A weakly compressible SPH method for RANS simulation of wall-bounded turbulent flows

TL;DR

This work develops a weakly compressible SPH (WCSPH) framework for two-equation RANS turbulence modeling of wall-bounded flows, addressing intrinsic Lagrangian challenges with Adaptive Riemann–eddy dissipation (ARD), Limited Transport Velocity Formulation (Limited TVF), and a Lagrangian particle-based wall model plus a boundary-offset technique. The approach couples the $k$–$\e$ model with a standard wall function, and implements a robust discretization (low-dissipation Riemann solver, RKGC, TVF) to achieve stability and accuracy for straight, curved, and mildly separated channels. Key contributions include: (i) an adaptive dissipation strategy (ARD) that bridges Riemann dissipation and eddy viscosity to prevent over-damping; (ii) a limited TVF to prevent artificial production of turbulent kinetic energy in plug-flow regions; (iii) a Lagrangian wall-model framework with wall-adjacent and extended layers, four wall-boundary conditions, a weighted velocity-gradient compensation, a constant-$y_p$ convergence strategy, and a boundary-offset technique for rigorous resolution studies; and (iv) demonstration of rigorous convergence and good agreement with DNS/experimental data across multiple wall-bounded turbulent flows. The method bridges particle-based and mesh-based RANS approaches, offering potential for efficient turbulent FSI simulations and extension to other turbulence models.

Abstract

This paper presents a Weakly Compressible Smoothed Particle Hydrodynamics (WCSPH) method for solving the two-equation Reynolds-Averaged Navier-Stokes (RANS) model. The turbulent wall-bounded flow with or without mild flow separation, a crucial flow pattern in engineering applications, yet rarely explored in the SPH community, is simulated. The inconsistency between the Lagrangian characteristic and RANS model, mainly due to the intense particle shear and near-wall discontinuity, is firstly revealed and addressed by the mainstream and nearwall improvements, respectively. The mainstream improvements, including Adaptive Riemann-eddy Dissipation (ARD) and Limited Transport Velocity Formulation (LTVF), address dissipation incompatibility and turbulent kinetic energy over-prediction issues. The nearwall improvements, such as the particle-based wall model realization, weighted near-wall compensation scheme, and constant $y_p$ strategy, improve the accuracy and stability of the adopted wall model, where the wall dummy particles are still used for future coupling of solid dynamics. Besides, to perform rigorous convergence tests, an level-set-based boundary-offset technique is developed to ensure consistent $y^+$ across different resolutions. The benchmark wall-bounded turbulent cases, including straight, mildly- and strongly-curved, and Half Converging and Diverging (HCD) channels are calculated. Good convergence is, to our best knowledge, firstly achieved for both velocity and turbulent kinetic energy for the SPH-RANS method. All the results agree well with the data from the experiments or simulated by the Eulerian methods at engineering-acceptable resolutions. The proposed method bridges particle-based and mesh-based RANS models, providing adaptability for other turbulence models and potential for turbulent fluid-structure interaction (FSI) simulations.

Figures (30)

• Figure 1: The profiles (from left to right) of the mean flow velocity $U$, turbulent kinetic energy $k$, turbulent dissipation rate $\epsilon$ and eddy viscosity $\mu_t$, obtained from $k-\epsilon$ RANS model in a fully-developed turbulent straight channel. Note the main stream and wall-adjacent region is separated by the dash lines indicating the first fluid layer thickness $y_p$.
• Figure 2: The typical flow field in a turbulent expanding channel obtained without Riemann dissipation.
• Figure 3: In a fully-developed turbulent straight channel, the results with the original and limited TVF on velocity gradient and turbulent kinetic energy. $N_f$ gives the number of fluid particles on the cross-section.
• Figure 4: Division of the near wall regions for the particle-based method. For all the fluid particles that are near the wall, the distance to wall $r^n_w$ is calculated. The particles that satisfy $r^n_w<R$, are defined in the $P_{ext}$ layer, where $R=2h-dp$. For those particle, whose $r^n_w<dp$, are defined in the $P$ layer.
• Figure 5: The particle-based wall model for complex geometry when $b=1$, $\mathbf{n}_b$ are the unit normal vectors evaluated at wall particles.