Machine-learning-based wake prediction and optimization of a circular cylinder with vortex generators using particle swarm optimization and musk ox optimization algorithms

Flow around a circular cylinder is a canonical bluff-body problem for tubular and offshore structures, where separation and vortex shedding govern unsteady loads. Here, experimental particle image velocimetry data at Reynolds number Re = 8 × 103 were used to train an artificial neural network (ANN) surrogate for the wake of a vortex-generators (VGs)-equipped circular cylinder. The surrogate learned a pointwise mapping from the VGs angles and spatial coordinates (α, β, x, y) to five flow statistics: mean velocities (⟨u⟩, ⟨v⟩), velocity fluctuations (urms′, vrms′), and Reynolds shear stress ⟨u′v′⟩. A comprehensive search of 1736 ANN candidates identified the best architecture as a multilayer perceptron with a 4-neuron linear projection layer followed by two 65-neuron hidden layers with hyperbolic tangent activation, trained with a mini-batch size of 2; near-unity accuracy was achieved across all five quantities (R2 ≈ 0.98–0.99) with low errors. SHapley Additive exPlanations (SHAP) and Sobol analyses showed that α is the primary control parameter: mean |SHAP| for α was 0.004 29 m/s for J(urms′), 0.007 88 m/s for J(vrms′), and 3.298 × 10−4 m2/s2 for J(⟨u′v′⟩), exceeding β by factors of 3.1, 2.9, and 2.6. The ANN was then coupled with Particle Swarm Optimization and Musk Ox Optimization, and both converged to the same optima for each target, yielding distinct (α, β) pairs for J(urms′), J(vrms′), and thereby demonstrating clear trade-offs among turbulence metrics.