Impact of truncation error and numerical scheme on the simulation of the early time growth of viscous fingering. (27th September 2018)

• Record Type: Journal Article
• Authors: Abdul Hamid, S.A.; Adam, A.; Jackson, M.D.; Muggeridge, A.H.
• Is Part Of: International journal for numerical methods in fluids. Volume 89:Number 1/2(2019)
• Page Start: 1
• Page End: 15
• Publication Date: 2018-09-27
• Subjects: advection‐diffusion -- error estimation -- numerical dispersion -- porous media -- truncation error -- viscous fingering; Fluid dynamics -- Mathematics -- Periodicals; 532
• DOI: 10.1002/fld.4680
• Languages: English
• ISSNs: 0271-2091

Abstract

Summary: The truncation error associated with different numerical schemes (first order finite volume, second order finite difference, control volume finite element) and meshes (fixed Cartesian, fixed structured triangular, fixed unstructured triangular and dynamically adapting unstructured triangular) is quantified in terms of apparent longitudinal and transverse diffusivity in tracer displacements and in terms of the early time growth rate of immiscible viscous fingers. The change in apparent numerical longitudinal diffusivity with element size agrees well with the predictions of Taylor series analysis of truncation error but the apparent, numerical transverse diffusivity is much lower than the longitudinal diffusivity in all cases. Truncation error reduces the growth rate of immiscible viscous fingers for wavenumbers greater than 1 in all cases but does not affect the growth rate of higher wavenumber fingers as much as would be seen if capillary pressure were present. The dynamically adapting mesh in the control volume finite element model gave similar levels of truncation error to much more computationally intensive fine resolution fixed meshes, confirming that these approaches have the potential to significantly reduce the computational effort required to model viscous fingering. Abstract : The truncation error associated with different numerical schemes and meshes is quantified in terms of apparent longitudinal and transverse diffusion in tracer displacements. Its impact on the early time growth of immiscible viscous fingers is also evaluated by comparison with the analytical growth rate. Transverse numerical diffusion is at least 1/3 apparent longitudinal diffusion and is negligible for parallel grids in first‐order finite volume schemes, whereas all fingers narrower than half the system width have similar, diminished growth rates.