Turbulent shear stresses and prime velocity distribution in compound channels

The turbulent stresses must be defined to calculate the velocity distributions in open channel flows. In the paper, the turbulent stresses are presented as a sum of the normal and shear turbulent stresses. The normal turbulent stresses act like pressure, i.e., they are isotropic and can be absorbed by the pressure-gradient term in the momentum equations. Therefore, only the shear stresses have to be defined to describe the velocity distribution, e.g., the prime velocity distribution in an open channel flow. The shear turbulent stresses are defined by the 3D mixing length hypothesis. This hypothesis is based on the mixing length tensor (MLT). It is shown how to define the components of MLT for compound channels and how to relate it to the turbulent stress tensor. The components of the MLT are defined based on the concept of the generic mixing length (GML). This concept is presented. Having calculated the generic mixing length, the main components of the MLT as well as the turbulent shear stresses can be calculated. The presented concept is applied to calculate the prime velocity distribution in laboratory open channels with two-stage cross-section. Two channels are considered, one with vertical sidewalls and one with inclined sidewalls. The basic hydrodynamics equations (parabolic approximation of Reynolds equations) together with the turbulence model are solved. The well-known Patankar-Spalding algorithm was used to solve these equations. Some numerical simulations were performed for different components of MLT, i.e. for different structure of turbulence. The results of numerical simulations were compared with the primary velocity distribution measured in the laboratory channel. These comparisons show that the model predicts the velocity field reasonably well

Lagrangian Model for a Single Saltating Grain in the Near-Wall Region of an Open-Channel Flow

A mathematical model for the continuous saltation of a particle near the granular bed in an open-channel flow is developed in detail. The model is based on the Lagrangian equations governing particle motion, and it takes into account the following forces: drag, lift, gravitation, virtual mass and the force responsible for particle-particle interactions. A model of particle-particle collisions is developed and used to determine the mean impulsive force acting upon a particle flowing and rebounding from the channel bed. The model can simulate the continuous saltation trajectories of a single particle in the near-bed region of turbulent flows, in which particle motion is controlled by collisions. The model has been calibrated and verified with available published data in a rather wide range of grain sizes from 0.53 mm to 15 mm. All parameters, such as lift, drag, restitution, friction coefficients and roughness height, have been set on the basis of a reanalysis of these published data

Spatially Averaged Log-Law for Flows over Rough Bed in Zero- and Non-Zero-Pressure Gradient Boundary Layers

Theoretical bases for building a logarithmic law for non-uniform flows over a large relative roughness are presented. In order to define the equivalent velocity distribution and to smooth out 3D flow irregularities, a special spatial averaging operation is defined. Basic equations are spatially averaged and double-averaged momentum equations for primary component velocity are derived for uniform flow over a gravel bed as well as for non-uniform flows. A new hypothesis is proposed, and some assumptions are introduced to solve these momentum equations. This results in a new version of the logarithmic velocity distribution (log law). To define this distribution, a full reconstruction of Nikuradse's graph for flows over an irregular gravel riverbed is considered. It is based on very precise measurements of velocity and other hydraulic parameters. In the case of non-uniform flows, the logarithmic velocity profile appears also in accelerating flows in a gravel bed channel, but the friction velocity should be re-defined according to Eq. (24). The same applies to decelerating flow with a positive pressure gradient, but only if the gravitational force exceeds the pressure gradient. For accelerating flows, the additive constant BP depends on the pressure gradient, and its values grow with a growing pressure gradient

Model of Particle-Particle Interaction for Saltating Grains in Water

A model of particle-particle interaction for bed sediment-laden flows, based on impulse equations, is presented. The model is applicable to dense flows in which particle motion is dominated by collisions. The model takes into account the possibility of sliding during the collision process. However, particle rotation is not considered in this model. The governing equations do not incorporate dimension of angular momentum. To verify this model, calculation of post-collision velocities was performed for several different collision simulations. The term of particle-particle interaction is implemented into a general Lagrangian model of trajectory of a sediment grain in a fluid flow. This general Lagrangian model is written according to Newton's second law; the rate of change of momentum of a particle is balanced against the surface and body forces

A turbulence model for 3-d flows with anisotropic structure of turbulence

A new turbulence model for flows in open channels with compound cross-sections is presented. The structure of turbulence in these channels can be anisotropic. This structure is described by the turbulent stress tensor that is presented here as the sum of two tensors, namely, normal and shear stress tensors. The normal and shear turbulent stresses are expressed by the turbulence intensities and the mixing length tensor (MLT), respectively. The turbulence intensities can be learned from measurements or another suitable approaches. One such approach that allows calculating the main component of the normal stresses is presented in the paper. The components of MLT are defined based on a new concept of generic mixing length (GML). The generic mixing length is assumed to depend on both distances; from the nearest wall and from the water surface. To demonstrate how the new model works the basic hydrodynamic equations (parabolic approximation of Reynolds equations) together with the turbulence model are solved. The well-known Patankar and Spalding (1972) algorithm was used when solving these equations. A series of numerical simulations were performed for different components of MLT and different channel geometries

Shear Stress Statistics in a Compound Channel Flow

The results of comprehensive measurements of three-dimensional turbulent velocities carried out in a laboratory compound channel are presented. Tests were performed in a two-stage channel with a smooth main channel bed consisting of concrete and rough floodplains and sloping banks. Instantaneous velocities were measured with the use of a three-component acoustic Doppler velocimeter. The main aim of the study is the recognition of structure of Reynolds stresses in turbulent open channel flows. Particular attention has been paid to bursting events such as ejections and sweeps. The bursting phenomenon occurs originally near the buffer layer and then shows a coherent or organized flow structure during its convection process. The probability density distributions of the turbulent velocities were measured at different distances from the bed in the main channel and also above the inclined walls. In the main channel, the lateral turbulent velocity is seen to follow the normal Gaussian distribution more closely than the remaining two components. Above the inclined walls, all distributions turned out to have greater skewness. The probability density distributions of correlations between velocity fluctuations were also calculated. These distributions have long tails and sharp peaks and fit the theoretical distributions very well. The structure of instantaneous Reynolds stresses was analyzed by a quadrant technique with an arbitrarily chosen threshold level. It has been shown that the largest contribution to turbulent stresses comes from the second quadrant (ejection) and the fourth quadrant (sweep). The basic temporal characteristics for quadrant events, like the average and maximum time for a zero hole size, have been determined in the study. Calculations of maximum duration times for all events reveal that times are greater for even quadrants than for odd quadrants

Measurements of 3D turbulence structure in a compound channel

The paper describes some turbulence measurements carried out in an experimental compound channel with flood plains. The surface of the main channel bed was smooth and made of concrete, whereas the floodplains and sloping banks were covered by cement mortar composed with terrazzo. Instantaneous velocities were measured be means of a three-component acoustic Doppler velocity meter (ADV) manufactured by Sontek Inc. This article presents the results of measurements of primary velocity, the distribution of turbulent intensities, Reynolds stresses, autocorrelation functions, turbulent scales, as well as the energy spectra

Simulation of three-dimension side discharge into an open channel

A three-dimensional computational model, solving Reynolds equations with the ke turbulence closure, has been presented to simulate the flow field in an open channel near a side-discharge channel. The purpose of this study was to exam this model's applicability for simulating the three-dimensional recirculation velocity field in the vicinity of the side discharge channel. The numerical simulations show that both the height and length of the recirculation zone were correctly predicted when compared with laboratory measurements. The predicted trend of the shape of the recirculation zone under different flow conditions agrees with experimental data. It was confirmed that the SMART upwinding scheme performs better than QUICK and HYBRID schemes, since it induces less numerical diffusion and no oscillations. It was found in this study that the SMART scheme needs some minor modifications for complex flow computations