LAUSR

The kinematics of a fully developed passive scalar is modelled using the hierarchical random additive process (HRAP) formalism. Here, ‘a fully developed passive scalar’ refers to a scalar field whose instantaneous fluctuations are statistically stationary, and the ‘HRAP formalism’ is a recently proposed interpretation of the Townsend attached eddy hypothesis. The HRAP model was previously used to model the kinematics of velocity fluctuations in wall turbulence: $u=\sum _{i=1}^{N_{z}}a_{i}$ , where the instantaneous streamwise velocity fluctuation at a generic wall-normal location $z$ is modelled as a sum of additive contributions from wall-attached eddies ( $a_{i}$ ) and the number of addends is $N_{z}\sim \log (\unicode[STIX]{x1D6FF}/z)$ . The HRAP model admits generalized logarithmic scalings including $\langle \unicode[STIX]{x1D719}^{2}\rangle \sim \log (\unicode[STIX]{x1D6FF}/z)$ , $\langle \unicode[STIX]{x1D719}(x)\unicode[STIX]{x1D719}(x+r_{x})\rangle \sim \log (\unicode[STIX]{x1D6FF}/r_{x})$ , $\langle (\unicode[STIX]{x1D719}(x)-\unicode[STIX]{x1D719}(x+r_{x}))^{2}\rangle \sim \log (r_{x}/z)$ , where $\unicode[STIX]{x1D719}$ is the streamwise velocity fluctuation, $\unicode[STIX]{x1D6FF}$ is an outer length scale, $r_{x}$ is the two-point displacement in the streamwise direction and $\langle \cdot \rangle$ denotes ensemble averaging. If the statistical behaviours of the streamwise velocity fluctuation and the fluctuation of a passive scalar are similar, we can expect first that the above mentioned scalings also exist for passive scalars (i.e. for $\unicode[STIX]{x1D719}$ being fluctuations of scalar concentration) and second that the instantaneous fluctuations of a passive scalar can be modelled using the HRAP model as well. Such expectations are confirmed using large-eddy simulations. Hence the work here presents a framework for modelling scalar turbulence in high Reynolds number wall-bounded flows.