An immersed boundary formulation incorporating a two-layer wall model approach for RANS simulations with complex geometry

We propose an immersed boundary (IB) formulation incorporating a two-layer wall model approach for Reynolds-averaged Navier-Stokes (RANS) simulations with complex geometry. The main objective of the present study is to broaden the capability of the IB method to high-Reynolds number turbulent flows with non-equilibrium effects. A two-layer wall model is applied to the IB method by considering a thin boundary layer approximation near the wall in order to reduce the computational cost of resolving the turbulent boundary layer. Tangential pressure gradient and convective terms are included in the thin boundary layer equation to make the wall model suitable for non-equilibrium flows, in which cross-derivative terms are modeled along the wall-normal direction to maintain the ordinary differential form of the equation. Turbulent viscosity in the inner layer is determined using a simplified k−ϵ−f_μ model, that performs better than the typical mixing-length model in the separation and reattachment regions, where the friction velocity u_τ becomes zero. To impose the mass conservation constraint near the IB surface with marginal computational cost, the IB approximated domain method is applied. The proposed IB formulation is validated using numerical simulations of two-dimensional turbulent channel flow, backward-facing step flow, turbulent flow around a circular cylinder, and three-dimensional turbulent flow over a wall-mounted circular sphere with a relatively coarse grid near the IB surface. The results demonstrate that the proposed IB formulation captures the flow patterns well and accurately predicts wall shear stress for both equilibrium and non-equilibrium flows, even with a relatively coarse grid near the wall.