From Courant to Saint Venant
View our May 2022 webinar in which we address the Courant-Friedrichs-Lewy Condition, the Saint Venant equations, and Eulerian flow field specifications:
As a bonus, you’ll get hear a Frenchman pronounce “Saint Venant” and find out that most of us have been mispronouncing the name of the famed Swiss mathematician, “Euler.” [Hint: it does not rhyme with the repeating name in this video clip]
This is the second free introductory webinar for our Hydrology and Hydraulics Essentials training course. The first webinar, addressing the Froude Number, took place in February 2022:
The following video lists the topics covered in the 8-part Hydrology and Hydraulics Essentials course series:
Background
In hydraulic modelling, the Courant Number essentially refers to the number of computational grid cells across which water travels per computational time step. For explicit solutions schemes, we tend to limit the Courant Number to 1.0 or less, meaning that flowing water would not skip any grid cells between computations. To add a bit more complexity to that definition, sometimes the velocity used to compute the Courant Number includes the velocity of the water itself, the wave celerity, or both. Here is the Courant Number equation as listed in the Australian Rainfall and Runoff guidelines:
We addressed the velocity “u” and the wave celerity (square root of gravity times depth) in our Froude Number webinar. To understand the remaining terms of the Courant Number equation, we’ll need to address delta T and delta X, which are covered nicely in Euler’s equation. Have a look at this brief clip from the movie “Hidden Figures”:
In this scene, Katherine Johnson, a real-life “computer” (which incidentally used to be a job description for humans rather than referring to a machine) arrives at the need to apply Euler’s equation for computing the Apollo Mission’s trajectories. UCLA professor Alan Garfinkel puts this moment into context here:
He provides an excellent, succinct explanation of Euler’s Method here:
For those wanting additional context, this video series includes some great explanations and animations of implicit and explicit time integration and a discussion of time step size:
Please contact us with any questions about this topic or the upcoming HH Essentials course series