Sharp-Interface Simulations of Heat-Driven Flows with Marangoni Effect

Transcription

1 th International Smposium on Turbulence and Shear Flow Phenomena (TSFP), Chicago, USA, Jul, 27 Sharp-Interface Simulations of Heat-Driven Flows with Marangoni Effect Johannes L. Pieper Felix S. Schranner Gurvan Hermange Nikolaus A. Adams Abstract A method to account for the effect of interfacial surfactant concentration on multi-phase flows has been proposed b Schranner & Adams (26). The therein validated method builds up on the the sharp-interface method of Hu et al. (26). This work advances the method to simulate and analse configurations of industrial applicabilit. In the majorit of these configurations temperature gradients, commonl due to local heat sources or sinks, are the dominant cause of the Marangoni flow. In order to account for the Marangoni effect properl, the temperature distribution within the two-phase flows under consideration has to be known. In this work, two numerical approaches for the calculation of the temperature are presented and evaluated. In the first approach the temperature transport equation, realized as a scalar transport equation, is coupled to the finite volume algorithm of Schranner & Adams (26). The underling method for the second approach is the weakl compressible high-resolution approach for incompressible flows (Schranner et al. (23)), which is generalized to compressible and incompressible flows. The extension to multi-phase flows with discontinuous phases is formulated in this work. The approaches are evaluated and compared on basis of the test flow proposed b Pendse & Esmaeeli (2). It is sufficientl complex in its evolution, et simple in its set-up and permits comparison to an analtical reference solution. The application of the methods to further thermocapillar flows is pursued. Both methods are robust and correctl predict the evolution of the Marangoni flows and temperature fields. Numerical experiments indicate that the passive scalar method is faster than the energ equation method. Yet, for two-phase flows with fluids of ver diverse properties the energ equation method is more accurate. Introduction In man classical two-phase flows, such as viscous fingering (Matar & Troian (999)), drop break-up and coalescence (Leal (24)), tip-streaming (Bruijn (993)), and buoanc-driven bubble-motion (Fdhila & Duineveld (996)), the effects of locall variable surface tension are significant. The causes are inhomogeneousl distributed interfacial surfactants and temperature gradients. The accurate and concurrentl robust two-phase flow simulation of immiscible fluids with large densit and viscosit ratios in combination with resolved interface dnamics can be numericall challenging. Hu et al. (26) have developed a full conservative, Eulerian, level-set based sharp-interface method (SIM) to simulate immiscible, compressible flows. The sharp-interface framework has demonstrated to accuratel predict two-phase flows, with each phase obeing different equations of state, and large densit and viscosit ratios (Lauer et al. (22); Schranner et al. (26)). A viscous extension of the SIM for incompressible two-fluid flows is due to Luo et al. (25). A robust, more accurate, simpler and eas to implement further-development of the SIM that incorporates inviscid, viscous, capillar and Marangoni interfacial stresses is due to Schranner & Adams (26). Generall, the SIM has demonstrated to be a reliable and simple approach to capture multi-fluid interfaces robustl even for complex interface topolog changes (Luo et al. (25); Hu et al. (26); Lauer et al. (22); Schranner et al. (26)). To account for interfacial surfactant transport a method has been developed and applied for the diffuse interface approach b Teigen et al. (2), Adalsteinsson & Sethian (23), and Xu & Zhao (23). It has been improved in terms of accurac, surfactant mass conservation and computational efficienc b Schranner & Adams (26), emploing the sharp-interface methodolog. To allow for the simulation of temperature-gradient driven Marangoni flows, the transport of mass, momentum and energ within each phase and across the interface needs to be considered appropriatel. Based on the accurate interfacial temperature distribution, the local capillar and Marangoni stresses as well as their contribution to the momentum and energ balance of the involved fluids can be computed straightforwardl. This generalization is pursued within the scope of this work. Therefore, two numerical approaches for advancing the temperature within the phases and exchanging energ across the interface are presented. The are evaluated on basis of a selection of test flows. Furthermore, the industrial application are sought. Basic Concepts The finite-volume algorithm of Schranner et al. (23) for weakl compressible turbulent and non-turbulent single-phase flows, further-developed to multi-phase configurations with capillar and Marangoni effects Schranner & Adams (26) is the underling numerical method. For incompressible flows the energ equation is decoupled from the transport of mass and momentum Panton (984) and becomes obsolete. Yet, it is not obsolete if internal energ or temperature distributions are of interest.

2 Passive scalar temperature transport equation (PS) For incompressible multi-phase flows with temperature gradients and negligible temporal pressure fluctuations the temperature equation ( ) T ρc p t + u T = λ T + λ T () 2 3 µ ( u)2 + 2µ (( u) : ( u)) + βtu p + ω holds within the two-fluid domain Ω = Ω ξ Ω ξ 2. The heat capacit c p, coefficient of thermal expansion β, thermal conductivit λ, dnamic viscosit µ = ρν are material properties of the fluids ξ and ξ 2. Numericall, within the multi-fluid cells, which are those cells containing both fluids, the material properties are determined from the volume fractions ζ ξ i(t) = V dv of the fluid phases Ω Ω ξ i ξ i and the individual fluid properties: e.g. c p = (ζ c p ) ξ + (ζ c p ) ξ 2. The derivatives of Eq. () are approximated with 4 th -order central (LHS: st term, RHS: st, 2 nd, 3 rd, 4 th term) and 5 th -order upwindbiased (LHS: 2 nd term, RHS: 5 th term) finite difference schemes, respectivel. Note, however, from numerical experiments we find that the order of convergence is independent from the choice of the order for the central differences, thus due to the order of accurac of the scheme for fluid transport. The 6 th term of Eq. () defines volume and area sources. To consider an area-distributed heat source (AHS) the power of a radiative heat source P is considered to act uniforml on a certain area AIF tot. The product with the ratio of interface-area AIF [i, j,k] to volume V [i, j,k] of a cell [i, j,k] is the heat added to each cell at the interface: ω = P AIF tot AIF [i, j,k] V [i, j,k]. Energ equation method (EEM) The method of Schranner & Adams (26) is extended to incorporate the energ equation for compressible as well as weakl compressible flows, see also Schranner (27). Hence, within each fluid phase the transport equations for the conservative states U = (ρ,ρu,ρe t ) are solved seperatel. e t = e kin + e i. B considering the interface-interaction flux in each multi-fluid cell X ξ i = {,X M,ξ i,x E,ξ i } (2) the two subdomains Ω ξ i containing one of the fluids ξ i are coupled and the sstem is globall conservative. The momentum flux is: X M,ξ i = {X,ξ i + X,ξ i} (Schranner & Adams (26)). In contrast to Schranner (27) conductive heat transfer is also considered in this work. Thus, the interfacial energ flux is X E,ξ i = X M,ξi u ξ i I + q I n ξ i I. (3) Thereb, the interface heat flux is q I = q I,α = λ I,α D(T ) I,α (4) with α =,2,3 denoting the Cartesian directions. Numericall, the temperature gradient across the interface in direction α, denoted as D(T ) I,α, is approximated with a 4 th -order central difference. The stencil is constructed of T = {(ζ T ) ξ +(ζ T ) ξ 2}. To account for the discontinuous λ within interface cells, let λ I,α = ). (5) D( xα λ(u) du The order of the approximation of D of Eq. (5) is consistent with the order of D(T ) I,α. The interface temperature T I is obtained from e i = c p T I. Temperature dependent material properties Temperature dependencies of the material properties are considered. For incompressible or weakl compressible fluids, the Boussinesq approximation is used to account for a temperature dependent densit in the buoant force, modeled as a source term: ρ = ρ re f ( β T ( T Tre f )), (6) where β T is the coefficient of thermal expansion. The temperature dependent surface-tension σ is modeled with a linear Langmuir constitutive equation: σ(t ) = σ re f + σ ( ) T Tre f. (7) T σ re f is the surface tension coefficient at a reference temperature T re f. These as well as the temperature gradient σ are constant T and material-specific. To consider solidification and melting phase changes the dnamic viscosit is considered constant and relativel high for temperatures lower than a specific the melting temperature T S. For higher temperatures, the dnamic viscosit obes an Arrhenius function. Continuous transition is achieved b means of a Heaviside function. µ max, T < T S 2 δt (µ max µ(t S )) µ(t ) = ( H (T T S + δt,δt)), T S 2 δt T S (8) µ exp( E T T S ), T T S. The temperature dependenc of the specific heat capacit c p includes a normall distributed latent heat h S around the melting point, the width is T : ( ) ) 2 exp ( T TS T c p (T ) = c p + h S. (9) π T Thermocapillar Flow To validate the two models and verif their implementation a thermocapillar driven creeping flow with two superimposed planar fluids is considered, see Pendse & Esmaeeli (2). The analtical solutions for the temperature T field, and the velocities u and v are available. The geometrical set-up is shown in figure. Figure. Geometrical Set-up for thermocapillar flow. δ = δ 2 =. and l = 4δ are used. The boundar condition are periodic in x -direction. In x 2 -directions a no-slip condition is assumed. The uniform temperature T (x,δ 2 ) = T c is imposed at the top boundar. Along the bottom boundar a sinusoidal temperature distribution with T (x, δ ) = T h + T cos( 2π l x ) is assumed. T h = 2, T = 4, T c =. The stead-state solution ma be found in Pendse & Esmaeeli (2) or Schranner & Adams (26). The kinematic viscosities are ν ξ = ν ξ 2 =.2. Both fluids obe the Tait equation of state, with γ =.. The speed of sound depends on the densit. For fluids with ρ =. the speed of sound is a =, for fluids with ρ = the speed of sound is a =.. Moreover, σ T = 5 4 and σ =.3 at T re f =. Four different cases are considered. For the upper fluid ρ ξ 2 =., c ξ 2 = 2/3 and λ ξ 2 =.2. The densit, heat capacit and the thermal conductivit of the lower fluid are case-dependent:

3 Case : ρ ξ = ρ ξ 2, c ξ = c ξ 2 and λ ξ = λ ξ 2, Case 2: ρ ξ = ρ ξ 2, c ξ = 5c ξ 2 and λ ξ = 5λ ξ 2, Case 3: ρ ξ = ρ ξ 2, c ξ = c ξ 2/ and λ ξ = λ ξ 2, Case 4: ρ ξ = ρ ξ 2, c ξ = c ξ 25/ and λ ξ = 5λ ξ 2. The reference length and velocit are L re f = δ =. and U re f = σ T T δ µ ξ l = 4, respectivel. Thus, one finds that the Renolds, Marangoni, Peclet, capillar, and Weber numbers are Re= U re f L re f =.25, Ma= U re f ρ ξ 2 c ξ 2 l = ν ξ 2 λ ξ 2 3, Pe= U re f L re f ρ ξ 2 c ξ 2 = λ ξ 2 2, Ca= ρξ 2U re f ν ξ 2 σ = 6 We= ρξ 2U re 2 f L re f σ = 48. These numbers compl Re<<, Ca<< and Ma<<, hence the analtical solution of Pendse & Esmaeeli (2) hold. To evaluate the simulation, the L - L - and L 2 - absolut errors are evaluated. The absolute error norms are calculated for u, v and T L = max M ( ε ), L = V ε dv { ε V } [i, j], M ( ( ) /2 L 2 = ε 2 dv V ) /2 { ε 2 V } [i, j]. M The is reduced in five steps from.2 to.625. (a) (b) (c) Case The first case serves as a basic verification and validation configuration. The temperature and flow field is depicted in Fig. 2. Figure 3 shows the L errors for the EEM and PS approaches. For both methods the results converge to the analtical solution independent from the chosen approach. Regarding T, L is below.49 for PS and.266 for EEM. The maximum error is lower than.2 for both methods at the largest resolution. The largest deviation of u and v, found on the interface, is (PS) and (EEM),.8 5 (PS) and.74 5 (EEM), respectivel. With respect to the maximum velocities of u max = 3 4 and v max = 6 5 these values indicate good results even for the lowest resolution. Spatial refinement leads to a reduction in the errors. The convergence order is between. and.3 for the velocities, and. < L (T ) <.6 for T x Figure 2. Results (PS) for Case with x 2 at stead state, t = 2. Case 2 The second case simulates two fluids with the same densit and different thermal conductivities and capacities. This corresponds to a highl simplified model of a laser beam welding process, as air and metal have a large difference in their thermal conductivities and capacities. Figure 4 depicts the L errors for the EEM and PS. L 2 and L are not shown for clarit. The simulation results for both methods are in a good agreement with the analtical solution. EEM leads to more accurate results than the results of PS. This is found from evaluating all three error norms. A possible explanation ma be that while the EEM calculates the T L x. x.6 x.3 x. x x.8.5 L,u,PS L,u,EEM L,v,PS L,v,EEM L,T,PS L,T,EEM Figure 3. L error norms for u, v and T for Case with PS and EEM at stead state, t = 2. energ for both fluids and models the heat flux across the sharp interface, the PS calculates the temperature for the entire domain without explicitl considering the interface. The deviations from the analtical solution are larger with the PS. Consequentl, larger deviations in the temperature gradient are found which lead to different results of the flow velocities. The maximum velocities of the analtical solution are u max,anl and v max,anl.2 4. The simulated maximum velocities are u max,ps , v max,ps.24 4, u max,eem , v max,eem.6 4 found at a mesh width of x.5. The accurac of the first case is not achieved for both methods. The temperature order of convergence is about.2 for PS and. for EEM. L x. x.2 x.3 x.3 x x..5 L,u,PS L,u,EEM L,v,PS L,v,EEM L,T,PS L,T,EEM Figure 4. L error norms for velocit u, v and temperature T for Case 2 with PS and EEM at stead state, t = 2. Case 3 This case extends Case b considering a densit of. Figure 5 depicts the L errors for the temperature at different resolutions. The velocit contours are depicted in figure 6 The temperature distribution approaches the analtical solution well. The maximum errors, e.g. L, are identical to Case for both methods and same resolutions. In the upper fluid, Ω ξ 2, which has a lower densit, significantl higher velocities are observed in the stationar state than in Ω ξ

4 L 2 3 x. x.6 L,T,PS L,T,EEM Case 4 Case 4 is a combination of Cases 2 and 3. It is a configuration that resembles laser beam welding better than the previous cases. Figure 7 depicts the L temperature error norms for when emploing EEM and PS. The temperature fields as well as the resolution dependent error norms are found to be identical to those of Case 2. The velocities are found to differ significantl from the analtical solution. A possible explanation is given for Case 3, which also holds for Case 4. x Figure 5. L error norms for T for Case 3 with PS and EEM at stead state, t = 2. L x. L,T,PS L,T,EEM independent from when emploing EEM or PS According to Pendse & Esmaeeli (2) the densit has no influence on the solution as long as Re<<, Ca<< and Ma<<. The ratio of the two dnamic viscosities, which is reduced from (Case ) to, has onl an influence on the maximum velocit in the analtical solution. Pendse & Esmaeeli (2) do not impose an restriction to the densit ratios. The question is raised, at this venue, whether the analtical solution does or does not appl to combinations of fluids with ver large densit ratios, although the creeping flow conditions are fulfilled for one of the two fluid. Within ξ (the lower fluid) Ca = 6 for Case 3, for Cases and 2 it is Ca = 6. Hence, ξ cannot be considered a creeping flow, even though ξ 2 can. Figure 7. L error norms for T for Case 4 with PS and EEM at stead state, t = 2. Thermocapillar motion of a Drop 5E 6 5E 6.5 5E 6 5E 6.5E 5.5E 5 5E 7 5E 7.5 E 7 E 7 E 8 E x 5E 7.5 5E 6 E 8 (a) u-velocit field 5E 7 3E 63E 6 3E 6 5E 6 E 7 E 7 E 8 E 8 E 8.5 E 8 5E 7 3E 6 E 7 E x (b) v-velocit field Figure 6. Velocities u and v with a densit-ratio of at stead state, t = 2. Figure 8. Principal sketch of the Set-up for Rising Bubble. Another test case for the validation of the implemented equations is provided b Young et al. (959), who have investigated the velocit of a two-dimensional rising drop. For this purpose, a drop with the radius R is set in a clinder with the height L and the radius 5R (see figure 8). Furthermore, a linear temperature field is specified over the height of the clinder. The temperature at the bottom of the clinder is T ( = ) = T H and at the top T ( = L) = T C. This results in a linear temperature profile. The surface tension changes linearl with the height according to the temperature distribution. For validation, the average ascent rate is calculated and compared to the ascent rate u Y GB according to Young et al. (959). The empiricall evaluated ascent rate is ) 2( σ T TH T C L R ρgr 2 µ +µ 2 µ u Y GB =. () 9µ + 6µ 2

5 The average ascent rate of the simulated drop is determined b ū 2 = u 2 dv = V 2 Ω 2 V 2 u 2 ζ ξ 2 V, (2) M where M is the number of cells, V 2 is the volume of the drop, and ζ ξ 2 is the volume fraction of the disperse phase. Furthermore, R =.5 and L = 7.5R. For the fluids, ρ ξ = ρ ξ 2 =.2, a ξ = 5., a ξ 2 = 4. The Tait equation of state holds for both fluids. The initial pressure for both fluids is equal to p =. With the dnamic viscosit µ ξ = µ ξ 2 =., g =, σ T = and the temperatures T C = 3 and T H = 3.5, a theoretical ascent rate of u Y GB = 45 4 is obtained. Thus, the Renolds number is Re= This configuration is simulated with EEM and PS. The results are nearl identical. For clarit, we onl provide results for PS in Figs. 9,. A convergence analsis is performed with a spatial resolution ranging from to 256, four different refinement levels are simulated. For a resolution of x = the deviation from the theoretical ascent rate is alread less than %. The results can be considered to be converged for a resolution of x = as these do not differ for a grid of x = The predicted ascent rates of the drop and the pressures agree well with the empirical solutions. ū2 uy GB pressure p Figure time t Normalised ascent velocit for different resolutions theoretical solution simulation 256 Figure. Simulated pressure at x =.25 and t = compared to theoreticall expected pressure Application Example - Laserdeep-Welding Based on Case 4, a simple set-up to simulate heat conduction welding is developed. Copper is chosen to be the welding material and air as the ambient gas. The geometrical set-up is l =. and δ = δ 2 =.25, comparable to figure. Moreover, ρ ξ 2 =.9, c ξ 2 = 4, µ ξ 2 = and λ ξ 2 =.262, ρ ξ = 896, c ξ = 385 and λ ξ = 394. The latent heat of copper is h ξ S = 27, it is distributed normall with T = 25 around the melting point T ξ S = 357, see Eq. (9). The dnamic viscosit is modeled according to Eq. (8), µ ξ max =, µ ξ (T S ) =.3 and E ξ = The heat source is a laser beam, modeled as an AHS. It acts within a small region with an focus diameter of d f =.3 and a powerdensit of 3 9. To account for thermal convection, ρ(t ) is modeled according to Eq. (6). To model the temperature dependent surface tension Eq. (7) holds. The set-up is shown in Figure. The feed Figure. Geometrical setup of the laser welding configuration. rate is zero in order to examine the convection in the molten pool. We simulate the initial heat-up process. The flow within the molten metal pool and air are visualized in Figs. 2 and 3, respectivel. Note that the molten metal pool is the area enclosed b the T = T S - line, the temperature within the enclosed area is higher than T S. The two-dimensional model with the selected parameters shows that the melting temperature is reached after s. A Marangoni flow develops, driving the liquid phase. The maximum velocities are observed at the interface where the melting temperature T S = 357 is reached. The comparabl higher velocities in the air are due to the comparabl low dnamic viscosit of the air. Figure 2. Fluid flows in molten metal pool at t =.5 3. Conclusion Two methods to simulate two-phase immiscible, incompressible flows with temperature dependent material properties have been presented. Both methods are applicable to simulate Marangoni flows with a sharp-interface. In the first method temperature is propagated as a passive scalar, multifluid cells are considered b mixing fluid properties, the transport equations of mass and momentum appl for the fluids in motion, the exchange of momentum across the interface couples the two phases. The second method emplos the transport equations for mass, momentum and energ and couples the immiscible phases b means of interface-fluxes, modeling the

6 Figure 3. Fluid flow in ambient gas at t =.5 3. exchange of momentum and energ across the interface. B comparing the results to analtical solutions of a selection of test flows the methods are verified and their implementations validated. Both methods are robust and correctl predict the evolution of the Marangoni flows and temperature fields. The simulation results of a thermocapillar flow are in excellent agreement with analtical data for both methods. Numerical experiments indicate that the passive scalar method is faster than the energ equation method. Yet, for two-phase flows with fluids of ver diverse properties the energ equation method is more accurate. As an industrial application example a two-dimensional base configuration of a laser beam welding process has been simulated. Acknowledgment We acknowledge the Deutsche Forschungsgemeinschaft (DFG) for funding. Felix S. Schranner is member of the TUM Graduate School. The Munich Centre of Advanced Computing (MAC) has provided the computational resources. REFERENCES Adalsteinsson, David & Sethian, J.A. 23 Transport and diffusion of material quantities on propagating interfaces via level set methods. Journal of Computational Phsics 85 (), Bruijn, R.A. De 993 Tipstreaming of drops in simple shear flows. Chemical Engineering Science 48 (2), Fdhila, R. Bel & Duineveld, P.C. 996 The effect of surfactant on the rise of a spherical bubble at high renolds and peclet numbers. Phsics of Fluids 8 (2), Hu, XY, Khoo, BC, Adams, Nikolaus A & Huang, FL 26 A conservative interface method for compressible flows. Journal of Computational Phsics 29 (2), Lauer, E., Hu, X. Y., Hickel, S. & Adams, N. A. 22 Numerical investigation of collapsing cavit arras. Phsics of Fluids 24 (5). Leal, L. G. 24 Flow induced coalescence of drops in a viscous fluid. Phsics of Fluids 6 (6), Luo, J., Hu, X.Y. & Adams, N.A. 25 A conservative sharp interface method for incompressible multiphase flows. Journal of Computational Phsics 284 (), Matar, Omar K. & Troian, Sandra M. 999 The development of transient fingering patterns during the spreading of surfactant coated films. Phsics of Fluids (), Panton, R.L. 984 Incompressible Flow. John Wile & Sons Australia, Limited. Pendse, Bhushan & Esmaeeli, Asghar 2 An analtical solution for thermocapillar-driven convection of superimposed fluids at zero renolds and marangoni numbers. International Journal of Thermal Sciences 49 (7), Schranner, Felix Sebastian 27 Weakl compressible models for complex flows. PhD thesis,. Schranner, Felix S. & Adams, Nikolaus A. 26 A conservative interface-interaction model with insoluble surfactant. Journal of Computational Phsics 327, Schranner, Felix. S., Hu, Xiangu & Adams, Nikolaus A. 23 A phsicall consistent weakl compressible high-resolution approach to underresolved simulations of incompressible flows. Computers & Fluids 86, Schranner, Felix S., Hu, Xiangu & Adams, Nikolaus A. 26 On the convergence of the weakl compressible sharp-interface method for two-phase flows. Journal of Computational Phsics 324, Teigen, Knut Erik, Song, Peng, Lowengrub, John & Voigt, Axel 2 A diffuse-interface method for two-phase flows with soluble surfactants. Journal of Computational Phsics 23 (2), Xu, Jian-Jun & Zhao, Hong-Kai 23 An eulerian formulation for solving partial differential equations along a moving interface. Journal of Scientific Computing 9 (-3), Young, N. O., Goldstein, J. S. & Block, M. J. 959 The motion of bubbles in a vertical temperature gradient. Journal of Fluid Mechanics 6 (3),