By Serene Tng (Ed.), Serene Tan (Ed.)
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Additional info for Designing Our City: Planning for a Sustainable Singapore
Mixing-length theory is strongly tied to the logarithmic “law” for the velocity profile of a turbulent boundary layer, and the concepts will be treated as nearly interchangeable. Mixing-length theory has been applied to simple free shear flows, but needs different constants and is slightly less accurate than the assumption of uniform eddy viscosity (also due to Prandtl) , whereas in wall-bounded flows it rests on only one primary constant and a secondary one, and has been dominant. Both approaches (mixing length and log law) have been described as “amounting only to dimensional analysis,” unfairly.
In the outer layer, the turbulent length scale is of the order of δ and in the inner layer, the turbulent length scale is ν/u. e. 2 Second order IBL Model According to the SCEM, we look for a uniformly valid approximation in the form U = u1 (x, y, ε) + εU1 (x, η, ε) + εU1 (x, yˆ, ε) + · · · (19a) εV1 (x, yˆ, ε) + · · · V = v1 (x, y, ε) + ε V1 (x, η, ε) + εˆ (19b) P = p1 (x, y, ε) + ε2 P1 (x, η, ε) + ε2 P1 (x, yˆ, ε) + · · · (19c) 2 Tij = ε τij,1 (x, η, ε) + ε τˆij,1 (x, yˆ, ε) + · · · 2 2 (19d) The flow defined by u1 , v1 and p1 is governed by the Euler equations and the second order generalized boundary layer equations are ∂U1 ∂V1 + ∂x ∂η ∂u1 ∂U1 ∂U1 ∂u1 v1 ∂U1 ∂U1 U1 + u1 + εU1 + εV1 + + εV1 ∂x ∂x ∂x ∂y ε ∂η ∂η ∂τxy,1 ∂τxx,1 ∂τyy,1 +ε − ∂η ∂x ∂x = 0 (20a) = (20b) ∂ U1 ∂ V1 + ∂x ∂ yˆ = 0 (20c) ε2 ∂ τˆxy,1 1 ∂ 2 U1 ε ∂ 2 U1 + + εˆ ∂ yˆ εR ∂η 2 εˆ2 R ∂ yˆ2 = 0 (20d) 36 J.
Significant gains ensue from bringing the boundary layer close to separation at the trailing edge and in other regions of “stress,” but the risks related to unforeseen separation are also very large, and neither the wind tunnel nor CFD can be completely trusted to predict flight. Another intellectual attraction is that the underlying mathematical technique of matched asymptotic expansions is more general than boundarylayer theory. It enters lifting-line theory , also due to Prandtl with the influence of Lanchester, which has similarly been displaced from CFD codes but not as a fundamental tool to understand and design wings.