Constraints on constitutive relations in turbulence modeling

Abstract The formulation of constitutive relations is central to turbulence modeling, yet their evolution has always been constrained by requirements of physical consistency, mathematical well-posedness and numerical stability. This review integrates these constraints into a unified framework, tracing how they have shaped the trajectory of turbulence closures from classical formulations to contemporary data-driven approaches. Fundamental principles including conservation laws, realizability, invariance, dimensional homogeneity, memory effects and asymptotic consistency delineate the admissible space of models while simultaneously guiding their refinement. Historical progress, from the Boussinesq approximation through nonlinear eddy-viscosity models to explicit algebraic stress closures, demonstrates the progressive incorporation of such constraints as corrective mechanisms that enhance robustness and predictive fidelity. In parallel, recent advances in physics-informed and machine learning-based models confirm that these same constraints remain indispensable for ensuring generalizability, physical admissibility and solver compatibility. By framing turbulence modeling through the lens of constraints, this review highlights their dual role as both limitations and design principles while underscoring their continuing relevance in shaping next-generation hybrid frameworks that unite physical rigor with data-driven adaptability.