Numerical investigation towards the improvement of hydraulic-jump prediction in rectangular open-channels

ABSTRACT

This study probed into the numerical characterization of a rapidly varied free-surface flow behavior including a hydraulic jump onset. The extended One-Dimensional St.-Venant Equations, embedding the Boussinesq add-on term, were employed to describe a free-surface wave behavior. The numerical computations were based on a second-order precision in time and fourth-order in space explicit finite-difference scheme ((2/4)-dissipative numerical scheme). The computed results were then compared with those issued from the McCormack – based alternative solver and experimental data quoted in the literature. The findings revealed that the developed solver allowed an accurate prediction of the amplitude and location characteristics of the hydraulic jump. In addition, they suggested that the (2-4)-dissipative scheme-based algorithm was more practical than the alternative shock capturing methods in terms of prediction accuracy, implementation simplicity, and calculation time consumption. Unlike the McCormack scheme – based solver, the proposed algorithm provided numerical signals free of numerical oscillations in the vicinity of the steep gradient involved by hydraulic jumps. Yet, it required more computational time than the McCormack scheme – based alternative.

KEYWORDS:

• (2/4)- dissipative scheme
• Boussinesq
• free-surface
• hydraulic jump
• open-channel
• wave

Notations

The following symbols are used in this paper:

$A$ = wetted cross-sectional area of the channel (${\mathrm{m}}^2$)

$\mathrm{B}$ = Boussinesq term (${\mathrm{m}}^4/{\mathrm{s}}^2$)

$c$ = wave-celerity ($\mathrm{m}/\mathrm{s}$)

$\mathrm{C}\mathrm{r}$ = Courant number (-)

$d$ = flow depth ($\mathrm{m}$).

$g$ = acceleration due to gravity ($\mathrm{m}/{\mathrm{s}}^2$)

$l$ = channel length ($\mathrm{m}$)

$n$ = Manning roughness coefficient ($\mathrm{s}/{\mathrm{m}}^{1/3}$)

$Q$ = discharge (${\mathrm{m}}^3/\mathrm{s}$)

$q_l$ = lateral inflow per unit width of the open-channel (${\mathrm{m}}^3/\mathrm{s}$)

$R_H$ = hydraulic radius ($\mathrm{m}$)

$s_0$ = longitudinal slope of the channel bottom ($\mathrm{m}/\mathrm{m}$)

$s_f$ = friction slope ($\mathrm{m}/\mathrm{m}$)

$T$ = free-surface width ($\mathrm{m}$)

$t$ = time ($\mathrm{s}$).

$u$ = depth-averaged velocity ($\mathrm{m}/\mathrm{s}$)

$v_w$ = surge-wave velocity ($\mathrm{m}/\mathrm{s}$)

$x$ = distance along the open-channel bed ($\mathrm{m}$)

$U$ = vector of unknown variables (-)

$\mathrm{G}$ = flux vector (-)

$H$ = second member vector (-)

Subscripts

$0$ = initial flow condition (-).

$i$ = mesh index in the $x$-direction (-).

$N_s$ = number of grid points of the open-channel (-)

Acronyms

(2/4)-Dissipative Scheme = explicit finite-difference scheme second-order in time and fourth-order in space

1-D-SVEqs = One-Dimensional St.-Venant Equations

Disclosure statement

The authors declare that there is no conflict of interest.