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Midterm-Review

Midterm-Review. GLY-5835. Basic Fluid Dynamics. Momentum. p = mv F = dp/dt = m dv/dt. Viscosity. Resistance to flow; momentum diffusion Low viscosity: Air High viscosity: Honey Kinematic viscosity. Poiseuille Flow.

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Midterm-Review

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  1. Midterm-Review GLY-5835

  2. Basic Fluid Dynamics

  3. Momentum • p = mv • F = dp/dt = m dv/dt

  4. Viscosity • Resistance to flow; momentum diffusion • Low viscosity: Air • High viscosity: Honey • Kinematic viscosity

  5. Poiseuille Flow • In a slit or pipe, the velocities at the walls are 0 (no-slip boundaries) and the velocity reaches its maximum in the middle • The velocity profile in a slit is parabolic and given by: u(x) • G can be gravitational acceleration or (linear) pressure gradient (Pin – Pout)/L x = 0 x = a

  6. Entry Length Effects

  7. Reynolds Number • The Reynolds Number (Re) is a non-dimensional number that reflects the balance between viscous and inertial forces and hence relates to flow instability (i.e., the onset of turbulence) • Re = v L/n • L is a characteristic length in the system • Dominance of viscous force leads to laminar flow (low velocity, high viscosity, confined fluid) • Dominance of inertial force leads to turbulent flow (high velocity, low viscosity, unconfined fluid)

  8. Re << 1 (Stokes Flow)

  9. Re = 30 Re = 40 Re = 47 Re = 55 Re = 67 Re = 100 Eddies and Cylinder Wakes Re = 41

  10. Laplace Law • There is a pressure difference between the inside and outside of bubbles and drops. • The pressure is always higher on the inside of a bubble or drop (concave side) – just as in a balloon. • The pressure difference depends on the radius of curvature and the surface tension for the fluid pair of interest: DP = g/r

  11. Young-Laplace Law • When solid surfaces are involved, in addition to the fluid1/fluid2 interface – where the interaction is given by the surface tension -- we have interfaces between both fluids and the surface • Often one of the fluids preferentially ‘wets’ the surface • This phenomenon is captured by the contact angle • Zero contact angle means perfect wetting; • DP = g cos q/r

  12. Basic Boltzmann Gas Concepts

  13. Kinetic Theory • Complete set of position (r) and momentum (p) coordinates for all individual particles gives exact dynamical state of system • Together with classical mechanics, allows exact prediction of future states • However, this level of description is essentially not possible

  14. Statistical Mechanics • Represent system by ensemble of many copies • Describe by distribution function f(N)(rN,pN,t); N is number of particles • Changes in f(N)(rN,pN,t) with time given by Liouville equation (6N variables) • Usually interested in low order distribution functions (N = 1, 2)

  15. First Order Distribution Function • f(1)(r,p,t) gives probability of finding a particular molecule with given position and momentum; positions and momenta of remaining N-1 molecules unspecified • No experiment can distinguish between molecules, so the choice of which molecule doesn’t matter • f(1) adequate for describing all gas properties that don’t depend on relative positions of molecules (dilute gas with long mfp)

  16. External force Xi (small relative to intermolecular forces) • For each component i there is an fi(1)(r,pi,t) such that probable number of type i molecules with position coordinates in the range r±dr and momentum coordinates pi±dpi is fi(1)(r,pi,t) dr dpi

  17. Expected Evolution w/o Collisions • If no collisions, then at time t + dt, the new positions of molecules starting at r are [r + (pi/mi)dt] • New momenta are pi = pi +Xidt • Thus,

  18. Collisions • But there are collisions that result in some phase points starting at (r,pi) not arriving at (r + pi/mi dt, pi+Xi dt) and some not starting at (r,pi) arriving there too • Set Gij(-)drdpidt equal to the number of molecules that do not arrive in the expected portion of phase space due to collisions with type j particles during time dt • Similarly, set Gij(+)drdpidt equal to the number of molecules that start somewhere other than (r, pi) and arrive in the portion of phase space due to collisions with type j particles during time dt

  19. Maxwell-Boltzmann Distribution

  20. Cellular Automata and Lattice Gas Cellular Automata

  21. CA Components Transition Rule: State of Neighbors/Self Last Time Step New State

  22. Wolfram’s Binary Rule Numbers • 3 neighbor, 2 state CAs 7 6 5 4 3 2 1 0 27 26 25 24 23 22 21 20 Block diagram from http://mathworld.wolfram.com/ElementaryCellularAutomaton.html

  23. FHP Lattice Gas Cellular Automaton • Fundamental basis is mass and momentum conservation • All particles have the same mass and speed so that momentum conservation reduces to conservation of the vector sum of the velocities S = Solid, R = Random Boolean variables n = (n1, n2, …, n8) 8 bits  256 possibilities Maximum of 1 particle per direction

  24. Zero net momentum, head-on, 2- and 3-particle collisions Possible post-collision configurations Pre-collision Choose based on random bit R Post-collision Pre - collision

  25. Pre-collision Post-collision Unchangeable configurations No configurations other than the original conserve mass and momentum All 5 and 6 particle collisions are similar; no configurations other than the original conserve momentum

  26. Collision ‘Look up’ Table • Configurations 64 through 127 and 192 through 255 (01000000 through 01111111 and 11000000 through 11111111) are on solids • Bounce back

  27. Collision ‘Look up’ Table • Two-particle head-on collisions

  28. Collision ‘Look up’ Table • Three-particle head-on collisions

  29. Remapings

  30. Noise • Averaging • Real thermodynamics

  31. e1 e2 e6 e3 e5 e4 Lattice Boltzmann Model Unit Vectors ea D2Q9 FHP

  32. Velocities 0,1 1,1 -1,1 1,0 -1,0 0,0 1,-1 -1,-1 0,-1 D2Q9

  33. f2 f1 e1 e2 f6 f3 f4 e6 f5 e3 e5 e4 Lattice Boltzmann Model Unit Vectors ea Direction-specific particle densities fa f7 (rest) Macroscopic flows Density Velocity

  34. Fundamental LBM Equations and their Implementation

  35. Collide and Stream Streaming Collision (i.e., relaxation towards local equilibrium) Collision and streaming steps must be separated if solid boundaries are present (bounce back boundary is a separate collision)

  36. Computation of macroscopic density and velocity

  37. Computation of feq

  38. Building an LBM Model

  39. D2Q9 6 2 5 e2 e6 e5 0 3 1 e3 e1 e4 e8 e7 7 4 8 Define velocity vectors • In MATLAB, can write ex(0+1)=0, ex(1+1)=1, etc. • %define es • ex(0)= 0; ey(0)= 0 • ex(1)= 1; ey(1)= 0 • ex(2)= 0; ey(2)= 1 • ex(3)=-1; ey(3)= 0 • ex(4)= 0; ey(4)=-1 • ex(5)= 1; ey(5)= 1 • ex(6)=-1; ey(6)= 1 • ex(7)=-1; ey(7)=-1 • ex(8)= 1; ey(8)=-1

  40. Problem definition LY=10 LX=20 tau = 1 g=0.00001 %set solid nodes is_solid_node=zeros(LY,LX) for i=1:LX is_solid_node(1,i)=1 is_solid_node(LY,i)=1 end LY LX % if ~is_interior_solid_node(j,i)

  41. Initialize density and fs (assuming zero velocity) %define initial density and fs rho=ones(LY,LX); f(:,:,1) = (4./9. ).*rho; f(:,:,2) = (1./9. ).*rho; f(:,:,3) = (1./9. ).*rho; f(:,:,4) = (1./9. ).*rho; f(:,:,5) = (1./9. ).*rho; f(:,:,6) = (1./36.).*rho; f(:,:,7) = (1./36.).*rho; f(:,:,8) = (1./36.).*rho; f(:,:,9) = (1./36.).*rho;

  42. // Computing macroscopic density, rho, and velocity, u=(ux,uy). for( j=0; j<LY; j++) { for( i=0; i<LX; i++) { u_x[j][i] = 0.0; u_y[j][i] = 0.0; rho[j][i] = 0.0; if( !is_solid_node[j][i]) { for( a=0; a<9; a++) { rho[j][i] += f[j][i][a]; u_x[j][i] += ex[a]*f[j][i][a]; u_y[j][i] += ey[a]*f[j][i][a]; } u_x[j][i] /= rho[j][i]; u_y[j][i] /= rho[j][i]; } } }

  43. On boundary, ‘neighboring’ point is on opposite boundary Periodic Boundaries ip = ( i<LX-1)?( i+1):( 0 ); in = ( i>0 )?( i-1):( LX-1); jp = ( j<LY-1)?( j+1):( 0 ); jn = ( j>0 )?( j-1):( LY-1); where LHS = (COND)?(TRUE_RHS):(FALSE_RHS); means if( COND) { LHS=TRUE_RHS;} else{ LHS=FALSE_RHS;}

  44. // Compute the equilibrium distribution function, feq. f1=3.; f2=9./2.; f3=3./2.; for( j=0; j<LY; j++) { for( i=0; i<LX; i++) { if( !is_solid_node[j][i]) { rt0 = (4./9. )*rhoij; rt1 = (1./9. )*rhoij; rt2 = (1./36.)*rhoij; ueqxij = uxij; //Can add forcing here; see MATLAB code ueqyij = uyij; uxsq = ueqxij * ueqxij; uysq = ueqyij * ueqyij; uxuy5 = ueqxij + ueqyij; uxuy6 = -ueqxij + ueqyij; uxuy7 = -ueqxij + -ueqyij; uxuy8 = ueqxij + -ueqyij; usq = uxsq + uysq;

  45. feqij[0] = rt0*( 1. - f3*usq); feqij[1] = rt1*( 1. + f1*ueqxij + f2*uxsq - f3*usq); feqij[2] = rt1*( 1. + f1*ueqyij + f2*uysq - f3*usq); feqij[3] = rt1*( 1. - f1*ueqxij + f2*uxsq - f3*usq); feqij[4] = rt1*( 1. - f1*ueqyij + f2*uysq - f3*usq); feqij[5] = rt2*( 1. + f1*uxuy5 + f2*uxuy5*uxuy5 - f3*usq); feqij[6] = rt2*( 1. + f1*uxuy6 + f2*uxuy6*uxuy6 - f3*usq); feqij[7] = rt2*( 1. + f1*uxuy7 + f2*uxuy7*uxuy7 - f3*usq); feqij[8] = rt2*( 1. + f1*uxuy8 + f2*uxuy8*uxuy8 - f3*usq); } } }

  46. Collide // Collision step. for( j=0; j<LY; j++) for( i=0; i<LX; i++) if( !is_solid_node[j][i]) for( a=0; a<9; a++) { fij[a] = fij[a] + ( fij[a] - feqij[a])/tau; }

  47. Bounceback Boundaries • Bounceback boundaries are particularly simple • Played a major role in making LBM popular among modelers interested in simulating fluids in domains characterized by complex geometries such as those found in porous media • Their beauty is that one simply needs to designate a particular node as a solid obstacle and no special programming treatment is required

  48. Two type of solids:

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