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
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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
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and Deformation of Porous Media. Luleå: Luleå University of Technology.
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Thesis. Mayagüez, Puerto Rico: University of Puerto Rico Mayagüez Campus.
Kundu, P. K. (1990). Fluid Mechanics. San Diego: Academic Press, Inc.
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Lacome, J. L. (2001, November 27). Smoothed Particle Hydrodynamics - Part II. FEA
Information International News For The World-Wide Engineering Community,
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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.
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Livermore Software Technology Corporation (LSTC).
Lucy, L. B. (1977). A numerical approach to testing of the fission hypothesis. The
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Murzyn, F., & Chanson, H. (2008). Experimental investigation of bubbly flow and
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Impact on Rotating Fan: The Influence of Bird Parameters. 11th International
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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
$# lcid sidr sfa sfo offa offo
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
*BOUNDARY_SPC_SET_ID
$# id
heading
2W1
46
$# nsid cid dofx dofy dofz dofrx
dofry dofrz
2 1 0 0 1 0 0
0
*SET_NODE_LIST_GENERATE_TITLE
W1
$# sid da1 da2 da3 da4 solver
2 0.000 0.000 0.000 0.000MECH
$# b1beg b1end b2beg b2end b3beg b3end
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
$# pid secid mid eosid hgid grav
adpopt tmid
2 2 2 2 0 0 0
0
*SECTION_SPH_TITLE
WATER
$# secid cslh hmin hmax sphini death
start
2 1.200000 0.950000 4.000000 0.0001.0000E+20 0.000
*SET_NODE_LIST_GENERATE_TITLE
GATE
$# sid da1 da2 da3 da4 solver
3 0.000 0.000 0.000 0.000MECH
$# b1beg b1end b2beg b2end b3beg b3end
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
*BOUNDARY_SPC_SET_ID
$# id
heading
1WATER
$# nsid cid dofx dofy dofz dofrx
dofry dofrz
1 1 0 0 1 0 0
0
*SET_NODE_LIST_GENERATE_TITLE
WATER
$# sid da1 da2 da3 da4 solver
1 0.000 0.000 0.000 0.000MECH
$# b1beg b1end b2beg b2end b3beg b3end
b4beg b4end
1 30300 0 0 0 0 0
0
*BOUNDARY_SPC_SET_ID
$# id
heading
2WALLS
$# nsid cid dofx dofy dofz dofrx
dofry dofrz
2 1 1 1 1 1 1
1
*SET_NODE_LIST_GENERATE_TITLE
WALLS
$# sid da1 da2 da3 da4 solver
2 0.000 0.000 0.000 0.000MECH
51
$# b1beg b1end b2beg b2end b3beg b3end
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
$# pid secid mid eosid hgid grav
adpopt tmid
1 1 1 1 0 0 0
0
*SECTION_SPH_TITLE
PKT
$# secid cslh hmin hmax sphini death
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
$# pid secid mid eosid hgid grav
adpopt tmid
2 2 1 1 0 0 0
0
*SECTION_SPH_TITLE
PKT
$# secid cslh hmin hmax sphini death
start
2 1.000000 1.000000 1.000000 0.0001.0000E+20 0.000
*SET_NODE_LIST_GENERATE_TITLE
Water 1
$# sid da1 da2 da3 da4 solver
3 0.000 0.000 0.000 0.000MECH
53
$# b1beg b1end b2beg b2end b3beg b3end
b4beg b4end
1 4375 0 0 0 0 0
0
*SET_NODE_LIST_GENERATE_TITLE
Water 2
$# sid da1 da2 da3 da4 solver
4 0.000 0.000 0.000 0.000MECH
$# b1beg b1end b2beg b2end b3beg b3end
b4beg b4end
4376 15100 0 0 0 0 0
0
*SET_NODE_LIST_GENERATE_TITLE
Water 3
$# sid da1 da2 da3 da4 solver
5 0.000 0.000 0.000 0.000MECH
$# b1beg b1end b2beg b2end b3beg b3end
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
$# lcid sidr sfa sfo offa offo
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
$# lcid sidr sfa sfo offa offo
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
$# lcid sidr sfa sfo offa offo
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
*DEFINE_VECTOR_TITLE
vector inflow x_dir
$# vid xt yt zt xh yh
zh cid
55
2 0.000 0.000 0.000 1.000000 0.000 0.000
1
*DEFINE_VECTOR_TITLE
vector inflow y_dir
$# vid xt yt zt xh yh
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
$# pid clid
2 1
*COMPONENT_END
56
Appendix B
Particle generation script, call.m and node_element_gen.m
% CALL
close all
clc
clear all
format long
m=8e-6;
z=0;
tc=0;
rc=0;
NIDs=1;
% WALLS
dx=0.001;
PID=1;
%Bottom
P1=[-0.002 0]; P2=[-0.002 -0.002]; P3=[2.302 0];
[Nodemax1,nx,ny]=node_element_gen(dx,PID,NIDs,m,P1,P2,P3,z,tc,rc)
%Left
P1=[-0.002 0.26]; P2=[-0.002 0.001]; P3=[0 0.26];
[Nodemax2,nx,ny]=node_element_gen(dx,PID,Nodemax1,m,P1,P2,P3,z,tc,
rc)
%Gate
P1=[1.2 0.26]; P2=[1.2 0.001]; P3=[1.202 0.26];
57
[Nodemax3,nx,ny]=node_element_gen(dx,PID,Nodemax2,m,P1,P2,P3,z,tc,
rc)
%Wier_1
P1=[2.3 0.02]; P2=[2.3 0.001]; P3=[2.302 0.02];
[Nodemax4,nx,ny]=node_element_gen(dx,PID,Nodemax3,m,P1,P2,P3,z,tc,
rc)
%Wier_2
P1=[2.303 0.02]; P2=[2.303 0.018]; P3=[2.5 0.02];
[Nodemax5,nx,ny]=node_element_gen(dx,PID,Nodemax4,m,P1,P2,P3,z,tc,
rc)
% Water
dx=0.002;
PID=2;
% Water1
P1=[0.002 0.2]; P2=[0.002 0.002]; P3=[1.198 0.2];
[Nodemax6,nx,ny]=node_element_gen(dx,PID,Nodemax5,m,P1,P2,P3,z,tc,
rc)
% Water3
P1=[1.204 0.02]; P2=[1.204 0.002]; P3=[2.298 0.02];
[Nodemax7,nx,ny]=node_element_gen(dx,PID,Nodemax6,m,P1,P2,P3,z,tc,
rc)
fidE = fopen('ELEMENT.k', 'a'); fprintf(fidE,'*END');
fclose(fidE);
fidN = fopen('NODE.k', 'a'); fprintf(fidN,'*END'); fclose(fidN);
58
function
[Nodemax,nx,ny]=node_element_gen(dx,PID,NIDs,m,P1,P2,P3,z,tc,rc)
% Node and element generation
dy=dx;
% Node generation
xlen=abs(P3(1)-P1(1));
nx=round(xlen/dx);
if ceil(nx)~=floor(nx), error('nx not an integer'),end
ylen=abs(P2(2)-P1(2));
ny=round(ylen/dy);
if ceil(ny)~=floor(ny), error('ny not an integer'),end
x(1)=P1(1);
y(1)=P1(2);
for k=2:nx+1
x(k)=x(k-1)+dx;
end
for l=2:ny+1
y(l)=y(l-1)-dy;
end
NID=NIDs:(NIDs+(nx+1)*(ny+1));
rek=1;
fidN = fopen('NODE.k', 'a');
59
if NIDs==1,fprintf(fidN,'*NODE\n$# 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);