Statistical properties of streamline geometry in turbulent wall-flows

Complex but coherent motions form and rapidly evolve within wall-bounded turbulent flows. Research over the past two decades broadly indicates that the momentum transported across the flow largely derives from the dynamics of these coherent motions. The associated spatial organization, and its inherent connection to the dynamics, motivates the present research on streamline curvature and torsion. All the present results have been calculated and compared using the existing direct numerical simulation databases for boundary layers and channel flows. Here we have investigated the statistical properties of the local curvature $\left(\kappa \right)$ and torsion $\left(\tau \right)$ of streamlines for the considered wall-bounded flows. The computation of $\kappa$ (deviation from a straight line-bending) and $\tau$ (out of plane motion-twisting) uses the local construction of the Frenet-Serret coordinate frame. The analysis shows that the statistics of these geometrical properties change significantly with wall-normal position. Even though the mean wall-normal velocity is zero (e.g., for channel flow), the wall-normal curvature component shows a notable positive peak close to the wall. The correlation coefficient and the conditional average of the wall-normal velocity corresponding to the wall-normal curvature exhibit an anticorrelation between them. The probability density function of the curvatures have been calculated across the flow and compared with the ${\kappa }^{-4}$ scaling proposed by Schaefer [J. Turbul., N28 (2012)] for both the total and fluctuating field. Although in isotropic turbulence this scaling of curvature pertains to scales that are near to and smaller than the Kolmogorov scale $\left(\eta \right)$, in wall-bounded turbulence we find the onset of this scaling to occur at slightly larger length scales of $\approx 10\eta$. In fact, the start of this scaling with wall distance coincides with the three-dimensionalization of the vorticity field and agrees with the stagnation point structure in the inertial domain observed by Dallas et al., [Phys. Rev. E 80, 046306 (2009)]. In this region, the mean radius of curvature scales like the Taylor microscale. The standard deviation of torsion exhibits a decreasing trend with distance from the wall. The torsion to curvature intensity ratio reveals that the out of plane motion of the streamlines exceeds in-plane bending. The joint pdf of curvature and velocity magnitude supports the notion that large curvature values correspond to the region near a stagnation point. Furthermore, the joint pdf results between curvature components provides information about the orientation of the streamlines at different wall-normal locations.

Figure 1

(a) Contours of the total velocity field; (b) contours of the fluctuating velocity field; (c) streamlines for the total velocity field; (d) streamlines for the fluctuating velocity field.

Figure 2

Mean statistics of curvature in case of the boundary layer (colored) (Red circle, for ${\delta }^{+}\approx 2530$ and blue triangle, for ${\delta }^{+}\approx 1660$) and the channel flow (black diamond, for ${\delta }^{+}\approx 934$). The vertical gray dash lines, represent the layer boundaries associated with the mean momentum equation (Table 1) corresponding to ${\delta }^{+}=1660$. Purple star are some positions where the pdfs have been calculated that are shown in Fig. 7. (a) Mean of curvature magnitude along with the inverse of Taylor microscale [dot symbol, using Eq. (9) and dash dot symbol, using Eq. (10)]. Pink square symbol is the Taylor microscale evaluated from the experimental data of Ref. [38] for ${\delta }^{+}\approx 6430$. Vertical dot line, denotes the wall-normal position at $0.2\delta$ for ${\delta }^{+}=1660$. (b) Mean streamwise curvature. (c) The mean of curvature magnitude in log-linear scale. (d) The mean streamwise curvature in log-linear scale. (e) Mean wall-normal curvature. The inset represents the correlation of the wall-normal curvature with the wall-normal velocity. (f) Mean spanwise curvature.

Figure 3

(a) Configuration that depicts positive ${\kappa }_y$ near $y^{+}=24$. The up and down arrows denote the local radius of curvature and the sign of ${\kappa }_y$ relative to the indicated minimum wall-normal location of the streamline. The inclined double arrows at “$A$” and “$B$” indicate the inflectional regions on this streamline. (b) The correlation coefficient of ${\tilde{v}}^{+}$ and ${\tilde{\kappa }}_y^{+}$ in case of the boundary layer (colored) (Red circle, for ${\delta }^{+}\approx 2530$ and blue triangle, for ${\delta }^{+}\approx 1660$) and the channel flow (black diamond, for ${\delta }^{+}\approx 934$). The vertical gray dash line, represent the layer boundaries associated with the mean momentum equation (Table 1) corresponding to ${\delta }^{+}=1660$. Purple star position denotes where the conditional averages are computed in Figs. 4 and 5.

Figure 4

(a) Data corresponding to $y^{+}\approx 3$ for channel flow at ${\delta }^{+}=934$. The color contours are conditional average of ${\tilde{v}}^{+}$ corresponding to positive ${\tilde{\kappa }}_y^{+}$, whereas the (pink) lines are contour of ${\tilde{\kappa }}_y^{+}$ ranging from $0.5\times {10}^{-3}$ to $4\times {10}^{-3}$, again conditioned on positive ${\tilde{\kappa }}_y^{+}$. (b) Same as (a); however, at $y^{+}\approx 17$ for boundary layer flow at ${\delta }^{+}=2530$, where the range of the ${\tilde{\kappa }}_y^{+}$ (pink) contour lines are now $0.5\times {10}^{-3}$ to $5\times {10}^{-3}$. Note that the color-bar range is not the same for the two figures owing to the different ranges associated with the color contours.

Figure 5

(a) Same flow configuration ($y^{+}\approx 17$ for boundary layer flow at ${\delta }^{+}=2530$) as Fig. 4, except now conditioned on negative ${\tilde{\kappa }}_y^{+}$. The (green) contour lines denote the negative values of ${\tilde{\kappa }}_y^{+}$ ranging from $-3.5\times {10}^{-4}$ to 0. (b) Sketch of representative streamlines with positive and negative curvatures, respectively, illustrating the conditional Figs. 4 and 5.

Figure 6

The ratio of the mean and standard deviation of the curvatures in case of the boundary layer (colored) (Red circle, for ${\delta }^{+}\approx 2530$ and blue triangle, for ${\delta }^{+}\approx 1660$) and the channel flow (black diamond, for ${\delta }^{+}\approx 934$). The vertical gray dash line, represent the layer boundaries associated with the mean momentum equation (Table 1) corresponding to ${\delta }^{+}=1660$. (a) Curvature magnitude ratio; (b) streamwise curvature ratio; (c) wall-normal curvature ratio and; (d) standard deviation of curvature in the spanwise direction.

Figure 7

Probability density function of the curvature magnitude and curvature components for boundary layer flow at ${\delta }^{+}=2530$ at varying wall-normal locations; diamond, $y^{+}=3$; triangle, $y^{+}=14$, star, $y^{+}=40$, square, $y^{+}=67$; plus, $y^{+}=174$. (a) Probability of the curvature magnitude computed from the instantaneous streamlines pattern; (b) probability of the curvature magnitude for the fluctuating streamlines pattern; (c) probability of the streamwise curvature for the instantaneous streamlines pattern; (d) probability of the streamwise curvature for the fluctuating streamlines pattern; (e) probability of the wall-normal curvature for the instantaneous streamlines pattern; (f) probability of the wall-normal curvature determined from the fluctuating streamlines pattern.

Figure 8

Compensated curvature pdfs for boundary layer flow at ${\delta }^{+}=2530$. (a) ${\kappa }^{-4}$ scaling for fluctuating velocity field; (b) ${\kappa }_y^{-4}$ scaling for velocity fluctuating field; (c) ${\tilde{\kappa }}_y^{-4}$ scaling for total velocity field. The $y$ locations start at $y^{+}=3$ from the top in (a) and (b) and decrease downwards. Pink stars denote the estimated onset of ${\kappa }^{-4}$ scaling.

Figure 9

(a) ${\kappa }^{+}$ (curvature magnitude for fluctuating streamlines) values at which ${\kappa }^{-4}$ behavior is first observed in the ${\kappa }^{+}$ pdf plot [Fig. 7 or Fig. 8] versus $y^{+}$; i.e., the onset of ${\kappa }^{-4}$ behavior, denoted by ${\kappa }_o^{+}$, as a function of $y^{+}$. Comparison is made between ${\kappa }_o^{-4}$ and the inverse of Kolmogorov length scale ($1/{\eta }^{+}$), where ${\eta }^{+}$ is computed from the total strain-rate ${\eta }_{\epsilon }^{+}$ and using the isotropic assumption. (b) Ratio of ${\lambda }^{+}$ [using the definitions from Eqs. (9) (dot symbols) and (10) (cross symbols)] and the inverse of curvature magnitude in case of the boundary layers (red symbols for ${\delta }^{+}\approx 2530$; blue symbols for ${\delta }^{+}\approx 1660$) and the channel flow (black symbols for ${\delta }^{+}\approx 934$). The vertical gray dash lines indicate the position of the momentum balance layer boundaries associated with ${\delta }^{+}=1660$ (Table 1).

Figure 10

PDF for positive and negative values of ${\tilde{\kappa }}_y^{+}$ at (a) the positive peak position of Fig. 2 i.e., at $y^{+}=3$, (b) at $y^{+}=10$, and (c) at $y^{+}=30$.

Figure 11

Statistics of curvature in case of the boundary layer (colored) (Red circle, for ${\delta }^{+}\approx 2530$ and blue triangle, for ${\delta }^{+}\approx 1660$) and the channel flow (black diamond, for ${\delta }^{+}\approx 934$) compared with the inverse of Taylor microscale [dot symbol, using Eq. (9) and dash dot symbol, using Eq. (10)]. Vertical gray dash lines, represent the layer boundaries associated with the mean momentum equation (Table 1) corresponding to ${\delta }^{+}=1660$. (a) Mean curvature statistics for total (lower curves) and fluctuating field (upper curves); (b) standard deviation of curvature for total (lower curves) and fluctuating field (upper curves).

Figure 12

Torsion statistics of curvature in case of the boundary layer (colored) (Red circle, for ${\delta }^{+}\approx 2530$ and blue triangle, for ${\delta }^{+}\approx 1660$) and the channel flow (black diamond, for ${\delta }^{+}\approx 934$). The vertical gray dash lines, represent the layer boundaries associated with the mean momentum equation (Table 1) corresponding to ${\delta }^{+}=1660$. (a) Mean; (b) standard deviation.

Figure 13

Ratio of $\tau$ and $\kappa$ for (a) the total velocity field and (b) the fluctuating velocity field in case of the boundary layer (colored) (Red circle, for ${\delta }^{+}\approx 2530$ and blue triangle, for ${\delta }^{+}\approx 1660$) and the channel flow (black diamond, for ${\delta }^{+}\approx 934$). The vertical gray dash lines, represent the layer boundaries associated with the mean momentum equation (Table 1) corresponding to ${\delta }^{+}=1660$.

Figure 14

Joint pdf of the fluctuating velocity magnitude with the inverse of curvature magnitude for fluctuating streamlines pattern in case of the Channel flow at (a) $y^{+}=3$, (b) $y^{+}=30$, (c) $y^{+}=63$, and (d) $y^{+}=911$.

Figure 15

(a) Joint pdf of the streamwise curvature component with wall-normal curvature component for fluctuating streamlines pattern in case of the Channel flow at (a) $y^{+}=3$, (b) $y^{+}=30$, (c) $y^{+}=63$, and (d) .$y^{+}=911$

Figure 16

Sketch of streamlines in the vicinity of a pair of stagnation regions for wall-turbulent flows within the inertial region. The spacing between these points is $\approx \lambda$, whereas the bounding turbulent motion has a size of $1/\overline{\kappa }$. The gray shaded region within the solid blue circle around the stagnation points of size $1/{\kappa }_0$ is where we expect the curvature pdf $P\left(\kappa \right)\sim {\kappa }^{-4}$ scaling. The streamlines curve sharply deflected within a smaller region of size $\eta$.

Figure 17

Probability density function of the curvature magnitude multiplied by Kolmogorov length scale and wall-normal distance for Channel flow at ${\delta }^{+}\approx 934$ at varying wall-normal locations; diamond, $y^{+}=3$; triangle, $y^{+}=14$, star, $y^{+}=40$, square, $y^{+}=67$; plus, $y^{+}=174$. (a) Probability of the curvature magnitude multiplied by Kolmogorov length scale computed from the instantaneous streamlines pattern. (b) Probability of the curvature magnitude multiplied by Kolmogorov length scale for the fluctuating streamlines pattern. (c) Probability of the curvature magnitude multiplied by the wall-normal distance for the instantaneous streamlines pattern. (d) Probability of the curvature magnitude multiplied by the wall-normal distance determined from the fluctuating streamlines pattern.