Steady two-dimensional free-surface flow over semi-infinite and finite-length corrugations in an open channel

Free-surface flow past a semi-infinite or a finite-length corrugation in an otherwise flat and horizontal open channel is considered. Numerical solutions for the steady flow problem are computed using both a weakly nonlinear and fully nonlinear model. The new solutions are classified in terms of a depth-based Froude number and the four classical flow types (supercritical, subcritical, generalized hydraulic rise, and hydraulic rise) for flow over a small bump. While there is no hydraulic fall solution for semi-infinite topography, we provide strong numerical evidence that such a solution does exist in the case of a finite-length corrugation. Numerical solutions are also found for the other flow types for either semi-infinite or finite-length corrugation. For subcritical flow over a semi-infinite corrugation, the free-surface profile is found to be quasiperiodic downstream.

Figure 1

Sketch of semi-infinite corrugation flow problem. The flow is from left to right with uniform flow as $x\to -\infty$.

Figure 2

Sketch of the four basic flow types for flow over a bump. (a) Supercritical flow. The solutions are nonunique with two types of solution for a given value of $F>1$ and a given bump. The solution with the largest value of $\eta$ is called a perturbation of a solitary wave. The other solution is referred to as a perturbation of a uniform stream. (b) Subcritical flow. The solution is unique for a given value of $F<1$ and a given bump. (c) Generalized hydraulic rise (solid curve). There is an infinite number of solutions for a given value of $F>1$ and a given bump. Hydraulic rise (broken curve). There is a unique solution for a given bump; the Froude number, $F>1$, cannot be chosen independently and comes as part of the solution.

Figure 3

The solid curves are weakly nonlinear solutions for supercritical flow with $F=1.10$, $\epsilon =0.1$, $k=3$, and $\lambda =1$. (a)-(e) Semi-infinite corrugation. (f) Finite-length corrugation, $\gamma =50$. The dotted curve in panel (a) is the linearized solution (12).

Figure 4

Fully nonlinear solutions for supercritical flow over finite-length corrugation with $F=1.10$, $\epsilon =0.1k=3$, $\lambda =1$, and $\gamma =10$.

Figure 5

The solid curves are weakly nonlinear solutions for subcritical flow with $F=0.90$, $\epsilon =0.05$, and $\lambda =1$. (a)–(d) Semi-infinite corrugation with (a) $k=2$, (b) $k=3$, (c) $k=5$, (d) $k=7$. The dotted curve in panel (a) is the linearized solution (11).

Figure 6

Subcritical flow solutions for a semi-infinite corrugation. Comparison between weakly (a), (c) and fully nonlinear (b), (d) results for $F=0.90$, $\epsilon =0.005$, $\lambda =10$. (a, b) $k=0.2$; (c, d) $k=0.1$.

Figure 7

Weakly nonlinear solutions for subcritical flow over a finite-length corrugation with $k=3$ for $F=0.90$, $\epsilon =0.05$, and $\lambda =1$. (a) $\gamma =8$, (b) $\gamma =8.42$.

Figure 8

Generalized hydraulic and hydraulic rise with $F=1.10$, $\epsilon =0.1$, and $k=3$. (a) Weakly nonlinear generalized hydraulic rise over a semi-infinite corrugation with $\lambda =10$. (b) Fully nonlinear generalized hydraulic rise over a semi-infinite corrugation with $\lambda =1$. (c) Weakly nonlinear generalized hydraulic rise over a finite-length corrugation with $\gamma =10$ and $\lambda =10$. (d) Weakly nonlinear hydraulic rise over a finite-length corrugation, $\gamma =10$ and $\lambda =10$.

Figure 9

Hybrid solutions on a finite-length corrugation with $F=1.10$, $\epsilon =0.1$, $k=3$, and $\lambda =10$. (a)–(c) Weakly nonlinear solutions with (a) a match between flow types 1 and 3 for $\gamma =20$ (globally the flow is type 3), (b) a match types 1 and 4 for $\gamma =20$ (globally the flow is type 4), and (c) a match between a type 3 generalized hydraulic rise and a type 3 generalized hydraulic fall for $\gamma =20$ (globally the flow is type 1). (d) Fully nonlinear solution for $\gamma =8$, $\lambda =1$ showing a type 3-type 3 match as in panel (c).