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.

Rayleigh–Taylor Instability  -  Occurs when a heavier fluid is supported above a lighter fluid in a gravitational field (or under acceleration).  The Atwood number governs the growth rate and the morphology of the resulting mixing layer, including the formation and rise of bubbles of the lighter fluid and the fall of spikes of the heavier fluid.  Higher Atwood numbers generally lead to faster growth rates and more asymmetric structures.

Richtmyer-Meshkov Instabilities  -  Develops when a shock wave passes through an interface between two fluids of different densities.  The Atwood number influences the post-shock growth rate of perturbations at the interface and the subsequent mixing.

Boussinesq Approximation  - A simplification commonly used in fluid dynamics, especially when dealing with buoyancy-driven flows, like natural convection.  The Boussinesq approximation assumes that density variations are negligible except where they appear in buoyancy (gravity) terms.  In many practical situations (like air or water flows with small temperature differences), the variation in density due to temperature is very small, and considering full compressible flow equations becomes unnecessarily complex.  The Boussinesq approximation simplifies the math while keeping the essential physics.

Atwood Number Interpretation

• Atwood Number (A = 0)  -  This indicates that the densities of the two fluids are equal (ρ1​ = ρ2​).  In this case, there is no buoyancy force arising from density differences, and phenomena driven by density gradients (like Rayleigh-Taylor or Richtmyer-Meshkov instabilities) will not occur due to this density difference.  The interface between the fluids behaves neutrally in terms of density stratification.
• Atwood Number (0 < A < 1)  -  This signifies a density difference between the two fluids, with ρ1​ being denser than ρ2​.  The magnitude of $A$ indicates the extent of this density contrast:
  • Small $A$ (close to 0)  -  The density difference is small.  Buoyancy forces are weak, and the dynamics of the interface are less pronounced in density driven instabilities.  The behavior of the flow can often be approximated using the Boussinesq approximation, where density variations are only considered in the buoyancy term.
  • Large $A$ (close to 1)  -  The density difference is significant (ρ1​ >> ρ2​).  Buoyancy forces are strong, leading to a more pronounced development of hydrodynamic instabilities.  In Rayleigh-Taylor instability, for example, the lighter fluid rises as broad bubbles, and the heavier fluid falls as narrow spikes.   As $A$ approaches 1, the behavior often resembles the case of a heavy fluid falling into a vacuum.
• Atwood Number (A < 0)  -  This indicates that the densities of the two fluids are equal (ρ1​ = ρ2​).  In this case, there is no buoyancy force arising from density differences, and phenomena driven by density gradients (like Rayleigh-Taylor or Richtmyer-Meshkov instabilities) will not occur due to this density difference.  The interface between the fluids behaves neutrally in terms of density stratification.