Computation of weakly-compressible highly-viscous liquid flows

We introduce a second-order accurate time-marching pressure-correction algorithm to accommodate weakly-compressible highly-viscous liquid flows at low Mach number. As the incompressible limit is approached (Ma 0), the consistency of the compressible scheme is highlighted in recovering equivalent incompressible solutions. In the viscous-dominated regime of low Reynolds number (zone of interest), the algorithm treats the viscous part of the equations in a semi-implicit form. Two discrete representations are proposed to interpolate density: a piecewise-constant form with gradient recovery and a linear interpolation form, akin to that on pressure. Numerical performance is considered on a number of classical benchmark problems for highly viscous liquid flows to highlight consistency, accuracy and stability properties. Validation bears out the high quality of performance of both compressible flow implementations, at low to vanishing Mach number. Neither linear nor constant density interpolations schemes degrade the second-order accuracy of the original incompressible fractional-staged pressure-correction scheme. The piecewise-constant interpolation scheme is advocated as a viable method of choice, with its advantages of order retention, yet efficiency in implementation.