Euler equations with phase transition Exact solutions to the Riemann problem for M. Hantke compressible isothermal Euler equations for Outline Introduction two phase flows with and without phase Model description transition Phase boundaries Classical waves Riemann problem for two phase flows, Case 1 Maren Hantke Riemann problem for two phase Otto-von-Guericke-University Magdeburg flows, Case 2 with Nucleation and cavitation Gerald Warnecke (Magdeburg) Wolfgang Dreyer (WIAS Berlin) 14th International Conference on Hyperbolic Problems: Theory, Numerics, Applications Padova, June 25 - 29, 2012 1 / 25
Outline Euler equations with phase transition M. Hantke Outline 1 Introduction Introduction Model description 2 Model description Phase boundaries 3 Phase boundaries Classical waves Riemann problem 4 Classical waves for two phase flows, Case 1 5 Riemann problem for two phase flows, Case 1 Riemann problem for two phase 6 Riemann problem for two phase flows, Case 2 flows, Case 2 Nucleation and 7 Nucleation and cavitation cavitation 2 / 25
Introduction Euler equations Models of Baer-Nunziato type with phase transition full Euler system to each phase M. Hantke Zein, Hantke, Warnecke. Modeling phase transition for compressible Outline two-phase flows applied to metastable liquids , J. Comput. Phys., 229 Introduction (2010), pp. 2964-2998. Model description Abeyaratne, Knowles. Kinetic relations and the propagation of Phase boundaries phase boundaries in solids , Arch. Rational Mech. Anal., 114 Classical waves (1991), pp. 119-154. Riemann problem for two phase Merkle, Dynamical phase transitions in compressible media , flows, Case 1 Doctoral thesis, Univ. Freiburg, 2006. Riemann problem for two phase flows, Case 2 M¨ uller, Voss. The Riemann problem for the Euler equations with Nucleation and nonconvex and nonsmooth equation of state: Construction of wave cavitation curves , SIAM J. Sci. Comput., 28 (2006), pp. 651-681. Hantke, Dreyer, Warnecke. Exact solutions to the Riemann problem for compressible isothermal Euler equations for two phase flows with and without phase transition , Quart. Appl. Math., to appear in print. 3 / 25
Introduction Euler equations Models of Baer-Nunziato type with phase transition full Euler system to each phase M. Hantke Zein, Hantke, Warnecke. Modeling phase transition for compressible Outline two-phase flows applied to metastable liquids , J. Comput. Phys., 229 Introduction (2010), pp. 2964-2998. Model description Abeyaratne, Knowles. Kinetic relations and the propagation of Phase boundaries phase boundaries in solids , Arch. Rational Mech. Anal., 114 Classical waves (1991), pp. 119-154. Riemann problem for two phase Merkle, Dynamical phase transitions in compressible media , flows, Case 1 Doctoral thesis, Univ. Freiburg, 2006. Riemann problem for two phase flows, Case 2 M¨ uller, Voss. The Riemann problem for the Euler equations with Nucleation and nonconvex and nonsmooth equation of state: Construction of wave cavitation curves , SIAM J. Sci. Comput., 28 (2006), pp. 651-681. Hantke, Dreyer, Warnecke. Exact solutions to the Riemann problem for compressible isothermal Euler equations for two phase flows with and without phase transition , Quart. Appl. Math., to appear in print. 3 / 25
Introduction Euler equations Models of Baer-Nunziato type with phase transition full Euler system to each phase M. Hantke Zein, Hantke, Warnecke. Modeling phase transition for compressible Outline two-phase flows applied to metastable liquids , J. Comput. Phys., 229 Introduction (2010), pp. 2964-2998. Model description Abeyaratne, Knowles. Kinetic relations and the propagation of Phase boundaries phase boundaries in solids , Arch. Rational Mech. Anal., 114 Classical waves (1991), pp. 119-154. Riemann problem for two phase Merkle, Dynamical phase transitions in compressible media , flows, Case 1 Doctoral thesis, Univ. Freiburg, 2006. Riemann problem for two phase flows, Case 2 M¨ uller, Voss. The Riemann problem for the Euler equations with Nucleation and nonconvex and nonsmooth equation of state: Construction of wave cavitation curves , SIAM J. Sci. Comput., 28 (2006), pp. 651-681. Hantke, Dreyer, Warnecke. Exact solutions to the Riemann problem for compressible isothermal Euler equations for two phase flows with and without phase transition , Quart. Appl. Math., to appear in print. 3 / 25
Introduction Euler equations Models of Baer-Nunziato type with phase transition full Euler system to each phase M. Hantke Zein, Hantke, Warnecke. Modeling phase transition for compressible Outline two-phase flows applied to metastable liquids , J. Comput. Phys., 229 Introduction (2010), pp. 2964-2998. Model description Abeyaratne, Knowles. Kinetic relations and the propagation of Phase boundaries phase boundaries in solids , Arch. Rational Mech. Anal., 114 Classical waves (1991), pp. 119-154. Riemann problem for two phase Merkle, Dynamical phase transitions in compressible media , flows, Case 1 Doctoral thesis, Univ. Freiburg, 2006. Riemann problem for two phase flows, Case 2 M¨ uller, Voss. The Riemann problem for the Euler equations with Nucleation and nonconvex and nonsmooth equation of state: Construction of wave cavitation curves , SIAM J. Sci. Comput., 28 (2006), pp. 651-681. Hantke, Dreyer, Warnecke. Exact solutions to the Riemann problem for compressible isothermal Euler equations for two phase flows with and without phase transition , Quart. Appl. Math., to appear in print. 3 / 25
Introduction Euler equations Models of Baer-Nunziato type with phase transition full Euler system to each phase M. Hantke Zein, Hantke, Warnecke. Modeling phase transition for compressible Outline two-phase flows applied to metastable liquids , J. Comput. Phys., 229 Introduction (2010), pp. 2964-2998. Model description Abeyaratne, Knowles. Kinetic relations and the propagation of Phase boundaries phase boundaries in solids , Arch. Rational Mech. Anal., 114 Classical waves (1991), pp. 119-154. Riemann problem for two phase Merkle, Dynamical phase transitions in compressible media , flows, Case 1 Doctoral thesis, Univ. Freiburg, 2006. Riemann problem for two phase flows, Case 2 M¨ uller, Voss. The Riemann problem for the Euler equations with Nucleation and nonconvex and nonsmooth equation of state: Construction of wave cavitation curves , SIAM J. Sci. Comput., 28 (2006), pp. 651-681. Hantke, Dreyer, Warnecke. Exact solutions to the Riemann problem for compressible isothermal Euler equations for two phase flows with and without phase transition , Quart. Appl. Math., to appear in print. 3 / 25
Balance equations Euler equations Isothermal Euler equations with phase transition ρ t + ( ρ v ) x = M. Hantke 0 ( ρ v ) t + ( ρ v 2 + p ) x = Outline 0 Introduction Model description Jump conditions across discontinuities Phase boundaries Classical waves � ρ ( v − W ) � = 0 Riemann problem ρ ( v − W ) � v � + � p � = 0 for two phase flows, Case 1 Riemann problem for two phase Mass flux across discontinuities flows, Case 2 Z = − ρ ( v − W ) Nucleation and cavitation with � � Q shock wave S shock wave Z = W = and z phase boundary w phase boundary 4 / 25
Equations of state Euler equations with phase transition M. Hantke Ideal gas law Outline kT 0 p V = ρ V 0 ≤ ρ V ≤ ˜ ρ Introduction m Model description Phase boundaries Liquid equation of state Classical waves Riemann problem � ρ L � for two phase p L = p 0 + K 0 − 1 ρ L ≥ ρ m flows, Case 1 ρ 0 Riemann problem for two phase flows, Case 2 Nucleation and cavitation 5 / 25
Euler equations Maxwell condition with phase transition � 1 /ρ V ( p 0 ) M. Hantke p ( ρ ) d 1 � ρ V ( p 0 ) − 1 1 � ρ = · p 0 ρ 0 Outline 1 /ρ 0 Introduction Model description Phase boundaries Equation of state Classical waves black: p ( 1 /ρ ) Riemann problem for T 0 = 573 . 15 K for two phase flows, Case 1 dashed red: Maxwell line Riemann problem for two phase flows, Case 2 Nucleation and cavitation a) red: ˜ ρ ( T ) a) black: ρ m ( T ) ρ ( T ) /ρ m ( T ) < 1 b) ˜ 4 6 / 25
Kinetic relation Euler equations with phase Case 1: trivial case transition M. Hantke Outline z = 0 Introduction Model description Phase boundaries Case 2 Classical waves Riemann problem for two phase � m � 3 / 2 flows, Case 1 p V = � g + e kin � z √ Riemann problem 2 π kT 0 for two phase flows, Case 2 � m � 3 / 2 � K 0 ln ρ L p V − kT 0 m ln p V Nucleation and = cavitation z √ kT 0 ρ 0 ρ 0 p 0 2 π � + 1 2 ( v L − w ) 2 − 1 2 ( v V − w ) 2 7 / 25
Kinetic relation Euler equations with phase Case 1: trivial case transition M. Hantke Outline z = 0 Introduction Model description Phase boundaries Case 2 Classical waves Riemann problem for two phase � m � 3 / 2 flows, Case 1 p V = � g + e kin � z √ Riemann problem 2 π kT 0 for two phase flows, Case 2 � m � 3 / 2 � K 0 ln ρ L p V − kT 0 m ln p V Nucleation and = cavitation z √ kT 0 ρ 0 ρ 0 p 0 2 π � + 1 2 ( v L − w ) 2 − 1 2 ( v V − w ) 2 7 / 25
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