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=∑ ᵢ₌₁^Nzaᵢ , 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 ( aᵢ ) and the number of addends is Nz~ log ([STIX]x1D6FF/z) . The HRAP model admits generalized logarithmic scalings including [STIX]x1D719² ~ log ([STIX]x1D6FF/z) , [STIX]x1D719(x)[STIX]x1D719(x+rₓ) ~ log ([STIX]x1D6FF/rₓ) , ([STIX]x1D719(x)-[STIX]x1D719(x+rₓ))² ~ log (rₓ/z) , where [STIX]x1D719 is the streamwise velocity fluctuation, [STIX]x1D6FF is an outer length scale, rₓ 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 [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.