## Abstract

This paper investigates dam-break problems with flows on one or two sides of zero or nonzero velocities over a mobile initially flat bed, and quasi-exact solutions are presented by solving the Riemann problems using the simple wave theory. The flow structures after dam collapse for nonzero velocities are much richer than those for zero velocities on both sides, although they are also a combination of waves of different characteristic families, which are consistent with Lax [CBMS-Regional Conference Series in Applied Mathematics, SIAM, 1973]. The wave can be a rarefaction, a shock, or a combination of a rarefaction and a semi-characteristic shock. The semi-characteristic shock is related to the morphodynamic characteristics. The relationship between morphodynamic and hydrodynamic characteristics is illustrated, along with types of waves (shock, rarefaction or a combination of these), and sediment convergence and type of characteristic. It is shown that the types of waves that may occur in the Riemann solution, and, in some cases, their possible approximate locations, can be determined prior to the construction of the Riemann solution itself. The Riemann solution presented here can be used to study shock–shock interactions.

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## Notes

We note that there is a misprint in Eq. (2.25) of [16]. The factor \((u-\lambda _i)\) should be \((\lambda _i-u)\).

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## Acknowledgements

This work is supported by the Natural Science Foundation of China with the Project Code 51509135, and the authors would like to acknowledge for their financial support. The authors would also like to acknowledge University of Nottingham UK and Ningbo China.

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Natural Science Foundation of China (Grant No. 51509135).

## Appendices

### Interpretation of wave structures in Sect. 3.1 by characteristics

Because \({\lambda ^\prime }={\lambda }/{\sqrt{h}}\) (see Fig. 2) is dependent only on *F* it is instructive to map each of the profiles in Fig. 5 onto it: see Fig. 8.

It can immediately be seen that each solution begins on the left at \(F = F_\mathrm{l}\) on the \({\lambda ^\prime }_1\) dispersion curve, and each solution will, at some point, jump to the \({\lambda ^\prime }_3\) curve, and then proceed to the right as \(F\rightarrow \infty \), apart from structure (i), which terminates at \(F = 0\).

The question is then at what point the jump occurs. In \({\lambda ^\prime }- F\) space, this, along with any other jump due to the presence of a shock, completely describes the solution. Note that for structures (i)–(iv), *h* (*u*) is monotonic decreasing (increasing) across the Riemann solution from left to right, notwithstanding the jump to zero *u* on the dry side. Therefore, *F* is monotonic increasing for positive *u*. *F* could be increasing or decreasing for negative *u*. However, for all the examined negative \(u_\mathrm{l}\) values, *F* is monotonic increasing (Fig. 8f). Further note that \({\mathrm{d}{\lambda ^\prime }_1}/{\mathrm{d}F}=0\) at \(F\approx 1.613\), \(\Rightarrow \) for \(F< 1.613\), \({\lambda ^\prime }_1\) increases for increasing *F* (Fig. 2).

For \(u_\mathrm{r}=0\) in structure (ii), the jump occurs for \(F<1.613\) (see Fig. 8). In this region \({\mathrm{d}{\lambda ^\prime }_1}/{\mathrm{d}F} > 0\), \(\Rightarrow {\mathrm{d}\lambda _1}/{\mathrm{d}F} > 0\) too, because *h*, as noted, is monotonic decreasing and \({\lambda ^\prime }_1<0\). Therefore, the \(\lambda _1\) wave is a rarefaction. We mentioned in Sect. 3.1 that the \(\lambda _1\) wave is a combination of a \(\lambda _-\) hydrodynamic wave and a morphodynamic wave, which can also be seen from Fig. 5b. The characteristics analysis that \({\mathrm{d}\lambda _1}/{\mathrm{d}F} > 0\) is consistent with the analysis from physical perspective that when water depth decreases across the \(\lambda _-\) wave, it is a rarefaction.

When \(F>1.613\), in contrast, we are in the region \({\mathrm{d}{\lambda ^\prime }_1}/{\mathrm{d}F} < 0\). Because \({\lambda ^\prime }_1\) and *h* are both decreasing it is not obvious whether \({\mathrm{d}\lambda _1}/{\mathrm{d}F} \lessgtr 0\) in each case. We notice that when \(F>1\), the \(\lambda _1\) (\({\lambda ^\prime }_1\)) characteristics behave as morphodynamic characteristics. It should be noted that \({\mathrm{d}{\lambda ^\prime }_1}/{\mathrm{d}F}\) is a small value around \(F=1.613\), and according to Eq. (21) in Sect. 2.5 it is possible that \({\mathrm{d}{\lambda ^\prime }_1}/{\mathrm{d}F}\) and \({\mathrm{d}\lambda _1}/{\mathrm{d}F}\) have different signs.

The results show that for \(1.613\lessapprox u_\mathrm{l}\lessapprox 1.83\), \(\lambda _1\) increases across the \(\lambda _1\) wave, and therefore the \(\lambda _1\) wave is a rarefaction fan (structure (ii)). In this case, \({\mathrm{d}{\lambda ^\prime }_1}/{\mathrm{d}F}<0\) and \({\mathrm{d}\lambda _1}/{\mathrm{d}F}>0\). However, for a further increase in \(u_\mathrm{l}\) , we have \({\mathrm{d}\lambda _1}/{\mathrm{d}F}>0\) in some part of the \(\lambda _1\) wave, and \({\mathrm{d}\lambda _1}/{\mathrm{d}F}<0\) in the other part. The results show that when \(1.83\lessapprox u_\mathrm{l}\lessapprox 1.848\), \(\lambda _1\) characteristics first diverge and then converge. Therefore, the \(\lambda _1\) wave is a combination of \(\lambda _1\) rarefaction and a \(\lambda _1\) semi-characteristic shock (structure (iii), e.g. \(u_\mathrm{l}=1.83\) in Fig. 5e–h). This behaviour can be seen in Fig. 8c. The solution traverses a small section of the \({\lambda ^\prime }_1\) dispersion curve before a jump along that curve (the semi-characteristic shock) and then the jump to the \({\lambda ^\prime }_3\) curve.

When \(u_\mathrm{l}\) increases further still, \({\mathrm{d}\lambda _1}/{\mathrm{d}F}<0\) has the same sign as \({\mathrm{d}{\lambda ^\prime }_1}/{\mathrm{d}F}\). When \(1.848\lessapprox u_\mathrm{l}\lessapprox 2.98\), the \(\lambda _1\) characteristics converge, resulting in a \(\lambda _1\) shock (structure (iv), e.g. \(u_\mathrm{l}=2.5\)). We can see this behaviour in Fig. 8d, in which there is an immediate jump along the \({\lambda ^\prime }_1\) curve, followed by the jump to the \({\lambda ^\prime }_3\) curve.

When \(u_\mathrm{l}\gtrapprox 2.98\), \(F_\mathrm{l}>1.613\) and \(h_*>1\). Across the \(\lambda _1\) wave, *h* increases and *u* decreases, and *F* therefore decreases. However, across the \(\lambda _1\) wave, \(F>1.613\). As *F* decreases, \({\lambda ^\prime }_1\) increases (Fig. 5e), and therefore \(\lambda _1\) also increases. Thus the \(\lambda _1\) wave is a rarefaction (structure (v), e.g. \(u_\mathrm{l}=3.5\)). The decreasing *F* results in the “reversal” of the path of the wave in \({\lambda ^\prime }-F\) space: see Fig. 8e.

Finally, the results show that the \(\lambda _3\) wave is always a rarefaction fan, although it is not immediately clear that \(\lambda _3\) increases from the relation \(\lambda _3={\lambda ^\prime }_3\sqrt{h}\). However, in the limit \(F\rightarrow \infty \), Eq. (12) can be factorised such that \({\lambda ^\prime }_3 \sim F\), \(\Rightarrow \lambda _3 \sim u\). The \(\lambda _3\) wave is a \(\lambda _-\) hydrodynamic wave, and it is a fan when *h* decreases from left to right.

### Interpretation of wave structures in Sect. 3.2.1 by characteristics

The Riemann solutions in Fig. 6 are mapped onto (\({\lambda ^\prime }\), *F*) space in Fig. 9, as the solutions are traversed from left to right.

The \(\lambda _1\) characteristics across the \(\lambda _1\) wave for varying \(u_\mathrm{l}\) are not analysed here, because they are very similar to those in the wet–dry problem in Sect. 3.1. However, we can see that there is a difference in the \(\lambda _1\) wave type because of the difference between \(\mathbf {U}_{\mathrm{l}*}\) and \(\mathbf {U}_{*}\).

Across the \(\lambda _3\) wave, we have \({\mathrm{d}{\lambda ^\prime }_3}/{\mathrm{d}F}>0\). In structure (i)–(ii), *h* and *F* both decrease. However, it is not clear whether \(\lambda _3={\lambda ^\prime }_3\sqrt{h}\) increases or decreases because \({\lambda ^\prime }_3<0\). The \(\lambda _3\) wave is a \(\lambda _+\) wave in structure (i) and a combination of the \(\lambda _+\) hydrodynamic wave and morphodynamic wave in structure (ii). In structure (ii), the \(\lambda _+\) wave is more dominant. Therefore, it is a shock because *h* decreases from left to right across the wave.

In structure (iii), *F* increases across the \(\lambda _3\) wave. The \(\lambda _3\) wave in structure (iii) is a morphodynamic wave, and because *F* is close to 0 where \({\mathrm{d}{\lambda ^\prime }_3}/{\mathrm{d}F}=0\), \({\mathrm{d}\lambda _3}/{\mathrm{d}F}\) could have different signs from \({\mathrm{d}{\lambda ^\prime }_3}/{\mathrm{d}F}\) (Sect. 2.5). The results show that \({\mathrm{d}\lambda _3}/{\mathrm{d}F}<0\), and the \(\lambda _3\) wave is a shock.

The \(\lambda _3\) wave in structure (iv)–(vi) is a \(\lambda _-\) wave or a combination of the \(\lambda _-\) hydrodynamic wave and morphodynamic wave (Fig. 9). In structure (iv)–(v), *h* decreases and *F* increases across the \(\lambda _3\) wave (Figs. 6e, 9g). Across the morphodynamic part of \(\lambda _3\) wave in structure (iv)–(v), \({\mathrm{d}\lambda _3}/{\mathrm{d}F}>0\) has the same sign with \({\mathrm{d}{\lambda ^\prime }_3}/{\mathrm{d}F}\) because it is not close to \({\mathrm{d}{\lambda ^\prime }_3}/{\mathrm{d}F}=0\) (i.e. \(F=0\)). Thus, the morphodynamic part is a rarefaction. The \(\lambda _-\) wave is also a rarefaction because *h* decreases. Therefore, the \(\lambda _3\) wave is a rarefaction in structure (iv)–(v). However, in structure (vi), *h* increases across the \(\lambda _3\) wave (Fig. 6 (e)), and the \(\lambda _3\) wave is a shock.

In structure (i)–(ii), the \(\lambda _2\) wave is a combination of \(\lambda _+\) hydrodynamic wave and morphodynamic wave. As water depth increases from left to right, the \(\lambda _+\) wave is a rarefaction. However, as *F* is close to -1.613 where \({\mathrm{d}{\lambda ^\prime }_2}/{\mathrm{d}F}=0\), the morphodynamic part \({\mathrm{d}\lambda _\mathrm{m}}/{\mathrm{d}F}\) changes its sign. The morphodynamic part is a combination of a rarefaction and a semi-characteristic in structure (i) and a rarefaction in structure (ii). Therefore, the \(\lambda _2\) wave is a combination of a rarefaction and a shock in structure (i), and a rarefaction in structure (ii).

We know that \({\mathrm{d}{\lambda ^\prime }_2}/{\mathrm{d}F}>0\) when \(F\gtrapprox -1.613\), and \({\mathrm{d}{\lambda ^\prime }_2}/{\mathrm{d}F}<0\) when \(F\lessapprox -1.613\). In structure (iii)–(vi), *h* and *F* decrease across the \(\lambda _2\) wave with \(F>-1.613\). Therefore, \({\lambda ^\prime }_2\) and \(\lambda _2={\lambda ^\prime }_2\sqrt{h}\) both decrease, and the \(\lambda _2\) wave is a shock. From the physical perspective, the \(\lambda _2\) wave in structure (iii)–(vi) is a \(\lambda _+\) wave, and it is a shock when *h* decreases.

### Interpretation of wave structures in Sect. 3.2.2 by characteristics

The Riemann solutions in Fig. 7 are mapped onto (\({\lambda ^\prime }\), *F*) space in Fig. 10, as the solutions are traversed from left to right. We can see the black solid line starts from \(F_\mathrm{l}\) along the \({\lambda ^\prime }_1\) curve, and ends at \(F_\mathrm{r}\) on the \(\lambda _2\) curve, with a jump between \({\lambda ^\prime }_1\) and \({\lambda ^\prime }_3\) (\({\lambda ^\prime }_3\) and \({\lambda ^\prime }_2\)) curves through the left (right) star region.

In structure (i)–(ii), across the \(\lambda _{1,3}\) wave, *h* increases, *u* decreases and *F* decreases (Fig. 10f). Therefore, \({\lambda ^\prime }_1\) decreases. We can deduce that \(\lambda _1\) decreases from \(\lambda _1={\lambda ^\prime }_1\sqrt{h}\) because \({\lambda ^\prime }_1<0\), and the \(\lambda _1\) wave is a shock. The \(\lambda _1\) wave is a \(\lambda _-\) hydrodynamic shock, and it is a shock when water depth increases from left to right.

In structure (i)–(ii), \({\lambda ^\prime }_3<0\) decreases as *F* decreases across the \(\lambda _3\) wave (Fig. 10a–c). In structure (i), the \(\lambda _3\) wave is a combination of \(\lambda _+\) hydrodynamic wave and morphodynamic wave. Because *h* decreases, the hydrodynamic wave in structure (i) is a shock. The morphodynamic part is overtaken by the hydrodynamic shock, and the \(\lambda _3\) wave in structure (i) is a shock. While in structure (ii), it is a morphodynamic wave, and \({\mathrm{d}\lambda _3}/{\mathrm{d}F}\) has the same sign as that of \({\mathrm{d}{\lambda ^\prime }_3}/{\mathrm{d}F}\) resulting in a \(\lambda _3\) shock.

The \(\lambda _2\) wave in structure (i) is a morphodynamic wave. Therefore \({\mathrm{d}\lambda _2}/{\mathrm{d}F}<0\) because \({\mathrm{d}{\lambda ^\prime }_2}/{\mathrm{d}F}<0\) and *F* is far from -1.613 where \({\mathrm{d}{\lambda ^\prime }_2}/{\mathrm{d}F}=0\). When *F* decreases, \(\lambda _2\) increases, and hence it is a rarefaction. In structure (ii), the \(\lambda _2\) wave is a combination of the \(\lambda _+\) wave and morphodynamic wave with \(\vert F_{*r}\vert \ll \vert F_\mathrm{r}\vert \) and \({{\lambda ^\prime }_2}_{*r} > {{\lambda ^\prime }_2}_{r}\) (Fig. 10c). Water depth decreases from left to right across the \(\lambda _+\) wave, so it is a shock.

In structure (iii)–(iv) (e.g. \(u_\mathrm{r}=0,\ 1.5\)), *h* decreases and *u* and *F* increase across the \(\lambda _1\) wave (Fig. 10d–e). Similar to the wet–dry dam-break solution, the \(\lambda _1\) and \(\lambda _3\) waves are both rarefactions.

In structure (iii), across the \(\lambda _2\) wave, if it is a rarefaction, *h* decreases, *u* decreases and *F* decreases (Figs. 7e, f, 10f). Therefore \({\lambda ^\prime }_2\) decreases because \(F\gtrapprox -1.613\) (Fig. 10), and therefore \(\lambda _2\) is a shock. However, in structure (iv), across the \(\lambda _2\) wave, *h* increases, *u* increases and *F* decreases (Fig. 10f), and therefore \({\lambda ^\prime }_2\) decreases. It is not clear whether \(\lambda _2\) should increase or decrease across the \(\lambda _2\) wave. From the physical perspective, the \(\lambda _2\) wave is a \(\lambda _+\) hydrodynamic wave, and it is a rarefaction if water depth increases from left to right. The results show that \(\lambda _2\) increases across the \(\lambda _2\) wave, and it is a rarefaction.

### Numerical investigation of wave structures for \(u_\mathrm{r}=-1.933\) and \(u_\mathrm{r}=-1.933\) in Sect. 3.2.2

We solve the Riemann problem from right to left. The difference (\(\delta B\)) between the calculated (\(B_\mathrm{lc}\)) and the already known (\(B_\mathrm{l}\)) bed levels in the left constant region, i.e. \(\delta B=B_\mathrm{lc}-B_\mathrm{l}\), is plotted against \(h_{\mathrm{r}*}\) for various \(u_\mathrm{r}\) values in Fig. 11. As mentioned, the two bed levels (\(B_\mathrm{lc}\) and \(B_\mathrm{l}\)) must agree within the desired accuracy, and \(\delta B=0\) corresponds to the possible physical solution for \(h_{\mathrm{r}*}\). From the plot, we can see that when \(u_\mathrm{r}\gtrapprox -1.933\), there are three roots, and the largest root is the physical solution to \(h_{\mathrm{r}*}\). As \(u_\mathrm{r}\) decreases, \(h_{\mathrm{r}*}\) also decreases (see Fig. 11), and the two roots coalesce around 0.65 when \(u_\mathrm{r}\approx -1.933\). When \(u_\mathrm{r}\lessapprox -1.933\), there is only one root, which is \(<0.1\). However, the \(\lambda _2\) shock corresponding to this root, is unphysical because of divergence of \(\lambda _2\) characteristics. Therefore, the \(\lambda _2\) wave becomes a rarefaction when \(u_\mathrm{r}\lessapprox -1.934\).

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Zhu, F., Dodd, N. Quasi-exact solution of the Riemann problem for generalised dam-break over a mobile initially flat bed.
*J Eng Math* **115**, 99–119 (2019). https://doi.org/10.1007/s10665-019-09994-6

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DOI: https://doi.org/10.1007/s10665-019-09994-6