Vector Identities THE UNIVERSITY OF NEW SOUTH WALES SCHOOL OF MECHANICAL AND MANUFACTURING ENGINEERING MECH4620 COMPUTATIONAL FLUID DYNAMICS Tutorial Problems (T2) (Due 4 pm Friday, Week7) 1. For an...

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Vector Identities THE UNIVERSITY OF NEW SOUTH WALES SCHOOL OF MECHANICAL AND MANUFACTURING ENGINEERING MECH4620 COMPUTATIONAL FLUID DYNAMICS Tutorial Problems (T2) (Due 4 pm Friday, Week7) 1. For an inviscid flow, the momentum equation for a Newtonian flow can be written as:     0i i j j i p u u u t x x            (1) where  is the density and p is the pressure. (a) In order to characterise a turbulent flow by the mean values of flow properties, apply the basic concepts of time-averaging along with the basic concepts of Reynolds averaging rule: ; ;            ; ; ds ds s s                    Derive the Reynolds-Averaged form of equation (1). Identify each term in the derived equation. (b) Based on the definition of the mass-weighted mean property :     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) Comment the difference between the Reynolds-Averaged and Favre-Averaged forms of equation (1). State the assumptions where these two equations are equivalent to each other. 2. Refering 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? -2- (c) In each of these expressions, what is the property that is required to be resolved and how is it being evaluated in turbulence modelling? (d) Subsituting these unknown expressions the Favre-Averaged form, express the revised Favre-Averaged form in terms of the property defined in part (c).