Notes Fire � [Nguyen, Fedkiw, Jensen � 02] � Gaseous fuel/air mix (from a burner, or a hot piece of wood, or …) heats up � When it reaches ignition temperature, starts to burn • “blue core” - see the actual flame front due to emission lines of excited hydrocarbons � Gets really hot while burning - glows orange from blackbody radiation of smoke/soot � Cools due to radiation, mixing • Left with regular smoke cs533d-term1-2005 1 cs533d-term1-2005 2 Defining the flow Interface speed � Inside and outside blue core, regular � Interface = flame front = blue core surface incompressible flow with buoyancy � D=V f -S is the speed of the flame front • It moves with the fuel flow, and on top of that, moves according � But an interesting boundary condition at the to reaction speed S flame front • S is fixed for a given fuel mix • Gaseous fuel and air chemically reacts to produce a � We can track the flame front with a level set � different gas with a different density � Level set moves by • Mass is conserved, so volume has to change • Gas instantly expands at the flame front � t + D � � = 0 � And the flame front is moving too � t + u LS � � � = 0 • At the speed of the flow plus the reaction speed � Here u LS is u f -Sn cs533d-term1-2005 3 cs533d-term1-2005 4
Numerical method Conservation of mass � Mass per unit area entering flame front is � f (V f -D) � For water we had to work hard to move interface accurately where • V f =u f •n is the normal component of fuel velocity � Here it � s ok just to use semi-Lagrangian method • D is the (normal) speed of the interface (with reinitialization) � Mass per unit area leaving flame front is � h (V h -D) � Why? • We � re not conserving volume of blue core - if reaction where is a little too fast or slow, that � s fine • V h =u h •n is the normal component of hot gaseous • Numerical error looks like mean curvature products velocity • Real physics actually says reaction speed varies with � Equating the two gives: mean curvature! ( ) = � h V h � D ( ) � f V f � D cs533d-term1-2005 5 cs533d-term1-2005 6 Velocity jump Ghost velocities � Plugging interface speed D into � This is a “jump condition”: how the normal component of velocity jumps when you go over the flame interface conservation of mass at the flame front � This lets us define a “ghost” velocity field that is gives: continuous • When we want to get a reasonable value of u h for semi- Lagrangian advection of hot gaseous products on the fuel side of ( ) the interface, or vice versa (and also for moving interface) � f S = � h V h � V f + S • When we compute divergence of velocity field � h V h = � h V f + � f S � � h S � Simply take the velocity field, add/subtract ( � f / � h -1)Sn � � V h = V f + � f � 1 � � S � h � � cs533d-term1-2005 7 cs533d-term1-2005 8
Conservation of momentum Simplifying � Momentum is also conserved at the interface � Make the equation look nicer by taking conservation of mass: � Fuel momentum per unit area “entering” the ( ) = � h V h � D ( ) + p f interface is ( ) � f V f V f � D � f V f � D multiplying both sides by -D: � Hot gaseous product momentum per unit area “leaving” the interface is ( ) = � h � D ( ) V f � D ( ) V h � D ( ) � f � D ( ) + p h � h V h V h � D and adding to previous slide � s equation: � Equating the two gives 2 + p f = � h V h � D 2 + p h ( ) ( ) ( ) + p f = � h V h V h � D ( ) + p h � f V f � D � f V f V f � D cs533d-term1-2005 9 cs533d-term1-2005 10 Pressure jump Temperature � This gives us jump in pressure from one side of � We don � t want to get into complex (!) chemistry the interface to the other of combustion � By adding/subtracting the jump, we can get a � Instead just specify a time curve for the reasonable continuous extension of pressure temperature from one side to the other • Temperature known at flame front (T ignition ) • For taking the gradient of p to make the flow • Temperature of a chunk of hot gaseous product rises incompressible after advection at a given rate to T max after it � s created • Then cools due to radiation � Note when we solve the Poisson equation density is NOT constant, and we have to incorporate jump in p (known) just like we use it in the pressure gradient cs533d-term1-2005 11 cs533d-term1-2005 12
Temperature cont’d Smoke concentration � For small flames (e.g. candles) can model initial � Can do the same as for temperature: initially temperature rise by tracking time since reaction: make it a function of time Y since reaction (rising Y t +u• � Y=1 and making T a function of Y from zero) • And ignore this regime for large flames � For large flames ignore rise, just start flame at T max (since transition region is very thin, close to � Then just advect without change, like before blue core) � Note: both temperature and smoke concentration play back into velocity equation � Radiative cooling afterwards: (buoyancy force) � � 4 T � T air T t + u � � T = � c T � � T max � T air � � cs533d-term1-2005 13 cs533d-term1-2005 14 Note on fuel SPH � We assumed fuel mix is magically being injected � Smoothed Particle Hydrodynamics into scene • A particle system approach • Just fine for e.g. gas burners � Get rid of the mesh altogether - figure out • Reasonable for slow-burning stuff (like thick wood) how to do � p etc. with just particles � What about fast-burning material? � Each particle represents a blurry chunk of • Can specify another reaction speed S fuel for how fast solid/liquid fuel turned into flammable gas (dependent fluid on temperature) (with a particular mass, momentum, etc.) • Track level set of solid/liquid fuel just like we did the � Lagrangian: advection is going to be easy blue core cs533d-term1-2005 15 cs533d-term1-2005 16
Mesh-free? SPH � Mathematically, SPH and particle-only methods � SPH can be interpreted as a particular way of are independent of meshes choosing forces, so that you converge to solving Navier-Stokes � Practically, need an acceleration structure to speed up finding neighbouring particles (to � [Lucy � 77], [Gingold & Monaghan � 77], figure out forces) [Monaghan…], [Morris, Fox, Zhu � 97], … � Most popular structure (for non-adaptive codes, � Similar to FEM, we go to a finite dimensional i.e. where h=constant for all particles) is… space of functions a mesh (background grid) • Basis functions now based on particles instead of grid elements • Can take derivatives etc. by differentiating the real function from the finite-dimensional space cs533d-term1-2005 17 cs533d-term1-2005 18 Kernel Cubic kernel � � x W ( x ) = 1 � Need to define particle � s influence in � Use where � � h 3 f surrounding space (how we � ll build the basis � � h functions) � Choose a kernel function W 2 s 2 + 3 � 1 � 3 0 � s � 1 4 s 3 , • Smoothed approximation to � � f ( s ) = 1 ( ) • W(x)=W(|x|) - radially symmetric � 4 2 � s 3 , 1 � s � 2 1 � • Integral is 1 � 2 � s � 0, • W=0 far enough away - when |x|>2.5h for example • Note: not good for viscosity (2nd derivatives � Examples: involved - not very smooth) • Truncated Gaussian • Splines (cubic, quartic, quintic, …) cs533d-term1-2005 19 cs533d-term1-2005 20
Estimating quantities Smoothed Particle Estimate � Say we want to estimate some flow variable q at � Take the “raw” mass estimate to get a point in space x density: � ( ) � ( x ) = m j W x � x j � We � ll take a mass and kernel weighted average � ( ) j Q ( x ) = m j q j W x � x j � Raw version: • [check dimensions] j • But this doesn � t work, since sum of weights is � Evaluate this at particles, use that to nowhere close to 1 � approximately normalize: • Could normalize by dividing by but that m j W j j involves complicates derivatives… ( ) m j W x � x j • Instead use estimate for normalization at each � q ( x ) = q j particle separately (some mass-weighted measure of � j j overlap) cs533d-term1-2005 21 cs533d-term1-2005 22 Incompressible Free Surfaces Continuity equation � Actually, I lied � Recall the equation is � t + u � � � = � � � � u • That is, regular SPH uses the previous formulation • For doing incompressible flow with free surface, we � We � ll handle advection by moving particles around have a problem � So we need to figure out right-hand side • Density drop smoothly to 0 around surface � Divergence of velocity for one particle is ( ) = v j � � W j ( ) � � v = � � v j W x � x j • This would generate huge pressure gradient, compresses surface layer � Multiply by density, get SPH estimate: � So instead, track density for each particle as a � � � � v i = m j v j � � i W ij primary variable (as well as mass!) j • Update it with continuity equation • Mass stays constant however - probably equal for all particles, along with radius cs533d-term1-2005 23 cs533d-term1-2005 24
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