# Axisymmetric Wake

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The behavior of the turbulent axisymmetric wake has been considered theoretically for years. However, the available experimental data did not support the theoretical predictions. Recently, equilibrium similarity theory (Johansson et al., 2003) has been applied to the axisymmetric wake, and the results have been validated using one computation (Gourlay et al., 2001) and one experiment (Johansson and George, 2006) that satisfied the criteria required by the theory. The theory produces two results: one for high Reynolds number wakes and one for moderate Reynolds number wakes (high enough to be turbulent, but low enough to have a limited range of turbulence scales). The high Reynolds number results are of the most relevance for wind turbine wakes. Below, the theory is briefly discussed and the important results are summarized. The data used to validate the theory is also presented and some discussion of the data is provided.

The axisymmetric wake is described by several parameters that are shown in Figure 1. A cylindrical coordinate system (*x, r, φ*) is used. An incoming flow of constant velocity *U∞* encounters a circular obstruction of diameter *D* producing an axisymmetric wake. The wake has a centerline velocity *UCL*with a corresponding wake defecit *U*0, where *U0* = *U∞* - *UCL*, that vary with downstream location *x*(the centerline velocity grows and the velocity deficit decreases). The wake also has a characteristic radial dimension *δ* that also varies with downstream location *x* (it grows). An important parameter describing the flow is the local Reynolds number *U0δ/ν*, where *ν* is the kinematic viscosity, that actually decreases with *x*. For all fluctuating quantities, a typical Reynolds decomposition is used (e.g., *û* = *U* + *u*, where *û* is the instantaneous velocity, *U* is the mean velocity, and *u* is the fluctuation about the mean).

**Figure 1:** Axisymmetric wake schematic showing important parameters.

**Theory**

Equilibrium similarity analysis requires use of conservation of momentum, as well as the continuity, momentum, and the Reynolds stress equations. Solutions of the form *H* = *G(x)f(η,*)* are sought, where *H* is the quantity of interest (e.g. velocity deficit, Reynolds stresses), *G(x)* is an amplitude dependent on downstream location *x*, and *f(η,*)* is a function dependent on a suitably normalized radius and possible dependence on the initial generation of the wake indicated by *. Substitution of these solutions into the equations described above yields a new set of equations. According to equilibrium similarity theory, each of the terms in these new equations must maintain the same relative importance as the flow evolves. Invoking this assumption yields relationships among the parameters that can be use to determine the *x* dependence of the wake width and centerline velocity decay. If the wake width is defined as

and the momentum deficit is defined as

then equilibrium similarity requires that

where *x0* is the virtual origin and *a* is a parameter that depends on the details of how the wake was created. This analysis also results in an expression for the centerline velocity deficit

where *b* is now a parameter that also depends on the initial cause of the wake. By its very nature, equilibrium similarity requires that the wake velocity deficit and Reynolds stress profiles collapse when suitably normalized. As discussed above, experimental and computational results have supported these results and show that they apply for *x/θ* > 100 and local Reynolds number of *U0δ*/ν* > 500. The experiment and computational results also show that, in the region where equilibrium similarity holds, the maximum turbulence intensity *umax*/*U0* is constant.

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**Remarks: **

It is important that the conditions where the similarity results apply be met in any computational simulation or experiment in order to make a fair comparison with theory.

**References: **

Gourlay, M.J., Arendt, S.C., Fritts, D.C., and Werne, J., Numerical modelling of initially turbulent wakes with net momentum,Physics of Fluids,13:3783-3801.

Johansson, P.B.V., George, W.K., and Gourlay, M.J., 2003, Equilibrium similarity, effects of inital conditions and local Reynolds number on the axisymmetric wake,Physics of Fluids15:603-617.

Johansson, P.B.V., and George, W.K., 2006, The far downstream evolution of the high-Reynolds-number axisymmetric wake behind a disk. Part 1. Single-point statistics,Journal of Fluid Mechanics,555:363-385.