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 = Sigma(Nz)(i=1) 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 ( ai) and the number of addends is N-z similar to log(delta/z). The HRAP model admits generalized logarithmic scalings including <phi(2)> similar to log(delta/z), <phi(x)phi(x + r(x))> similar to log(delta/r(x)), <(phi(x) - phi(x + r(x)))(2)> similar to log(r(x)/z), where phi is the streamwise velocity fluctuation, delta 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 phi 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.