Thesis defense

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

2

Objective
Introduction
SPH General View
Literature Survey
Numerical Model
Results
Conclusion
Future Work
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• 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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6

• Numerical solution merits:
1. Fast Performance
2. Cheapness
3. Compromising results
• Famous Numerical Method
Prediction
&
Validation
Mesh Based Methods
CSM, CFD & CHT
7

Mesh deformation
Results inaccuracy
Huge memories & processors
High computational time
Meshed Methods Simulation Problems
BREAKDOWN
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• SPH - Smoothed particle hydrodynamics
• Mesh-less Lagrangian numerical method
• Firstly used in 1977
• Developed for Solid mechanics, fluid dynamics
• Competitive to traditional numerical method
11

Mesh MethodMeshless Method
12

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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Math
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Momentum equation
Energy equation
Continuity equation Density summation
22

Heat Conduction equation
Equation of state
Adiabatic sound speed equation
23

• 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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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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2- Particles interpenetration treatment
Sharp variations in the flow & wave discontinuities
Particles interpenetration and system collapse
SOLUTION
a) Artificial Viscosity b) Average Velocity
27

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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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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• 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.
31

• 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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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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• 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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I. Shock tube
1- Validation
Pressure and internal energy distribution inside shock tube after 0.2s (2 solution)
36

I. Shock tube
1- Validation
Density and velocity distribution inside shock tube after 0.2s (2 solution)
37

I. Shock tube
2- Progressive time
Properties distribution inside shock tube after wave reflection
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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 ( )
41
Comparison of maximum percentage error for different smoothing length

II. 1-D heat conduction
2- Error Analysis
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C) 2-D transient conduction with isothermal boundaries
ti=100 oC a=10 cm
aAnalytical solution
44

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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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

Temperature Contours after 8s (3 solutions)
47
1- Smoothing Length and Virtual Particles effect

2- Virtual Particles effect
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2- Virtual Particles effect
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D) 1-D/ 2-D adiapatic compression stroke
Specification: D=0.1285 m, Ls=1.2D = 0.15842 m, N= 1000 rpm, rc = 6
Medium (Air): Pi= 1*105 Pa, Ti= 300 K, ρi = 0.973 kg/m3, ui=0 m/s
Cv= 717.5 J/kg, γ= 1.4
Time step: dτ=0.00001 sec
Virtual
particles
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1-D 2-D
Discretization
Total: Nx
Boundary: (2)
Interior: (Nx-2)
Total: Na= (Nx) x (Ny)
Boundary: 2Ny + 2(Nx-2)
Interior: (Nx-2) x (Ny-2)
Smoothing length
(h)
Smoothing Kernel
function
B-spline kernel function B-spline kernel function
Boundary
repulsive force
D=0.01 m2/sec2,
r0= 1.25x10-5 m, n1=12 & n2=4
D = 2.75x10-3 m2/sec2,
r0= 0.15 dx, n1 = 12, n2 = 4
Artificial Viscosity απ=0.1, βπ= 0 & φ=0. 1h απ= 0.005, βπ= 0.005 & φ=0. 1h
Average Velocity ϵ = 0.9
Reference Isentropic relation Isentropic relation
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• Virtual Particles & boundary conditions
- Variable no.
- Symmetry conditions at cylinder wall
- Moving piston boundary conditions:
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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
31 41 51 61 71 81
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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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
41 51 61 71
Absolutemaximumpercentageerror(%)
Percentage error for different particles
numbers
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
41 51 61 71
computationaltime(sec)
calculation time for different particles
numbers
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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

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V. 2-D Compression stroke
1- Transient Period
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Cylinder properties variation inside the cylinder with crank angle

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• 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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• 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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• 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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