Cospectral budget of turbulence explains the bulk properties of smooth pipe flow

Connections between the wall-normal turbulent velocity spectrum $E_{ww}\left(k\right)$ at wave number $k$ and the mean velocity profile (MVP) are explored in pressure-driven flows confined within smooth walls at moderate to high bulk Reynolds numbers (Re). These connections are derived via a cospectral budget for the longitudinal ($u^{\prime }$) and wall-normal ($w^{\prime }$) velocity fluctuations, which include a production term due to mean shear interacting with $E_{ww}\left(k\right)$, viscous effects, and a decorrelation between $u^{\prime }$ and $w^{\prime }$ by pressure-strain effects [$=\pi \left(k\right)$]. The $\pi \left(k\right)$ is modeled using a conventional Rotta-like return-to-isotropy closure but adjusted to include the effects of isotropization of the production term. The resulting cospectral budget yields a generalization of a previously proposed “spectral link” between the MVP and the spectrum of turbulence. The proposed cospectral budget is also shown to reproduce the measured MVP across the pipe with changing Re including the MVP shapes in the buffer and wake regions. Because of the links between $E_{ww}\left(k\right)$ and the MVP, the effects of intermittency corrections to inertial subrange scales and the so-called spectral bottleneck reported as $k$ approaches viscous dissipation eddy sizes ($\eta$) on the MVP shapes are investigated and shown to be of minor importance. Inclusion of a local Reynolds number correction to a parameter associated with the spectral exponential cutoff as $k\eta \to 1$ appears to be more significant to the MVP shape in the buffer region. While the bulk shape of the MVP is reasonably reproduced in all regions of the pipe, the solution to the cospectral budget systematically underestimates the negative curvature of the MVP within the buffer layer.

Figure 1

Schematic of the smooth-walled pipe flow configuration showing the pipe length $L$, pipe radius $R$, distance from the wall $y=R-r$, distance from the pipe center $r$, and the resulting total stress distribution when the pressure gradient ($\partial P/\partial x$) is assumed constant.

Figure 2

The idealized vertical velocity spectrum $E_{ww}\left(k\right)$ assumed in the calculations of $F_{wu}\left(k\right)$. Much of the energy in the vertical velocity variance (${\sigma }_w^2$) is contained in two regimes, a near constant regime for $k<k_a$ and inertial subrange scaling for $k>k_a$ given by K41 scaling. The exponential corrections $\text{exp}(-{\beta }_d\eta k)$ shown here for ${\beta }_d=5.2$ become significant near the viscous subrange as $k\eta \to 1$, while very-low-wave-number modulations reduce $E_{ww}\left(k_a\right)$ from its constant value to ${\varphi }_p\left(kR\right)$ as $kR\to 1$. Here, ${\varphi }_p\left(kR\right)={[1+{\left(Rk\right)}^{-2}]}^{-{\gamma }_p}$ for the case ${\gamma }_p=8/3$ is shown. The idealized two-regime spectrum (constant and inertial) used in the analysis of the intermediate region is for ${\beta }_d=0$ and ${\gamma }_p=0$.

Figure 3

Variations of ${\beta }_d$ with ${\text{Re}}_{\lambda }$ reported across several simulation studies. Circles and squares are taken from [43], pluses are taken from high-resolution direct numerical simulations in [44], and diamond is from a wind-tunnel experiment [39]. The line ${\beta }_d\approx 9.1{\text{Re}}_{\lambda }^{-0.1}$ is also shown and used in the cospectral calculations of the MVP.

Figure 4

Comparison between measured (symbols) and modeled (solid line) profiles of $U^{+}$ as a function of wall distance $y^{+}$ (left) using Eq. (28) across selected $\text{Re}=7.43\times {10}^4$, $1.45\times {10}^5$, $1.80\times {10}^6$, and $3.57\times {10}^7$. Measurements are from the Superpipe experiment [5, 6]. The parabolic (dots) and logarithmic (dashed) profiles with $\kappa =0.44$ as reported by [5, 6] are shown for reference.

Figure 5

Same as Fig. 4 but zooming in on the buffer region. Since no measurements are reported in the buffer region for $\text{Re}=3.57\times {10}^7$, it is not included in the zoom-in.

Figure 6

Left panel: Same as Fig. 4 but for all profiles reported in the Superpipe experiment [5, 6]. Right panel: measure ${\sigma }_w^{+}={\sigma }_w/u_{\tau }$ (plus symbol) for the highest and lowest Re and modeled ${\sigma }_w^{+}$ for all the reported profiles [5, 6] are also shown. The arrow direction indicates increasing Re.