Outline Introduction Previous results Model description - nonisothermal case Phase boundaries Creation of new phases Riemann solver On the compressible Euler equations for two phase flows with phase transition Maren Hantke Otto-von-Guericke-University Magdeburg with Ferdinand Thein (Magdeburg) MNMCFF 2014 Sino-German Symposium Modern Numerical Methods for Compressible Fluid Flows and Related Problems Beijing, May 21 - 27, 2014 1 / 24
Outline Introduction Previous results Model description - nonisothermal case Phase boundaries Creation of new phases Riemann solver Outline 1 Introduction 2 Previous results 3 Model description 4 Phase boundaries 5 Creation of new phases 6 Riemann Solver for the isothermal Euler system 2 / 24
Outline Introduction Previous results Model description - nonisothermal case Phase boundaries Creation of new phases Riemann solver Introduction Models of Baer-Nunziato type full Euler system to each phase Zein, Hantke, Warnecke. Modeling phase transition for compressible two-phase flows applied to metastable liquids , J. Comput. Phys., 229 (2010), pp. 2964-2998. Models using one set of Euler equations Dumbser, Iben, Munz. Efficient implementation of high order unstructured WENO schemes for cavitating flows , Computers & Fluids, 86 (2013), pp. 141-168. Hantke, Dreyer, Warnecke. Exact solutions to the Riemann problem for compressible isothermal Euler equations for two phase flows with and without phase transition , Quarterly of Applied Mathematics, vol. LXXI 3 (2013), pp. 509-540. 3 / 24
Outline Introduction Previous results Model description - nonisothermal case Phase boundaries Creation of new phases Riemann solver Isothermal Euler equations Nonisothermal Euler equations no pt pt no pt pt (I) pt (II) structure of the solution characterization of phase boundaries existence results uniqueness results Creation of new phases Solver 4 / 24
Outline Introduction Previous results Model description - nonisothermal case Phase boundaries Creation of new phases Riemann solver Model description - isothermal case Isothermal Euler equations ρ t + ( ρ v ) x = 0 ( ρ v ) t + ( ρ v 2 + p ) x = 0 Jump conditions across discontinuities � ρ ( v − W ) � = 0 ρ ( v − W ) � v � + � p � = 0 Mass flux across discontinuities � Q , S shock wave Z = − ρ ( v − W ) with Z , W = z , w phase boundary Kinetic relation � m � 3 / 2 p V z = 0 z = √ � g + e kin � or kT 0 2 π 5 / 24
Outline Introduction Previous results Model description - nonisothermal case Phase boundaries Creation of new phases Riemann solver Initial data Riemann initial data Equations of state ideal gas law, Tait equation stiffened gas law, generalized stiffened gas law, IAPWS Results structure of the solution characterization of phase boundaries existence results uniqueness results relationship of solutions with / without phase transition creation of new phases 6 / 24
Outline Introduction Previous results Model description - nonisothermal case Phase boundaries Creation of new phases Riemann solver Isothermal case - structure of the solution selfsimilar solution two classical waves, one phase boundary phase boundary: contact discontinuity or nonclassical discontinuity 7 / 24
Outline Introduction Previous results Model description - nonisothermal case Phase boundaries Creation of new phases Riemann solver Example Solution structure at time t = 0.001 s Velocity in m/s Solution structure at time t = 0.001 s Velocity in m/s −50 −100 W V* W L* W V* W L* −100 −200 −300 −150 −400 −200 −500 −0.5 0 0.5 1 −0.5 0 0.5 1 −1 −0.5 0 0.5 1 −1 −0.5 0 0.5 1 Density in kg/m 3 Density in kg/m 3 4 Pressure in Pa 5 Pressure in Pa x 10 x 10 2 10 800 800 600 600 1.5 8 400 400 1 6 200 200 0 0 4 0.5 −0.5 0 0.5 1 −0.5 0 0.5 1 −1 −0.5 0 0.5 1 −1 −0.5 0 0.5 1 8 / 24
Outline Introduction Previous results Model description - nonisothermal case Phase boundaries Creation of new phases Riemann solver Model description - nonisothermal case Euler equations Equations of state Jump conditions across discontinuities 1 ∂ρ I 0 = ∂ t − � z � ∂ ( ρ I w ) 0 = − � zv � + � p � ∂ t ∂ e I ∂ t + � − z ( e + p ρ + 1 2 ( v − w ) 2 )+ q � = 0 Kinetic relation z = 0 or z = ? Entropy condition 0 ≤ ζ = ∂ s I ∂ t + � − zs + q T � 1 Dreyer, On Jump Conditions at Phase Boundaries for Ordered and Disordered Phases, WIAS Preprint, 869, 2003 9 / 24
Outline Introduction Previous results Model description - nonisothermal case Phase boundaries Creation of new phases Riemann solver Case I: z = 0 - no phase transition Balances across the phase boundary simplify v L = v V = w p L = p V Phase boundary is a contact wave The exact solution is selfsimilar and can be constructed easily. 10 / 24
Outline Introduction Previous results Model description - nonisothermal case Phase boundaries Creation of new phases Riemann solver Case II: z � = 0 - phase transition may occur ∂ s I Simplifying assumptions: ρ I = 0, e I = 0 ⇒ ∂ t = 0 Balances across the interface 0 = � z � ⇔ 0 = � ρ ( v − w ) � 0 = − z � v � + � p � z � ( e + p ρ + 1 2 ( v − w ) 2 ) � 0 = Entropy condition 0 ≤ ζ = − z � s � Kinetic relation z ∼ − � s � 11 / 24
Outline Introduction Previous results Model description - nonisothermal case Phase boundaries Creation of new phases Riemann solver Case IIa: z � = 0 Entropy condition 0 ≤ ζ = − z � s � Kinetic relation z ∼ − � s � The exact solution is selfsimilar and can be constructed easily. Two classical shock or rarefaction waves, contact wave, phase boundary Problem: Thermal equilibrium cannot occur. Only evaporation processes are possible. 12 / 24
Outline Introduction Previous results Model description - nonisothermal case Phase boundaries Creation of new phases Riemann solver Case IIb: z � = 0 We need more general assumptions! Balances across phase boundaries ∂ρ I 0 = ∂ t − � z � ∂ ( ρ I w ) 0 = − � zv � + � p � ∂ t ∂ e I ∂ t + � − z ( e + p ρ + 1 2 ( v − w ) 2 )+ q � = 0 Entropy condition 0 ≤ ζ = ∂ s I ∂ t + � − zs + q T � Taking into account heat conduction ∂ s I ∂ t � = 0 13 / 24
Outline Introduction Previous results Model description - nonisothermal case Phase boundaries Creation of new phases Riemann solver Case IIb: z � = 0 Simplifying assumptions: ρ I = 0 Balances across phase boundaries = � z � 0 = � ρ ( v − w ) � 0 ⇔ = − z � v � + � p � 0 ∂ e I ∂ t − z � ( e + p ρ + 1 2 ( v − w ) 2 ) � = 0 Entropy condition 0 ≤ ζ = ∂ s I ∂ t − z � s � e I = e I ( T I ) Kinetic relation � m 0 � 3 / 2 p V � g + Ts + 1 2 ( v − w ) 2 − sT I � z = √ kT I 2 π 14 / 24
Outline Introduction Previous results Model description - nonisothermal case Phase boundaries Creation of new phases Riemann solver Case IIb: z � = 0 - surfacial tension ∂ σ ( T I ) T I e I = − T 2 � = const ! I ∂ T I σ ≡ const cannot be used! os rule? Also a linear relation for σ cannot be used! E¨ otv¨ Katayama-Guggenheim rule � 11 / 9 � 1 − T I σ = σ 0 T c may be used. Problem: Selfsimilarity of the solution is lost! 15 / 24
Outline Introduction Previous results Model description - nonisothermal case Phase boundaries Creation of new phases Riemann solver Behavoir of the phase boundary System, that has to be solved balances across the interface kinetic relation Initial data: liquid state, initial interface temperature Closure conditions ideal gas law for the vapor phase Katayama-Guggenheim rule Result system runs into steady state for higher temperatures the process is much faster 16 / 24
Outline Introduction Previous results Model description - nonisothermal case Phase boundaries Creation of new phases Riemann solver Creation of new phases From now on the fluid under consideration is water. Situation 1: pure water vapor is compressed Situation 2: pure liquid water is expanded Isothermal case For sufficiently high compression of water vapor liquid water is created. By sufficiently strong expansion of liquid water water vapor can be created. Nonisothermal case? 17 / 24
Outline Introduction Previous results Model description - nonisothermal case Phase boundaries Creation of new phases Riemann solver Creation of liquid water Theorem 1: Nonexistence result (MH, Ferdinand Thein, 2014) 2 Using the real equations of state for water or any good approximation of the real equation of state condensation by compression cannot occur. This result holds for compressible Euler equations, phase transitions modeled by a kinetic relation compressible Euler equations, phase transition modeled using an equilibrium assumption models of Baer Nunziato type, phase transition modeled using relaxation terms 2 Hantke, Thein, Why condensation by compression in pure water vapor cannot occur in an approach based on Euler equations, accepted for publication in Quarterly of Applied Mathematics 18 / 24
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