L ICE N T IAT E T H E S I S
ISSN: 1402-1757
ISBN 978-91-7439-678-2 (print)
ISBN 978-91-7439-679-9 (pdf)
Luleå University of Technology 2013
Patrick Jonsson Smoothed Particle Hydrodynamics in Hydropower Applications Modelling of Hydraulic Jumps
Department of Engineering Sciences and Mathematics
Division of Fluid and Experimental Mechanics
Smoothed Particle Hydrodynamics in
Hydropower Applications
Modelling of Hydraulic Jumps
Patrick Jonsson
Smoothed Particle Hydrodynamics in
Hydropower Applications
Modelling of Hydraulic Jumps
Patrick Jonsson
Division of Fluid and Experimental Mechanics
Department of Engineering, Sciences and Mathematics
Luleå University of Technology
SE-971 87 Luleå, Sweden
[email protected]
Luleå, May 2013
Printed by Universitetstryckeriet, Luleå 2013
ISSN: 1402-1757
ISBN 978-91-7439-678-2 (print)
ISBN 978-91-7439-679-9 (pdf)
Luleå 2013
www.ltu.se
PREFACE
This work has been carried out at the Division of Fluid and Experimental Mechanics,
Department of Engineering, Sciences and Mathematics at Luleå University of
Technology during 2010-2013.
The research presented in this thesis was carried out as a part of “Swedish Hydropower
Center -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, Karlstad 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.
I would like to thank my supervisors Prof. Staffan Lundström, adjunct Prof. Patrik
Andreasson (Vattenfall Research and Development) and Dr. Gunnar Hellström for their
guidance and support and, perhaps most significantly, for believing in me.
A special thanks to Dr. Pär Jonsén at the division of Mechanics and Solid Materials for
his encouragement and support in the intricate field of Smoothed Particle
Hydrodynamics.
I also want to thank all of the colleagues at the University who contributes to a creative
and stimulating workplace and, especially, to my office companion Simon Johansson for
the fun and inspiring working environment.
Finally, I want to express my deepest appreciation to my family and all of my friends,
without you, it would never have been possible.
Patrick Jonsson, Luleå, May 2013.
i
ii
SUMMARY OF CONTENTS
In present thesis, the Lagrangian particle based method Smoothed Particle
Hydrodynamics (SPH) is used to model two-dimensional problems associated with
hydropower applications such as dam break evolution and hydraulic jumps. In the SPHmethod, the fluid domain is represented by a set of non-connected particles which possess
individual material properties such as mass, density, velocity, position and pressure.
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
neighbouring particles. The adaptive nature of the SPH-method together with the nonconnectivity between the particles results in a method that is able to handle very large
deformations as is the case for highly disordered free-surface flows such as hydraulic
jumps.
The dam break case was used as a model validation test case where the response of
different parameter settings was explored. The SPH spatial resolution and the choice of
artificial viscosity (a term in the momentum equation) constants had a major impact on
the results. Increasing the spatial resolution increased the number of flow features
resolved and setting the constants equal to unity resulted in a highly viscous and
unphysical solution.
Following the parameter study, the work focused on SPH simulations of hydraulic jumps.
A hydraulic jump is a rapid transition from supercritical flow to subcritical flow
characterized by the development of large scale turbulence, surface waves, spray, energy
dissipation and considerable air entrainment. Several features of the jump were explored
using the SPH method and good agreement with theory and experiments was obtained for
e.g. the conjugate depth and the mean free surface elevation in the roller section.
However, the free surface fluctuation frequencies were over predicted and the model
could not capture the decay of fluctuations in the horizontal direction.
iii
iv
APPENDED PAPERS
Paper A
Jonsson, Patrick; Jonsén, Pär; Andreasson, Patrik; Lundström, T. Staffan; Hellström, J.
Gunnar I. (2011), Smoothed Particle Hydrodynamics Modelling of Hydraulic Jumps, II
International Conference on Particle-based Methods – Fundamentals and Applications
(PARTICLES 2011), Barcelona, Spain.
Paper B
Jonsson, Patrick; Jonsén, Pär; Andreasson, Patrik; Lundström, T. Staffan; Hellström, J.
Gunnar I. (2013), Modelling Dam Break Evolution Over a Wet Bed with Smoothed
Particle Hydrodynamics: A Parameter Study, Manuscript.
Paper C
Jonsson, Patrick; Jonsén, Pär; Andreasson, Patrik; Lundström, T. Staffan; Hellström, J.
Gunnar I. (2013), Smoothed Particle Hydrodynamics Modelling of Hydraulic Jumps:
Bulk Parameters and Free Surface Fluctuations, Manuscript.
v
vi
PAPER ABSTRACTS
Paper A
This study focus on Smoothed Particle Hydrodynamics (SPH) modelling of twodimensional hydraulic jumps in horizontal open channel flows. Insights to the complex
dynamics of hydraulic jumps in a generalized test case serves as a knowledgebase for real
world applications such as spillway channel flows in hydropower systems. In spillways,
the strong energy dissipative mechanism associated with hydraulic jumps is a utilized
feature to reduce negative effects of erosion to spillway channel banks and in the old river
bed. The SPH-method with its mesh-free Lagrangian formulation and adaptive nature
results in a method that handles extremely large deformations and numerous publications
using the SPH-method for free-surface flow computations can be found in the literature.
Hence, the main objectives with this work are to explore the SPH-methods capabilities to
accurately capture the main features of a hydraulic jump and to investigate the influence
of the number of particles that represent the system. The geometrical setup consists of an
inlet which discharges to a horizontal plane with an attached weir close to the outlet. To
investigate the influence of the number of particles that represents the system, three initial
interparticle distances were studied, coarse, mid and fine. For all cases it is shown that the
SPH-method accurately captures the main features of a hydraulic jump such as the
transition between supercritical- and subcritical flow and the dynamics of the highly
turbulent roller and the air entrapment process. The latter was captured even though a
single phase was modelled only. Comparison of theoretically derived values and
numerical results show good agreement for the coarse and mid cases. However, the fine
case show oscillating tendencies which might be due to inherent numerical instabilities of
the SPH-method or it might show a more physically correct solution. Further validation
with experimental results is needed to clarify these issues.
Paper B
When investigating water flow in spillways and energy dissipation, it is important to
know the behaviour of the free surfaces. To capture the real dynamic behaviour of the
free surfaces is therefore crucial when performing simulations. Today, there is a lack in
the possibility to model such phenomenon with traditional methods. Hence, this work
focuses on a parameter study for one alternative simulation tool available, namely the
meshfree, Lagrangian particle method Smoothed Particle Hydrodynamics (SPH). The
parameter study includes the choice of equation-of-state (EOS), the artificial viscosity
constants, using a dynamic versus a static smoothing length, SPH particle spatial
resolution and the finite element method (FEM) mesh scaling of the boundaries. The two
vii
dimensional SPHERIC Benchmark test case of dam break evolution over a wet bed was
used for comparison and validation. The numerical results generally showed a tendency
of the wave front to be ahead of the experimental results, i.e. to have a greater wave front
velocity. The choice of EOS, FEM mesh scaling as well as using a dynamic or a static
smoothing length showed little or no significant effect on the outcome. The SPH particle
resolution and the choice of artificial viscosity constants had a major impact though. A
high particle resolution increased the number of flow features resolved for both choices of
artificial viscosity constants but on the expense of increasing the mean error.
Furthermore, setting the artificial viscosity constants equal to unity for the coarser cases
resulted in a highly viscous and unphysical solution and thus the relation between the
artificial viscosity constants and the particle resolution and its impact on the behaviour of
the fluid need to be further investigated. In addition to the investigation of the above
parameters, a new post processing method for tracing the free surface and interpolate
SPH data is proposed also.
Paper C
A hydraulic jump is a rapid transition from supercritical flow to subcritical flow
characterized by the development of large scale turbulence, surface waves, spray, energy
dissipation and considerable air entrainment. Hydraulic jumps can be found in waterways
such as spillways connected to hydropower plants and are an effective way to eliminate
problems caused by high velocity flow, e.g. erosion. Due to the importance of the
hydropower sector as a major contributor to the Swedish electricity production, the
present study focuses on Smoothed Particle Hydrodynamic modelling of 2D hydraulic
jumps in horizontal open channels. Four cases with different spatial resolution of the SPH
particles were investigated by comparing the conjugate depth in the subcritical section
with theoretical results. These showed good agreement and an increased oscillation
tendency for the most resolved case. The coarsest case was run for a longer time than the
others due to computational time limitations and a quasi-stationary state was achieved.
Thus for the coarsest case, additional variables were explored. The mean vertical velocity
distribution in the horizontal direction compared favourably with experiments and the
maximum velocity for the SPH-simulations indicated a too rapid decrease in the
horizontal direction and poor agreement to experiments was obtained. Furthermore, the
mean and the standard deviation of the free surface fluctuation showed generally good
agreement with experimental results even though some discrepancies were found
regarding the peak in the maximum standard deviation. The free surface fluctuation
frequencies were over predicted and the model could not capture the decay of the
fluctuations in the horizontal direction.
viii
DIVISION OF WORK
Paper A
“Smoothed Particle Hydrodynamics Modelling of Hydraulic Jumps”
Jonsson P., Jonsén P., Andreasson P., Lundström T. S., Hellström J. G. I.
All simulations and analysis was performed by Jonsson under the supervision of Jonsén,
Andreasson, Lundström and Hellström. The paper was written by Jonsson, Jonsén,
Andreasson, Lundström and Hellström.
Paper B
“Modelling Dam Break Evolution Over a Wet Bed with Smoothed Particle
Hydrodynamics: A Parameter Study”
Jonsson P., Jonsén P., Andreasson P., Lundström T. S., Hellström J. G. I.
All simulations and analysis was performed by Jonsson under the supervision of Jonsén,
Andreasson, Lundström and Hellström. The paper was written by Jonsson, Jonsén,
Andreasson, Lundström and Hellström.
Paper C
“Smoothed Particle Hydrodynamics Modelling of Hydraulic Jumps: Bulk Parameters and
Free Surface Fluctuations”
Jonsson P., Jonsén P., Andreasson P., Lundström T. S., Hellström J. G. I.
All simulations and analysis was performed by Jonsson under the supervision of Jonsén,
Andreasson, Lundström and Hellström. The paper was written by Jonsson, Jonsén,
Andreasson, Lundström and Hellström.
ix
Paper A
Smoothed Particle Hydrodynamics Modelling of
Hydraulic jumps
SMOOTHED PARTICLE HYDRODYNAMICS MODELLING OF
HYDRAULIC JUMPS
*
†
*
PATRICK JONSSON , PÄR JONSÉN , PATRIK ANDREASSON , T. STAFFAN
*
*
LUNDSTRÖM AND J. GUNNAR I. HELLSTRÖM
*
Division of Fluid and Experimental Mechanics
Luleå University of Technology (LTU)
SE-971 87, Luleå, Sweden
e-mail: [email protected], www.ltu.se
†
Division of Mechanics of Solid Materials
Luleå University of Technology (LTU)
SE-971 87, Luleå, Sweden
e-mail: [email protected], www.ltu.se
Key words: hydraulic jump, smoothed particle hydrodynamics, sph, particle study.
Abstract. This study focus on Smoothed Particle Hydrodynamics (SPH) modelling of
two-dimensional hydraulic jumps in horizontal open channel flows. Insights to the
complex dynamics of hydraulic jumps in a generalized test case serves as a
knowledgebase for real world applications such as spillway channel flows in hydropower
systems. In spillways, the strong energy dissipative mechanism associated with hydraulic
jumps is a utilized feature to reduce negative effects of erosion to spillway channel banks
and in the old river bed. The SPH-method with its mesh-free Lagrangian formulation and
adaptive nature results in a method that handles extremely large deformations and
numerous publications using the SPH-method for free-surface flow computations can be
found in the literature. Hence, the main objectives with this work are to explore the SPHmethods capabilities to accurately capture the main features of a hydraulic jump and to
investigate the influence of the number of particles that represent the system. The
geometrical setup consists of an inlet which discharges to a horizontal plane with an
attached weir close to the outlet. To investigate the influence of the number of particles
that represents the system, three initial interparticle distances were studied, coarse, mid
and fine. For all cases it is shown that the SPH-method accurately captures the main
features of a hydraulic jump such as the transition between supercritical- and subcritical
flow and the dynamics of the highly turbulent roller and the air entrapment process. The
latter was captured even though a single phase was modelled only. Comparison of
theoretically derived values and numerical results show good agreement for the coarse
and mid cases. However, the fine case show oscillating tendencies which might be due to
inherent numerical instabilities of the SPH-method or it might show a more physically
correct solution. Further validation with experimental results is needed to clarify these
issues.
1
1 INTRODUCTION
The main components of a large scale hydropower station are a large reservoir dam where
rain and melt water is stored and the turbine/generator assembly which converts the
potential energy stored in the water into electricity1. The hydropower sector contributes to
almost half of the total power production in Sweden2. However, the overall contribution
from the hydropower sector varies throughout the year due to variation in energy demand
from the consumers which in turn is due to seasonal variation in temperature and
maintenance shutdown of industrial processes. As production varies throughout the year
and the amount of precipitation is uncontrollable the need to find ways to handle the
water head in the reservoir is crucial to obtain optimal working conditions. If the inherent
flow regulation by generation is insufficient, spillways are engaged. When spillways are
used the potential energy of the water is converted to kinetic energy potentially causing
erosion problems to structures in the spillway channel as well as in other water-ways
downstream the spillway like old river beds. An effective way to reduce the high kinetic
energy levels is to design the spillway channels to trigger a hydraulic jump which is a
natural occurring phenomenon in free flowing fluids characterized by large energy
dissipation mechanisms3. The main feature of a hydraulic jump is a sudden transition of
shallow and fast moving flow into a relatively slow moving flow with rise of the fluid
surface to keep continuity. The transition phase is known as the roller where the free
surface is highly disturbed and air entrapment occurs.
Modelling of highly disturbed free surface flows such as the hydraulic jump is complex
when grid based method is used. Severe problems with mesh entanglement and
determination of the free surface have been encountered4. Meshfree methods such as the
Smooth Particle Hydrodynamic (SPH) method have been shown to be a good alternative
to grid based methods to overcome the above stated problems5,6. The SPH-method has
matured rapidly during the last decade or even years and was thus chosen as the
computational method.
The aim with this work is thus to explore the capability of the SPH-methods to accurately
capture the main features of a hydraulic jump and to investigate the influence of the
number of particles that represent the system.
2 METHOD
2.1 Smoothed particle hydrodynamics
Smoothed Particle Hydrodynamics (SPH) is a meshfree, adaptive, Lagrangian particle
method for modelling fluid flow. The technique was first invented independently by
Lucy7 and Gingold and Monaghan8 in the late seventies to solve astrophysical problems
in three-dimensional open space. Movement of astronomical particles resembles the
2
motion of fluids, thus it can be modelled by the governing equations of classical
Newtonian hydrodynamics. The method did not attract much consideration 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).
In the SPH-method, the fluid domain is represented by a set of non-connected particles
which possess individual material properties such as mass, density, velocity, position and
pressure9. 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 neighbouring particles, hence making the technique a pure adaptive, mesh-free
Lagrangian method. With adaptive is meant that at each time step the field approximation
is done based on the local distribution of neighbouring 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 as is the case for highly disordered
free-surface flows such as hydraulic jumps.
To further clarify the methodology of the SPH-method, the following key steps is
employed to reduce the partial differential equation (PDEs) governing the problem at
hand to a set of ordinary differential equations (ODEs).
1. The problem domain is represented by a set of non-connected particles.
2. The integral representation method is used for field function approximation,
known as 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 neighbouring particles in the so
called support domain.
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.
A more detailed description of the methodology and the kernel- and particle
approximations can be found in the textbook by Liu and Liu9.
3
As concluded in previous work10,11 the SPH-method do not necessarily behave in the
same manner as mesh-based computational methods where refinement of the mesh is
anticipated to yield better solutions of the PDEs. Instead, further refinement or more
particles in the SPH-method might lead to deterioration of solutions or even divergent
behaviour. This may be caused by inherent numerical instabilities of the SPH-method.
2.2 Hydraulic jump
The main characteristic of a hydraulic jump is the sudden transition of rapid shallow flow
to slow moving flow with rise of the fluid surface also known as a transition from
supercritical to subcritical flow3. The transition is strongly dissipative which is favourable
when energy should be consumed as when kinetic energy levels should be reduced in
spillway flows. Further characteristics of hydraulic jumps is the development of a largescale highly turbulent zone known as the “roller” with surface waves and spray, energy
dissipation and air entrapment.
As stated above the hydraulic jump is characterized by a supercritical and a subcritical
region where the depths are significantly different. These depths ݀ଵ and ݀ଶ are referred to
as conjugate depths and can be seen in the schematic Figure 1.
Fluid
surface
Wall boundary
Roller
݀ଶ
‫ݒ‬ଵ ݀ଷ ݀ଵ Figure 1: Schematic figure of the hydraulic jump showing the conjugate depths ࢊ૚ and ࢊ૛ and
the depth ࢊ૜ past the weir.
A dimensionless relation between the conjugate depths can easily be derived from
continuity, momentum and energy equations for a rectangular channel by assuming
hydrostatic pressure distribution and uniform velocity distribution at the up- and
downstream end of the control volume. Furthermore, the friction between the bottom and
the fluid and the slope of the bottom is both assumed to be zero. With these assumptions
4
the conservations equations yield the dimensionless relation between the conjugate depths
for a rectangular channel as,
݀ଶ ͳ
ൌ ቆටͳ ൅ ͺ‫ݎܨ‬ଵଶ െ ͳቇ
݀ଵ ʹ
(1)
where ‫ݎܨ‬ଵ is the upstream Froude number,
‫ݎܨ‬ଵ ൌ
‫ݒ‬ଵ
(2)
ඥ݃݀ଵ
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 types of hydraulic
jumps3. Furthermore, the depth ݀ଷ is derived in a similar manner with the same
assumptions as above, further details can be found in the textbook by Chanson3.
2.3 Numerical setup
To reduce the complexity of modelling a three dimensional spillway channel with
adherent hydraulic jump a two dimensional, horizontal and single phase (water) model is
studied here. Following previous work12,13 the material model MAT_NULL is used to
model water with density ߩ ൌ ͳͲͲͲ ݇݃Τ݉ଷ and dynamic viscosity ߤ ൌ ͲǤͲͲͳͷܲܽ‫ݏ‬.
The null material has no shear stiffness or yield strength and behaves in a fluid-like
manner14. As the dynamic viscosity ߤ is nonzero, a deviatoric viscous stress of the form,
ᇱሶ
ߪ௜௝ᇱ ൌ ߤߝపఫ
(3)
ᇱሶ
is computed where ߝపఫ
is the deviatoric strain rate15. Furthermore, the null material must
also be used together with an equation of state (EOS) defining the pressure in the
material. Varas et al.13 used the Gruneisen equation of state which employs the cubic
shock velocity-particle velocity, which also was used in this work. The wall boundaries
were modelled as rigid shell finite elements and the interaction between the boundaries
and the SPH-particles was governed by a penalty based node to surface contactalgorithm.
The geometrical setup of the problem can be seen in Figure 2 where the fluid enters the
domain using the BOUNDARY_SPH_FLOW inlet condition with an prescribed inlet
velocity of ‫ݒ‬ଵ ൌ ͳǤͷ ݉Τ‫ ݏ‬and depth ݀ଵ ൌ ͲǤͲʹ݉. The horizontal plane situated between
5
the inlet and the weir measures ͳǤͳ݉ and is prefilled to a depth equal to weir height, i.e.
݀ଵ . Both the prefill of the horizontal plane and the weir assembly was introduced to
trigger the hydraulic jump faster. Furthermore, at ͳǤͻ݉ downstream of the inlet the outlet
is situated.
Fluid surface
Wall boundary
‫ݒ‬ଵ
݀ଵ
Weir
Horizontal plane
Figure 2: Geometrical setup at ࢚ ൌ ૙࢙.
Initial results indicated that after an initial transient phase of roughly ͵ǤͲ‫ ݏ‬the hydraulic
jump was fully developed. However, the roller region was non-stationary even after the
initial transient phase and travelled upstream toward the gate with very low velocity,
hence all subsequent simulation were run for five seconds to include such phenomena.
Three different initial interparticle spacing ο݈ሺο݈ ൌ ο‫ ݔ‬ൌ ο‫ݕ‬ሻ of ͲǤͲͲͷ݉ሺܿ‫݁ݏݎܽ݋‬ሻ,
ͲǤͲͲͶ݉ሺ݉݅݀ሻ, and ͲǤͲͲʹ݉ሺ݂݅݊݁ሻ were used to investigate the influence of the
number of particles that represent the system. The total number of particles and the
overall computational time for each case is summarized in Table 1.
Table 1: The total number of particles and the overall computational time for each case.
Case
Total number of particles
Computational time ሾࢎሿ
‫݁ݏݎܽ݋ܥ‬ሺο݈ ൌ ͲǤͲͲͷ݉ሻ
͸ͺͺͲ
ͳͻ ሺͳʹ ܿ‫ݏ݁ݎ݋‬ሻ
‫݀݅ܯ‬ሺο݈ ൌ ͲǤͲͲͶ݉ሻ
ͳͲ͹ͷͲ
ʹͻ ሺͺ ܿ‫ݏ݁ݎ݋‬ሻ
‫݁݊݅ܨ‬ሺο݈ ൌ ͲǤͲͲʹ݉ሻ
Ͷ͵ͲͲͲ
ͳͺ͸ ሺͳʹ ܿ‫ݏ݁ݎ݋‬ሻ
All simulations where done using the commercial available software package LSTC LSDYNA v. 971 R5.1.1 on HP Z600 Linux machines with eight to twelve cores.
6
3 RESULTS AND DISCUSSION
This part will start with a qualitative comparison of the three cases at successive time
steps and end with a quantitative comparison of theoretically derived values and results
from the simulations.
At ‫ ݐ‬ൌ ͲǤͲ‫ ݏ‬the fast incoming fluid begins to flow into the initially stationary fluid which
starts to move together downstream, i.e. in the positive x-direction. A wave forms and
breaks as more water flows onto the horizontal plane and at roughly ͳǤͲ‫ ݏ‬it reaches the
weir and starts to spill over. The fluid reaches the outlet at roughly ͳǤͷ‫ ݏ‬and in the
meantime the roller region has moved closer to the weir. The roller continues to move
downstream towards the weir until roughly ͵ǤͲ‫ ݏ‬when the velocity of the roller declines
rapidly and change direction. Past ͵ǤͲ‫ ݏ‬a quasi-stationary state is attained as the velocity
of roller upstream is very low. Figures 3-5 shows the three cases using the above stated
number of particles with colour coded velocities in the positive x-direction.
7
Figure 3: Visualization of the coarse ሺο࢒ ൌ ૙Ǥ ૙૙૞࢓ሻcase at successive time steps with colour
coded velocities in the positive x-direction.
8
Figure 4: Visualization of the mid ሺο࢒ ൌ ૙Ǥ ૙૙૝࢓ሻ case at successive time steps with colour
coded velocities in the positive x-direction.
9
Figure 5: Visualization of the fine ሺο࢒ ൌ ૙Ǥ ૙૙૛࢓ሻ case at successive time steps with color
coded velocities in the positive x-direction.
Comparing the results in Figures 3-5 in a qualitative manner both the coarser cases
produces smooth hydraulic jumps with well-defined free-surfaces especially past the
roller region. The fine case behaves in a more chaotic manner with large oscillation of the
free-surface which is clearly seen in the subcritical region past the roller. However, all
three cases capture the main features of a hydraulic jump such as the high velocity small
depth supercritical- and low velocity large depth subcritical region. Furthermore, the
highly turbulent roller where air entrapment occurs is clearly visible in all cases even
though a single phase has been modelled only.
10
As stated above, the hydraulic jump is fully developed at roughly ͵ǤͲ‫ ݏ‬hence all data
from the simulations was obtained past this time at intervals of ͲǤͷ‫ݏ‬. In this work, the
depth ݀ଶ in the subcritical section and the depth ݀ଷ in the contraction past the weir have
been evaluated only. However, as can be seen in Figures 3-5 the velocity field can be
obtained as well as other quantities such as pressure distribution and acceleration of
individual particles. To determine the position of the free-surface and hence the depths ݀ଶ
and ݀ଷ , the average value of the y-coordinate of particles located at the assumed freesurface with half the initial interparticle distance added was used. Particles in the interval
of ሾͳǤͲͳǤͳሿ and ሾͳǤͷͳǤ͸ሿ in the x-direction was used to determine ݀ଶ and ݀ଷ respectively.
With the present inlet condition ‫ݒ‬ଵ ൌ ͳǤͷ ݉Τ‫ ݏ‬and depth ݀ଵ ൌ ͲǤͲʹ݉, ‫ ݎܨ‬ൌ ͵ǤͶ and
consequently ݀ଶ ൌ ͲǤͲͺ͸݉ and ݀ଷ ൌ ͲǤͲͷͺ݉. Results obtained from the simulations of
the depths ݀ଶ and ݀ଷ are summarized in Tables 2-3.
Table 2: The theoretically derived depth ࢊ૛ and numerical results obtained for the three cases
coarse, mid and fine.
࢚࢏࢓ࢋሾ࢙ሿ
૜Ǥ ૙
૜Ǥ ૞
૝Ǥ ૙
૝Ǥ ૞
૞Ǥ ૙
ࢊ૛ǡ࢚ࢎࢋ࢕࢘࢟ ሾ࢓ሿ
ͲǤͲͺ͸
ͲǤͲͺ͸
ͲǤͲͺ͸
ͲǤͲͺ͸
ͲǤͲͺ͸
ࢊ૛ǡࢉ ሾ࢓ሿ
ͲǤͲͺ͵
ͲǤͲͺʹ
ͲǤͲͺ͵
ͲǤͲͺͶ
ͲǤͲͺͳ
ࢊ૛ǡ࢓ ሾ࢓ሿ
ͲǤͲͺ͹
ͲǤͲͺʹ
ͲǤͲͺ͵
ͲǤͲͺ͵
ͲǤͲͺ͸
ࢊ૛ǡࢌ ሾ࢓ሿ
ͲǤͲ͸͹
ͲǤͲͺͺ
ͲǤͲͺ͵
ͲǤͲͺʹ
ͲǤͲͺͺ
Table 3: The theoretically derived depth ࢊ૜ and numerical results obtained for the three cases
coarse, mid and fine.
࢚࢏࢓ࢋሾ࢙ሿ
૜Ǥ ૙
૜Ǥ ૞
૝Ǥ ૙
૝Ǥ ૞
૞Ǥ ૙
ࢊ૜ǡ࢚ࢎࢋ࢕࢘࢟ ሾ࢓ሿ
ͲǤͲͷͺ
ͲǤͲͷͺ
ͲǤͲͷͺ
ͲǤͲͷͺ
ͲǤͲͷͺ
ࢊ૜ǡࢉ ሾ࢓ሿ
ͲǤͲ͸Ͳ
ͲǤͲͷ͸
ͲǤͲͷͺ
ͲǤͲͷͺ
ͲǤͲͷͺ
ࢊ૜ǡ࢓ ሾ࢓ሿ
ͲǤͲͷ͹
ͲǤͲͷͻ
ͲǤͲ͸ͳ
ͲǤͲͷͺ
ͲǤͲͷͺ
ࢊ૜ǡࢌ ሾ࢓ሿ
ͲǤͲ͸Ͳ
ͲǤͲͷͷ
ͲǤͲͷ͵
ͲǤͲ͸ͳ
ͲǤͲͷʹ
As derived from Tables 2-3 both the coarse and mid cases agrees well with the
theoretically derived values of both depths. However, the depths for the fine case
oscillates heavily compared to other two. This chaotic behaviour might be due to inherent
numerical instabilities of the SPH-method satisfying the conclusions of previous
works10,11. However, it might be the most physically correct solution implying that the
threshold of refinement when a divergent behaviour is obtained is not reached or it might
not be applicable to the current problem. Further validation through experiments is
needed in order to confirm which of the above statements is correct. Experiments of
current or similar geometrical setup can be found in the literature16 hence such analysis
has great potential for future work efforts.
11
As mentioned above, the pressure distribution is easily obtained from simulation data.
However, as the pressure was greatly over predicted by the Gruneisen EOS, no such
figure is shown. This behaviour is at this stage not known to the authors but the validity
of the Gruneisen EOS and the parameters used for current problem might be
questionable.
5 CONCLUSIONS
The capabilities of the SPH-method to accurately capture the main features of a hydraulic
jump such as the transition between supercritical- and subcritical flow has been
demonstrated. Furthermore, the dynamics of the highly turbulent roller and the air
entrapment process has been visualized even though a single phase has been modelled
only. Comparison of theoretically derived values and numerical results of the depth ݀ଶ in
the subcritical section and the depth ݀ଷ in the contraction past the weir show good
agreement for the coarse and mid cases. The fine case show oscillating tendencies which
might be due to inherent numerical instabilities of the SPH-method or it might show a
more physically correct solution. Further validation through experiment is needed to
clarify these issues.
6 ACKNOWLEDGEMENT
The research presented 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. www.svc.nu
REFERENCES
[1] Hellström, J. G. I. (2009). Internal Erosion in Embankment Dams - Fluid Flow
Through and Deformation of Porous Media. Luleå: Luleå University of
Technology.
[2] Sandberg, E. (2009, June 3). Svensk energi. Retrieved January 30, 2011, from
http://www.svenskenergi.se/sv/Om-el/Elproduktion/
[3] Chanson, H. (2004). The Hydraulics of Open Channel Flow: An Introduction.
Oxford: Elsevier Ltd.
[4] Scardovelli, R., & Zaleski, S. (1999). Direct numerical simulation of free-surface and
interfacial flow. Annual Review of Fluid Mechanics, 567-603.
[5] 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.
[6] 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.
[7] Lucy, L. B. (1977). A numerical approach to testing of the fission hypothesis. The
Astronomical Journal, 1013-1024.
12
[8] 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.
[9] Liu, G. R., & Liu, M. B. (2009). Smoothed Particle Hydrodynamics : a meshfree
particle method. Singapore: World Publishing Co. Pte. Ltd.
[10] Quinlan, N. J., Basa, M., & Lastiwka, M. (2006). Truncation error in mesh-free
methods. International Journal for Numerical Methos in Engineering, 66:20642085.
[11] Graham, D. I., & Hughes, J. P. (2008). Accuracy of SPH viscous flow models.
International Journal for Numerical Methods in Fluids, 56:1261-1269.
[12] Vesenjak, M., Müllerschön, H., Hummel, A., & Ren, Z. (2004). Simulation of Fuel
Sloshing - Comperative Study. LS-DYNA Anwenderforum. Bamberg: DYNAmore
GmgH.
[13] Varas, D., Zaera, R., & López-Puente, J. (2009). Numerical modeling of
hydrodynamic ram phenomenon. International Journal of impact Engineering 36,
363-374.
[14] LSTC. (2010). LS-DYNA Keyword User´s Manual Version 971 Rev 5. Livermore:
Livermore Software Technology Corporation (LSTC).
[15] Hallquist, J. O. (2006). LS-DYNA Theory Manual. Livermore Software Technology
Corporation.
[16] Lennon, J. M., & Hill, D. F. (2006). Particle Image Velocity Measurements of
Undular and Hydraulic Jumps. Journal of Hydraulic Engineering, 1283-1294.
13
Paper B
Modelling Dam Break Evolution Over a Wet
Bed with Smoothed Particle Hydrodynamics:
A Parameter Study
Modelling Dam Break Evolution Over a Wet Bed with Smoothed Particle
Hydrodynamics: A Parameter Study
P. Jonsson*, P. Jonsén¤, P. Andreasson*, T.S. Lundström* and J.G.I. Hellström*
*
Division of Fluid and Experimental Mechanics
Luleå University of Technology (LTU)
SE-971 87, Luleå, Sweden
e-mail: [email protected], www.ltu.se
¤
Division of Mechanics and Solid Materials
Luleå University of Technology (LTU)
SE-971 87, Luleå, Sweden
Key words: SPH, dam break, post processing
Abstract
When investigating water flow in spillways and energy dissipation, it is important to
know the behaviour of the free surfaces. To capture the real dynamic behaviour of the
free surfaces is therefore crucial when performing simulations. Today, there is a lack in
the possibility to model such phenomenon with traditional methods. Hence, this work
focuses on a parameter study for one alternative simulation tool available, namely the
meshfree, Lagrangian particle method Smoothed Particle Hydrodynamics (SPH). The
parameter study includes the choice of equation-of-state (EOS), the artificial viscosity
constants, using a dynamic versus a static smoothing length, SPH particle spatial
resolution and the finite element method (FEM) mesh scaling of the boundaries. The two
dimensional SPHERIC Benchmark test case of dam break evolution over a wet bed was
used for comparison and validation. The numerical results, generally showed a tendency
of the wave front to be ahead of the experimental results, i.e. to have a greater wave front
velocity. The choice of EOS, FEM mesh scaling as well as using a dynamic or a static
smoothing length showed little or no significant effect on the outcome. The SPH particle
resolution and the choice of artificial viscosity constants had a major impact though. A
high particle resolution increased the number of flow features resolved for both choices of
artificial viscosity constants but on the expense of increasing the mean error.
Furthermore, setting the artificial viscosity constants equal to unity for the coarser cases
resulted in a highly viscous and unphysical solution and thus the relation between the
artificial viscosity constants and the particle resolution and its impact on the behaviour of
the fluid need to be further investigated. In addition to the investigation of the above
parameters, a new post processing method for tracing the free surface and interpolate
SPH data is proposed also.
1
Introduction
In Sweden hydropower plays a significant role for the supply of energy and it generates
roughly 45 % (66.0 TWh in 2011) of the total electricity power production (Svensk
Energi, 2012). The first larger hydropower plant in Sweden commenced its operation for
almost 100 years ago and the latest was put into service during the 1970s. Hence, most of
the hydropower plants in Sweden are old which has initiated a number of projects on
refurbishments.
Due to the deregulated energy market hydropower plants are more and more run at off
design conditions. Consequently, to have an optimal hydropower that, for instance, can
run at the best efficiency point when the winnings are high, the plants must be
modernized and adapted to the energy market. Other key aspects are new requirements on
renewable and “green” energy sources and that it will rain more than previously expected.
This makes the prospect of redesigning old hydropower stations even higher. Finally the
demand on hydropower as a short term regulating source is increasing to take care of
more frequent and larger fluctuations in the energy system, mostly due to the on-going
expansion of wind power.
Regarding the Swedish hydropower sector as a whole focus is set on the service and
administration as well as refurbishments of existing plants, to meet the new demands. To
assess hydraulic properties in these projects, physical scale models are often used.
Performance, accuracy and reliability of mathematical models are in these cases
considered to be too poor. Hydropower hydraulics is characterized by large scales and
flow rates. The geometry is usually partly formed by nature, i.e. to some extent chaotic at
a scale of roughness. Furthermore, highly aerated and disturbed free surface flows are
frequently encountered, for instance, in spillway channel flows and in energy dissipaters.
This constitutes significant mathematical simulation challenges. Thus, there is a need of
computationally robust and reliable numerical methods to handle these fundamentally
complex problems.
Modelling of highly disturbed aerated free surface flows is complex when grid based
method is used (Violeau, 2012). Severe problems with mesh entanglement and
determination of the free surface have been encountered (Scardovelli & Zaleski, 1999).
The meshfree, Lagrangian particle based method Smooth Particle Hydrodynamic (SPH)
has shown to be a good alternative to grid based methods to overcome such problems
(Gomez-Gesteria et al., 2010) and (Monaghan, 2012). The technique was first proposed
independently by (Lucy, 1977) and (Gingold & Monaghan, 1977) in the late seventies to
solve astrophysical problems in a three-dimensional open space. Today, the SPH method
has been applied to a number of fields and problems (Monaghan, 2012) and the maturity
2
of the method has increased significantly. Thus, SPH was chosen as the computational
method in the present work.
In the SPH-method, the fluid domain is represented by a set of non-connected particles
which possess individual material properties such as density, velocity 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 their material properties changes as a function of time
and spatial co-ordinate due to interactions with neighbouring particles. A major
advantage of SPH is that the method is mesh-free, thus considerable time is saved as
compared to methods that need a predefined mesh. However, as the SPH is relatively
unexplored as compared to traditional FVM/FEM methods there are some areas that still
need considerable attention. For instance, wall boundary conditions are generally difficult
to set in SPH as they do not appear in a natural way within the SPH formalism.
Furthermore, the SPH method is typically slower computational wise since the time step
depends on the speed of sound and the explicit integration techniques used.
The idealized dam break problem, where a given volume of initially stationary water is
released onto a dry channel bed by a sudden removal of a gate is a well-studied problem
(see e.g. (Ritter, 1892), (Schoklitsch, 1917) and (Whitham, 1955)). After the gate has
been removed a negative surge propagates upstream and a dam break wave moves rapidly
downstream. If however, the dam break wave propagates over a wet bed (an initial fluid
layer at rest) the flow field become considerably different, see for instance (Stansby et al.,
1998) and (Jánosi et al., 2004). (Jánosi et al., 2004) studied the influence of adding a
polymer in order to reduce the turbulence drag using a dam break setup. However, clean
water test was conducted as well and the experimental results has been used by many
authors as a validation test case such as (Crespo et al., 2008), (Gomez-Gesteira et al.,
2012) and (Lee et al., 2008). The SPH European Research Interest Community
(SPHERIC), ERCOFTAC special interest group for SPH, has adopted the work by
(Jánosi et al., 2004) and digitized wave profiles, wave velocity data and snapshots are
avalible at (wiki.manchester.ac.uk/spheric).
3
The aim of this paper is to study the effects of several test parameters when applied to the
SPHERIC test case based on the work by (Jánosi et al., 2004) of dam break evolution
over a wet bed. The test parameters are;
x
x
x
x
x
choice of equation-of-state
choice of the artificial viscosity constants
choice of using a dynamic versus static smoothing length
choice of SPH particle resolution, and
choice of FEM mesh scaling as thin finite shell elements are used as wall
boundary conditions
Furthermore, a new post processing method to detect the free surface and interpolate SPH
data is presented.
Method
Governing equations
The superscripts ߙ and ߚ are used to denote coordinate directions and the subscripts ݅ and
݆ denotes particle indices. The continuity and momentum equations can be written in the
SPH formalism according to,
ே
݉௝
݀ߩ௜
ൌ ߩ௜ ෍ ࢜௜௝ ‫ܣ‬௜௝
݀‫ݐ‬
ߩ௝
(1)
௝ୀଵ
ே
ߪ ఈǡఉ ൫࢞௝ ൯
݀࢜ఈ
ߪ ఈǡఉ ሺ࢞௜ ሻ
ሺ࢞௜ ሻ ൌ ෍ ݉௝ ቆ
‫ܣ‬௜௝ െ
‫ܣ‬௝௜ ቇ
ߩ௜ ߩ௝
݀‫ݐ‬
ߩ௜ ߩ௝
(2)
௝ୀଵ
where ߩ௜ is the density, ݉௝ is the particle mass and ࢞௜ and ࢜௜௝ ൌ ࢜௜ െ ࢜௝ is the position
and velocity vector respectively. ߪ ఈǡఉ is the total stress tensor and
‫ܣ‬௜௝ ൌ
ͳ
݄ௗ௜௠ାଵ
ฮ࢞௜ െ ࢞௝ ฮ
ߠᇱ ቆ
ቇǤ
݄
(3)
In equation 3, ݀݅݉ is the number of space dimensions and ߠ ᇱ is the derivative of the
function used to define the kernel function ܹ, i.e.
4
ܹሺ࢞ǡ ݄ሻ ൌ
ͳ
ߠሺ࢞ሻ
݄ሺ࢞ሻௗ௜௠
(4)
and ݄ is the smoothing length. The commonly used cubic B-spline kernel is obtained by
choosing ߠ as,
͵
͵
‫ ͳۓ‬െ ܴଶ ൅ ܴଷ
ʹ
Ͷ
ۖ
ߠൌ‫ܥ‬
ͳ
ሺʹ െ ܴሻଷ
‫۔‬
ۖ Ͷ
‫ە‬
Ͳ
ܴ൏ͳ
ͳ൑ܴ൏ʹ
(5)
ܴ൒ʹ
where ‫ ܥ‬is a constant of normalization defined for one-, two- and three dimensional
రఱ
వ
and ‫ܥ‬ଷ஽ ൌ రഏ
. Here, ܴ is the normalized distance between
spaces as; ‫ܥ‬ଵ஽ ൌ యమ, ‫ܥ‬ଶ஽ ൌ భరഏ
two particles, i.e. ܴ ൌ
ฮ࢞೔ ି࢞ೕ ฮ
௛
. The total stress tensor ߪ ఈǡఉ in the momentum equation is
made up from two parts, the isotropic pressure ‫ ݌‬and the deviatoric viscous stress
߬ ఈǡఉ ,according to
ߪ ఈǡఉ ൌ െ‫ ߜ݌‬ఈǡఉ ൅ ߬ ఈǡఉ Ǥ
(6)
The NULL material model will be applied that defines the deviatoric viscous stress as
(LS-DYNA keyword user's manual, 2012),
߬ ఈǡఉ ൌ ߤߝ ఈǡఉ
(7)
where ߝ ఈǡఉ is deviatoric strain rate. By setting the dynamics viscosity ߤ to zeros the total
stress tensor reduces to the isotropic pressure which is defined by an equation-of-state
(EOS). In the present work, three different EOSs found in literature are compared. These
are,
Gruneisen EOS:
The Gruneisen EOS employs the cubic shock velocity-particle velocity and defines the
pressure for compressed materials as (LS-DYNA keyword user's manual, 2012)
ߛ଴
ܽ
ቁ ߤ஽ െ ߤ஽ଶ ቃ
ʹ
ʹ
ሺߛ଴ ൅ ܽߤ஽ ሻ‫ܧ‬଴
‫݌‬௜ ൌ
ଶ൅
ଶ
ߤ஽
ߤ஽ଷ
൤ͳ െ ሺܵଵ െ ͳሻߤ஽ െ ܵଶ
൨
െ ܵଷ
ߤ஽ ൅ ͳ
ሺߤ஽ ൅ ͳሻଶ
ߩ଴ ܿ଴ଶ ߤ஽ ቂͳ ൅ ቀͳ െ
and for expanded materials as,
5
(8)
‫݌‬௜ ൌ ߩ଴ ܿ଴ଶ ߤ஽ ൅ ሺߛ଴ ൅ ܽߤ஽ ሻ‫ܧ‬଴
(9)
where ܵଵ , ܵଶ and ܵଷ are coefficients of the slope of the ‫ݑ‬௦ െ ‫ݑ‬௣ curve, ‫ݑ‬௦ and ‫ݑ‬௣ are the
chock and particle velocity respectively. The variable ܿ଴ is the intercept of the curve
corresponding to the adiabatic speed of sound, ߛ଴ is the Gruneisen gamma, ܽ is the first
volume correction to ߛ଴ , ‫ܧ‬଴ is the initial internal energy and ߤ஽ ൌ ߩ௜ Τߩ଴ െ ͳ where ߩ଴ is
the initial density. Properties and constants for water given by (Boyd et al., 2000) are
applied, see table 1.
Table 1: Gruneisen EOS properties and constants for water (Boyd et al., 2000).
࢓
ࡶ
ࡿ૚
ࡿ૛
ࡿ૜
ࢉ૙ ቂ ࢙ ቃ
ࢽ૙
ࢇ
ࡱ૙ ቂ࢑ࢍቃ
ͳǤͻ͹ͻ
Ͳ
Ͳ
ͳͶͺͶ
ͲǤͳͳ
͵ǤͲ
͵ǤͲ͹ʹ ‫Ͳͳ כ‬ହ
࣋૙ ቂ
࢑ࢍ
ቃ
࢓૜
ͳͲͲͲ
Linear Polynomial EOS:
The Linear Polynomial EOS is defined as (LS-DYNA keyword user's manual, 2012),
‫݌‬௜ ൌ ‫ܥ‬ଵ ൅ ‫ܥ‬ଶ ߤ஽ ൅ ‫ܥ‬ଷ ߤ஽ଶ ൅ ‫ܥ‬ସ ߤ஽ଷ ൅ ሺ‫ܥ‬ହ ൅ ‫ߤ ଺ܥ‬஽ ൅ ‫ߤ ଻ܥ‬஽ଶ ሻ‫ܧ‬Ǥ
(10)
(Johnson and Holzapfel, 2006) and (Selezneva et al., 2010) used the following
expressions for the constants in this equation,
‫ܥ‬ଵ
‫ܥ‬ଶ
‫ܥ‬ଷ
‫ܥ‬ସ
ൌ Ͳ
ൌ ܿ଴ଶ ߩ଴ ൌ ‫ܥ‬ଵ ሺʹ݇ െ ͳሻ
ൌ ‫ܥ‬ଵ ሺ݇ െ ͳሻሺ͵݇ െ ͳሻ
(11)
where ܿ௢ is the adibatic speed of sound and ߤ஽ ൌ ߩ௜ Τߩ଴ െ ͳ, where ߩ଴ is the initial
density. (Selezneva et al., 2010) used ݇ ൌ ʹ and ‫ܥ‬ହ ൌ ‫ ଺ܥ‬ൌ ‫ ଻ܥ‬ൌ Ͳ.
Morris EOS:
The Morris EOS defines the pressure as (Federico et al., 2012)
‫݌‬௜ ൌ ܿ଴ଶ ሺߩ௜ െ ߩ଴ ሻ
(12)
which can be obtained by setting the constants ‫ܥ‬ଵ ൌ ‫ܥ‬ଷ ൌ ‫ܥ‬ସ ൌ ‫ܥ‬ହ ൌ ‫ ଺ܥ‬ൌ ‫ ଻ܥ‬ൌ Ͳ and
‫ܥ‬ଶ ൌ ܿ଴ଶ ߩ଴ in the Linear Polynomial EOS. The response of the above EOSs can be seen in
figure 1.
6
9
5
Equation−of−state response
x 10
EOS Gruneisen
EOS Linear Polynomial
EOS Morris
4
Pressure P [Pa]
3
2
1
0
−1
−2
0.5
0.6
0.7
0.8
0.9
1
ρi/ρ0
1.1
1.2
1.3
1.4
1.5
Figure 1: The response of the Gruneisen, Linear Polynomial and Morris Equation-of-state
when altering the density ratio ࣋࢏ Ȁ࣋૙ .
Artificial Viscosity:
The artificial viscosity proposed by (Monaghan, 1992) is implemented as,
ଶ
െ‫ݍ‬ଵ ܿҧ௜௝ ߶௜௝ ൅ ‫ݍ‬ଶ ߶௜௝
ȫ௜௝ ൌ ቐ
തതതത
ߩ
పఫ
Ͳ
࢜௜௝ ࢘௜௝ ൏ Ͳ
(13)
࢜௜௝ ࢘௜௝ ൐ Ͳ
where
߶௜௝ ൌ
݄࢜௜௝ ࢘௜௝
ǡ
࢘ଶ௜௝ ൅ ߟଶ
(14)
‫ݍ‬ଵ and ‫ݍ‬ଶ are constants and ࢘௜௝ ൌ ࢘௜ െ ࢘௝ and ࢜௜௝ ൌ ࢜௜ െ ࢜௝ are the position and velocity
vector, respectively. Furthermore, ܿҧ௜௝ ൌ
௖೔ ା௖ೕ
ଶ
7
and ߩҧ௜௝ ൌ
ఘ೔ ାఘೕ
ଶ
are the mean speed of
௛೔ ା௛ೕ
sound and density, respectively. If the smoothing length ݄ is dynamic, ݄ ൌ ݄ത௜௝ ൌ
.
ଶ
The ߟଶ ൌ ͲǤͲͳ݄ଶ is used to avoid the denominator to go to zero. According to
(Monaghan, 1992), the linear term associated with ‫ݍ‬ଵ in the artificial viscosity expression
produces both bulk and shear viscosity. The second quadratic term is incorporated to
handle high Mach number shocks and is intended to suppress particle penetrations
(Monaghan, 1992) and (Liu & Liu, 2003). Furthermore, the values of the constants ‫ݍ‬ଵ and
‫ݍ‬ଶ is according to (Crespo et al., 2008) problem dependent. In literature, many authors
suggest setting ‫ݍ‬ଶ ൌ Ͳ and ‫ݍ‬ଵ ൌ ͲǤͲͳ‫Ͳ݋ݐ‬Ǥͳ (Dalrymple & Rogers, 2006), (Monaghan,
1994). However, (Liu & Liu, 2009) states that the values of ‫ݍ‬ଵ and ‫ݍ‬ଶ is usually set equal
to unity. The impact of the choice of the constants will be another parameter to be studied
in the present work.
With the reduction of the total stress tensor, the inclusion of the artificial viscosity term
together with the symmetric properties of the ‫ܣ‬௜௝ ൫‫ܣ‬௜௝ ൌ െ‫ܣ‬௝௜ ൯ term, the momentum
equation reduces to the familiar expression,
ே
ܲ௝
ܲ௜
݀࢜ఈ
ሺ࢞௜ ሻ ൌ െ ෍ ݉௝ ቆ
൅
൅ ȫ௜௝ ቇ ‫ܣ‬௜௝ Ǥ
݀‫ݐ‬
ߩ௜ ߩ௝ ߩ௜ ߩ௝
(15)
௝ୀଵ
A first order time integration scheme is applied and the time step is determined according
to (Hallquist, 2006) as,
ߜ‫ ݐ‬ൌ ‫ܥ‬஼ி௅ ‫݊݅ܯ‬௜ ൬
݄௜
൰
ܿ௜ ൅ ‫ݒ‬௜
(16)
where ‫ܥ‬஼ி௅ is the Courant number usually set to unity. The smoothing length ݄ is either
static or dynamic. For the dynamic case ݄ is obtained from,
݄݀ ͳ
ൌ ݄݀݅‫ݒ‬ሺ‫ݒ‬௜ ሻǤ
݀‫͵ ݐ‬
(17)
For numerical reasons, a minimum and maximum value is imposed on the smoothing
length in the following manner ‫ܪ‬௠௜௡ ݄଴ ൏ ݄ ൏ ‫ܪ‬௠௔௫ ݄଴ . In the present study ‫ܪ‬௠௜௡ and
‫ܪ‬௠௔௫ was set equal to ͲǤͷ and ͳǤͷ respectively while ݄଴ is the initial smoothing length
given by ݄଴ ൌ ͳǤʹ݀଴ , where ݀଴ is the initial distance between particles. The impact on
the results from using a static or dynamic smoothing length will also be studied in the
current study.
8
Boundary Condition
As earlier mentioned, boundary conditions do not appear in a natural way in the SPH
formalism. When a fluid particle comes close to a solid wall boundary it is affected by the
lack of interpolation points outside the boundary. This give rise to a truncated kernel and
thus unwanted and unphysical effects emerge. To overcome these problems, virtual
particles are placed outside the domain of interest to mimic the wall boundary in a proper
way. Three kinds of virtual particles or boundary conditions have been presented in the
literature. The first concept is based on ghost particles (Randles & Libersky, 1996) where
mirror particles are generated outside the boundary when a fluid particle is within a
distance less than the kernel smoothing length. The generated particle has the same
properties as the fluid particle except the velocity which is in the opposite direction
normal to the wall creating a repulsive force. Two major drawbacks have been identified
being the variable number of the boundary particles for each time step and that the
method is difficult to implement for complex geometries (Gomez-Gesteria et al. 2010). In
the second method a central force is applied to fluid particles close to the boundary, the
repulsive particle method (Monaghan, 1994). The form of the force applied is similar to
forces between molecules. For a fluid and boundary particle separated by a distance ‫ ݎ‬the
force per unit mass has the form given by the Lennard-Jones potential. Several
modifications to the repulsive force method is found in literature, see for instance
(Monaghan & Kos, 1999). A third method is called the dynamic particle (Crespo et al.,
2007) and (Gomez-Gesteira et al., 2009) where boundary particles satisfy the same
equations as fluid particles except that their positions are fixed or externally imposed.
Two major drawbacks have been identified for this boundary condition as well. Firstly, as
density decreases locally and hence pressure for particles moving away and separating
from the wall there is a tendency for particle to get stuck at the wall and too large
boundary layers may form. Secondly, fluid particles may penetrate the boundary in an
unphysical way. In the present work yet another method is used where wall boundaries
are modelled as rigid shell finite elements and the coupling between the boundaries and
the SPH-particles are governed by a penalty based “node to surface” contact-algorithm
(Jonsén et al., 2012). This is a so-called one-way contact where the SPH particles are
defined as the “slave side” and the FEM elements as the “master side”. In applying the
penalty method, the slave nodes are checked for penetration through the master surface. If
a slave node has penetrated, an interface spring is placed between the master surface and
the node. The spring stiffness is chosen approximately in the order of magnitude of the
stiffness of the interface element normal to the interface. The resultant force applied to
the SPH particle in the normal direction of the FEM element is proportional to the
amount of penetration. The wall boundaries used here can be compared with the LennardJones-type boundary condition where a central force is applied to a fluid particle
mentioned above. However, the normal component is only considered. Hence, the
9
boundary condition used here is truly frictionless. For more information regarding the
contact algorithm see (LS-DYNA keyword user's manual, 2012).
Geometrical setup
To validate the SPH model experiment, the SPHERIC Benchmark test case of dam break
evolution over a wet bed is used. Numerous authors have adopted this test case for
validation proposes, see e.g. (Crespo et al., 2008), (Gomez-Gesteira et al., 2012) and (Lee
et al., 2008). The test case is based on the experimental work by (Jánosi et al., 2004)
where water initially stored in a tank is released through a gate into a prefilled rectangular
channel. The schematic arrangement and geometry is illustrated in figure 2.
‫ݕ‬
Gate
‫ݔ‬
‫் ܮ‬஽ ‫ܮ‬஼஽
‫் ܮ‬௅
‫ܮ‬஼௅
Figure 2: Schematic arrangement and geometry for the dam break test case.
In the experiment by (Jánosi et al., 2004), several tank and channel depths were
investigated. However, in the present work only one combination of tank and channel
depth were considered, namely the tank depth ‫் ܮ‬஽ ൌ ͲǤͳͷ݉ and channel depth ‫ܮ‬஼஽ ൌ
ͲǤͲͳͺ݉. The tank length ‫் ܮ‬௅ ൌ ͲǤ͵ͺ݉ was set as in the experiment and the channel
length ‫ܮ‬஼௅ ൌ ͳǤ͸ʹ݉ was smaller than in the experiment (‫ܮ‬஼௅ǡா௑௉ ൌ ͻǤͷͷ݉) in order to
reduce the overall number of particle and hence computational time. According to (Jánosi
et al., 2004), a propagating bore develops after the removal of the gate. The stationary
water in the channel resists the oncoming water and, as a consequence, a unstable
”mushroom-like” wave forms and breaks in both forward and backward directions. In the
experiment, the kinetic energy was usually enough to build up several breaking events
until a smooth propagating bore was formed. Two CCD-cameras recorded the events,
thus digitized snapshots, wave profile data and average wave front velocities is available
for comparison. Here, wave profiles are compared only. According to (Violeau, 2012), a
general deferens of approximately ͲǤͲͶ͵‫ ݏ‬is observed between experiments and SPH
simulations, thus this timeshift is accounted for in the result section. The gate velocity
was constant and measured to ͳǤͷ݉Ȁ‫ݏ‬. According to the Benchmark test case instruction,
it is vital to include the gate movement in simulations.
10
Numerical setup
As stated above, the three EOSs including the different values of the artificial viscosity
constants ‫ݍ‬ଵ and ‫ݍ‬ଶ are to be tested and compared with experimental results. Thus, six
cases is set up, see table 2.
Table 2: Test case setup, EOS and artificial viscosity constants.
EOS
Gruneisen
Artificial viscosity constants
‫ݍ‬ଵ ൌ ͲǤͳ ‫ݍ‬ଶ ൌ Ͳ
‫ݍ‬ଵ ൌ ͳ ‫ݍ‬ଶ ൌ ͳ
‫ݍ‬ଵ ൌ ͲǤͳ ‫ݍ‬ଶ ൌ Ͳ
‫ݍ‬ଵ ൌ ͳ ‫ݍ‬ଶ ൌ ͳ
‫ݍ‬ଵ ൌ ͲǤͳ ‫ݍ‬ଶ ൌ Ͳ
‫ݍ‬ଵ ൌ ͳ ‫ݍ‬ଶ ൌ ͳ
Linear Polynomial
Morris
Furthermore, for each case additional parameters were considered. Firstly, the spatial
resolution of the SPH particles which was defined by dividing the channel depth ‫ܮ‬஼஽ ,
here defined as the characteristic length scale of the problem, by Ͷ, ͷ, ͸, and ͳͲ. All SPH
particles were initially placed on a structured grid with zero initial velocity. Secondly, the
scaling of the FEM boundaries was tested where the side of a FEM element was set equal
to ʹ‫ݔ‬, ͳ‫ ݔ‬and ͲǤͷ‫ ݔ‬the initial interparticle spacing. Finally, the dynamic (D) and static (S)
smoothing length was compared. All parameters are summarized in table 3.
Table 3: Additional parameters tested for each case; SPH resolution, FEM scaling and
smoothing length behaviour (dynamic (D) and static (S)).
ࡲࡱࡹȀࡿࡼࡴ
૛࢞
૚࢞
૙Ǥ ૞࢞
ࡸ࡯ࡰ Ȁ૝
ܵͳȀ‫ͳܦ‬
ܵͷȀ‫ܦ‬ͷ
ܵͻȀ‫ͻܦ‬
ࡸ࡯ࡰ Ȁ૞
ܵʹȀ‫ʹܦ‬
ܵ͸Ȁ‫ܦ‬͸
ܵͳͲȀ‫Ͳͳܦ‬
ࡸ࡯ࡰ Ȁ૟
ܵ͵Ȁ‫͵ܦ‬
ܵ͹Ȁ‫ܦ‬͹
ܵͳͳȀ‫ͳͳܦ‬
ࡸ࡯ࡰ Ȁ૚૙
ܵͶȀ‫ܦ‬Ͷ
ܵͺȀ‫ܦ‬ͺ
ܵͳʹȀ‫ʹͳܦ‬
The initial density ߩ଴ of the SPH particles was set equal to ͳͲͲͲ݇݃Ȁ݉ଷ and all
simulations were stopped at ͲǤͶͺͺ‫ ݏ‬due to the time shift proposed by (Violeau, 2012).
As the tank and channel depths was fixed at ͲǤͳͷ݉ and ͲǤͲͳͺ݉ respectively, the total
number of SPH particles depend on the resolution only and ranges from about ͶͷͷͲ to
ʹͺ͸ͲͲ. A total of 144 simulations were conducted using the nonlinear finite element
code LSTC LS-DYNA v. 971 R7.0.0 on multicore Linux workstations.
Post processing
To the author’s knowledge, there is no unified method on representing or visualizing SPH
results. However, the post processing method of (Price, 2007) should be mentioned as
one alternative. Here, we present a new post processing method that incorporates the
detection of the free surface as well as interpolation of data. The method is divided into
11
two steps. Firstly, all nodes representing the boundary must be detected and secondly a
Delaunay triangulation and Barycentric interpolation is performed, for a detailed
description of Delaunay triangulation and Barycentric interpolation see (Berg et al.,
2000), (Ungar, 2010) and (Hellström et al., 2010). The major assumption made here is
that any arbitrary point overlapped by one or more circles (2D) defining the “volume” of
a SPH particle is considered to be situated inside the material (in this case water). The
boundary node identification technique is based on the “ARC”-method presented in
(Dilts, 2000). A boundary node is defined as a node with a non-overlapped part of its
circumference. Thus, the position of the boundary nodes itself or its circumference based
on the particle volume can be used as a definition of the free-surface. The boundary node
identification part is also crucial later during the Delaunay triangulation as “dummy
nodes” are placed on the circumference of SPH particle representing the boundary or free
surface. The Delaunay triangulation creates what is known as a convex hull, i.e. the outer
boundary of the triangulation is always convex. This poses a problem when an arbitrary
free surface is to be detected. Finally, the Barycentric interpolation using the triangulation
onto a structured grid is performed. Traditional techniques for data visualization such as
streamline plotting can then be used when the data is mapped to a structured grid. See
Appendix A for further details.
Apart from data visualization, the post processing method can be used to statistically
quantify the difference between numerical and experimental results. If grid points located
inside the water are given a value of unity and points outside a value of zero for both
numerical and experimental results a measure of the overlap can be defined. By imposing
both the numerical and experimental results on the same grid (ͲǤͶ ൑ ‫ ݔ‬൑ ͳǤͲʹ and
Ͳ ൑ ‫ ݕ‬൑ ͲǤͳͷ with ݀‫ ݔ‬ൌ ݀‫ ݕ‬ൌ ͷ݁ ିହ ݉) the non-overlapped grid points can be identified.
The error is here defined as number of non-overlapped grid points divided by the total
number of points in the experiment for each time step. The mean error is then the average
of all time step errors. Non-overlapped grid points can be seen as an area and hence when
the non-overlapped area goes to zero the mean error goes to zero as well.
Results & Discussion
The post processing method described in the section above was used to visualize the
results. To the left in figure 3, the absolute velocity field from one SPH case together with
the digitized experimental data points (red) of the wave profile for the successive time
steps are shown. Note that the time step given includes the time shift proposed by
(Violeau, 2012). To the right in figure 3, the dummy nodes (red) representing the free
surface together with 35 randomly distributed streamlines (blue) are shown. The data is
12
taken from the D12 case using the Gruneisen EOS with artificial viscosity constants ‫ݍ‬ଵ
and ‫ݍ‬ଶ set equal to unity.
Figure 3: Left – Interpolated absolute velocity field [m/s] together with experimental data (red).
Right - dummy nodes (red) and streamlines (blue).
One of the most significant discrepancies when comparing the experimental and
numerical results is that the wave front obtained from the SPH simulation is ahead of the
experimental wave front, see figure 3. This is evident for all studied cases and especially
evident in the early stages when the “mushroom” jet mentioned by (Jánosi et al., 2004)
starts to form and break. The numerical over prediction of the speed of the wave front is
not isolated to the early stages as the predicted secondary splash in the later stages is
ahead of the experimental results and the splash height is over predicted. One possible
explanation that the SPH wave front is ahead of the experiments is due to the frictionless
contact between the SPH particles and the FEM mesh, i.e. the free-slip boundary
condition. There is a possibility to include a friction force in the contact formulation.
However, the proposed friction formulation available is not applicable for these types of
problems (LS-DYNA keyword user's manual, 2012). (Crespo et al., 2008) used the
dynamic particle boundary condition and obtained good results with no tendency of the
13
SPH wave front to be ahead of the experimental results. It may be possible to implement
the dynamic particle boundary condition without major alteration to the code (LSDYNA). However, no such tests have been conducted in present work.
In order to better refer to the different flow features in the later stages of the dam break
evolution, the domain is divided into four zones, see figure 4.
ZONE 1
ZONE 2
ZONE 3
ZONE 2
ZONE 4
ZONE 3
ZONE 1
ZONE 4
Figure 4: The later stages of the dam break evolution divided into four zones.
Having these zones in mind figures 5 to 8 yields the effects of particle resolution on the
outcome for the three EOSs and artificial constants ‫ݍ‬ଵ and ‫ݍ‬ଶ . Dummy nodes are used to
visualize the free surface and the shaded grey area is the experimental results.
Furthermore, the choice of setting ‫ݍ‬ଵ ൌ ͳ and ‫ݍ‬ଶ ൌ ͳ is shown in blue and ‫ݍ‬ଵ ൌ ͲǤͳ and
‫ݍ‬ଶ ൌ Ͳ in red respectively. Few flow features are visible for the case with lowest
resolution ሺ‫ܮ‬஼஽ ȀͶሻ, see figure 5. The mushroom jet is almost not detected and zone four
is not resolved at all. The simulations seem to behave too viscous to capture the
significant flow features, regarding both the combinations of the artificial constants ‫ݍ‬ଵ
and ‫ݍ‬ଶ . When increasing the resolution to ‫ܮ‬஼஽ Ȁͷ , figure 6, few flow features are seen for
cases with artificial viscosity constants sett equal to unity. For the other choice, more
features are visible such as a minor breaking wave. However, neither the secondary
splash nor zone four is detectible with this resolution. Regarding the cases with resolution
‫ܮ‬஼஽ Ȁ͸ the mushroom jet is more pronounced than in the cases with lower resolution, see
figure 7. Breaking wave and secondary splash as well as the initial stages of zone four are
visible for all EOSs using artificial constants set equal to ‫ݍ‬ଵ ൌ ͲǤͳ and ‫ݍ‬ଶ ൌ Ͳ. For the
cases with finest resolution ሺ‫ܮ‬஼஽ ȀͳͲሻ most of the flow features shown in the experiment
are now visible in the numerical results for both choices of the artificial viscosity
constants, see figure 8. Thus, the resolution of the SPH particles has a major impact on
the outcome.
Generally, a good agreement between simulations and experimental data for zone one
being closest to the dam is shown, see figures 5-8. The bump in the second zone stems
from the mushroom jet that broke down in the upstream direction. Although the bump
often appears in the simulations the horizontal position of the bump is generally poorly
predicted. Cases with low resolution of the SPH particles and artificial constants set equal
14
to unity seems to be unable to resolve the bump as exemplified in figure 5. The third zone
or the secondary splash zone show large differences between the different cases, compare
figure 5 and 8. The final zone, zone four is resolved for ‫ܮ‬஼஽ ȀͳͲ and to some extent in the
‫ܮ‬஼஽ Ȁ͸ cases, see figures 7 and 8. The dynamic versus a static smoothing length as well as
the scaling of the FEM mesh showed no significant effects on the results.
15
ࡸ
Figure 5: Comparison of experimental (grey) and numerical results. Case D1 ( ࡯ࡰ ), Gruneisen,
૝
Linear Polynomial and Morris EOS and artificial viscosity constants; ࢗ૚ ൌ ૚ , ࢗ૛ ൌ ૚ (blue)
and ࢗ૚ ൌ ૙Ǥ ૚,ࢗ૛ ൌ ૙ (red).
16
ࡸ
Figure 6: Comparison of experimental (grey) and numerical results. Case D2 ( ࡯ࡰ ), Gruneisen,
૞
Linear Polynomial and Morris EOS and the artificial viscosity constants; ࢗ૚ ൌ ૚ , ࢗ૛ ൌ ૚
(blue) and ࢗ૚ ൌ ૙Ǥ ૚,ࢗ૛ ൌ ૙ (red).
17
ࡸ
Figure 7: Comparison of experimental (grey) and numerical results. Case D3 ( ࡯ࡰ ), Gruneisen,
૟
Linear Polynomial and Morris EOS and the artificial viscosity constants; ࢗ૚ ൌ ૚ , ࢗ૛ ൌ ૚
(blue) and ࢗ૚ ൌ ૙Ǥ ૚,ࢗ૛ ൌ ૙ (red).
18
ࡸ
Figure 8: Comparison of experimental (grey) and numerical results. Case D4 ( ࡯ࡰ ), Gruneisen,
૚૙
Linear Polynomial and Morris EOS and the artificial viscosity constants; ࢗ૚ ൌ ૚ , ࢗ૛ ൌ ૚
(blue) and ࢗ૚ ൌ ૙Ǥ ૚,ࢗ૛ ൌ ૙ (red).
19
As shown above there is a qualitative agreement between simulations and experiments if
the set-up is fine enough. Using the method proposed in the post-processing section a
quantitative comparison can also be carried out in the form of the mean error that is
summarised for all cases in figure 9.
Mean Error for each case setup
0.21
LCD/4
LCD/5
0.2
LCD/6
LCD/10
Mean Error
0.19
0.18
0.17
=1
=0
q
2
2
=1
1
or
M
ris
or
M
ris
1
q
1
q
=0
.
=1
1
lq
ia
m
no
ly
Po
Li
ne
ar
Po
ar
Li
ne
q
=1
2
2
q
q
.1
=0
1
q
ia
l
m
no
ly
ru
G
G
ru
ne
is
ne
en
is
e
1
n
q
1
q
=0
=1
.1
2
2
q
q
=1
=0
0.15
=0
0.16
Figure 9: The calculated mean error for each EOS and artificial constants setup.
The error for all cases is within ͷΨ of each other and showing a total of roughly ͳͷΨ to
ʹͲΨ discrepancy between experimental and numerical results. Three significant trends
are evident in the above figure. Firstly, the major difference between the six cases is the
choice of artificial viscosity coefficients ‫ݍ‬ଵ and ‫ݍ‬ଶ not the EOS. Selecting ‫ݍ‬ଵ and ‫ݍ‬ଶ
equal to unity seems to produce the better result. However, as can be seen in figure 5 and
6 for the two lower resolutions, this choice of constants results in a highly viscous and
unphysical solution, i.e. the artificial viscosity term is greatly over predicted. For the two
finest resolutions, the contrary seems to be true, i.e. a more dynamic and fluid like
behaviour is obtained. In the authors opinion, the lower error obtained by setting ‫ݍ‬ଵ and
‫ݍ‬ଶ equal to unity is due to the highly viscous solution which reduces the wave front
velocity and hence lowering the mean error. This makes it cumbersome to find a good
relation of spatial resolution and artificial viscosity constants to obtain a trustworthy
solution. Secondly, the results connected to the ‫ܮ‬஼஽ ȀͳͲ cases or the case with the highest
resolution seems to produce the worst results. This is in sharp contrast to the expected
outcome, as generally more nodes in any numerical method should produce a better result
20
and also the qualitative investigation indicated this. One plausible explanation to this
behaviour is that more features are resolved using more particles such as the breaking
wave and zone four. Also, the height of the secondary splash produced mainly in the
‫ܮ‬஼஽ ȀͳͲ-cases are generally over predicted thus increasing the number of none
overlapping grid points, i.e. increasing the mean error, as can be seen in figure 8. Thirdly,
all EOSs produce similar results which are expected as the density differences seen in the
simulations are very small. However, the density differences are not small enough and
unphysical pressure overestimation and large pressure oscillations are obtained
ሺേͳͲ଺ ܲܽሻ, especially close to wall boundaries as well as near the free-surface. It is
likely that the over prediction of pressure is due to the speed of sound ܿ଴ which was set
equal to the physical speed of sound of ͳͶͺͶ݉Ȁ‫ݏ‬. Usually, ܿ଴ is reduced to at least 10
times the expected maximum velocity in the flow (Monaghan, 1994) and (Monaghan &
Kos, 1999). Initial tests with reduced speed of sound ܿ଴ ൌ ͳͲඥ݃‫ܪ‬௥௘௙ were performed,
where ‫ܪ‬௥௘௙ is the maximum water height in the tank ൫‫ܪ‬௥௘௙ ൌ ͲǤͳͷ݉൯ as according to
(Crespo et al., 2008). However, severe particle collapse and contact instabilities were
obtained and, hence, the physical speed of sound was used. Generally, the pressure
oscillations near wall boundaries and free surfaces can be avoided by use of either density
filters or correction of the kernel and/or kernel gradient. Considerable efforts have been
devoted to overcome these problems in the SPH research community (Gomez-Gesteria et
al., 2010). Several correction and filtering techniques have been proposed and are
available in literature, see for instance (Bonet & Lok, 1999). In the zeroth order Shepard
density filter, the simplest and quickest type of density filter, oscillations are smoothed
out by filtering density and then re-assigning a new density to each particle (Colagrossi &
Landrini, 2003), (Belytschko et al., 1998) and (Dilts, 1999). Kernel and kernel gradient
correction techniques are generally included to handle the truncated kernel function close
to free-surfaces and boundaries where conditions of consistency and normalization fail
(Belytschko et al., 1998), (Bonet & Lok, 1999), (Vila, 1999) and (Chen & Beraun, 2000).
The increased accuracy and stability gained (Oger et al., 2007) when using kernel
gradient correction techniques triumph the loss of momentum conserving abilities
(Vaughan et al, 2008). Furthermore, the development of a truly incompressible SPH
(ISPH) solver have shown great promise in estimation of pressure and show less
oscillations, see for instance (Lee et al., 2008). As with boundary conditions no filtering
or correction techniques are currently implemented in LS-DYNA and the effects of such
methods are not possible to test.
21
Conclusions
The two dimensional (2D) SPHERIC Benchmark test case of dam break evolution over a
wet bed has been used to investigate the impact of several parameters when performing
SPH simulations of free surface flows. The choice of EOS, the artificial viscosity
constants, a dynamic versus a static smoothing length, SPH particle resolution and the
FEM mesh scaling of the boundaries were investigated. The numerical results showed
generally a tendency to be ahead of the experimental results, i.e. to have a greater wave
front velocity. This was argued to be an effect of using a frictionless contact algorithm
between the SPH particles and the FEM boundaries. The choice of EOS, FEM mesh
scaling as well as using a dynamic or a static smoothing length showed little or no
significant improvement to the result. However, all EOSs showed larger than physical
pressure overestimation and pressure oscillation in the order of ͳͲ଺ ܲܽ. This is believed
to be an effect of not reducing the speed of sound in the calculations, which is generally
done. The SPH particle resolution and the choice of artificial viscosity constants had a
major impact. The method used for comparing numerical and experimental results
showed increased mean error for both highly resolved cases and artificial viscosity
constants set equal to ͲǤͳ and Ͳ. However, by visual observation it was noted that
increasing the number of particles representing the system increased the number of flow
features resolved and a highly viscous solution was obtained by setting the artificial
viscosity constant to unity. Thus, the relation between the artificial viscosity constants
and the particle resolution and its impact on the behaviour of the fluid need to be further
investigated.
Apart from the investigation of the above parameters, a new post processing method was
proposed. It was shown that a smoothed velocity field was obtained as well as the ability
to do streamline plotting and visualization of the free surface using the dummy nodes.
Acknowledgement
The research presented 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. www.svc.nu
22
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26
Appendix A:
In order to obtain the free surface from the SPH simulations a post processing program is
implemented. The program can also visualize streamlines and interpolate the velocity
field. The boundary node identification routine is summarized in the generic script below.
fori=1:Totalnumberofnodes
Findneighbouringnodesinsidetheradiusofʹ‫ݎ‬௠ ,seefigure10
forj=1:Totalnumberofneighbouringnodes
ifͲ ൏ ݀ ൏ ‫ݎ‬ெ ൅ ‫ݎ‬ே஻ (Partialoverlap)
Calculatetheclosedoverlappingangleinterval,ሾߛ௦௧௔௥௧ ǡ ߛ௘௡ௗ ሿ
elseif݀ ൌ Ͳ(Totaloverlap)
Stopcomputationandproceedtonextnode
else(Nooverlap)
Continuewithnextneighbouringnode
end
end
Calculatethefinaloverlappingintervals(differentintervalscanoverlapeachother)
CalculatethenoneͲoverlappedorthefreeintervals(ifexciting)
iffreeinterval==empty(Totaloverlap)
Stopcomputationandproceedtonextnode
elseiffreeinterval==[02pi](Nooverlap)
Placedummynodesintheinterval[02pi]withstepsdr
else(Partially)
Placedummynodesontheboundaryinthefreeintervalsexceptattheintervalendpoints
end
end
The algorithm loops sequentially over all SPH nodes in the chosen domain. For each
node (M), the neighbouring (NB) particles within a circle of ʹ‫ݎ‬ெ are considered. This can
be done by various techniques such as linked lists (Gomez-Gesteira, 2009). Each NBnode must then be checked whether it is totally-, partially- or non-overlapping the main
node. If the volumes of the SPH nodes are varying, different conditions must be used.
However, in this study only constant volumes are considered. The closed overlapping
angle interval ሾߛ௦௧௔௥௧ ǡ ߛ௘௡ௗ ሿ, measured in a counter clockwise direction from the x-axis
of a local coordinate system with origin located at the centre of the main node needs to be
obtained. Firstly, by identifying the two triangles created by the overlapping circles, see
figure 10, the angle ߙ is calculated by the law of cosines as the sides of the triangles are
known, i.e.
27
rNB
Neighbour node
rM
d
α
α
Main node
Figure 10: Geometry of the two triangles created by the overlapping nodes defining the angle ࢻ.
ଶ
‫ݎ‬௠ଶ ൅ ݀ ଶ െ ‫ݎ‬ே஻
ߙ ൌ …‘• ିଵ ቆ
ቇ
ʹ‫ݎ‬ெ ݀
(18)
where ݀ is the distance between the main and the neighbouring nodes and ‫ݎ‬ெ and ‫ݎ‬ே஻ are
the radiuses. Secondly, the angle ߚ from the local x-axis to the line joining the main and
neighbouring node is determined by inferring an “imaginary” particle or point at distance
݀ from the main node in the positive x-direction, i.e. ሺ‫ݔ‬ெ ൅ ݀ǡ ‫ݕ‬ெ ሻ. Once again, the law
of cosine can be used to determine ߚ,
Neighbour
β
d
im
Imaginary
d
β
Main
d
Figure 11: Geometry of the included “imaginary” node defining the angle ࢼ.
28
ଶ
ʹ݀ ଶ ൅ ݀௜௠
ߚ ൌ …‘• ିଵ ቆ
ቇ
ʹ݀ ଶ
(19)
If however ‫ݕ‬ெ ൐ ‫ݕ‬ே஻ , then ߚ ൌ ʹߨ െ ߚ. The closed overlapping angle interval is then
determined as ሾߛ௦௧௔௥௧ ǡ ߛ௘௡ௗ ሿ ൌ ሾߚ െ ߙǡ ߚ ൅ ߙ ሿ. If the ߛ௦௧௔௥௧ ൐ ߛ௘௡ௗ the angel interval is
split into two intervals, i.e. ሾߛ௦௧௔௥௧ ǡ ʹߨሿ and ሾͲǡ ߛ௘௡ௗ ሿ. The final set of overlapping
intervals from each neighbouring particle needs further transformation as individual
intervals can overlap each other. This can be done by several sorting techniques and is not
given here. The free intervals or non-overlapped intervals based on the final sorted set of
overlapping intervals are then used to determine if the main node is a boundary node or
not. For boundary nodes and non-overlapped nodes, dummy nodes are placed in the free
intervals on the circumference of the main node to better follow the free surface, see
figure 12. Dummy nodes are separated by an arbitrary angle ݀‫ݎ‬, here set equal to ߨȀʹͲ.
Furthermore, dummy nodes are given the values of velocities and other parameters as the
main node, since this is needed later on in the interpolation stage.
Figure 12: Dummy nodes (+) and SPH nodes (*).
29
The Delaunay triangulation including both dummy and SPH nodes where created using
the built-in function DelaunayTRI in the commercial software package Mathworks
MatLab v 8.1.0.604 (R2013a). The inclusion of the dummy nodes is both a way to detect
the free surface as well as detecting which triangles is located inside the material and thus
valid for interpolation, see figure 13.
Figure 13: Delaunay triangulation including both dummy (+) and SPH nodes (*).
30
The second step is to create the interpolation points. Here, a structured grid with
dimensions ͲǤͶ ൑ ‫ ݔ‬൑ ͳǤͲʹ and Ͳ ൑ ‫ ݕ‬൑ ͲǤͳͷ with ݀‫ ݔ‬ൌ ݀‫ ݕ‬ൌ ͷ݁ ିହ ݉ was used. For
each interpolation point the following computations are done:
x
x
x
Check which triangle the interpolation point is inside and whether the triangle is
valid for interpolation, i.e. if less than three triangle vertices are dummy nodes.
Based on the coordinates of the triangle vertices, calculate the barycentric
coordinate ܾ௫ of the interpolation point (Ungar, 2010).
Perform the barycentric interpolation,
݂ሺ݅݊‫ݐ݊݅݋݌ ݊݋݅ݐ݈ܽ݋݌ݎ݁ݐ‬ሻ ൌ ܾଵ ݂ଵ ൅ ܾଶ ݂ଶ ൅ ܾଷ ݂ଷ
(20)
where ܾ௫ is the barycentric coordinate and ݂௫ are the function values such
velocity at the triangle vertices.
The above calculations can be made by using vectorization techniques hence not written
as a for-loop. Finally, the data is mapped on a structured grid where traditional techniques
for data visualization such as streamline plotting can be used.
31
Paper C
Smoothed Particle Hydrodynamics Modelling of
Hydraulic Jumps: Bulk Parameters and Free
Surface Fluctuations
Smoothed Particle Hydrodynamic Modelling of Hydraulic Jumps: Bulk
Parameters and Free Surface Fluctuations
P. Jonsson*, P. Jonsén¤, P. Andreasson*, T. S. Lundström* and J. G. I. Hellström*
*
Division of Fluid and Experimental Mechanics
Luleå University of Technology (LTU)
SE-971 87, Luleå, Sweden
e-mail: [email protected], www.ltu.se
¤
Division of Mechanics and Solid Materials
Luleå University of Technology (LTU)
SE-971 87, Luleå, Sweden
Key words: SPH, hydraulic jump, conjugate depth, free surface fluctuation, vertical
velocity distribution
Abstract
A hydraulic jump is a rapid transition from supercritical flow to subcritical flow
characterized by the development of large scale turbulence, surface waves, spray, energy
dissipation and considerable air entrainment. Hydraulic jumps can be found in waterways
such as spillways connected to hydropower plants and are an effective way to eliminate
problems caused by high velocity flow, e.g. erosion. Due to the importance of the
hydropower sector as a major contributor to the Swedish electricity production, the
present study focuses on Smoothed Particle Hydrodynamic modelling of 2D hydraulic
jumps in horizontal open channels. Four cases with different spatial resolution of the SPH
particles were investigated by comparing the conjugate depth in the subcritical section
with theoretical results. These showed good agreement and an increased oscillation
tendency for the most resolved case. The coarsest case was run for a longer time than the
others due to computational time limitations and a quasi-stationary state was achieved.
Thus for the coarsest case, additional variables were explored. The mean vertical velocity
distribution in the horizontal direction compared favourably with experiments and the
maximum velocity for the SPH-simulations indicated a too rapid decrease in the
horizontal direction and poor agreement to experiments was obtained. Furthermore, the
mean and the standard deviation of the free surface fluctuation showed generally good
agreement with experimental results even though some discrepancies were found
regarding the peak in the maximum standard deviation. The free surface fluctuation
frequencies were over predicted and the model could not capture the decay of the
fluctuations in the horizontal direction.
1
Introduction
Fluid mechanics of large hydropower plants are characterized by very high flow rates and
large physical dimensions both in production and spill waterways. The hydraulic head
harvested in production needs to be handled when spillways are engaged as well. In open
spillways, this is done by accelerating the flow and then using the dissipative features of a
hydraulic jump. A hydraulic jump is a rapid transition from high velocity supercritical
flow to low velocity subcritical flow with rise of the free surface, in order to keep
continuity. The hydraulic jump is characterized by the development of large scale
turbulence, surface waves, spray, energy dissipation and considerable air entrainment.
The highly turbulent transition zone is usually referred to as the “roller” and the start of
the roller the “jump toe” or the “impingement point”. A hydraulic jump is typically
characterised by its inflow Froude number ‫ݎܨ‬ଵ ൌ ‫ݒ‬ଵ Τඥ݃݀ଵ , where ‫ݒ‬ଵ is the depth
average upstream flow velocity, ݀ଵ is the upstream depth and ݃ is the acceleration of
gravity, see figure 1. Based on the ‫ݎܨ‬, the jump may be classified into five types: undular
ሺ‫ݎܨ‬ଵ ൌ ͳ െ ͳǤ͹ሻ, weak ሺ‫ݎܨ‬ଵ ൌ ͳǤ͹ െ ʹǤͷሻ, oscillating ሺ‫ݎܨ‬ଵ ൌ ʹǤͷ െ ͶǤͷሻ, steady ሺ‫ݎܨ‬ଵ ൌ
ͶǤͷ െ ͻሻ and strong ሺ‫ݎܨ‬ଵ ൐ ͻሻ, each with its own distinct characteristics (Chanson,
2004).
Roller
Jump toe
Shear layer
‫ݒ‬ଵ
݀ʹ
݀ଵ
‫ݕ‬
‫ܮ‬௥
‫ݔ‬
Figure 1: Schematic figure of a hydraulic jump.
By applying the continuity and momentum equation in integral form, a dimensionless
relationship of the upstream depth ݀ଵ and downstream depth ݀ଶ can be obtained. For a
horizontal and rectangular channel this procedure yields the Bélanger equation,
݀ଶ ͳ
ൌ ቆටͳ ൅ ͺ‫ݎܨ‬ଵଶ െ ͳቇǤ
݀ଵ ʹ
2
(1)
(Hager et al., 1990) reviewed a broad range of data and correlations and proposed a semiempirical relationship for the length of the roller ‫ܮ‬௥ as,
‫ݎܨ‬ଵ
‫ܮ‬௥
ൌ ͳ͸Ͳ –ƒŠ ൬ ൰ െ ͳʹ
݀ଵ
ʹͲ
ሺʹ ൏ ‫ݎܨ‬ଵ ൏ ͳ͸ሻǤ
(2)
The external geometry of hydraulic jumps has been studied for a long time and only
recently, the internal flow structures have been considered. Depending on the type of
jump, a number of regions in the roller can be identified (Chanson & Brattberg, 2000)
and (Lin et al., 2012). For the steady jumps, a boundary layer is seen close to the bottom
which transitions into a high velocity jet or core region. Continuing in the vertical
direction from the bottom, a highly turbulent air-water shear layer is observed where
bubble break up and coalescences occurs and momentum is transferred to the region
above. This final region is a highly aerated recirculation region with significant free
surface wave and splash production, see figure 1.
Several authors have investigated the velocity field in the above-mentioned regions using
different experimental techniques, e.g. (Rajaratnam, 1965), (Hornung et al., 1995),
(Lennon & Hill, 2006), (Chanson & Brattberg, 2000) and (Lin et al., 2012). (Chanson,
2010) investigated the vertical velocity distribution in the downstream direction using
double-tip phase-detection probes of hydraulic jumps with ͷǤͳ ൏ ‫ݎܨ‬ଵ ൏ ͳͳǤʹ. It was
shown that the velocity profiles in the developing shear layer followed a wall jet pattern
proposed earlier by (Rajaratnam, 1965) and (Chanson & Brattberg, 2000). Furthermore,
the maximum velocity ܸ௠௔௫ in the horizontal direction had a longitudinal decay with
increasing distance from the jump toe and the following empirical function
ሺ‫ ݔ‬െ ‫ݔ‬ଵ ሻ
ܸ௠௔௫
ൌ ‡š’ ቆെͲǤͲʹͺ
ቇǡ
݀ଵ
‫ݒ‬ଵ
(3)
where ‫ ݔ‬is the distance downstream of the inlet and ‫ݔ‬ଵ is the location of the jump toe,
revealed the best correlation based on the data of (Chanson, 2010), (Chanson &
Brattberg, 2000), (Kucukali & Chanson, 2008) and (Murzyn & Chanson, 2009b).
(Long et al., 1991) investigated hydraulic jumps with inflow ‫ ݎܨ‬Ͷ െ ͻ using a high speed
video camera in a horizontal rectangular channel. (Long et al., 1991) postulated that at
any given time when the roller is at its most upstream location, the maximum number of
vortices exists and their size increase in the downstream direction. The vortices showed
an anticlockwise rotational tendency and vortex-pairing of smaller vortex structures was
observed. Eventually, the smaller vortices merged with a semi-stationary larger vortex at
the end of the roller causing a forward spill of water enabling further vortex production at
the impingement point. At this stage, the jump toe was at the most downstream location
3
observed and (Long et al., 1991) argued that the vortex production is linked with the
jump toe fluctuations. (Mossa, 1999) investigated the oscillating behaviour of several
different hydraulic jump using point gauge and concluded that a correlation between
jump toe fluctuations and vortex structures in the roller is evident.
Besides affecting the jump toe, the vortex structures influence the free surface. (Mouaze
et al., 2005) investigated the behaviour of the free surfaces using resistivity probes for
four different hydraulic jumps ሺͳǤͻͺ ൏ ‫ݎܨ‬ଵ ൏ ͶǤͺʹሻ with partially developed inflow
conditions. At the jump toe, where bubbles were generated and entrained, there was a
slight increase in the free surface elevation while a sudden large gradient in fluctuation
was seen. Downstream the jump toe, a maximum in free surface fluctuations indicating
strong turbulence was observed. (Mouaze et al., 2005) argued that the fluctuations were
caused by large coherent structures reaching the free surface generated in the shear and
roller regions, generally characterized by large recirculation vortices and bubbles. Further
downstream, a large dissipative zone emerged as fluctuations decreased significantly and
the free surface became almost horizontal. (Murzyn et al., 2007) continued the work of
(Mouaze et al., 2005) but with a different experimental technique (wire gauges) and came
to a similar conclusion. (Kucukali & Chanson, 2008) investigated free surface
fluctuations in hydraulic jumps with slightly larger inflow Froude number ሺͶǤ͹ ൏ ‫ݎܨ‬ଵ ൏
ͺǤͷሻ than previous studies using ultrasonic displacement meters, see the extended report
(Kucukali & Chanson, 2007). Yet again, data of the mean free surface elevation showed a
gradual increase towards the theoretical value in the downstream direction and the peak in
standard deviation was typically found at ሺͳͲ ൑ ሺ‫ ݔ‬െ ‫ݔ‬ଵ ሻȀ݀ଵ ൑ ͳͷሻ. (Murzyn &
Chanson, 2009a) (see extended report (Murzyn & Chanson, 2007)) used the same set-up
and measuring apparatus as (Kucukali & Chanson, 2008) and showed that the maximum
standard deviation of free surface fluctuations was best correlated by the function,
ߟᇱ
ൌ ͲǤͳͳ͸ሺ‫ݎܨ‬ଵ െ ͳሻଵǤଶଷହ
ቆ ቇ
݀ଵ ௠௔௫
(4)
where ߟᇱ is the standard deviation of free surface fluctuations. Fast Fourier Transform
(FFT) analysis of the displacement meter output signal indicated dominant frequencies in
the range of ͳ െ Ͷ‫ ݖܪ‬and a minor tendency of the frequency to decrease in the
downstream direction was noted. (Murzyn & Chanson, 2009a) showed also that the
dimensionless Strouhal number ܵ‫ ݐ‬ൌ ሺ݂݀ଵ ሻΤ‫ݒ‬ଵ decreased rapidly with increasing
Reynolds number ܴ݁ ൌ ሺ‫ݒ‬ଵ ݀ଵ ሻΤߥ , where ݂ is the frequency. More recently, (Chachereau
& Chanson, 2011) (see extended report (Chachereau & Chanson, 2010)) investigate
hydraulic jumps with relatively small ‫ ݎܨ‬ሺʹǤͶ ൏ ‫ݎܨ‬ଵ ൏ ͷǤͳሻ and large ܴ݁ ሺ͸Ǥ͸ ‫Ͳͳ כ‬ସ ൏
ܴ݁ ൏ ͳǤ͵ ‫Ͳͳ כ‬ହ ሻ with acoustic displacement meters as the previously two studies. Similar
conclusions were made as for the above studies regarding the mean and standard
4
deviation of the free surface elevation as well as the free surface fluctuation frequencies.
However, in this study the ܵ‫ ݐ‬showed a minor decrease with increasing ‫ ݎܨ‬and data was
best correlated by the following function,
ܵ‫ݐ‬௙௦ ൌ ͲǤͳͶ͵ ‡š’ሺെͲǤʹ͹‫ݎܨ‬ଵ ሻ
ሺʹǤͶ ൏ ‫ݎܨ‬ଵ ൏ ͸Ǥͷሻ
(5)
Modelling of highly disturbed aerated free surface flows such as hydraulic jumps, is
complex when grid based method is used (Violeau, 2012). Severe problems with mesh
entanglement and determination of the free surface have been encountered (Scardovelli &
Zaleski, 1999). The meshfree, Lagrangian particle based method Smooth Particle
Hydrodynamic (SPH) has shown to be a good alternative to grid based methods to
overcome such problems (Gomez-Gesteria et al., 2010) and (Monaghan, 2012). The
technique was first proposed independently by (Lucy, 1977) and (Gingold & Monaghan,
1977) in the late seventies to solve astrophysical problems in three-dimensional open
spaces. Today, the SPH method has been applied to a number of fields and problems
(Monaghan, 2012) and the maturity of the method has increased significantly. A major
advantage of SPH is that the method is mesh-free, thus considerable time is saved as
compared to methods that need a predefined mesh. However, as the SPH is relatively
unexplored as compared to traditional FVM/FEM methods there are some areas that still
need considerable attention. For instance, wall boundary conditions are generally difficult
to set in SPH as they do not appear in a natural way within the SPH formalism.
Furthermore, the SPH method is typically slower computational wise since the time step
depends on the speed of sound and the explicit integration techniques used.
A few papers have been devoted to SPH modelling of hydraulic jumps. (López et al.,
2010) investigated the capability of the SPH method to reproduce mobile hydraulic jumps
with different inflow ‫ݎܨ‬. The 2D geometrical set-up composed of a tank and a gate,
through which water discharged into a stilling basin where a weir triggered the jumps was
used. Good agreement with experimental and theoretical results was shown for ‫ ݎܨ‬less
than five. Furthermore, the ݇ െ ߝ turbulence closure model was implemented to improve
the solution in the turbulent roller region for ‫ ݎܨ‬more than five. However, due to the
increased computational cost a less complex model were later used. (Jonsson et al. 2011)
investigated the effects of altering the spatial resolution of the SPH particles and its
impact on hydraulic jump behaviour. Good agreement for the conjugate depth ݀ଶ when
comparing with the theoretical value was obtained for all cases. However, the most
resolved case seemed to oscillate more than the others. (Federico et al., 2012) developed a
2D SPH model to enforce in and outflow boundary conditions and demonstrated through
three cases, one being a hydraulic jump test case, the general ability of the SPH method to
handle uniform, non-uniform and unsteady flows. By implementing in- and outflow
boundary conditions, the use of a tank and gate as well as a weir was no longer necessary
5
hence improving the efficiency of the computation. Several jumps with different ‫ ݎܨ‬were
investigated and good agreement when compared with theoretical values was found for
the conjugate depth ݀ଶ . Furthermore, good agreement was obtained for the internal
velocity field when compared with the experimental work by (Hornung et al., 1995)
although the size of the vortex structures was generally over predicted. (Chern &
Syamsuri, 2012) displayed the possibility to investigate the effects of a corrugated bed on
the hydraulic jump characteristics using SPH. The geometrical setup was similar to the
one used by (López et al., 2010) but a more sophisticated method to model turbulence
was employed, namely the laminar and subparticle scale (SPS) technique. Several
corrugated bed types were investigated and evaluated using the conjugate depth ratio,
jump length, bottom shear stress distribution and the energy dissipation. It was shown
that the sinusoidal bed produced the highest bottom shear stresses and largest energy
dissipation as well as the lowest conjugate depth ratios and shortest jump length, which is
in agreement with experimental results. (De Padova et al., 2013) investigated 3D undular
hydraulic jumps influenced by waves generated at the channel walls. The results showed
a reasonable agreement in terms of time-average water depths and longitudinal velocity
components as well as the trapezoidal wave pattern. However, poor agreement was found
for the prediction of the inclination of the oblique wave front and (De Padova et al., 2013)
argued that this is possibly an effect of the wall boundary conditions used.
Based on the previous studies presented above, this paper focus on the impact of the
spatial resolution of the SPH particles to accurately reproduce the conjugate depth ݀ଶ .
Furthermore, the internal velocity field as well as free surface fluctuations will be
investigated and compared with experimental results using the SPH method.
Method
Governing equations
In the SPH-method, the fluid domain is represented by a set of non-connected particles
which possess individual material properties such as density, velocity 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 their material properties changes as a function of time
and spatial co-ordinate due to interactions with neighbouring particles.
The superscripts ߙ and ߚ are used to denote coordinate directions and the subscripts ݅ and
݆ denotes particle indices. The continuity and momentum equations can be written in the
SPH formalism according to,
6
ே
݉௝
݀ߩ௜
ൌ ߩ௜ ෍ ࢜௜௝ ‫ܣ‬௜௝
݀‫ݐ‬
ߩ௝
(6)
௝ୀଵ
ே
ߪ ఈǡఉ ൫࢞௝ ൯
݀࢜ఈ
ߪ ఈǡఉ ሺ࢞௜ ሻ
ሺ࢞௜ ሻ ൌ ෍ ݉௝ ቆ
‫ܣ‬௜௝ െ
‫ܣ‬௝௜ ቇ
ߩ௜ ߩ௝
݀‫ݐ‬
ߩ௜ ߩ௝
(7)
௝ୀଵ
where ߩ௜ is the density, ݉௝ is the particle mass and ࢞௜ and ࢜௜௝ ൌ ࢜௜ െ ࢜௝ is the position
and velocity vector respectively. ߪ ఈǡఉ is the total stress tensor and
‫ܣ‬௜௝ ൌ
ͳ
݄ௗ௜௠ାଵ
ฮ࢞௜ െ ࢞௝ ฮ
ߠᇱ ቆ
ቇǤ
݄
(8)
In equation 8, ݀݅݉ is the number of space dimensions and ߠ ᇱ is the derivative of the
function used to define the kernel function ܹ, i.e.
ܹሺ࢞ǡ ݄ሻ ൌ
ͳ
ߠሺ࢞ሻ
݄ሺ࢞ሻௗ௜௠
(9)
and ݄ is the smoothing length. The commonly used cubic B-spline kernel is obtained by
choosing ߠ as,
͵
͵
‫ ͳۓ‬െ ܴଶ ൅ ܴଷ
ʹ
Ͷ
ۖ
ߠൌ‫ܥ‬
ͳ
ሺʹ െ ܴሻଷ
‫۔‬
ۖ Ͷ
‫ە‬
Ͳ
ܴ൏ͳ
ͳ൑ܴ൏ʹ
(10)
ܴ൒ʹ
where ‫ ܥ‬is a constant of normalization defined for one-, two- and three dimensional
రఱ
వ
and ‫ܥ‬ଷ஽ ൌ రഏ
. Here, ܴ is the normalized distance between
spaces as; ‫ܥ‬ଵ஽ ൌ యమ, ‫ܥ‬ଶ஽ ൌ భరഏ
two particles, i.e. ܴ ൌ
ฮ࢞೔ ି࢞ೕ ฮ
௛
. The total stress tensor ߪ ఈǡఉ in the momentum equation is
made up from two parts, the isotropic pressure ‫ ݌‬and the deviatoric viscous stress ߬ ఈǡఉ ,
according to
ߪ ఈǡఉ ൌ െ‫ ߜ݌‬ఈǡఉ ൅ ߬ ఈǡఉ Ǥ
(11)
The NULL material model defines the deviatoric viscous stress as (LS-DYNA keyword
user's manual, 2012),
߬ ఈǡఉ ൌ ߤߝ ఈǡఉ
7
(12)
where ߝ ఈǡఉ is deviatoric strain rate. By setting the dynamics viscosity ߤ to zero the total
stress tensor reduces to the isotropic pressure which is defined by an equation-of-state
(EOS). In present work, the Morris EOS (Federico et al., 2012) is used,
‫݌‬௜ ൌ ܿ଴ଶ ሺߩ௜ െ ߩ଴ ሻ
(13)
where ܿ଴ is the adiabatic speed of sound and ߩ଴ is the initial density. By using the Linear
Polynomial EOS implemented in LS-DYNA as,
‫݌‬௜ ൌ ‫ܥ‬ଵ ൅ ‫ܥ‬ଶ ߤ஽ ൅ ‫ܥ‬ଷ ߤ஽ଶ ൅ ‫ܥ‬ସ ߤ஽ଷ ൅ ሺ‫ܥ‬ହ ൅ ‫ߤ ଺ܥ‬஽ ൅ ‫ߤ ଻ܥ‬஽ଶ ሻ‫ܧ‬
(14)
where ߤ஽ ൌ ߩ௜ Τߩ଴ െ ͳ and by setting the constants ‫ܥ‬ଵ ൌ ‫ܥ‬ଷ ൌ ‫ܥ‬ସ ൌ ‫ܥ‬ହ ൌ ‫ ଺ܥ‬ൌ ‫ ଻ܥ‬ൌ Ͳ
and ‫ܥ‬ଶ ൌ ܿ଴ଶ ߩ଴ the Morris EOS is obtained.
The artificial viscosity proposed by (Monaghan, 1992) is implemented as,
ଶ
െ‫ݍ‬ଵ ܿҧ௜௝ ߶௜௝ ൅ ‫ݍ‬ଶ ߶௜௝
ȫ௜௝ ൌ ቐ
തതതത
ߩ
పఫ
Ͳ
࢜௜௝ ࢘௜௝ ൏ Ͳ
(15)
࢜௜௝ ࢘௜௝ ൐ Ͳ
where
߶௜௝ ൌ
݄࢜௜௝ ࢘௜௝
ǡ
࢘ଶ௜௝ ൅ ߟଶ
(16)
‫ݍ‬ଵ and ‫ݍ‬ଶ are constants and ࢘௜௝ ൌ ࢘௜ െ ࢘௝ and ࢜௜௝ ൌ ࢜௜ െ ࢜௝ are the position and velocity
vector, respectively. Furthermore, ܿҧ௜௝ ൌ
௖೔ ା௖ೕ
ଶ
and ߩҧ௜௝ ൌ
ఘ೔ ାఘೕ
ଶ
are the mean speed of
sound and density, respectively. If the smoothing length ݄ is dynamic, ݄ ൌ ݄ത௜௝ ൌ
௛೔ ା௛ೕ
ଶ
.
The ߟଶ ൌ ͲǤͲͳ݄ଶ is used to avoid the denominator to go to zero. In literature, many
authors suggest setting ‫ݍ‬ଶ ൌ Ͳ and ‫ݍ‬ଵ ൌ ͲǤͲͳ‫Ͳ݋ݐ‬Ǥͳ (Dalrymple & Rogers, 2006),
(Monaghan, 1994). Following (Jonsson et al., 2013), the artificial viscosity constants are
here set equal to ‫ݍ‬ଵ ൌ ͲǤͳ and ‫ݍ‬ଶ ൌ Ͳ.
With the reduction of the total stress tensor, the inclusion of the artificial viscosity term
together with the symmetric properties of the ‫ܣ‬௜௝ ൫‫ܣ‬௜௝ ൌ െ‫ܣ‬௝௜ ൯ term, the momentum
equation reduces to the familiar expression,
ே
ܲ௝
ܲ௜
݀࢜ఈ
ሺ࢞௜ ሻ ൌ െ ෍ ݉௝ ቆ
൅
൅ ȫ௜௝ ቇ ‫ܣ‬௜௝
݀‫ݐ‬
ߩ௜ ߩ௝ ߩ௜ ߩ௝
௝ୀଵ
8
(17)
A first order time integration scheme is used in LS-DYNA and the time step is determined
according to (Hallquist, 2006),
ߜ‫ ݐ‬ൌ ‫ܥ‬஼ி௅ ‫݊݅ܯ‬௜ ൬
݄௜
൰
ܿ௜ ൅ ‫ݒ‬௜
(18)
where ‫ܥ‬஼ி௅ is the Courant number usually set equal to unity.
Boundary Conditions
The wall boundaries were modelled as rigid shell finite elements and the coupling
between the boundaries and the SPH-particles are governed by a penalty based “node to
surface” contact-algorithm (Jonsén et al., 2012) and (Jonsson et al., 2013). This is a socalled one-way contact where the SPH particles are defined as the slave side and the FEM
elements as master side. In applying the penalty method, the slave nodes are checked for
penetration through the master surface. If a slave node has penetrated, an interface spring
is placed between the master surface and the node. The spring stiffness is chosen
approximately in the order of magnitude as the stiffness of the interface element normal
to the interface. The resultant force applied to the SPH particle in the normal direction of
the FEM element is proportional to the amount of penetration. The wall boundaries used
here can be compared with the Lennard-Jones-type boundary condition where a central
force is applied to fluid particles (Monaghan, 1994). However, the normal component is
considered only. Thus, the boundary condition used here is truly frictionless. For more
information regarding the contact algorithm see (LS-DYNA keyword user's manual,
2012).
Geometrical and Numerical setup
A two dimensional horizontal spillway channel and hydraulic jump is investigated in
present work with a single phase (water) model. The geometrical setup is shown in figure
2, where the water enters the domain with an prescribed inlet velocity of ‫ݒ‬ଵ ൌ ͳǤͷ ݉Τ‫ݏ‬.
9
Computational Box
ͳǤͳ݉
ͲǤͺ ݉
Gate
‫ݒ‬ଵ
Inlet
݀ଵ
Outlet
Weir
‫ݕ‬
Stilling Basin
‫ݔ‬
Figure 2: Geometrical setup.
All particles entering the domain must be placed in an ordered configuration outside the
point of entry, i.e. the inlet (LS-DYNA keyword user's manual, 2012). The depth ݀ଵ of
the incoming jet is set equal to ʹ ‫ିͲͳ כ‬ଶ ݉, thus generating a supercritical flow with
Froude number ‫ݎܨ‬ଵ ൎ ͵ǤͶ. Furthermore, above the inlet a gate is placed to hinder
particles from leaving the domain in an unphysical manner. The stilling basin located
between the inlet and the weir measures ͳǤͳ݉ and is initially filled to a depth equal to the
weir height, i.e. ݀ଵ . Both the prefilled zone and the weir are introduced in order to trigger
the hydraulic jump faster. The computational box illustrated in figure 2 (dotted line) is
included to determine which SPH particles are activated and deactivated. Nodes located
inside the box are activated and the governing equations are solved. Deactivated particles
or nodes outside the box maintain the state from previous active time step or the
externally imposed state, thus the governing equations are not solved for these particles.
The boundary of the computational box can be used as an outlet if placed properly. Here,
the computational box dimensions was set according to Ͳ ൑ ‫ ݔ‬൑ ͳǤͻ and െͲǤͳ ൑ ‫ ݕ‬൑
ͲǤͷ.
All SPH particles were initially placed on a structured grid and the initial density ߩ଴ was
set equal to ͳͲͲͲ݇݃Ȁ݉ଷ . Usually, ܿ଴ is reduced to ͳͲ times the maximum flow velocity
(Monaghan, 1994). However, initial results showed severe compressible effects and
contact instabilities hence ܿ଴ was set equal to the adiabatic speed of sound of ͳͶͺͶ݉Ȁ‫ݏ‬.
In order to investigate the influence of altering the number of nodes representing the
system, four cases with different spatial resolution was set up. Here, the characteristic
length scale was chosen to the incoming jet depth ݀ଵ and consequently, the SPH spatial
resolution was set equal to ݀ଵ ȀͶ, ݀ଵ Ȁͷ, ݀ଵ Ȁ͸ and ݀ଵ ȀͳͲ. The length of the rigid shell
10
finite elements is set equal to half the initial SPH spatial resolution and thus varies for
each case. A static smoothing length of ͳǤʹ times the initial interpartical spacing was used
for all cases, see (Jonsson et al., 2013). Due to prohibiting computational time (in excess
of 2500 hours on a 12 core Linux machine for the ݀ଵ ȀͳͲ case) all cases was run for 5s
except the coarsest which was run for ͵Ͳ‫ݏ‬, see table 1. Furthermore, data was saved at
each ͲǤͲͳ‫ݏ‬.
Table 1: Properties of the simulation cases.
Case
݀ଵ ȀͶሺ͵Ͳ ‫ݏ‬ሻ
݀ଵ Ȁͷሺͷ‫ݏ‬ሻ
݀ଵ Ȁ͸ሺͷ‫ݏ‬ሻ
݀ଵ ȀͳͲሺͷ ‫ݏ‬ሻ
Total Number of
Particles
36879
10749
15479
42999
Number of
Cores
8
8
16
12
Total Computational
Time [h]
114
9
26
114
All simulations were conducted using the nonlinear finite element code LSTC LS-DYNA
v. 971 R7.0.0 on multicore Linux workstations. The post processing method proposed by
(Jonsson et al., 2013) was used to visualize and interpolate the SPH data. Furthermore,
statistical measurements of the deviations of conjugate depth ݀ଶ were determined by the
following formulas,
ଶ
‫ܣ‬ൌඩ
ௌ௉ு
σே
௜ୀଵ൫݀ଶǡ௜ ൯
(19)
ଶ
்ுாைோ௒
σே
൯
௜ୀଵ൫݀ଶǡ௜
and
ଶ
ௌ௉ு
்ுாைோ௒
σே
൯
௜ୀଵ൫݀ଶǡ௜ െ ݀ଶǡ௜
Ǥ
ܲൌඩ
ଶ
்ுாைோ௒
ே
σ௜ୀଵ൫݀ଶǡ௜
൯
(20)
A perfect agreement of numerical and theoretical results implies that ‫ ܣ‬՜ ͳ and ܲ ՜ Ͳ,
(Crespo et al., 2008).
11
Results and Discussion
In figure 3-7, the absolute velocity field for the four cases is presented and, as a general
result, more flow features is resolved when increasing the spatial resolution of the SPH
particles.
During the initial time steps, the stationary fluid in the stilling basin resist the fast
incoming jet thus creating a “mushroom-like” wave similar to the wave created in dam
break flows over a wet bed, see (Stansby et al., 1998) and (Jánosi et al., 2004). The wave
then breaks into both forward and backward direction and both the incoming and the
initially stationary water starts to translate downstream. After roughly ͳ‫ݏ‬, the part of the
mushroom wave that broke in the downstream direction reaches the weir and the part that
broke in the upstream direction has translated ͲǤͶ݉ in the downstream direction. The
position and time of those events are similar in all cases, compare figure 3-7 at ͳ‫ݏ‬. The
flow characteristics can now be described as a movable hydraulic jump, thus the part
located at ͲǤͶ݉ can be seen as the jump toe, see figure 3-7 at ͳ‫ݏ‬.
Past ͳ‫ݏ‬, the jump toe translates further downstream until it reaches the weir. It should be
noted that some variance for the jump toe velocity in the downstream direction is seen,
compare figure 3-7 at ʹ‫ݏ‬. The minor differences of the jump toe position (figure 5 and 6
at ʹ‫ )ݏ‬is a consequence of the change in jump toe propagation direction. The direction
shift is caused by the bulking effect with linear increase of the free surface as the jump
toe approaches the weir, see figure 8. When the toe has reached the weir, a maximum
depth is attained and a release of fluid in the upstream direction is observed, compare
figure 3-8.
Past ʹ‫ݏ‬, the jump toe propagates in the upstream direction and yet again a difference in
toe velocity is observed, compare figure 3-7 at ͵‫ ݏ‬to ͷ‫ݏ‬. A plausible reason for the
minor difference for cases ݀ଵ Ȁ͸ and ݀ଵ ȀͳͲ is that the number of flow features is almost
the same for these cases even though the refinement ratio is large. For the ݀ଵ ȀͶ case
which was set up to run until ͵Ͳ‫ݏ‬, the jump toe continues to propagate upstream until
approximately ͳͳ‫ ݏ‬when it reaches the gate and inlet. Past ͳͳ‫ݏ‬, a quasi-stationary state
is obtained, see figure 7.
Generally, for experimental studies (e.g. (Kucukali & Chanson, 2008) and (Murzyn &
Chanson, 2009a)) the jump toe becomes stationary at a certain distance downstream the
inlet after an initial transient phase and not at the inlet itself. The continuing propagation
of the jump toe in the upstream direction shown here might be explained by the
frictionless contact algorithm used in the model. In all cases, especially the coarser ones,
a boundary layer close to the bottom seems to affect the jet behaviour, see figure 3 and 4
at ͳ‫ ݏ‬and figure 9. However, this should not be interpreted as an actual boundary layer. It
12
is more likely an effect of the truncated kernel domain due to the lack of SPH nodes
outside the boundary. Considerable research has been devoted to this topic and a possible
solution is the use of a different boundary condition such as the ghost particle (Randles &
Libersky, 1996), the repulsive particle (Monaghan, 1994) or the dynamic particle method
(Crespo et al., 2007) and (Gomez-Gesteira et al., 2009). Furthermore, the various density
filters and kernel and kernel gradient correction methods proposed in literature (e.g.
(Bonet & Lok, 1999)) may also be applied to reduce these unwanted effects.
13
Figure 3: Absolute velocity field ሾ࢓Ȁ࢙ሿ for case ࢊ૚ Ȁ૝, ૙࢙ to ૞࢙.
14
Figure 4: Absolute velocity field ሾ࢓Ȁ࢙ሿ for case ࢊ૚ Ȁ૞, ૙࢙ to ૞࢙.
15
Figure 5: Absolute velocity field ሾ࢓Ȁ࢙ሿ for case ࢊ૚ Ȁ૟, ૙࢙ to ૞࢙.
16
Figure 6: Absolute velocity field ሾ࢓Ȁ࢙ሿ for case ࢊ૚ Ȁ૚૙, ૙࢙ to ૞࢙.
17
Figure 7: Absolute velocity field ሾ࢓Ȁ࢙ሿ for case ࢊ૚ Ȁ૝, ૙࢙ to ૚૞࢙.
In figure 8, the conjugate depth ݀ଶ located at ‫ ݔ‬ൌ ͳ݉ downstream of the inlet as a
function of time is shown. The position of ݀ଶ was chosen to not include the effects from
the weir as the flow depth is reduced to meet continuity, see for instance figure 7. Good
agreement is obtained for the conjugate depth which is comparable to the results of
(López et al., 2010), (Jonsson et al., 2011) and (Federico et al., 2012). At roughly ͲǤͺ‫ݏ‬,
݀ଶ increases rapidly followed by the linear increase, as mentioned above in the result
section. The maximum depth is also visible followed by the release of water in the
upstream direction. Beyond ʹǤͷ‫ݏ‬, there is an oscillatory behaviour of the numerical depth
18
d2 SPH / d2 THEORY
for all cases around the theoretical value, determined by equation 1. It should be noted
that the finer cases tend to oscillates more than the coarser due to the increased number of
flow features resolved, compare figures 3-7 and figure 8.
(A)
1
d1/4
d1/5
0.5
d1/6
d1/10
0
d2 SPH / d2 THEORY
d2 THEORY
0
1
2
3
Time [s]
4
5
(B)
d2 THEORY
1
d1/4
0.5
0
0
5
10
15
Time [s]
20
25
30
Figure 8: Dimensionless conjugate depth ࢊ૛ǡࡿࡼࡴ Ȁࢊ૛ǡࢀࡴࡱࡻࡾࢅ as a function of time. Above (A) all
cases ૙࢙ to ૞࢙ and below (B) case ࢊ૚ Ȁ૝ ૙࢙ to ૜૙࢙.
In figure 8 B, the conjugate depth ݀ଶ for the ݀ଵ ȀͶ case is shown only. As mentioned
above, the jump toe reaches the gate and the inlet at approximately ͳͳ‫ ݏ‬which is
indicated by a slight increase of the depth ݀ଶ in figure 8 B. Statistical measures of the
errors A and P, induced by the oscillations of the free surface for the four cases is
obtained by applying equation 19 and 20, see table 2.
19
Table 2: Statistical measures A and P (equation 19 and 20) of the conjugate depth ࢊ૛ at
location ࢞ ൌ ૚࢓.
Case (Time interval)
D1/4 (2.5s-5s)
D1/5 (2.5s-5s)
D1/6 (2.5s-5s)
D1/10 (2.5s-5s)
D1/4 (15s-30s)
A
0.969
0.986
0.968
0.936
1.041
P
0.049
0.041
0.077
0.119
0.045
To exclude the initial transient phase, data was collected in the time interval ʹǤͷ‫ ݏ‬to ͷǤͲ‫ݏ‬
for all cases and in the interval ͳͷ‫ ݏ‬to ͵Ͳ‫ ݏ‬for the ݀ଵ ȀͶ case. Generally, good agreement
is obtained and it seems that the ݀ଵ Ȁͷ case produce the best result. Otherwise, higher
resolution seems to increase the error which yet again is likely to be coupled with the
number of flow features resolved, (Jonsson et al., 2011) and (Jonsson et al., 2013)
reported a similar behaviour.
Henceforth, data from the coarsest case ݀ଵ ȀͶ in the time interval ‫ ݐ‬ൌ ͳͷ‫ ݏ‬to ‫ ݐ‬ൌ ͵Ͳ‫ݏ‬
(the quasi-stationary state) will be discussed. Using the post processing method proposed
by (Jonsson et al., 2013), average velocity fields can be created. In figure 9, the
dimensionless average vertical velocity distribution in the downstream direction is plotted
where black lines represents data each ͲǤͲʹͷ݉ in the interval Ͳ ൏ ‫ ݔ‬൏ ͳ݉. The
incoming jet together with the recirculation zone represented by the negative values for
the first steps in the downstream direction is visible. These results compare favourably
with experimental results, see for instance (Lin et al., 2013) and (Chanson, 2010). Further
downstream, the jet weakens and the velocity profile becomes almost uniform. However,
two significant discrepancies are detectible. Firstly, the effects of the “boundary layer”
discussed above are seen close to the bottom. This was concluded to be an effect of the
truncated kernel and not an actual boundary layer. Secondly, the velocity profiles collapse
to zero velocity close to the free surface. This is an effect of the averaging process where
values outside the water are given a value of zero. Thus, as the free surface is oscillating
the average value close to or at the free surface tend to go to zero. However, this
discrepancy has a minor effect for present study.
20
Vertical velocity distribution x=[0,1]
First non−negative vertical velocity at x=0.2235 m
Maximum velocity Vmax
5
←x=1m
y/d1
4
3
2
←x=0m
1
0
−0.2
0
0.2
0.4
0.6
vx/v1
0.8
1
1.2
Figure 9: Dimensionless average vertical velocity distribution ࢜࢞ Ȁ࢜૚ as a function of ࢟Ȁࢊ૚ .
The red line in figure 9 represent the first position downstream the jump toe with nonnegative average velocity in the x-direction. Hence, this position is interpreted as the end
of the roller section and the length of the jump as the jump toe is at ‫ ݔ‬ൌ Ͳ݉. Comparing
the numerical value of the jump length ‫ܮ‬௥ǡௌ௉ு ൌ ͲǤʹʹ͵ͷ݉ with theoretical value
‫ܮ‬௥ǡ்ுாைோ௒ ൌ ͲǤʹͻͺͺ݉ by using equation 2 a ʹͷΨ discrepancy is obtained. The blue
star markers in figure 9 represent the average maximum velocity ܸ௠௔௫ in the downstream
direction which can be compared with the empirical function (equation 3) obtained by
(Chanson, 2010), see figure 10.
21
exp(−0.028*(x−x1)/d1) (Chanson, 2010)
SPH d1/4 (Present Study)
1.2
Vmax/v1
1
0.8
0.6
0.4
0.2
0
10
20
30
(x−x1)/d1
40
50
60
Figure 10: Longitudinal distribution of dimensionless average maximum velocity ࢂ࢓ࢇ࢞ Ȁ࢜૚ as
function of dimensionless distance ሺ࢞ െ ࢞૚ ሻȀࢊ૚ from the jump toe.
The jump toe position ‫ݔ‬ଵ for the SPH results was assumed to be zero although small
oscillations were observed in the horizontal direction. As can be seen in figure 10, ܸ௠௔௫
decreases rapidly downstream indicating a too dissipative zone as compared with the
empirical function. Beyond ሺ‫ ݔ‬െ ‫ݔ‬ଵ ሻȀ݀ଵ ൌ ʹͲ, ܸ௠௔௫ stabilizes and a more uniform
velocity field is achieved, compare figure 7 and 10. A small tendency of ܸ௠௔௫ to increase
is also observed which can be explained by the influence of the weir, as discussed above.
As mentioned in the introductory section, several authors have commented on the
existence and the implication of large vortex structures and its effect on the free surface in
the roller region. The vortex paring mechanism and the merging with the stationary
vortex reported in (Long et al., 1991) was not observed in present results. However,
individual vortices translated with increasing size in the downstream direction in
agreement with (Long et al., 1991). Figure 11 present a snapshot of the flow structures
found for the ݀ଵ ȀͶ case run until ͵Ͳ‫ݏ‬. The blue lines represents streamlines and an active
vortex are seen close to the jump toe which is translating downstream as well as a
decaying vortex further downstream. The effects on the free surface of the vortices are
marked in red and green.
22
0
0.1
0.2
0.3
X [m]
0.4
0.5
0.6
Figure 11: Snapshot of vortices in the roller region downstream of the inlet (inlet position at
࢞ ൌ ૙࢓) visualized by streamlines (blue) and free surface waves caused by active vortex (red)
and decaying vortex (green).
In figure 12, the dimensionless average free surface profile ߟȀ݀ଵ is shown and good
agreement between numerical and experimental results are seen, (Kucukali & Chanson,
2007), (Murzyn & Chanson, 2007) and (Chachereau & Chanson, 2010). A minor over
prediction of the average free surface profile is however noted which can be coupled to
the jump toe reaching the inlet and the gate, hence, increasing the depth as discussed
above, compare figure 8 B and figure 12.
23
η/d1
8
Fr1=4.7 ref 1
7
Fr1=5.0 ref 1
6
Fr1=3.1 ref 2
5
Fr1=4.2 ref 2
Fr1=3.2 ref 3
4
Fr1=3.4 ref 3
3
Fr1=3.9 ref 3
Fr1=3.4 ref 4
2
1
0
−10
0
10
20
(x−x1)/d1
30
Figure 12: A comparison of numerical and experimental results of the dimensionless timeaverage free surface profile ࣁȀࢊ૚ as function of dimensionless distance ሺ࢞ െ ࢞૚ ሻȀࢊ૚ from the
jump toe. Ref 1 (Kucukali & Chanson, 2007), ref 2 (Murzyn & Chanson, 2007) and ref 3
(Chachereau & Chanson, 2010).
Figure 13 presents the dimensionless standard deviation of the free surface fluctuations
ߟᇱ Ȁ݀ଵ which indicate a minor under prediction as compared to the experimental results.
Several authors have noted that the maximum standard deviation is obtained downstream
of the jump toe and not at the jump toe itself.
24
Fr1=4.7 ref 1
η′/d1
0.8
Fr1=5.0 ref 1
0.7
Fr1=3.1 ref 2
0.6
Fr1=4.2 ref 2
0.5
Fr1=3.2 ref 3
0.4
Fr1=3.4 ref 3
Fr1=3.9 ref 3
0.3
Fr1=3.4 ref 4
0.2
0.1
0
−10
0
10
20
(x−x1)/d1
30
Figure 13: A comparison of numerical and experimental results of the dimensionless standard
deviation of the free surface fluctuations ࣁᇱ Ȁࢊ૚ as function of dimensionless distance ሺ࢞ െ ࢞૚ ሻȀ
ࢊ૚ from the jump toe. Ref 1 (Kucukali & Chanson, 2007), ref 2 (Murzyn & Chanson, 2007)
and ref 3 (Chachereau & Chanson, 2010).
For the present model and spatial resolution chosen, a shift in the downstream direction
of the location of the maximum standard deviation compared with experimental results is
seen in figure 13. A plausible explanation is that an “inviscid” (artificial viscosity only)
single phase model is used where the dissipative contribution from the air bubbles and
turbulence is not accurately included and thus vortex breakup and diffusion occurs further
downstream. However, when investigating the dimensionless maximum standard
deviation ሺߟᇱ Ȁ݀ଵ ሻ௠௔௫ as function of Froude number ‫ݎܨ‬ଵ a good agreement with
experimental results and the empirical function (equation 4) is observed, see figure 14.
25
(Kucukali & Chanson, 2007)
(Murzyn & Chanson, 2007)
(Chachereau & Chanson, 2010)
0.116(Fr1−1)1.235 (Chachereau & Chanson, 2010)
SPH d1/4 (Present study)
(η′/d1)max
1.5
1
0.5
0
1
2
3
4
5
6
7
8
9
10
Fr1
Figure 14: Numerical and experimental results of maximum dimensionless standard deviation
ሺࣁᇱ Ȁࢊ૚ ሻ࢓ࢇ࢞ as a function of the Froude number ࡲ࢘૚ .
A Fast Fourier Transform (FFT) analysis of the numerical depth at several positions
downstream the jump toe was performed. Figure 15 presents a typical output which can
be compared with outcomes from similar analysis of experimental data, see for instance
(Chachereau & Chanson, 2010).
26
−3
4
x 10
Frequency = 3.54 Hz
3.5
Amplitude [m]
3
2.5
2
1.5
1
0.5
0
0
10
20
30
Frequency [Hz]
40
50
Figure 15: Typical output of the FFT analysis based on the SPH data. Here, case ࢊ૚ Ȁ૝ at
ሺ࢞ െ ࢞૚ ሻΤࢊ૚ ൌ ૡǤ ૠ૞.
The spectral analysis showed dominant frequencies in the range ʹ‫ ݖܪ‬to ͷ‫ ݖܪ‬and a
significant peak at approximately ͵Ǥͷ‫ ݖܪ‬which is in agreement with experimental
observations, see (Kucukali & Chanson, 2007), (Murzyn & Chanson, 2007) and
(Chachereau & Chanson, 2010). Figure 16 presents the characteristic free surface
fluctuation frequency ‫ܨ‬௙௦ as function of the dimensionless distance downstream the jump
toe ሺ‫ ݔ‬െ ‫ݔ‬ଵ ሻȀ݀ଵ . In the experiments, the FFT analysis indicated several characteristic
frequencies which here are shown as both individual dots and lines. For the numerical
results, the characteristic frequency is shown only.
27
Fr1=5.1 (Chachereau & Chanson, 2010)
Fr1=3.9 (Chachereau & Chanson, 2010)
Fr1=3.4 (Chachereau & Chanson, 2010)
Fr1=3.2 (Chachereau & Chanson, 2010)
Fr1=3.4 SPH d1/4 (Present study)
4
Ffs [Hz]
3.5
3
2.5
2
1.5
0
5
10
15
20
(x−x1)/d1
25
30
35
Figure 16: Numerical and experimental results of free surface fluctuation frequency ࡲࢌ࢙ ሾࡴࢠሿ
as function of the dimensionless distance downstream of the jump toe ሺ࢞ െ ࢞૚ ሻȀࢊ૚ .
Two trends are observed in figure 16. Firstly, the numerical result over predicts the
significant frequency. Still it should be noted, that the intensity or amplitude was greatly
reduced in the horizontal direction for the FFT outcome but the most significant
frequency was still at approximately ͵Ǥͷ‫ݖܪ‬. Secondly, the general trend of decreasing
frequency in the downstream direction observed for the experimental data is not captured
by the SPH model. Yet again, the “inviscid” single phase model could be a plausible
explanation to this behaviour causing elevated vortex generation rates and as a
consequence too high free surface fluctuation frequencies.
Figure 17 presents the dimensionless Strouhal number ܵ‫ݐ‬௙௦ as function of the inflow
Froude number ‫ݎܨ‬ଵ . A minor under prediction of ܵ‫ݐ‬௙௦ is observed when comparing with
the empirical function (equation 5) proposed by (Chachereau & Chanson, 2010),
however, numerical results are within the experimental range.
28
SPH d1/4 (Present study)
(Murzyn & Chanson, 2009)
(Chachereau & Chanson, 2010)
0.143exp(−0.27Fr1) (Chachereau & Chanson, 2010)
0.12
0.1
Stfs
0.08
0.06
0.04
0.02
0
2
3
4
5
6
7
Fr1
Figure 17: Numerical and experimental results of free surface fluctuation Strouhal number
ࡿ࢚ࢌ࢙ as function of Froude number ࡲ࢘૚ .
Conclusions
Two dimensional hydraulic jumps have been investigated in present work using the
meshfree, Lagrangian particle based method Smoothed Particle Hydrodynamics. Four
cases with different spatial resolution of the SPH particles was set up and as a general
result more flow features was observed for the highly resolved cases. It was also noted
that the propagations of the jump toe differed among the cases and as a consequence of
the frictionless boundary condition used, the jump toe propagated all the way to the inlet
for the coarsest case. Furthermore, a fictitious boundary layer was observed close to the
bottom which affected the incoming jet and was likely caused by the truncated kernel due
to the lack of SPH nodes outside the boundary. The conjugate depth ݀ଶ was compared
with the theoretical value for the various spatial resolutions and good agreement was
achieved. Some increased oscillations were however seen for the most resolved case. For
the coarsest case which was run for ͵Ͳ‫ݏ‬, additional features was investigated as a quasi29
stationary state was reached past ͳͷ‫ݏ‬. The vertical velocity distribution compared fairly
good with experimental results even though the length of the jump was under predicted
by roughly ʹͷΨ. The maximum velocity in horizontal direction indicated a too
dissipative zone past the roller which is likely caused by the “inviscid” and single phase
model used. The investigation of vortex structures and its effect on the free surface
showed generally good agreement for the mean and the standard deviation, even though
the peak in the standard deviation occurred further downstream as compared to the
experimental results. However, when comparing the maximum standard deviation as
function of the Froude number a positive result was obtained. The investigation of free
surface fluctuation frequencies indicated a general over prediction and that the
longitudinal decay was not captured by the SPH model. Also, a minor under estimation of
the Strouhal number was obtained even though the outcome was within the range of
experiments.
The SPH method is interesting for hydropower applications in general and especially for
hydropower hydraulics. However, boundary conditions and resolution scaling behaviour
needs to be further developed.
Acknowledgement
The research presented 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. www.svc.nu
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