Use of Thomas Algorithm for Shock Wave Analysis in Bubbly Flow

A numerical approach is developed for simulation of pressure wave propagation in a tube containing a dilute concentration of small gas bubbles. The two-phase fluid is considered homogeneous and spatial distribution of bubbles is assumed to be uniform. Bubble oscillations are modeled using the Keller equation which accounts for liquid compressibility. Heat transfer between liquid and gas is included in the analysis through solution of the radial conduction equation for a spherical gas bubble with moving interface. An energy balance over the bubble surface determines bubble internal pressure, which is assumed to be uniform. Continuity and momentum relations for the homogenous mixture along with the Keller equation are used to derive an alternate set of equations, which are more amenable to application of elementary numerical methods. These alternate equations include a diffusion equation, which is linear in the homogeneous mixture pressure. Two additional equations define the bubble radius and gas-liquid interface speed in terms of the local spatial variation in the homogeneous pressure field. The diffusion equation is solved easily using the second-order accurate Crank-Nicolson method in conjunction with the Thomas algorithm for the discretized tridiagonal algebraic system. The remaining equations comprising the fluid model are solved with an explicit, second-order accurate predictor-corrector scheme. The present approach avoids the need for staggered grids and iterative pressure correction methods used in previous work. Numerical calculations are carried out for a shock wave in a liquid column containing gas bubbles. Results show good agreement with experimental data available in the literature.