1. Reynolds averaging, and the modeling of the resulting averages of fluctuating product terms, makes it...

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1. Reynolds averaging, and the modeling of the resulting averages of fluctuating product terms, makes it possible to simulate incompressible turbulent flows. To reduce the number of extra terms that require modeling in reacting flows, with variable density, a density weighted averaging procedure known as Favre averaging is used. = pui P a. Show that the Favre averaged Continuity, Momentum, Energy and Species Mass Fraction equations are then given by: op (pu) at + axi = 0 pu) pu at + ax j aph) pun) a + = -(1)- -(3)- -(2)- ap + - puu") dxi dx ah Th +5h PD it ak-pkuj + PD axj axj axj (7) +k -(4a)- axj at axj axj "ax j (p)) at + Which, with pixuj = becomes ax j 140 k Ox j (p) (pu) + at axj = a afk a + PD k = pk axj axj ax j +7k ax j -(4b)- b. Show that the terms h and I are functions of Turbulent Prandtl number () and Turbulent Schmidt number () respectfully. c. Describe the source/sink terms and explicitly [write out the typical formulas used]. d. Briefly describe how the average fluctuating product term puu"; is typically modeled numerically. e. Briefly describe how the species source/sink term can be modeled and the main difficulties associated with modeling this term. 1. Reynolds averaging, and the modeling of the resulting averages of fluctuating product terms, makes it possible to simulate incompressible turbulent flows. To reduce the number of extra terms that require modeling in reacting flows, with variable density, a density weighted averaging procedure known as Favre averaging is used. = pui P a. Show that the Favre averaged Continuity, Momentum, Energy and Species Mass Fraction equations are then given by: op (pu) at + axi = 0 pu) pu at + ax j aph) pun) a + = -(1)- -(3)- -(2)- ap + - puu") dxi dx ah Th +5h PD it ak-pkuj + PD axj axj axj (7) +k -(4a)- axj at axj axj "ax j (p)) at + Which, with pixuj = becomes ax j 140 k Ox j (p) (pu) + at axj = a afk a + PD k = pk axj axj ax j +7k ax j -(4b)- b. Show that the terms h and I are functions of Turbulent Prandtl number () and Turbulent Schmidt number () respectfully. c. Describe the source/sink terms and explicitly [write out the typical formulas used]. d. Briefly describe how the average fluctuating product term puu"; is typically modeled numerically. e. Briefly describe how the species source/sink term can be modeled and the main difficulties associated with modeling this term.