On Air Bubbles Sliding through a Thermal Boundary Layer

Transcription

1 On Air Bubbles Sliding through a Thermal Boundary Layer A comparative study of Predicions by OpenFoam and Fluent s VOF Schemes Dr Yan Delauré and Abdulaleem Albadawi School of Mechanical Engineering, Dublin City University 5th OpenFOAM Workshop June 21, June 24, 2010

2 Table of contents 1 Formulation Governing Equations VOF Formulation Solution 2 Problem Setup Sample OpenFOAM Sample Fluent Comparison 3 Summary

3 Formulation Governing Equations Conservation Equations Incompressible Continuity and Variable Flow properties (ρu) t U = 0 + (ρu U) = p + τ + ρg + F sv Incompressible Energy Equation with uniform density and heat capacity but variable thermal conductivity ρc p T t + ρc p (T U) = k (T ) + k T Volume Fraction Conservation α t + (αu) = 0

4 Formulation VOF Formulation Continuum Surface Force Model Continuum Surface Force a OpenFOAM Implementation F sv (x s ) = σ [c] κ(x s)n(x s ) κ(x s ) = ˆn(x s ) ˆn(x s ) = c(x s) c(x s ) c = α α κ = V p ˆn f dv (σκ) f = interpolate( σκ) n f = sngrad(α) ˆn f = interpolate( α α ) F p = reconstruct( (σκ) f n f ) a Brackbill,J.U et al, 1992

5 Formulation VOF Formulation Static Contact Angle ˆn sf User Specified Contact Angle : θ Interface θ ˆn c ˆn { Target Interface a 12 = ˆn ˆn sf = cos(β) b 1 = ˆn c ˆn sf = cos(θ) b 2 = ˆn ˆn c = cos (β θ) { b = b 2 a 12 b 1 1 a 12 a 12 a = b 1 a 12 b 2 1 a 12 a 12 ˆn makes an angle β with ˆn sf : ˆn ˆn sf = cos(β) Boundary condition specifies the angle θ: ˆn c ˆn sf = cos(θ) ˆn c = aˆn sf + bˆn can be solved for a and b

6 Formulation VOF Formulation Energy Equation Class twophaseheatedmixture: public transportmodel transportmodel modified to allow: viscosity as function of temperature ρ(t ) = [ (T 273) ] kg m 3 density and conductivity as functions of temperature µ(t ) = exp[ T 1 + (690T 1 ) 2 3 kg ] 10 m.s k(t ) = [ 8.01(T 273) (T 273) ] 10 6 W m.k Energy Conservation Equation T ρc p t + ρc p (T U) = k (T ) + k T fvscalarmatrix TEqn ( rhocp* fvm::ddt( T) + rhocp*fvm::div(phi, T) - fvm::laplacian(kthf, T) - (fvc::grad(t) & fvc::grad(kthf)) )

7 Formulation Solution Solution Algorithm Algorithm (A PISO Formulation with outer iterations) while t < endtime do Set t := t + δt and Set initer := 0 repeat Set nouteriter := ter + 1 correct the two-phase mixture properties Solve the volume fraction transport equation for α p Solve the momentum prediction equation for U p Set piter := 0 repeat Set piter := piter + 1 Solve the pressure correction equation for p p Correct the flux φ f and Velocity U p Until piter > niter Solve the energy equation for T p Until nouteriter > NOuterIter end while

8 Formulation Solution Solution Parameters Discretisation Schemes ddtschemes: default backward gradschemes: default Gauss linear divschemes div(rho*phi,u) Gauss limitedlinearv 1.0 div(phirb,alpha) Gauss interfacecompression div(phi,alpha), div(phi,t) Gauss vanleer laplacianschemes: default Gauss linear corrected Solution Parameters momentumpredictor no ncorrectors 3; nnonorthogonalcorrectors 0 nalphacorr 2; nalphasubcycles 0; calpha 2 NOuterIter 3

9 Problem Setup Benchmarking 1 Fluent methods: HRIC and Geo-Reconstruct (PLIC) 2 OpenFOAM: alpha scheme parameters {c α, α sub, α cor } = {{1, 0, 2}, {2, 0, 2}{2, 2, 4}} thermal property k = {constant, variable} 3 Parameters Bubble injection after 15s of heating. t=0s at injection Parameters Values Units Surface tension N/m Contact angle 25 - Plate inclination angles 20 - Plate dimensions {x,y,z}={0.073,0.01,0.002} m Tank dimensions {x,y,z}={0.08,0.01,0.02} m Bubble diameter m Boundary heat flux q =4889 W/m 2

10 Sample OpenFOAM OpenFOAM 3D Temperature Contour Parameters {c α, α sub, α cor } = {2, 0, 2}, k = var

11 Sample OpenFOAM OpenFOAM Temperature Profiles along Plate Centerline Parameters {c α, α sub, α cor } = {2, 0, 2}, k = var

12 Sample OpenFOAM OpenFOAM Temperature D bubble /2 from plate Parameters {c α, α sub, α cor } = {2, 0, 2}, k = var

13 Sample OpenFOAM OpenFOAM Temperature Profiles along Plate Centerline Parameters {c α, α sub, α cor } = {1, 0, 2}, k = var

14 Sample OpenFOAM OpenFOAM Temperature D bubble /2 from plate Parameters {c α, α sub, α cor } = {1, 0, 2}, k = var

15 Sample Fluent Fluent Temperature Profiles along Plate Centerline HRIC VOF Scheme

16 Sample Fluent Fluent Temperature Profiles along Plate Centerline PLIC VOF Scheme

17 Sample Fluent Fluent Temperature D bubble /2 from plate PLIC VOF Scheme

18 Comparison Temperature 0ms

19 Comparison Temperature

20 Comparison Velocity of Bubble Centre of Gravity

21 Comparison Stream t = 210ms 3D View Fluent Geo-Reconstruct Scheme OFOAM {c α, α sub, α cor } = {2, 0, 2}

22 Comparison Stream t = 210ms top View Fluent Geo-Reconstruct Scheme OFOAM {c α, α sub, α cor } = {2, 0, 2}

23 Summary Summary Fluent The HRIC scheme does not preserve the sharp interface THE PLIC scheme uses 1-3 cells to represent the interface OpenFOAM alpha Scheme The interface spreads over 2-3 cells for both c α = 2 and c α = 1 Shedding of fractions of gas phase increases noticeably when c α is reduced to 1. this could be due to differences in the bubble dynamics the dynamics of the sliding bubble also changes significantly with a reduction in the slide velocity but an increase in the bouncing of the bubble when c α is reduced to 1 (approx. 30 % difference in average bubble slide velocity) Significant mixing between the two phases is occurring at the wall with c α = 1 Sub Cycle Iterations of Alpha Scheme are not necessary when Courant number is less than 0.25

24 Summary Summary Continued Outer iteration to update fluid properties after energy solution are required for robust solution. Comparison between PLIC VOF and alpha Scheme with α = 2 Small differences in the temperature profiles and thermal boundary layer characteristics where observed before injection of bubble (max. 2 % of temperature range) The average bubble velocity predicted by Fluent is approx. 13 % higher Significant differences in temperature profiles are observed in the wake of bubble, both at the plate surface and at a distance of half the bubble diameter from the plate. Bubble wake shows significant differences in pathlines and bubble shape/dynamics

25 Summary Acknowledgment This work was part supported by Science Foundation Ireland under its Research Frontier Programme - Grant no 09/RFP/ENM2151.