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Tubulent kinetic energy equation (k-equation) for non-buoyant high Reynolds number: ak --utensions (-) dudul Ох, Ох, Eq. 3.1 (a) Explain what each term represents in Eq. 3.1. (15 marks) (6) Write the modelled form of the k-equation, indicating which terms are being modelled (10 marks)

Answer 1

Solution Turbulent Kinetic Energy Equation (k- equation) for non-bougant high reynolds number! ок ч; ч. Әці - чі 4; че Әя; ок + Oі әк ОХ лі + ә plu Ә» і 2 ? ar au aui (і) әуі дні Where Local Derivative ЭК Я 4; әк ді; Advection – чі чі əчі Production (P) Ә) Әч; ч, Чи 21 Turbulent Transport (T) 2 «le чр ді Pressure Diffusion -r dui dui Әхээх, Dissipation (EX) Scanned with CamScanner

(b) The first assumption made in modeling the equation is that the Boussinesa eddy viscosity approximation is valid and that the Reynolds stresses can be modeled by Tij = Mt 3x ) – 3 Pk Sis avi G + OU; for an incompressible fluid. The eddy vis cosity in equation 6 is based on a similiar concept that was used for algebric modds and is generally given by MA skal 1/2 Where l is some twebultme seale. This length scale, &, may he similiar to the mixing length, but is written without the subscript "mix" to avoid cenfu- sion with mixing length models. The characteristic velocity scale in egr ③ is takich to be the square root of'k', since the turbulent fluctuations at a point in the flow field are representative of the turbulent transport of momentum. The prescription for Mi given by equation ® is very similiar to that given by : Mt & Vmix lmix A first approximation for modeling the 'dissipation (CE)! may be obtained by considering a steady, homogenous thin shear layer where it can be shown that spanned with camscanner and disaipation of turbulent Kinetic energy

balance (inc. flows is in local equilibrium). Mathe- matically this may be written as - 4i u aui v dui dui E dxj AXK ask Since homogen tous, shear generated turbulence can be characterized by one length scale I and one velocity seale u, the left side of equation ③ may be taken to seale approximately as 4%. Hence, for turbulent Hows where production balances dissipation then! Ex 43 é If an eddy of mass size I is moving with speed a its energy dissipated per unit mass should be approx. imated by En (drag) velouty) (Su²1²) u 43 213 l mass The dissipation is gonerally modeled by E = k/ l The turbulent trangport of k, and the pressure dibbusion terms are generally modeled by assuming a gradient diffusion transport mechanism. With this assumptions, the twebulent transport of k and the Scanned with CamScanner

pressure fluctuation term are given by! - ( - 2 sui ni u p'uj Mt ak ook aaj Whert The = closure Coefficient known as the prandtl- Schmidt number for k. Combining the modeled terms for the Reynolds stress, dissipation, turbulent diffusion and pressure diffusion with the transient, convective and viscong terms gives the modeled k equation as ! s ak tsa (Ujk) = Tij dui - seta (ut Mt ak 2x dxj TA 2x at with the dissipation and eddy viscosity is given by E Co k 3/2 I Where Co and Cu are constants My = Su Sat CScanned with CamScanner