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Air Compressibility and Aeration Effects in Coastal Flows

Zhihua Mo, [1] [2] Ling Qian and Derek Causon

Introduction

Multiphase flow impact problems are frequently encountered in natural events and industrial applications. Violent water waves may cause severe damage to ships, offshore platforms and coastal defences. Recent examples are the winter storms that occurred in the United Kingdom from December 2013 to February 2014, during which huge waves destroyed railway lines and coastal walls (Kay, 2014; Rodgers and Bryson, 2014). These disasters have resulted in huge economic loss. Scientific investigations from engineering, environmental and other perspectives are a necessity to understand and mitigate these naturally occurring events.

In the past several decades, great efforts have been made to study wave impacts on breakwaters, sea walls and liquid storage tanks, etc. through carefully controlled experiments (Bullock et ah, 2007: Lugni et ah, 2010), field measurements (Crawford, 1999) and theoretical analysis (Peregrine, 2003; Korobkin, 2006). The extreme impulsive pressures recorded in violent wave impact events can be tens or even hundreds of times those of impacts induced by ordinary non-breaking waves (Lugni et ah, 2010). Intentions to classify the impact events into different types can be found in the works of Kirkgoz (1982), Schmidt et ah (1992) and Oumeraci et ah (1993). Here, we follow the ideas of Lugni et ah (2010) to divide these into three modes including (a) impact of an incipient breaking wave, (b) impact of a broken wave with an air pocket and (c) impact of a broken wave with water-air mixing. The second impact mode of an overturning wave enclosing an air cavity is of particular interest in the present study.

Traditionally the influence of trapped air pockets on plunging waves was ignored due to the small density of air compared to water. However, laboratory and field observations disclose that air may play an important role in the impact process (Bullock et ah, 2007; Lugni et ah, 2010; Peregrine, 2003; Chan et ah, 1988; Lugni et ah, 2010). During the transition of a plunging wave, an air pocket (or pockets) may be trapped in the body of the wave and compressed by the water mass; thus a portion of the wave energy will be transferred to the pocket. Once the wave front impacts the surface of the structure, the air pocket starts expanding to release the stored energy. The strongest pressure peak in the form of a “cathedral-roof” shape and subsequent pressure oscillations will be experienced by the structure. This distinct phenomenon has been discovered in experiments (Bullock et ah, 2007; Chan et ah, 1988; Lugni et ah, 2010). In addition, the slamming of blunt bodies could also trap air into water beneath the lower surface of the structures. The air is extensively compressed and forms a very thin layer as the structure approaches the water surface. It may also expand afterwards when the pressure drops. The trapped air layer may repeat the expansion and contraction cycle leading to pulsating loads on the structure.

The problem becomes even more complicated when air bubbles are present in the water. In the ocean, bubbles are entrained/generated in the water by a number of different processes including biological production, entrapment of air by capillary waves, white capping and wave breaking (Crawford, 1999). These bubbles generally persist for many wave periods in seawater and do not tend to coalesce and hence remain small, rising slowly through the water (Scott et ah, 1975). The peak pressure and impact duration are strongly influenced by trapped large air pockets as well as entrained small air bubbles. This might be expected to be closely related to the compressibility of the air and water-air mixture (Bredmose et ah, 2009; Peregrine et ah, 2005; Plumerault, 2009). Furthermore, negative pressures essentially gauge values below the atmospheric pressure have been recorded in field measurements (Crawford, 1999) and laboratory experiments (Bullock et ah, 2007; Lugni et ah, 2010; Oumeraci et ah, 1993). Bullock et ah stated that negative pressures have the potential to induce large seaward forces resulting in the removal of blocks from masonry structures (Bullock et ah, 2007). Medina-Lopez et ah suggested that cavitation may have made a significant contribution to the failure of the Mutriku Breakwater Wave Plant, particularly causing localised damages at joints between the staked cell units (Medina-Lopez, 2015). Crawford pointed out that such large forces could produce a sufficient overturning moment to cause overall failure of important structures like breakwaters (Crawford, 1999). Additionally, Lugni et ah indicated that even small pressure fluctuations might induce local flow cavitation for conditions close to the cavitation threshold (Lugni et ah, 2010). Therefore, it is necessary to be aware of these issues and the need to include all relevant physics when theoretical analysis, experiments or numerical computations are used to investigate wave impact problems.

Compared to experimental investigations of plunging wave impacts, which have made significant progress regarding measurement of peak pressures and forces (Bullock et ah, 2007; Lugni et ah, 2010; Chan et ah, 1988; Lugni et ah, 2010), numerical simulations are not yet adequate to fulfil industrial and academic requirements due to the extreme complexity of these problems. Not to mention the challenge of resolving the free surface, which might overturn, break and experience further strong deformations, the compressibility of air and the water-air mixture and possible flow cavitation make the problem much more difficult. Earlier efforts for developing 3D numerical wave tanks (NWTs) were mostly based on single-fluid incompressible potential flow theory (De Divitiis and De Socio, 2002; Riccardi and Iafrati, 2004; Semenov and Iafrati, 2006; Xu et ah, 2010; Zhang et ah, 1996). Since air is not explicitly considered in the mathematical model, computation of entrapped air pockets and/or entrained air bubbles in waves cannot directly be achieved with these single-fluid NWTs. Two-fluid NWTs based on the incompressible Navier-Stokes equations have been proposed to simulate both the liquid and gas phases for violent wave breaking problems (Christensen and Deigaard, 2001; Greaves, 2006; Greaves, 2004; Lubin et ah, 2006; Qian et ah, 2006; Ma et ah, 2011). However, these treat both the water and air as incompressible fluids, which means the density of each fluid remains constant throughout the process. Unfortunately, compressibility effects in the air pocket and water-air mixture cannot be handled properly by these models, nor, importantly, cavitation effects.

More recently, researchers have started to explore the importance of the compressibility of air and the water-air mixture for wave impact problems. Peregrine et ah (2005) and Bredmose et ah (2009) proposed a weakly compressible flow model combined with a single-phase potential flow solver to compute wave impact events. A fully conservative flow model based on the compressible Euler equations was adopted in the impact zone to describe the water wave with entrapped air pocket or wave with entrained air hubbies. In the energy equation, only the compressibility of the gas phase was included without considering compressibility in the liquid phase. Systematic numerical analysis of wave impacts on vertical walls were conducted and promising results were presented. However, model tests of a one-dimensional shock wave passing through a water-air interface exhibited strong nonphysical pressure oscillations at the material interface. This Gibbs phenomenon is a well known numerical artefact when using a conservative variable scheme and measures should be taken to preclude them. Plumerault et al. (2012) also carried out studies of aerated-water wave problems. They described the flow with a three-fluid model for gas, liquid and gas-liquid components in the flow. Energy conservation was not enforced explicitly with the corresponding equation removed from the equation set in their mathematical model. Instead, a pressure relaxation method (Saurel and Abgrall, 1999) was employed to solve the three-fluid model. Strong nonphysical oscillations also arose at the material interface in their water-air shock tube results (Plumerault et ah, 2012). They presented results for a deep water breaking Stokes wave in the incompressible limit. We observe that these works still have deficiencies in treating the material interface between water and air. In addition, it is unclear whether they have the capability explicitly to deal with local flow cavitation, as pointed out by Lugni et al. (2010).

The compressibility of air is also a crucial factor influencing the performance and resonance characteristics of coastal oscillating water columns (OWCs) even for non-violent flows (Dimakopoulos et ah, 2017). Due to scaling effects, data obtained laboratory experiments cannot be properly extrapolated to full scale prototypes, where significant discrepancy in phase between air pressure and hydrodynamic pressure could change the peak load and performance of OWCs (Weber et ah, 2007).

It is of great importance to develop an appropriate wave modelling tool, which could deal with trapped air pockets and a water-air mixture and to produce physical solutions in both high and low pressure regions fully accounting for compressibility and cavitation effects. This chapter summarises the recently developed compressible multiphase hydrocode AMAZON-CW. The compressible wave model is presented in Section 2. The numerical method, which utilises a finite volume approach, a third-order MUSCL reconstruction and the HLLC approximate Riemann solver to compute the convective fluxes in the governing equations, is presented in Section 3. Numerical results for benchmark test cases are presented in Section 4. Final conclusions are drawn in Section 5.

  • [1] Centre for Mathematical Modelling and Flow Analysis, The Manchester Metropolitan University. ManchesterMl 5GD. UK.
  • [2] Corresponding author: This email address is being protected from spam bots, you need Javascript enabled to view it
 
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