# Analysis of the Interconnections between Classic Vortex Models of Coherent Structures Based on DNS Data

^{*}

## Abstract

**:**

## 1. Introduction

_{*}and spanwise distance D = 100y

_{*}in the boundary layer region [3,4,5]. Note that y

_{*}= v/u

_{*}defines the inner scale, where v is kinematic viscosity and u* is friction velocity, which represents the shear stress velocity. Lin et al. [6] used particle image velocimetry (PIV) to capture the flow fields. Their results show that the spatial distribution of high-speed streaks is similar to that of low-speed streaks.

## 2. Materials and Methods

#### 2.1. Closed Channel Flow: DNS

_{x}, L

_{y}, and L

_{z}are the spatial domains along the x, y and z directions, respectively; Δx and Δz are the grid resolutions in the x and z directions, respectively; N

_{x}and N

_{z}correspond to the grid numbers; Δy

_{c}is wall-normal grid spacing at the channel center; N

_{y}represents the grid numbers along the y direction; the superscript

^{+}denotes normalization by the inner scale (u* and v); and u* is the friction velocity and represents the shear stress velocity, for example, Δx

^{+}= Δxu*/v.

#### 2.2. Detection of Streaky Structures

#### 2.2.1. Detection Function

_{std}(y

^{+}) is the standard deviation of the streamwise velocity at y

^{+}; C

_{t}(y

^{+}) is the water depth threshold at y

^{+}; F

_{d}is the dimensionless value of detection function; C is a constant, equal to 0.6, as recommended by Lin et al. [6]; and max[u

_{std}] is the maximum value of u

_{std}in the flow domain. F

_{d}> Ct (high-speed) and F

_{d}< −Ct (low-speed) identify the streaks. Justification for the two equations and specific details are provided in Wang et al. [10].

_{d}for low-speed streaks at y

^{+}= 21.05. The positive and negative values of F

_{d}indicate the existence of instantaneous streamline fluctuations, forming the low- and high-speed streaks. Low-speed regions (brown), high-speed regions (blue), and other flow regions (green) can be recognized by applying a threshold value of C

_{t}(y

^{+}) to the contour map, as shown in Figure 2b.

#### 2.2.2. Image Processing

_{t}were assigned the value 1, whereas values greater than –C

_{t}were assigned a value 0. Figure 3 shows the image processing procedure for extracting low-speed streaks. The procedure for extracting high-speed streaks is similar, but uses a different C

_{t}threshold value.

_{d}image was binarized (Figure 3a), and a basic morphological transformation was used to filter out noise in the binary images. This transformation was done in two steps [23]. First, the opening operator (Figure 3b) and closing operator (Figure 3c) were used to delete some isolated regions and fill some holes. The opening operator is derived from the fundamental morphological operations of dilation as well as erosion and was used to break the adhesion between objects and remove small particle noise; the closing operator combines the operations of erosion and dilation and can be used to connect neighboring regions and fill in small holes. The area of the streak graph does not change significantly during calculation when using the opening and closing operators.

#### 2.2.3. Model of Streaky Structures

_{1}, …, w

_{i−}

_{1}, w

_{i}, w

_{i+}

_{1}, …, w

_{ns}when the streak distances are denoted by d

_{1}, …, d

_{i}

_{−1}, d

_{i}, d

_{i}

_{+1}, …, d

_{ns−}

_{1}.

_{(r)}, and the mean streak width of the whole velocity field at each y

_{+}, W

_{(y+)}, were obtained by Equations (3) and (4):

_{(r)}, and the mean spanwise distance of the whole velocity field at each y

^{+}, D

_{(y}

^{+}

_{)}, were calculated by

^{+}= Du*/v and W

^{+}= Wu*/v. The 30 instantaneous x–z velocity fields (DNS data) were captured at each y position in all cases.

## 3. Results

^{+}) and outer region (H) scales. The relationship between two streaky structures of different scales is unclear. We calculated the spanwise distances over the entire water depth continuously for both the inner and outer regions. The results show that the development of streaky structures along the entire flow depth is a continuous process.

#### 3.1. Streamwise Vortex Model

#### 3.2. Attached Eddy Vortex Model

#### 3.3. Hairpin Vortex Model

#### 3.3.1. Vortex Extraction in the X–Z Plane

_{ci}-criterion [25]. Streaky structures form in the in x–z plane, so a brief introduction to extracting a vortex in the x–z plane is given.

_{ci}is given by

_{ci}as ${\Lambda}_{ci}={\lambda}_{ci}{\omega}_{z}/\left|{\omega}_{z}\right|$, where ω

_{z}is the fluctuating spanwise vorticity and λ

_{ci}and Λ

_{ci}are swirling strength discriminators. ${\Lambda}_{ci}^{rms}(y)$ is the local root mean square of Λ

_{ci}at the wall-normal position y, and we defined the normalized swirling strength Ω

_{ci}by

_{ci}in Equation (8) represent a clockwise or counterclockwise vortex, respectively.

#### 3.3.2. Vortex Density

^{+}is calculated by

_{vortex}is the spanwise number of vortices at position y

^{+}.

^{+}. Figure 7 shows that the population density of 2D vortices varies with water depth, reaching a maximum in the near-wall region at y

^{+}= 40.22. This result may be partly due to the number of streamwise vortices in the deeper water. In the outer region, vortex density decreases gradually as y

^{+}increases. As the shear stress in the x–z plane is approximately zero, the population density of prograde vortices is equal to that of retrograde vortices at each y

^{+}position. These results agree with those obtained by Chen et al. [27], which confirms the logic of our vortex extraction method.

#### 3.3.3. Location of Vortices and Streaks

^{+}and z

^{+}represent the dimensionless length along the streamwise and spanwise directions, respectively.

_{i}

_{,j}) velocity and the eight surrounding streamwise velocities (Figure 9) were averaged using Equation (10). Equation (11) was then used to calculate F

_{d}at the core of each vortex. We use C

_{t}(y) to identify the region of the distribution of vortex cores.

_{t}(y

^{+}): low-speed streaks, high-speed streaks, and in-between regions.

#### 3.3.4. Vortex Density in Different Streaks

_{t}) and are difficult to recognize relative to the spanwise distance. If C

_{t}is too large, the streak width will not include the whole with of the streak; nevertheless, if C

_{t}is too small, the streak width will contain parts of the in-between region. However, even with different threshold values, the change tendencies are basically consistent. Here, we used the threshold value suggested by Lin et al. [6] to identify the width of low- and high-speed streaks. Figure 11 shows that as y

^{+}increases, streak width first increases and then decreases. As the water depth increases from inner region to outer region, the streak scale also increases. When water depth is close to the surface (about 0.7H), the weak boundary layer restrains the streak scale, and the streak width will decrease. This trend is stable and clearly demonstrated.

_{s}, can be obtained by

_{s}is the total area of low- or high-speed streaks and A

_{t}is the total area of the flow field (2048 × 1536). Figure 12 shows that the percentage area of both low- and high-speed streaks decreases as y

^{+}increases. In the near-wall region (y/h < 0.1), the gradients are steep; in the outer region (y/h > 0.3), the decreasing trends become less steep. This result shows that the streaks occur mainly in the near-wall region where there is high shear stress. Mean shear stress decreases as y/h increases, and so does its effect on the streaks. Streaks in the outer region decrease in number and so the percentages of both low- and high-speed streaks in the x–z plane also decrease.

#### 3.3.5. Calculation of Vortex Density

## 4. Discussion and Conclusions

#### 4.1. Discussion

#### 4.2. Conclusions

_{t}. Statistical methods were used to calculate the characteristic dimensions of both low- and high-speed streaks. We investigated three models of coherent structures (streamwise vortex, attached eddy vortex, and hairpin vortices) and demonstrated their application. Analysis of the characteristic dimensions of streaky structures and vortices and further analysis of the relationships among the three vortex models led us to suggest a straightforward hypothesis. The results we obtained are summarized as follows.

- (1)
- The average width of streaks and the average distance between adjacent streaks that we observed are consistent with the results of previous studies, which indicates the suitability of our method of identifying and calculating both low- and high-speed streaky structures.
- (2)
- The development of streaks from the inner turbulent region to the outer region is a continuous process. The length of streaky structures increases linearly with the water depth, and it is approximately twice the water depth. This result also shows the suitability of both the streamwise vortex and the funnel vortex models.
- (3)
- The spanwise vortex density in the x–z plane is greatest within low-speed streaks, intermediate in the in-between region, and least in the high-speed streaks. We infer that the legs of the hairpin vortices envelop the low-speed streaky structures to move in the streamwise direction and conclude that the hairpin vortex model provides a suitable representation.
- (4)
- The theoretical model of the locations in the x–z plane of streamwise vortices, attached-eddy vortices and hairpin vortices established the possibility of the coexistence of three vortex structures; this recognition increases our understanding of the mechanics of coherent structures in turbulent flows.

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

Parameter Description Unit | |

A_{s} | Total area of low- or high-speed streaky structures |

A_{t} | Total area of the flow field (2048 × 1536) |

Ct_{(y+)} | Streak threshold at y^{+} |

d_{(r)} | Dimensionless spanwise distance at each y_{+} |

D | Nondimensional spanwise distance |

$\overline{{F}_{d}}$ | Detection function value of average vortex velocity |

F_{d} | Dimensionless value of detection function |

m | Total number of rows of the flow field |

N_{x} | Grid numbers in the x direction |

N_{y} | Grid numbers in the y direction |

N_{z} | Grid numbers in the x direction |

N_{vortex} | Number of spanwise vortices at position y^{+} |

P_{s} | Area percentages of low- and high-speed streaky structures |

u | Instantaneous velocity in the x direction m/s |

u_{*} | Friction velocity m/s |

uij | Velocity at the position of vortex core |

ū | Average velocity of spanwise vortices in the x–z plane m/s |

ū′ | Fluctuating velocity m/s |

u′ | Streamwise velocity fluctuation m/s |

v | Instantaneous velocity in the y direction m/s |

w | Instantaneous velocity in the z direction m/s |

W | Average nondimensional width |

x | Streamwise direction |

Δx | Grid resolution in the x directions |

y* | Inner scale |

y | Vertical direction |

Δyc | Wall-normal grid spacing at the channel left |

z | Spanwise direction |

Δz | Grid resolution in the z directions |

(m, n) | Grid position in the x–z plane |

ustd(y^{+}) | Standard deviation of the streamwise velocity at y^{+} |

λci | Two-dimensional swirling-strength 1/s |

Λci | Dimensionless swirling strength |

ω_{z} | Fluctuating spanwise vorticity 1/s |

${\Lambda}_{ci}^{rms}(y)$ | Local root mean square of Λci at the wall-normal position y |

Ωci | Normalized swirling strength |

Π^{+} | Vortex population density |

υ | Kinematic viscosity cm^{2}/s |

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**Figure 2.**Visualization of streaks represented by the dimensionless value of detection function F

_{d}: (

**a**) original F

_{d}with the range of the color bar set from −2.5 to 2.5; (

**b**) after applying the threshold value to F

_{d}.

**Figure 3.**Image processing: (

**a**) binary image, (

**b**) opening operator, (

**c**) closing operator, and (

**d**) clean image.

**Figure 6.**Streaky structures described in terms of the attached eddy hypothesis: (

**a**) A pair of vortices and the (

**b**) corresponding high- and low-speed streaky structures for each slice.

**Figure 14.**Representation of hairpins and hairpin packets by Adrian [24].

**Figure 16.**Theoretical models of a streamwise vortex, an attached eddy vortex, and a hairpin vortex in the x–z plane.

**Table 1.**Parameters of the DNS (data from Del Alamo et al. [21]).

Parameter | L_{x}/H | L_{z}/H | L_{y}/H | Δx^{+} | Δz^{+} | Δyc^{+} | N_{x} | N_{z} | N_{y} |
---|---|---|---|---|---|---|---|---|---|

Original | 8π | 3π | 2 | 7.6 | 3.8 | 7.6 | 3072 | 2304 | 385 |

Present study | 16π/3 | 2π | 1 | 7.6 | 3.8 | 7.6 | 2048 | 1536 | 193 |

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**MDPI and ACS Style**

Wang, H.; Peng, G.; Chen, M.; Fan, J. Analysis of the Interconnections between Classic Vortex Models of Coherent Structures Based on DNS Data. *Water* **2019**, *11*, 2005.
https://doi.org/10.3390/w11102005

**AMA Style**

Wang H, Peng G, Chen M, Fan J. Analysis of the Interconnections between Classic Vortex Models of Coherent Structures Based on DNS Data. *Water*. 2019; 11(10):2005.
https://doi.org/10.3390/w11102005

**Chicago/Turabian Style**

Wang, Hao, Guoping Peng, Ming Chen, and Jieling Fan. 2019. "Analysis of the Interconnections between Classic Vortex Models of Coherent Structures Based on DNS Data" *Water* 11, no. 10: 2005.
https://doi.org/10.3390/w11102005