[Paper Review] A compressible two-fluid model for the finite volume simulation of violent aerated flows. Analytical properties and numerical results

Frédéric Dias, Denys Dutykh+1 more|arXiv (Cornell University)|May 16, 2008

Computational Fluid Dynamics and Aerodynamics10 references3 citations

TL;DR

This paper proposes a compressible two-fluid model for simulating violent aerated flows, such as wave impacts on coastal structures, by assuming shared velocity, pressure, and temperature between air and water phases. The model uses a finite volume method to solve a hyperbolic system of conservation laws, demonstrating its ability to capture wave dynamics and dispersion behavior, with numerical results showing its effectiveness for fast qualitative analysis of air-entrained flows.

ABSTRACT

In the study of ocean wave impact on structures, one often uses Froude scaling since the dominant force is gravity. However the presence of trapped or entrained air in the water can significantly modify wave impacts. When air is entrained in water in the form of small bubbles, the acoustic properties in the water change dramatically and for example the speed of sound in the mixture is much smaller than in pure water, and even smaller than in pure air. While some work has been done to study small-amplitude disturbances in such mixtures, little work has been done on large disturbances in air-water mixtures. We propose a basic two-fluid model in which both fluids share the same velocities. It is shown that this model can successfully mimic water wave impacts on coastal structures. Even though this is a model without interface, waves can occur. Their dispersion relation is discussed and the formal limit of pure phases (interfacial waves) is considered. The governing equations are discretized by a second-order finite volume method. Numerical results are presented. It is shown that this basic model can be used to study violent aerated flows, especially by providing fast qualitative estimates.

Motivation & Objective

• To develop a simplified two-fluid model that captures the effects of air entrainment in violent wave impacts on coastal structures.
• To analyze the analytical properties of the model, including hyperbolicity and dispersion relations, under compressible and isentropic assumptions.
• To provide a numerically stable and second-order accurate finite volume discretization for practical simulation of aerated flows.
• To validate the model through numerical tests, including falling water column and water drop impact, demonstrating its capability for qualitative yet fast simulation of complex wave dynamics.

Proposed method

• The model assumes homogeneous equilibrium between air and water phases, with shared velocity, pressure, and temperature, reducing the system to a single set of conservation laws for mass, momentum, and energy.
• The equation of state is derived from the isentropic assumption, relating pressure to volume fraction, density, and internal energy via a closed-form expression involving adiabatic indices and constants.
• The governing equations are rewritten in terms of physical variables: volume fraction α, velocity u, and pressure p, leading to a hyperbolic system with a well-defined flux Hessian.
• A second-order finite volume scheme is applied using a Rusanov-type numerical flux for conservation and stability, with reconstruction based on cell averages.
• The model is validated through numerical simulations of benchmark cases, including a falling water column and a water drop impact, to assess wave formation and dynamics.
• The dispersion relation is analytically derived for small perturbations, showing how wave speed depends on phase properties and volume fraction, with a formal limit to pure fluid behavior.

Experimental results

Research questions

• RQ1 Can a compressible two-fluid model with shared velocity, pressure, and temperature accurately simulate violent aerated flows such as wave impacts?
• RQ2 What are the analytical properties of the model, particularly its hyperbolicity and wave propagation characteristics?
• RQ3 How does the model’s dispersion relation behave under small perturbations, and how does it reduce to the pure fluid limit?
• RQ4 Can the model reproduce key physical phenomena like wave formation and air entrainment effects in a numerically stable and second-order accurate manner?

Key findings

• The model successfully captures wave dynamics in air-water mixtures, even without a sharp interface, due to the compressibility and volume fraction dependence of the effective wave speed.
• The dispersion relation derived from the linearized system shows that wave propagation depends on the effective sound speed, which is significantly reduced in air-entrained mixtures compared to pure water.
• Numerical simulations of the falling water column and water drop test cases demonstrate the model’s ability to reproduce complex flow features such as splashing and jet formation with second-order accuracy.
• The model exhibits correct asymptotic behavior in the limit of pure phases, recovering the known wave speeds of pure water and air, validating its consistency with single-phase theory.
• The finite volume discretization with Rusanov flux ensures stability and conservation, enabling robust simulation of large disturbances typical in wave impact scenarios.