d Is = (v+) dy (11-14) Convective Turbulent Flow over Horizontal Flate Plate Let's consider a turbulent flow over a long flat plate. The Ox axis is taken along the plate in the direction of the flow with a uniform external velocity ue. The Oy axis is normal to the plate. T is th temperature at the wall and Te the temperature in the external flow supposed to be uniform. The Prandtl number of the fluid is Pr = uCp/k, and the turbulent Prandtl number Prt=tCp/kt is constant but different from the unity. -Considering the Couette Flow assumption and going from eq. (11.14), give the expression of EM as a function of v and out. Then the expression of H as a function of Pr, and ut By+ ay+ In this Couette flow region the heat flux p on the Oy direction is considered constant and equal to its value at the wall pw. 8T dy+ Write the total apparent heat flux expression. From this differential equation, calculate the expression of as function of Pr, Prt, ut, ur, P, Cp and w Calculate the distribution of temperature in the laminar viscous sublayer (0 < y+ 5). Calculate the expression of the temperature at y = 5, T5 as a function of Tw, w, Pr, p, Cp, and ur. Calculate the distribution of temperature in the intermediate zone (Buffer layer) (5 y 30) considering the velocity profile proposed by Von Karman in this region u+ = 5lny+ 3.05 or equivalently u+5+5ln(y+/5) (see figure 1). - Develop an expression for Tw - T30 where T30 is the temperature at y+= 30. Suppose that the Prandtl number of the fluid is sufficiently high such in the turbulent zone (y+ > 30) where is negligible compared to t, k can be considered negligible compared to kt. Write the expression of the heat flux and the shear stress 7 in this zone. Deduce the expression of aT then aT Considering that the ratio in the turbulent zone is constant and equal to its value at the wall, calculate the expression of t as a function of w, UT, Prt, p and Cp. T Jut Considering that this expression is valid from y+= 30 to the edge of the boundary layer, Integrate it and calculate the expression of T30 - Te where Te is the temperature at the edge of the boundary layer where the non-dimensional velocity is ut. Finally calculate the expression of T-Te and deduce the expression of Nusselt number as a function of Re, C, Prt and Pr. 20 15 10 Laminar layer Buffer layer Turbulent layer u+-3.05+5.00 Iny+ 1 2 5 10 20 + Nikuradse Reichardt +5.5+2.5 Iny+ * Reichardt-Schuh 100 200 500 1,000 100 50 00 FIGURE 1 - Universal velocity distribution for turbulent boundary layer.