Atwood Number

Atwood number, abbreviated as A or At, a dimensionless number, describes density difference between two adjacent fluids with a common interface.  It is used in fluid dynamics to describe the flow behavior and stability of a two-phase system with a density difference.  The Atwood number represents the ratio of the density difference between the two phases to the average density of the system.  The Atwood number is commonly used in the study of multiphase flows, such as the behavior of bubbles in a liquid, the flow of oil and water in pipelines, or the motion of liquid droplets in gas environments.  It helps characterize and predict the interfacial dynamics, mixing, and stability of such two-phase systems.

In Rayleigh–Taylor instability, the penetration distance of heavy fluid bubbles into the light fluid is a function of acceleration time scale.

Atwood Number Interpretation

• Small Atwood Number (A  ≈  0)  -  The densities of the two fluids are very close \(( \rho_1 \approx \rho_2  )\)​.  Buoyancy effects are weak, and any instability or mixing (Rayleigh-Taylor or Richtmyer-Meshkov) grows slowly.  Think of two similar liquids, like saltwater and slightly less salty water, motion is sluggish.
• Large Atwood Number (A  ≈  1)  -  The densities differ greatly \(( \rho_1 >> \rho_2 )\)​.  Buoyancy dominates, driving rapid motion or instability.  For example, a heavy fluid like water \(( \rho_1 = 1000\; kg/m^3 )\)​ over air \(( \rho_2 \approx 1.2\; kg/m^3 )\) gives \(A = \frac{1000 - 1 }{ 1000 + 2} \approx 0.998\), nearly 1, so the heavy fluid plunges aggressively into the light one.
• Intermediate Atwood Number (0  <  A  <  1)  -  A moderate density contrast exists, leading to noticeable but not extreme buoyancy effects.  The behavior depends on the specific value, oil \(( \rho_1 = 800\; kg/m^3 )\) over water \(( \rho_2 = 1000 \;kg/m^3 )\) isn’t typical (since heavier is usually below), but swapping them gives \(A = \frac{1000 - 800 }{ 1000 + 800 } \approx 0.11\), a modest drive.