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Flow model for dispersed water waves

In hydrodynamics, ordinary (non-breaking) water waves are usually considered as separated- phase flows, in which the free-surface separates the gas- and liquid-components completely. Neither the water sprays ejected from the wave front nor the air bubbles entrained into the water are considered in the separated flow model. However for breaking waves as mentioned in the above section, the water mass may trap large air pockets and/or small air bubbles when the crest starts to curl. At the same time, water droplets may also be ejected into the air as illustrated in Figure 1. In this case, the flow is of dispersed gas- and liquid- phases. In order to construct the mathematical model for wave impact problems, we adopt the homogeneous equilibrium approach to make the following assumptions:

• 1. The bubbly fluid is assumed to be a homogeneous mixture of air and water.
• 2. Each component obeys the conservation laws of mass, momentum and energy.
• 3. The mixture obeys the conservation laws of mass, momentum and energy.

Figure 1: The inherent multi-phase nature of an overturning wave.

• 4. The pressure and velocity of all the phases and components are identical within a small fluid particle.
• 5. The fluid undergoes an adiabatic thermodynamic process during the compression and expansion of the trapped air pockets.

These assumptions are based on the belief that differences in the thermodynamic and mechanical variables will promote momentum, energy and mass transfer between the phases rapidly enough so that equilibrium is reached (Corradini, 1997; Bernard-Champmartin, 2014). The equilibrium model is usually considered an appropriate approach to treat free- surface flows (Saurel and Lemetayer, 2001).

It is important to pay attention to the thermodynamics of air enclosed wave impact events. Simply assuming the fluids (especially the gas phase) to be isothermal could be problematic, because an isothermal process usually should occur slowly enough to allow the system to continually adjust its temperature through heat exchange with the surrounding environment or an outside thermal reservoir. However, a wave impact event usually lasts for only a few seconds or less. Within such a short time, it is difficult for the whole system to adjust its temperature to reach a uniform and constant state. The recent work of Abraham- sen and Faltinsen (2011) reveals that the gas phase undergoes an adiabatic process during air enclosed plunging wave impacts.

Mathematical model

For each individual fluid component i (i = 1 for air, i = 2 for water), its basic material properties can be described as follows

• 1. Density pi = where mi is mass and Q; volume.
• 2. Pressure p = p = p2, both components have the same pressure.
• 3. Velocity V = ' = V'2. both components have the same velocity.
• 4. Internal energy p,c.
• 5. Kinetic energy p^eF = 4pt V2.
• 6. Total energy pej = pie + pp;^.

To determine the internal energy of fluid component i, an appropriate equation of state should be adopted. Since the ideal gas equation of state is not suitable for liquid, the stiffened-gas equation of state is utilised for both fluid components in the present study.

Therefore, the internal energy of component г can be described as and the total energy is

where 7* is a polytropic constant and pc t is a pressure constant . The parameters Г, and П, are defined as

Additionally, the speed of sound for each component can be calculated as

The formulae (1), (2) and (4) given to calculate the internal/total energy and speed of sound comply with the adiabatic assumption.

In order to describe the material properties of the homogeneous mixture, we introduce the volume fraction function aj for air, defined as

Accordingly, we have a2 = 1 — 07 for water. Based on these values, the material properties of the water-air mixture can be expressed by the following

• 1. Density p = Y,iLi aiPi
• 2. Momentum pV = ai(Pi^0
• 3. Kinematic energy peK = <*,(/?£eb)
• 4. Internal energy pc1 = аг(Р«е!)
• 5. Total energy peT = pc1 + peK = ai(pieJ)

in which N = 2. Substituting Equation (2) into the formulation of mixture total energy, the pressure can be computed as

The speed of sound for the bubbly water-air mixture can be estimated by Wood’s formula (Wood, 1941)

The mathematical model used here for the flow of the water-air mixture consists of the mass, momentum and energy conservation laws for the mixture. A conservation law of mass for each component is also included. In particular, gravitational effects should be considered and included for water wave problems. Consequently, the underlying conservative part of the flow model can be expressed in the following form

in which U is the vector of conservative variables, F is the flux function, G are the source terms and these are defined as

where u, v and w are the velocity components along x, у and z axes; g is the gravitational acceleration; h is the enthalpy given by

In addition to the conservative part, the advection of volume fraction function Dqj/Dt also needs to be considered. Here, we adopt Kapila et aids one-dimensional advection equation (Kapila et ah, 2001)

extended to three dimensions

where К is a function of the volume fraction and sound speed given by

Equation (11) is derived from the pressure equilibrium assumption, and its right hand side term assures that the material derivatives of the phase entropy are zero in the absence of shock waves. If we neglect the right hand side of Equation (11), then this is a standard transport equation for a as pointed out by Murrone and Guillard (2005).

The overall flow model includes Equations (8) and (13) and we write it in the following form

The underlying reason we do not choose a fully conservative or primitive model but a quasi- conservative model is due to the following factors. Fully conservative flow models have the aforementioned difficulties at material interfaces where nonphysical oscillations inevitably occur even for first order schemes (Saurel and Abgrall, 1999) due to a nonphysical pressure update (Ivings et ah, 1998; Johnsen and Colonius, 2006) or negative volume fraction (Abgrall and Kami, 2001) during numerical computations. Although a primitive variable flow model can avoid these oscillations, difficulties arise when resolving strong shock waves to maintain the correct shock speed (Ivings et ah, 1998). For complicated problems consisting of both material interfaces and shock waves, combining the fully conservative and primitive variable model formulations has previously been found to be an effective strategy (Ivings et ah, 1997). However this method is quite intricate as switching is required between the two models for the different regions (Ivings et al., 1998). Quasi-conservative flow models, which combine the conservation laws with a non-conservative scalar (volume fraction or other material property) advection equation, have proved proficient and much simpler in the past (Saurel and Abgrall, 1999; Johnsen and Colonius, 2006).

If gravitational effects are excluded and only the x direction is considered, Equation (14) will reduce to a five-equation system which can be named the five-equation reduced model Murrone and Guillard (2005) or Kapila et al. model (Saurel et al., 2009). For the one-dimensional five-equation reduced model, Murrone and Guillard proved that it can be derived from the two-pressure and two-velocity Baer-Nunziato equations in the limit of zero relaxation time (Murrone and Guillard, 2005). The)' also proved that the reduced system has five real eigenvalues (three eigenvalues are equal) and is strictly hyperbolic with five linearly independent eigenvectors. Important information about other mathematical properties of the one-dimensional system, which include the structure of the waves, expressions for the Riemann invariants and the existence of a mathematical entropy, can also be found in their work. The three-dimensional two-pressure two-velocity Baer-Nunziato model has eleven equations in total (Tokareva and Toro, 2010). The model (14) is computationally less expensive as it deals with only seven equations.

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