Random field reconstruction of inhomogeneous turbulence. Part I: Modeling and analysis

We develop and analyze a random field model for the reconstruction of turbulent velocity fluctuations from inhomogeneous characteristic flow quantities provided by RANS simulations that is accessible to both a rigorous analytical validation of the model properties and efficient numerical simulation. The model is fully continuous and based on an explicit representation formula in terms of stochastic integrals combining moving average and Fourier-type representations in time and space, respectively. The structure of the model is systematically derived from spectral representations of homogeneous fields by means of suitable stochastic integral transformations that ensure the preservation of consistency properties when progressing to the case of inhomogeneous flow characteristics. Moreover, we employ a two-scale approach that separates a macro scale related to the variations of the characteristic flow quantities from a representative turbulence scale on which the fluctuations are modeled, allowing to assess the model properties by asymptotic analysis w.r.t. the scale ratio. In particular, a novel inhomogeneous ergodicity result establishes the recovery of the inhomogeneous characteristic flow quantities by means of local averages of a single sample path in time and space.