Timescale interpolation and no-neighbour discretization for a 1D finite-volume Saint-Venant solver

ABSTRACT

A new finite-volume numerical method for the one-dimensional (1D) Saint-Venant equations for unsteady open-channel flow is developed and tested. The model uses a recently-developed conservative finite-volume formulation that is inherently well-balanced for natural channels. A new timescale interpolation approach provides transition between 1st-order upwind and 2nd-order central interpolation schemes for supercritical and subcritical flow, respectively. This interpolation meets a proposed “no-neighbour” criterion for simplicity in future parallel implementation. Tests with a highly-resolved transitional flow and a coarsely-resolved natural channel show that the method is stable and accurate when applied with a flowrate damping algorithm that limits propagation of energy down to subgrid scales.

Keywords:

• Finite volume method
• hydraulic jumps
• one-dimensional models
• open channel flow
• Saint-Venant equations
• shallow flows
• streams and rivers

Acknowledgments

We thank the City of Austin for use of the Waller Creek survey data.

ORCID

Ben R. Hodges http://orcid.org/0000-0002-2007-1717

Frank Liu http://orcid.org/0000-0001-6615-0739

Notation

A	=	cross-sectional area (m${}^2$)
C	=	information speed (m s${}^{-1}$)
$f_c$	=	semi-discrete source-term function for continuity (m${}^3$ s${}^{-1}$)
F	=	Froude number (–)
F	=	geometry function (m)
g	=	gravitational acceleration (m s${}^{-2}$)
H	=	hydraulic depth (m)
$\hat{H}$	=	maximum depth (m)
k	=	RK4 step (m${}^3$ or m${}^3$ s${}^{-1}$)
K	=	geometry source term (m${}^4$ s${}^{-2}$)
L	=	length of element (m)
$\hat{n}$	=	Gauckler–Manning–Strickler roughness (m${}^{-1/3}$ s)
Q	=	flowrate (m${}^3$ s${}^{-1}$)
P	=	wetted perimeter (m)
R	=	hydraulic radius (m)
$S_f$	=	friction slope (–)
$S_0$	=	bottom slope (–)
t	=	time (s)
T	=	timescale (s)
u	=	continuum velocity (m s${}^{-1}$)
U	=	discrete average velocity (m s${}^{-1}$)
V	=	volume (m${}^3$)
W	=	channel top width (m)
x	=	along-channel distance (m)
Z	=	bottom elevation (m)
β	=	flowrate damping coefficients (–)
δ	=	discrete geometry coefficient (–)
${ϵ}_J$	=	small value for hydraulic jump discrimination (–)
${ϵ}_{\mathrm{\Lambda }}$	=	flow oscillation melding coefficient (–);
η	=	free-surface elevation (m)
λ	=	bottom geometry function (m${}^{-1}$)
Λ	=	grid-scale oscillation indicator (–)
φ	=	interpolated variable (–)
ψ	=	free-surface slope (–)

Data availability statement

SvePy is available as an open-source code on GitHub at https://github.austin.utexas.edu/hodgesbr/SvePy. The archival version of the code (used in this paper) is available at https://doi.org/10.18738/T8/EHHBNB. Supplementary explanations of code details and all model data are available for public access through the University of Texas Data Repository under the SvePy dataverse at https://dataverse.tdl.org/dataverse/SvePy.

Additional information

Funding

This article was developed under Cooperative Agreement No. 83595001 awarded by the US Environmental Protection Agency to The University of Texas at Austin. It has not been formally reviewed by EPA. The views expressed in this document are solely those of the authors and do not necessarily reflect those of the Agency. EPA does not endorse any products or commercial mentioned in this publication.