A NEW LIQUID-VAPOR PHASE- TRANSITION TECHNIQUE FOR THE LEVEL SET METHOD Nathaniel R. Morgan George W. Woodruff School of Mechanical Engineering Georgia Institute of Technology Dr. Smith (Advisor) Dr. Ghiaasiaan Dr. Sotiropoulos Dr. Mucha Dr. Dolbow [CSGF Alumni] Acknowledgments • This work was supported by the Department of Energy Computational Science Graduate Fellowship Program of the Office of Scientific Computing and Office of Defense Programs in the Department of Energy under contract DE-FG02-97ER25308. Research Objective Improve the liquid-vapor phase-transition capabilities of the level set method Account for different properties in the respective phases Capture the Latent heat absorbed or released Capture the different temperature gradients across the interface Handle temperature gradients in both phases Level Set Method Use a higher-dimensional function to represent the interface [Osher (1988)] φ<0 is phase 1 φ>0 is phase 2 φ=0 is interface The interface is advected according to: ∂φ G +V •∇φ = 0 ∂t INT Capturing the Interface The interface can be implicitly captured ⎧ 0 if φ < −ε ⎪1 ⎛ φ 1 ⎛ πφ ⎞⎞ H (φ) = ⎜1+ + sin ⎟, if φ <= ε ε ⎨ ⎜ ε π ⎜ ε ⎟⎟ ⎪2 ⎝ ⎝ ⎠⎠ φ > ε ⎩⎪ 1 if ⎧ 0 if φ < −ε ⎪ 1 ⎛ ⎛ πφ ⎞⎞ δ (φ) = ⎜1+ cos ⎟, if φ <= ε ε ⎨ ε ⎜ ⎜ ε ⎟⎟ ⎪2 ⎝ ⎝ ⎠⎠ φ > ε ⎩⎪ 0 if Properties and Geometric Quantities The properties of each respective phase can be represented using the Heaviside function γ = γ φ + γ ()− φ 1H( ) 2 1 H( ) Interface geometry G ∇φ G ∇φ n = κ = ∇ • n = ∇ • ∇φ ∇φ 2 2 Governing Equations Conservation of Mass ∂ρ G + ∇ • ()ρV = 0 ∂t The respective phases are incompressible G G ∇ • = ∇ • = Vl 0 Vv 0 The continuity equation corresponding to phase transition [Juric (1998)] G G G V = V H +V (1− H) G l G v G G ∇ • = − • ∇ ∇ • = Γ V (Vl Vv ) H or V MASS Governing Equations Mass G ∇ • = Γ V MASS Momentum [Brackbill (1992)] K ∂V G G ∇P K ∇ •τ σκ∇H + ()V •∇ V = − + g + − ∂t ρ ρ ρ G G T τ = µ(∇V + ()∇V ) Energy ∂cT G ∇ • ()k∇T Γ +V •∇cT = + ENERGY ∂t ρ ρ Interface Physics The heat flux is continuous across the interface, but the properties and the gradient are not continuous k Liquid Vapor T Liquid Vapor Thermal Conductivity Temperature Interface Jump Conditions Mass and energy conservation across the interface [Welch (2000)] ζ = ζ −ζ [[ ]] v l G G ρ()− • G = Mass [[ V VINT ]] n 0 G G ρ ()− • G = − G • G Energy [[ h V VINT ]] n [[q]] n Interface Jump Conditions Interface velocity G G ()− k ∇T + k ∇T V = V + v l INT ρ()− hv hl Mass source term ⎛ 1 1 ⎞ ()− k ∇T + k ∇T •∇H (φ) Γ = ⎜ − ⎟ v l MASS ⎜ ρ ρ ⎟ − ⎝ v l ⎠ hv hl Energy Source Term Discrete energy equation n + (c(φ)T) ∆t G T n 1 = + (−V •∇()c(φ)T n + c(φ)n+1 c(φ)n+1 ∇ •(k(φ)∇T)n Γn+1 ⎞ + + ENERGY ⎟ n n ⎟ ρ(φ) ρ(φ) ⎠ The source term corrects the temperature field to satisfy the interface boundary condition (c(φ)T)n ∆t ⎛ G ∇ • (k(φ)∇T)n ⎞ T * = + ⎜−V •∇()c(φ)T n + ⎟ n+1 n+1 ⎜ n ⎟ c(φ) c(φ) ⎝ ρ(φ) ⎠ ⎛ Γ n+1 ⎞ T n+1 = T * + ∆t⎜ ENERGY ⎟ n+1 = * + ∆ ⎜ n n+1 ⎟ so T T TPC ⎝ ρ(φ) c(φ) ⎠ Interface Boundary Condition Extrapolation equation G T = T −φ n •∇T n sat i, j i, j i, j Liquid φ>0 i, j i+1, j φ=0 Vapor φ<0 i, j-1 Liquid Vapor Interface Boundary Condition Extrapolation equation T − B T = sat i, j A ⎛ min(0,−φ n x ) max(0,−φ n x ) min(0,−φ n y ) max(0,−φ n y ) ⎞ A =1+ ⎜− i, j i, j + i, j i, j − i, j i, j + i, j i, j ⎟ ⎜ ∆ ∆ ∆ ∆ ⎟ ⎝ x x y y ⎠ ⎛ min(0,−φ n x ) max(0,−φ n x ) B = ⎜ i, j i, j T − i, j i, j T + ⎜ ∆ i+1, j ∆ i−1, j ⎝ x x min(0,−φ n y ) max(0,−φ n y ) ⎞ + i, j i, j T − i, j i, j T ⎟ ∆ i, j+1 ∆ i, j−1 ⎟ y y ⎠ Interface Temperature Gradients (I) Use ghost nodes Extrapolate temperature values to the ghost nodes in the other respective phase along the interface Liquid Red nodes = vapor ghost nodes Vapor Blue nodes = liquid φ=0 ghost nodes Interface Temperature Gradients (II) Derivative stencil Vapor ghost nodes i, j+1 Liquid i, j i+1, j Liquid i-1, j i, j i, j-1 φ=0 φ=0 Vapor Vapor Ghost Node Construction Ghost node construction equation T − B T = sat GHOST i, j A ⎛ min(0,φ n x ) max(0,φ n x ) min(0,φ n y ) max(0,φ n y ) ⎞ A =1+ ⎜− i, j i, j + i, j i, j − i, j i, j + i, j i, j ⎟ ⎜ ∆ ∆ ∆ ∆ ⎟ ⎝ x x y y ⎠ ⎛ min(0,φ n x ) max(0,φ n x ) B = ⎜ i, j i, j T − i, j i, j T + ⎜ ∆ i+1, j ∆ i−1, j ⎝ x x min(0,φ n y ) max(0,φ n y ) ⎞ + i, j i, j T − i, j i, j T ⎟ ∆ i, j+1 ∆ i, j−1 ⎟ y y ⎠ Solution Strategy Calculate the projected velocity field (Chorin 1968) Calculate the new level set field Calculate the projected temperature field Apply the interface temperature condition Construct the new ghost node values Evaluate the continuity equation source term Calculate the new pressure field Calculate the new velocity Now, go back to step one Test Problems (I) 1-D phase-change [Özişik (1993)] ∂T ∂2T v = α v T ∂t v ∂x2 wall Tsat dδ ∂T ρ h = − k v v fg v ∂ dt x x=δ (t) 0 δ(t) y = δ = 2 Tv (x (t),t) Tsat ()− ∑ Ti, j TExact L = i, j T (x = 0,t) = T 2 v wall N x N y Test Problems (II) 2-D phase-change [Özişik (1993)] ∂ ∂ ∂ Tv = α 1 ⎛ Tv ⎞ v ⎜ ⎟ ∂t r ∂r ⎝ ∂r ⎠ T dδ ∂T sat ρ h = − k v v fg v ∂ Heat Source dt r x=δ (t) 0 δ(t) r = δ = Tv (x (t),t) Tsat 2 ()T −T ∂ ∑ i, j Exact ⎛ Tv ⎞ = i, j lim 2πk = Q L2 → ⎜ v ⎟ r 0⎝ ∂r ⎠ N x N y 1D Test Problem Results (I) properties Ja = 9 α -3 v = 10 1D Test Problem Results (II) 1D Test Problem Results (III) 2D Test Results (I) 2 sec 3 sec 2D Test Results (II) 2-D Test Results (III) Film Boiling Movie Conclusions The new liquid-vapor phase-transition technique extends the modeling capabilities of the level set method (1) It handles different properties in each respective phase (2) It captures the liberation or absorption of latent heat (3) It captures the discontinuous temperature gradients across the interface (4) It can handle temperature gradients in both phases.