# Study of Vortical Structures in Turbulent Near-Wall Flows

## Abstract

Streamwise and spanwise vortices are investigated in a database of near-wall turbulence constituted of SPIV data of boundary layer covering a large range of Reynolds numbers (*Re* _{ θ }∈[1300;18950]).The detection algorithm is based on a fit of the velocity field surrounding extrema of swirling strength to an Oseen vortex. Some statistical results on the characteristics of the vortices (radius, vorticity, convection velocity, density) are investigated, giving some new insight into the organization of near-wall turbulence.

## Keywords

Particle Image Velocimetry Turbulent Boundary Layer Interrogation Window Streamwise Vortex Wall Unit## 1 Introduction

Near-wall turbulence is organized into coherent structures which are believed to play a key role in the maintenance of turbulence. In particular, the streamwise and spanwise oriented vortices, through their ability to transport mass and momentum across the mean velocity gradient, are a fundamental feature of near-wall turbulence. The description, scaling laws and generation mechanism of these structures have been the focus of many studies, but still remain unclear. The present study aims at providing new insight into these outstanding issues through the analysis of a SPIV database of turbulent boundary layer covering a large range of Reynolds numbers (*Re* _{ θ }∈[1300;18950]).

## 2 Description of the Database

Characteristics of the LML and LTRAC database

Facility | Plane (1–2) | | | Domain | | Mesh step Δ | 1st point | n° records |
---|---|---|---|---|---|---|---|---|

LTRAC | XY | 1300 | 820 | 4 | 15.2 | 4.7 | | 605 |

LTRAC | XY | 2200 | 1390 | 2.6 | 14.2 | 4.3 | | 1815 |

LML | XY/YZ | 7630 | 2590 | 0.15 | 16.8 | 3.9 | | 3840/2048 |

LML | XY/YZ | 10140 | 3620 | 0.08 | 11.2 | 3.2 | | 4096/4096 |

LML | XY/YZ | 13420 | 5020 | 0.08 | 15.4 | 4.5 | | 4096/4352 |

LML | XY/YZ | 18950 | 6860 | 0.08 | 20.6 | 6.0 | | 3840/4608 |

The measurements in the LTRAC water-tunnel were realized in a streamwise/wall-normal (XY) plane of a turbulent boundary layer at moderate Reynolds numbers (*Re* _{ θ }=1300 and 2200). The free-stream turbulence intensity in the water-tunnel is relatively high, on the order of 2.6%*U* _{∞} and 5.4%*U* _{∞} for the measurements at *Re* _{ θ }=1300 and *Re* _{ θ }=2200 respectively. Extensive details on the experimental procedure and qualification of the data can be found in [6, 7]. The data feature low measurement uncertainty (0.75% of the free stream velocity on the in-plane velocity components, and 1.5% on the out-of plane component), large spatial dynamic range (thanks to the large CCD array of the PCO 4000 camera), and high spatial resolution (with an interrogation window size of 15^{+} and a mesh spacing of 4.5^{+} on average).

The measurements in the LML wind-tunnel were realized both in a streamwise/wall-normal (XY) plane and in a spanwise/wall-normal (YZ) plane of a turbulent boundary layer at high Reynolds numbers (*Re* _{ θ }=7800, 10140, 13420 and 18950). The description of the setup as well as the qualification of the data can be found in [6]. At all Reynolds numbers and in both planes, the measurement feature low uncertainty (0.7%*U* _{∞} for all velocity components) and high spatial resolution (with an IW size varying from 11.2^{+} to 20.6^{+} and a mesh spacing from 3.2^{+} to 6^{+} depending on the Reynolds number).

## 3 Average Properties of the Database

The wall normal evolution of the mean streamwise velocity was computed by averaging the data over the number of sample acquired and over the homogeneous directions (*x* for the XY plane, *z* for the YZ plane). Only the profiles obtained in the XY plane are shown; the profiles in the YZ plane show the same overall behavior (except for the position of the first valid point in the wall-normal direction, see Table 1).

*U*

_{∞}and the momentum thickness

*θ*) in Fig. 1(a) and in wall units with a logarithmic scale in the near-wall region (scaled with the friction velocity

*U*

_{ τ }and the viscosity ν) in Fig. 1(b).

In the outer region (*y*/*δ*≥0.2), the boundary layer data display a good collapse in external scaling (Fig. 1(a)). In the inner region (*y*/*δ*≤0.2), all the data display a very good collapse in wall-units, and is in excellent agreement with the law of the wall defined by the Van Driest law (for 0<*y* ^{+}<55) and the logarithmic law (for 55<*y* ^{+} and *y*/*δ*<0.2).

*E*

_{11}(

*k*) along the streamwise and spanwise direction, at

*y*

^{+}=100. It is computed as the product of the Fourier transform of the

*u*velocity (after a periodization of the field for the SPIV data) times its complex conjugate, divided by the length of the field, and averaged over the number of samples and the homogeneous direction. The streamwise spectra

*E*

_{11}(

*k*

_{ x }) are shown in Fig. 2(a), and the spanwise spectra

*E*

_{11}(

*k*

_{ z }) in Fig. 2(b), in an inner scaling (scaled with the distance to the wall

*y*and the friction velocity

*U*

_{ τ }, see [9]). The SPIV spectra are compared with a spectrum computed on the DNS of channel flow from [3] at

*Re*

_{ τ }=2000. As it can be seen, the SPIV spectra and the DNS spectra are in excellent agreement in the low (\(\bar{k}<1\)) and intermediate (\(1<\bar{k}<10\)) wavenumbers range. In particular, all spectra tend to a ‘−5/3’ power law in the inertial subrange. In the high wavenumber domain, a spurious lift-up of the PIV spectra with respect to the DNS spectra is visible: it indicates that the effect of measurement noise dominates the effect of spatial averaging over the interrogation window [5]. Taking the DNS spectrum as the reference spectrum of the flow, it is possible to compute the wavenumber \(\bar{k}_{\mathit{SNR}=1}\) at a signal-to-noise-ratio of 1 (and the associated structure size, e.g. radius of a vortex that would be resolved with

*SNR*=1, \(r^{+}_{\mathit{SNR}=1}=\frac {1}{2}\frac{2\varPi y^{+}}{\bar{k}_{\mathit{SNR}=1}}\)) [7]. The cut-off wavenumbers are reported on the spectra. Their values read: \(\bar{k}_{\mathit{SNR}=1}=12.7\) (\(r^{+}_{\mathit{SNR}=1}=25\)) for the LTRAC datasets, \(\bar {k}_{\mathit{SNR}=1}=15.2\) (\(r^{+}_{\mathit{SNR}=1}=21\)) for the LML datasets in the XY plane, and \(\bar{k}_{\mathit{SNR}=1}=16.6\) (\(r^{+}_{\mathit{SNR}=1}=19\)) for the LML dataset in the YZ plane.

## 4 Detection Technique

Local detection techniques based on the 3D velocity gradient tensor \(A_{ij}=\frac{\partial u_{i}}{\partial x_{j}}\) (such as the Δ criterion [2]) present the advantage of being Galilean invariant. They have been successfully used to detect vortices in turbulent boundary layer. Because the database used in the present contribution is planar, the detection technique employed is based on the 2D velocity gradient tensor. This tensor is computed using a second order least-square derivative scheme that minimizes the propagation of measurement noise [4]. When complex eigenvalues of the tensor exist, their imaginary part, called the swirling strength, is used as a detection function of vortex cores. This function is first normalized by the wall-normal profile of its standard deviation (as suggested by DelAlamo et al. [3] and Wu and Christensen [12]), and then smoothed using a 3×3 sliding average to remove the remaining noise. Extrema exceeding a fixed threshold (*λ* _{ ci }(*x* _{1/3},*x* _{2})>1.5*λ* _{ ci,RMS }(*x* _{2})) are retained as center of vortex cores.

## 5 Results: Characteristics of the Vortices

The detection technique described above is employed to detect streamwise vortices (in the YZ planes) and spanwise vortices (in the XY planes) in the SPIV boundary layer database. Some statistics are computed on the characteristics of the detected vortices. The wall-normal evolution of the mean radius, vorticity and density are obtained by taking into account eddies contained in layers of 25^{+} in height.

### 5.1 Density of the Vortices

*Re*

_{ θ }=10140 (featuring the highest spatial resolution) and

*Re*

_{ θ }=18950 (with the lowest spatial resolution) are shown in Figs. 3(a) and 3(b) respectively. The total density of spanwise vortices (detected in the XY plane), the density of prograde (spanwise vortices with rotation in the same senses as the mean shear

*w*

_{0}<0) and retrograde (

*w*

_{0}>0) vortices, and the total density of streamwise vortices (detected in the YZ plane) are represented.

In the region *y* ^{+}<150, the streamwise oriented vortices (detected in the YZ plane) are more numerous than the spanwise oriented vortices (detected in the XY plane). The fact that quasi-streamwise vortices are the major constituent of the near-wall region was also observed using a *λ* _{2} criterion by Jeong et al. [8] in a DNS of channel flow at *Re* _{ τ }=180 and by Sheng et al. [10] in holographic 3D velocity field of a turbulent boundary layer at *Re* _{ τ }=1400. The wall-normal location where the density of the streamwise vortices reaches a maximum is obtained at *y* ^{+}≈60. After this maximum is reached, the density of streamwise vortices decreases rapidly with increasing wall-normal distance. This behavior of the streamwise vortices density in the near-wall region is in good agreement with the findings of Stanislas et al. [11] in SPIV measurements of a turbulent boundary layer at *Re* _{ θ }=7800. The density of the spanwise vortices, in contrast, continuously increases from the wall up to *y* ^{+}≈150 where a maximum is reached. Among the spanwise vortices, the predominance of the prograde vortices is overwhelming. Both prograde and retrograde forms increase with increasing wall-normal distance, but at very different rates: the increase is very fast for the prograde vortices, and rather slow for the retrograde ones. This suggests that different mechanisms are responsible for their formation.

In the outer region (*y* ^{+}>150), the vortex densities behave differently. First, it is of interest to note that the vortices population is now almost equally constituted of streamwise and spanwise vortices. The density of both vortices decreases at a medium rate with increasing wall-normal distance. Again, the prograde and the retrograde vortices follow different evolutions: the density of the prograde vortices continuously decreases with increasing wall-normal distance, while the density of the retrograde vortices stabilizes until *y*/*δ*=0.5.

### 5.2 Radius of the Vortices

*Re*

_{ θ }=1300, 2200, 7630 and 13420 (which feature the same spatial resolution, see Table 1) are in excellent agreement. On a different note, the dataset at

*Re*

_{ θ }=10140 (with the highest spatial resolution) displays smaller values of mean radius, and the dataset at

*Re*

_{ θ }=18950 (with the lowest spatial resolution) higher values of mean radius.

In the XY plane, the radius appears to be slowly increasing with wall normal distance over the whole field for all Reynolds numbers, except for the dataset at *Re* _{ θ }=1300 where a steep increase of mean radius is observed for *y* ^{+}>400⇔*y*/*δ*>0.5. This may be due to intermittency with the free stream; in any case the spanwise vortex density in this region is quite low, cf. previous paragraph. Close to the wall (*y* ^{+}≈50), the mean radius is about 20^{+} on the dataset with the highest spatial resolution (*Re* _{ θ }=10140) which is comparable to the findings of [1] at *Re* _{ θ }=7500.

In the YZ plane, the behavior of the mean radius is quite different. Two regions must be distinguished: the region *y* ^{+}<100, where the radius increases strongly with wall-normal distance, and the region *y* ^{+}>100, where the radius increases slowly with wall-normal distance. This composite behavior in the YZ plane was also observed in [11] at *Re* _{ θ }=7800 and *Re* _{ θ }=13420. Close to the wall (*y* ^{+}≈50), the radius is as small as 15^{+} on the dataset with the highest spatial resolution (*Re* _{ θ }=10140), which is in excellent agreement with the estimation of [10] for the radius of the streamwise vortices in the upper buffer layer.

Therefore the streamwise vortices (detected in the YZ plane) are found to be smaller than the spanwise vortices (detected in the XY plane) in the near wall region. This may indicate that, being inclined to the wall, the streamwise vortices are stretched by the mean velocity gradient. In the region *y* ^{+}>100, the mean radius of the streamwise and spanwise vortices are comparable and slowly increase with increasing wall-normal distance toward a value of 25^{+}.

*y*

^{+}>150 and equal to 8

*η*, except for the channel flow data. In the YZ plane, the behavior is the same except that the mean radius is found to be

*y*independent over the full region of investigation (and not only for

*y*

^{+}>150).

*y*in the logarithmic region are shown in Fig. 6(a) for the spanwise vortices (XY plane) and in Fig. 6(b) for the streamwise vortices (YZ plane). It can be seen that the distribution of radius is also universal in both

*Re*and

*y*in Kolmogorov scaling.

### 5.3 Vorticity of the Vortices

*y*

^{+}<250) a good collapse of the vorticity in all datasets and in both planes is observed. The peak of vorticity at the wall is slightly higher in the YZ plane than in the XY plane. This may indicate that the streamwise vortices (detected in the YZ plane) are more intensified by the mean velocity gradient than the spanwise vortices (detected in the XY plane). A good universality is observed for the streamwise vortices detected in the YZ plane, but some differences are visible for the spanwise vortices detected in the outer region of the XY plane, maybe owing to Reynolds number effects.

*w*

_{0}〉

*τ*appears to be quite constant in the upper buffer layer and in the logarithmic region of the SPIV datasets (〈

*w*

_{0}〉

*τ*≈1.4).

*y*in the logarithmic region are shown in Fig. 9(a) for the spanwise vortices (XY plane) and in Fig. 9(b) for the streamwise vortices (YZ plane). It can be seen that the distribution of vorticity is also universal in both

*Re*and

*y*in Kolmogorov scaling.

## 6 Conclusion

A coherent structure detection was undertaken on the SPIV database of boundary layer flow at *Re* _{ θ }∈[1300;18950]. At all Reynolds numbers, the near-wall region is the most densely populated region, predominantly with streamwise vortices that are on average smaller and more intense than spanwise vortices. In contrast, the logarithmic region is equally constituted of streamwise and spanwise vortices having equivalent characteristics. In the outer region, some differences between the datasets are observed depending on the scaling employed: the wall-units scaling or the Kolmogorov scaling. In wall-unit scaling, a good universality in Reynolds numbers is observed in the near-wall and logarithmic region: the vorticity is found to be maximum at the wall, decreasing first rapidly and then slowly with increasing wall-normal distance; the radius is increasing slowly with wall-normal distance in both regions, except for the streamwise vortices for which a sharp increase in radius is observed in the near-wall region. The wall-units scaling is found to be deficient in the outer region, where Reynolds number effects are observed. In contrast, the Kolmogorov scaling appears to be universal both in Reynolds number and wall-normal distance across the three regions investigated, with a mean radius on the order of 8*η* and a mean vorticity on the order of 1.5*τ* ^{−1}. A good universality of the PDF of the vortex radius and vorticity is also observed in Kolmogorov scaling in the logarithmic region.

## References

- 1.Carlier, J., Stanislas, M.: Experimental study of eddy structures in a turbulent boundary layer using particle image velocimetry. J. Fluid Mech.
**535**, 143–188 (2005) MathSciNetzbMATHCrossRefGoogle Scholar - 2.Chong, M.S., Perry, A.E.: A general classification of three-dimensional flow fields. Phys. Fluids
**5**, 765–777 (1990) MathSciNetCrossRefGoogle Scholar - 3.DelAlamo, J.C., Jiménez, J., Zandonade, P., Moser, R.D.: Self-similar vortex clusters in the turbulent logarithmic region. J. Fluid Mech.
**561**, 329–358 (2006) CrossRefGoogle Scholar - 4.Foucaut, J.M., Stanislas, M.: Some considerations on the accuracy and frequency response of some derivative filters applied to particle image velocimetry vector fields. Meas. Sci. Technol.
**13**, 1058–1071 (2002) CrossRefGoogle Scholar - 5.Foucaut, J.M., Carlier, J., Stanislas, M.: PIV optimization for the study of turbulent flow using spectral analysis. Meas. Sci. Technol.
**15**, 1046–1058 (2004) CrossRefGoogle Scholar - 6.Herpin, S.: Study of the influence of the Reynolds number on the organization of wall-bounded turbulence. Ph.D. thesis, Ecole Centrale de Lille and Monash University (2009) Google Scholar
- 7.Herpin, S., Wong, C.Y., Stanislas, M., Soria, J.: Stereoscopic PIV measurements of a turbulent boundary layer with a large spatial dynamic range. Exp. Fluids
**45**, 745–763 (2008) CrossRefGoogle Scholar - 8.Jeong, J., Hussain, F., Schoppa, W., Kim, J.: Coherent structures near the wall in a turbulent channel flow. J. Fluid Mech.
**332**, 185–214 (1997) zbMATHGoogle Scholar - 9.Perry, A.E., Henbest, S., Chong, M.S.: A theoretical and experimental study of wall turbulence. J. Fluid Mech.
**165**, 163–199 (1986) MathSciNetzbMATHCrossRefGoogle Scholar - 10.Sheng, J., Malkiel, E., Katz, J.: Using digital holographic microscopy for simultaneous measurements of 3d near wall velocity and wall shear stress in a turbulent boundary layer. Exp. Fluids
**45**, 1023–1035 (2008) CrossRefGoogle Scholar - 11.Stanislas, M., Perret, L., Foucaut, J.M.: Vortical structures in the turbulent boundary layer: a possible route to a universal representation. J. Fluid Mech.
**602**, 327–342 (2008) zbMATHCrossRefGoogle Scholar - 12.Wu, Y., Christensen, K.T.: Population trends of spanwise vortices in wall turbulence. J. Fluid Mech.
**568**, 55–76 (2006) zbMATHCrossRefGoogle Scholar