dot {m} \, \boldsymbol {i}$, the contribution of the normal component to the pressure on the frontal surface of the obstacle would be $- \dot {m} \, \boldsymbol {i} \boldsymbol {\cdot } \boldsymbol {n}=\dot {m} \cos \phi$. That is, $f_{hs} \propto \dot {m} [\cos \phi ]$, because the pressure is proportional to the component of the incident and reflected waves perpendicular to the oblique surface. Then the force $F_w$ is the component of the pressure force in the direction of the wave. The result is the following approximation for the wave force:

(7.1)\begin{equation} F_w ={-} \oint_{s} f_{hs} \, \boldsymbol{i} \boldsymbol{\cdot} \boldsymbol{n} \, {\rm d}s \propto \oint_{s} \dot{m} \, [\cos \phi]^2 \,{\rm d}s. \end{equation}

As shown in figure 8, the back end of the obstacle remains dry. Therefore, the contribution of the rear surface to the wave force is negligible.

On the oblique surface of the RP, the direction-cosine square is $[\cos 45^\circ ]^2= 0.5$. Therefore, the roll-wave impact force on the RP prism would be 50 % of the impact force on the SP if the cosine square were the only factor. On the other hand for the oblique surface of the TP, $[\cos 60^\circ ]^2= 0.25$. The wave impact force on the TP would be 25 % of the force on the SP.

This heuristic interpretation does not account for the different lengths of the line integration between RP and TP. It also does not consider the possible difference in the pressure distribution over the oblique surfaces. Nevertheless, the cosine-square correction has brought the data of the impact-force coefficients for the RS prism denoted by the diamond symbols to closely follow the data for the TP represented by the triangular symbols as shown in figure 11.

Figure 11. The relative wave force coefficient ${\hat {C}_{HU^2}}/ \{ [\hat {C}_{HU^2}]_{SP} [\cos \phi ]^2 \}$ for the RP and TP, corrected by the direction-cosine square. The dependence of the relative coefficient on (a) the position $S_o x_b/H$ and undisturbed-flow Froude number $Fr$, and (b) the amplitude of the front runner $\hat {h}_{FR}/H$. The symbols $\diamond$ and $\triangleleft$ denote the data for RP and TP, respectively.

7.2. Impact by front runner of finite amplitude

A similar radiation pressure dependence on the cosine square is known for electromagnetic waves impinging on oblique plane surfaces (Wright Reference Wright1992). Unlike electromagnetic waves, the roll-wave dynamics is nonlinear. Hence, the reduction is further dependent on the amplitude of the roll waves. Figure 11(a) shows the reduction in terms of the relative coefficient that depends on the obstacle's position and undisturbed-flow Froude number, and figure 11(b) on the amplitude of the front runner, $\hat {h}_{FR}/H$. Note that, as shown in figure 11(b), the ratio ${\hat {C}_{HU^2}}/{ [\hat {C}_{HU^2}]_{SP} [\cos \phi ]^2} \simeq 1$ as front runner's amplitude $\hat {h}_{FR}/H$ approaches unity for small-amplitude roll waves. Therefore, the cosine square will be the only reduction to the force on the oblique surface if the waves’ amplitude is sufficiently small.

We may use the relationship in figure 11(b) to find the wave impact on the structure with a sharp front pointing to the incoming waves of finite amplitude. As an example, for a front runner with an amplitude of $\hat {h}_{FR}/H= 2$, the relative coefficient varies in the range

(7.2)\begin{equation} \frac{\hat{C}_{HU^2}}{ [\hat{C}_{HU^2}]_{SP} [\cos \phi]^2} \simeq 0.72 \ {\rm to} \ 0.83 \end{equation}

in the range of Froude number varying from $Fr = 5.60$ to 3.71. The selection of a TP would have $[\cos \phi ]^2 = 0.25$ and hence a relative coefficient $[\hat {C}_{HU^2}]_{TP} / [\hat {C}_{HU^2}]_{SP} \simeq 0.18$ to 0.21. This calculation shows the effect of the front runner's amplitude and the structure's shape and orientation. The wave force on the TP pointing at the incoming wave is $[\hat {C}_{HU^2}]_{SP} / [\hat {C}_{HU^2}]_{TP} \simeq 4.8$ to 5.6 times smaller compared with the impact on a blunt object of the SP having the same width in this example with a front-runner amplitude of $\hat {h}_{FR}/H= 2$.

8. Summary and conclusion

In this paper, we studied the spatial development of roll waves produced by a local disturbance focusing on the impact force of the waves against obstacles. Numerical simulations were conducted using the shallow-water model and an adaptive quadtree mesh to determine the evolution of the roll-wave packet and the impact force against obstacles of various shapes and orientations. We found the wave impact force raised sharply to a peak on the arrival of the front runner of the wave packet.

The peak force coefficient depends on two dominant influencing parameters: the Froude number of the undisturbed flow $Fr$ and the dimensionless obstacle distance from the local disturbance $S_o x_b/H$. The front-runner amplitude increases toward the long-wavelength limit of Dressler's solution, which are $\hat {h}_{FR}/H= 4.20$, 6.48 and 9.25 and $c_{FR}/U= 1.89$, 2.08 and 2.26 for $Fr = 3.71$, 4.63 and 5.60, respectively. The corresponding asymptotic limits of the wave force coefficient for the impact on the CP are ${\hat {C}}_{HU^2} \simeq 22.3$, 46.9 and 88.4 for $Fr = 3.71$, 4.63 and 5.60, respectively. These wave force coefficients are much greater than the steady force coefficient without the roll waves. Therefore, determining the structural loading from the steady uniform flow and ignoring the roll-wave instability would significantly underestimate the impact.

The roll waves are destructive for the waves’ large momentum flux compared with the uniform flow. The wave impact against blunt objects like the SP and CP is most significant. On the other hand, pointy objects such as the RP and TP can re-direct the incident waves to reduce the impact force. For example, the rotation of a SP by 45$^\circ$ reduces the impact force coefficient by more than a factor of two. We suggest using a pointy front with a sharp angle to minimize the impact and propose a heuristic interpretation to evaluate the impact reduction. The impact force reduces proportionately to the direction-cosine square $[\cos \phi ]^2$ of the oblique surface for small-amplitude waves. A further reduction is related to the finite amplitude of the roll waves.

The simulation for the impact force is computationally demanding because a large computation domain is required to fully accommodate the roll-wave packet's development to the nonlinear stage. Limited by the finite computational resources, we have ignored the viscous and turbulence stresses in our shallow-water 2-D modelling of the wave force. The present simulation is the first-ever numerical study of the roll-wave impact force on structures. However, our 2-D simulation results are comprehensive, covering a good range of conditions for various influencing factors, setting the stage for future studies of the 3-D aspects of the problem.