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# 1. For an inviscid flow, the momentum equation for a Newtonian flow can be written as: -(puu;)+ Jxi (pu)+ at 2x j where p

## 1. For an inviscid flow, the momentum equation for a Newtonian flow can be written as: -(puu;)+ Jxi (pu)+ at 2x j where p is the density and is the pressure. = = 0 (1) (a) In order to characterise a turbulent flow by the mean values of flow properties, apply thebasic concepts of time-averaging along with the basic concepts of Reynolds averaging rule: = =, a+b=a+b; a=a a= a+a''; Sads = Sds ds (b) Based on the definition of the mass-weighted mean property : b = and along with the basic concepts of Reynolds averaging rule above, derive the Favre- Averaged form of equation (1). Identify each term in the derived equation. (c) Assuming incompressible flows and constant properties, briefly comment on the difference between the Reynolds-Averaged and Favre-Averaged form of equation (1). 2. Referring to the Favre-Averaged form derived for equation (1) in Question (1), (a) How many unknowns that are required to be resolved? What are these unknowns commonly known as in the time-averaged momentum equations? (b) Based on the eddy viscosity concept, write and expand separately each of these unknown expressions? (c) In each of these expressions, what are the properties that are required to be resolved and how is it being evaluated in turbulence modelling? (d) Substituting these unknown expressions the Favre-Averaged form, express the revised Favre-Averaged form in terms of the property defined in part (c).

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