A Cartesian adaptive level set method for two-phase flows

Transcription

1 Center or Turbulence Research Annual Research Bries A Cartesian adaptive level set method or two-phase lows By F. Ham AND Y.-N. Young 1. Motivation and objectives Simulations o lows involving ree suraces have become ubiquitous in the literature. The evolution o both simulation methods and computer speed have allowed the investigation o many problems o increasing complexity and engineering relevance. For 2D simulations, it is reasonably common to use grid resolutions o or larger. See or example the 2D planar breaking wave simulations o Chen et al. (1999), or the axisymmetric drop breakup simulations o Han and Tryggvason (1999a, 1999b). Many such ree surace problems are undamentally three dimensional and the absence o threedimensional eects such as the pinch-o o stretched ilaments by surace tension makes it diicult to make valid physical conclusions or use 2D results to develop models. Achieving similar grid resolution in 3D, however, is outside the computational reach o most research centers. For many problems o interest, a ine grid is required in only a relatively small portion o the domain (e.g. around regions o high surace curvature) and much coarser grids are suicient to capture the smoothly evolving velocity ields ar rom the surace. It is this type o problem that motivates the present Cartesian adaptive method, where the inest resolution in the neighbourhood o the ree surace may be o order (sometimes called the eective resolution), but the coarsening away rom the ree surace yields an actual number o control volumes that is less than using the ine grid everywhere by at least an order o magnitude. A number o adaptive Cartesian methods or reesurace calculations have been proposed in the literature. For example, Haj-Hariri et al. (1997) used Cartesian local grid reinement to investigate thermocapillary motion o 3D drops. Their method used an octree-based structure to store the grid heirarchy. More recently Sussman et al. (1999) proposed an adaptive levelset method based on the embedded grid approach, where the global computational domain is divided into uniormly reined rectangular patches. This approach has the advantage o allowing a traditional staggered or semi-staggered structured discretization throughout most o the domain, with special interpolations required only at the interace between grid levels. The embedded grid approach may not, however, lead to the smallest Cartesian grids possible because o the restriction that reined regions be rectangular and isotropic. Additionally, embedded grid approaches can be diicult to parallelize and load balance. In the present contribution we develop a level set method based on local anisotropic Cartesian adaptation as described in Ham et al. (2002). Such an approach should allow or the smallest possible Cartesian grid capable o resolving a given low. The remainder o the paper is organized as ollows. In section 2 the level set ormulation or ree surace calculations is presented and its strengths and weaknesses relative to the other ree surace methods reviewed. In section 3 the collocated numerical method is described. In section 4 the method is validated by solving the 2D and 3D drop oscilation problem. In section

2 228 Ham & Young 5 we present some results rom more complex cases including the 3D drop breakup in an impulsively accelerated ree stream, and the 3D immiscible Rayleigh-Taylor instability. Conclusions are given in section Mathematical Formulation For recent reviews o the level set method and its many applications, the interested reader is reerred to Osher and Fedkiw (2001) or Sethian (2001). Here we briely present its ormulation or incompressible 2-luid problems. The incompressible, immiscible, two-luid system is treated as a single luid with strong variations in density and viscosity in the neighborhood o the interace. The continuity and momentum equations or such a low can be written in conservative orm as: ρ t + ρu j = 0, x j (2.1) ρu i + ρu ju i = p + τ ij + ρg i + σκδ(d)n i, t x j x i x j (2.2) where u i is the luid velocity, ρ the luid density, p the pressure, τ ij the viscous stress tensor, g i the acceleration due to gravity, σ the surace tension coeicient, κ the local ree surace curvature, δ the Dirac delta unction evaluated based on d the normal distance to the surace, and n i the unit normal at the ree surace. It is well known that solving equations 2.1 and 2.2 directly in the presence o the discontinuities at the interace will lead to the excessive smearing o the interace over long time integration, and/or numerical ringing in the region o the discontinuities due to dispersive errors in the numerical discretization. In the present work, we capture the interace as the zero-level isosurace o a higher-order unction φ, the level set unction (eectively an additional transported scalar). On either side o the interace, the sign o the level set unction can be used to determine which luid is present. This allows the luid properties to be written as a unction o the level set only, speciically ρ = ρ(φ). This allows the continuity equation to be written: ( ) dρ φ dφ t + u φ j + ρ u j = 0 (2.3) x j x j For the case o constant density except in the neighborhood o the zero level set, which represents the luid interace, the solution o the continuity equation can thus be decomposed into the solution o the ollowing: φ t + u φ j = 0 x j (2.4) φ=0 u j x j = 0 (2.5) Eq. 2.4 is the standard evolution equation or the level set unction, and eq. 2.5 is the incompressible continuity equation. Note that the level set equation need only be solved near the interace, which we have deined as the zero level set. Away rom the interace, it is common to make φ equal to a signed distance unction. In a similar way, the application o chain rule to the momentum equation allows the

3 Cartesian adaptive level set method 229 U u i, ρ, µ, p, φ PSrag replacements Figure 1. Spatial location o variables or collocated discretization. decomposition o the momentum equation into the solution o the level set equation, eq. 2.4, and the solution o the ollowing momentum equation: ρ u i t + ρ u ju i = p + τ ij + ρg i + σκδ( φ )n i (2.6) x j x i x j Equations comprise our governing system or the present work. Note that the level set ormulation results in a system o governing equations that can no longer be written in conservative orm. Thus, when the inite volume method is applied to this system, we cannot expect to achieve discrete conservation o mass and momentum (in the region o the interace). It is the hope o the level set ormulation that these errors in conservation are mitigated by the more accurate capturing o the interace that is possible with the smoothly-varying level set unction. 3. Numerical Method The governing equations are solved on a Cartesian adaptive grid. The grid arrangement and adaptation are described more completely in Ham et al. (2002), and here we only discuss details speciic to the simulation o multiphase lows using level set methods on such a grid. Figure 1 shows the spatial arrangement o variables on the Cartesian adaptive grid. All variables are stored at the control volume (CV) centers with the exception o a ace-normal velocity, located at the ace centers, and used to enorce the divergenceree constraint in each time step. The variables are staggered in time so that they are located most conveniently or the time advancement scheme. Denoting the time level by a superscript index, the velocities are located at time level t n and t n+1, and pressure, density, viscosity, and the level set at time levels t n 1 2 and t n The semi-descretization o the governing equations in each time step is then as ollows. Step 1. Advance the level set: φ n+ 1 2 φ n 1 2 t + u n 1 ( ) j φ n φ n+ 1 2 = 0 (3.1) 2 x j

4 230 Ham & Young Here the advantage o time-staggering is evident i.e. we know the velocity u n i rom the previous time step, so this equation is decoupled rom the other equations, and can be advanced on its own to get the level set value at the mid-point o the new time level, φ n Here we have used an implicit Crank-Nicholson time advancement scheme, which requires an iterative solution method. In practice the hyperbolic system is not sti, and can be quickly converged by a simple iterative scheme such as Gauss-Seidel. The spatial derivatives required in eq. 3.1 are approximated by a 5 th -order WENO scheme (Jiang & Peng 2000). While eective at recovering the viscosity solution to the level set equation, the WENO discretization requires that the grid be locally reined in a uniorm way in a band about the zero level set, and thus does not take ull advantage o the lexibility in the adaptive grid. Step 2. Reinitialize the level set by solving to steady state: φ = sgn (φ) (1 φ ). (3.2) τ The reinitialization step is perormed every time step, and ensures that the level set is a signed distance unction away rom φ = 0, without (theoretically) changing the location o the zero-level set. Spatial derivatives required in eq. 3.2 are once again approximated using a 5 th -order WENO scheme (Jiang & Peng 2000). For this equation we use explicit third order TVD Runge-Kutta method or time integration. In practice, it is not neccessary to solve to steady state, and we have ound that 5 iterations with a pseudo-time step o τ = x/4 is suicient when perormed every computational time step. Step 3. Calculate the new density and viscosity at the midpoint o the time step: ( = ρ1 H µ n+ 1 2 = µ1 H φ n+ 1 2 ( φ n+ 1 2 ) ( ( + ρ 2 1 H ) ( + µ 2 1 H φ n+ 1 2 ( φ n+ 1 2 )) )) (3.3) (3.4) where ρ 1, µ 1 are the constant properties o the luid occupying the region φ > 0, and ρ 2, µ 2 are the constant properties o the luid occupying the region φ 0, and H is a smoothed Heaviside step unction (Sussman et al. 1994). Step 4. Project the momentum equations: The remaining steps are a variant o the collocated ractional step method described, or example, by Kim and Choi (2000). First, calculate a projected velocity ield û i using the momentum equations: û i u n i t = 1 ( p x i n R n+ 1 2 i ) (3.5) where R i contains all other terms in the momentum equation, eq 2.6. We discretize both the convective and viscous terms implicitly using second-order symmetric discretizations, and the surace tension explicitly based on the known level set at the midpoint o the current time step, φ n Some care must be taken in the discretization o the surace tension terms when treated as source terms in the momentum equations o a collocated scheme, as described below.

5 Cartesian adaptive level set method 231 Step 5. Subtract the old pressure gradient: u i n+1 = û i n+1 + t 1 δp δx i Step 6. Interpolate the starred velocity ield to the aces: U n+1 = u i n+1 t R i n+ 1 2 n 1 2 Rn+ 1 2 (3.6), (3.7) where () is a second-order interpolation operator that yields a ace-normal component rom two CV-centered vectors. Step 7. Solve the variable coeicient Poisson equation or the new pressure: 1 t U n+1 A = 1 Step 8. Update the ace velocities to the new divergence-ree ield: n+ δp 1 2 A. (3.8) δn U n+1 U t = 1 n+ p 1 2 n (3.9) Step 9. Reconstruct the pressure gradient at the CV centers: 1 n+ p = x i n+ p 1 x i 2 (3.10) n where () xi represents a reconstruction operator. In this case we use a ace-area weighted least squares reconstruction. Step 10. Update the CV velocities: u n+1 i u i t = 1 p x i n+ 1 2 (3.11) By adding eqs. 3.5, 3.6, and 3.11, it is clear that a second-order time advancement o the momentum equation is recovered. A critical dierence between the present ormulation and the ormulation o Kim and Choi is in the calculation o the starred ace-normal velocities (eq. 3.7). Kim and Choi assume that: n+1/2 R i ρ n+1/2 Rn+1/2. (3.12) ρ n+1/2

6 232 Ham & Young This is an O( x 2 ) approximation, seemingly consistent with the overall accuracy o the method, and signiicantly simpliies the calculation o the Poisson equation source term. In the present investigation, however, it was ound that when surace tension orces were introduced in the region o the zero level set, this approximation could lead to large nonphysical oscillations in the CV-centered velocity ield. To solve this problem, the surace tension orces must be calculated at the aces, and then averaged to the CV centers, i.e.: Ri σ ρ Rσ x i (3.13) ρ With this calculation o the surace tension orces, we can no longer make the assumption o eq. 3.12, and the additional terms must be included in the calculation o U and thus in the Poisson equation source term. 4. Validation As validation cases or the method, we solve the 2D column and 3D drop oscillation problems D column oscillation From Lamb (1932), or a 2D column o liquid in the limit o zero viscosity perturbed according to: r = a (1 + εcos (nθ)) (4.1) where a is the mean radius, ε a small perturbation, and n the mode, the oscillation requency will be: ω 2 = n ( n 2 1 ) σ ρa 3. (4.2) Figure 2(a) shows the initial condition or a drop oscillation calculation or mode n = 5. Here we have exaggerated the perturbation amplitude to ε = 0.15 or illustrative purposes. In the actual computations, ε = 0.02 was used or the initial condition. Figure 2(b) compares the calculated results to the theory (period T = 2π/ω), with generally good agreement. For these calculations, we used periodic boundary conditions in a square domain o size L = 4a, and set the density o the surrounding luid to ρ 2 = 0.001ρ to minimize its eect on the oscillations. The luid viscosity was adjusted to keep the decay in amplitude o the oscillations to less that 5% per cycle. For a 3D drop perturbed according to D drop oscillation where the surace harmonics o order n, S n, are deined: r = a (1 + ɛs n ) (4.3) S 1 = cosθ (4.4) π

7 Cartesian adaptive level set method theory computation Period T 5 PSrag replacements mode n (a) (b) Figure 2. a) Sample grid and initial zero-level set or 2D drop oscillation problem with n = 5. b) Comparison o computed and theoretical 2D drop oscillation periods or modes n = 2 through 5. the oscillation requency is given by: S 2 = 1 5 ( 3cos 2 θ 1 ) (4.5) 4 π S 3 = 1 7 ( 5cos 3 θ 3cosθ ) (4.6) 4 π S 4 = 3 1 ( 35cos 4 θ 30cosθ + 3 ) (4.7) 16 π S 5 = 1 11 ( 63cos 4 θ 70cos 2 θ + 15 ) (4.8) 16 π ω 2 = n (n 1) (n + 2) σ ρa 3. (4.9) Figure 3 (a) shows the drop surace or the mode n = 5. Figure 3 (b) compares the calculated results to the theory, also with good agreement. 5. Results In the ollowing subsections we present some results rom simulations o more complex lows. For the purposes o this paper describing the numerical method, these cases are meant to simply illustrate the potential o the method. Consequently we do not present any analysis o the associated low physics, which will be the subject o uture investigations.

8 234 Ham & Young 10 theory computation Period T 5 PSrag replacements mode n (a) (b) Figure 3. a) Initial zero-level set or 3D drop oscillation problem with n = 5. b) Comparison o computed and theoretical 3D drop oscillation periods or modes n = 2 through Secondary breakup o spherical drops The secondary breakup o drops is an important problem in spray atomization, and the subject o the 2D axisymmetric numerical investigation o Han and Tryggvason (1999a, 1999b). We used the present Cartesian adaptive method to calculate 3D drop breakup in an impulsively accelerated ree stream. Figure 4 shows the calculated ree surace or a case run with equivalent resolution o with a domain size o 5D 5D 5D Rayleigh-Taylor Problem The Rayleigh-Taylor instability reers to the instability o the plane interace between two luids o dierent density superimposed one over the other and subject to gravity. When the upper luid has a density greater than the lower luid, the interace can be unstable to small perturbations whose ampliication is well described by linear theory with dependence on the density ratio, gravity, surace tension coeicient, and viscosity (Chandrasekhar 1961). As the ampliication continues and the problem enters the nonlinear regime, the luid mixing can become chaotic, characterized by bubbles o lighter luid rising into the heavier luid, and spikes o heavier luid alling into the lighter luid, with regions o high vorticity near the spike/bubble interace. The resulting mixing zone is known experimentally (Schneider et al. 1998) to broaden in a way that depends linearly on the gravitational acceleration g and the Atwood number A = (ρ 1 ρ 2 )/(ρ 1 +ρ 2 ), and quadratically on the time, i.e. h b,s = α b,s Agt 2, (5.1) where h b or h s represents the penetration length or bubbles or spikes respectively, and α has been introduced as a constant o proportionality, sometimes called the acceleration constant. Recently, the miscible Rayleigh-Taylor problem has been investigated numerically by

9 Cartesian adaptive level set method 235 (a) 0.5 (b) 1.5 (c) 2.5 (d) 3.5 (e) 4.5 () 5.5 (g) 6.5 (h) 8.5 Figure 4. Breakup o an initially spherical drop in an impulsively accelerated ield. The number below each igure indicates the non-dimensional time tu/d. Free stream velocity is rom top-let-back to lower-right-ront. Properties or this simulation are as ollows: Re = ρ gdu/µ g = 1000, W e = ρ gu 2 D/σ = 20, Oh = µ l / ρ l σd = Young et al. (2001), who calculate α b Experimental results or immiscible luids yield higher values o α b A recent review o experimental and simulation results or the immiscible case is given by Glimm et al. (2001). Figure 5 presents some results rom the application o the present Cartesian adaptive method to this problem. These computations were perormed in a box with the interace perturbed randomly with a maximum amplitude o Properties o the simulation were selected to give a wavelength o maximum instability o λ = These

10 236 Ham & Young Figure 5. Evolution o the interace or the 3D Rayleigh-Taylor problem with multi-mode perturbation: top panels plan view looking down on interace; bottom panels elevation view rom side. preliminary simulations were made with the relatively coarse equivalent resolution in the region o the zero-level set o , or about 10 CVs per wavelength or the most unstable mode. Figure 5 illustrates the evolution o the interace at 3 dierent times or the Atwood number A = Conclusions A simulation tool that integrates Cartesian adaptive grids with the level set method has been developed. The method as described is particularly suited to problems where the smallest scales are associated with the ree surace motions, and signiicant savings can thus be realized by coarsening the grid away rom the surace. The method has been validated and its potential demonstrated by solving several test problems, including 2D and 3D drop oscillations, drop breakup in an impulsively accelerated ree stream, and the immiscible 3D Rayleigh-Taylor problem. Our ongoing work is ocused on changing the level set solver to the particle level set method o Enright et al. (2002). The improved conservation properties o this method may allow us to move away rom the higher-order WENO discretizations presently used in the level set advancement and reinitialization equations. With lower-order methods, adaptation can be added in the region o the zero-level set, resulting in a urther signiicant reduction in computational cost.

11 Cartesian adaptive level set method 237 REFERENCES Chandrasekhar, S Hydrodynamic and Hydromagnetic Stability. Oxord University Press. Chen, G., Khari, C, Zaleski, S., & Li, J Two-dimensional Navier-Stokes simulation o breaking waves. Phys. Fluids 11 no. 1, Enright, D., Fedkiw, R., Ferziger, J. & Mitchell, I A hybrid particle level set method or improved interace capturing. J. Comput. Phys. 183, Glimm, J., Grove, J. W., Li, X. L., Oh, W. & Sharp, D. H A critical analysis o Rayleigh-Taylor growth rates. J. Comput. Phys. 169, H. Haj-Hariri, Q. Shi, and A. Borhan 1997 Thermocapillary motion o deormable drops at inite reynolds and marangoni numbers. Phys. Fluids 9, 845. Ham, F. E., Lien, F. S. & Strong, A. B A Cartesian grid method with transient anisotropic adaptation. J. Comput. Phys. 179, Han, J. & Tryggvason, G. 1999a Secondary breakup o axisymmetric liquid drops. I. Acceleration by a constant body orce. Phys. Fluids 11 no. 12, Han, J. & Tryggvason, G. 1999b Secondary breakup o axisymmetric liquid drops. II. Impulsive acceleration. Phys. Fluids 13 no. 6, Jiang, G-.S. & Peng, D Weighted ENO schemes or Hamilton-Jacobi equations. SIAM J. Sci. Comput. 21 (6), Kim, D. & Choi, H A Second-Order Time-Accurate Finite Volume Method or Unsteady Incompressible Flow on Hybrid Unstructured Grids. J. Comput. Phys. 162, Lamb, H Hydrodynamics. Cambridge University Press. Osher, S. & Fedkiw, R Level set methods: an overview and some recent results. J. Comput. Phys. 169, 463. Schneider, M. B., Dimonte, G. & Remington, B Large and small structure in Rayleigh-Taylor mixing. Phys. Rev. Lett. 80, Sethian, J Evolution, implementation, and application o level set and ast marching methods or advancing ronts. J. Comput. Phys. 169, 503. Sussman, M., Smereka, P. & Osher, S A level set approach or computing solutions to incompressible two-phase low. J. Comput. Phys. 114, Mark Sussman, Ann S. Almgren, John B. Bell, Phillip Colella, Louis H. Howell, and Michael L. Welcome 1999 An Adaptive Level Set Approach or Incompressible Two-Phase Flows J. Comput. Phys. 148, Young, Y.-N., Tuo, H., Dubey, A. & Rosner, R On the miscible Rayleigh- Taylor instability: two and three dimensions. J. Fluid Mech. 447,