Numerical Simulations Of Basic Interfacial Instabilities With the Improved Two-Fluid Model

1. Introduction Mathematical models Numerical models Simulations Conclusions Numerical Simulations Of Basic Interfacial Instabilities With the Improved Two-Fluid Model Luka Štrubelj, Iztok Tiselj “Jožef Stefan” Institute, Reactor Engineering Division – R4 International Conference NNNuuucccllleeeaaarrr EEEnnneeerrrgggyyy fffooorrr NNNeeewww EEEuuurrrooopppeee 222000000999 Bled / Slovenia / September 14-17 Luka Štrubelj, Iztok Tiselj “Jožef Stefan” Institute

2. Introduction Mathematical models Numerical models Simulations Conclusions Outlook of presentation 1 Introduction 2 Mathematical models 3 Numerical models 4 Simulations Kelvin-Helmholtz instability Rayleigh-Taylor instability 5 Conclusions Open questions Luka Štrubelj, Iztok Tiselj “Jožef Stefan” Institute

3. Introduction Mathematical models Numerical models Simulations Conclusions Introduction Two-phase flows Stratified flows (flows with large interfaces) Dispersed flows (flows with many small droplets/bubbles/particles) Mixed or transitional flows Luka Štrubelj, Iztok Tiselj “Jožef Stefan” Institute

6. Introduction Mathematical models Numerical models Simulations Conclusions Introduction – Motivation Cold leg Cold water ECC injection Downcomer Steam Hot water Mixing of hot and cold water DCC on the jetJet instabilities Turbulence in the liquid phase Turbulence and momentum transfer at the interface Heat and mass transfer at the interface Bubble entrainment Bubble migration Mixed water Heat transfer to the walls Turbulence production by the jet and entrained bubbels Figure: Most important ﬂow phenomena during a pressurized thermal shock situation with partially ﬁlled cold leg of primary system in nuclear power plant. Luka Štrubelj, Iztok Tiselj “Jožef Stefan” Institute

7. Introduction Mathematical models Numerical models Simulations Conclusions Mathematical models – Two-ﬂuid model Mass balance equation ∂αk ∂t + vk · αk = 0 (1) Momentum balance equation ∂ (ρkαkvk) ∂t + (ρkαkvk · ) vk = − αk p + · (µkαk vk) + ρkαkg + FD,k + FS,k (2) Luka Štrubelj, Iztok Tiselj “Jožef Stefan” Institute

8. Introduction Mathematical models Numerical models Simulations Conclusions Mathematical models – Two-ﬂuid model Mass balance equation ∂αk ∂t + vk · αk = 0 (1) Momentum balance equation ∂ (ρkαkvk) ∂t + (ρkαkvk · ) vk = − αk p + · (µkαk vk) + ρkαkg + FD,k + FS,k (2) Luka Štrubelj, Iztok Tiselj “Jožef Stefan” Institute

9. Introduction Mathematical models Numerical models Simulations Conclusions Mathematical models – Interfacial forces General drag force for stratiﬁed ﬂow FD,1 = α1α2 (v2 − v1) ρmix τr (3) τr = ∆t 100 Splitted surface tension force FS,k = βkFS = αkσκ α (4) Luka Štrubelj, Iztok Tiselj “Jožef Stefan” Institute

10. Introduction Mathematical models Numerical models Simulations Conclusions Mathematical models – Interfacial forces General drag force for stratiﬁed ﬂow FD,1 = α1α2 (v2 − v1) ρmix τr (3) τr = ∆t 100 Splitted surface tension force FS,k = βkFS = αkσκ α (4) Luka Štrubelj, Iztok Tiselj “Jožef Stefan” Institute

11. Introduction Mathematical models Numerical models Simulations Conclusions Mathematical models – Interface sharpening Why to use a special interface tracking model? Luka Štrubelj, Iztok Tiselj “Jožef Stefan” Institute

12. Introduction Mathematical models Numerical models Simulations Conclusions Mathematical models – Interface sharpening Why to use a special interface tracking model? Luka Štrubelj, Iztok Tiselj “Jožef Stefan” Institute

13. Introduction Mathematical models Numerical models Simulations Conclusions Mathematical models – Interface sharpening Conservative level set (Olsson, 2005) ∂α1 ∂τ + · [α1 (1 − α1) n] = ε∆α1 (5) ε = ∆x 2 ∆τ = ∆x 32 Luka Štrubelj, Iztok Tiselj “Jožef Stefan” Institute

14. Introduction Mathematical models Numerical models Simulations Conclusions Mathematical models – Interface sharpening The interface is still smeared over 3 cells Figure: Void fraction smeared over several cells using conservative level set method. The smearing is not increasing during time Luka Štrubelj, Iztok Tiselj “Jožef Stefan” Institute

15. Introduction Mathematical models Numerical models Simulations Conclusions Mathematical models – Interface sharpening The interface is still smeared over 3 cells Figure: Void fraction smeared over several cells using conservative level set method. The smearing is not increasing during time Luka Štrubelj, Iztok Tiselj “Jožef Stefan” Institute

16. Introduction Mathematical models Numerical models Simulations Conclusions Numerical models Spatial discretization: finite difference Conservative calculation of the fluxes with the high resolution scheme for advection Staggered grid Time scheme: Euler explicit Pressure correction: SIMPLE Solver: CGSTAB Operator splitting for the drag force Luka Štrubelj, Iztok Tiselj “Jožef Stefan” Institute

20. Introduction Mathematical models Numerical models Simulations Conclusions Kelvin-Helmholtz instability Rayleigh-Taylor instability Simulations Several test cases were used to validate the improved two-ﬂuid model with interface sharpening for ﬂows with large interfaces: Rayleigh-Taylor instability, dam break (the interface sharpening, the drag force) Pressure jump over a droplet, droplet oscillations, rising bubble, wetting angle, Kelvin-Helmholtz and Rayleigh-Taylor instability (the surface tension force implementation) Luka Štrubelj, Iztok Tiselj “Jožef Stefan” Institute

23. Introduction Mathematical models Numerical models Simulations Conclusions Kelvin-Helmholtz instability Rayleigh-Taylor instability Kelvin-Helmholtz instability Fluid properties ρ1 = 780 kg/m3 ρ2 = 1000 kg/m3 µ1 = 0.0015 Pa·s µ2 = 0.001 Pa·s σ = 0.04 N/m g = −9.81 m/s2 Proposed by Tiselj, 2004 Figure: Tilted tube: initial conditions. Luka Štrubelj, Iztok Tiselj “Jožef Stefan” Institute

24. Introduction Mathematical models Numerical models Simulations Conclusions Kelvin-Helmholtz instability Rayleigh-Taylor instability Kelvin-Helmholtz instability Luka Štrubelj, Iztok Tiselj “Jožef Stefan” Institute

25. Introduction Mathematical models Numerical models Simulations Conclusions Kelvin-Helmholtz instability Rayleigh-Taylor instability Kelvin-Helmholtz instability Luka Štrubelj, Iztok Tiselj “Jožef Stefan” Institute

26. Introduction Mathematical models Numerical models Simulations Conclusions Kelvin-Helmholtz instability Rayleigh-Taylor instability Kelvin-Helmholtz instability λc = 2π σ g(ρ2−ρ1) Table: Onset of instability tonset and the most unstable wavelength λc : experiment, theory and simulations on various grids. nx × ny tonset [s] λc [mm] Theoretical 1.7 27 Experimental 1.88 ± 0.07 25 – 45 Numerical 1830×30 1.93 42.5 2440×40 2.04 40.4 3660×60 2.12 39.1 Luka Štrubelj, Iztok Tiselj “Jožef Stefan” Institute

27. Introduction Mathematical models Numerical models Simulations Conclusions Kelvin-Helmholtz instability Rayleigh-Taylor instability Kelvin-Helmholtz instability Figure: Kelvin-Helmholtz instability amplitude growth: theoretical, in experiment and in numerical simulations on diﬀerent grids. Luka Štrubelj, Iztok Tiselj “Jožef Stefan” Institute

28. Introduction Mathematical models Numerical models Simulations Conclusions Kelvin-Helmholtz instability Rayleigh-Taylor instability Rayleigh-Taylor instability Fluid properties ρ1 = 3 kg/m3 ρ2 = 1 kg/m3 µ1 = 0.03 Pa·s µ2 = 0.01 Pa·s σ = 0.01; 0.04; 0.16 N/m g = −9.81 m/s2 Dimensions L = 4 m H = 1 m Most unstable wavelength λ∗ = λc √ 3 = 2π √ 3 σ g(ρ2−ρ1) Luka Štrubelj, Iztok Tiselj “Jožef Stefan” Institute

29. Introduction Mathematical models Numerical models Simulations Conclusions Kelvin-Helmholtz instability Rayleigh-Taylor instability Rayleigh-Taylor instability Luka Štrubelj, Iztok Tiselj “Jožef Stefan” Institute

30. Introduction Mathematical models Numerical models Simulations Conclusions Kelvin-Helmholtz instability Rayleigh-Taylor instability Rayleigh-Taylor instability Luka Štrubelj, Iztok Tiselj “Jožef Stefan” Institute

31. Introduction Mathematical models Numerical models Simulations Conclusions Kelvin-Helmholtz instability Rayleigh-Taylor instability Rayleigh-Taylor instability Table: Most unstable wavelength of Rayleigh-Taylor instability: comparison between the theory and numerical simulation. σ = 0.01 N/m σ = 0.04 N/m σ = 0.16 N/m nx × ny λ∗ [mm] λ∗ [mm] λ∗ [mm] Th. 243 487 973 Num. 128×32 358-437 492-787 787-1313 256×64 331-496 496-793 992-1323 512×128 362-498 498-797 996-1328 Luka Štrubelj, Iztok Tiselj “Jožef Stefan” Institute

32. Introduction Mathematical models Numerical models Simulations Conclusions Open questions Conclusions Two-fluid model for flows with large interfaces was improved The interface was sharpened with the conservative level set model and the numerical diffusion was eliminated The surface tension force was implemented The interfacial drag force was modified to more universal formulation The improved two-fluid model was validated on test cases: Rayleigh-Taylor instability Kelvin-Helmholtz instability It was shown that the developed two-fluid model with interface sharpening can be used as an interface tracking model Luka Štrubelj, Iztok Tiselj “Jožef Stefan” Institute

37. Introduction Mathematical models Numerical models Simulations Conclusions Open questions Open questions Development of the coupled model Implementation of the energy equation with energy and mass transfer between phases Implementation of the turbulence model Luka Štrubelj, Iztok Tiselj “Jožef Stefan” Institute

40. Luka Štrubelj, Iztok Tiselj “Jožef Stefan” Institute

Numerical Simulations Of Basic Interfacial Instabilities With the Improved Two-Fluid Model

Recommended

Recommended

More Related Content

What's hot

What's hot (20)

Viewers also liked

Viewers also liked (20)

Similar to Numerical Simulations Of Basic Interfacial Instabilities With the Improved Two-Fluid Model

Similar to Numerical Simulations Of Basic Interfacial Instabilities With the Improved Two-Fluid Model (20)

Recently uploaded

Recently uploaded (20)

Numerical Simulations Of Basic Interfacial Instabilities With the Improved Two-Fluid Model