# Dominant features in three-dimensional turbulence structure: comparison of non-uniform accelerating and decelerating flows

## Abstract

The results are presented from an experimental study to investigate three-dimensional turbulence structure profiles, including turbulence intensity and Reynolds stress, of different non-uniform open channel flows over smooth bed in subcritical flow regime. In the analysis, the uniform flow profiles have been used to compare with those of the non-uniform flows to investigate their time-averaged spatial flow turbulence structure characteristics. The measured non-uniform velocity profiles are used to verify the von Karman constant κ and to estimate sets of log-law integration constant B_{r} and wake parameter П, where their findings are also compared with values from previous studies. From κ, B_{r} and П findings, it has been found that the log-wake law can sufficiently represent the non-uniform flow in its non-modified form, and all κ, B_{r} and П follow universal rules for different bed roughness conditions. The non-uniform flow experiments also show that both the turbulence intensity and Reynolds stress are governed well by exponential pressure gradient parameter β equations. Their exponential constants are described by quadratic functions in the investigated β range. Through this experimental study, it has been observed that the decelerating flow shows higher empirical constants, in both the turbulence intensity and Reynolds stress compared to the accelerating flow. The decelerating flow also has stronger dominance to determine the flow non-uniformity, because it presents higher Reynolds stress profile than uniform flow, whereas the accelerating flow does not.

### Keywords

Non-uniform flow Accelerating flow Decelerating flow Uniform flow Smooth bed Turbulence intensity Reynolds stress Turbulence structure### List of symbols

- B
_{r} Log-law integration constant

- D
_{uw} Empirical exponential constants for Reynolds stress profile

- Fr
Froude number

- g
Gravitational acceleration

- h
Water flow depth

- k
_{s} Nikuradse roughness

- P
Pressure

- R
Hydraulic radius

- Re
_{k} Roughness Reynolds number

- S
_{o} Channel slope

- u
Flow velocity

- u
_{*} Shear velocity

- u′
Fluctuation of streamwise velocity

- v′
Fluctuation of lateral velocity

- w′
Fluctuation of vertical velocity

- x
Longitudinal distance

- y
Lateral distance

- z
Vertical distance

- z
_{o} Reference zero-plane displacement level

- β
Pressure gradient parameter

- δ
Water depth where maximum velocity occurs

- κ
Von Karman constant

- λ
_{uw} Empirical exponential constant for Reynolds stress profile

- ν
Kinematic viscosity

- П
Wake parameter

- ρ
Water density

- τ
_{o} Bed shear stress

## 1 Introduction

The analysis of flow turbulence is commonly performed on the time-averaged velocity, turbulence intensity and Reynolds stress in two-dimensional (2D) flow domain [19]. However, studying 3D flow characteristics can provide more descriptive flow information about its turbulence structure, which is useful for various hydraulic engineering applications. The time-averaged flow velocity is often reproduced by logarithmic profile that is normalised by the wall shear velocity. To systematically represent flow velocity, the Prandtl van Karman type velocity distribution’s logarithmic-wall law was utilised by Keulegan [10] in his investigation on rectangular open channel flow. To improve Keulegan’s study, Coles [6] proposed the log-wake law with a wake correction to more precisely represent velocity distribution at the outer flow region where the ratio of flow vertical location to full flow depth (z/h) is bigger than 0.2. Coles’ method has been proven to give better accuracy compared to the log-wall law, as concluded by Song and Graf [29] and Dey and Raikar [7]; as well as in the modified log-wake law study by Yang [32].

The turbulence structure, including the time-averaged turbulence intensity and Reynolds stress, is produced from the Reynolds decomposed elements of instantaneous flow velocity. The same as the velocity log profile, the wall shear velocity is often used to normalise the turbulence intensity and Reynolds stress. This normalisation allows the turbulence representation in a scale benchmarked by the wall value so to permit comparison between different data sets collected under different hydraulic and wall boundary conditions. There are several ways to determine the shear velocity [27], in which two common approaches are: (1) the extrapolation method from the measured Reynolds stress profile, and (2) the energy gradient method. These methods have been shown to give reasonable estimation of the wall shear velocity [7, 28] providing that the wall boundary is relatively uniform through longitudinal space, i.e. particularly suitable for smooth bed flows. In comparison, the Reynolds stress profile extrapolation method has been found to be more prone to error as it is more dependent to the near bed/inner flow region measurements. For the measuring technique utilised in this study, Acoustic Doppler Velocimeter (ADV), the quality of the measured signal-to-noise ratio (SNR) can be sensitive to the reflective signals from wall boundary [35]. On the other hand, the energy gradient method involves the use of basic flow parameters, such as the hydraulic radius and bed slope; hence it is more error-resistant in calculating shear velocity [24, 25].

_{r}. Kironoto and Graf [14] and Song and Chiew [28] further detailed the change of B

_{r}and Π values using a pressure gradient parameter β, which is defined by

_{o}is the bed shear stress, and \( \partial P/\partial x \) is the flow pressure gradient. The work on β-effects on flow non-uniformity has been further expanded in the study by Onitsuka et al. [22]. All these studies showed that the non-uniform flow turbulence intensity and Reynolds stress can be well-represented by expression of β.

In this study, we attempt to compare the non-uniform flow turbulence patterns in subcritical flow regime over smooth bed. It aims to investigate the non-uniformity impact to the time-averaged spatial flow velocity, turbulence intensity and Reynolds stress profiles and to find their respective relationship to flow pressure gradient. To this end, these non-uniform flow profiles are also compared with previous literature findings to investigate their flow properties and to identify the flow behaviour under different types of non-uniform flows (i.e. accelerating and decelerating flows). Compared to previous works, this study identifies clearer dominant characteristics between the tested accelerating and decelerating flows which can add to the existing knowledge and tests of the non-uniform flow.

## 2 Experimental description

### 2.1 Experimental instrumentations

The flume presented in Fig. 1 has dimensions of 12 m × 0.50 m × 0.45 m, and it is a recently refurbished flume located at the Hydraulic Laboratory, University of Bradford [26]. The flume is operated by a circulating system, where the outlet discharge is directed into a filtering tank then to a water pump system to be re-circulated back into the flume. The flume consists of glass walls and a smooth stainless-steel base. A flat gate is located at the channel end to control the flow depth in the flume. Two parallel tracks are utilised on top of the flume for attaching measuring trolley used for holding and securing the ADV equipment.

The employed ADV has down-looking probes—product of the Nortek Ltd. (Vectrino ADV). It has a limitation of 5 cm measuring distance downward from the probe location, which restricts the data collection at 5 cm vertical distance near to the water flow free surface. The ADV is equipped with the four-probe-receiver, which can significantly reduce the noise signal of the measurements as compared with the three-probe-receiver ADV [3].

### 2.2 Experimental conditions

_{k}, to indicate and confirm the smooth bed property used in this study [34]. The velocity measurements for the non-uniform flows (Test 2, 3, 4 and 5) are taken at four streamwise locations (at 3, 5, 6 and 7 m from the flume inlet). For the uniform flow test (Test 1), the measurements are conducted at five different streamwise locations from upstream to downstream to ensure its uniformity characteristic. At each streamwise location, the velocity measurements are recorded at 15–20 vertical positions. Each sampling point can have a minimum sampling volume size of 1 mm

^{3}; however, for the measurement point that has low SNR ratio (lower than 18 dB), the sampling volume is increased. In all tests, all the point velocity measurements are conducted at a sampling frequency of 100 Hz for 5 min.

Summary of experimental conditions in uniform and non-uniform flows

Test no. | Channel slope (× 10 | Discharge (m | Fr | Re | Flow characteristic | Bed condition | Measuring locations presented |
---|---|---|---|---|---|---|---|

1 | 1.25 | 0.0315 | 0.55 | 28 | Uniform | Smooth | Multiple |

2 | 0 | 0.0270 | 0.42–0.66 | 21–22 | Non-uniform | Smooth | Multiple |

3 | 0 | 0.0315 | 0.41–0.57 | 23–24 | Non-uniform | Smooth | Multiple |

4 | 0 | 0.0360 | 0.44–0.77 | 22–29 | Non-uniform | Smooth | Multiple |

5 | 2.50 | 0.0315 | 0.63–0.87 | 20–32 | Non-uniform | Smooth | Multiple |

## 3 Uniform flow results and analysis

For the von Karman constant κ in Eq. (2), relatively consistent values have been found in different literature studies. A range of κ = 0.40–0.42 was proposed for the flows over smooth bed investigated by Coles [6] and Cardoso et al. [4]; while, similar value (κ = 0.40) was also suggested for the rough bed flow by Song et al. [30]. More recently, Auel et al. [2] summarised from various smooth and rough bed flow studies that κ = 0.385–0.435, universally. For the log-law integration constant B_{r} of smooth bed uniform flows, it was proposed as B_{r} = 4.9 in Mellor and Gibson [15], and Anwar and Atkins [1]; and B_{r} = 5.1 in Coles [6], and Cardoso et al. [4]. In comparison to different rough bed flow studies (i.e. B_{r} = 8.47 ± 0.90 in [13]; B_{r} = 8.42 ± 0.22 in [30]; and B_{r} = 7.80 ± 0.37 in [7]), the smooth bed flow was found to have lower B_{r} value.

For the wake parameter П, different estimations were made in various literature studies for the smooth bed uniform flow. In those studies, some have suggested higher values, namely Nezu and Rodi [20]—П = 0.20; while others proposed lower values, namely Kirkgoz [11]—П = 0.10, Steffer et al. [31]—П = 0.08–0.15, and Cardoso et al. [4]—П = 0.079 ± 0.093. When compared to rough bed flow studies, i.e. Π = 0.09 by Kironoto and Graf [13], Π = 0.08 by Song et al. [30], and Π = 0.110 ± 0.026 by Dey and Raikar [7], Π does not show separate distinct values for flow over different bed roughness.

### 3.1 Discussion

^{+}distribution profiles for Test 1 are used to empirically verify κ, and to calculate B

_{r}and П constants appeared in Eq. (2). The shear velocity in this test has been obtained using the bed shear stress calculated from the uniform pressure gradient approach as follows

_{o}is the bed slope. The measurements at five different locations across up- to downstream for Test 1 show the unchanged coefficient values of κ, B

_{r}and П to be 0.43, 4.7, and 0.0754, respectively. These parameters are found to give consistent values with most of the other smooth bed uniform flow studies discussed in this section; however, 0.43 is found to be at the higher end of κ range proposed by Auel et al. [2].

_{1}, D

_{2}, D

_{3}, λ

_{1}, λ

_{2}and λ

_{3}are all empirical constants for the 3D turbulence intensities. It can be observed from Fig. 3 that the measured turbulence intensity profiles (symbols) at all five measured locations correspond reasonably with Eqs. (4)–(6) (lines) in which their regression coefficients R

^{2}falls between 0.75 and 0.83. For the lateral turbulence intensity profile, Papanicolaou and Hilldale [23] reported that D

_{2}and λ

_{2}should be slightly larger than D

_{3}and λ

_{3}, which is in agreement with our finding. All turbulence intensity empirical constants from this study have been compared to literature findings in Table 2. The measured streamwise and vertical profiles in this study are slightly higher than others’ data presented in Table 2, which is resulted from the higher u

_{*}= 2.83 cm/s employed in this study. This finding shows that even in hydraulically smooth flow condition the wall shear stress may affect turbulence intensity profiles within a high flow depth in streamwise and vertical directions provided that R and h are not large. Auel et al. [2] has consistently suggested this same conclusion in their supercritical flow tests with relatively small R and h settings.

Empirical constants from present study and literature for turbulence intensity exponential law

## 4 Non-uniform flow results and analysis

*ρ*is the density of water, and

*dh/dx*is the flow streamwise water level gradient.

Depth-averaged u-velocity data of different locations for Tests 2–5

Test no. | Flow characteristic | u-velocity | |||
---|---|---|---|---|---|

3 m | 5 m | 6 m | 7 m | ||

1 | Uniform | 0.566, 11.0 | 0.566, 11.1 | 0.566, 11.0 | 0.566, 11.0 |

2 | Accelerating | 0.482, 11.2 | 0.502, 10.8 | 0.505, 10.7 | 0.510, 10.6 |

3 | Accelerating | 0.509, 12.3 | 0.517, 12.1 | 0.519, 12.0 | 0.528, 11.9 |

4 | Accelerating | 0.551, 13.0 | 0.555, 12.8 | 0.564, 12.6 | 0.599, 12.2 |

5 | Decelerating | 0.788, 8.0 | 0.757, 8.3 | 0.695, 8.8 | 0.686, 9.2 |

_{r}and П of the non-uniform flow are in the function of β, the non-uniform flow’s log-wake law should therefore be expressed as

### 4.1 Discussion

_{r}and П constants for Test 2–5 are also presented in Table 4. These values have been used to produce Fig. 9, where the accelerating flow falls in the region of β < − 1; while the decelerating flow falls in the region of β > − 1.

Summary of data and parameters of the non-uniform flow tests (Tests 2–5)

Test no. | Streamwise location (m) | \( \partial P/\partial x \) (N/m | β (–) | κ (–) | B | П (–) | u | u |
---|---|---|---|---|---|---|---|---|

2 | 3 | − 13.734 | − 3.098 | 0.441 | 8 | − 0.397 | 2.17 | – |

5 | − 13.93 | − 2.729 | 0.429 | 8.9 | − 0.180 | 2.10 | 2.09 | |

6 | − 14.028 | − 2.590 | 0.427 | 8.4 | − 0.188 | 2.13 | 2.11 | |

7 | − 14.126 | − 2.153 | 0.425 | 8.6 | − 0.181 | 2.12 | – | |

3 | 3 | − 15.5 | − 4.941 | 0.437 | 8.6 | − 0.328 | 2.30 | – |

5 | − 15.598 | − 4.798 | 0.431 | 7.1 | − 0.302 | 2.33 | 2.36 | |

6 | − 15.696 | − 5.164 | 0.429 | 7 | − 0.296 | 2.35 | – | |

7 | − 15.794 | − 4.343 | 0.439 | 8.1 | − 0.290 | 2.32 | 2.31 | |

4 | 3 | − 16.088 | − 6.809 | 0.44 | 6.9 | − 0.572 | 2.23 | – |

5 | − 16.187 | − 6.223 | 0.431 | 8.9 | − 0.431 | 2.34 | 2.29 | |

6 | − 16.285 | − 5.886 | 0.433 | 8.9 | − 0.427 | 2.75 | 2.78 | |

7 | − 16.383 | − 6.160 | 0.435 | 7.7 | − 0.392 | 2.91 | – | |

5 | 3 | 37.278 | 0.443 | 0.431 | 7.5 | 0.043 | 2.06 | – |

5 | 37.180 | 0.641 | 0.433 | 8 | 0.050 | 2.51 | – | |

6 | 37.082 | 0.728 | 0.431 | 8.4 | 0.058 | 3.10 | 3.06 | |

7 | 36.984 | 0.743 | 0.429 | 10.5 | 0.060 | 3.19 | 3.17 |

From the results at Figs. 5, 6, 7 and 8, an analysis has been conducted to find out the influence of non-uniformity towards log-wake law. To this end, Fig. 9 is produced to investigate the impact of non-uniformity on κ, B_{r} and П constants. The measurements in Fig. 9a show that the von Karman constant κ remains unchanged across the investigated β range at about 0.43 with regression coefficient R^{2} of 0.92 compared to measured data, and this consistency remains for both accelerating and decelerating flow tests. The constant κ = 0.43 is also within the κ range proposed in Auel et al. [2]. In Fig. 9b, this study’s B_{r} remains constant at around B_{r} = 8.1 with regression coefficient R^{2} of 0.84 compared to measured data. When compared to rough bed non-uniform flow B_{r}, such as 8.5 proposed by Kironoto and Graf [14] and 8.21–8.61 proposed by Song and Chiew [28], B_{r} proposed in this study agrees well with them. In their rough bed flow’s log-wake law, their utilised parameter z^{+} was affected by bed roughness. This comparison shows the universality of non-uniform flow B_{r} in different bed roughness. Figure 9c shows that the wake parameter П is varying with β in non-uniform flows. The experimental data here is compared with non-uniform flow formulae proposed for the smooth bed flow by Nezu et al. [18] and for the rough bed flow by Kironoto and Graf [14], in which the data shows reasonably good agreement with both formulae. This comparison further suggests that the β-expression of П should be universal for both hydraulically rough and smooth non-uniform flows. From these κ, B_{r} and П findings, it can be concluded that the log-wake law can sufficiently represent the non-uniform flow without needing any modification. The detailed analysis also reveals that all κ, B_{r} and П follow universal rules for different bed roughness conditions.

_{1}, D

_{2}, D

_{3}, λ

_{1}, λ

_{2}, and λ

_{3}are in the function of β. Figures 10 and 11 are produced from the empirical coefficients in Eqs. (10)–(12), where the measured data are investigated across a range of β from accelerating to decelerating flow regimes. The thin dash-lines in Figs. 10 and 11 represent the uniform flow region at β = − 1. The proposed empirical quadratic relationships with β are presented in Eq. (13) for D

_{1}, D

_{2}and D

_{3}with regression coefficient R

^{2}of 0.88, 0.85 and 0.84, respectively. In contrast, constant relationships can be seen for λ

_{1}, λ

_{2}, and λ

_{3}(with regression coefficient R

^{2}of 0.64, 0.61 and 0.61, respectively) as in Eq. (14).

Two points can be observed from Eqs. (13)–(14): (a) all D_{1}, D_{2} and D_{3} have consistent tendency to decrease from higher values at decelerating flow region to lower values at accelerating flow region in the investigated β range; and (b) it is presented in Eq. (14) that λ_{1} > λ_{2} > λ_{3}. These findings suggest that the decelerating flow has higher 3D turbulence intensity profiles than the accelerating flow; and the turbulence intensity characteristics are more dominantly dictated by the streamwise flow. The fitted relationships of Eqs. (13)–(14) also represent the uniform flow data well, suggesting they are working across both the uniform and non-uniform flows.

_{uw}and λ

_{uw}are the empirical constants for Reynolds stress profile. Figure 12 presents the measured Reynolds stress’ constants in Eq. (15), which their quadratic relationship with β is presented in Eqs. (16)–(17). In Fig. 12, the fitted D

_{uw}and λ

_{uw}from the measured data have respective regression coefficient R

^{2}of 0.76 and 0.58 when compared to Eqs. (16)–(17).

Theoretically for uniform flow, the Reynolds stress near bed should be almost equal to \( u_{*}^{2} \) where D_{uw} = 1 (dotted line in Fig. 12a). From Fig. 12a, we can clearly observe that D_{uw} remains around unity at the accelerating flow region (where β < − 1), which shows a close resemblance to the uniform flow characteristic. At the decelerating flow region, stronger non-uniformity characteristic has been observed where all the measurements are higher than the uniform flow’s D_{uw}. This finding suggests that the decelerating flow is more influential in determining the non-uniform characteristic for Reynolds stress compared to the accelerating flow. This suggestion is further supported by the fact that three different strengths of accelerating pressure gradients were tested in this study (from β = − 2.6 to β = − 6.3) and none of them give significant alteration to D_{uw} from uniform flow characteristic. In the full range of β, D_{uw} shows a weak quadratic function suggesting its slow change across uniform and non-uniform flows. λ_{uw} is also found to be representable by a quadratic polynomial function with β in Eq. (17). However, unlike D_{uw} in Eq. (17), λ_{uw} shows a stronger expression in β^{2} proposing its fast change across uniform and non-uniform flows. The non-constant λ_{uw} also displays different characteristic from the constant uniform flow’s λ_{1}, λ_{2} and λ_{3}.

As discussed above, by summarising the findings at Figs. 9, 10, 11 and 12 we can draw useful conclusions to the studied non-uniform flows. The insights gathered from these figures are also useful for hydraulic flow applications, such as flow through control structures which experiences rapidly varied flow between accelerating and decelerating features. For example, in flow phenomena passing weir or sluice, such as hydraulic jump or drop, sudden acceleration or deceleration in flow usually takes place. This rapid variation not only changes the flow characteristic but also affects other associated flow issues, i.e. sediment transport. Thus, a good understanding on accelerating and decelerating flows’ characteristics and their dominance features can help in design such structures, i.e. in controlling the flow turbulence within these structures.

## 5 Measurement limitations and cautions

### 5.1 Flow regime

In present study, the main concentration is put on measuring the flow characteristics at centreline flow region. This is because when a flow is in ‘wide’ channel, i.e. with high width-to-flow-depth aspect ratio (b/h), velocity dip should not take place. This will cause the velocity and turbulence structure profiles to follow similar pattern regardless the measurements at centreline or region relatively nearer to sidewalls within the same longitudinal position.

There are several assumptions used in previous studies to describe the occurrence of velocity dip and the division between wide and narrow flows. Nezu and Nakagawa [19] described the b/h ratio of 4.0–5.0 as the threshold from narrow to wide open channel flow, where flow regime under this threshold limit can be impacted by velocity dip effect. However, studies focused on non-uniform flows, i.e. Graf and Song [8] and Song [27], found that for flows with and above aspect ratio of b/h = 3.5, no velocity dip was recorded, which suggested they possessed a wide open-channel flow characteristic. Auel et al. [2] also described that strong velocity dip occurred up to b/h < 3, and weak velocity dip can take place up to b/h < 5.0.

In this study, flows with minimum aspect ratio of b/h = 3.57 were tested. Our assumption for centreline data measurements was based on the tested aspect ratio would not cause dip phenomenon to the measured profile, which this assumption cannot be utilised if lower aspect ratio flows are going to be tested in any further work. In other words, if velocity dip caused by the secondary current from the side-walls takes place, then it can affect the measured data across channel width. For the flows with low aspect ratio, it will be essential to measure lateral flow profiles at various transverse locations within a longitudinal position to determine the effect from side-walls’ secondary current.

### 5.2 Flow non-uniformity

This study presents five flow tests for uniform, accelerating and decelerating flows to analyse their velocity and turbulence structure profiles. The uniform flow characteristic is utilised to compare with the non-uniform flow findings, in order to validate the proposed relationships in this study (i.e. at Figs. 10, 11). The measured data in this study are also used to perform analysis to identify the key behaviour of accelerating and decelerating flows. However due to this wide research topic, further studies will be needed to test and identify the wider range of flow non-uniformity in order to further explore present study’s findings. For example, more laboratory studies need to be conducted to provide more extreme non-uniform flow tests. More specifically, further tests with wider flow discharge and velocity ranges can create more extreme accelerating and decelerating flow conditions for advancing this study’s flow tests. The added flow tests should also concentrate on supercritical flow condition to add on to the current obtained subcritical flow knowledge. Within a flow, the higher velocity increment (for accelerating flow) and decrement (for decelerating flow) can also be tested to identify the flow impacts from more extreme non-uniformity. In turns of the flow measurements, to obtain better accuracy convergence for time-averaged data, an ADV with higher sampling frequency power can be used for recording longer sampling time’s data.

## 6 Conclusions

In this study, 3D turbulence characteristics of different accelerating and decelerating flows were investigated. Uniform flow test was also conducted for comparison to the non-uniform flows. The employed non-uniform flows measured a set of κ, B_{r} and П. κ and B_{r} for the non-uniform flows remained almost constant; while П was found to change in a linear relationship with the pressure gradient parameter β. Both κ and B_{r} constants found from this study corresponded to previous studies suggested range, in which κ fell within the suggested range in Auel et al. [2] and B_{r} showed similarity with the proposed values in rough bed flow studies by Kironoto and Graf [14] and Song and Chiew [28]. The B_{r} finding further suggested its universality in different bed roughness conditions. The measured П was also found to correspond well to both smooth and rough bed flow formulae proposed by Nezu et al. [18] and Kironoto [12]; which suggested that П can be represented by universal rule for both rough and smooth bed non-uniform flows.

The experiments also showed that both non-uniform flow’s 3D turbulence intensities and Reynolds stress were governed relatively well by exponential equations, where their exponential constants were well-described by quadratic functions in the investigated β range. It was found that the decelerating flow showed higher turbulence intensity profile than the accelerating flow. For the accelerating flow tests, the normalised Reynolds stress distribution found to have similar magnitude (measured by coefficient D_{uw}) to the uniform flow; whereas this D_{uw} magnitude was deviated in the decelerating flow. From the finding of non-uniform flows’ turbulence intensity and Reynolds stress profiles, we can conclude that the decelerating flow has more dominant impact towards the flow’s non-uniformity than the accelerating flow, due to its greater influence to alter the flow’s turbulence structure. This has also been concluded from the comparison with the uniform flow profiles.

## Notes

### Acknowledgements

The first and fourth authors acknowledge the support of the Major State Basic Research Development Grant No. 2013CB036402. The support from the Major State Basic Research Development Program (973 program) of China is also greatly appreciated. Comments and suggestions made by anonymous reviewers have greatly improve the quality of the manuscript.

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