A new physically based model for cumulative wakes and application to wind farms

Accurate representation of cumulative wake interactions remains a central challenge in wind farm aerodynamics. Most existing multiple-wake models rely on empirical superposition rules. This paper develops a physically consistent analytical framework for multiple wakes, termed the cumulative cosine model, derived directly from mass and momentum conservation under the assumption of a cosine-shaped velocity deficit distribution. The proposed model extends the physically based wake superposition concept to cosine-based wake formulations. A theoretical budget analysis clarifies the relationship between the cumulative cosine formulation and the linear sum–local velocity (LS–LV) methods, demonstrating that the latter can be interpreted as a simplified approximation. In contrast, conventional root-square-sum (RSS) methods are shown to underestimate velocity deficits due to implicit violation of momentum balance. Validation against high-fidelity large-eddy simulations (LES) for turbine arrays demonstrates that the cumulative cosine model accurately reproduces wake velocity deficits across multiple downstream interactions. The model is further applied to power prediction and layout optimization of offshore wind farms. While both the cumulative cosine and LS-LV models accurately predict power output, the former exhibits a higher computational cost due to the increased complexity of its analytical structure. Layout optimization demonstrates that the reduced computational efficiency of the cumulative cosine model limits the number of objective function evaluations within a fixed time budget, thereby resulting in lower optimization performance compared with the cosine model (LS–LV). The proposed model and findings of this study provide valuable insights for theoretical wake modeling and practical wind farm applications.