A hierarchical random additive model for passive scalars in wall-bounded flows at high Reynolds numbers

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 ~ log(δ/z). The HRAP model admits generalized logarithmic scalings including 〈ϕ ²〉~log(δ/z), 〈ϕ(x)ϕ(x+r _x )〉 ~ log(δ/r _x ), 〈(ϕ(x) - ϕ(x+r _x ))²〉 ~ log(r _x /z), where ϕ is the streamwise velocity fluctuation, δ is an outer length scale, r _x is the two-point displacement in the streamwise direction and 〈·〉 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 ϕ 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.