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 METHODحرارية لحاالتسريان عددية نمذجة

الهيدروديناميكية الجزيئات طريقة باستخدامالمتقاربة

Under supervision ofProf. Essam E. Khalil Dr. Essam Abo-Serie

Dr. Hatem Haridy

Presented byEng. Tarek M. ElGammal



الرحيم الرحمن الله بسمد+ر*ي ص, ل*ي ح+ ر, اش+ ب1 ر, ال, ق,ل6ل+ و,اح+ م+ر*ي

أ, ل*ي ر+ ي,س1 و,وا ه6 ق, ي,ف+ ان*ي ل1س, م1ن د,ة: ع6ق+

و+ل*ي ق,العظيم الله صدق

2



OutlineObjective

Introduction

SPH General View

Literature Survey

Numerical Model

Results

Conclusion

Future Work3



OBJECTIVE

4



• 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.

Objective

5



INTRODUCTION

6



Introduction

• Numerical solution merits:1. Fast Performance2. Cheapness 3. Compromising results• Famous Numerical Method

Prediction &

Validation

Mesh Based Methods

CSM, CFD & CHT7



Introduction

Mesh deformation Results inaccuracy

Huge memories & processors

High computational time

Meshed Methods Simulation Problems

BREAKDOWN

8



Introduction

9



SPH GENERAL VIEW

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SPH General view

• 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 Method

SPH General view

Meshless Method

12



Fluid is continuum and not discrete

Properties of particlesV, P, T, etc. haveto take into accountthe properties ofneighbor particles

SPH General view

13



Math

SPH General view

14

Math

Math

Math

Math



SPH General view

Momentum equation

Energy equation

Continuity equation Density summation

22



SPH General viewHeat Conduction equation

Equation of state

Adiabatic sound speed equation

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• Important Additions1- Boundary deficiency treatments:

Truncation of the particle kernel zone by the solid boundary (or the free surface)

SPH General view

Inaccurate results for particles near the boundary and unphysical penetrations.

SOLUTIONa) Boundary Particles b) Virtual Particles

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1a) Boundary ParticlesParticles are located at the boundaries to produce a

repulsive force for every fluid particle within its kernel.

SPH General view

1b) Virtual (Ghost) ParticlesThese particles have the same values depending on the interior real particles nearby the boundaries which act as mirrors.

25



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2- Particles interpenetration treatment

SPH General view

Sharp variations in the flow & wave discontinuities

Particles interpenetration and system collapse

SOLUTIONa) Artificial Viscosity b) Average Velocity

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2a) Artificial viscosityComposed 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.

SPH General view

28



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.

SPH General view

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LITERATURE SURVEY

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• Shock tubeLiu 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 conductionFinite 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.

Literature Survey

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• 2-D Heat conductionR. 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 StrokeFazio 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.

Literature Survey

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NUMERICAL MODEL/ RESULTS

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A) Shock tube

Numerical Model

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/kgu= 0 m/s u= 0 m/sNx=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

Numerical Model

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. 1h35



I. Shock tube1- Validation

Results

Pressure and internal energy distribution inside shock tube after 0.2s (2 solution)36



I. Shock tube1- Validation

Results

Density and velocity distribution inside shock tube after 0.2s (2 solution)37



I. Shock tube2- Progressive time

Results

Properties distribution inside shock tube after wave reflection38



B) 1-D Heat Conduction

Numerical Model

ti=0 oCtb1=100 oCtb2=0 oC

L =1 cm

ρ=2700 kg/m3 α = 0.84 cm2/sec

F.D. (C.N.) SPH

dx=0.1 cm, dτ=0.01 secAnalytical solution

39



• Smoothing length (h)

• Smoothing Kernel function

Numerical Model

B-spline kernel function

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II. 1-D heat conduction1- Optimum Smoothing lengthpercentage error ( )

41

Results

Comparison of maximum percentage error for different smoothing length



II. 1-D heat conduction2- Error Analysis

42

Results



43

Results



C) 2-D transient conduction with isothermal boundaries

ti=100 oC a=10 cm

a

tb= 0 oC

Analytical solution

Numerical Model

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1521 particles

BoundaryParticles160

dx

Smoothing length (h): h= C . dx (parametric study)

Kernel Function: Cubic B-Spline, dt=0.001 s

Virtual Particles

Numerical Model

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III. 2-D Heat Conduction

Minimum errorat the centre region

Error = Tref – Tc

Tref is the analytical solution temperature

Tc is computed SPH temperature

Results

46

1- Smoothing Length effect



III. 2-D Heat ConductionResults

Temperature Contours after 8s (3 solutions)47

1- Smoothing Length and Virtual Particles effect



III. 2-D Heat Conduction2- Virtual Particles effect

Results




III. 2-D Heat Conduction 2- Virtual Particles effect

Results




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Results



D) 1-D/ 2-D adiapatic compression stroke

Numerical Model

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

DiscretizationTotal: 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 ـــــــــ ϵ = 0.9

Reference Isentropic relation Isentropic relation

Numerical Model

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• Virtual Particles & boundary conditions

Numerical Model

- Variable no. - Symmetry conditions at cylinder wall- Moving piston boundary conditions:

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IV. 1-D Compression stroke1- Optimum Smoothing Length

Results

Optimum smoothing length of different particle number based on minimum error of pressure

31 41 51 61 71 810

0.5

1

1.5

2

2.5

3

f(x) = − 3.88578058618805E-18 x² + 0.0400000000000004 x − 0.240000000000012

optimized factor of smoothing length at different particles numbers

hopt/dx

Polynomial (hopt/dx)

Number of Particles Nx

optim

um s

moo

thin

g le

ngth

fact

or

(h_o

pt/d

x)

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IV. 1-D Compression stroke1- Optimum Smoothing Length

Results

Percentage error and time consumption of different particles number

41 51 61 710

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

Percentage error for different particles numbers


Abs

olut

e m

axim

um p

erce

ntag

e er

ror

(%)

41 51 61 710

0.51

1.52

2.53

3.54

4.5

calculation time for different particles numbers


com

puta

tiona

l tim

e (s

ec)

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IV. 1-D Compression stroke2- Transient Period

Results

Properties variation inside the cylinder at different times (compared to the reference value)

56

Pis

ton

loca

tion

Cylin

der h

ead

Pis

ton

loca

tion Cy

linde

r hea

d

Pis

ton

loca

tion

Cylin

der h

ead

Pis

ton

loca

tion

Cylin

der h

ead



57

ResultsIV. 2-D Compression stroke



V. 2-D Compression stroke1- Transient Period

58

Results

Cylinder properties variation inside the cylinder with crank angle



CONCLUSIONS

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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.

Conclusions

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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.

Conclusions

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FUTURE WORK

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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

Future Work

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Education

Thesis defense