CFL Number and its significance in CFD

The CFL number, also known as the Courant-Friedrichs-Lewy number, is a dimensionless number used in fluid dynamics and numerical simulations to assess the stability and accuracy of numerical solutions to partial differential equations, in particular those that describe wave propagation phenomena.

It’s named after Richard Courant, Kurt Friedrichs, and Hans Lewy, who were mathematicians instrumental in its development and first introduced in 1928. This derivation is one of the most influential works for the development of CFD techniques.

The CFL number is defined as:

Where:
⦁ is the characteristic velocity of the system
⦁ is the time step used in the numerical simulation
⦁ is the spatial grid spacing
⦁ is usually 1

The CFL number essentially quantifies the ratio of the time step to the spatial grid spacing relative to the velocity of waves in the system.

Significance:

Fig 1. Movement of Fluid

The above figure helps us in understating the importance of CFL numbers. If CFL <=1, then the fluid particles move from one cell to another within one time step (at most). While if CFL >1 a fluid particle moves through two or more cells at each time step, and this can affect convergence negatively.

If CFL number goes over the critical value, the solution becomes unstable, leading to the solution that grow uncontrollably over time, providing unrealistic and inaccurate results.

While CFL number being on the lower side or close to 1, though the solution is highly stable, realistic, and accurate, the spatial resolution of the domain should be as refined as possible to capture the fluid movement. This eventually leads to the smaller time steps and has the effect on computational costs like HPC requirements, higher data storage capacity and the time required to solve the analysis.

The following is an example of Volume of Fluid (VoF) method in the filling of Water by displacing Air has been performed using 3DEXPERIENCE Fluid Dynamics Engineer (FMK) role.

The following images show the clear difference between the interface definition between the two different CFL numbers.

On the left side of the image, one can observe the intricate dispersion of the water medium displacing the initially occupied air space. This visual display captures the dynamic development of bubble patterns influenced by surface tension over a specific time frame. In contrast, the right side of the image presents an approximation of the interface between the two media, resulting in the absence of any discernible physical dispersion of water.

The significance of the CFL (Courant-Friedrichs-Lewy) number becomes evident in scenarios involving time-dependent analysis. The CFL number acts as a crucial parameter, influencing the accuracy and stability of simulations, particularly when considering the evolving dynamics of fluid interactions and the impact of surface tension on bubble formation over time.

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