Interaction between turbulent flow and free surfaces

Transcription

1 Center or Turbulence Research Annual Research Bries Interaction between turbulent low and ree suraces By Y.-N. Young, F. E. Ham, AND N. N. Mansour 1. Motivation and objectives Fluids with interaces are ubiquitous in ields ranging rom geophysics and engineering to applied physics and biology. In various setups, the interace instability has been studied by masters such as Faraday, Fermi, Lighthill, Miles, Rayleigh and Taylor, just to name a ew. Interaction o turbulent low with a luid interace, however, is much less understood compared to the stability problem. This is due to the complexity o wave-turbulence interaction, and the wide dynamical range to be covered. A classic example is the wind over water problem in oceanography, where a wind (turbulent or laminar) blows over the sea surace. In a solitary moment at the beach, one immediately sees that the dynamical range is at least our decades as large waves o length meters break into small droplets o millimeter sizes. The ampliication o ree-surace waves driven by a mean shear low can be dominated by a critical layer instability, caused by the resonance between the surace waves and the wind (Miles 1957; Alexakis, Young & Rosner 2002a). Non-linear analysis can illuminate the initial ampliication period (Alexakis, Young & Rosner 2002b), however, the breaking o growing waves and the ensuing mixing cannot be easily understood in this manner. Thus our motivation in this project is to careully investigate the challenging problem o turbulence interaction with luid interaces. Speciically, we will develop and utilize tools and models to examine how the mixing o luids and momentum proceed and partition among the dierent luids. The ollowing interim objectives are purposed: development o underlying numerical method or LES based on the second-order Cartesian adaptive inite-volume method o Ham et al. (2002), integration o ree-surace methodology based on the level set method, sub-grid scale model implementation and development 2. Mathematical ormulation We treat the incompressible, immiscible two-luid system 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 variable density low can be written in conservative orm as: ρu i t ρ t + ρu j = 0, (2.1) + ρu ju i = p x i + τ ij + ρg i + σκδ(d)n i, (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

2 302 Y.-N. Young, F. E. Ham & N. N. Mansour 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. When the density can be written ρ = ρ(φ) where φ(x i, t) is a level set unction that is evolved to describe the location o the interace, the application o chain rule to the continuity equation yields: ( dρ φ dφ t + u j ) + ρ u j = 0 (2.3) φ For the case o constant density except in the neighborhood o the zero level set e.g. ρ = ρ + H (φ) + ρ (1 H (φ)) then the solution o the continuity equation can be decomposed into the solution o the ollowing: φ t + u φ j = 0 (2.4) φ=0 u 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. In a similar way, the application o chain rule to the momentum equation allows the 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(d)n i (2.6) x i 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 (ρ) and momentum (ρu i ) 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 tracking o the interace that is possible with the smoothly-varying level set unction. Recent developments to the level set method that improve its conservation properties speciically the particle level set method o Enright, Fedkiw, Ferziger & Mitchell (2002) will also be investigated and are described in a later section o this brie. 3. Numerical Method The system described by eqs is commonly solved on a structured staggered grid using inite dierence methods, where coupling between the velocity ield and pressure occurs naturally. In the present work, we will develop a inite volume method suitable to solve the system on a collocated grid. This will allow the use o both Cartesian structured grids with local reinement, and ully unstructured grids. The spatial arrangement o variables is shown in Figure 1. The arrows at the aces in the igure represent the location o the ace-normal velocity, U. The ollowing numerical method is proposed. First, the level set unction is advanced in time by solving the ollowing semi-implicit second-order discretization o the level set equation:

3 Free surace turbulence 303 U U PSrag replacements u i, p, φ PSrag replacements u i, p, φ (a) Cartesian grid with local reinement (b) Unstructured grid Figure 1. Spatial location o variables or collocated discretization. φ n+1 φ n + 1 ( 3 t V 2 U n 1 ) ( 1 2 U n 1 A 2 φn + 1 ) 2 φn+1 = 0 (3.1) where A is the ace area. Note that the ace-normal velocities, U, are used to advance the level set unction. The required ace values o the level set unction, φ, are interpolated rom the cell-centered φ values using a second order ENO scheme (Sussman, Smereka and Osher 1994). At this point, the level set unction is re-initialized to a signed distance unction by solving the ollowing equation to steady-state: φ + sgn(φ) ( φ 1) = 0. (3.2) τ In practice, only a ew time steps are required, and the equation need only be solved in a band about the zero level set (Peng, Merriman, Osher, Zhao & Kang 1999). With the level set advanced, properties can be calculated based on the level set at the mid-point o the time interval: ( 1 ( = ρ φ n + φ n+1)) (3.3) 2 ( 1 µ ( = µ φ n + φ n+1)) (3.4) 2 In the present investigation, we use a smoothed property variation in the region o the zero level set as described by Sussman, Smereka & Osher (1994). The momentum and continuity equations are then solved using a ractional-step method similar to that described in Kim & Choi (2000). Speciic attention must be paid, however, to the discrete orm o orce terms that have rapid spatial variation, speciically the surace tension orces added to the momentum equation in the neighborhood o the interace. A ractional step discretization o the momentum equation proceeds as ollows. Advance the momentum equation to solve or CV-centered pseudo-velocities û i n+1 using:

4 304 Y.-N. Young, F. E. Ham & N. N. Mansour ( ) û n+1 i u n i = δp n 1/2 + R i, (3.5) t δx i where all other terms in the momentum equation have been incorporated into the right hand side term R i, approximated at the midpoint o the current time step. Following Kim and Choi (2000), a second pseudo-velocity ield denoted by a superscript * is then calculated: n 1/2 u n+1 i = û n+1 1 δp i + t δx i This starred velocity ield is then interpolated to the aces: U n+1 = u i n+1 t R i R (3.6), (3.7) where () is a second-order interpolation operator that yields a ace-normal component rom two CV-centered vectors. The divergence o these ace velocities is then used as the source term in a variable-coeicient Poisson equation or the pressure: 1 t U n+1 A = 1 δp A. (3.8) δn With the Poisson equation solved, the ace-normal and CV-centered velocity ields are corrected as ollows, completing the time advancement: U n+1 = U n+1 t u n+1 1 i = u n+1 1 δp i t δx i δp δn (3.9) (3.10) Following this correction, the ace-normal velocity components will exactly conserve mass. The CV-centered pressure gradient required in eq is reconstructed rom the ace-normal components using a second-order reconstruction operator, δp δx i = 1 δp xi. (3.11) δn In the present work we use a ace-area weighted average. At this point the time step is completed, and the algorithm would cycle back to eq 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: R i R. (3.12)

5 Free surace turbulence 305 (a) eq assumed or surace tension (b) eq assumed or surace tension Figure 2. Comparison o calculated velocity ield around a cylindrical drop or two dierent ormulations o the surace tension orces. 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.: R σ i 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. Results 4.1. Surace tension ormulation To illustrate the importance o a proper handling o the surace tension orces, igure 2 compares the calculated velocity ield around a cylindrical drop with surace tension using the two dierent ormulations. Figure 3 compares the normalized pressure and velocity along a horizontal line through the center o the drop or the same two calculations. Clearly the ormulation that assumes eq is superior. Although the steady velocity ield or this problem should be exactly zero, even with the new ormulation we do observe eight small vortices arranged symmetrically around the perimeter o the drop. These parasitic currents have been observed by other investigators (Tryggvason et al. 2001). For the improved ace-based surace-tension ormulation, the maximum induced velocity is on the order o 0.001σ/µ, consistent with the observations o others using staggered structured codes.

6 306 Y.-N. Young, F. E. Ham & N. N. Mansour PSrag replacements p/σκ PSrag replacements x u1µ/σ x (a) pressure along horizontal centerline (b) x-velocity along horizontal centerline Figure 3. Cylindrical drop computation: Comparison o a) calculated pressure and b) calculated CV-centered velocity through the horizontal centerline o the drop: eq assumed or surace tension; eq assumed or surace tension Adaptive mesh reinement and coarsening Because o the substantial variations in resolution requirement, some sort o local reinement capability is considered essential to the success o this investigation. In the present work, the adaptive Cartesian method o Ham et al. (2002) is being developed to solve the system described by eqs To illustrate the potential o the method, igure 4 shows the adapted mesh that results when the Rayleigh-Taylor instability is calculated with a simple isotropic adaptation criteria in the region o the zero level set. The development o a more optimal low and level set adaptation criterion is an ongoing part o this investigation. 5. The particle level set method As a compliment to the numerical development presented in the previous sections, the potential o the particle level set method o Enright, Fedkiw, Ferziger & Mitchell (2002) to improve the conservation properties in the region o the ree surace is also being investigated. In the particle level set method, massless Lagrangian particles are placed in a band around the zero level set and used to correct the level set evolution. Initially, particles o the same signs as the level set unction values are placed within this band. Each particle (with coordinate x p ) is advanced according to the local velocity d x p = u( x p ), (5.1) dt where u( x p ) is the interpolated velocity at the particle position. As in Enright, Fedkiw, Ferziger & Mitchell (2002), each particle is given a sign (s p ), a radius (r p ), and an individual level set unction φ p deined as

7 Free surace turbulence 307 Figure 4. 2D Rayleigh-Taylor instability: Calculated evolution o the mesh in the region o the ree surace. φp (~x) = sp (rp ~x ~xp ). (5.2) The sign o each particle is assigned once initially, and the radius o each particle is determined according to the algorithm in Enright, Fedkiw, Ferziger & Mitchell (2002). At each time step, errors made in the level set unction leads to displacement o particles; some may end up on the wrong side relative to the zero level set. Explicitly, a particle is ound to be on the wrong side i φ(~xp ) sp < 0. (5.3) Once the particle is ound to be on the wrong side, it will be deined as an escaped particle i the ollowing condition is met φ(~xp ) sp > rp. (5.4) Following the algorithm in Enright, Fedkiw, Ferziger & Mitchell (2002), only the escaped particles contribute to correcting the level set. The correction is made by reassigning the values o the level set φ according to the ollowing rules: or a given set o escaped positive particles (E+ ) and a level set φ, the maximum o the level set values rom all escaped positive particles (E+ ) is picked as the projected level set value φ+ (~x) φ+ (~x) = max+ (φp (~x), φ+ (~x)) p in E (5.5) with φ+ (~x) irst initialized with φ(~x). Similarly or escaped negative particle, given a set o escaped negative particles (E )

8 308 Y.-N. Young, F. E. Ham & N. N. Mansour (a) (b) Figure 5. (a) Initial level set on a uniorm grid o resolution Particles are also included. (b) Ater one revolution: with particle correction; without particle correction. and a level set φ, the minimum o the level set values rom all escaped negative particles is the projected level set value φ ( x) φ ( x) = min p in E (φ p( x), φ ( x)) (5.6) with φ ( x) irst initialized with φ( x). The corrected level set value is then determined as the value that is closer to the interace φ( x) = { φ + i φ + φ φ i φ + > φ. (5.7) To ensure that the level set remains as close to the boundary between positive and negative particles as possible, the particle correction is conducted ater both the transport o the level set and the re-initialization. Results rom the standard test o Zalesak s disk are shown in igure 5. Ater one revolution, the level set with particle correction still retains the sharp corners compared with the results without particle correction where the corners are severely smoothed and eroded. Such results are consistent with those presented in Enright, Fedkiw, Ferziger & Mitchell (2002) or equivalent numerical resolution Additional geometrical inormation A potential improvement to the particle level set method is to incorporate more geometric inormation into each particle. From dierential geometry, the geometric properties such as the line element/surace element o the level set also evolve as the level set is transported. A simple idea then is to associate each particle with such geometric properties o the level sets that the particles reside on. Speciically, at time t = 0, with each particle a line element vector l is deined as the derivative o the distance r on the curve parameterized by λ

9 Free surace turbulence 309 where r is related to the luid low by l r λ, (5.8) d r = u(r,t). (5.9) dt It ollows that the line element l satisies the ollowing equation l i = W ij l j, (5.10) where W ij ui and the dot is the time derivative. Higher order derivatives are needed or computation o curvature. At t = 0, we deine a as a = l λ. (5.11) This leads to the ollowing equations or each component o a a i = W ij a j + W ijk l j l k, (5.12) where W ijk 2 u i x k. Denoting the length o l as ξ l, we can calculate the curvature o the level set associated with each particle as κ ξ 3 ɛ ijk n i l j a k, where the Einstein summation is assumed or repeated indexes, ɛ ijk is the delta unction, and n i is the normal vector. With this additional inormation, we may modiy how we correct the level set according to the escaped particles. For example, we may incorporate the normal vectors and the curvature into the particle level set unction φ p. The simplest modiication, just using the line element l, is to write the particle level set unction as φ p( x) = s p (r p n p ( x x p ) ), (5.13) where n p is the normal vector to the level set where the particle belongs. Figure 6 shows how φ p (equation 5.2) compares with φ p or a positive particle (s p = 1). The dashed line is where φ p = 0 and the solid line is where φ p = 0. Whenever a particle escapes and is used to correct the level set unction, the shortest distance between the grid point and all the zero particle levels will be used to correct the value o the level set unction. To test this idea, we advect a circular level set by a single vortex in two-dimensions. Inside a unit box, the mixing low is described by the stream unction and the corresponding velocity Ψ = 1 π sin2 (πx) sin 2 (πy), (5.14) u = 1 2 (sin(2πy) sin(2πy) cos(2πx)), v = 1 ( sin(2πx) + cos(2πy) sin(2πx)). (5.15) 2 The initial condition is a circle o radius 0.15 located at (0.5, 0.75) (c Enright, Fedkiw, Ferziger & Mitchell (2002)), and the numerical resolution is The improvement is signiicant when two level sets move toward each other within distances less than two grid spacings. More small-scale structures are retained using φ p

10 310 Y.-N. Young, F. E. Ham & N. N. Mansour + + np + Figure 6. Particle level set φ p and φ p or a positive particle inside a cell. (a) (b) Figure 7. Advection o a circle by a single vortex. (a) The level set at t = 2.8. The solid line is using equation 5.12 or particle correction, and the dashed line is based on equation 5.2. (b) Detail o the level sets with particles superimposed. or particle correction (solid line in ig. 7b) than the correction based on φ p (dashed line in ig. 7b). Eventually the numerical limitation sets in and annihilation o level sets occurs or both simulations using φ p or φ p. 6. Future plans At the time o writing, the Cartesian adaptive code is undergoing a period o validation where a series o 2D and 3D ree surace problems will be investigated, including bubble oscillation, interaction and breakup problems, and the 2D and 3D Rayleigh-Taylor in-

11 Free surace turbulence 311 stability. In the context o this validation, a low and level set based adaptation criterion similar to the interpolation error based criterion described in Ham et al. (2002) will be developed. Eventually it is planned to incorporate the particle level set into the Cartesian adaptive code, and compute the interaction between a turbulent boundary layer and ree surace with overturning waves. REFERENCES Alexakis, A., Young, Y.-N. & Rosner, R Shear instability o luid interaces: stability analysis. Phys. Rev. E Alexakis, A., Young, Y.-N. & Rosner, R Weakly non-linear analysis o winddriven gravity waves. J. Fluid Mech. submitted. Enright, D., Fedkiw, R., Ferziger, J. & Mitchell, I A Hybrid Particle Level Set Method or Improved Interace Capturing. J. Comput. Phys., to appear. Ham, F. E., Lien, F. S. & Strong, A. B A Cartesian grid method with transient anisotropic adaptation. J. Comput. Phys. 179, Kim, D. & Choi, H A Second-Order Time-Accurate Finite Volume Method or Unsteady Incompressible Flow on Hybrid Unstructured Grids J. Comput. Phys. 162, Miles, J On the generation o surace waves by shear lows. J. Fluid Mech. 3, 185. Peng, D., Merriman, B., Osher, S., Zhao, H. & Kang M A PDE-based ast local level set method. J. Comput. Phys. 155, Rhie, C.M., & Chow, W.L Numerical study o the turbulent low past an airoil with trailing edge separation. AIAA J. 21, Sussman, M., Smereka, P. & Osher, S A level set approach or computing solutions to incompressible two-phase low. J. Comput. Phys. 114, Tryggvason, G., Bunner, B. Esmaeeli, A., Juric, D., Al-Rawahi, N., Tauber, W., Han, J., Nas, S. & Jan, Y.-J A Front-Tracking Method or the Computations o Multiphase Flow J. Comput. Phys. 169,