Scaling and self-similarity in two-dimensional hydrodynamics

Area of Science:

• Fluid Dynamics
• Computational Hydraulics
• Applied Mathematics

Background:

• Depth-averaged two-dimensional (2D) hydrodynamic equations are crucial for modeling large-scale water bodies.
• Understanding self-similarity in these systems simplifies complex flow behavior and enables scaling.
• The k-ε turbulence model is widely used but requires conditions for self-similar application.

Purpose of the Study:

• To investigate the conditions for self-similarity in depth-averaged 2D hydrodynamic equations.
• To explore self-similarity conditions specifically for the 2D k-ε turbulence model.
• To establish scaling relations for hydraulic model development.

Main Methods:

• Utilized one-parameter Lie group of point scaling transformations.
• Analyzed self-similarity conditions for flow variables and turbulence parameters.
• Performed numerical simulations to validate self-similarity between prototype and scaled domains.

Main Results:

• Derived self-similarity conditions for numerous flow variables, including velocities, stresses, and turbulence parameters.
• Demonstrated that initial-boundary value problems (IBVP) of depth-averaged 2D hydrodynamic flow can achieve self-similarity.
• Showcased that prototype domains can be self-similar with scaled domains through parameter adjustments.

Conclusions:

• The Lie group scaling approach provides a framework for achieving self-similarity in 2D hydrodynamic models.
• Adjusting scaling parameters and exponents allows for the creation of multiple scaled domains.
• The derived scaling relations offer significant spatial, temporal, and economic flexibility for physical hydraulic modeling.