Turbulence Dispersion Force Physics, Model Derivation and Evaluation

Transcription

1 Turbulence Dispersion Force Physics, Model Derivation and Evaluation J.-M. Shi, T. Frank, A. Burns 3 Institute of Safety Research, FZ Rossendorf shi@fz-rossendorf.de ANSYS CFX Germany 3 ANSYS CFX FZR ANSYS CFX Workshop on Multiphase Flow , Dresden FORSCHUNGSZENTRUM ROSSENDORF Institut für Sicherheitsforschung

2 Contents Physics Modeling approaches Eulerian approach Lagrangian approach A new model derivations Evaluation

3 Turbulent dispersion (TD A result of the (dispersed phase particle eddy (continuous phase interaction Important for dispersed phase with small St Turbulent mass diffusion (α k u k and interphase momentum transfer Turbulent mass diffusion is usually modeled by a mass diffusion term The interphase momentum transfer due to turbulent dispersion is often separated as an interfacial force, to be calculated from the temperal correlation of the interfacial force fluctuations. Usually, only the drag contribution is essential.

4 Eulerian model mass diffusion Reynolds averaging t (ρ kα k + (ρ k α k U k = applying φ = φ + φ ( t (ρ kα k + (ρ k α k U k = (ρ k α k u k with α k u k ν k,t α k ( σ k Adopting Favré-averaged velocity Ũ k = α ku k α k = U k + α k u k (3 α k U k = Ũ k + u k + u k, u k = α k u k α k (4 t (ρ kα k + (ρ k α k Ũ k = (5 Using Favré-averaged velocity simplifies the equation system 3

5 Eulerian model turbulent dispersion force The Favré-Averaged Drag Force (FAD model, Drag: F D = C fp (U p U f = 8 C Dρ f U p U f }{{} D fp 6 α f d p }{{} A fp (U p U f (6 Reynolds averaging: F D = C fp (Ũp Ũ f + FTD (7 F TD = C fp α f u f α f α pu f EDH ν f,t C fp α p σ f ( αp α f α p α f (8 For details refer to A. Burns, T. Frank, I. Hamill and J.-M. Shi, ICMF4, Paper No. 39 4

6 Lagrangian method tracking trajectories of each particle, parcel V n+ p δt V n p = x n+ p = x n p + Vn pδt (9 ρ n f C D V p V 4ρ p d f (V p V f + p ρ p δv F other ( ( 3 construct fluid fluctuating velocity for particle-eddy interaction V n+ f = U f (x n+ p, t + v n+ f, v n+ f = av n f + ben, where ( a = exp ( δt T L, b = ( kf 3 a, e n N(, ( P n+ F n+ p Lagrangian time step P n = F n F n+ Eulerian time step 5

7 A new TD force model derivation ( Ensemble average for a cell V at time t j F D (t j = V n(t j i= f i D (t j = V n(t j i= 3 4 ρ f C i D d i p V i p Vi f (Vi p Vi f δv i (3 Time averaged drag F D = V T E N j= N n(t j i= 3 4 ρ f C i D d i p V i p Vi f (Vi p Vi f δv i δt (4 T E (macro time scale τ e = C k ǫ (eddy life time; δt τ e n(t N n(t j n(t n(t j n(t N V t N = t T E / t j = t jδt t = t t j = t + jδt t N = t + T E / 6

8 Model derivation ( Converting into the Eulerian frame Eulerian variables α p (t j = V n(t j i= δv i (5 Φ(t j = n(t j i= Φi δv i = n(t j i= δv i V n(t j i= Φi δv i α p (t j (6 Reynolds averaging operator Φ(t = T E t+te / t T E / Φ(θdθ = T E N j= N Φ(t j δt (7 Averaging the drag force F D = α p f D = α p f }{{ D + α } pf }{{ D} mean drag turbulent dispersion force (8 7

9 Derivation details ( F D = 3 4 ρ f 3 4 ρ C D f U p U d f ( U p U f αp p C D d p } {{ } D ( U p U f αp ( Up U f = D U p U f ( α p U p α p U f = D U p U f = D U p U f ( α p Ũ p α p U f α pu f α p (Ũp Ũ f + αp α f u f = Dα p U p U f ( Ũ p Ũ f + D Up U f ( αp α f α pu f α f α f u f α pu f } {{ } turbulent dispersion force F TD (9 8

10 Derivation details ( Eddy viscosity hypothesis (EVH α u f = ν f,t α u σ f = ν f,t α ( f σ f α Model expression (for the continuous phase C D F TD 3 4 ρ ν f,t f U p U d p σ f } f {{} C TD ( α p α p α f α f ( Results for two-fluid model, α p + α f = F TD 3 4 ρ f C D d p ν f,t σ f U p U f α p α f ( 9

11 Remarks The present derivation illustrates the physics of the turbulent dispersion The present derivation explains why double average makes sense Lagrangian evaluation of turbulent dispersion is very expensive. This derivation might provide a theoretical foundation for a deterministic TD force model for the Lagrangian solver

12 Evaluation

13 FZR MTLoop test facility Air-water system, isothermal Inner pipe diameter D = 5. mm Wire mesh sensor measurements Test section from gas injection: L =.3 to 3.3 m Injection nozzle arrangement:

14 Test case definition 3

15 Two-fluid model Evaluation FZR-38 experiment FAD-SST RPI-SST, C TD =.35.5 FZR-39 experiment FAD-SST RPI-SST, C TD =.35 α g..8.6 α g For details refer to J.-M. Shi, T. Frank, E. Krepper, D. Lucas, U. Rohde, H.-M. Prasser, ICMF4, Paper No.4 Th. Frank, J.-M. Shi, and A. Burns, 3rd International Symposium on Two-Phase Flow Modeling and Experimentation, Pisa, Italy, -4 September, 4 4

16 Poly-dispersed model Evaluation Stationary, axisym., bubbly flow at the upper test section (L/D = 59. Data from measurements: superficial velocities, mean bubble diameter, local gas volume fraction Index Ul Ug Air Air Air Air 3 [m/s] [m/s] VF[%] dp VF[%] dp VF[%] dp VF[%] Ul, Ug superficial velocity, dp diameter [mm], VF gas volume fraction. 5

17 Results from poly-dispersed model.5 FZR-7 exp. sum d= 4.8mm d= 7.mm.5 FZR-7 exp. sum d= 4.6mm d= 6.4mm α g α g FZR-83 exp. sum d= 3.7mm d= 5.mm d= 6.7mm.5 FZR-84 exp. sum d= 4.6mm d= 6.4mm α g α g

18 TD coefficient, FZR7, C TD = 3 4 ρ f C D dp ν f,t σ f U p U f 4 d= 4.8mm, CTD, FZR7, Grace d= 6.6mm, CTD 4 d= 4.8mm, CTD, FZR7Ishii&Zuber d= 6.6mm, CTD Turb. Disp. Coeff 8 6 Turb. Disp. Coeff CD, Slip Velocity d= 4.8mm, CD, FZR7Grace d= 7.mm, CD d= 4.8mm, Slip V d= 7.mm, Slip V CD, Slip Velocity d= 4.8mm, CD, FZR7Ishii&Zuber d= 7.mm, CD d= 4.8mm, Slip V d= 7.mm, Slip V

19 TD coefficient, FZR, C TD = 3 4 ρ f C D dp ν f,t σ f U p U f 35 3 d=.4mm, CTD, FZR, Grace d= 3.4mm, CTD 35 3 d=.4mm, CTD, FZRIshii&Zuber d= 3.4mm, CTD Turb. Disp. Coeff 5 5 Turb. Disp. Coeff CD, Slip Velocity d=.4mm, CD, FZR, Grace d= 3.4mm, CD d=.4mm, Slip V d= 3.4mm, Slip V CD, Slip Velocity d=.4mm, CD, FZRIshii&Zuber d= 3.4mm, CD d=.4mm, Slip V d= 3.4mm, Slip V