Turbulence dissipation and the role of coherent structures in the near wake of a square prism

Between streamwise distances $4d$ and at least $10d$ in the planar turbulent wake of a square prism of side length $d$, the turbulent fluctuating velocities are highly non-Gaussian, the turbulent energy spectrum has a close to $-5/3$ power law range, and the turbulence dissipation rate obeys the nonequilibrium dissipation scaling if the energy of the coherent structures is not included in the scaling. In this same range of streamwise distances, the coherent structure dissipation rate decays proportionally to the stochastic turbulence dissipation rate and there is a strong tendency of alignment or antialignment between fluctuating velocities and fluctuating vorticities which appears to coincide with the presence of coherent structures.

Figure 1

Evolution of ${\tilde{C}}_L$ and ${\tilde{C}}_D$ along the phase $\varphi$ normalized by $\pi$.

Figure 2

Isovorticity (black lines) and streamlines of the phase-averaged fluctuating velocity field (color coded, blue for negative spanwise vorticity and red for positive spanwise vorticity) in the streamwise-transverse plane of the flow for an arbitrary phase. The arrow indicates the direction of the free-stream velocity.

Figure 3

PDFs of (a), (c), and (d) $u_i={\tilde{u}}_i+u_i^{\prime }$ (with $i=1,2,3$, respectively) and (b) and (e) $u_i^{\prime }$ (with $i=1,2$, respectively) at $x_1/d=2,4,6,8$ on the geometric centerline. The PDFs are offset by a decade between consecutive locations. The solid lines show a Gaussian fit to the data.

Figure 4

Stochastic and coherent contributions to the turbulent kinetic energy (normalized by $U_{\infty }^2)$ and turbulent dissipation (normalized by $U_{\infty }^3/d)$.

Figure 5

Ratio between different definitions of the velocity scale $\mathcal{U}$ and the mean turbulent kinetic energy associated with the stochastic motions.

Figure 6

Evolution of $C_{\varepsilon }$ and ${\text{Re}}_{\lambda }$ along the centerline using $\mathcal{U}={k^{\prime }}^{1/2}$.

Figure 7

Evolution of $C_{\varepsilon }{\text{Re}}_{\lambda }$ along the centerline using $\mathcal{U}={k^{\prime }}^{1/2}$.

Figure 8

Ratio between the dissipation ${\varepsilon }^{\prime }$ associated with the stochastic component and the dissipation $\tilde{\varepsilon }$ associated with the phase component as a function of $x_1/d$ on the centerline.

Figure 9

PDFs of the relative helicity $\widehat{h}$ at different locations on the centerline.

Figure 10

Joint PDFs of $\widehat{h}$ with both ${\widehat{h}}_{ss}$ (left column) and ${\widehat{h}}_{cs}$ (right column) for (a) and (b) $x_1/d=2$, (c) and (d) $x_1/d=4$, and (e) and (f) $x_1/d=10$.