Axisymmetric wavy core flow in a heavy viscous oil

A direct numerical simulation of spatially periodic wavy core flows is carried out under the assumption that the densities of the two fluids are identical and that the viscosity of the oil core is so large that it moves as a rigid solid which may nevertheless be deformed by pressure forces in the water. The waves which develop are asymmetric with steep slopes in the high pressure region at the front face of the wave crest and shallower slopes at low pressure region at lee side of the crest. The simulation gives excellent agreement with the experiments of Bai et al.[1992] on up flow in vertical core flow where axisymmetric bamboo waves are observed. We define a threshold Reynolds number and explore its utility; the pressure force of the water on the core relative to a fixed reference pressure is negative for Reynolds numbers below the threshold and is positive above. The wave length increases with the holdup ratio when the Reynolds number is smaller than a second threshold and decreases for larger Reynolds numbers. We verify that very high pressures are generated at stagnation points on the wave front. It is suggested that the positive pressure force is required to levitate the core off the wall when the densities are not matched and to center the core when they are. A further conjecture is that the principal features which govern wavy core flows cannot be obtained from any theory in which inertia is neglected.