# 3D SPH Simulation of Dynamic Water Surface and Its Interaction with Underlying Flow Structure for Turbulent Open Channel Flows Over Rough Beds

01 Jun 2018-Vol. 01, Iss: 02, pp 1840003

Abstract: In this study, a fully 3D numerical model based on the Smoothed Particle Hydrodynamics (SPH) approach has been developed to simulate turbulent open channel flows over a fixed rough bed. The model f...

Topics: Open-channel flow (61%), Smoothed-particle hydrodynamics (53%), Turbulence (52%)

## Summary (4 min read)

Jump to: [Introduction] – [3.1 SPHysics code] – [3.2 Model setup and computational parameters] – [3.3 Treatment of turbulence and roughness elements in 3D model] – [4.1 Aims of the experiments] – [4.2 Hydraulic flume setup] – [4.3 Water surface measurement] – [4.4 Water surface data collection] – [5.1 Water surface pattern] – [5.2 Spatial distribution of the computed mean water level] – [5.3 Propagation of water surface pattern] and [5.5 Correlation function of the underlying vertical flow velocity]

### Introduction

- In this study a fully 3D numerical model based on the Smoothed Particle Hydrodynamics (SPH) approach has been developed to simulate turbulent open channel flows over a fixed rough bed.
- The comparison has demonstrated that the proposed 3D SPH model can simulate well the complex free surface flows over a fixed rough bed.
- Savelsberg and Van de Water [2009] reported that although there are several appealing relations between subsurface flow field and water surface gradient, the water surface of fully developed turbulent flow exhibits a dynamic behaviour of its own.
- This feature is implemented to recognize the free surface parti l s. Lee et al. [2008] and Farhadi et al. [2016] suggested that a threshold criterion rangig from 1.2 to 1.5 can be used to determine which particles belong to the water surface.

### 3.1 SPHysics code

- //www.sphysics.org) is a free open-source SPH code that was released in 2007 and developed jointly by researchers at the Johns Hopkins University (U.S.A.), the University of Vigo , the University of Manchester (U.K.) and the University of Roma La Sapienza, also known as SPHysics code (http.
- It is programmed in the FORTRAN language, and has been developed specifically for free surface hydrodynamics [Gómez-Gesteira et al., 2012].
- In SPHysics, four different time integration schemes are implemented, i.e. the PredictorCorrector, Verlet algorithm, Symplectic algorithm and Beeman algorithm.
- One is called the Shepard filter and the other is the Moving Least Squares (MLS) filter.
- The dynamic wall particle treatment is advantageous mainly because of its computational simplicity, since the wall particles are computed inside the same loop as the fluid particles, and thus the computational time is reduced.

### 3.2 Model setup and computational parameters

- To be dimensionally consistent with the experiment, the numerical flume width was taken as 0.46 m wide for the four flow conditions listed in Table 1.
- The real water viscosity ( 60 10 m2/s) was used and the MLS filter was applied every 30 time steps to smooth out the density and pressure fluctuations.
- The computational time step was automatically adjusted to follow the Courant stability requirement [Gómez-Gesteira et al., 2012].
- To reduce the time of simulation and to reach the stable flow quicker, an analytical solution based on the power law )/1(max )/( mHyUU was initially imposed within the fluid block for each flow condition.
- Similar to the previous 2D model [Gabreil et al., 2018], the bed reference level 0y was taken 4.0 mm below the top of the spheres , from which the mean flow depth wh is measured.

### 3.3 Treatment of turbulence and roughness elements in 3D model

- Czernuszenko and Rylov [2000] proposed a simple analytical model based on the generalisation of Prandtl’s mixing length approach that could be used to obtain the mean velocity and shear stress distributions in 3D nonhomogeneous turbulent flows.
- This simple model was also implemented in the current 3D SPH model by modifying the original SPS model.
- (c) In 3D turbulent open channel flow, the flow is not only influenced by the existence of the roughness element on the channel bed, but it is also influenced by the vertical side walls.
- The vertical drag forces were only computed on the sidewalls where high vertical velocities occur due to the interaction between the flow and sidewall corners.

### 4.1 Aims of the experiments

- The aim of these experiments was to measure the temporal change in water surf c elevations at different locations in the streamwise and lateral directions.
- These measurements are then used to support the application of the SPH approach for use in open channel shallow, turbulent free surface flows.
- This will allow examination of the underlying flow patterns and the water surface spatial pattern.

### 4.2 Hydraulic flume setup

- Measurements were carried out in a 0.459 m wide and 12.6 m long rectangular open channel flume including a recirculating water system.
- At the upstream end the hydraulic flume is supported on a fixed pivot joint, and on a pivot joint attached to an adjustable jack at the downstream end.
- The sidewalls of the flume were composed of glass to enable flow observation.
- These flow conditions were selected to investigate the influence of rough bed elements on the water surface patterns of the turbulent flows.
- The experimental Reynolds Numbers (R ) ranged from approximately 11000 to 43000, so all the flows were fully turbulent.

### 4.3 Water surface measurement

- The temporal changes in the water surface were measured using conductance wave probes.
- The wave probes consisted of two thin wires, which were laterally separated by a istance of 13.0 mm.
- At the bottom of the flume, the upper layer of spheres were drilled with 1.0 mm diameter holes, and each probe was carefully attached into these holes.
- All the probes were connected to wave monitor modules provided by Churchill Controls.
- This procedure was repeated for a number of six flow depths ranged from 30 mm to 130 mm, so that a linear trend between the water depth and voltage was achieved for each wave probe.

### 4.4 Water surface data collection

- Before water surface measurements were taken, the uniform steady flow condition was first achieved and was allowed to stabilise for at least one hour.
- For all flow conditions the water temperature change was within 5.0% of the mean measured value.
- It can be seen that the behaviour of the PDF closely follows a Gaussian distribution.
- The measured data here will be used to support the development of the 3D SPH numerical model which is demonstrated in the following section.
- Fig.4. Probability Density Function (PDF) of the measured water surface fluctuations for all flow conditions in Table 1.

### 5.1 Water surface pattern

- Similar to the 2D SPH model, the water surface elevations were extracted from the SPH particle data using the principle of the divergence of particle position [Gabreil et al., 2018].
- Therefore this was used to compute the instantaneous water surface elevations in the streamwise and lateral directions as follows.
- The particle divergence r was then computed at each of these locations.
- Here it should be noted that the proposed 3D SPH model still predicts the s andard deviation of water surface fluctuations smaller than that in the experiments.
- In the current SPH model, the drag force was used to model the rough bed rather than modelling the real roughness geometry.

### 5.2 Spatial distribution of the computed mean water level

- The time averaged water surface elevations at each grid point were computed and plotted in Fig.6.
- Contour plots of the computed mean free surface elevations for the four flow condition (dashed lines: flume centreline) Condition (3) Condition (4) Condition (2) wh (mm) wh (mm) wh (mm) Table 2 presents a comparison of the computed mean water surface elevations along the flume centreline with the experimental data.
- Additionally the mean water surface elevations measured by the two lateral wave probe arrays were compared with the computed data as presented in Figure 7.

### 5.3 Propagation of water surface pattern

- This section looks at the dynamic behaviour of the water surface along the flume centrelin .
- The black-dashed lines in Figure 8 represent the depth averaged streamwise velocity U listed in Table 1.
- It should be noted that using a much more refined particle size, longer simulation time and longer flume length would allow for more accurate water surface patterns to be simulated.
- For conditions 3 and 4, the behaviour of the computed cross correlation function does not fluctuate as observed in the experiments.

### 5.5 Correlation function of the underlying vertical flow velocity

- In the previous section, it has been shown that the proposed 3D SPH model can initially simulate the free surface behaviour which was found to be closely related to the underlying main flow velocity.
- This section applies the spatial correlation function to the computed vertical velocity along the flume centreline and throughout the flow depth.
- All of these numerical findings provide evidence that SPH model has the capability in simulating such flows if a suitable SPH particle size is selected.
- By comparing with the previous 2D simulations [Gabreil et al., 2018], it was found that the 2D model was not able to show the change in the water surface standard deviation for the different flow conditions.
- Also the particle size used in both models is about four times larger than the measured standard deviation of water surface, which may suggest that the magnitude of water surface fluctuation was underestimated.

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interaction with underlying flow structure for turbulent open channel flows over rough beds.

White Rose Research Online URL for this paper:

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Version: Accepted Version

Article:

Gabreil, E.O., Tait, S., Nichols, A. et al. (1 more author) (2019) 3D SPH simulation of

dynamic water surface and its interaction with underlying flow structure for turbulent open

channel flows over rough beds. International Journal of Ocean and Coastal Engineering, 1

(2). 1840003. ISSN 2529-8070

https://doi.org/10.1142/S2529807018400031

Electronic version of an article published as International Journal of Ocean and Coastal

Engineering, Vol. 01, No. 02, 1840003 (2018) https://doi.org/10.1142/S2529807018400031

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1

3D SPH Simulation of Dynamic Water Surface and its Interaction with Underlying Flow

Structure for Turbulent Open Channel Flows over Rough Beds

Eslam Gabreil, Lecturer in Water Engineering, Department of Civil Engineering, Al-Jabal Al-Gharbi University,

Gharian, Libya, Email: eslamgabreil@yahoo.com

Simon Tait, Professor of Water Engineering, Department of Civil Engineering, University of Sheffield, Sheffield

S1 3JD, UK, Email:

s.tait@sheffield.ac.uk

Andrew Nichols, Lecturer in Water Engineering, Department of Civil and Structural Engineering, University of

Sheffield, Sheffield S1 3JD, UK, Email: a.nichols@sheffield.ac.uk

Giulio Dolcetti, Research Associate, Department of Civil and Structural Engineering, University of

Sheffield, Sheffield S1 3JD, UK, Email. g.dolcetti@sheffield.ac.uk

Abstract

In this study a fully 3D numerical model based on the Smoothed Particle Hydrodynamics

(SPH) approach has been developed to simulate turbulent open channel flows over a fixed

rough bed. The model focuses on the study of dynamic free surface behaviour as well as its

interaction with underlying flow structures near the rough bed. The model is improved from

the open source code SPHysics (http://www.sphysics.org) by adding more advanced

turbulence and rough bed treatment schemes. A modified sub-particle-scale (SPS) eddy

viscosity model is proposed to reflect the turbulence transfer mechanisms and a modified drag

force equation is included into the momentum equations to account for the existence of

roughness elements on the bed as well as on the sidewalls. The computed results of

variousfree surface patterns have been compared with the laboratory measurements of the

fluctuating water surface elevations in the streamwise and spanwise directions of a

rectangular open channel flow under a range of flow conditions. The comparison has

demonstrated that the proposed 3D SPH model can simulate well the complex free surface

flows over a fixed rough bed.

Keywords: SPHysics; free surface flow; rough bed; flow surface dynamics; underlying flow

structure; flow turbulence; bed drag

2

1. Literature Review on Dynamic Water Surface Patterns in Turbulent Open

Channel Flow

Flows with a free surface in civil engineering applications are mainly turbulent. These include

flows in man-made channels and rivers. The study of turbulent flow structures for flows with

a free surface is essential to understanding the fluid dynamics for such civil engineering

applications. All turbulent flow structures in the natural environment are inherently three-

dimensional (3D). These types of flow are characterised by turbulent structures at a range of

scales, intense energy dissipation, and random vorticity [Mathieu and Scot, 2000].

When a fluid flows over a solid boundary, the fluid-air interface is often observed to be

wrinkled. In open channel flows with the absence of the wind, the vertical velocities must

dissipate at the surface, generating horizontal velocities and deforming the surface.

Furthermore, turbulent eddies can never die inside the flow; they must end perpendicularly at

the free surface, causing temporal changes in water surface elevation above these vortices

[Smolentsev and Miraghaie, 2005; Savelsberg and Van de Water, 2009]. Studies of the

dynamic behaviour of water surfaces requires the measurement of the instantaneous

elevations of the water surface and the instantaneous velocities of the underlying flow.

Several techniques have been used to provide a means of measuring these instantaneously and

synchronously [Dabiri, 2003; Savelsberg and Van de Water, 2006; Cooper et al, 2006;

Nichols et al, 2010; Nichols et al, 2016]. Horoshenkov et al. [2013] measured the

instantaneous water surface elevations in turbulent open channel flows using conductance

wave probes. The advantage of using conductance wave probes is that, they are easy to set up

and calibrate compared to the early mentioned techniques. Conductance wave probes can also

be operated at different frequencies to avoid mutual interaction between two or more closely

spaced probes, and generally provide high dynamic accuracy.

A number of experimental studies have been conducted and reported in the literature on

understanding the linkage between the dynamic behaviour of the water surface and the

turbulent flow structures underneath it [e.g., Smolentsev&Miraghaie, 2005; Cooper at al.,

2006; Savelsberg et al., 2006; Savelsberg & Van de Water, 2009; Fujita et al., 2011;

Horoshenkov et al., 2013; Krynkin et al., 2014]. Kumar et al. [1998] performed an

experimental investigation of the characteristics of free surface turbulence in horizontal glass

channel flow with Reynolds Numbers ranging from 2800 to 8800. Their results indicated that

the persistent structure of the water-air interface can be classified into three types: upwellings,

downwellings and spiral eddies. Statistical analyses of Dabiri [2003] have shown that the free

surface deformation is strongly correlated with the near surface vorticity field with a

correlation coefficient of about 0.7 to 0.8. Smolentsev and Miraghaie [2005] performed an

experimental study of flow conditions ranging from weak to strong turbulence in very wide

open channel having an aspect ratio (flow width/flow depth) higher than 40. They observed

that three types of disturbance are always presented on the free surface at the same time:

capillary waves, gravity waves and turbulent waves that are generated due to the interactions

between the bulk flow and the water surface. The turbulent waves were found by [Smolentsev

3

and Miraghaie, 2005] to be the most dominant type, having a characteristic size (in the free

surface plane)of approximately half the mean flow depth. An interesting feature has also been

observed on the free surface is that these turbulent waves have celerity very close to the

average flow velocity, while the speed of capillary and gravity waves were different. This

feature was also observed by Fujita et al. [2011]and Nicholas [2014] who stated that the water

surface waves travel with velocity close to the mean flow velocity. Savelsberg and Van de

Water [2009] reported that although there are several appealing relations between subsurface

flow field and water surface gradient, the water surface of fully developed turbulent flow

exhibits a dynamic behaviour of its own. They attributed this to the large eddies of subsurface

turbulent flow exciting random gravity and capillary waves which move in all directions

across the water surface. Fujita et al. [2011] showed that there is a correlation between the

vertical velocity components and the boil vortices on the surface that are not due to the

gravity waves. Horoshenkov et al. [2013] experimentally studied the free surface dynamic

behaviour and its interactions with the underlying turbulence of shallow open channel flows

over a gravel bed. The temporal change in water surface elevations was measured using

conductance wave probes in the centre of the channel at different streamwise positions. They

found that the free surface roughness patterns are strongly controlled by bulk flow properties

and are not strongly influenced by gravity waves. Horoshenkov et al. [2013] also showed that

the free surface roughness patterns can be described by a well-correlated analytical formula

and established a number of empirical relationships between the water surface parameters and

the corresponding hydraulic parameters. Nichols et al. [2016] determined the free surface

profile for several flow conditions by using the LIF technique and showed that the

independent surface behaviour noted by [Savelsberg and Van de Water, 2009] was not due to

travelling waves, but due to each individual water surface feature oscillating vertically in time

as it is carried in space by the bulk flow. It was concluded that this complex behaviour of

oscillating surface features, overlapping and out of phase in space and time, is responsible for

decorrelating the surface pattern from the turbulence field that generates it. The spatial period

of the oscillation was shown to match the characteristic spatial period of the spatial

correlation functions of [Horoshenkov et al., 2013], giving a physical explanation for the

oscillatory form of spatial correlation function observed.

2. Literature Review on SPH Applications in Open Channel Flows

Numerical simulations are used as a very valuable tool in the field of hydrodynamics and

hydraulic engineering to solve complex problems that are impractical to examine

experimentally. They also have the advantage of disclosing details of flow structures without

the spatial-temporal limitations of laboratory instruments. Thus they can provide an

economical and flexible tool to study flows of practical interest. In numerical simulations, the

physical governing equations are described by one of two main approaches. The first one is

the mesh-based approach in which the fluid domain is decomposed into a fixed grid.

Examples of this approach are Finite Volume (FV), Finite element (FE) and Finite difference

(FD). However, simulating complex flows with large deformations is limited and difficult

with these methods due to the numerical diffusions raised from the advection terms in the

Navier-Stokes (N-S) equations [Gotoh and Sakai, 1999]. The second approach is mesh-free,

where the fluid domain is decomposed into moving points of space commonly called

4

“particles”. The Finite Points [Onate et al., 1996], Free Mesh [Yagawa and Yamada, 1996],

and Moving Particle Semi-implicit (MPS) [Koshizuka et al., 1998] techniques are all

examples of mesh-free approaches. Such techniques are inherently well suited for the

simulation of flows with complex boundaries. In recent years, the most popular Lagrangian

mesh-free method to have been used is Smoothed Particle Hydrodynamics (SPH). Although

the SPH method has been widely used in coastal hydrodynamics, using this method for the

simulation of open channel flow problems has received little attention, especially for the

simulation of turbulent free surface flows over rough beds. The SPH technique, originally

formulated by [Gingold and Monaghan, 1977], initially focussed on the provision of solutions

to astrophysics problems related to the formation and eventual evolution of galaxies [Li and

Liu, 2004]. It finds wide use in solving applied mechanics problems due to its advantage of

using a discretization method to approximate a continuum as a set of particles. The most

compelling advantage of the application of the SPH method is its inherent ability to use the

set of particles to predict the behaviour of highly strained motions without the need for grids

or meshes [Violeau, 2012]. Due to its meshless nature, SPH can handle complex solid

boundaries and can also define free surface flows without the typical problems of grid-based

methods that they need to be coupled with a suitable technique such as volume of fluid (VOF)

to capture the air-water interface.

The treatment of inflow and outflow boundaries in SPH is the key for the successful

simulation of open channel flow problems. In recent years, different inflow and outflow

boundaries have been implemented. For example, Lee et al. [2008] used a periodic open

boundary by which the fluid particles that leave the computational domain through the

outflow boundary are instantly re-inserted at the inflow boundary, and the fluid particles close

to one open lateral boundary interact with the fluid particles close to the complementary open

lateral boundary on the other side of the computational domain. However, this boundary

treatment is not suitable for applications in which the fluid volume leaving the computational

domain does not have the same fluid volume that needs to be generated to enter the

computational domain at the same time. In the technique developed by Shakibaeinia and Jin

[2010], the fluid particles leaving and entering the computational domain are added to and

subtracted from an additional type of particles called ’storage particles’ which exist before the

inlet and after the outlet of the domain of interest. With the method used by Federico et al.

[2012], the desired pressure and velocity conditions are imposed at the inflow region to the

inflow particles and water depth time series are determined by increasing or decreasing the

number of particles in the vertical direction. Meister et al. [2014] performed the same

numerical technique and the analytical solution of the main velocity and the corresponding

pressure distribution were initially imposed. Moreover, Tan et al. [2015] performed an

incompressible SPH (ISPH) technique to simulate open channel turbulent flows over a

smooth bed. The comparisons indicated that the velocity trend in the upper region is quite

promising, but the error becomes larger near the channel bed as the flow depth becomes

shallower. Kazemi et al. [2017] used similar techniques of Federico et al. [2012] and Tan et al.

[2015], but with the difference in that the inflow particle velocities are linked with those of

the inner fluid particles, so that the flow is evolved naturally without any prescription of the

inflow velocity.

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