Compared with the local correlation-based shear stress transport (SST) $\gamma -{\text{Re}}_{\theta }$ transition model (where SST $k-\omega$ transport equations are coupled with intermittency $\gamma$ and transitional momentum-thickness Reynolds number ${\text{Re}}_{\theta }$ transport equations), relatively simple and convenient modifications are applied to the parent SST $k-\omega$ model for computing natural and separation-induced transitions over the hydrofoil at a low-Reynolds number (LRN). The curiosity toward hydrofoil performance at an LRN has been enhanced by increasing attention to autonomous marine systems, deserving numerical simulations for transitional flow using computational fluid dynamics. With the newly devised transitional SST (T-SST) model, the viscous sublayer blending function $F_2$ is slightly modified, and a stress-intensity parameter as a function of eddy-to-laminar viscosity ratio $R_T$ is introduced; intended formulations are plausible and have significant impacts on the transition prediction. Owing to the inherent potential for predicting bypass transition, two anisotropic versions of the ${\overline{v}}^2-f$ (V2F) turbulence model are selected to evaluate their competencies in capturing separation-induced and natural transitions. Results demonstrate that natural transition prediction is more challenging than separation-induced transition for the V2F model. Nonetheless, the T-SST model performs consistently well in replicating both transitional phenomena.

An approach to devising a consistency formulation for $P_k/\epsilon$$P_k/\epsilon$ (production-to-dissipation ratio) is proposed to obtain a non-singular $C_{\mu }$$C_{\mu }$ (coefficient of eddy-viscosity) embedded in the one-equation model based on the turbulent kinetic energy k. The dissipation rate ε is evaluated with an algebraically prescribed length scale having only one adjustable coefficient, accompanied by an anisotropic function $q_{\epsilon }$$q_{\epsilon }$ enhancing the dissipation in non-equilibrium flow regions. The model accounts for the distinct effects of low Reynolds number (LRN) and wall proximity. The stress-intensity ratio $R_{\mathrm{b}}=\overline{u_1{u}_2}/k$$R_{\mathrm{b}}=\overline{u_1{u}_2}/k$ is formulated as a function of local variables without resorting to a constant $\sqrt{C_{\mu }^{\ast }}=0.3$$\sqrt{C_{\mu }^{\ast }}=0.3$. The parameters $R_{\mathrm{b}}$$R_{\mathrm{b}}$ and $P_k/\epsilon$$P_k/\epsilon$ entering the turbulence production $P_k$$P_k$ prevents presumably the overestimation of $P_k$$P_k$ in flow regions where non-equilibrium effects could result in a misalignment between turbulent stress and mean strain rate with a linear eddy-viscosity model. A comparative assessment of the present model with the Spalart–Allmaras (SA) one-equation model and the shear stress transport (SST) k–ω model is provided for well-documented simple and non-equilibrium turbulent flows. Finally, the current model provides a proposal to compute free shear flows.

An algebraic intermittency function is developed for “laminar-to-turbulent” transition flow within the framework of Bradshaw stress–intensity factor (ratio of principal shear-stress over turbulent kinetic energy in the boundary layer), which is parameterized with a “flow-structure-adaptive” variable (eddy-to-laminar viscosity ratio). Naturally, the intermittency inherits the “flow-structure-adaptive” character and captures various transition phenomena like bypass, separation-induced, and natural transitions when incorporated in an undamped eddy-viscosity transport equation. An additional viscous-production term is added with the eddy-viscosity transport equation to ensure proper generation of eddy-viscosity at the viscous sublayer when computing separation-induced transition over a low-Reynolds number airfoil. Splitting the intermittency into low and elevated free-stream turbulence intensities has the potential to avoid the “trial-and-error” inconsistency involved in most of the correlation-based transition models for precise computations. The results demonstrate that the proposed algebraic intermittency model is rational and feasible.

The SIMPLE algorithm is devised by interpolating the mass continuity and non-advective momentum equations, provoking apparent simplicity and clarity in the formulation. The SIMPLE variant entitled as the SIMPLE-AC scheme convokes an artificial compressibility (AC) parameter to augment the diagonal dominance of discretized pressure-correction equation. Both methods are characteristically pressure-based, employing a cell-centered finite-volume Δ-formulation on a non-orthogonal collocated grid. A dual-dissipation scheme accompanied by limiting factors, repressing local extrema into the cell-face velocity and pressure, is used to unravel the issue of pressure-velocity decoupling; both SIMPLE and SIMPLE-AC schemes maintain an equivalent scaling (e.g., primary and auxiliary pseudo-time steps remain the same) with the cell-face dissipation and nodal influence coefficients. The phenomenal progress embedded in both contrivances facilitate an avoidance of the pervasive velocity/pressure under-relaxation. However, the SIMPLE-AC algorithm is benefited with using a higher CFL number, enhanced robustness, and convergence compared with the SIMPLE method.