The effect of obstacle length and height in subcritical free-surface flow

Abstract

Two-dimensional free-surface flow past a submerged rectangular disturbance in an open channel is considered. The forced Korteweg–de Vries model of Binder et al. (Theor Comput Fluid Dyn 20:125–144, 2006) is modified to examine the effect of varying obstacle length and height on the response of the free-surface. For a given obstacle height and flow rate in the subcritical flow regime an analysis of the steady solutions in the phase plane of the problem determines a countably infinite set of discrete obstacle lengths for which there are no waves downstream of the obstacle. A rich structure of nonlinear behaviour is also found as the height of the obstacle approaches critical values in the steady problem. The stability of the steady solutions is investigated numerically in the time-dependent problem with a pseudospectral method.

1 Introduction

Free-surface flow past topographic disturbances in an otherwise flat and horizontal impermeable channel bottom is a well-studied problem in hydrodynamics with many practical applications, including open channel flow past infrastructure such as weirs, dams and spillways. A common occurrence downstream of the disturbance are large surface waves which can damage infrastructure and degrade or erode waterway banks with negative impacts for the flora and fauna that reside along them [1,2,3,4]. When a frame of reference is taken moving with the disturbance, a two-dimensional flow approximation can also model the local behaviour along the downstream centre-line of an underwater craft (e.g. submarine) moving close to the surface. In this physical situation surface waves are undesirable as they contribute to the wave drag or resistance on the moving submerged disturbance [5,6,7,8,9]. Thus a key goal in these applications is to understand how to reduce, dampen or eliminate the amplitude of these unwanted surface waves, motivating many studies that go as far back to the seminal work of Lamb [10].

Lamb [10] used linear theory with an ideal or potential two-dimensional flow approximation to model steady flow past a semi-ellipse disturbance in the bottom of an open channel. They predicted waveless solutions for a countably infinite set of ellipse-lengths when the length of the ellipse is a constant multiple of a Bessel function’s zero, for fixed ellipse height and flow rate. Forbes [7, 8] considered the non-linear steady flow problem to compute drag free solutions that depended on both ellipse length and height. More recently, Hocking et al. [11] revisited the non-linear problem and with improvements in the efficiency of numerical computations was able to confirm the waveless predictions up to the sixth harmonic, for a fixed flow rate.

Binder et al [12,13,14,15] used both a weakly and fully nonlinear approach to model steady flow past a rectangular or boxcar disturbance in the channel bottom. The weakly nonlinear model is a forced Korteweg–de Vries equation (fKdV) with a piecewise constant forcing term representing the boxcar disturbance in the channel topography [12]. The piecewise constant forcing term allows for the use of phase plane analysis in what would otherwise be a non-autonomous problem for other types of topographic forcing (e.g. an ellipse). The weakly nonlinear model is then essentially described by two autonomous problems with one (unforced) phase plane representing the flow upstream and downstream of the boxcar and a second (forced) phase plane representing the flow directly above the boxcar. Weakly nonlinear solutions in the phase plane can be constructed by transitioning from trajectories and fixed points in one phase plane to intersecting trajectories and fixed points in the other.

Solutions identified with the weakly nonlinear phase plane analysis have been computed in the corresponding fully nonlinear problem using boundary integral methods [12,13,14,15,16,17,18,19,20] and good quantitative agreement is found between the two approaches when the height of the boxcar is small and the dimensionless flow rate is close to unity. Importantly, the qualitative agreement between the weakly and fully nonlinear theories is good provided the maximum wave amplitude on the free-surface does not approach a Stokes limiting configuration of a stagnation point with an enclosed angle of $120^{\text {o}}$ at the wave crest [21, 22]. Therefore, in the present work we choose not to compute numerical solutions to the fully nonlinear problem and instead focus on the analytical predictions of the weakly nonlinear theory to gain further insight into the fundamental nature of the problem.

In this paper, we use the phase plane analysis to enable a systematic approach to determining the qualitatively different types of steady solution when the dimensionless flow rate is less than unity with uniform subcritical flow far upstream. A comprehensive map of the steady solutions in the boxcar height versus length space is presented and discussed to further our understanding on the effect of varying obstacle length and height on the waveless solutions. In addition, and in terms of the weakly nonlinear theory, the choice of disturbance also allows us to connect the shortening of a long-length boxcar to that of a short-length bump [23]. Shen [24] established the use of a Dirac delta function, or point forcing to approximate a short-length disturbance on the channel bottom. This approximation allows one to find solutions in a single unforced phase plane with a vertical jump from a fixed point or trajectory to another trajectory or fixed point [25,26,27].

For the weakly nonlinear problem with boxcar forcing there are in general four fixed points and two phase planes, which gives rise to a rich and complicated structure of solution behaviour [12, 22]. Therefore, we first linearise the fKdV model to reduce the number of fixed points from four to two, one point for each of the two (unforced and forced) phase planes. In the linearised problem, the solution behaviour downstream of the obstacle is described by the superposition of two cosines, corresponding to waves produced at each end of the boxcar, which cancel each other out for discrete lengths of the boxcar in the waveless case. Similar to Lamb’s linear result of flow past an ellipse (i.e. the zeros of Bessel functions) when the length of the boxcar is a constant multiple of a sine function’s zero we have waveless flow downstream of the boxcar. The linearised fKdV also allows us to isolate the limiting behaviour as the length of the boxcar tends to zero while the area of the boxcar remains constant. We show that the two phase plane boxcar model is consistent with the one phase plane point forcing model for short-length disturbances, and find that rather than vanishing, the downstream amplitude instead approaches the value of the amplitude of the corresponding point forcing solution.

We find that, when the height of the boxcar is less than a determined critical value, the sinusoidal wave cancellation found in the linearised fKdV is essentially replaced with cnoidal wave cancellation in the weakly nonlinear fKdV problem, for a countably infinite set of discrete boxcar lengths. We find behaviours similar to those for the fully nonlinear problem of flow past an ellipse [11], and comment on the main differences. As the height of the boxcar approaches and passes critical values, we find bifurcations in the solution behaviour with an increasing maximum downstream wave amplitude replaced by a constant supremum downstream wave amplitude.

To conclude our study, we examine the stability of the steady solutions numerically using an established pseudospectral method [16,17,18,19, 21, 28, 29]. We show that steady solutions with waves downstream (for sufficiently small boxcar heights) are stable to perturbations in the time dependent problem. Steady solutions that are waveless with subcritical flow downstream of the boxcar are unstable. However, given a sufficient duration in time, the perturbed solution approaches another stable steady solution with very small amplitude waves downstream of the boxcar. Therefore, the map of stationary waveless solutions provided in this work may inform engineering designs in practical applications which aim to reduce the amplitude of unwanted surface waves downstream of a disturbance.

2 Forced Korteweg–de Vries model

We take a frame of reference with the disturbance in the channel bottom moving at constant speed U from right-to-left with the assumption that the flow far upstream approaches a uniform flow with constant velocity U and depth H. The conditions of the flow can be characterised by the dimensionless flow rate, or upstream depth-based Froude number

$$\begin{aligned} F = \frac{U}{\sqrt{gH}}, \end{aligned}$$

(1)

where g is the acceleration due to gravity. The Froude number also represents the ratio of the characteristic flow speed to the speed of linear water waves in shallow water. When $F<1$ the flow far upstream is called subcritical and when $F>1$ the flow far upstream is called supercritical [10]. In this work, we shall only consider flows that are subcritical and uniform far upstream with $F<1$.

We consider the two-dimensional irrotational flow of an incompressible inviscid fluid bounded by the free surface and impermeable channel bottom topography. Dimensionless quantities can be made by taking the uniform velocity U as the velocity scale and uniform depth H as the length scale. With these assumptions the time dependent forced Korteweg–de Vries (fKdV) equation [12, 22, 27]

$$\begin{aligned} 6\eta _{t} - \eta _{xxx} - 9 \eta \eta _{x} + 6(F-1)\eta _{x}&= 3\sigma _x, \end{aligned}$$

(2)

can be used to model the response of the free-surface elevation $\eta$ due to the topographic forcing $\sigma$. The uniform flow far upstream gives the conditions

$$\begin{aligned} \eta (x,t) \rightarrow \, 0\quad \eta _{x}(x,t) \rightarrow \, 0 \quad \eta _{xx}(x,t) \rightarrow \, 0\quad \text {and}\quad \sigma (x) \rightarrow \, 0 \quad \text {as} \quad x\rightarrow - \infty . \end{aligned}$$

(3)

The free-surface is located at $y=1+\eta$ (e.g. see Fig. 1).

The fKdV is a weakly nonlinear approximation to the full problem and the second and third terms in the left-hand side of Eq. (2) model the effect of dispersion and nonlinearity. Hence the wave speed depends on both the wavelength and wave amplitude, which is not predicted by the linear theory of waves in shallow water. Previous studies (see review of Binder  [22]) have shown both the qualitative and quantitative agreement between the weakly nonlinear approximation and results from the fully nonlinear problem is good when the value of the Froude number is close to unity and the height of the disturbance is small, and qualitative agreement is excellent provided the maximum wave amplitude on the free-surface doesn’t approach a Stokes limiting configuration with a stagnation point and enclosed angle of 120$^{\text {o}}$ at the wave crest. An advantage of the fKdV approximation is its amenability to analytical analysis whereas only numerical solutions are available to analyse in models of the full problem. Thus, the parametric maps of the steady solution space presented in this work can be readily determined without recourse to time consuming numerical computations needed in the full problem.

3 The steady problem

If the flow is steady (2) can be integrated to obtain

$$\begin{aligned} \eta _{xx}+\frac{9}{2}\eta ^2-6(F-1)\eta= & {} -3\sigma , \end{aligned}$$

(4)

where the boundary conditions (3) for uniform flow far upstream have been used to evaluate the constant of integration.

Our primary aim is to understand the response of the free-surface to changes in the length of a rectangular obstacle. We also wish to connect or relate this behaviour to the known point forcing results for flow past a short-length bump [24]. Therefore we consider a boxcar disturbance in the bottom of the channel described by a linear combination of unit step functions, u(x),

$$\begin{aligned} \sigma (x)=\frac{A}{2 L}(u(x+L)-u(x-L))=h(u(x+L)-u(x-L)), \end{aligned}$$

(5)

where $L>0$ gives the length of the boxcar, $l=2L$, with height $h=A/(2 L)$ and amplitude of forcing, A. Note the height of the boxcar can be negative or positive, for $A<0$ or $A>0$, respectively.

Crucially, the boxcar disturbance (5) allows us to perform an analytical analysis of the problem in the weakly nonlinear phase plane of the problem. Binder et al. [12, 22] showed that (4) with forcing (5) can be expressed as two autonomous systems on $| x | >L$ and $|x|<L$. The fixed points and trajectories that describe all the possible solutions in the $(\eta ,\eta _x)$ phase plane are given by

$$\begin{aligned} \eta ^{\pm }=\frac{2}{3}(F-1)\pm \sqrt{\frac{4}{9}(F-1)^2-\frac{2}{3}h} \quad \text{ and } \quad \eta _x=0, \end{aligned}$$

(6)

and

$$\begin{aligned} \eta _{x}^2=6(F-1)\eta ^2-3\eta ^3-6h\eta +C, \end{aligned}$$

(7)

for different values of the constant, C. Equations (6) and (7) describe an unforced phase portrait (A), for $|x|>L$ when $h=0$ [e.g. see black broken curves (Figs. 1e & 2e)]; and a forced phase portrait (B), for $|x|<L$ when $h\ne 0$ [e.g. see black solid curves (Fig. 2e)]. For subcritical flow with $F<1$, in both portraits (A) and (B), the fixed point $(\eta ^{+},0)$ is a centre and the fixed point $(\eta ^{-},0)$ is a saddle. We note that the fixed points do not exist for all values of F in portrait (B) when $h>0$. Their existence requires a positive discriminant in (6), thus defining the maximum height,

$$\begin{aligned} h_{\text {max}}=\frac{2}{3}(F-1)^2, \end{aligned}$$

(8)

for which the fixed points of portrait (B) coalesce into a degenerate node at $(\frac{2}{3}(F-1),0)$. In the setting of a semi-infinite step this is the maximum height for which solutions exist [12]. However, we will see that boxcar solutions continue to exist for larger values of the boxcar height.

Analysis of the trajectory equations provides two additional heights,

$$\begin{aligned} h^{+}=\frac{1}{2}(F-1)^2, \end{aligned}$$

(9)

and

$$\begin{aligned} h^{-}=-\frac{8}{9}(F-1)^2. \end{aligned}$$

(10)

We will see that solution behaviour undergoes a bifurcation at these values.

As previously stated, we wish to examine the behaviour as the boxcar decreases in length, and as noted by Grimshaw [23], when taking the limit $L\rightarrow 0$ in (5), we find

$$\begin{aligned} \sigma (x)=A\delta (x), \end{aligned}$$

(11)

where $\delta (x)$ is the Dirac delta function. This recovers Shen’s [24] point forcing model for a short-length compact bump, where the amplitude of forcing A is the area of the bump. Short-length bump solutions in the phase plane are described by portrait (A) (with $h=0$ and $x\ne 0$) along with a discontinuity in the derivative

$$\begin{aligned} \eta _x(0^+)-\eta _x(0^-)=-3A, \end{aligned}$$

(12)

at the location of the point of forcing, $x=0$.

The idea now is to use an analysis of the phase plane to identify physically realistic solutions for the boxcar forcing that satisfy the uniform flow conditions (3). The conditions require that all solutions begin their journey in the phase plane at the centre (0, 0) of phase portrait (A), for $F<1$. The flow in the phase plane is in the clockwise direction because the flow is from left to right in the physical plane, relative to the boxcar. In the case of boxcar forcing, the solutions in the phase plane consist of two transitions at intersection points between the trajectories (and fixed points) of portraits (A) and (B), the first from (A) to (B) and the second from (B) back to (A). In the physical plane, these transitions correspond to the location of the two edges of the boxcar at $|x|=L$. In contrast, for point forcing the movement in the phase plane is solely between the trajectories and fixed points of portrait (A) via the vertical jump condition (12), giving a discontinuity in the slope of the free-surface at the location of the bump, $x=0$, in the physical plane (see Fig. 1).

To compute numerical solutions to the steady problem, we discretise Eq. (4) with a second order, central finite difference scheme. This results in a nonlinear system of algebraic equations which are solved using matlab’s fsolve for the prescribed forcing, $\sigma$, and subject to the boundary conditions (3).

Fig. 1

Point forcing, $F=0.8$. a–d Solutions in the physical plane. e–h Corresponding solutions (red curves) in the phase plane (broken curves, portrait (A)). a and b Subcritical flow, $A=0.03$ and $A=-0.03$. c and d Hydraulic falls, $A=A_{\text {max}}$ and $A=-A_{\text {max}}$

Fig. 2

Negative boxcar forcing, $F=0.8$, $h<0$. a–d Subcritical flow, $h=-0.02$. Left-right $l=\{20, 18.75, 17.50, 16.25\}$. e–h Corresponding solutions (red curves) in the phase plane, for a–d. i–k Hydraulic falls. i $h=-0.036$, $l=22.10$. j $h=-0.05$, $l=19$. k $h=-0.05$, $l=22.05$. l–n Corresponding solutions in the phase plane, for i–k. The black broken and solid curves belong to portraits (A) and (B), respectively

For uniform subcritical flow far upstream with $F<1$ there are three basic solution types which can be characterised by the behaviour of the flow far downstream as $x\rightarrow \infty$. The first is non-uniform flow with waves far downstream [see Figs. 1a,b, 2b–d, 3b–d,j,k,r–t, 4a,c]. The second is uniform subcritical flow with $\eta \rightarrow \eta ^{+}=0$ (portrait (A)) far downstream [see Figs. 2a, 3a,i,l]. In this case the downstream Froude number is equal to the upstream Froude number, $F<1$. The third is uniform flow with $\eta \rightarrow \eta ^{-}=-4/3|F-1|$ (portrait (A)) far downstream [see Figs. 1c,d, 2i–k, 3q, 4b,d], and in this case we can define an additional downstream Froude number

$$\begin{aligned} F_d=2-F, \end{aligned}$$

(13)

in terms of the upstream Froude number, $F<1$ [22]. This third basic type of solution, with $F<1$ and $F_d>1$, is called a hydraulic fall [30]. When there is a hydraulic fall in the point forcing problem the magnitude of the amplitude of forcing is

$$\begin{aligned} |A|= A_{\text {max}}=\frac{4\sqrt{2}}{9}|F-1|^{\frac{3}{2}}, \end{aligned}$$

(14)

which cannot be chosen independently of the Froude number, F, [22, 25].

In addition to the downstream Froude number, $F_d>1$, and for the boxcar forcing only, it is possible to define an approximation for the local Froude number directly above the support of forcing, $|x|<L$ [22]

$$\begin{aligned} F_b=1\mp \sqrt{(F-1)^2-\frac{3}{2}h} \quad \text{ for } \quad \eta ^{\pm }, \end{aligned}$$

(15)

where $\eta ^{\pm }$ is a fixed point of portrait (B). The validity of (15) depends on the length of the boxcar and the local behaviour of the flow above the boxcar. If the flow is locally subcritical / supercritical then the corresponding fixed point in (15) to examine is $\eta ^+$ / $\eta ^-$, respectively. In the case of maximum height (8), where there is a single fixed point in portrait (B), the discriminant of (15) is zero and so the local Froude number is critical with $F_b=1$.

Fig. 3

Positive boxcar forcing, $F=0.8$, $h>0$. Rows 1, 3, 5. Solutions in the physical plane. Rows 2, 4, 6. Corresponding solutions in the phase plane, for Rows 1, 3, 5. a–d $h=0.011$, $l=\{20, 18.33, 16.66,14.99\}$. i–l $h\approx 0.02$, $l=40$. q–t $h\approx 0.02$, $l=20$. The black broken and solid curves belong to portraits (A) and (B), respectively

Fig. 4

Positive boxcar forcing with $h\ge h_m$, $F=0.8$. a–d Solutions in the physical plane. e–h Corresponding solutions in the phase plane. a and b $h=h_m=0.27$, $l=\{2, 2.72\}$. c and d $h=0.032$, $l=\{1.5, 2\}$. The black broken and solid curves belong to portraits (A) and (B), respectively

The solution space for the point forcing problem has received considerable attention in many previous studies. Figure 1 shows all the qualitatively different types of solutions for upstream subcritical flow (see Binder [22] for a more comprehensive review). However, classification of the solutions to the boxcar forcing problem has received much less attention and the solution behaviour as the boxcar shortens in length is not as well understood. The complexity of the boxcar solutions can be seen in the phase plane (see Figs.  2 and 3, for example) where there is in general four fixed points and two portraits (A) and (B) of trajectories to consider when trying to identify solutions that satisfy the uniform flow condition at $x=-\infty$.

Analysing the localised linear behaviour is difficult on the nonlinear phase diagram because the movement around and between the fixed points under examination are close to each other. This is further complicated by the limiting behaviour of the boxcar as it shortens in length. In order to gain insight into the nonlinear boxcar problem we therefore consider the linearised problem and omit the nonlinear term in (4). As we shall see, this enables us to blow-up the localised linear behaviour of the solutions and examine / analyse closed form solutions as the length of the boxcar tends to zero.

4 Linearised problem

The fixed points and trajectories that describe the linear solutions for the boxcar in the phase plane are then

$$\begin{aligned} \eta ^e= & {} \frac{3h}{6(F-1)} \quad \text{ and } \quad \eta _x^e=0, \end{aligned}$$

(16)

$$\begin{aligned} \eta _{x}^2= & {} 6(F-1)\eta ^2-6h\eta +\frac{3}{2}\frac{h^2}{(F-1)}+ C. \end{aligned}$$

(17)

Equations (16) and (17) describe an unforced phase plane (A*), for $|x|>L$ when $h=0$ [e.g. see black broken curves Fig. 5e]; and a forced phase plane (B*), for $|x|<L$ when $h\ne 0$ [e.g. see black solid curves Fig. 5e]. The two fixed points are both centres when $F<1$ and saddles when $F>1$. In the case of point forcing, we have (A*) (with $h=0$) and the jump condition (12) with a single fixed point at the origin. There are now only two fixed points in the linear boxcar forcing problem and a single fixed point in the linear point forcing problem. Moreover, the solutions in the physical plane can be solved using elementary methods once a viable solution has been identified in the linearised phase plane of the problem.

For the linear subcritical flow problem, with $F<1$, we have the boxcar forcing solution

$$\begin{aligned} \eta _b(x)=\frac{3h}{\mu ^2} {\left\{ \begin{array}{ll} 0 &{} x<-L\\ \cos \mu (x+L) -1&{}-L<x<L\\ \cos \mu (x+L)-\cos \mu (x-L)=-2\sin \mu x \sin \mu L &{}x>L, \end{array}\right. } \end{aligned}$$

(18)

and for the point forcing we find

$$\begin{aligned} \eta _p(x)=-\frac{3A}{\mu }u(x)\sin {\mu x}, \end{aligned}$$

(19)

where $\mu =\sqrt{|6(F-1)|}$.

This leads us to define the wave amplitude directly above the boxcar, downstream of the boxcar, and downstream of the point forcing as

$$\begin{aligned} a^e_b=\frac{3h}{\mu ^2}, \quad a_b=\frac{6h\sin \mu L}{\mu ^2},\quad a_p=\frac{3A}{\mu }, \end{aligned}$$

(20)

respectively. The wavelength in both the boxcar and point forcing problems is $\lambda =\frac{2\pi }{\mu }$.

We examine three sets of discrete values of L for the boxcar forcing problem,

$$\begin{aligned} L^m(n)=\frac{(2n+1)\pi }{2\mu }, \quad L_{\pm }^e(n)=\frac{(6n\pm 1)\pi }{6\mu }, \quad L^w(n)=\frac{n\pi }{\mu }, \end{aligned}$$

(21)

where $n\in \mathbb {Z^+}$. The first set, $L^m(n)$, maximises the wave amplitude,

$$\begin{aligned} a^m_b=\frac{6h}{\mu ^2}, \end{aligned}$$

downstream of the boxcar. In this case, the two cosines in (18) are in phase and resonate with each other. The second set, $L_{\pm }^e(n)$, is when the amplitude downstream of the boxcar is equal to the amplitude above the boxcar, i.e. $a_b=a^e_b$. The third set is for waveless flow downstream of the boxcar and in this case the two cosines are out of phase and cancel each other out.

The results shown in Fig. 5 are for a long-length boxcar as it shortens in length (from left to right), for the $n=2$ members of each set in (21). We choose to focus our discussion on the analysis in the phase plane because the qualitative behaviour of the linear problem is applicable to the weakly non-linear problem. In the phase plane, the entire upstream portion of the uniform free-surface, $-\infty<x<-L$, is represented by the centre located at the origin, or the fixed point of portrait (A*). At the upstream edge of the boxcar, $x=-L$, there is a transition from this fixed point onto the intersecting red orbit belonging to (B*), which corresponds to the waves above the boxcar, $-L<x<L$, with amplitude $a^e_b$. This behaviour is common to all the results shown in the second row of Fig. 5. At the downstream edge of the boxcar there is a second transition back to (A*) and the downstream amplitude depends on the length of the boxcar. This transition occurs at a trough of both waves directly above and downstream of the boxcar, with the downstream waves attaining their maximum amplitude (larger red orbit), $a^m_b$, for $L^m(2)$ in Fig. 5a,e. As the boxcar shortens in length the downstream amplitude decreases, and for the results shown in Fig. 5b,f it has decreased to the same amplitude of the waves above the boxcar, $a_b=a^e_b$, for $L_+^e(2)$. The transition occurs at the intersection of the two red orbits in the lower half of the phase plane of Fig. 5f. The downstream amplitude continues to decrease as the boxcar continues to shorten in length, and in Fig. 5c,g we observe waveless flow where the red orbit intersects (for the second time) with the fixed point of (A*), for $L^w(2)$. There is now a change in the solution behaviour with the amplitude of the waves downstream of the boxcar increasing as the boxcar decreases in length. In Fig. 5d,h, we once again see waves with the same amplitude above and downstream of the boxcar, $a_b=a^e_b$, for $L_-^e(2)$. However, the profile in Fig. 5d is qualitatively different to the profile in Fig. 5b. This is because the transition from portrait (B*) to (A*) occurs at the intersection of the two red orbits in the upper half of the phase plane of Fig. 5h. The same downstream behaviour seen in Fig. 5 is found for longer length boxcars $L_-^e(n) \le L\le L^m(n)$, for $n\ge 3$, with an additional wave appearing directly above the boxcar for each integer increment in n.

Fig. 5

Linearised model, long-length boxcar, $F=0.8$, $h=0.1$. a–d Physical plane, $L=\{L^m(2)=7.17, L^e_{+}(2)=6.21, L^w(2)=5.74, L^e_{-}(2)=5.26\}$. e–h Corresponding solutions in the phase plane, for a–d

Fig. 6

Linearised model, medium-length boxcar, $F=0.8$, $h=0.1$. a–d Physical plane. $L=\{L^m(1)=4.30, L^e_{+}(1)=3.35, L^w(1)=2.87, L^e_{-}(1)=2.39\}$. e–h Corresponding solutions in the phase plane, for a-d

Fig. 7

Linearised model, short-length boxcar and point forcing, $F=0.8$. a, b, d Boxcar forcing, $h=0.1$, $L=\{L^m(0)=1.43, L^e_{\pm }(0)=0.48 (A\approx 0.1=h), L=0.1 (A=0.02)\}$. c Point forcing, $A=0.1$. e–h Corresponding solutions in the phase plane, for a–d

The results for $L^m(1)$ are shown in Fig. 6a in which the downstream waves have once again reached their maximum amplitude, with the length of the boxcar being a single wavelength $\lambda$ shorter than the boxcar length in Fig. 5a. We return to examining the solution behaviour as the boxcar continues to decrease in length (see Figs. 6 and 7). In Fig. 6c the length of the boxcar is equal to the wavelength, $l=2L=\lambda$. Boxcar lengths less than a wavelength $\lambda$ can no longer support waves directly above the boxcar (see Figs. 6d,h and 7), and in the phase plane this is shown by the solution trajectory traversing only a section of the orbit of portrait (B*). It is interesting to examine the boxcar and corresponding point forcing problem for approximately the same value of $A=0.1$ in Fig. 7b,c. We see the point forcing approximation is very good when

$$\begin{aligned} L<\frac{\lambda }{2}=\frac{\pi }{\mu }, \end{aligned}$$

(22)

thus defining a short-length bump in the linear subcritical flow problem. The approximation improves as the bump continues to decrease in length for a fixed h (see almost vertical transition along a small section of the orbit belonging to portrait (B*) in Fig. 7d,h), and moreover in the limiting process as $L\rightarrow 0$ for fixed A (with $h=A/(2L)$) we have

$$\begin{aligned} \lim _{L\rightarrow 0} \eta _b=\eta _p, \end{aligned}$$

and hence, we recover the point forcing solution for subcritical flow. In the phase plane, the limiting behaviour corresponds to the fixed point of (B*) $\eta ^e\rightarrow -\infty$, and this is illustrated in Fig. 8. In particular, we observe that the movement along the orbit of (B*) that intersects with the centre of (A*) is almost vertical, thus providing a graphical representation of the consistency between the point and boxcar forcing models.

A similar set of results is found when $h,A<0$, for the linear problem. In the physical plane the free-surface elevation is simply reflected about the x-axis with the corresponding behaviour in the phase plane being a reflection about the $\eta$ axis followed by a reflection about the $\eta _x$ axis, for the results shown in Figs. 5–7 with $h\rightarrow -h$. An animation showing both the physical and phase plane results for the boxcar as the length varies for fixed |h| is provided in the supplementary material.

Fig. 8

Limiting behaviour in linearised model, $F=0.8$, $A=0.1$. a and b Boxcar forcing, $L=0.1$, $h=0.5$ (i.e. indicative of $L\rightarrow 0$ and $|\eta ^e|\rightarrow \infty$). c and d Point forcing

5 Weakly nonlinear problem

In the linear problem for fixed boxcar height, h, and Froude number, F, with varying boxcar length, l, a maximum and zero downstream wave amplitude is found as the boxcar shortens or increases in length by a surface wavelength, $\lambda$. This behaviour is explained by the two cosines in the solution (18) with each one being switched on at the upstream and downstream edge of the boxcar. When the cosines are in phase there is a maximum amplitude and when they are out of phase there is zero amplitude. Qualitatively similar behaviour is found in the weakly nonlinear problem but with the orbits now being described by Jacobi elliptic functions [31] instead of the two sinusoids in the linear problem (see first two rows of Figs. 2 and 3). Good quantitive agreement between the linear and weakly nonlinear problems is found when |h| is small (see Fig. 9a). This is because the two centres are close together in the weakly nonlinear phase plane and the movement between the portraits (A) and (B) result in small deviations of the free surface elevation $\eta$ in the physical plane, consistent with the linearised problem. However, the additional two saddle points and larger |h| in the weakly nonlinear problem provide a much richer structure of solutions as is seen in Figs. 2i–n, 3i–x and 4.

We use a weakly nonlinear phase plane analysis to categorise the solutions into eight cases I–VIII, which depend on the value(s) of the box height, h, given in Table 1, for a fixed value of $F<1$.

Table 1 Value of h for the cases I–VIII

The results are summarised in a plot of the supremum downstream amplitude against the height of the boxcar shown in Fig. 9a (red curve and markers). For each of the cases I–VIII, we plot a phase portrait to help with our discussion of the solution behaviour (second and fourth columns, Fig. 9). The blue trajectory corresponds to the free-surface directly above the boxcar and is determined by h. For cases I–IV the blue trajectory is periodic and the trajectory Eq. (7) can be used to determine its amplitude,

$$\begin{aligned} a_{\text {blue}}=\frac{1}{2}\left| (F-1)+\sqrt{(F-1)^2-2h}\right| . \end{aligned}$$

(23)

All solutions transition from the centre equilibrium point located at the origin in portrait (A) to the blue intersecting trajectory in portrait (B), regardless of the length of the boxcar, l. Solutions then traverse along the blue trajectory before transitioning to an intersecting trajectory belonging to portrait (A), which depends on the value of l. We illustrate two of these trajectories in red and green. The red trajectory corresponds to the solution with supremum or maximum downstream amplitude, whereas the green periodic orbit corresponds to a typical solution with downstream waves similar to those obtained in the linear problem. To the left of each phase portrait panel (with fixed h and F) is an associated plot showing the effect of varying the boxcar length, l, on the downstream wave amplitude (solid black curve). Results for the linear problem are also presented for comparison in the supremum amplitude verses h and downstream amplitude verses l plots (grey broken curves).

Fig. 9

Weakly nonlinear analysis of wave amplitude downstream of the boxcar, $F=0.8$. a Maximum / supremum amplitude of downstream waves versus h. Weakly nonlinear, solid red curve. Linear, broken grey lines. b–q Columns 1 and 3: Downstream amplitude versus l, for each of the cases I–VIII with fixed h (see panels for values of h). Weakly nonlinear, solid curves. Linear, broken grey curves. Columns 2 , 4: Corresponding phase planes for the plots in columns 1 and 3. Free-surface directly above boxcar, blue curves. Typical downstream wave-train, green curves. Maximum/supremum downstream amplitude, red curves

We begin the discussion by considering cases III and IV when |h| is small and the behaviour is qualitatively similar to the linear problem for all values of l (see first two rows of Figs. 2 and 3). For these two cases we see that the blue trajectory belonging to portrait (B) (solid curves) and red trajectory belonging to portrait (A) (broken curves) are both bounded by the homoclinic orbit of portrait (A) (Fig. 9g,i). As a result all solutions are constructed in the same manner as the linear solutions with a maximum and zero amplitude solution obtained when the length of the boxcar shortens or increases by a wavelength of the blue trajectory. We can determine the value of this maximum amplitude (red trajectory) from the trajectory Eq. (7)

$$\begin{aligned} a_{\text {red}}=a_{\text {blue}}+\frac{1}{4}\left| \kappa +\sqrt{\kappa ^2+8h}\right| , \end{aligned}$$

(24)

where $\kappa =(F-1)-\sqrt{(F-1)^2-2h}$.

Similar plots of wave height verses length have previously been produced by Forbes and Hocking [7, 11] to examine subcritical flow past a semi-ellipse in the linear and the fully nonlinear problem. The locations of the linear waveless solutions depend only on the Froude number, F, as we find with the linearised KdV equation. The locations of the fully nonlinear waveless solutions depend on both the Froude number, F, and the height of the obstacle, as we find with the weakly nonlinear KdV equation. A notable difference between the weakly nonlinear KdV equation for a boxcar and fully nonlinear flow over an ellipse is observed as the length of the obstacle vanishes. We find as the length vanishes the downstream amplitude vanishes. For the fully nonlinear problem Forbes instead finds that the amplitude downstream approaches a small nonzero value. For both the linear and fully nonlinear flow past an ellipse they find that the value of the maximum wave height decreases as the length increases whereas we find for flow past a box car that the maximum amplitude remains the same.

Next we consider case II where the box height takes the single value recorded in Table 1. At this precise value of $h=h^-$ the red trajectory in the phase plane, corresponding to the supremum downstream amplitude, is the homoclinic orbit of portrait (A) (Fig. 9e), which gives a hydraulic fall in the physical plane (Fig. 2i). Orbits infinitesimally close to the homoclinic orbit result in large amplitude waves that approach a supremum value of $|F-1|$. This results in a bifurcation in solution behaviour where the increasing maximum amplitude of case III is replaced with the constant supremum amplitude in case II, as seen in Fig. 9a.

In case I, with values of $h<h^-$, we observe that the blue trajectory now intersects the red homoclinic orbit at two points (Fig. 9c). These two intersections both give hydraulic fall solutions (Fig. 2j,k). However, between these two intersections the blue trajectory passes outside the red homoclinic orbit. In this exterior region the trajectories in portrait (A) are unbounded and thus correspond to boxcar lengths with no steady solutions. This can be seen as gaps in the solid curves of Fig. 9b. On either side of the gaps we can see that the downstream amplitude approaches the supremum value $|F-1|$ as the length approaches the values that give the hydraulic falls. This gives wedges of typical wave solutions centred on the waveless solutions and approaching the supremum amplitude at the edges where there are hydraulic fall solutions.

The boxcar height once again takes a single value $h=h^+$ in case V (see Table 1) where the blue trajectory is now the homoclinic orbit in portrait (B) (see Fig. 9k). We remark that the trajectory intersecting with the unforced centre of (A) in Fig. 3i,j,k,l,t also appears to be the homoclinic orbit in portrait (B). However, they are in fact inner periodic orbits extremely close to the homoclinic orbit and thus are classified as case IV solutions and exhibit behaviour not observed in the linearised problem. One difference is the absence of periodic waves directly above the long-length boxcar. Returning to the discussion on the blue homoclinic trajectory of Fig. 9k, for case V; the trajectory is not periodic and so the profile of the free-surface directly above the boxcar no longer has waves and instead falls to a lower level of uniform flow (Fig. 3s). As the length of the boxcar increases the second transition point approaches the saddle point in portrait (B) along its stable manifold. This results in the trajectory corresponding to the downstream waves approaching the red periodic trajectory that intersects the saddle point in portrait (B). Hence, as the length of the boxcar increases the downstream amplitude approaches the supremum

$$\begin{aligned} a_{\text {red}}\rightarrow \frac{1+\sqrt{5}}{4}|F-1|, \end{aligned}$$

(25)

given by the red trajectory (24). The main difference between this case and the preceding cases is that there are no longer waveless solutions for any value of $l>0$, instead we see the amplitude increases monotonically as the boxcar increases in length.

We now consider case VI with values of $h>h^+$ (see Table 1), which result in the homoclinic orbit in portrait (B) being located to the left of the centre in portrait (A). The blue trajectory is now unbounded (Fig. 9m). Similar to case V the free-surface profile directly above the box does not have any waves (Fig. 3r). However, a key difference is that the solutions in case VI are no longer confined by the periodic trajectory in portrait (A) which intersects the saddle point in portrait (B). Instead the blue unbounded trajectory allows for solutions which can connect to any periodic trajectory and the homoclinic orbit in portrait (A). The intersection of the blue trajectory with the red homoclinic orbit gives a hydraulic fall (Fig. 3q) with a maximum boxcar length. There are no steady solutions for lengths of the boxcar greater than that for the hydraulic fall. This can be seen in the downstream amplitude verses length plot (Fig. 9l) where solutions only exist for a sufficiently short-length boxcar with the amplitude increasing towards the supremum $|F-1|$ at which the hydraulic fall occurs with the largest length of boxcar.

Similar behaviour to case VI is seen in the downstream amplitude verses length plot of Fig. 9n for case VII when the boxcar height $h=h_m$. For this single value of the boxcar height (see Table 1) the discriminant in Eq. (15) is zero and the two equilibrium points in portrait (B) merge into a single degenerate node (see Fig. 9o). This gives critical flow [29, 32, 33] above boxcar with a local Froude number, $F_b=1$ (see Fig. 4a,b). Furthermore, in the last case VIII, as the height increases with $h>h_m$, portrait (B) no longer has real equilibrium points instead consisting purely of unbounded trajectories (see Fig. 9p). Despite this solutions still exist for sufficiently small boxcar lengths with behaviour similar to cases VI and VII (see Fig. 4c,d).

Of particular interest is the limiting behaviour as $l\rightarrow 0$ in the hydraulic fall solutions, for fixed Froude number, F. When the length of the boxcar is long with positive height, $h>0$, we have a hydraulic fall similar to the one shown in Fig. 3q. The height increases as the length decreases, and when the value of h is close to $h_m$, we have solutions similar to that shown in Fig. 10a,e with solution in the phase plane similar to the hydraulic fall in Fig. 4f, for the value of $h=h_m$. Thus, when $h=h_m$, we have the critical solution where there is only a single value of boxcar length corresponding to the given values of $h_m>0$ and $F<1$. As the length continues to decrease we have $h>h_m$ with a phase plane similar to Fig. 4h, but the height h can now be chosen independently of the Froude number F, which is paired with a corresponding value of l for the hydraulic fall. Ultimately, the length vanishes and height becomes infinite with the solution behaviour in the boxcar phase plane (Fig. 10b,f) approaching the point forcing phase plane shown in Fig. 1c,g. The limiting process of $l\rightarrow 0$ when $h<0$ is shown in Fig. 10c,d and this also compares well with the point forcing hydraulic fall in Fig. 1d,h, for $h<0$.

Fig. 10

Limiting behaviour of boxcar length in hydraulic falls, $F=0.8$. a–d Physical plane. a and b Positive boxcar height, $h>0$. a $l=4$, $h=0.022$. b $l=0.25$, $h=0.188$. c and d Negative boxcar height, $h<0$. c $l=2.5$, $h=-0.036$. d $l=0.25$, $h=-0.188$. e–h Corresponding solutions in the phase plane, for a–d. The black broken and solid curves belong to portraits (A) and (B), respectively

The downstream amplitude can also be examined with fixed area of boxcar or amplitude of forcing A versus boxcar length (see Fig. 11). We present results from six cases $|A|<A_{\text {max}}$, $|A|=A_{\text {max}}$, $|A|>A_{\text {max}}$. We find waveless solutions at discrete values of l and regions of wavy solutions with decreasing maximum wave amplitude as l increases. We remark that the waveless solutions are not evenly spaced and that the amplitudes are right skewed.

Fig. 11

Downstream amplitude versus boxcar length with fixed area of boxcar, $F=0.8$. Weakly nonlinear: solid curves. Linear: broken curves. a–f Amplitude of forcing or area of boxcar, $A=\{-0.02, -A_{\text {max}}, -0.5, 0.02, A_{\text {max}}, 0.08 \}$

Next we examine the limiting behaviour as the boxcar length $l\rightarrow 0$. In cases $|A|<A_{\text {max}}$ (Fig. 11a,d) the downstream amplitude approaches the point forcing subcritical solution (see Fig. 1a,b). When $|A|=A_{\text {max}}$ (Fig. 11b,e) the downstream amplitude approaches the supremum value $|F-1|$ with the wavelength becoming unbounded which corresponds to the point forcing hydraulic falls (see Fig. 1c,d). As expected, solutions are bounded away from $l=0$ in the cases $|A|>A_{\text {max}}$ as there are no steady point forcing solutions for these values of the amplitude of forcing.

To complement our analysis of the wave amplitude downstream of the boxcar we also present results for the downstream wavelength versus boxcar length with fixed height, for cases I–VIII (Fig. 12). In cases I–IV the wavelength approaches the constant wavelength, $\lambda$, in the linearised problem when the length of the boxcar approaches the lengths corresponding to the waveless solutions. In these four cases, the maximum/unbounded wavelength corresponds to boxcar lengths with maximum/supremum wave amplitude. The unbounded wavelengths in case I and II correspond to hydraulic fall solutions. Similar to the wave amplitude plot in Fig. 9k, the wavelength increases monotonically and approaches the wavelength of the red orbit in case V. For the remaining cases VI–VIII, the wavelength increases (for sufficiently small lengths) and ultimately becomes unbounded at the hydraulic fall solution.

Fig. 12

Downstream wavelength versus boxcar length, $F=0.8$. The values of the height of the boxcar is given in the individual panels for each of the cases I–VIII. Weakly nonlinear: solid curves. Constant linear wavelength, $\lambda$: broken lines

The results discussed above are summarised in Fig. 13a where the hydraulic fall solutions (broken black curves) bound a region of steady solutions in $L-h$ parameter space. The maximum downstream amplitude solutions are illustrated with red solid curves, and as the height decreases the maximum downstream amplitude solutions bifurcate into two negative height hydraulic fall solutions (see intersection of red and broken curves along $h=h^-$ in Fig. 13a). In the physical plane, these solutions differ by a crest height at the leading edge of the front (e.g. see Fig. 2j,k). Waveless solutions (black solid curves) can be compared to the results of Hocking [11] for a semi-ellipse with positive height in Fig. 13b. We remark that the waveless (solid black curves) and maximum downstream amplitude (solid red curves) solutions with $h>0$ are disconnected and both approach $h=h^+$ as $l\rightarrow \infty$ (see upper broken black curve for positive height hydraulic fall solutions in Fig. 13a).

Fig. 13

Summary of weakly nonlinear boxcar solutions in $L-h$ parameter space and comparison with nonlinear results of Hocking [11] for a (positive height) semi-ellipse. a Weakly nonlinear boxcar solutions, $F=0.8$. The hydraulic fall solutions (broken black curves) bound a region of steady solutions. The marker on the upper broken black curve is for the value of $h=h_m$. Solutions with maximum downstream amplitude are illustrated with solid red curves. Waveless solutions are given by the solid black curves, which can be compared to the curves in b for a (positive height) semi-ellipse. The upper and lower horizontal broken light grey lines are for $h=h^+$ and $h=h^-$, respectively. b Fully nonlinear waveless solutions for a semi-ellipse, $F=0.5$. Modified image taken from Hocking [11]

Fig. 14

Froude number contours of the first two waveless solutions in $L-h$ space. a Weakly nonlinear boxcar contours. b Nonlinear semi-ellipse contours taken from Hocking [11]

A further comparison of the results for the boxcar and semi-ellipse is given by contours of the Froude number in $L-h$ space (Fig. 14a,b). For the fully nonlinear flow past a semi-ellipse Hocking [11] found that the two adjacent waveless solutions merge as the height increases with waveless solutions only existing in a bounded range of lengths. In contrast, for the boxcar we instead find that the two solutions remain seperate and asymptote to the height $h=h^+$ for arbitrarily large boxcar lengths.

6 Time-dependent problem

Following Grimshaw and others [21, 28, 29, 34,35,36,37], we investigate the time evolution of the steady-state solutions using an established numerical pseudospectral method. Stationary solutions are found numerically, perturbed by multiplying the steady-state profile by a factor, k, and are substituted as the initial condition. We modify the time dependent equation with the addition of a sponge layer, as found in [29, 34,35,36,37],

$$\begin{aligned} 6\eta _{t} - \eta _{xxx} - 9 \eta \eta _{x} + 6(F-1)\eta _{x} +s(x)\eta = 3\sigma _x. \end{aligned}$$

The sponge layer is defined by

$$\begin{aligned} s(x)=a\big [2+\tanh \big (b(x+c-D/2)\big )+\tanh \big (b(c-x-D/2)\big ) \big ], \end{aligned}$$

with domain length D and constants a, b and c. The values used in computations are $D= \{500, 800,1000\}$, $a=30$, $b=0.1$ and $c=50$. The sponge layer is implemented to absorb waves approaching the boundary, overcoming the typical limitation of periodic boundary conditions.

To avoid issues associated with unbounded spatial derivatives at the edges of the boxcar we use smooth functions to approximate the forcing in (5) with

$$\begin{aligned} \sigma (x)=\frac{h}{2}\left[ \tanh \Big (d(x+l/2)\Big )+\tanh \Big (d(-x+l/2)\Big )\right] , \end{aligned}$$

where d controls the steepness of the edges [32].

In the pseudospectral method the number of Fourier modes used is determined by the number of grid points, N. We tested that all solutions are independent of both the number of grid points and the truncation of the spatial domain (thus also of the spacial grid mesh sizes). In the computations below we used 8192 grid points. The resulting system of equations is solved with matlab’s ode45 with a relative tolerance of $10^{-9}$ and an absolute tolerance of $10^{-12}$. The time steping is chosen automatically by the ode solver.

A selection of time-dependent results are shown in Figs. 15 and 16 in which the perturbation factor, k, is slightly less than one. Similar results are found for $k>1$.

Fig. 15

Evolution of perturbed stationary solution, $F=0.8$. The initial profile is 0.98 times the steady-state profile. First column, steady-state profiles. Second & third column, contour and waterfall plots of perturbed steady-state evolution. a–c $h=-0.011$ and $l=56$. d–f $h=0.019$ and $l=58$. g–i $h=0.020$ and $l=58$. j–l $h=0.020$ and $l=20$

We find that steady-state solutions with downstream waves from cases I–IV (e.g. Figs. 2b–d, f–h, 3b–d, f–h) are numerically stable (Fig. 15a–c, $h<0$). In contrast, stationary waveless solutions (Figs. 2a,e, 3a,e,i,l,m,p) are unstable (Fig. 15d–i, $h>0$). However, given sufficient duration, these unstable waveless solutions are observed to approach nearby stationary solutions with small amplitude waves (not shown). We note that for Fig. 15g–i the surface profile can be described as a matching of a semi-infinite step hydraulic fall and rise. The instability can be seen to develop from the hydraulic rise, this is consistent with the results of Chardard et al. [28] who found that a rising front is unstable and a falling front is stable.

We observe that, as the height of the boxcar in case IV approaches the height in case V, solutions with small amplitude waves downstream (e.g. Fig. 3 j-k,n-o) become unstable. We conjecture that solutions with large waves downstream such as Fig. 3t,x are also unstable however the time needed for the instability to develop increases rapidly as the amplitude downstream increases.

We find that the steady-state solutions for case VI appear numerically stable (Fig. 15j–l). Steady-state solutions for cases VII and VIII appear numerically stable (not shown).

Fig. 16

Evolution of perturbed stationary solution, $F=0.8$. The initial profile is 0.98 times the steady-state profile. First column, steady-state profiles. Second & third column, contour and waterfall plots of perturbed steady-state evolution. a–c $h=0.020$, $l=20$. d–f $h=0.033$, $l=2$. g–i $h=-0.062$, $l=22.6$. j—l $h=-0.061$, $l=24.9$

Hydraulic falls with positive height $h>0$ are shown to be stable to perturbation in Fig. 16a–f. However, and as expected, we find a stable and unstable branch of stationary solutions for hydraulic falls with negative height $h<h^-$. Stationary solutions with a noticeable difference in crest height at the location of the front are unstable (Fig. 16g–i), whereas solutions with no difference in crest height are stable (Fig. 16j–l). The stable / unstable branches of stationary hydraulic falls with $h<h^-$ are shown by the broken curves left / right of the bifurcation point(s) in Fig. 13a. We note that it is the unstable branch of these solutions that limits to the point forcing negative hydraulic fall.

7 Discussion

We have investigated the effect of varying the height and length of a rectangular obstacle in the bottom of a channel on the free-surface elevation in open channel flow, with $F<1$. When the height of the boxcar is small, $|h|\ll 1$, a linearised fKdV model shows the behaviour downstream of the boxcar can be described by the superposition of two cosines, providing closed-form expressions for the wavelength and amplitude. Thus a countably infinite set of discrete boxcar lengths is determined by examining the zeros of a sine function with waveless flow downstream of the disturbance. The result is analogous to Lamb’s [10] linearised problem of flow past a semi-ellipse where waveless flow is determined by the zeros of Bessel’s functions. The boxcar forcing allows us to examine solutions in the phase plane and the simpler linearised problem helps us to interpret more complicated behaviour in the weakly nonlinear problem. Notably, sinusoidal orbits in the linearised problem are replaced with orbits described by Jacobi elliptic functions in the weakly nonlinear problem.

The weakly nonlinear fKdV model provides us with the means to study the qualitative behaviour of the full problem but without the need for time consuming computations associated with solving fully nonlinear models numerically. Guided by an analysis in the weakly nonlinear phase plane, we provide analytic parametric maps of the solutions, which we categorise into eight cases I–VIII. The cases are defined by the values of the negative and positive boxcar height in hydraulic falls, $h^-<0$ and $h^+>0$, and the maximum height, $h_m>0$. In the linearised problem waveless solutions depend only on the value of the Froude number, F, while in the weakly nonlinear problem they depend on both F and h. Dependence on both obstacle height and Froude number was also found by Forbes and Hocking [7, 11] in their studies on waveless nonlinear solutions for flow past a semi-ellipse, for a positive height semi-ellipse. However, in the obstacle length and height parameter space, two adjacent branches of waveless solutions for the nonlinear ellipse problem with $|h|\ll 1$ merge into a loop as the height increases, whereas adjacent branches of waveless solutions in the weakly nonlinear boxcar problem remain disconnected and approach $h^+$ as the length increases (see Fig. 13).

In addition to the waveless boxcar solutions, in length and height parameter space, we find adjacent branches of solutions with maximum downstream amplitude given by Eq. (24) also remain disconnected and approach $h^+$ as the length increases, for $h>0$. When $h<0$, all the boxcar solution branches for the maximum downstream amplitude bifurcate into two hydraulic falls at the constant value of $h=h^-<0$. Hence the curves describing boxcar hydraulic fall solutions determine a region in which steady solutions exist in length and height parameter space (see Fig. 13).

We also examine the effect of the boxcar shortening in length with fixed boxcar height, and established that boxcar lengths less than the linearised fKdV wavelength, $\lambda$, can be considered short in length. The linear wavelength, $\lambda$, is also a good measure of the downstream wavelength in the weakly nonlinear problem when the amplitude of the waves is small (see first row of Fig. 12). When the area of the boxcar is fixed and as the length tends to zero, we recover the point forcing model of Shen [24]; we show both wavy and hydraulic fall boxcar solutions approach those found in the point forcing model.

To conclude our study we perturbed our stationary solutions and used a pseudospectral method to investigate the stability. For case I the upper branch hydraulic falls (Fig. 2k,n) were found to be numerically stable while the lower branch hydraulic falls (Fig. 2j,m) were found to be unstable. Steady solutions with waves downstream, cases I–III (Fig.2b–d, f–h) and case IV (for sufficiently small heights, Fig. 3b–d, f–h) were found to be numerically stable. Perturbed, steady, waveless solutions (Figs. 2a,e, 3a,e,i,l,m,p) were found to approach nearby stable solutions with small amplitude waves. Therefore the results from our study may be of use in practical settings or applications where there is a desire to reduce the amplitude of waves downstream of a topographic disturbance.

Solutions for case IV (Fig. 3i–p, t–x) were found to become unstable as the boxcar height approaches that of case V. Solutions for cases V–VIII were found to be numerically stable, including the hydraulic falls with positive height, $h>h^+$.

While the focus in this study was on the subcritical flow problem with uniform flow and $F<1$ far upstream, our future research involves working on the supercritical problem when $F>1$. Similar to subcritical flow, the point forcing problem is well understood in the supercritical flow regime, and flow types not found in the subcritical problem include flows that are uniform both far upstream and downstream with supercritical flow and generalised hydraulic rises with supercritical flow far upstream and waves downstream of the disturbance. Solutions to these two flow types are known to be non-unique for the point forcing problem. Therefore, we anticipate a rich structure of non-uniqueness in the weakly nonlinear solutions of the supercritical boxcar forcing problem.

Future research also includes investigating stronger nonlinear effects in the fully nonlinear problem. While previous work has shown good qualitative and quantitative agreement between the weakly and fully nonlinear models for a small boxcar height, h, and for F close to unity [12, 22], highly nonlinear behaviour such as large amplitude waves approaching a Stokes limiting configuration with a stagnation point at the crest(s) can only be examined with a fully nonlinear model [21, 22].