Influence of wall-attached structures on the boundary of the quiescent core region in turbulent pipe flow

The entrainment phenomena of the quiescent core region in a turbulent pipe flow are examined by characterizing the tall wall-attached structures of the streamwise velocity fluctuations $\left(u\right)$ extracted from direct numerical simulation data for $\mathrm{R}{\mathrm{e}}_{\tau }=926$. The quiescent core region is the uniform momentum zone with the highest streamwise velocity magnitude. The conditionally averaged turbulence statistics undergo an abrupt jump across the boundary of the core region, which indicates that there is a viscous shear layer at this interface, and that it is similar to the turbulent-nonturbulent interface. The two-point correlation of the interface height fluctuations extends along the streamwise direction in a manner similar to that of $u$ at the core boundary. The magnitudes of the local entrainment velocity were conditionally sampled by measuring the local curvatures of the core boundary, which are associated with large-scale structures. The local entrainment velocity $\left(v_{\mathrm{n}}\right)$ is enhanced at saddle concave and elliptic convex regions of the interface. This behavior mainly arises from small-scale motions due to viscous diffusion, which indicates that the large-scale features modulate the small scales associated with the entrainment velocity. In particular, the saddle concave shape formed by the bulgelike low-speed structures significantly strengthens the viscous component of $v_{\mathrm{n}}$. The turbulence statistics near the core boundary were obtained based on the projection area of the tall wall-attached structures that were extracted from the instantaneous flow field. Sharp changes in the turbulence statistics are evident; the magnitude of these changes increases with increases in the height $\left(l_y\right)$ of the identified structures. We reconstructed the turbulence statistics by using the tall structures with ${l_y}^{+}>100$ and found that the reconstructed profiles are remarkably similar to the turbulence statistics across the core boundary. These results confirm that the sudden changes in the averaged turbulence statistics and the modulation of $v_{\mathrm{n}}$ are associated with the instantaneous wall-attached structures that reach the core region.

Figure 1

(a) 3D-contour representation of the instantaneous streamwise velocity for 1.8 < x/R < 1.9 and 0 < y/R < 1. (b) The PDF of the instantaneous streamwise velocity was constructed by using $1.2R$ as the streamwise domain length of the subzone (inset). (c) The distribution of the modal velocities as a function of the streamwise domain length of the subzone (Δx). (d) The PDF of the modal velocities obtained from all the instantaneous flow fields with $\mathrm{\Delta }x=1.2R$.

Figure 2

(a) Instantaneous flow field of the streamwise velocity in the $x\text{−}y$ plane. The white lines are the contour $\tilde{u}=0.95U_{\mathrm{CL}}$. (b) The ${\mathrm{\Omega }}_{\mathrm{C}}$ and ${\mathrm{\Omega }}_{\mathrm{N}}$ regions with bubbles $\left({\mathrm{\Omega }}_{\mathrm{B}}\right)$ and drops $\left({\mathrm{\Omega }}_{\mathrm{D}}\right)$ in the cross-stream plane at $x/R=1.5$ (see the vertical dashed line in (a)). ${\mathrm{\Omega }}_{\mathrm{C}}$ is the core region where $\tilde{u}>0.95U_{\mathrm{CL}}$ (gray), and ${\mathrm{\Omega }}_{\mathrm{N}}$ is the noncore region where $\tilde{u}<0.95U_{\mathrm{CL}}$ (white) respectively. (c) Boundary of the core $\left({\mathrm{\Omega }}_{\mathrm{I}}\right)$ obtained from Eq. (1). In (a), the thick white line indicates the identified core boundary $\left({\mathrm{\Omega }}_{\mathrm{I}}\right)$ and the inset shows the x-ζ plane $\left(2<x/R<3\right)$ of the instantaneous streamwise velocity.

Figure 3

Conditional mean profiles of (a) the streamwise velocity, (b) the wall-normal velocity, (c) the turbulence intensity, (d) the Reynolds shear stress, (e) the spanwise vorticity ${\omega }_z$, and (f) the population densities of the prograde $\left({\mathrm{\Pi }}^{\mathrm{p}}\right)$ and retrograde $\left({\mathrm{\Pi }}^{\mathrm{r}}\right)$ vortices as functions of ζ.

Figure 4

Two-point correlations of (a) the boundary height fluctuations, and (b) the streamwise velocity fluctuations; the reference wall-normal location is at $y={\tilde{y}}_{\mathrm{I}}$.

Figure 5

(a) Percolation diagram for the identified structures. Isosurfaces of the identified structures in the instantaneous flow field when (b) $\alpha =0.5$, (c) 1.5, and (d) 1.9. The blue and red isosurfaces are the low- and high-speed structures, respectively.

Figure 6

(a) The number of clusters per unit wall-parallel area $\left(=2\pi R{L}_x\right)$ as a function of $y_{\min }$ and $y_{\max }$. (b) The population densities of the wall-attached low-speed (blue) and high-speed (red) structures as functions of ${\zeta }_{\min }$.${\zeta }_{\min }$ is the minimum wall-normal distance of the object from the core boundary. The arrow denotes ${\zeta }_{\min }=-0.066R\left(\approx 60\nu /u_{\tau }\right)$. Isosurfaces of the (c) wall-attached and (d) wall-detached structures. The red and blue isosurfaces correspond to positive and negative fluctuations respectively, and the brightness of the color indicates the distance from the wall.

Figure 7

JPDFs of the relative positions of the wall-attached structures near the core boundary: (a) pairs of high-speed structures, (b) pairs of high- and low-speed structures, and (c) pairs of low-speed structures.

Figure 8

Enstrophy flux $\left(-v{w_z}^2\right)$ for a saddle concave geometry with a wall-attached low-speed structure in the instantaneous flow field. (ai) Isosurface of the core boundary. The color indicates the height of the boundary. (bi) A wall-attached low-speed structure. The brightness of the color indicates the distance from the wall. (aii, bii) Instantaneous enstrophy flux in the cross-stream plane. The black (aii) and blue (bii) lines are the core boundary and the wall-attached structure, respectively.

Figure 9

(a) JPDF of the local curvatures at the boundary of the quiescent core region as a function of the mean and Gaussian curvatures $H$ and $K$. Conditional averages as a function of $H$ and $K$ of (b) the local entrainment velocity, (c) the viscous contribution, and (d) the inviscid contribution.

Figure 10

(a) JPDF of the local curvatures at the boundary of the quiescent core region obtained by projecting the wall-attached high-speed structures onto the core boundary as a function of the Gaussian and mean curvatures $K$ and $H$. Conditional averages as a function of $H$ and $K$ of (b) the local entrainment velocity, (c) the viscous contribution, and (d) the inviscid contribution.

Figure 11

(a) JPDF of the local curvatures at the boundary of the quiescent core region obtained by projecting the wall-attached low-speed structures onto the core boundary as a function of the Gaussian and mean curvatures $K$ and $H$. Conditional averages as a function of $H$ and $K$ of (b) the local entrainment velocity, (c) the viscous contribution, and (d) the inviscid contribution.

Figure 12

Conditional averages of (a) the streamwise velocity, (b) the streamwise turbulence intensity, (c) the wall-normal turbulence intensity, and (d) the Reynolds shear stress contained within the projection area of the wall-attached structures as a function of ζ. The heights of the wall-attached structures vary from ${l_y}^{+}=100$ to ${l_y}^{+}=900$. The horizontal line in (a) is $\tilde{u}=0.95U_{\mathrm{CL}}$.

Figure 13

Conditional averages of the local entrainment velocity on the projection area of (ai) the wall-attached high-speed structures and (bi) the wall-attached low-speed structures with respect to $l_y$. Conditional averages of the viscous and inviscid contributions to the local entrainment velocity on the projection area of (aii) the wall-attached high-speed structures and (bii) the wall-attached low-speed structures with respect to $l_y$. The dashed line indicates the mean value.

Figure 14

The conditional profiles (lines) in Fig. 3 and the reconstructed profiles (circles) obtained from Eq. (11): (a) the streamwise velocity, (b) the streamwise turbulence intensity, (c) the wall-normal turbulence intensity, and (d) the Reynolds shear stress.