Smoothed Particle Hydrodynamic Modeling of Hydraulic Jumps1020914/FULLTEXT02.pdf · 2016-10-04 · The hydraulic jump is a natural occurring phenomenon in flowing fluids characterized

MASTER'S THESIS

Smoothed Particle HydrodynamicModeling of Hydraulic Jumps

Patrick Jonsson

Master of Science in Engineering TechnologyMechanical Engineering

Luleå University of TechnologyDepartment Engineering Sciences and Mathematics

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Smoothed Particle Hydrodynamic Modeling

of Hydraulic Jumps

Patrick Jonsson

Master of Science Programme

Mechanical Engineering

Luleå University of Technology

Department of Applied Physics and Mechanical Engineering

Division of Fluid Mechanics

Master’s Thesis

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Cover picture: Copyright © Maria Gustavsson.

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Abstract

In this thesis the capabilities of the Smooth Particle Hydrodynamic (SPH) method to

accurately capture the main features of a hydraulic jump have been investigated. Two

conceptually different modeling approaches were tested, the Tank and Inflow approach.

The Tank approach incorporated the modeling of a large reservoir tank which in the other

case was replaced with an inlet condition. Successful outcomes were achieved for the

Tank case but not for the more efficient and less computationally costly Inflow case due

to poorly implemented boundary conditions in the software. Comparison of numerical

results with theoretical derived values for the Tank case showed systematic under

predicted of the velocity in the fast moving jet just after the gate opening. Furthermore,

the depth past the turbulent roller region showed a continuously decreasing error when

compared with theory which indicates a non-fully developed hydraulic jump at the early

stages of the simulation. Comparison with previous work showed both under and over

estimation of specific parameters which indicates that the number of particles chosen to

represent the system affects the outcomes as roughly seven times more particles was used

in this thesis as compared to previous work. Despite these deviations from theory and

previous work, the main conclusion is that the SPH-method is a viable tool when

performing free-surface flow and particularly hydraulic jump simulations.

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Acknowledgement

First I like to thank my supervisors Prof. Staffan Lundström, Dr. Gunnar Hellström at

Luleå University of Technology and Prof. Patrik Andreasson at Vattenfall Research &

Development AB for giving me the opportunity to do my master thesis as an introduction

to my PhD-studies, and also for giving me support when problems occurred.

I would like to thank Dr. Pär Jonsén and Lic. Gustaf Gustafsson for invaluable guidance

in the intricate subject of Smooth Particle Hydrodynamics and I look forward to future

collaboration.

I also would like to thank all of my coworkers at the Division of Fluid Mechanics at

Luleå University of Technology and especially my office companion Simon Johansson

for constructive criticism and a creative working environment.

The research presented in this thesis was carried out as a part of "Swedish Hydropower

Centre - SVC". SVC has been established by the Swedish Energy Agency, Elforsk and

Svenska Kraftnät together with Luleå University of Technology, The Royal Institute of

Technology, Chalmers University of Technology and Uppsala University.

Participating hydro power companies are: Alstom, Andritz Hydro, E.ON Vattenkraft

Sverige, Fortum Generation, Holmen Energi, Jämtkraft, Karlstads Energi, Linde Energi,

Mälarenergi, Skellefteå Kraft, Sollefteåforsens, Statkraft Sverige, Statoil Lubricants,

Sweco Infrastructure, Sweco Energuide, SveMin, Umeå Energi, Vattenfall Research and

Development, Vattenfall Vattenkraft, VG Power and WSP.

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Table of Contents

Abstract .............................................................................................................................. iii

Acknowledgement .............................................................................................................. v

1 Introduction ................................................................................................................. 1

2 Theory ......................................................................................................................... 3 2.1 The Governing Equations ................................................................................... 3 2.2 Smooth Particle Hydrodynamics ........................................................................ 6

2.2.1 The Kernel Approximation ......................................................................... 8 2.2.2 The Particle Approximation ...................................................................... 11

2.3 Hydraulic Jump ................................................................................................. 13

3 Method ...................................................................................................................... 19 3.1 Boundary Condition .......................................................................................... 20

3.1.1 Inlet Boundary Condition ......................................................................... 20 3.1.2 Outlet Boundary Condition ....................................................................... 22 3.1.3 Wall Boundary Condition ......................................................................... 22

3.2 Material Modeling ............................................................................................ 23 3.3 Modeling Approaches ....................................................................................... 25

3.3.1 TANK Method .......................................................................................... 25 3.3.2 INFLOW Method ..................................................................................... 28

4 Results & Discussion ................................................................................................ 29 4.1 TANK Result .................................................................................................... 29 4.2 INFLOW Result ................................................................................................ 35

5 Conclusion ................................................................................................................ 37

6 Future Work .............................................................................................................. 39

7 References ................................................................................................................. 41

Appendix A ....................................................................................................................... 43

Appendix B ....................................................................................................................... 56

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1 Introduction

The hydropower sector contributes to almost half of the total power production in Sweden

(Sandberg, 2009). The remaining part is mainly nuclear and small contributions come

from the bioenergy and wind power sectors. The hydropower contribution varies

throughout the year peaking in the winter months when subzero temperatures puts high

demand on power supply. Further, the contribution varies also from year to year due to

the dependence of the amount of rain and melted water collected in the reservoirs. In an

average year the hydropower sector produces of electricity comparable to

and for high and low production years respectively. The main features

of a hydropower station are the large reservoir dam where melt water and rain is stored

and the turbine/generator assembly which converts the potential energy stored in the

water into electricity (Hellström, 2009). As production varies throughout the year and the

amount of precipitation is uncontrollable the need to control the water head in the

reservoir is crucial to obtain optimal working conditions. If the inherent regulation by

generation is insufficient, this is achieved by the use of spillways. When spillways are

used large quantities of water is unleashed with high potential energy levels which are

converted to kinetic energy potentially causing erosion problems to structures in the

channel as well as in the old river bed. An effective way to reduce the high kinetic energy

levels is to design the spillway channels to trigger a hydraulic jump. The hydraulic jump

is a natural occurring phenomenon in flowing fluids characterized by large energy

dissipation mechanism and is found not only in manmade structures (Chanson, 2004). In

rapids the hydraulic jump can be seen and felt downstream of rocks when cannoning or

rafting. Additional features of the hydraulic jump is the sudden transition of shallow and

fast moving flow into slow moving flow with rise of the fluid surface. The transition

phase is known as the roller where the free surface is highly disturbed and air entrapment

occurs.

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Modeling of highly disturbed free surface flows such as the hydraulic jump is very

complex when grid based method is used. Severe problems with mesh entanglement and

determination of the free surface have been encountered. The use of meshfree methods

such as the Smooth Particle Hydrodynamic (SPH) method has been shown in previous

studies (López, Marivela, & Garrote, 2010), (Federico, Marrone, Colagrossi, Aristodemo,

& Veltri, 2010) to be a good alternative to traditional methods to overcome above stated

problems. The SPH-method is a meshfree, adaptive, Lagrangian particle method for

modeling fluid flow and as the maturity of method has increased rapidly during the last

decade or even years it was chosen as the computational method in this thesis. To reduce

the complexity of modeling a three dimensional spillway channel with adherent hydraulic

jump a two dimensional model was chosen. Further, two different approaches on how to

model the jump have been investigated.

The aim for this thesis has been to explore the SPH-methods capabilities to accurately

capture the main features of the hydraulic jump and to reproduce findings in previous

studies (López, Marivela, & Garrote, 2010), (Federico, Marrone, Colagrossi, Aristodemo,

& Veltri, 2010) in the commercial available software package LSTC LS-DYNA.

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2 Theory

The following section contains a description of the governing equation for the fluid flow

as well as the fundamentals of the SPH-method. Furthermore, an introduction to the

phenomena of hydraulic jump is also presented.

2.1 The Governing Equations

The fundamental equations in fluid dynamics are based upon the following three

fundamental physical laws of conservation,

Conservation of mass

Conservation of momentum

Conservation of energy

There are two fundamentally different approaches when describing the above

conservation relations, the Eulerian description and Lagrangian description. The most

common form, the Eulerian is a spatial description where a finite volume, called a control

volume (CV), is defined through which fluid flows in and out. There is no need to track

the position and velocity of individual fluid particles, instead a set of field variables are

defined at any location and at any time within the C.V., e.g. the pressure field and velocity

field variables. The Lagrangian form is a material description where individual fluid

particles are followed in space and time. The major difference between the two

descriptions is that the Lagrangian form employs the total time derivative as the

combination of the local derivative and convective derivative. Physically, the total time

derivative could be interpreted as the time rate of change following a moving fluid

element. Further, the local time derivative could be interpreted as the time rate of change

at a fixed point implying that the flow field property itself might be fluctuating with time.

Finally, the convective derivative could be interpreted as the change due to the motion of

the fluid element to another location where the flow properties are spatially different.

Due to the Lagrangian nature of the SPH-method, all subsequent conservation equations

will be defined on Lagrangian form. For further information regarding the conservation

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equations on Eulerian form the reader is referred to standard textbooks in fluid dynamics

such as (Cengel & Cimbala, 2006).

For a Lagrangian infinitesimal fluid cell, the continuity equation which is based on the

idea of conservation of mass states that the mass contained in the control volume

is,

( 2.1 )

where is the fluid density. As no mass is able to cross the control volume boundary, i.e.

the mass is conserved, the time rate of mass change is zero implying that

( 2.2 )

Rearranging and simplifying with the velocity divergence, the continuity equation on

Lagrangian form is obtained according to

( 2.3 )

The momentum equation is based on conservation of momentum and is represented by

Newton’s second law, which state that the net force on a fluid element is equal to the

mass times the acceleration. The net forces consist of both body forces, i.e. gravity,

magnetic, etc., and surfaces forces such as pressure imposed by surrounding fluids and

shear and normal stresses which result in shear deformation and volume change.

All forces in the x-direction affecting a fluid element and corresponding velocity

components are depicted in Figure 1.

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Figure 1: Forces in x-direction acting on a fluid element and velocity components (Liu &

Liu, 2009).

Summarizing all components and after some rearrangement and simplifications the

momentum equation in , and direction is,

( 2.4 )

where the shear stress is proportional to the shear strain rate through the dynamic

viscosity ,

( 2.5 )

where,

( 2.6 )

and where is the Dirac delta function.

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The energy equation derived from the conservation of energy states that the time rate of

energy change inside a fluid element is equal to the net heat flux into or out of the fluid

element and the time rate of work done by the body and surface forces acting on the

element. Neglecting the heat flux and the body forces, the time rate of internal energy

change consists of the work done by the isotropic pressure multiplying the volumetric

strain and the energy dissipation due to viscous shear forces. Hence, the energy equation

can be written as,

( 2.7 )

The set of partial differential equations (PDEs) defined above are the well-known Navier-

Stokes equations on Lagrangian form which governs the fluid flow.

2.2 Smooth Particle Hydrodynamics

Smoothed Particle Hydrodynamics (SPH) is a meshfree, adaptive, Lagrangian particle

method for modeling fluid flow. The technique was first invented independently by Lucy

(Lucy, 1977) and Gingold and Monaghan (Gingold & Monaghan, 1977) in the late

seventies to solve astrophysical problems in three-dimensional open space. Movement of

astronomical particles resembles the motion of a liquid or a gas, thus it can be modeled by

the governing equations of classical Newtonian hydrodynamics. The method did not

attract much attention in the research community until the beginning of the 1990 when

the method was successfully applied to other areas than astrophysics. Today, the SPH-

method has matured even further and is applied in a wide range of fields such as solid

mechanics (e.g. high velocity impact and granular flow problems) and fluid dynamics

(e.g. free-surface flows, incompressible and compressible flows).

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In the SPH-method, the fluid domain is represented by a set of non-connected particles

which possesses individual material properties such as mass, density, velocity, position

and pressure (Liu & Liu, 2009). Besides representing the problem domain and acting as

information carriers the particles also act as the computational frame for the field function

approximations. As the particles move with the fluid the material properties changes over

time due to interaction with neighboring particles, hence making the technique a pure

adaptive, meshless Lagrangian method. With adaptive is meant that at each time step the

field approximation is done based on the local distribution of neighboring particles. The

adaptive nature of the SPH-method together with the non-connectivity between the

particles results in a method that is able to handle very large deformations which are of

the essence when simulating highly disordered free-surface flows such as hydraulic

jumps.

To further clarify the intricate methodology of the SPH-method, one has to be aware of

the basic idea behind any numerical method. This is to reduce the set of partial

differential equations governing the problem at hand, see Section 2.1, into a set of

ordinary differential equations (ODEs) in a discretized form with respect to time only.

The set of ODEs can then be solved using standard explicit integration routines. The

SPH-method employs the following key steps in order to achieve the above,

1. The problem domain is represented by a set of non-connected and preferably

structured distribution of particles.

2. The integral representation method is used for field function approximation,

known as the kernel approximation. See Section 2.2.1 The Kernel

Approximation.

3. The kernel approximation is then further approximated using particles, i.e. the

particle approximation. The particle approximation replaces the integral in the

kernel approximation by summations over all neighboring particles in the so

called support domain. See Section 2.2.2 The Particle Approximation.

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4. The summations or the particle approximation are performed at each time step,

hence the adaptive nature of the SPH-method as particle position and the

magnitude of the individual properties varies with time.

5. The particle approximation is employed to all terms of the field functions and

reduces the PDEs to discretized ODEs with respect to time only.

6. The ODEs are solved using standard explicit integration algorithms.

As stated in item two above, the fundamental mathematical formulations in SPH are

based on integral interpolation theory (Gomez-Gesteira, Rogers, Dalrymple, & Crespo,

2010) and below follows derivation of the kernel- and particle approximation, key

features in SPH.

2.2.1 The Kernel Approximation

The derivation of the kernel approximation where an arbitrary field function is

represented by the integration of the function multiplied with the so called smoothing

kernel function begins with the following identity,

( 2.8 )

where is the arbitrary function of the three dimensional position vector , is the

volume containing and is the Dirac delta function given by,

( 2.9 )

where is the distance between the particle of evaluation and any arbitrary

particle in . The integral representation in eq. ( 2.8 ) is at this stage exact as long as

is defined and continuous in . Replacing the Dirac delta function in eq. (

2.8 ) with the smoothing kernel function yields,

( 2.10 )

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where is the so called smoothing length defining the influence volume of the smoothing

function, see below for greater detail. The integral representation in eq. ( 2.10 ) is the

kernel approximation as long as is not the Dirac delta function. A number of different

smoothing functions might be used, but they all should decrease with increasing distances

away from the point of evaluation, be an even function, be sufficiently smooth and satisfy

the below stated mathematical conditions (Liu & Liu, 2009),

The normalization condition,

( 2.11 )

The Delta function property,

( 2.12 )

The positivity condition,

( 2.13 )

The compact condition,

( 2.14 )

where is a constant defining the effective non-zero area of the smoothing function

called the support domain. The smoothing kernel function used by LS-DYNA and hence

used in this thesis is the cubic B-spline defined as,

( 2.15 )

where,

( 2.16 )

and is the constant of normalization equal to , and for one-,

two- and three dimensional space respectively. As mentioned above there exists a

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multitude of different smoothing kernel functions and the choice is thus significant as to

which problem are to be solved. Further information regarding the currently available and

the construction of smoothing functions the reader is referred to (Liu & Liu, 2009).

As stated above the smoothing length defines the influence volume or area of the

smoothing function depended of the dimensionality of the problem to be solved, three and

two dimensions respectively. To the author’s knowledge most of the present work

employs a static smoothing length but LS-DYNA use the concept of variable smoothing

length developed by W. Benz in the late 80’s in order to avoid problems due to

compression and expansion of the material (Lacome, 2001). The key idea is to keep the

same number of particles in the neighborhood meaning keeping the same mass of

particles in the neighborhood. Thus, in expansion when the particles are moving away

from each other the smoothing length is allowed to increase and in compression decrease.

If a static smoothing length is employed, numerical fracture can break down the

calculations or significantly slow down the overall processes. The derivation of the

equation governing the smoothing length begins with recognizing that the total mass of

particles enclosed in a sphere of radius is,

( 2.17 )

Differentiating with regard to time yields,

( 2.18 )

Realizing that in order to conserve mass, the second term on the right hand side must be

zero and with some simplification the equations that govern the smoothing length is,

( 2.19 )

where is the divergence of the velocity hence the smoothing length varies in both

space and time.

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In order to apply the methodology of the SPH-method the spatial derivative of a function

is needed. The integral representation of the spatial derivative of an arbitrary

function is,

( 2.20 )

See (Liu & Liu, 2009) for information regarding the deriving processes in greater detail.

2.2.2 The Particle Approximation

The second key feature in the SPH-method is the particle approximation where the

continuous integrals in eq. ( 2.10 ) and eq. ( 2.20 ) are converted to discretized forms of

summation over all particles in the support domain, see Figure 2.

Figure 2: The particle approximation of particle in the support domain of radius

(Liu & Liu, 2009).

The infinitesimal volume in eq. ( 2.10 ) and eq. ( 2.20 ) are replaced by the finite

volume of particle , i.e. which is related to the mass and density in the following

way,

( 2.21 )

Applying the above relationship to eq. ( 2.10 ) and eq. ( 2.20 ) yields,

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( 2.22 )

where is the number of particles in the support domain. The particle approximation

equations for a function is,

( 2.23 )

where

( 2.24 )

Equation ( 2.23 ) states that the value of the function at particle is approximated by the

function value of all the particles in the support domain weighted by the smoothing kernel

function and particle volume. The particle approximation for the spatial derivative is,

( 2.25 )

where

( 2.26 )

Equation ( 2.25 ) states that the value of the gradient of a function at particle is

approximated by the function value of all the particles in the support domain weighted by

the gradient of the smoothing kernel function and particle volume.

Applying the above to the Navier-Stokes equations presented in section 2.1 is out of

scope for present thesis but the reader is yet again referred to (Liu & Liu, 2009) for in-

depth knowledge.

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2.3 Hydraulic Jump

The occurrence of hydraulic jumps is a natural occurring phenomenon in flowing fluids

such as water. The main characteristic is the sudden transition of rapid shallow flow to

slow moving flow with rise of the fluid surface better known as a transition from

supercritical to subcritical flow (Chanson, 2004), see below for more details. Transition

between supercritical and subcritical flow are characterized by strong dissipative

mechanism which is favorable when high kinetic energy levels is unwanted such as in

spillway flows. Further characteristics of hydraulic jumps are the development of the

large-scale highly turbulent zone known as the “roller” where surface waves and spray,

energy dissipation and air entrapment is present, see Figure 3. As stated by Chanson

(2004) the overall flow field is extremely complicated and varies rapidly hence the bulk

features of the hydraulic jump are usually considered only.

Figure 3: Hydraulic jump in flume, flow left to right (Murzyn & Chanson, 2008).

Before defining the super- and subcritical flow conditions one must define the critical

flow condition. The derivation of the critical flow condition begins with the specific

energy which is a function of the flow depth and is defined as,

( 2.27 )

where is the pressure at the bottom, is the depth-averaged velocity, is the elevation

and is the bottom elevation. The specific energy is similar to the energy per unit mass,

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measured with the channel bottom as reference datum. For a slow moving flow the

velocity is small and the depth is large, thus the kinetic energy term is small. For a

rapid flow where the velocity is large and, by continuity the flow depth is small hence the

pressure term is small. For a constant discharge and a rectangular channel with

constant width (Chanson, 2004) the critical flow condition is attained when eq. ( 2.27 )

assumes its minimum value i.e.,

( 2.28 )

After transformation of eq. ( 2.28 ) the minimum specific energy is,

( 2.29 )

where the critical depth is,

( 2.30 )

Another important parameter in hydraulic applications and in the definition of critical

flow conditions is the dimensionless Froude number defined as,

( 2.31 )

where is the characteristic dimension which for open rectangular channels is the

depth . Further, the Froude number is a non-dimensional parameter for the balance of

internal to gravitational (hydrostatic pressure) forces. Froude number is proportional to

the square root of the ratio of the internal forces over the weight of the fluid i.e.,

( 2.32 )

In any flow situation where a free-surface is present and thus gravity effects are

significant the Froude number must be taken into consideration. When model studies of

open channel flows and hydraulic structures are performed one uses the concept of

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Froude similarity, which states that the Froude number must be equal for both the model

and the prototype,

( 2.33 )

Table 1 below summarizes the relevant characteristics of subcritical, critical and

supercritical flow in rectangular channels.

Table 1: Characteristic of subcritical, critical and supercritical flow in rectangular

channels.

Subcritical Critical Supercritical

Depth of flow

Velocity of flow

Froude number

The definition of the critical velocity is a direct consequence of the definition of the

Froude number at critical flow, i.e. .

As will be evident in the Method section below, one of the modeling approaches used

encompasses the use of a large reservoir tank with a gate opening to model hydraulic

jumps. A relation between the surface height in the tank and the velocity in the

supercritical section is easily derived, see Figure 4.

Figure 4: Schematic figure showing depth relation at position one and two.

d1

a d2

q

1

2

Fluid surface

Wall boundary

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Following a streamline for frictionless, incompressible and steady flow the Bernoulli

equation state,

( 2.34 )

where the depth-averaged velocity, the gravitational constant, is the bottom

elevation, the free-surface height measured from the bottom, i.e. . For

the horizontal and frictionless case depicted in the above figure, the Bernoulli equation

can be written as,

( 2.35 )

By continuity, the flow must be the same at position one and two yielding,

( 2.36 )

Substituting eq. ( 2.36 ) into eq. ( 2.35 ) and after some manipulation a relation between

the flow and the free-surface heights and can be written as,

( 2.37 )

where,

( 2.38 )

Eq. ( 2.37 ) is a well-known equation in hydraulic engineering and for sufficiently large

ratios of the free-surface height at position two can be approximated as,

( 2.39 )

where

( 2.40 )

and is the gate opening height. Substituting eq. ( 2.38 )-( 2.40 ) into eq. ( 2.37 ) and by

the continuity relation eq. ( 2.36 ), an explicit relationship between the velocity in the

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supercritical section, the gate opening height and the free-surface height in the reservoir

tank is obtained as,

( 2.41 )

As stated above the hydraulic jump is characterized by a supercritical and a subcritical

region where the depths are significantly different. These depths and are referred to

as conjugate depths and can be seen in the schematic Figure 5.

Figure 5: Schematic figure of the hydraulic jump showing the conjugate depths and

.

A dimensionless relation between the conjugate depths can easily be derived from

continuity, momentum and energy equations for a rectangular channel. Here is assumed

hydrostatic pressure distribution and uniform velocity distribution at the up- and

downstream end of the control volume. Further, the friction between the bottom and the

fluid is assumed to be zero. With these assumptions the conservations equations yield the

dimensionless relation between the conjugate depths for a rectangular channel as,

( 2.42 )

d2

d3

v2

v3

Roller

Fluid surface

Wall boundary

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where is the upstream Froude number, , which by definition must be

greater than one. The upstream Froude number is also used as an indicator of the general

characteristics of the jump in a rectangular horizontal channel as different upstream

Froude numbers produce different hydraulic jumps. The below table summarizes the

different types of hydraulic jumps and the main features observed through experiments,

see Table 2.

Table 2: Classification of hydraulic jumps (Chanson, 2004).

Type Characteristics

1 Critical flow No hydraulic jump.

1-1.7 Undular jump Free-surface undulations developing downstream of

jump, negligible energy loss.

1.7-2.5 Weak jump Low energy loss.

2.5-4.5 Oscillating jump Unstable oscillating jump with large waves of irregular

period. To be avoided.

4.5-9 Steady jump

Steady jump with 45-70% energy dissipation. Insensitive

to downstream condition and is considered as Best

economical design.

>9 Strong jump Rough jump with up to 85% energy dissipation. To be

avoided due to the risk of channel bed erosion.

As stated by Chanson, the above classification of hydraulic jump is considered as rough

guidelines only as other researcher have produced for instance undular hydraulic jump

with Froude number as high as three.

The Froude number is analogous to the Mach number in compressible gas flows, defined

as the ratio of the speed of flow and the sound speed of the medium. Further, supercritical

and subcritical flow is comparable to supersonic and subsonic where information is

unable to travel upstream in the super-sonic/critical case and able to travel upstream in

the other case. One might also compare the hydraulic jump with the shockwave as both

are recognized as strong energy dissipaters (Kundu, 1990).

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3 Method

The commercial available software LSTC LS-DYNA version ls971d R5.0 where used on

an HP Z600 with eight cores in order perform the simulation of two dimensional (2D)

hydraulic jumps. LS-DYNA is perhaps better known for the capabilities to solve complex

solid mechanic problems such as crash test simulation in the automotive industry, but the

capabilities reach even further to highly nonlinear and explicit multi-physical problems

(Hallquist, 2006). The implementation of the SPH-solver has made the LS-DYNA

package one of the few commercial software packages available at the market today. To

the authors knowledge most of today’s researcher in the SPH community uses highly

experimental codes developed for research only. However, some attempts have been

made in order to converge the research to one or two codes such as the SPH-FLOW

developed by a consortium composed of industry and academic partners and the freely

available SPHysics code developed jointly by universities in the U.S. and Europe. The

developments of the SPH-method have also been encouraged by the ERCOFTAC Special

Interest Group SPHERIC.

Previous studies has shown that the SPH-method is well suited to simulate hydraulic

jumps (López, Marivela, & Garrote, 2010), (Federico, Marrone, Colagrossi, Aristodemo,

& Veltri, 2010). Both works were done using in-house research codes not available on the

open market and to the author’s knowledge no work on hydraulic jumps has been

conducted using the LS-DYNA SPH-solver. Thus, the work conducted during this thesis

has been of trial-and-error character. However, there are some work done using the LS-

DYNA SPH-solver to model water such as the modeling of fuel sloshing in a tank

(Vesenjak, Müllerschön, Hummel, & Ren, 2004) and modeling of bird-substitute

impacting on a rotating fan (Selezneva, Stone, Moffat, Behdinan, & Poon, 2010).

To begin with, the LS-DYNA SPH-solver uses the concept of a computational box

(LSTC, 2010). Which is a box shaped volume defined in three dimensional space where

particles inside is activated, i.e. the particle approximation are computed for each particle

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at each time step. When a particle is outside the computational box it is deactivated

implying that no computation is done which saves both time and computational effort.

Further, as the particle crosses the boundary of the box field variables are conserved

meaning that particles follow the exit trajectories with maintained velocities. To define

the box in LS-DYNA one uses the DEFINE_BOX k-word or if one requires a moving

box the DEFINE_BOX_SPH k-word. The computational box is a vital tool when

performing SPH computations as stated above, however some issues concerning the box

have been encountered as explained below.

Another key feature is the modeling of the gravitational field, , done by

the LOAD_BODY_Y k-word. As will be evident further below one of the two modeling

approaches used is primarily gravitational driven.

Below follows the boundary condition and the most significant k-words used together

with the two conceptually different modeling approaches used to model the hydraulic

jump.

3.1 Boundary Condition

In the field of fluid dynamics boundary conditions such as inlets, outlets and walls are

standard features, one might even claim that they are mandatory in order to perform

Computational Fluid Dynamic (CFD) simulations successfully. As the implementation in

LS-DYNA of the SPH-method is in its early stages, rather few boundary conditions is

currently implemented and working properly.

3.1.1 Inlet Boundary Condition

The inlet boundary condition is implemented and is applied by using the

BOUNDARY_SPH_FLOW k-word. Initially, the user defines all particles representing

the inflow and has the ability to choose whether the particles move according to a

prescribed velocity, displacement or acceleration (LSTC, 2010). At time , all

particles are deactivated in a similar fashion as particles outside the computational box

but move according to the prescribed motion defined. The boundary of activation where

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the SPH-particles are activated and the particle approximation is computed is defined by a

predefined node and a vector. The ability of the BOUNDARY_SPH_FLOW k-word is

promising but it is unfortunately not working properly as two distinct problems arose

during the simulations. Firstly, deactivated particles were affected by gravity outside the

boundary of activation and the computational box which resulted in unwanted vertical

motion. Secondly, random particles outside the boundary of activation which were

supposed to move according to the prescribed motion remained stationary in initial

positions which resulted in a discontinuous inflow of particles. The first problem where

resolved by the support office at the Nordic official distributor of LS-DYNA, Engineering

Research AB (ERAB). The update was implemented in the development version of LS-

DYNA which the author was able to try but the second problem still remained.

Efforts to get around the problems experienced by using the BOUNDARY_SPH_FLOW

k-word were conducted by applying the BOUNDARY_PRESCRIBED_MOTION_SET

together with the DEFINE_DEATH_TIMES_SET k-words. The

BOUNDARY_PRESCRIBED_MOTION_SET k-word has similar capabilities as the

BOUNDARY_SPH_FLOW k-word but with the drawback that the termination of the

prescribed motion of the particles/nodes are controlled by the complimentary k-word

DEFINE_DEATH_TIMES_SET. Yet again, problem arose during the simulations due to

gravitational effects on constrained nodes outside the computational box with the

unwanted result of vertical displacement. One remedy proposed to solve the issue of

gravity effects on constrained nodes were to including yet another

PRESCRIBED_MOTION k-word defined to counteract the effects of gravity by applying

a prescribed acceleration equal in magnitude but opposite in direction to the gravity itself.

Unfortunately, this idea cut short as the DEATH_TIMES k-word is able to terminate one

prescribed motion only. The addition of another DEATH_TIMES k-word might be seen

as a possible remedy but only one DEATH_TIMES k-word is acceptable in order to

perform simulations successfully.

22

3.1.2 Outlet Boundary Condition

No particular outlet boundary condition is currently implemented in LS-DYNA.

However, a crude version of an outlet has been applied in this thesis namely the boundary

of the computational box. As mentioned above each particle crossing the boundary of the

box conserves the field variables and follows the exit trajectories with maintained

velocities in a deactivated state. The shortcomings are evident as the user has no control

of which information from downstream particles are affecting upstream particles.

3.1.3 Wall Boundary Condition

The wall boundary condition is used to model solid walls where fluid penetration is

prohibited. In traditional CFD modeling a multitude of different additional condition is

applicable such as free-slip and no-slip conditions where fluid particles have a finite and

zero velocity relative to the boundary respectively. Such conditions are currently

available in experimental codes only. The most common and accepted method in the SPH

community is to model solid walls as stationary SPH-particles locked in space and time.

Three distinct boundary particles have been proposed; the Ghost particle, Repulsive

particle and the Dynamic particle all with special features on how to treat effects at or

close to the boundary. As the theory of these particles is out of scope for this thesis see

(Gomez-Gesteira, Rogers, Dalrymple, & Crespo, 2010) or (Liu & Liu, 2009) for greater

details. SPH-particles locked in space and time with the same material properties and

features as the freely flowing fluid has been used in this thesis as no specially treated

boundary particles are available in LS-DYNA. Furthermore, wall particles have been

modeled with half the interparticle distance in both x- and y-direction as the free flowing

fluid,

( 3.1 )

Initialy, both fluid and wall particles where generated with the builtin particle generator

in LSPrePost 3.1 but later in MathWorks MatLab for increased controll and faster setup

time as wall particles are modeled in three layers, a combersome and teadious task in the

builtin particle generator, see Figure 6. See Appendix B for MatLab script.

23

Figure 6: Fluid (blue) and wall (red) particles.

Questions regarding the smoothness of the wall boundaries when SPH-particles are used

have been raised. The RIGIDWALL_PLANAR_FINITE k-word were proposed as a

remedy but showed at an early stage problem with both unstable and unphysical

solutions. A new contact algorithm CONTACT_2D_NODE_TO_SOLID have been

implemented in the development version of LS-DYNA which control contact between

SPH-nodes and solids or shell elements composed of a traditional mesh. Due to lack of

time, no work has been conducted with the new contact algorithm even though it seems

as a promising alternative.

3.2 Material Modeling

Previous studies (Vesenjak, Müllerschön, Hummel, & Ren, 2004), (Huertas-Ortecho,

2006), (Varas, Zaera, & López-Puente, 2009) showed that the material model

MAT_009_NULL have been used to model water with density and

dynamic viscosity . The null material has no shear stiffness or yield

strength and behaves in a fluid-like manner (LSTC, 2010). As the dynamic viscosity is

nonzero, a deviatoric viscous stress of the form,

( 3.2 )

is computed where is the deviatoric strain rate. Furthermore, the null material must

also be used together with an equation of state (EOS) defining the pressure in the

material. Varas et al. (2009) used the Gruneisen equation of state which employs the

24

cubic shock velocity-particle velocity and defines the pressure for compressed materials

as,

( 3.3 )

and for expanded materials,

( 3.4 )

where , and are coefficients of the slope of the curve, and being

the shock and particle velocity respectively. is the intercept of the curve which

corresponds to the adiabatic speed of sound of water. the Gruneisen gamma and is

the first volume correction to . the initial internal energy and where

is the initial density. Properties and constant given by (Boyd, Royles, & El-Deed,

2000) are summarized below in Table 3.

Table 3: Water properties and constants for the Gruneisen equation of state.

Property

Value

As very low time steps and thus large overall computational time was experienced with

the NULL material model the much faster and simpler Drücker-Prager based material

model MAT_005_SOIL_AND_FOAM used by (Gustafsson, Cante, Jonsèn, Weyler, &

Häggblad, 2009) were proposed. Even though Gustafsson et al. (2009) employed the

material model successfully it was discarded due to severe penetration issues at the

boundaries.

As described above the hydraulic jump phenomenon includes air entrapment in the roller

region as one of the significant features. However, no simulations incorporating the air as

25

a second phase have been conducted due to the increased complexity of multiphase

modeling.

3.3 Modeling Approaches

Two conceptually different modeling approaches, the Tank and Inflow approach have

been used in this thesis. The main difference between the two approaches is the modeling

of a large reservoir tank which by definition is less efficient as the movement of a huge

number of particles in the reservoir is computed, which for present work, is of less

importance.

3.3.1 TANK Method

The Tank approach is based on the work done by (López, Marivela, & Garrote, 2010)

whom with an in-house research code conducted SPH simulations on a set of hydraulic

jumps with different upstream Froude numbers. They also conducted a physical

experiment on a downscaled setup, fifty times smaller than the numerical, for validation

purposes. The main features of the simulation setup is the modeling of a large reservoir

tank initially filled with water which discharges through a gate opening onto a horizontal

bed with an attached weir close to the outlet, see schematic in Figure 7 below.

Figure 7: Schematic figure of the TANK-simulation setup.

The scale and hence the measurements used in this thesis were chosen to represent the

physical experiment and are summarized in Table 4 below.

Tank

Weir

Gate

Roller

Fluid surface Wall boundary

Bed

26

Table 4: TANK-simulation measurements.

Measurement Length [m]

Initial tank height 0.2

Tank width 1.2

Gate opening 0.02

Bed length 1.1

Weir height 0.02

Weir width 0.2

To achieve Froude similarity, the simulation time was multiplied by a factor of in

order to compare numerical results. Hence, the simulation time in this thesis was just

below three seconds.

The weir assembly close to the outlet was used to trigger the onset of the hydraulic jump,

a widely used feature in spillway channels. To trigger the hydraulic jump even faster the

bed region were initially filled with water to a depth equal to the weir height, i.e. .

The initial interparticle distance in the free fluid was set to and in the

wall boundaries. The above measurements and interparticle distances yielded a total of

particles with a mass of each.

Initial simulation runs indicated problems with severe pressure fluctuations and free-

surface oscillation in the reservoir tank when full gravitational acceleration was applied

instantaneous at time . To reduce the unwanted oscillations, gravitational load was

continuously increased from zero at to full gravitational load at . The use of

a movable gate which opened when full gravitational load were achieved with the

opening speed of eliminated the unphysical effects of reduced gravity. See

Appendix A for how this was implemented.

No currently accepted or widely used method to determine the fluid surface has yet been

proposed. Hence the post processing method has been conducted using average depths

values of particle centers close to the assumed free surface and point of interest. The

procedure to determine the velocity in the flow has been conducted in a similar manner

27

with the alteration that all particles from the bottom to the surface and close to the point

of interest have been used, see Figure 8.

Figure 8: Particles included in the depth (left) and velocity (right) determination are

shown in red.

28

3.3.2 INFLOW Method

The inflow approach is based on the work done by (Federico, Marrone, Colagrossi,

Aristodemo, & Veltri, 2010) who proposed a new algorithm for in- and outflow

conditions and showed its applicability to different types of hydraulic jumps. The

simulation setup composed of an inlet through which water discharged onto a horizontal

bed and continued to flow until it reached the outlet. Federico et al. (2010) concluded that

the use of a weir was unnecessary as the triggering of a hydraulic jump could be done by

initially position the SPH-nodes in two domains with depths based on the conjugate

depths theory and assigning appropriate velocities in the bed region. To trigger the onset

faster the author proposed the bed region to be divided further into three domains, see

Figure 9 below.

Figure 9: Schematic figure of the INFLOW-simulation setup.

As the implementation of the inlet boundary condition in LS-DYNA is at this stage not

working properly less effort has been put into this method. However, one successful run

was obtained. See Appendix A for reduced k-word input file.

I

N

L

E

T

O

U

T

L

E

T

Fluid surface Wall boundary

29

4 Results & Discussion

In this section the main results of present study together with comparison with previous

studies will be presented. As previously stated the work was run as a numerical

experiment thus several simulations runs were conducted and only the final runs will be

presented.

4.1 TANK Result

The Tank simulation results are presented with figures showing the hydraulic jump in the

bed region at successive time steps and simulation data such as the depths in the super-

/subcritical sections.

Figure 10 below shows the hydraulic jump at time steps indicated in the upper left corner

following the time steps used by López et al. (2010) with an added second to account for

the time until full gravitational load was obtained. Different colors indicate different

velocities in the positive x-direction measured in .

Subsequent to opening the gate at a fast moving jet shoots out of the gate opening

into the initially stationary fluid in the bed region. A wave forms and breaks as the fluid

starts to move together with the jet in the positive x-direction and the moving hydraulic

jump starts to form. The highly turbulent and disturbed roller region moves further

towards the weir and outlet section with decreased velocity as time proceeds. The

tendencies for the jump section to decelerate indicates that a quasi-stationary state might

be attained as time proceeds further but as no simulations were run past four seconds no

such state were ever assumed. One can argue that a quasi-stationary state do not exist for

the present configuration as the tank height decreases over time causing the velocity in

the supercritical section to decline enabling the roller and subcritical section to move

upstream.

30

Figure 10: Visualization of the tank simulation at successive time steps with color coded

velocities in the positive x-direction.

31

The overall result of the simulations is that the capability of the SPH-method to capture

the main structures in the flow field, namely the formation of a shallow and fast moving

jet and its transition into a deeper and slow moving section is demonstrated. However,

some shortcomings were detected such as the over prediction of pressure by a factor of

ten at the bottom in the tank section. This needs to be further investigated. Furthermore,

the use of the boundary of the computational box as an outlet might influence the

outcome of the simulation adversely as can be seen in Figure 10 for time step

and . The flow seems at both time steps to accelerate and reduce in depth

which is unrealistic but most certainly depend on the lack of a outlet boundary condition.

The use of SPH-nodes proved to be the most successful way to model solid walls.

However, issues of wall smoothness and the effects on flow properties have been raised

as mentioned above. Another drawback with the SPH-method is the overall

computational time which for the present result was almost hours.

To be able to compare and validate the numerical results with theory and previous work a

number of parameters were derived, see Table 5.

Table 5: Simulation results using the TANK method.

1.707 0.174 0.0118 1.606 0.055

2.414 0.165 0.0112 1.499 0.053

3.121 0.155 0.0110 1.489 0.066

3.828 0.146 0.0110 1.365 0.064

The explicit relationship eq. ( 2.41 ) derived in the theory section state that knowing the

reservoir tank height and the gate opening height , the velocity in the

supercritical section is easily determined. Based on above results the theoretically

velocity were then compared to the value obtained from the simulation , see

Table 6. Furthermore, the conjugate depths relation eq. ( 2.42 ) and the Froude number

definition eq. ( 2.31 ), also derived in the theory section states that knowing the velocity

and the depth in the supercritical section the upstream Froude number

32

and the depth is easily determined. Based on the results presented in Table 5 the

theoretical depth was then compared with the depth obtained from the

simulation, see Table 6. Knowing the Froude number one is able to determine which type

of jump is present, see Table 2.

Table 6: Comparison of simulation and theoretical results using the TANK method.

1.707 1.788 4.729 0.073 -10.2 -25.0

2.414 1.734 4.519 0.066 -13.5 -20.6

3.121 1.679 4.541 0.065 -11.3 1.6

3.828 1.628 4.153 0.059 -16.2 8.3

The upstream Froude number shows that a hydraulic jump classified as a steady jump is

produced for the first three time steps and an oscillating jump for the final time step. The

velocity comparison shows a systematic under prediction of the velocity and when

compared with the unphysical over prediction of pressure seen in the reservoir tank the

result is ambiguous. Two possible explanations to this behavior are given. Firstly, the

pressure is correct indicating a higher particle mass than specified is used during

computations yielding too much resistance thus lowering the velocity. Secondly, the

pressure is incorrectly presented during post-processing thus a lower pressure is used

during computations yielding the lower velocity. As stated above, the over prediction of

pressure and its effect on overall results should be investigated further. Comparison of the

depth indicate large differences for the first two time steps and almost non for the

second two. This randomness might be explained with the relatively inaccurate method

used to determine the position of the free-surface presented in the method section.

However, the under prediction for the first two time step might indicate that the hydraulic

jump is not fully developed at the concerned time steps.

Comparison of numerical results from the present study and numerical results obtained by

López et al. (2010) are summarized in Table 7 below.

33

Table 7: Comparison of numerical results for present study and numerical results

obtained by (López, Marivela, & Garrote, 2010).

1.707 -4.2 -30.9 8.1 -8.9

2.414 -7.5 -34.8 1.9 -31.1

3.121 -8.8 -29.7 5.5 -8.1

3.828 -2.4 -27.6 3.77 -4.0

The above data shows good agreement for all parameters except for the depth in the

supercritical section and the second time step for the depth in the subcritical section.

The large deviation for depth at the second time step might be explained as above by

the method used to determine the free-surface. However, the relative large and systematic

under prediction of the depth could in the author’s opinion not be explained by the

inherent arbitrariness of the method used. No distinct explanation to why this relative

large under prediction occurs is at this stage known. However, by realizing that continuity

must be maintained in the supercritical section one can argue that the results presented in

this thesis more accurately represent the problem due to the larger under prediction of the

depth compared to the relative small over prediction of the velocity

. The differences shown in the above table not only indicate the importance of

choosing the appropriate number of particles representing the problem but also that the

number of particles chosen most certainly affect the outcome as roughly seven times

more particles were used in this thesis compared to the previous work. Furthermore, the

differences found motivate a particle study investigating the effects of varied number of

particles as a future work effort.

Comparison of numerical results for the present study and experimental results obtained

by López et al. (2010) are summarized in Table 8 below.

34

Table 8: Comparison of numerical results for present study and experimental results

obtained by (López, Marivela, & Garrote, 2010).

1.707 -1.8 -18.9

2.414 -6.7 -32.5

3.121 -7.6 -17.9

3.828 3.1 -6.3

Good agreement for the reservoir tank depth is shown but only moderate to poor for

the depth in the subcritical section. However, as no method was presented in (López,

Marivela, & Garrote, 2010) of how experimental data was obtained and the arbitrary

method used in this thesis, less significance is dedicated to the under prediction of the

depth .

35

4.2 INFLOW Result

The Inflow results will be presented with figures showing the hydraulic jump at

successive time steps only. No comparison with previous studies will be presented due to

the randomness of the error experienced when using the inlet condition as previously

stated. However, the results show a very effective and inexpensive way to model

hydraulic jumps hence the author have chosen to include the results even though no

comparison or validation have been done. Figure 11 below shows the hydraulic jump at

time steps indicated in the upper left corner with color coded velocities in the x-direction

measured in . At time the initial velocities in the domain were chosen as

(red) , (yellow) and (blue) which are

based on theoretical assumptions presented in the theory section.

At the inlet the velocity constraint is terminated and the particle approximation is

computed. The hydraulic jump initiates almost instantaneous at and moves with

very low velocity in the negative x-direction indicating the onset of a quasi-stationary

state. Just as in the Tank simulation the use of the boundary of the computational box

adversely affect the flow close to the outlet at . As no simulation were run

passed one second the effects might have worsen or even destroyed the hydraulic jump

thus the results are non-trustworthy.

36

Figure 11: Visualization of the inflow simulation at successive time steps with color

coded velocities in the x-direction.

37

5 Conclusion

The present study has shown that the SPH-method is able to capture the main structures

of the hydraulic jump flow field. Successful outcomes were achieved with the Tank

approach but not for the Inflow case due to difficulties with the implementation of the

inlet boundary condition used. However, the Inflow approach were more efficient as the

number of activated particle at any given time step were less compared to the Tank

approach implying less computational cost. Both approaches suffered from the use of the

boundary of the computational box as outlet condition especially the Inflow case. The

commonly used and accepted method to model wall boundaries as stationary SPH-nodes

worked very well when the interparticle distance were half compared to the free flowing

fluid. However, the use of SPH-nodes as walls imply wall smoothness issues not

addressed in present thesis.

Comparisons of numerical results with theoretically derived values for the Tank case

showed a systematic under prediction of the velocity in the supercritical section which

is an ambiguous result when compared with the over prediction of the pressure in the

reservoir tank. Furthermore, comparison of the theoretically derived depth in the

subcritical section showed when compared with the numerical results a large under

estimation for the first two time steps. The conclusion is that the hydraulic jump is not

fully developed at the mentioned time step as the error continuously decreased with time.

Comparison of simulation result with the previous work showed generally under

estimation, especially for the depth . However, the velocity showed an

over prediction by roughly five percent. The conclusion is that the number of particles

used to represent the system influence the outcomes as roughly seven times more

particles were used in this thesis compared to the previous work. Finally, the comparison

of present numerical results with experimental results from the work by López et al.

(2010) showed good agreement for the tank depth but only moderate for the depth

in the subcritical section.

38

As concluded earlier in (López, Marivela, & Garrote, 2010) and (Federico, Marrone,

Colagrossi, Aristodemo, & Veltri, 2010), the Smoothed Particle Hydrodynamic method is

a viable tool when performing free-surface flow and particularly hydraulic jump

simulations, but the maturity of the implementation of the SPH-method in the commercial

software LS-DYNA is at this stage insufficient. However, recent developments indicate

that problems experienced in this work might be solved in future releases of the software.

39

6 Future Work

There are numerous areas in which future work efforts could be directed. The first issue

to be solved is the pressure in the reservoir tank which was over predicted by a factor of

ten. As mentioned in the result section plausible errors might have been inferred in the

particle generating script which should undergo careful scrutiny. The Gruneisen equation-

of-state determining the pressure in the material might contain inaccuracies as well and

should be investigated thoroughly. Another area of interest dealing with pressure is the

possibility to assign a hydrostatic pressure distribution in the fluid which might reduce

the sever pressure oscillations when full gravitational load is applied instantaneous. If this

approach should work the use of both the continuously increasing gravity and the

movable gate is obsolete saving computational time.

The influences of particle distribution and the number of particle representing the

problem have not been thoroughly explored and is one of the strong candidates for future

work efforts. A particle study investigating the effects of particle size and hence wall

smoothness has been proposed. The wall smoothness influences the overall energy losses

in the fluid affecting the outcome of the simulations. A particle study might reveal

convergence or divergence tendencies as the level of discretization is varied. In traditional

CFD, grid independent solutions are attained through the use of Richardson extrapolation

which might be possible to implement in the SPH-community as well introducing the

concept of “particle independent solutions”. The newly implemented k-word

CONTACT_2D_NODE_TO_SOLID should be thoroughly investigated as it yields

perfectly smooth wall boundaries.

The outcome of present study was seriously affected by the method used to determine the

position of the free-surface which is another interesting area of future work. As discrete

particles is used to represent the system the approach to determine the surface might best

be done be applying statistical measurements. Another approach is to borrow ideas from

the established multiphase flow techniques using traditional grids such as the Volume Of

40

Fluid (VOF) or Level Set (LS) methods. Already established methods might also give

insights of how to model an extra phase such as air.

Additional future work possibilities are to conduct a physical experiment in order to

better validate the simulations. The use of Particle Imaging Velocimetry (PIV) or Laser

Doppler Velocimetry (LDV) might give insights to the inner flow field of the hydraulic

jump a subject not addressed here. A study exploring the inner flow field would be most

beneficial.

The author’s future research work will be focused on solving the issue of the over

predicted pressure and perform the particle study state above.

41

7 References

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Phenomena Relating to Axisymmetric Structures. Sixth International LS-DYNA

Users Conference Simulation, (pp. 15-36). Dearborn, Michigan.

Cengel, Y. A., & Cimbala, J. M. (2006). Fluid Mechanics: Fundamentals and

Applications. New York: McGraw-Hill.

Chanson, H. (2004). The Hydraulics of Open Channel Flow: An Introduction. Oxford:

Elsevier Ltd.

Federico, I., Marrone, S., Colagrossi, A., Aristodemo, F., & Veltri, P. (2010). Simulation

of hydraulic jump through sph model. IDRA XXXII Italian Conference of

Hydraulics and Hydraulic Construction. Palermo, Italy.

Gingold, R. A., & Monaghan, J. J. (1977). Smoothed particle hydrodynamics: theory and

application to non-spherical stars. Monthly Notices of the Poyal Astronomical

Society, 181, 375-389.

Gomez-Gesteira, M., Rogers, B. D., Dalrymple, R. A., & Crespo, A. J. (2010). State-of-

the-art of classical SPH for free-surface flows. Journal of Hydraulic Research

Vol. 48 Extra Issue, 6-27.

Gustafsson, G., Cante, J. C., Jonsèn, P., Weyler, R., & Häggblad, H.-Å. (2009).

Comparison of Smoothed Particle Method and Particle Finite Element Method in

Applied Granular Flow Problems. International Conference on Particle-Based

Methods - Particles 2009. Barcelona: CIMNE.

Hallquist, J. O. (2006). LS-DYNA Theory Manual. Livermore Software Technology

Corporation.

Hellström, G. I. (2009). Internal Erosion in Embankment Dams - Fluid Flow Through

and Deformation of Porous Media. Luleå: Luleå University of Technology.

Huertas-Ortecho, C. A. (2006). Robust Bird-Strike Modeling Using LS-DYNA. Master

Thesis. Mayagüez, Puerto Rico: University of Puerto Rico Mayagüez Campus.

Kundu, P. K. (1990). Fluid Mechanics. San Diego: Academic Press, Inc.

42

Lacome, J. L. (2001, November 27). Smoothed Particle Hydrodynamics - Part II. FEA

Information International News For The World-Wide Engineering Community,

pp. 6-11.

Liu, G. R., & Liu, M. B. (2009). Smoothed Particle Hydrodynamics : a meshfree particle

method. Singapore: World Publishing Co. Pte. Ltd.

López, D., Marivela, R., & Garrote, L. (2010). Smoothed particle hydrodynamics model

applied to hydraulic structure: a hydraulic jump test case. Journal of Hydraulic

Research Vol. 48Extra Issue, 142-158.

LSTC. (2010). LS-DYNA Keyword User´s Manual Version 971 Rev 5. Livermore:

Livermore Software Technology Corporation (LSTC).

Lucy, L. B. (1977). A numerical approach to testing of the fission hypothesis. The

Astronomical Journal, 1013-1024.

Murzyn, F., & Chanson, H. (2008). Experimental investigation of bubbly flow and

turbulence in hydraulic jumps. Enviromental Fluid Mechanics Vol. 9 Iss. 2, 143-

159.

Sandberg, E. (2009, June 3). Svensk energi. Retrieved January 30, 2011, from

http://www.svenskenergi.se/sv/Om-el/Elproduktion/

Selezneva, M., Stone, P., Moffat, T., Behdinan, K., & Poon, C. (2010). Modeling Bird

Impact on Rotating Fan: The Influence of Bird Parameters. 11th International

LS-DYNA User Conference (pp. 37-46). Dearborn, Michigan USA: LSTC.

Varas, D., Zaera, R., & López-Puente, J. (2009). Numerical modeling of hydrodynamic

ram phenomenon. International Journal of impact Engineering 36, 363-374.

Vesenjak, M., Müllerschön, H., Hummel, A., & Ren, Z. (2004). Simulation of Fuel

Sloshing - Comperative Study. LS-DYNA Anwenderforum. Bamberg:

DYNAmore GmgH.

43

Appendix A

Reduced input k-word file TANK-method.

$# LS-DYNA Keyword file created by LS-PrePost 3.1 (Beta) -

14Sep2010(17:24)

$# Created on Dec-5-2010 (13:38:06)

*KEYWORD

*TITLE

$# title

LS-DYNA keyword deck by LS-PrePost

*CONTROL_SPH

$# ncbs boxid dt idim memory form

start maxv

1 11.0000E+20 2 100 5

0.0001.0000E+15

$# cont deriv iact

0 0 0 0

*CONTROL_TERMINATION

$# endtim endcyc dtmin endeng endmas

4.000000 0 0.000 0.000 0.000

*CONTROL_TIMESTEP

$# dtinit tssfac isdo tslimt dt2ms lctm

erode ms1st

4.0000E-6 0.900000 0 0.000 0.000 0

0 0

$# dt2msf dt2mslc imscl

0.000 0 0

*DATABASE_BINARY_D3PLOT

$# dt lcdt beam npltc psetid

0.001000 0 0 0 0

$# ioopt

0

44

*DATABASE_HISTORY_SPH_SET

$# id1 id2 id3 id4 id5 id6

id7 id8

1 2 0 0 0 0 0

0

*DEFINE_BOX_TITLE

BOX for pkt

$# boxid xmn xmx ymn ymx zmn

zmx

1 -0.010000 2.502000 -0.050000 2.000000 -0.200000 0.200000

*DEFINE_COORDINATE_SYSTEM_TITLE

cord_sys

$# cid xo yo zo xl yl

zl

1 0.000 0.000 0.000 1.000000 0.000 0.000

$# xp yp zp

0.000 1.000000 0.000

*DEFINE_CURVE_TITLE

Gravity curve 2

$# lcid sidr sfa sfo offa offo

dattyp

4 0 1.000000 1.000000 0.000 0.000 0

$# a1 o1

0.000 0.000

1.0000000 9.8100004

100.0000000 9.8100004

*DEFINE_CURVE_TITLE

DISP curve gate


dattyp

5 0 1.000000 1.000000 0.000 0.000 0

$# a1 o1

0.000 0.000

45

1.0000000 0.000

1.0190001 0.0190000

5.0000000 0.0190000

*ELEMENT_SPH

$# nid pid mass

*BOUNDARY_PRESCRIBED_MOTION_SET_ID

$# id

heading

1Moving gate

$# nsid dof vad lcid sf vid

death birth

3 2 2 5 1.000000 01.0000E+28

0.000

*BOUNDARY_SPC_SET_ID

$# id

heading

1Walls

$# nsid cid dofx dofy dofz dofrx

dofry dofrz

1 1 1 1 1 1 1

1

*SET_NODE_LIST_GENERATE_TITLE

Walls

$# sid da1 da2 da3 da4 solver

1 0.000 0.000 0.000 0.000MECH

$# b1beg b1end b2beg b2end b3beg b3end

b4beg b4end

1 9129 0 0 0 0 0

0


$# id

heading

2W1

46


dofry dofrz

2 1 0 0 1 0 0

0


W1


2 0.000 0.000 0.000 0.000MECH


b4beg b4end

9130 74509 0 0 0 0 0

0

*EOS_GRUNEISEN_TITLE

EOS water 2

$# eosid c s1 s2 s3 gamao

a e0

2 1484.0000 1.979000 0.000 0.000 0.110000 3.000000

3.0720E+5

$# v0

1.000000

*LOAD_BODY_Y

$# lcid sf lciddr xc yc zc

cid

4 1.000000 0 0.000 0.000 0.000 1

*MAT_NULL_TITLE

water NULL 2

$# mid ro pc mu terod cerod

ym pr

2 1000.0000 -1.000E+6 8.9000E-4 0.000 0.000 0.000

0.000

*NODE

$# nid x y z tc

rc

47

*PART

$# title

WALLS

$# pid secid mid eosid hgid grav

adpopt tmid

1 1 2 2 0 0 0

0

*SECTION_SPH_TITLE

WALLS

$# secid cslh hmin hmax sphini death

start

1 1.000000 1.000000 1.000000 0.0001.0000E+20 0.000

*PART

$# title

WATER


adpopt tmid

2 2 2 2 0 0 0

0

*SECTION_SPH_TITLE

WATER


start

2 1.200000 0.950000 4.000000 0.0001.0000E+20 0.000


GATE


3 0.000 0.000 0.000 0.000MECH


b4beg b4end

7696 8475 0 0 0 0 0

0

*END

48

*COMPONENT

$# clid color1 color2 color3 color4

1 0.769000 0.004000 0.110000 0.000 0 0

0

$# name

Part 1

*COMPONENT_PART

$# pid clid

1 1

$# pid clid

2 1

*COMPONENT_END

49

Reduced input k-word file INFLOW-method.

$# LS-DYNA Keyword file created by LS-PrePost 3.1 (Beta) -

14Sep2010(17:24)

$# Created on Nov-5-2010 (13:52:33)

*KEYWORD

*TITLE

$# title

LS-DYNA keyword deck by LS-PrePost

*CONTROL_SPH

$# ncbs boxid dt idim memory form

start maxv

1 11.0000E+20 2 1000 5

0.0001.0000E+15

$# cont deriv iact

0 0 0 0

*CONTROL_TERMINATION

$# endtim endcyc dtmin endeng endmas

2.000000 0 0.000 0.000 0.000

*CONTROL_TIMESTEP

$# dtinit tssfac isdo tslimt dt2ms lctm

erode ms1st

4.0000E-6 0.800000 0 0.000 0.000 0

0 0

$# dt2msf dt2mslc imscl

0.000 0 0

*DATABASE_BINARY_D3PLOT

$# dt lcdt beam npltc psetid

0.001000 0 0 0 0

$# ioopt

0

*DATABASE_HISTORY_SPH_SET

$# id1 id2 id3 id4 id5 id6

id7 id8

50

1 0 0 0 0 0 0

0

*ELEMENT_SPH

$# nid pid mass


$# id

heading

1WATER


dofry dofrz

1 1 0 0 1 0 0

0


WATER


1 0.000 0.000 0.000 0.000MECH


b4beg b4end

1 30300 0 0 0 0 0

0


$# id

heading

2WALLS


dofry dofrz

2 1 1 1 1 1 1

1


WALLS


2 0.000 0.000 0.000 0.000MECH

51


b4beg b4end

30301 33303 0 0 0 0 0

0

*BOUNDARY_SPH_FLOW

$# id styp dof vad lcid sf

death birth

5 3 4 2 3 1.0000001.0000E+20

0.000

$# nid vid

30301 2

*EOS_GRUNEISEN_TITLE

EOS water

$# eosid c s1 s2 s3 gamao

a e0

1 1480.0000 2.560000 1.986000 1.226800 0.500000 0.000

0.000

$# v0

0.000

*INITIAL_VELOCITY

$# nsid nsidex boxid irigid

4 0 1 0

$# vx vy vz vxr vyr vzr

0.930274 0.000 0.000 0.000 0.000 0.000

*LOAD_BODY_Y

$# lcid sf lciddr xc yc zc

cid

1 1.000000 0 0.000 0.000 0.000 1

*NODE

$# nid x y z tc

rc

*PART

$# title

52

SphNode


adpopt tmid

1 1 1 1 0 0 0

0

*SECTION_SPH_TITLE

PKT


start

1 1.300000 1.000000 5.000000 0.0001.0000E+20 0.000

*MAT_NULL_TITLE

water NULL

$# mid ro pc mu terod cerod

ym pr

1 1000.0000 -1.000E+6 0.001000 0.000 0.000 0.000

0.000

*PART

$# title

SphNode


adpopt tmid

2 2 1 1 0 0 0

0

*SECTION_SPH_TITLE

PKT


start

2 1.000000 1.000000 1.000000 0.0001.0000E+20 0.000


Water 1


3 0.000 0.000 0.000 0.000MECH

53


b4beg b4end

1 4375 0 0 0 0 0

0


Water 2


4 0.000 0.000 0.000 0.000MECH


b4beg b4end

4376 15100 0 0 0 0 0

0


Water 3


5 0.000 0.000 0.000 0.000MECH


b4beg b4end

15101 30300 0 0 0 0 0

0

*DEFINE_BOX_TITLE

BOX for pkt

$# boxid xmn xmx ymn ymx zmn

zmx

1 0.000 2.000000 -0.050000 2.000000 -0.200000 0.200000

*DEFINE_COORDINATE_SYSTEM_TITLE

cord_sys

$# cid xo yo zo xl yl

zl

1 0.000 0.000 0.000 1.000000 0.000 0.000

$# xp yp zp

0.000 1.000000 0.000

*DEFINE_CURVE_TITLE

54

Gravity curve


dattyp

1 0 1.000000 1.000000 0.000 0.000 0

$# a1 o1

0.000 9.8100004

1000.0000000 9.8100004

*DEFINE_CURVE_TITLE

Motion curve


dattyp

2 0 1.000000 1.000000 0.000 0.000 0

$# a1 o1

0.000 1.2170908

10.0000000 1.2170908

*DEFINE_CURVE_TITLE

Disp curve


dattyp

3 0 1.000000 1.000000 0.000 0.000 0

$# a1 o1

0.000 0.000

10.0000000 12.1709080

*DEFINE_VECTOR_TITLE

vector sph-flow

$# vid xt yt zt xh yh

zh cid

1 -0.052000 0.126000 0.000 0.000 0.126000 0.000

1


vector inflow x_dir


zh cid

55

2 0.000 0.000 0.000 1.000000 0.000 0.000

1


vector inflow y_dir


zh cid

3 0.000 0.000 0.000 0.000 1.000000 0.000

1

*END

*COMPONENT

$# clid color1 color2 color3 color4

1 0.769000 0.004000 0.110000 0.000 0 0

0

$# name

Part 1

*COMPONENT_PART

$# pid clid

1 1

$# pid clid

2 1

*COMPONENT_END

56

Appendix B

Particle generation script, call.m and node_element_gen.m

% CALL

close all

clc

clear all

format long

m=8e-6;

z=0;

tc=0;

rc=0;

NIDs=1;

% WALLS

dx=0.001;

PID=1;

%Bottom

P1=[-0.002 0]; P2=[-0.002 -0.002]; P3=[2.302 0];

[Nodemax1,nx,ny]=node_element_gen(dx,PID,NIDs,m,P1,P2,P3,z,tc,rc)

%Left

P1=[-0.002 0.26]; P2=[-0.002 0.001]; P3=[0 0.26];

[Nodemax2,nx,ny]=node_element_gen(dx,PID,Nodemax1,m,P1,P2,P3,z,tc,

rc)

%Gate

P1=[1.2 0.26]; P2=[1.2 0.001]; P3=[1.202 0.26];

57


rc)

%Wier_1

P1=[2.3 0.02]; P2=[2.3 0.001]; P3=[2.302 0.02];


rc)

%Wier_2

P1=[2.303 0.02]; P2=[2.303 0.018]; P3=[2.5 0.02];


rc)

% Water

dx=0.002;

PID=2;

% Water1

P1=[0.002 0.2]; P2=[0.002 0.002]; P3=[1.198 0.2];


rc)

% Water3

P1=[1.204 0.02]; P2=[1.204 0.002]; P3=[2.298 0.02];


rc)

fidE = fopen('ELEMENT.k', 'a'); fprintf(fidE,'*END');

fclose(fidE);

fidN = fopen('NODE.k', 'a'); fprintf(fidN,'*END'); fclose(fidN);

58

function

[Nodemax,nx,ny]=node_element_gen(dx,PID,NIDs,m,P1,P2,P3,z,tc,rc)

% Node and element generation

dy=dx;

% Node generation

xlen=abs(P3(1)-P1(1));

nx=round(xlen/dx);

if ceil(nx)~=floor(nx), error('nx not an integer'),end

ylen=abs(P2(2)-P1(2));

ny=round(ylen/dy);

if ceil(ny)~=floor(ny), error('ny not an integer'),end

x(1)=P1(1);

y(1)=P1(2);

for k=2:nx+1

x(k)=x(k-1)+dx;

end

for l=2:ny+1

y(l)=y(l-1)-dy;

end

NID=NIDs:(NIDs+(nx+1)*(ny+1));

rek=1;

fidN = fopen('NODE.k', 'a');

59

if NIDs==1,fprintf(fidN,'*NODE\n$# nid x

y z tc rc\n');end

fidE = fopen('ELEMENT.k', 'a');

if NIDs==1, fprintf(fidE,'*ELEMENT_SPH\n$# nid pid

mass\n');end

for j=1:length(x)

for n=1:length(y)

NO(rek,:)=[NID(rek) x(j) y(n) z tc rc];

fprintf(fidN, '%8.0f%16.7f%16.7f%16.7f%8.0f%8.0f\n',

NO(rek,1),NO(rek,2),NO(rek,3),NO(rek,4),NO(rek,5),NO(rek,6));

El(rek,:)=[NID(rek),PID,m];

fprintf(fidE, '%8.0d%8.0d%16.7e\n',

El(rek,1),El(rek,2),El(rek,3));

rek=rek+1;

end

end

fclose(fidN);

fclose(fidE);

Nodemax=max(NID);

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