Scaling patch analysis of planar turbulent wakes

A scaling patch approach is used to investigate the proper scales in planar turbulent \textcolor{black}{wakes}. A proper scale for the mean axial flow is the well-known maximum velocity deficit $U_\mathrm{ref} = U_\infty - U_\mathrm{ctr}$, where $U_\infty$ is the free stream velocity and $U_\mathrm{ctr}$ is the mean axial velocity at the wake centerline. From an admissible scaling of the mean continuity equation, a proper scale for the mean transverse flow is found as $V_\mathrm{ref} = (d\delta/dx) U_\mathrm{ref}$, where $d\delta/dx$ is the growth rate of the wake width. From an admissible scaling of the mean momentum equation, a proper scale for the kinematic Reynolds shear stress is found as $R_{uv,\mathrm{ref}}= U_\infty V_\mathrm{ref}$, which is a mixed scale of the free stream velocity and the mean transverse flow scale. Expressions are derived for the scaled mean transverse velocity and Reynolds shear stress in the far field of planar turbulent wakes. Using a Gaussian function for the mean axial velocity deficit, approximate functions for the scaled mean transverse velocity and Reynolds shear stress are developed and found to agree well with experimental and simulation data. This work reveals that the mean transverse flow, despite its small magnitude, plays an important role in the scaling and understanding of the planar turbulent wake.