This chapter presents the mathematical formulation of two-dimensional, steady-state, and turbulent airflow and its coupling with multi-mode heat transfer over building envelopes. The governing equations for fluid motion are derived from the fundamental conservation laws of mass, momentum, and energy, incorporating the RNG K–ε turbulence model to represent Reynolds-averaged turbulent fluctuations. These equations are expressed in generalized curvilinear coordinates, allowing simulation over complex architectural geometries such as sloped or non-orthogonal roof structures. Assumptions of Newtonian fluid behavior, constant thermophysical properties, and negligible body forces are employed to simplify the modeling framework. The mathematical formulation includes coordinate transformations using covariant and contravariant basis vectors, as well as Jacobian and metric tensors, facilitating control volume construction for arbitrary geometries frequently encountered in architectural and urban contexts. The heat transfer model integrates three distinct mechanisms: conduction through solid roof layers, forced convection induced by wind–surface interactions, and radiative exchange with both solar and sky sources. Each mechanism is incorporated through dedicated source terms within the energy equation, providing a physically grounded representation of building-scale thermal processes. Designed with architects and urban planners in mind, this formulation equips the reader with the theoretical tools necessary to simulate the interaction between wind and building surfaces under realistic environmental conditions. The models developed in this chapter form the computational foundation for the numerical methods introduced in Chap. 3 .