Computational Fluid Dyna mics

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Computational Fluid Dyna mics

Real-time animation of low-Reynolds-number flow using Smoothed Particle Hydrodynamics

presentation by 薛德明 Dominik Seifert B97902122

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Visualization of my results

http://csie.ntu.edu.tw/~b97122/archive/fluid_dyn amics/capture.mp4

My code

http://csie.ntu.edu.tw/~b97122/archive/fluid_

dynamics/OpenTissue_backup.rar

My motivation: From Dust

http

://www.youtube.com/watch?v=CfKQCAxizrA

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The framework OpenTissue

 OpenTissue is an open source simulation framew ork with a simple, working SPH implementation

 I added some features to their implementation

 Their original implementation is described in this 88 page document:

 “Lagrangian Fluid Dynamics Using Smoothed Parti cle Hydrodynamics”

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Smoothed Particle Hydrodynamics Quick review

 Note: SPH particles are not actual particles!

 They are really fluid samples of constant mass

 Particles are placed inside a container to represen t a fluid

 Every particle is then assigned a set of initial pr operties

 After every time step (a few milliseconds), update the properties of all particles

 Use interpolation methods to solve the Navier-Stoke s equation to find force contributions, then integr ate to find new velocity and position

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SPH

Particle Properties

Support Radius (constant)

Minimum interaction distance between particles

Mass (constant; ensures Conservation of Mass explicitly)

Position (varying)

Surface normal (varying)

Velocity (varying)

Acceleration (varying)

Sum of external forces (varying)

Density (varying due to constant mass and varying volum e)

Pressure (varying)

Viscosity (constant)

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Smoothed Particle Hydrodynamics

Solver pseudocode

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Smoothed Particle Hydrodynamics The pressure problem

 Fluids cannot be accurately incompressible

 Pressure value approximated by Ideal Gas Law:

 k called “gas-stiffness”

 Entails assumptions of gas in steady state

 Does not consider weight pressure

 Causes “pulsing” because of lagging interplay between gravity a nd pressure force

 Large gas-stiffness can reduce/eliminate the lag and the pulsing

 Alternatively, take density to the power of heat capacity ratio

 But high pressure requires a smaller time-step and thus makes the simulation more expensive

pV = nRT ⇒ p= ( k ' ) V = k ρ

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My contributions I

Fluid-fluid interactions

 In OpenTissue, a system can be comprised of only a single fluid

 I changed the code to support more than one fluid at a tim e

 The math and physics are mostly the same, except for:

 Viscosity

 Kernel support radius

 Still missing:

 Surface tension interfaces between fluids of different polarity

 But I still spent three days on changing the framework due to heavily templated C++ code

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My contributions II

Fluid-solid interactions

 In OpenTissue only supports static (immovable) obje cts

 I wanted to add the ability to add objects that int eract with the fluid, objects that can float, sink etc.

 Solid dynamics are very complicated! What are…

 Tensile Strength?

 Compressive Strength?

 Young’s modulus?

 I came up with an intuitive but not quite correct a pproach

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My contributions II Restoring force

 Place an invisible spring between every two pa rticles that are close to each other initially

 Store the initial distance between every two

“neighboring” particles

 Add a new spring force component that contribu tes:

k * abs(current_distance – original_distance)

 Works for very few particles, but not for many

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My contributions III

Control Volumes - Overview

 Control volumes are used in the analysis of fl uid flow phenomena

 They are used to represent the conservation la ws in integral form

 Conservation of mass over a given (control) vo lume c.v. with surface area c.s.:

 = density; u = velocity; n = surface normal



0= d

dt

∫

c.v.

ρ dV +

∮

c.s.

ρ (u⋅̂n ) dA

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My contributions III

Control Volumes in SPH

 Volume integrators are easy: Simply accumulate all contributions of all particles in volume

 Area integrators are trickier

 Time derivative can be obtained via difference quotients: for any property

 Fluid properties at some point in the field ca n be obtained by interpolation

d η

d t ≈Δ η

Δ t = η_{T0+ 1}− η_T0

Δ t η

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My contributions III

Field evaluator function

ValueType evaluate(vector pos, real radius, ParticleProperty A):

ParticleContainer particles;

search(pos, radius, particles);

ValueType res = ValueType(0);

foreach particle p in particles:

real W = DefaultKernel.evaluate(pos - p.position);

res += (p.*A)() * p.mass/p.density * W;

return res;

 This required me to change the spatial partioning grid to support queries at arbitrary locations

 After that, I only had to implement the famous SPH field evaluation template for some property A:

 Translates to:

A( x)=∑

j∈Ω

A_jV _jW ( x− x _j , h)=∑

j ∈Ω

A_j m_j

ρ_j W ( x −x_j , h)

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My contributions III Area Integrator

 Goal: Find average of property at discretely sa mpled points

 I went for an evenly distributing sampler

 Aliasing is not an issue, don’t need random samp ling

 # of samples ns should be proportional to # of particles that can fit into the surface:

 So we get:

D ̄f =

∫

D

f ( A)dA

∮

c.s.

ρ (u⋅̂n) dA≈ A_C 1

ns

∑

j ns

q_j= A_C ns

∑

j ns

ρ _j(u _j⋅ ̂n_j)

ns=ceil

(

^A^A^C^p

)

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My contributions III Disk Integrator

real integrateDiskXZ(real ns, vector2 p_center, real r, field_evalutor f):

real q_total = 0

real ds = sqrt(PI / ns) * r //

int samples_in_diameter = sqrt(ns * 4/PI) //

vector2 min = p_center – r

for (int i = 0; i < samples_in_diameter; i++) for (int j = 0; j < samples_in_diameter; j++)

vector2 pos = (min.x + i * ds, min.z + j * ds) if (length(pos - p_center) > r) continue

q_total += f(pos) return q_total / ns

 In this approach, every kind of surface needs their own integrator

 I only have to consider disks in my pipe flow example

 The disk integrator iterates over the cells of an imaginary grid that we lay over the disk to find the average of fluid property f

A_C= π

4 (2 r )²=ns⋅ds²⇒ 2 r

ds =

√

^π⁴ ^ns

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My contributions III

Area Integrator - Considerations

 No need to sample over an already sampled set

 Can use spatial selection instead:

 Find all particles in distance d from the surface

 Use scaled smoothing kernel to add up contributions

 I was not quite sure how to mathematically scale the k ernel, so I went for the sampling approach

 I also used the integrator to place the cylindrically- shaped fluid inside the pipe

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My contributions III Conservation of mass

M _{T0+ 1}−M _T0

Δ t + A_C

ns

∑

j ns

ρ _j(u _j⋅ ̂n _j)

0= d

dt

∫

c.v.

ρ dV +

∮

c.s.

ρ (u⋅̂n ) dA

 The integral form:

 Becomes:

 The first term is the time derivative of Mass inside the c.v.

 The second term is the mass flux through the c.v.’s surface area

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Particle boundary deficiency, Holes in the fluid and

Control Volume - Correctness

 Boundary deficiency:

 Since atmosphere and structure are not represented in this mod el, computations have to cope with a pseudo-vacuum ( 真空 )

 Governing equations are adjusted to cope with the deficiency

 e.g. Level set function for surface tension considers:

 Inside fluid = 1

 Outside fluid = 0

C.v.’s must always be completely filled!

 Fluid volume is never correct which causes “holes” in the fluid

 Think: What is the space between the particles/samples?

C.v. computations can also never be 100% correct!

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Bibliography

 [1] OpenTissue @ http://www.opentissue.org/ “ OpenTissue is a collection of generic algorith ms and data structures for rapid development of interactive modeling and simulation.”

 [2] Smoothed Particle Hydrodynamics – A Meshf ree Particle Method (book)

 [3] Lagrangian Fluid Dynamics Using Smoothed P article Hydrodynamics

 [4] Particle-Based Fluid-Fluid Interaction

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Summary

 Given high enough gas stiffness, SPH model is OK to simulat e visually appealing real-time flow but is quite inaccurate

 OpenTissue SPH implementation is not very mature, lacks a l ot of features

 I really miss:

 Accurate pressure values

 Correct fluid-solid interaction

 Arbitrary geometry

 I still cannot create real-time interactive applications in volving fluid flow but it was still an insightful endeavour .

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What’s next?

 Surface rendering until next week

 Then choose one from the list…

 Improve SPH implementation

 Add generic surfaces (currently only supports implicit primitives)

 Make it adaptive (choose sample size dynamically)

 Learn solid dynamics and work on Fluid-Structure interaction

 More work on surface rendering

 Optimized OpenGL/DX/XNA implementation that runs on the GPU

 Work on Computational Galactic Dynamics ( 計算星系動力學 )

 with professor 闕志鴻 from the Institute of Astronomy ( 天文所 )

 Simulation of dark matter & fluid during galaxy formation

 Work on level sets and the level set method in CFD

 with professor Yi-Ju 周 from the Institute of Applied Mechanics ( 應力所 )