QUESTION 2 [50 Marks] (i) Considering a two-dimensional turbulent flow, the instantaneous time-averaged equation governing the conservation of a scalar property can be written as: (p) 0 (p) 0 (pv) 2 (p) + (D)+s, + + dy ax (7) Applying the Reynolds averaging rule such as =; +=+; =; a=+a''; and derive the Reynolds-Averaged Navier-Stokes (RANS) form of the above equation (7): [8 Marks] (ii) If the mean property can be defined as = form of the above equation (7): derive the Favre-Averaged Navier-Stokes (FANS) [8 Marks] (iii) Now, provided that is given as cpT, D is given as k, S, is given as and the specific heat at constant pressure is assumed to be a non-fluctuating fluid property, write down the RANS and FANS equations for temperature. [4 Marks] (iv) Considering the x-momentum equations in (i) and (ii). From a numerical perspective, what is the advantage of using FANS? [4 Marks] (v) Comment on the advantages and disadvantages in terms of prediction accuracy, computational costs, and modelling features of the following turbulence models: RNG k-e, Realizable k-e, Reynolds Stresses Model, SST k-w. [8 Marks]