Adaptive Dissipation in the Smagorinsky Model for Turbulence in Boundary-Driven Flows

Abstract: This paper enhances the classic Smagorinsky model by introducing an innovative, adaptive dissipation term that adjusts dynamically with distance from boundary regions. This modification addresses a known limitation of the standard model over dissipation near boundaries thereby improving accuracy in turbulent flow simulations in confined or wall-adjacent areas. We present a rigorous theoretical framework for this adaptive model, including two foundational theorems. The first theorem guarantees existence and uniqueness of solutions, ensuring that the model is mathematically well-posed within the adaptive context. The second theorem provides a precise bound on the energy dissipation rate, demonstrating that dissipation remains controlled and realistic even as boundary effects vary spatially. By allowing the dissipation coefficient to decrease near boundary layers, this approach preserves the finer turbulent structures without excessive smoothing, yielding a more physically accurate representation of the flow. Future work will focus on implementing this adaptive model in computational simulations to empirically verify the theoretical predictions and assess performance in scenarios with complex boundary geometries.