Study of local isotropy in a turbulent pipe flow using longitudinal and transverse structure functions

The scaling exponents of the longitudinal $\left\langle \Delta u_z^n\right\rangle$ and the two transverse structure functions, $\left\langle \Delta u_r^n\right\rangle$ and $\left\langle \Delta u_{\theta}^n\right\rangle$ with $n \leq 7$ are studied in a fully developed incompressible turbulent pipe flow at $Re_D = 24580$ and $50000$ using direct numerical simulation flow fields. The scaling exponents for $\left\langle \Delta u_r^n\right\rangle$ and for $\left\langle \Delta u_{\theta} ^n\right\rangle$ increase with the turbulent Reynolds number $R_ {\lambda}$. However, the scaling exponents for $\left\langle \Delta u_z^n\right\rangle$ remain nearly unchanged. The Kolmogorov universal constants in both of the dissipative range and inertial range for the longitudinal structure functions show a smaller increase with $R_{\lambda}$ than those for the transverse structure functions. The present results are compared with previous experimental and DNS data for channel and duct flows (Antonia \textit{et al.} (1997). Phys. Fluids, 9 (11), 3465)