Courant Number

on . Posted in Dimensionless Numbers

Courant number, abbreviated as Co, a dimensionless number, also called Courant-Friedrichs-Lewy number (CFL), is used in numerical simulations and computational fluid dynamics to determine the stability and accuracy of numerical solutions for time dependent partial differential equations, particularly those that involve convection or wave propagation.  The Courant Number represents the ratio of the distance a wave or disturbance travels during a time step to the distance between grid points.  It essentially describes how far a wave can travel relative to the grid spacing in one time step.  If the Courant Number is too large, it can lead to instability in the numerical solution, causing the simulation to produce unrealistic or incorrect results.

The choice of Co number depends on the specific numerical scheme being used.  Different schemes have different stability and accuracy characteristics associated with certain ranges of Co values.  In practice, the Co number is often kept below a certain threshold to ensure stable and accurate simulations.

Courant Number formula

$$Co = \Delta t \; U\;/\; \Delta x$$     (Courant Number)

$$\Delta t = C0 \; \Delta x \;/\; U$$

$$U = Co \; ( \Delta x \;/\; \Delta t )$$

$$\Delta x = \Delta t \; U\;/\; Co$$

Symbol English Metric
$$Co$$ = Courant number $$dimensionless$$
$$\Delta t$$ = time interval $$sec$$ $$s$$
$$U$$ = magnitude of the velocity (whose dimension is length/time) $$ft\;/\;sec$$ $$m\;/\;s$$
$$\Delta x$$ = distance between any two consecative points $$ft$$ $$m$$

Tags: Fluid Wave