For the last decades, there have been a growing interest in the development of accurate and robust numerical algorithms for the multi-material flows. A number of numerical methods had been proposed in an effort to study the dynamics of the fluid interface. Basically, these methods can be classified as the interface tracking method and the interface capturing method. Both approaches have their merits and drawbacks. The interface tracking method, such as the volume of fluid (VOF) method, the level set method and the arbitrary Lagrangian-Euler (ALE) method has the advantage that it can exactly define the location of fluid interface. However, sophisticated and usually complicated reconstruction procedure is needed, especially when there is breaking up or merging of the fluid interface, and loss of mass or energy conservation is usually found. In contrast, the interface capturing method treats the fluid interface as a weak solution in the computation domain. Additional fluid variables, such as the mass fraction, void fraction or specific heat ratio of fluid are used to identify the different fluid. Because the interface capturing method obeys the Rankine-Hugoniot relation, it can successfully preserve the conservation law. But it may take several mesh cells to capture the fluid interface and smearing of the interface is usually found in the simulation result.
The fluid interface problem becomes much more challenging when materials of disparate compressibility and density are considered. For example, the density of water is about 800 times the density of air, and the sound speed of water is 5 times larger than that of air. It’s difficult to capture a shock wave in the water, since the computation is very sensitive to the water density and any small variation in the density field will cause large oscillations in the pressure field. The problem will become much more stiff if the shock wave and the fluid interface are present simultaneously.
In our previous work we proposed a numerical method for the solution of the single-pressure compressible multifluid model. First, the AUSM+-up scheme, first introduced for single phase flow at all speeds by adding pressure and velocity diffusion terms to the mass and pressure fluxes, was adopted to handle the compressible liquid fluid. It was shown that the new AUSM+-up scheme could accurately capture the underwater shock wave. Then, we proposed a stratified flow model to construct the governing equation for the multifluid flow. The basic idea of the stratified flow model is to consider that different fluids within the same cell are separated from each other. So we can apply the conservation law and obtain the governing equation for both the gas and liquid fluids. Then, interfaces between the same and different fluid can be defined on the cell boundary. Numerical fluxes associated with each type of the interface can be calculated. In the actual implementation, the numerical flux through the interfaces between the same fluid (gas-gas and liquid-liquid) is calculated by the AUSM+-up scheme, and the numerical flux on the interfaces between different fluids is calculated by the Exact Riemann solver, respectively. It can be shown that our method can capture the fluid interface exactly and is robust and accurate to simulate shock-interface interaction problems.
Numerical method
Assuming that the fluids are inviscid, inter-penetrating to each other and in non-homogeneous and non-equilibrium state, the single-pressure compressible multifluid model proposed by Ishii can be written as:
where α is the void fraction of fluid. The subscript ”i” = ”g” or ”l”, representing gas or liquid phase fluid respectively. Other notations are standard. The problem for the above equation is that it is non-linear and not in conservative form, hence presenting a numerical challenge in numerical simulation. In this work, we introduce a stratified flow model which defines liquid and gas flow in distinct control volume. So we can rewrite the governing equation as:
One advantage of the stratified flow method is that the above equation is now in conservation form. When cooperate with the AUSM+-up scheme designed for the compressible liquid flow, our method can successfully simulate the phenomena of shock-fluid interface interaction problem with large pressure and density disparity.
Shock-water droplet interaction problem
In this case we have a planar water column (diameter 6.4 mm) at the origin and the incoming air shock wave at the position x = −4.0mm in the beginning. The fluid states before the shock is p = 1.0×105Pa, u = 0.0m/sec, T = 347.0◦K; and after the shock is p = 2.3544 × 105Pa, u = 246.24m/sec, T = 451.2◦K. The simulation result is demonstrated below. The pressure field is represented by the color contours, and the void fraction function is outlined by the white lines.
Underwater shock-air bubble interaction problem
The interaction of underwater shock and the air bubble is studied. A air bubble with diameter 6.0 mm is immersed in the water with its center in the origin. The incoming shock is initially located in x = −4.0mm. The fluid states before the shock is p = 1.013250×105Pa, u = 0.0m/s and T = 292.98◦K; and the fluid states behind the shock is p = 1.6 × 109Pa, u = 661.81m/s and T = 595.14◦K. The Mach number of the shock wave is M = 1.509.
