Investigating the unidirectional flow behavior in trapezoidal open-channel

ABSTRACT

This paper applies a Multiple-Grid technique -based Finite Element algorithm to predict the free-surface wave behavior in trapezoidal open-channel. This technique aims to limit the computational time consumed in the standard Finite Element procedure. The spatial integration of the flow model uses a Discontinuous Galerkin approach; while the time integration employs a Semi Implicit Euler method. The test case relates to the transient flow regime arising from the failure of a hydraulic power plant placed at the downstream extremity of an open-channel. Results show better performances of the Multiple-Grid technique compared to the conventional one in terms of consumed computational time criterion. Besides, the results obtained by the proposed algorithm illustrate close agreement with those obtained by the Finite Difference Method -based alternative algorithm.

KEYWORDS:

• Finite element method
• free-surface
• galerkin
• multiple-grid
• open-channel
• one-dimensional
• transient flow
• trapezoidal

Notations

The following symbols are used in this paper:

A = wetted cross-sectional area of the channel (m²)

b = channel bottom width (m).

c = wave-celerity (m/s)

Cr = Courant number (-)

d = flow depth (m).

g = acceleration due to gravity (m/s³)

L = channel length (m)

n = Manning roughness coefficient (s/m^1/3)

q = discharge (m³/s)

q1 = lateral inflow per unit width of the open-channel (m³/s)

R_H = hydraulic radius (m)

s₀ = longitudinal slope of the channel bottom (m/m)

s_f = friction slope (m/m)

U = free-surface width (m)

t = time (s).

u = depth-averaged velocity (m/s)

x = distance along the open-channel bed (m)

U = vector of unknown variables (-)

K = stiffness matrix (-)

f = second member vector (-)

β = side slope of the open-channel

Subscripts

0 = initial flow condition (-).

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

N = number of grid points of the open-channel within the coarse mesh scale (-)

n_G = number of grid points of the open-channel within the fine mesh scale (-)

Acronyms

MG-FEM = Multiple-Grid technique –based Finite Element Method

Disclosure statement

No potential conflict of interest was reported by the author(s).