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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).

Thesis defense Thesis defense Presentation Transcript

  • 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
  • MSc. Thesis Defense 2013 Faculty of Engineering Cairo University 2
  • MSc. Thesis Defense 2013 Faculty of Engineering Cairo University Objective Introduction SPH General View Literature Survey Numerical Model Results Conclusion Future Work 3
  • MSc. Thesis Defense 2013 Faculty of Engineering Cairo University 4
  • 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. 5
  • MSc. Thesis Defense 2013 Faculty of Engineering Cairo University 6
  • MSc. Thesis Defense 2013 Faculty of Engineering Cairo University • Numerical solution merits: 1. Fast Performance 2. Cheapness 3. Compromising results • Famous Numerical Method Prediction & Validation Mesh Based Methods CSM, CFD & CHT 7
  • MSc. Thesis Defense 2013 Faculty of Engineering Cairo University Mesh deformation Results inaccuracy Huge memories & processors High computational time Meshed Methods Simulation Problems BREAKDOWN 8
  • MSc. Thesis Defense 2013 Faculty of Engineering Cairo University 9
  • MSc. Thesis Defense 2013 Faculty of Engineering Cairo University 10
  • 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 11
  • MSc. Thesis Defense 2013 Faculty of Engineering Cairo University Mesh MethodMeshless Method 12
  • MSc. Thesis Defense 2013 Faculty of Engineering Cairo University Fluid is continuum and not discrete Properties of particles V, P, T, etc. have to take into account the properties of neighbor particles 13
  • MSc. Thesis Defense 2013 Faculty of Engineering Cairo University Math 14
  • MSc. Thesis Defense 2013 Faculty of Engineering Cairo University Momentum equation Energy equation Continuity equation Density summation 22
  • MSc. Thesis Defense 2013 Faculty of Engineering Cairo University Heat Conduction equation Equation of state Adiabatic sound speed equation 23
  • MSc. Thesis Defense 2013 Faculty of Engineering Cairo University • 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 24
  • MSc. Thesis Defense 2013 Faculty of Engineering Cairo University 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. 25
  • MSc. Thesis Defense 2013 Faculty of Engineering Cairo University 26
  • MSc. Thesis Defense 2013 Faculty of Engineering Cairo University 2- Particles interpenetration treatment Sharp variations in the flow & wave discontinuities Particles interpenetration and system collapse SOLUTION a) Artificial Viscosity b) Average Velocity 27
  • 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. 28
  • 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. 29
  • MSc. Thesis Defense 2013 Faculty of Engineering Cairo University 30
  • 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. 31
  • 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. 32
  • MSc. Thesis Defense 2013 Faculty of Engineering Cairo University 33
  • 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 34
  • 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 35
  • 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) 36
  • 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) 37
  • MSc. Thesis Defense 2013 Faculty of Engineering Cairo University I. Shock tube 2- Progressive time Properties distribution inside shock tube after wave reflection 38
  • 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 39
  • MSc. Thesis Defense 2013 Faculty of Engineering Cairo University • Smoothing length (h) • Smoothing Kernel function B-spline kernel function 40
  • MSc. Thesis Defense 2013 Faculty of Engineering Cairo University II. 1-D heat conduction 1- Optimum Smoothing length percentage error ( ) 41 Comparison of maximum percentage error for different smoothing length
  • MSc. Thesis Defense 2013 Faculty of Engineering Cairo University II. 1-D heat conduction 2- Error Analysis 42
  • MSc. Thesis Defense 2013 Faculty of Engineering Cairo University 43
  • MSc. Thesis Defense 2013 Faculty of Engineering Cairo University C) 2-D transient conduction with isothermal boundaries ti=100 oC a=10 cm aAnalytical solution 44
  • MSc. Thesis Defense 2013 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 45
  • 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 46 1- Smoothing Length effect
  • MSc. Thesis Defense 2013 Faculty of Engineering Cairo University III. 2-D Heat Conduction Temperature Contours after 8s (3 solutions) 47 1- Smoothing Length and Virtual Particles effect
  • MSc. Thesis Defense 2013 Faculty of Engineering Cairo University III. 2-D Heat Conduction 2- Virtual Particles effect Temperature Contours after 8s (3 solutions) 48
  • MSc. Thesis Defense 2013 Faculty of Engineering Cairo University III. 2-D Heat Conduction 2- Virtual Particles effect Temperature Contours after 8s (3 solutions) 49
  • MSc. Thesis Defense 2013 Faculty of Engineering Cairo University 50
  • MSc. Thesis Defense 2013 Faculty of Engineering Cairo University 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 51
  • MSc. Thesis Defense 2013 Faculty of Engineering Cairo University 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 52
  • MSc. Thesis Defense 2013 Faculty of Engineering Cairo University • Virtual Particles & boundary conditions - Variable no. - Symmetry conditions at cylinder wall - Moving piston boundary conditions: 53
  • 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 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) 54
  • 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 41 51 61 71 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 41 51 61 71 computationaltime(sec) Number of Particles Nx calculation time for different particles numbers 55
  • 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) 56 Pistonlocation Cylinderhead Pistonlocation Cylinderhead Pistonlocation Cylinderhead Pistonlocation Cylinderhead
  • MSc. Thesis Defense 2013 Faculty of Engineering Cairo University 57 IV. 2-D Compression stroke
  • MSc. Thesis Defense 2013 Faculty of Engineering Cairo University V. 2-D Compression stroke 1- Transient Period 58 Cylinder properties variation inside the cylinder with crank angle
  • MSc. Thesis Defense 2013 Faculty of Engineering Cairo University 59
  • 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. 60
  • 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. 61
  • MSc. Thesis Defense 2013 Faculty of Engineering Cairo University 62
  • 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 63
  • MSc. Thesis Defense 2013 Faculty of Engineering Cairo University 64
  • MSc. Thesis Defense 2013 Faculty of Engineering Cairo University 65