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y x U∞ U∞ δ(x) u δ Figure 3.5.: The boundary layer on a flat plate. Acknowledgment of Marcus Thern. Where ρe and ue is the inviscid density and velocity respectively at the boundary layer edge. This can be rewritten according to [14] and [15] which is cited in [2] . cf L θ = 2un unedx (3.14) e0 Where ue is the boundary edge velocity, L is the flow path length through the com- ponent. Depending on the nature of the flow (i.e. if it is accelerating or defusing) the exponent n can be varied. Pai makes no variation and says that n = 5 provides a general solution, this is cited in [2]. Aungier makes use of the following approximation of equa- tion (3.14) which describes the boundary layer momentum thickness at the discharge station. u15 u25 L θ = cfρave u +2 u +1 8ρ 333 ρave = ρ1 + 2ρ2 + ρ3 4 (3.15) (3.16) The density average value is included to model the compressible effects. The velocities are taken as the absolute velocities in a stationary coordinate system and as the relative velocities in a rotational one. The boundary layer displacement thickness is derived from the definition of shape factor, H. δ∗ = Hθ (3.17) From the 71th power-law and for turbulent flow at discharge it can be shown that, [2]. H = 1.2857 (3.18) The skin friction coefficient is modeled according to Aungier’s empirical model, quoted below. 22PDF Image | A Detailed Analysis of Radial Turbines
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