Numerical Simulation of Viscous Flow by Smoothed Particle Hydrodynamics

TLDR

Smoothed particle hydrodynamics (SPH) is an effective numerical method for many problems, especially in astrophysics, but has been limited to inviscid flows because it does not readily solve fluid equations with second‑order derivatives. This work introduces a new SPH formulation that can solve the Navier‑Stokes equations for fluids with constant viscosity. The method is applied to two‑dimensional Poiseuille flow, three‑dimensional Hagen‑Poiseuille flow, and two‑dimensional isothermal flow around a cylinder, assuming a linear temperature dependence along the flow direction, and its density and velocity fields are compared with finite‑difference results. The numerical results agree closely with analytic solutions, producing nearly uniform density and expected parabolic velocity profiles, and match finite‑difference results for Reynolds numbers 6–55 with drag coefficient differences of only 2–4%.

Abstract

Smoothed particle hydrodynamics (SPH) is an effective numerical method to solve various problems, especially in astrophysics, but its applications have been limited to inviscid flows since it is considered not to yield ready solutions to fluid equations with second-order derivatives. Here we present a new SPH method that can be used to solve the Navier-Stokes equations for constant viscosity. The method is applied to two-dimensional Poiseuille flow, three-dimensional Hagen-Poiseuille flow and two-dimensional isothermal flows around a cylinder. In the former two cases, the temperature of fluid is assumed to be linearly dependent on a coordinate variable x along the flow direction. The numerical results agree well with analytic solutions, and we obtain nearly uniform density distributions and the expected parabolic and paraboloid velocity profiles. The density and velocity field in the latter case are compared with the results obtained using a finite difference method. Both methods give similar results for Reynolds number Re = 6, 10, 20, 30 and 55, and the differences in the total drag coefficients are about 2 ∼4