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QUESTION 3 Question 3 (Total 40 marks) Turbulent kinetic energy equation (k-equation) for non-buoyant high Reynolds number: дk at .+ Viaxi uu'un p'u a axi ou ou ax; x; Eq. 3.1 (a) Explain what each term represents in Eq. 3.1. (15 marks) (b) Write the modelled form of the k-equation, indicating which terms are being modelled. (10 marks) c) When turbulence is in a state of local equilibrium, convective and diffusive transport terms are negligible in the k-equation, then the production of k is equal to the a dissipation. For a two-dimensional flow with the k-equation becomes дх ay au 2 UT Cy)" k2 =CD Eq.3.2 where the eddy viscosity v7 = CuVKL and L is the turbulence length scale. Use Eq. 3.2 to show that UT L2 Very (15 marks)

Answer 1

a.

- ək + ūj at диј 3 © Equation for turbulent Kinetic energy is as follows:- 1411, -vui' ou' ak Uj'P' (44jullie tē daj axj. Ui'uj aui + диі әлі arj azj changed axj 3 Vidutie rate of dissipation transport of K turbulence teht by advection production Tate of T of k

diffusive transport ofk 6 y transport of K k by Reynolds streven @=> transport of K * -Y transport of K by presne viscous strelje oy

above eq? convection-diffusions from is CFD pt of view, e91: - Transient term on LHS, advection tem → RHS: diffwjon term & source tem

b.

a model through OK-E model Last two terms in above equation can be modeled as the those terms giver rise to additional unknowns. them in a approximate form model eq? for turbulent kinetic energy k!- DK ak tuj ak Vt ak + DE ət axi rate of increase Convective rate of rate of transport Diffusive of K Production transport destruction - Representing them axj lok axj

tur © Diffusive term has been modelled O Production :- because of interaction Geth eddies with the mean flowy, the velocity gradient of mean flow interact with eddies and dissipation because of Vis cous effects model eqn for turbulent dissipation :- je production Vt JE + U a aj aj de д€ 66 axj it Cei PE - CE2 82 K L DE Dt K at y rate of rate of production destruction Diffiyive transport rate of increase ofE of convective transport dissipation

this model eq?, we can write in standard convection Diffusion form without incuming any additional to unknowns → Overa unknowns49 with involve additional averaging → In order to model this eq?, we have to consider Arad Coef Ficients. some fitting model constants have been benchmark experiments. → Standard values of these obtained by with fitted by experimental e nomen'cal results. → Values are Constants: Cu CE, Cez, 6k6€

can be 3 axj 's strej calculated as follows: Reynold's aui + auj Pui'ui' = -2P k dij tut axj K&ε are obtained through 2 transport egn. Therefore, expresing ut in terms of K&E in order the value of ut.

Please post the question c separately. Thank you!!