This paper presents an iteration-based multi-fidelity framework to reduce the computational cost of CFD-in-the-loop turbulence model discovery for steady Reynolds-averaged Navier–Stokes (RANS) simulations. The framework builds on tensor-based gene expression programming (TB-GEP), which generates symbolic turbulence models embedded directly into CFD solvers and evaluated a posteriori during training. Instead of advancing every newly generated candidate to a practical high-fidelity endpoint, the framework first evaluates all candidates at a partially converged CFD state. The resulting fitness estimates guide multi-objective preselection of a small subset of promising candidates for continuation to the high-fidelity endpoint. The framework is applied to coupled Reynolds-stress anisotropy and turbulent kinetic energy production corrections for rectangular-duct flow. This is a challenging benchmark in which baseline k–ω SST fails to predict Prandtl’s secondary flow of the second kind. Compared with conventional single-fidelity CFD-in-the-loop training, the proposed approach reduces nominal search-stage solver work by approximately 71% to reach the same training-score target. The representative coupled model improves secondary-flow and turbulent kinetic energy predictions relative to baseline SST in all six test duct cases. The results support partially converged CFD states as a practical source of fitness guidance for more efficient turbulence model discovery in the steady RANS configurations considered.
A DNS-calibrated framework is developed for reconstructing mean velocity profiles in turbulent flow over rough walls using a two-layer reference-plane approach. The model divides the near-wall domain into a roughness-dominated lower region, represented by a parabolic velocity profile, and an upper region resolved using the standard RANS equations starting from a reference plane located a distance ks above the adopted wall-normal origin. To parameterize the lower region, three velocity profile coefficients, a1, a2, and a3, are extracted from DNS data and correlated with the roughness function and roughness geometric metrics, including effective slope and root-mean-square roughness height. A new scaling variable, γ, is introduced to quantify the contribution of the lower region to the total volume flow rate. The framework is assessed against additional DNS cases of turbulent channel and pipe flows with three-dimensional roughness, demonstrating its ability to reconstruct mean velocity profiles under conditions where the required roughness information is prescribed. The formulation maintains roughness-geometry sensitivity without requiring additional fitting for the assessment cases and offers a path forward for embedding geometric effects into RANS wall treatments through physically meaningful parameters. The framework requires the hydrodynamic roughness information, ks and ΔU+, to be prescribed or obtained from an independent roughness-function or drag relation; it is therefore not intended as a standalone predictor of rough-wall drag.
