Solutions for the turbulent classical wake using Lie symmetry methods

Prandtl’s extended mixing length model applied to the two-dimensional turbulent classical far wake

“…Various closure models—which in general may depend on x , y , ufalse¯, normal∂ufalse¯/normal∂y, and higher partial derivatives of ufalse¯—are discussed in §3. A detailed derivation of the governing equations is given in [32]. For completeness, an outline is provided below.…”

“…Let L be the length in the x -direction, beyond which the reduction in velocity is small enough to be neglected, and E C the characteristic effective kinematic viscosity. The turbulent or modified Reynolds number is then ReT=ULEC, and is related to the Reynolds number, Re = U L / ν , by ReT=ReνEC. A turbulent region downstream of the object can develop for large Re , but in order for a turbulent boundary layer to exist, terms of order 1/ Re T must be neglected [32]. This is similar to the condition for a laminar boundary layer to exist, which requires terms of order 1/ Re to be neglected [31].…”

“…Previous work investigates the class of models that can be described by a kinematic eddy viscosity of the form νT=νTfalse(x,y,normal∂ufalse¯/normal∂yfalse) [23,32,34,35]. In this paper, we extend the work conducted in [23] by considering effective kinematic viscosities of the form E=E(x,∂u¯∂y,normal∂2u¯∂y2). This form is convenient not only in that it extends the range of closure models that can be applied, but it also incorporates the CEV model, the PML model, the EPML model, and any variations thereof, as special cases, allowing for a unified derivation and a direct comparison.…”

“…However, from (2.17) and (2.18), the eddy viscosity vanishes at the wake boundary and on the wake axis, and this is no longer true. It was shown in [32] that including the kinematic viscosity leads to an infinite wake boundary, and the form of the mixing length can be obtained without imposing additional restrictions. While idealized boundary conditions on unbounded domains may seem a long way from realistic experiments, we shall see in §5 that in practice the computed velocity profiles decay rapidly—exponentially in some cases—indicating that the assumption of an unbounded domain has little effect on the behaviour of the model.…”

Prandtl's extended mixing length model applied to the two-dimensional turbulent classical far wake

“…Various closure models -which in general may depend on x, y, ū, ∂ ū/∂y, and higher partial derivatives of ū -are discussed in Section 3. A detailed derivation of the governing equations is given in [32]. For completeness, an outline is provided below.…”