Numerical Computation of High Reynolds Number Cavity Flow Using SPH Method with Stream Function and Vorticity Formulation
Yusuke Imoto
TL;DR
This work tackles resolving small-scale vortices in high Reynolds number cavity flow by formulating the incompressible Navier–Stokes equations in the stream function–vorticity framework and applying SPH discretization. The method advances $\omega$ with a forward-time scheme $(\omega^{n+1}_i-\omega^n_i)/\Delta t^n=(1/Re)\langle\Delta \omega^n_i\rangle+\xi^n_i$ and solves $\langle\Delta \phi^{n+1}_i\rangle=\omega^{n+1}_i$, followed by updating particle positions via $\Delta t^n\langle\nabla \phi^{n+1}_i\rangle$. Results show that the SV formulation reproduces the small vortices observed in high-order finite-difference benchmarks for $Re=10^2,10^3,10^4$, whereas the velocity–pressure SPH formulation underpredicts these features. Limitations include reduced accuracy at $Re=10^4$ with about $10^4$ particles and the inability to extend the stream-function formulation to three dimensions, motivating further work on higher-Re regimes and alternative 3D-capable formulations.
Abstract
When numerically computing high Reynolds number cavity flow, it is known that by formulating the Navier-Stokes equations using the stream function and vorticity as unknown functions, it is possible to reproduce finer flow structures. Although numerical computations applying methods such as the finite difference method are well known, to the best of our knowledge, there are no examples of applying particle-based methods like the SPH method to this problem. Therefore, we applied the SPH method to the Navier-Stokes equations, formulated with the stream function and vorticity as unknown functions, and conducted numerical computations of high Reynolds number cavity flow. The results confirmed the reproduction of small vortices and demonstrated the effectiveness of the scheme using the stream function and vorticity.