1. MSc. Thesis Defense 2013
Faculty of Engineering Cairo University
MSc. THESIS DEFENSE
on
NUMERICAL SIMULATION FOR THERMAL
FLOW CASES USING SMOOTHED PARTICLE
HYDRODYNAMICS METHOD
Under supervision of
Prof. Essam E. Khalil Dr. Essam Abo-Serie
Dr. Hatem Haridy
Presented by
Eng. Tarek M. ElGammal
3. MSc. Thesis Defense 2013
Faculty of Engineering Cairo University
Objective
Introduction
SPH General View
Literature Survey
Numerical Model
Results
Conclusion
Future Work
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5. MSc. Thesis Defense 2013
Faculty of Engineering Cairo University
• Introducing the mesh-less method (Smoothed
Particle Hydrodynamics: SPH) as a promising
alternative for computing engineering
problems.
• Comparison with the meshed approach based
on the accuracy and time consumption.
• Optimizing the solution parameters to
maintain stability and reduce error.
• Trying to make a good start to develop a
software package for solving engineering
cases.
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11. MSc. Thesis Defense 2013
Faculty of Engineering Cairo University
• SPH - Smoothed particle hydrodynamics
• Mesh-less Lagrangian numerical method
• Firstly used in 1977
• Developed for Solid mechanics, fluid dynamics
• Competitive to traditional numerical method
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Mesh MethodMeshless Method
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Fluid is continuum
and not discrete
Properties of particles
V, P, T, etc. have
to take into account
the properties of
neighbor particles
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Momentum equation
Energy equation
Continuity equation Density summation
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Heat Conduction equation
Equation of state
Adiabatic sound speed equation
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• Important Additions
1- Boundary deficiency treatments:
Truncation of the particle kernel zone by the solid
boundary (or the free surface)
Inaccurate results for particles near the boundary
and unphysical penetrations.
SOLUTION
a) Boundary Particles b) Virtual Particles
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18. MSc. Thesis Defense 2013
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1a) Boundary Particles
Particles are located at the boundaries to produce a
repulsive force for every fluid particle within its
kernel.
1b) Virtual (Ghost) Particles
These particles have the same values depending on the
interior real particles nearby the boundaries which act
as mirrors.
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20. MSc. Thesis Defense 2013
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2- Particles interpenetration treatment
Sharp variations in the flow & wave discontinuities
Particles interpenetration and system collapse
SOLUTION
a) Artificial Viscosity b) Average Velocity
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21. MSc. Thesis Defense 2013
Faculty of Engineering Cairo University
2a) Artificial viscosity
Composed of shear and bulk viscosities to transform
the sharp kinetic energy into heat.
It’s represented in a form of viscous dissipation term in
the momentum & energy equations.
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22. MSc. Thesis Defense 2013
Faculty of Engineering Cairo University
2b) Average velocity (XSPH ):
It makes velocity closer to the average velocity of the
neighboring particles. In incompressible flows, it can
keep the particles more orderly. In compressible
flows, it can effectively reduce unphysical
interpenetration.
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24. MSc. Thesis Defense 2013
Faculty of Engineering Cairo University
• Shock tube
Liu G. R. and M. B. Liu (2003): Introduction of SPH
solution for shock wave propagation inside 1-D shock
tube and comparison to G. A. Sod finite difference
solution (1978).
Limitation: Incomplete solution due to boundary deficiency
• 1-D Heat conduction
Finite Difference solution based on (Crank Nicholson)
solution for time developed function in 1-D space.
Limitation: Solution in SPH for transient period doesn’t exist.
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25. MSc. Thesis Defense 2013
Faculty of Engineering Cairo University
• 2-D Heat conduction
R. Rook et al. (2007): Formula for Laplacian derivative.
2-D heat conduction within a square plate of
isothermal walls compared to the analytical solution.
Limitation: Simple value of (h) besides boundary deficiency
• Compression Stroke
Fazio R. & G. Russo (2010) Second order boundary
conditions for 1-D piston problems solved by central
lagrangian scheme
Limitation: Solution in SPH for transient period of compression
stroke doesn’t exist.
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27. MSc. Thesis Defense 2013
Faculty of Engineering Cairo University
A) Shock tube
P= 1 N/m2 P= 0.1795 N/m2
ρ= 1 kg/m3 ρ= 0.25 kg/m3
e= 2.5 kJ/kg e= 1.795 kJ/kg
u= 0 m/s u= 0 m/s
Nx=320 Nx=80
m= 0.00187 kg, Cv= 0.715 kJ/kg.K, γ= 1.4, dτ=0.005 s
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28. MSc. Thesis Defense 2013
Faculty of Engineering Cairo University
• Smoothing length (h)
• Smoothing Kernel function
• Virtual Particles & boundary conditions
• Boundary force
• Artificial Viscosity
B-spline kernel function
Fixed no./ symmetry conditions (except the velocity)
D=0.01, r0= 1.25x10-5 m, n1=12 & n2=4
απ=βπ= 1 & φ=0. 1h
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29. MSc. Thesis Defense 2013
Faculty of Engineering Cairo University
I. Shock tube
1- Validation
Pressure and internal energy distribution inside shock tube after 0.2s (2 solution)
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30. MSc. Thesis Defense 2013
Faculty of Engineering Cairo University
I. Shock tube
1- Validation
Density and velocity distribution inside shock tube after 0.2s (2 solution)
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31. MSc. Thesis Defense 2013
Faculty of Engineering Cairo University
I. Shock tube
2- Progressive time
Properties distribution inside shock tube after wave reflection
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32. MSc. Thesis Defense 2013
Faculty of Engineering Cairo University
B) 1-D Heat Conduction
ti=0 C
tb1=100 Ctb2=0 C
L =1 cm
ρ=2700 kg/m3 α = 0.84 cm2/sec
F.D. (C.N.) SPH
dx=0.1 cm, dτ=0.01 sec
Analytical solution
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• Smoothing length (h)
• Smoothing Kernel function
B-spline kernel function
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II. 1-D heat conduction
1- Optimum Smoothing length
percentage error ( )
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Comparison of maximum percentage error for different smoothing length
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II. 1-D heat conduction
2- Error Analysis
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Faculty of Engineering Cairo University
C) 2-D transient conduction with isothermal boundaries
ti=100 oC a=10 cm
aAnalytical solution
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Faculty of Engineering Cairo University
1521 particles
Boundary
Particles
160
dx
Smoothing length (h): h= C . dx (parametric study)
Kernel Function: Cubic B-Spline, dt=0.001 s
Virtual
Particles
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39. MSc. Thesis Defense 2013
Faculty of Engineering Cairo University
III. 2-D Heat Conduction
Minimum error
at the centre region
Error = Tref – Tc
Tref is the analytical
solution temperature
Tc is computed
SPH temperature
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1- Smoothing Length effect
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III. 2-D Heat Conduction
Temperature Contours after 8s (3 solutions)
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1- Smoothing Length and Virtual Particles effect
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III. 2-D Heat Conduction
2- Virtual Particles effect
Temperature Contours after 8s (3 solutions)
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42. MSc. Thesis Defense 2013
Faculty of Engineering Cairo University
III. 2-D Heat Conduction
2- Virtual Particles effect
Temperature Contours after 8s (3 solutions)
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47. MSc. Thesis Defense 2013
Faculty of Engineering Cairo University
IV. 1-D Compression stroke
1- Optimum Smoothing Length
Optimum smoothing length of different particle number
based on minimum error of pressure
y = 0.04x - 0.24
0
0.5
1
1.5
2
2.5
3
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optimumsmoothinglengthfactor
(h_opt/dx)
Number of Particles Nx
optimized factor of smoothing length
at different particles numbers
hopt/dx
Poly. (hopt/dx)
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48. MSc. Thesis Defense 2013
Faculty of Engineering Cairo University
IV. 1-D Compression stroke
1- Optimum Smoothing Length
Percentage error and time consumption of different particles number
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
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Absolutemaximumpercentageerror(%)
Number of Particles Nx
Percentage error for different particles
numbers
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
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computationaltime(sec)
Number of Particles Nx
calculation time for different particles
numbers
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49. MSc. Thesis Defense 2013
Faculty of Engineering Cairo University
IV. 1-D Compression stroke
2- Transient Period
Properties variation inside the cylinder at different times (compared to the
reference value)
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Pistonlocation
Cylinderhead
Pistonlocation
Cylinderhead
Pistonlocation
Cylinderhead
Pistonlocation
Cylinderhead
50. MSc. Thesis Defense 2013
Faculty of Engineering Cairo University
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IV. 2-D Compression stroke
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Faculty of Engineering Cairo University
V. 2-D Compression stroke
1- Transient Period
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Cylinder properties variation inside the cylinder with crank angle
53. MSc. Thesis Defense 2013
Faculty of Engineering Cairo University
• SPH is (Real Flow) solution.
• Adaptive nature is a merit for solving complex
problems.
• SPH converges better than F.D. In some case.
• Every solution has an optimum smoothing length
(hopt ) .
• hopt changes at different number of discretizing
particles (N). May other parameters affect it like the
initial gradients and material properties.
• Virtual Particles are capable of solving boundary
inconsistency and improper penetrations.
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54. MSc. Thesis Defense 2013
Faculty of Engineering Cairo University
• Boundary conditions should be carefully treated at
the virtual particle to obtain the adequate results.
• Two Techniques of virtual particles are: fixed or
variable number.
• Suitable small value coefficients in SPH solution
controlling terms.
• For well simulating the discontinuity waves, reviewed
artificial viscosity and DSPH are recommended in
such cases.
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56. MSc. Thesis Defense 2013
Faculty of Engineering Cairo University
• Working on more complex cases (industry).
• Introducing Laminar shear term/turbulence
models.
• Relating between (hopt) and initial physical
quantities (e.g. temperature gradient and
particles spacing).
• Using variable smoothing length based on the
problem gradients is an important issue.
• Coding using more efficient software
products: e.g. Python, Octave
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Engineering Problem solutions:Analytical, Experimental, Numerical
Cases: High deformable bodies, Free surface, Waves discontinuities, multi-phase flows.
High deformations, free surface flow and material interfaces
Mesh-less Lagrangian numerical methodFirstly used in 1977Has many versions: SPH, Incompressible SPH (ISPH), Weakly compressible SPH (WCSPH), Discontinuous SPH (DSPH), Corrective SPH (CSPH) & Adaptive SPH (ASPH). ISPH to solve the possion pressure equation for incomp. Flow…WCSPH proposes pressure equation of state depending the density and speed of soundRSPH for magneto-hydrodynamics…..CSPH add some corrective and normalizing terms…..ASPH introduces anisotropic kernel function
Discretize the domain into unconnected particles (Adaptive nature).Each particle has its own properties: m, dv, ρ, T, P, u, v, w
Property & gradients are approximated with the help of the neighbor particles (Smoothing Kernel Function).Neighbor particles are determined by Smoothing length (h)
Skipping many detailing and boring math (as we are engineers not mathematician), only we will stress on some
1- The SPH mathematics starts with the integral representation of a function at a point using the Dirac delta function2- An approximation done in the integration using a weighting function (W) instead of the Dirac delta to evaluate the function from all neighboring points (particles)
1-Continuing to the 1st and 2ndgradients (by using 2nd order truncated Taylor series), we get the gradient forms in integral representation2- It should be noted that in case of 2D and 3D, the (r= radial distance) replaces the horizontal 1D distance (x)
1-So for a function (i.e. property like density or temperature which is our case) in a 2D varying space, the function and derivative integral calculations at a point (or particle) are approximated in a discrete weighted summation from neighboring particles (including the particle of interest)2- These neighbors are determined within a limited zone, even kernel function W. This function area is determined by a predefined distance called smoothing length (h)
The smoothing kernel function: is the function that relates the effect of the neighboring particles on the particle of interest. This happens in a limited zone to neglect the far particles effect without losing the accuracyThe number (2) is a function dependent factor (K) to define the neighboring effect
Based on the mentioned properties, the smoothing kernel function is limited upon a group of particles, taking a positive bell shape normalizing curve and tends to be delta function on a small differential particle. In addition, the kernel function has merit of logical share for particles such that the amount of contribution is proportional with the distance from the center even if the relative position is reversed. This should be represented in a well organized way to show how the kernel function and its derivative is smoothly change from one point to another (i.e. no discontinuities) especially in case of particles disorder.
Gaussian type of kernel function that has the privilege of stability and fast smoothness even for disorder particles despite of the missed real compactness which implies an increase in the support domain and consequently the computation time.Cubic is Gaussian compact and also the quadratic and quintic but the are very long
The first two solution are presenting better results than CSPH beside they perfectly reflect the boundary conditions .
Repulsive force is added to the momentum equation as an external force, while virtual particles affects like the real neighboring particles
DSPH is Discontinuous SPH approach which uses edited terms at the particles of the wave front
αΠ , βΠ are constants that are all typically set around 1.0 , ϕ = 0.1hijThe viscosity associated with αΠ produces a bulk viscosity, while the second term associated with βΠ, which is intended to suppress particle interpenetration at high Mach number.
This term is added to the velocity not the momentum (acceleration) equation
The following will show the cases modeled and their results
MatLab coded from Fortran….Adiabatic
Sharp variation at the contact discontinuity in SPH solution (especially in the internal energy….error accumulation)…..after period of time these variation will disappear
Here I began to check if there is a best value for the smoothing length to minimize the solution error
Because of the very close and almost zero errors at the steady state, it’s preferred to make the comparison of (h) based on error of the first time step
The computational time for SPH calculations is tc= 0.503 sec while it is tc= 0.76 sec in C-N calculations.From graphs, SPH generates results of low accuracy at the very early transient period (i.e. first time steps) which is fortunately inconsiderable. After some time steps the percentage error enters the reasonable margin. Compared to (C-N) solution, SPH errors at first is much higher but they turn to be minimized to such negligible values meanwhile C-N results are in the same range of error during the whole time progress
1- MatLab software was the helpful tool to use (it has many predefined important functions and doesn’t need to variables declaration like Fortran)2- Using MatLab, the domain is discretized to equally spaced points which represent particles of equal differential volumes, densities and masses and have temperatures of specific places (internal and boundary)
Some inconsistency arose in the no-virtual solution. This error may be larger if we have more than one governing equation to calculate
The low value of properties at the piston head during the first period may be because of the rarefaction wave moves opposite to the pressure waves or due the applied boundary condition. Short stroke prevented that due high velocity wave propagating
1- (Real Flow) as it simulates motion and interaction of fluid masses not interpolation points as in meshed methods2- Adaptive nature= particles move freely without any commitment to another particles (no connectivity)/ Smooth gradients in the results of compression stroke and heat diffusion are strong proof of the efficiency of the SPH approach. 3- optimum smoothing length (hopt ) which minimize the error. Other values disorient the smoothing function from the actual results.5- This appeared in the 1-D shock tube where good results of wave reflections and acceptable result at the boundaries.
1- F.D. scheme is preferable to discretize these conditions./ Isothermal boundaries can skip the virtual particles because of low error propagation2- Also fixed no. of virtual particles can be used in some case like Neumann boundary conditions in CHT but they may not help well in case of CFD (variable virtual particles are better)3- to prevent any inconvenient results and not dominating the numerical solution
1- with more comparison with meshed methods especially F.E. and F.V.1- Solving some industrial problems (e.g. air bubbles detection in casting process).
