Mathematical Problems in Engineering

Volume 2017 (2017), Article ID 1458591, 9 pages

https://doi.org/10.1155/2017/1458591

## Investigation of Velocity Distribution in Open Channel Flows Based on Conditional Average of Turbulent Structures

^{1}College of Water Resources & Civil Engineering, China Agricultural University, Beijing 100083, China^{2}School of Civil, Mining & Environmental Engineering, University of Wollongong, NSW 2522, Australia

Correspondence should be addressed to Yu Han; ua.ude.liamwou@619hy and Liu-Chao Qiu; nc.ude.uac@oahcuiluiq

Received 4 January 2017; Accepted 24 April 2017; Published 29 May 2017

Academic Editor: Jian G. Zhou

Copyright © 2017 Yu Han et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We report the development of a new analytical model similar to the Reynolds-averaged Navier-Stokes equations to determine the distribution of streamwise velocity by considering the bursting phenomenon. It is found that, in two-dimensional (2D) flows, the underlying mechanism of the wake law in 2D uniform flow is actually a result of up/down events. A special experiment was conducted to examine the newly derived analytical model, and good agreement is achieved between the experimental data in the inner region and the model’s prediction. The obtained experimental data were also used to examine the DML-Law (dip-modified-log-law), MLW-Law (modified-log-wake law), and CML-Law (Cole’s wake law), and the agreement is not very satisfactory in the outer region.

#### 1. Introduction

Velocity distribution in open channel flows offers a wide range of applications in the fields of hydrometry, sediment transport, river restoration, power plant design, and so forth [1–3]. Accurate investigation of the velocity distribution in open channel flows has been conducted over the past century. The law of the wall, developed for pipe flows, has often been applied to uniform flows in open channel flows [4]; this “universal wall function” or log-law of the wall is expressed asand herein is the von Karman constant, is the mean friction velocity, and is the vertical direction. In (1), the boundary condition is at [5], and the parameter is the constant of integration that must be determined using the boundary condition [6]. It is expected that the parameter in (1) should be a variable as well [7], for a smooth boundary should be related to the thickness of the local viscous sublayer; that is, where is the coefficient to be determined experimentally, is the kinematic viscosity, and denotes the local friction velocity, which may be different from .

The velocity vertical profile is well described by the classical log-law in the inner region, , but the log-law normally deviates from the experimental data in the outer ration, [8]. Since this method is unable to represent the outer region of a velocity profile, Coles [9] suggested an improvement, called the wake law. Many researchers have discussed the validity of such a law [10, 11] and have improved both the log and wake laws for smooth or rough flows [12, 13].

Coles [9] extended the log-law by introducing a purely empirical correction function. He developed an additional term (i.e., the wake term) as a supplement to the log-law, to account for the velocity deviation caused by wake. This model is known as the wake law:where denotes the “profile” parameter; Coles [9] gave = 0.55. For open channel flows, Coleman [14] obtained an average value of 0.19. Nezu and Rodi [15] yielded a value of 0 to 0.20, while Kirkgöz [16] reported a value of 0.1. Cardoso et al. [10] obtained a Π value of −0.077 over a smooth bed. Kironoto and Graf [11] found value in the range of −0.08 to 0.15 for water flows over a gravel bed. However, the values of are established by experimentation, with large variations reported in the literatures. While significant advances have been made by using Cole’s wake law, the mechanism of Cole’s wake law (*CWL-Law*) and the associated wake strength parameter are not fully understood. These investigations often ascribe the large variations to either smooth or rough boundaries and few researchers have taken into account the velocity-dip phenomenon caused by secondary currents [17] (Guo 2015).

Different from the positive deviation of measured velocity from the log-law’s prediction or the CWL-Law, Yang et al. [18] discussed the negative deviation of measured velocity from the log-law’s prediction or the dip phenomenon, in which the maximum longitudinal velocity occurs below the water surface. They suggested that Cole’s wake law is not able to describe the entire velocity profile when the dip phenomenon exists. Yang et al. [18] proposed a dip-modified-log-law (*DML-Law*) based on the analysis of the Reynolds-averaged Navier-Stokes (RANS) equations. This law, involving two logarithmic distances, one from the bed (i.e., the log-law) and the other from the free surface, has the advantage that it contains only one parameter for dip-correction. The DML-Law reverts to the classical log-law for . Yang et al. [18] modified the log-law by adding a term to express the dip phenomenon instead of Cole’s wake law based on Reynolds equations:Guo and Julien [12] proposed a modified-log-wake law (*MLW-Law*) which fits velocity profiles with a dip phenomenon. However, this law cannot be used for predictive applications since it requires fitting the near-free-surface velocities to the parabolic law to obtain dip position and maximum velocity. As Guo and Julien indicate, the MLW-Law can be used only in flow measurements since it requires measured velocities; that is,where and is the distance from the bed to the point where the velocity is maximum.

Nezu and Nakagawa [19] found that in the outer region the mean velocity data deviate systematically from the log-law distribution; at sufficiently high Reynolds numbers, this deviation cannot be neglected in the free-surface region; that is, . It is thus worth investigating the underlying mechanism of velocity deviation from the log-law. Bonakdari et al. [20] also proposed a new formulation of the vertical velocity profile in open channel flows based on an analysis of the Navier-Stoke equations. On the basis of this law and of the suggestions of Absi [21], a new analytical procedure was presented by Pu et al. [22].

With the advent of nonintrusive instrumentation, visualisation techniques reveal that turbulence is dominated by forceful structures that are well organised and ordered eddies with a certain lifespan, now termed as coherent structures. Coherent structures can be placed into two categories: bursting phenomena that occur in the wall region and large-scale vertical motion in the outer region. For coherent structural analysis, the method of quadrants has been developed based on measured velocity fluctuations, that is, quadrant I, and ; quadrant II, and ; quadrant III, and ; and quadrant IV, and [23]. Cellino and Lemmin [24] pointed out that it would be meaningful to concentrate on and in quadrants because the most significant feature of turbulence is fluctuations in the wall-normal direction. They also developed an approach to calculate conditionally averaged upward/downward velocity based on velocity fluctuations. However, the linkage of quadrant analysis and upward/downward events with velocity deviation from the log-law has not been established.

The objective of this research is to present a new analytical approach to determine the streamwise velocity profile and establish a relationship between the log wake function and up/down events. This leads to the present research aim to develop a universal model to express the velocity profile in uniform flows. Such a novel analytical approach will be used to explain that streamwise velocity profiles in the outer region depart from the universal logarithmic law of the wall. The proposed analytical approach and those previously proposed mathematical methods will be examined by using experimental data.

#### 2. Theoretical Consideration

The RANS momentum and continuity equations can be written for steady state flow as where , , and denote the mean velocity in the streamwise (), lateral (), and vertical () directions, respectively, is the gravitational acceleration, is the energy slope, and are the Reynolds stress tensor components. For a uniform and fully developed flow, (7) becomesAlternatively, (9) becomeswhere is the density of the fluid, is the dynamic viscosity of the fluid, and , , and are the turbulent velocity fluctuation in the , , and directions. For 2D flow, (10) can be simplified as Integration of (13) yieldswhere is the integration constant. Using the boundary condition, , , , and , and thus . For water surface, , , and and then the shear velocity can be rewritten asTherefore, the momentum equation has the following form by dividing (15):where and .

Cellino and Lemmin [24] realised that it would be meaningful to develop an approach to calculate the distribution of streamwise velocity by considering the bursting phenomenon. However, the above time averaged method cannot be used to express the bursting phenomenon. Yang [25] proposed an approach to calculate the conditionally averaged upward/downward velocity based on velocity fluctuations. He first defines the occurrence probabilities based on the direction of vertical velocity. For upflow (subscript ) and downflow (subscript ), is expressed aswhere is the time interval between two consecutive velocity acquisitions in an experiment; is the total observed time period; that is, . The upflow and downflow velocity definitions are given in the following way:where and denote upflow and downflow events, respectively, and the symbol “~” represents instantaneous velocities. Yang [25] obtained the Reynolds shear stress for upflow and downflow as follows, and both deviate from the standard linear distribution; that is, where , , , and ; is water depth; and and are friction velocities for upflow (subscript ) and downflow (subscript ), respectively.

The expression of eddy viscosity iswhere and (24) can be expressed as The velocity gradient in upflow and downflow can then be expressed bySimilarly, the gradient of mean velocity can be expressed byInserting (26) into (27) yields where , and then Here, we denote Then Integration of (31) with respect to yields In nonslip boundary conditions and then and thus

Finally, we obtained the corrected log equation asThe value of used in (30) corrects the deviation of velocity from the log-law by considering upward and downward motion adequately. Thus, we can find the existence of the wake function from the newly derived log-law. Upward and downward motion can decompose turbulent flow into two distinct events, and it also indicates that if , then the classic log-law can be obtained. Many researchers determine the different and parameters in from their experiments. Equation (33) implies that the main reason for such different results is not from different experimental conditions, but it is rather from upflow and downflow effects. It is necessary to verify (33) using experimental data.

#### 3. Experimental Set-Up

The experiments were undertaken in a large-scale flow loop at the University of Wollongong, Australia, in which water was supplied from a head tank. The main components of this flume were the head tanks tail tank, glass water channel, recirculation pipe system, and two pumps as shown in Figure 1.