Accurate Solution Algorithms for Incompressible Multiphase Flows William J. Rider, Douglas B. Kothe, S. Jay Mosso, John H. Cerutti, Los Alamos National Laboratory Los Alamos, New Mexico, 87545, USA and John I. Hochstein Mechanical Engineering Department Memphis State University Memphis, TN, 38152, USA January 7, 1995
Abstract
ities [4, 5, 6, 7, 8, 9]. An example of a MAC successor is the SOLA-VOF method, which coupled the SOLA (solution algorithm) methodology with a volume tracking method for uid interfaces [10]. A more recent example is the RIPPLE code that implements the CSF (continuum surface force) model discussed in detail later in this paper [11, 12]. In the nearly thirty years since the inception of these algorithms, numerous advances have been made. We discuss the impact of some of these advances on the accurate solutions of incompressible multiphase ows. In particular we focus upon recent advances with projection methods coupled with high-order Godunov advection and interface tracking. Additionally, we discuss improvements in computing the solutions to linear systems of equation's that are often the most expensive portion of the solution algorithm. First, we describe an algorithm designed to improve the accuracy of the ow solutions. This method is a semi-implicit projection method that uses high-order Godunov methods for advection [13, 14, 15, 16, 17]. The method is second-order accurate in both time and space, and does not su er from cell Reynolds numberbased stability restrictions. The linear systems of equations arising from the projection and di usion steps are solved via multigrid methods (with a preconditioned conjugate gradient method as an accelerator). The algorithm is constructed for a wide range of ow regimes, including multiphase ows where density ratios across interfaces can be arbitrarily large (typically as large as 1000:1). This feature is greatly facilitated by the use
A number of advances in modeling multiphase incompressible ow are described. These advances include high-order Godunov projection methods, unsplit piecewise linear interface reconstruction and tracking and the continuum surface force model. Various aspects of projection methods and high-order Godunov methods are expanded upon where they are important to modeling multiphase ow phenomena. Alternative approaches to interface tracking are presented including an innovative particle based approach. Examples are given that show the strengths and weaknesses of the methodology.
I. Introduction Los Alamos National Laboratory (LANL) has a long history in computational uid dynamics. A particularly important contribution came in 1965 with the Marker-and-Cell (MAC) method [1, 2, 3] for incompressible multiphase ows, which is a method along with its successors that remains a popular choice to this day. This method enabled researchers to investigate a number of multiphase phenomena numerically including drop and splash dynamics and Rayleigh-Taylor instabil All correspondence should be sent to William J. Rider, Mail Stop B256, Los Alamos National Laboratory, Los Alamos, NM 87545, USA. E-mail:
[email protected]. This work performed under the auspices of the U.S. Department of Energy by Los Alamos National Laboratory under Contract W-7405-ENG-36. This paper is declared work of the U.S. Government and is not subject to copyright protection in the United States.
1
AIAA{95{0699
II. Godunov-Projection Methods of volume tracking with this method. In conjunction with approximate projection methods [15, 16, 17], low error formulations [18] and the ltering of nonsolenoidal We solve the equations of incompressible multiphase modes [19, 16, 20], the method remains robust for di- ow. The equations of motion is cult problems. This method also merges quite naturally with physical models such as the CSF model for surface @ u + r (uu)+ 1 r = 1 r ?ru + ruT + F; (1a) @t tension [21]. A similar e ort based on the approximate projection method described in [15] is being made by E. and the conservation of mass is G. Puckett et. al [22]. r u = 0; (1b) with density and viscosity transport
One notable di erence between our approach and the MAC method is that our method solves for the entire
ow eld even in cases with large density jumps. The MAC method is often used in a manner where only the heavy uid's motion is found and lighter uid's is neglected.
and
@ +u @t
r = 0;
@ +u @t
r = 0:
(1c)
(1d) In these equations, u is the velocity vector, is the incompressible pressure, the uid density, the visCapturing uid interfaces with volume tracking techcosity and F is the momentum (velocity) source term. niques remains quite e ective, but only when subseF can include the e ects of gravity or surface tension for quent improvements to the original method (developed instance. In our calculations, the transport of density 15 years ago) are taken into consideration. An example and viscosity is solved with one equation, is the use of piecewise linear or planar approximations to the interface geometry [23]. Volume tracking allows @f + u rf = 0; (1e) interfaces to be captured and maintained compactly in @t one cell without imposing restrictions on the topological complexity or the number of interfaces that can be rep- where f is a volume fraction of a uid having certain resented. We will also present particle based approaches properties. This volume fraction is then used to debased on the work of Brackbill and others [24] extending termine the average cell-centered values for density and the PIC approach developed at Los Alamos forty years viscosity (as well as de ning the location of a front). Density is recovered from f as follows (for two uids a ago by Harlow [25, 26]. and b. (f ) = fa + (1 ? f ) b ; Physical processes speci c to and localized at uid and similarly for viscosity interfaces (e.g, surface tension, phase change) are mod (f ) = fa + (1 ? f ) b : eled by applying the process to uid elements everywhere within the interface transition regions. Surface processes are thereby replaced with volume pro- These equations will be then solved via a projection cesses whose integral e ect properly reproduces the de- method described next. We will also discuss several imsired interface physics. This methodology is the un- portant aspects of the methodology: lters, unsplit adderlying feature in the CSF method for surface ten- vection and the solution of linear systems of equations. sion, which has proven successful in a variety of studies [27, 28, 11, 12, 29, 30, 31, 32, 33]. The CSF method lifts II.1 Projection all topological restrictions (typically inherent in models Our basic goal with projection methods is to advance a for surface tension) without sacri cing accuracy, robust- velocity eld, V = (V x; V y )T by some convenient means ness, or reliability. It has been extensively veri ed and disregarding the solenoidal nature of V, then recover the tested in two-dimensional ows through its implementa- proper solenoidal velocity eld, Vd (r Vd = 0). The tion in a classical algorithm for free surface ows, where means to this end is a projection, P , which has the e ect complex interface phenomena such as breakup and coalescence have been predicted. This success led to our Vd = P (V) : current focus on extensions and enhancements to the CSF method, namely three-dimensionality, an implicit The projection accomplishes this through the decompoformulation, and the addition of variable surface ten- sition of the velocity eld into parts that are divergencesion e ects. Computational examples are shown of its free and curl-free. This is known as a Hodge or capabilities. Helmholtz decomposition [34]. The curl-free portion 2
AIAA{95{0699 will be denoted by the gradient of a potential, r'. This decomposition can be written V = Vd + r'; (2) where = 1=. Taking the divergence of (2) gives r V = r Vd + r r' ! r V = r r': Once ' has been computed, then Vd can be found through Vd = V ? r': (3) We will be concerned with a class of solution algorithms known as approximate projection methods. These methods are extensions of the classic work of Chorin [35], and its modernization by Bell, Colella and Glaz [13] for solving the incompressible Navier-Stokes equations. These methods are also used to solve equations with varying density [14]. The approximate projection methods were introduced by Almgren, Bell and Szymczak [15] (also see [17, 18, 19]). These methods also allow us to use collocated grids rather than the staggered grids (often called MAC grids after their introduction by Harlow and Welch [1]). The collocated grids allow advection to be more easily handled along with the entire data structure (Tau has derived a staggered grid high-order Godunov method with promise [36]). Projection methods can be de ned in several ways. In the classical approach the discrete equations behave similarly to the analytic projection operators. Unfortunately, these methods have a number of problems in practice. We will be de ning discrete methods based on the continuous projections rather than demanding that the discrete system algebraically match the conditions for being a projection. Thus, the most straightforward means to discretize each operator (r, r and r r) will be chosen (not quite true, but nearly [15, 18]). We have found that the formulation of the Poisson pressure equation can have a profound impact on the solutions' quality. The most robust formulation we found is formulated in the following fashion for a fractional step method r n1+ 21 r = 1t r u;n+1 ; where is the pressure and u;n+1 is the predicted velocity de ned by ?1 1 t 1 T n + ;n +1 2 r+r u = I ? 2 n+ 1 r 2 un ? tr (uu)n+ 12 ? n+t12 r n? 21 ? t 1 1 1 +tFn+ 2 + 2 n+ 1 r n+ 2 ru + ruT n 2 + n+t1 r n? 21 : 2
This option controls the presence of the discrete divergence without sacri cing the accuracy of the method. The important step to note is that the old time pressure is used for the predictor viscous solution, then it is subtracted away making the pressure equation solve for pressure rather than a pressure increment. A full exposition on this subject is given in [17, 18].
II.2 Filters
Because the discrete divergence is nonzero it is important to control the growth of the discrete divergence in the solution. The divergence errors are a measure of the solution's quality. Because the discrete divergence still has a central di erence form, it does not detect all non-solenoidal modes in the solution thus leaving some divergence error largely una ected by the projection. For this reason when using approximate projections it is important to control the growth of divergence errors. This can be accomplished through the use of ltering. We have designed two types of lters: one based upon the derivation of second-order projection operators that sense the divergent modes in the solution and a second based upon physical arguments. The projection lters then use a single iterative sweep to damp the error modes. The physically based lters we call velocity lters are based upon removing non-physical modes from the solution working only on the velocity eld. All of these lters are always applied at the end of a computational cycle. The lters used here are described more fully in [17, 20] (also see [19] and [37] for a discussion of lters in the context of combustion) or more generally in [16].
II.3 Advection
The use of non-operator split high-order Godunov advection is also important to the method's success. This method avoids the use of operator splitting and allows the consistent use of high-order monotone advection in incompressible ows. The method also has excellent phase error properties and compares favorably with higher order Runge-Kutta methods on typical grids. The basic idea is sketched out in Figure 1. Space does not permit an adequate description of the method and good descriptions appear in a number of sources [38, 13, 14, 39, 15, 17]. Our method is nearly identical to that described in [39]. The basic point of the method is to implement a second-order Taylor series expansion in both time and space for the dependent variables in multidimensional form. Variables are constructed to cell-edges (where they will be multi-valued) and then the upwind values are propagated in the solution. During this procedure the complete e ect of the governing equations is used to approximate the time derivatives (source terms, pressure gradients and the incompressibility constraint). The use 3
AIAA{95{0699 proaches are equally expensive in terms of operation count although they are cheaper in terms of storage. For banded Gaussian elimination and standard iterative methods like SOR, the operation count scales with the i,j i-1,j number of equations, N , squared. Because of the nonconstant coecients of the operators, FFT or cyclic reduction methods are also ruled out. We are left with two good choices: preconditioned conjugate gradient methi-1,j-1 i,j-1 ods and multigrid methods. We have a symmetric system of equations thus enabling the use of a standard preconditioned conjugate gradient (PCG) method [41]. For a preconditioner we have experimented with the usual incomplete Cholesky method, but we have found a composite Jacobi iteration (multi-pass) to be more cost ecient and e ective. Figure 1: This diagram shows a geometric representaOur other (and more favored) approach is a multition of the corner-transported upwind method that is grid method [42, 43]. As opposed to the PCG method the basis of our advection method. The method alge- (which scales at best N 5=4 ), the multigrid method's opbraically computes the area swept out by the character- eration count scales with N . We use a multigrid method istics as shown above. based upon an approach nearly identical to that described in [44] or [37]. this method has not proved itself to of the incompressibility constraint leads to the \MAC" beUnfortunately ecient in all cases. Certain ow topologies cause projection where time-centered normal velocities at cell the method to converge at a slow rate thus destroying edges are used to compute a potential eld that corrects the advantages of the multigrid algorithm. In order to them to be divergence-free [39, 19, 17]. This allows the maintain the eciency of the multigrid algorithm and method to use a divergence-free velocity eld for the integrate the robustness of the PCG method have purpose of advection (even with an approximate pro- combined them into a multigrid preconditionedweconjujection) and maintain second-order accuracy for CFL gate gradient (MGCG) algorithm [45]. Care must be numbers greater than one-half [17] taken to insure that the multigrid algorithm produces a Several points are important to emphasize. This symmetric iteration matrix, but this is simple in pracmethod is not bound by a stability restriction related to tice. the Reynolds number. Thus this method can be used to We have found this to be an ecient and robust compute both viscous and inviscid ows. Results show method even in our most challenging problems. The that the method can resolve ows up to a grid Reynolds method is faster than the pure multigrid method on number of 40. Beyond this the solver will degrade in a most problems (where the multigrid worked well before) graceful fashion and produce physical, but not neces- and seems to scale like a multigrid method with probsarily accurate results (in the classical sense, see Brown lem size. This holds for problems where the multigrid and Minion [40]). The advection method is also robust previously had diculties. in cases where the ow contains discontinuities (such as a shear).
III. Interface Kinematics
II.4 Numerical Linear Algebra
We wish to follow the motion of an interface embedded in the overall motion of the ow eld. This can be accomplished through the solution of the equation
The most computationally expensive portion of the algorithm is the solution of the linear system of equations arising from the discretization of the projection/pressure equation. For periodic or solid wall boundaries this system of equations is semi-positive de nite, but solvable. It is vital to solve this problem as eciently as possible. Furthermore, this problem has been rendered more dicult by solving for the entire ow eld (thus producing a non-constant coecient Poisson problem). Again, the ante is raised by allowing these coecients to typically vary in a discontinuous manner. Direct solution (i.e. Gaussian elimination) is too expensive for further consideration. Classical iterative ap-
@f +u @t
rf = 0;
(4a) where f is some scalar carrying interface or \color" information. An equivalent equation for incompressible
ows is @f + r (uf ) = 0; (4b) @t since r u = 0. A particle method is perhaps the most straightforward, drawing upon recent advances in PIC algorithms [46]. Particles are assigned a mass according to 4
AIAA{95{0699 the density of the uid in which they reside and a volume (hence size) according to the interpolation function chosen to interpolate quantities to and from the computational grid. While this method provides a superior multidimensional grid-independent advection scheme, there are as a result some practical diculties, namely the cost and accuracy associated with interpolating the particle information to an Eulerian grid [47, 48]. This problem often exhibits itself as a noisy density eld. Later, we will describe a new approach that makes a particle method quite feasible for interface tracking. Simply discretizing (4) with a high resolution nite di erence scheme is quite appealing. An advection algorithm is typically an integral part of a ow solver. This is done in the methods presented in [14, 49]. From the advances in high speed ows in the last decade, there are a number of methods that minimize numerical dissipation. An example of this are high-order Godunov methods in particular PPM [50, 51]. Problems with the numerical dissipation (leading to a thickening interface) led researchers to propose an ingenious compromise. The level set methods could be implemented with the same di erence techniques already well developed for advection, but without allowing the interface to smear. The interface is de ned as the zero level set of a distance function, , from that interface. Instead of (4), the following equation is solved @ +u @t
r
Interfaces 0.7756
Y
0.7
0.6
0.5533 0.5333
0.6
0.7
0.7644
X
Figure 2: This diagram shows the reconstruction of a portion of a circle by a PLIC method. ner free of oscillatory modes [61]. Later, we describe a manner to improve particle method's behavior. These results become quite obvious when nontrivial, pure advection problems are used to evaluate the methods. By nontrivial we mean that the ow eld varies in space and perhaps time and contains vorticity. For the purposes of solving multiphase ow we favor a method that preserve symmetries in the solution naturally. For this reason we have developed an unsplit (not operator split) version of the PLIC method that will be used below for our example calculations.
= 0:
Improvements for incompressible ows are suggested by Sussman, Smereka and Osher [52]. VOF methods have been used for several decades starting at the national laboratories (Livermore [53] and Los Alamos [54] and later Sandia [55]). The earlier work is typi ed by the SLIC [56] algorithm and the original method with the name VOF [10]. In each of these methods the interface is designated as a straight line in a cell de ned by the volume of a given uid in that cell. Youngs [23] improved the general method greatly by allowing the reconstruction of the interface to be multidimensional and linear in nature. Youngs further extended his method to three dimension in [57]. Recently, Pilliod and Puckett have improved the accuracy [58, 59, 60]. Here we refer to this method as the piecewise linear interface calculation (PLIC). We show this approach on an example interface in Figure 2. Capturing methods although simple are expensive when implemented to achieve the best resolution of interfaces and still do not match the resolution of PLIC. Level set methods although conceptually appealing suffer from a number of detriments. First among these is the lack of volume (mass) conservation. Cost is also an issue. Particle methods o er excellent, numerical di usion-free results, but have two problems: cost and the problem of mapping the particles to a grid in a man-
III.1 Unsplit PLIC Method
The unsplit PLIC method was developed with the knowledge that the nature of incompressible ows is inherently multidimensional. The operator split approach to advecting interfaces has diculties. An incompressible velocity eld containing signi cant vorticity can appear quite compressible in one direction thus leading to signi cant conservation errors in an operator split approach. In addition this method requires a reconstruction of the interface on each directional sweep. As with our advection of the velocity eld we have built upon the work of Colella's unsplit advection scheme [38, 13] to advect interfaces. Rather than algebraically determine uxes, the uxes are determined using the cell-edge-centered velocity eld and a geometric uxing algorithm. Our reconstruction algorithm produces second-order results on smooth interfaces [61, 58]. In addition we have implemented an algorithm that allows the method to conserve volume to a high degree 5
AIAA{95{0699 phase change and surface tension. In this section we describe our method for modeling interfacial surface tension, which is formulated with a localized volume force as prescribed by the recent CSF model [21]. Although originally developed for surface tension, the basic approach of the CSF model lends itself quite well to interfacial physics in general, i.e., surface phenomena other III.2 Colored Particle Method than surface tension can be encapsulated easily within Another particularly promising approach for interface the CSF model. Typical examples are phase change tracking has been named the \colored particle" method. and momentum exchange [31], where the surface physics The aspect of this method that makes it similar to the are mass and momentum ux, respectively, transferred earlier MAC method is that the particles do not carry ei- across the interface. ther volume or mass, but rather identity. The entire ow In this section, we brie y review the theory of the CSF domain is lled with particles each carrying an identity. These identities or colors are then used to produce a model, then discuss some important issues pertaining weighted average for a physical property on the grid. to the accurate formulation of the necessary discrete This allows the density and viscosity to be smoothly operators. We also outline the CSF enhancements and reconstructed on the Eulerian grid. With mass or vol- improvements that are currently being studied. These ume carrying particles, the smoothness of the transfer developments should allow a more ecient modeling of to the Eulerian grid is a diculty in maintaining quality a wider variety of interfacial ows. solutions without a large number of particles (with the The central theme of the CSF model is formulation problem still present, but lessened in severity). of interface dynamics into a localized volumetric force, In our calculation, the particles are transported using which is quite di erent from earlier numerical models a second-order predictor-corrector method. First, the of interfacial The basic premise of the position of the particles is estimated at a time centered CSF model is phenomena. to replace interfacial surface phenomena position (normally applied via a discrete boundary condition) as xn+ 21 = xn + 2t un (xn ) ; smoothly varying volumetric forces derived from a product of the appropriate interfacial physics per unit area then the advanced time position is computed and an approximation to the surface (interface) delta function. The CSF formulation makes use of the fact xn+1 = xn + tun+ 12 xn+ 21 : that numerical models of discontinuities in nite volume and nite di erence schemes are really continuous tranVelocities are computed at the particle location via bi- sitions within which the uid properties vary smoothly linear interpolation. With the particles advanced in from one uid to another (over a distance of O(h), where time, bilinear interpolation is used to transfer the par- h is a length comparable to the resolution a orded by ticle identities to the grid as densities and viscosities. the computational mesh). It is not appropriate, thereThis is done in a particle-weighted fashion. According fore, to apply in these schemes a boundary condition at to the bilinear interpolation, the particles each have a an interface \discontinuity", which in the case of surface k . For instance, to get a density weight in a grid cell of !i;j tension is a pressure jump across the interface. Surface in a cell tension should instead act on uid elements everywhere P k k within the transition region. k !i;j i;j = P k : ! k i;j In the case of surface tension, the relevant surface This method will not conserve mass in the usual fash- physics is a force per unit area arising from local inion, but because particles are used, information about terface curvature and local (tangential) variations in the interface is not lost. Unlike other approaches (cap- the surface tension coecient. Application of interfaturing, level sets, or VOF) there is no di usion or dis- cial physics using the CSF model then reduces to appersion to destroy information about the interface as plication of a localized force in the momentum equathe particles are advected as Lagrangian markers. This tion, regardless of interface topology. The CSF model is therefore ideally suited for dynamical interfaces of is evident in the results we present. arbitrary topology. The resulting simplicity, accuracy, In each calculation we use 16 particles per cell. and robustness has led to its widespread and popular use [27, 28, 11, 12, 29, 30, 31, 32, 33] in modeling comIV. Interface Dynamics plex interfacial ows that were in many cases previously intractable. Algorithms for interface dynamics must model In the CSF model, surface tension is reformulated as physics speci c to and localized at interfaces, such as of accuracy. Any overshoot or undershoot in volume fraction is redistributed to neighboring interface cells that can either accept or give volume. The cells that accept or give volume in the redistribution have volume fractions between zero and one.
6
AIAA{95{0699 a volume force Fs satisfying Z
Given (6{8), the normal component of the CSF volume force Fs is easily estimated from rst and second order spatial derivatives of the uid volume fractions f . The interface normal vector n,
Z
lim Fs(x)dV = fs(xs )dS ; (5) h!0 V S where xs is a point on the interface and the area integral is over the portion S of the surface lying within the small volume of integration V . The surface tension force per unit interfacial area, fs (x), is given by [21]:
n = rf ; is rst computed, n^ follows from normalization, and follows from (7). The molli ed delta function in our formulation is the magnitude of n. The force, which resides at cell centers in our scheme, will be nonzero only within the interface transition region. As shown in (5), it is normalized to recover the conventional description of surface tension as the local product h ! 0. Its line integral directed normally through the interface transition region is approximately equal to the pressure jump . As stated in reference [21], a wide stencil in general leads to a better estimate of curvature. However, in contrast to the discretizations presented in [21], we have chosen a conservative discretization for the curvature and the force Fs . This is motivated by the need to preserve an important physical property of surface tension, namely that the net normal surface tension force (and also the curvature ) should vanish over any closed surface. A conservative discretization of is therefore used, given by: Z 1 1 X n^ dA ; (9) =? r n^ dV ? f V V f f where V is the control (cell) volume, dAf the area of face f (pointing outward) on V , and n^ f the unit interface normal on face f of V . For the unit interface normal, a six-cell stencil is used to compute both components of n^ f at each face. The cell-centered curvature then results by summing over cell faces, bringing the e ective stencil to nine cells. Without such a conservative discretization, it is possible to induce arti cial horizontal motion of bubbles that should otherwise rise vertically due to buoyant forces. As suggested in [21], smoother variations in generally result if a molli ed VOF function fe is used to compute the face normals in (9) above. A variety of smoothing algorithms (e.g., B-spline or point-Jacobi) have been found to give the desired results, which is the mitigation of high wavenumber contributions to (resulting possibly from discretization errors). This should, however, be used with caution because the actual interface geometry could be molli ed unphysically. Nevertheless, use of a molli ed f in estimating was the preferred choice in recent numerical studies detailed in [27, 28]. Although some examples of the e ects of smoothing can be found in [21], it still warrants further investigation, especially as it relates to convergence and consistency. The representation of surface tension in the CSF model as an explicit force is linearly stable only for time
fs (xs ) = (xs )^n(xs ) + rs (xs ) ; (6) where is the surface tension coecient, rs is the sur-
face gradient [21], n^ is the interface unit normal, and is the mean interfacial curvature, given by [62]: (xs ) = ? (r n^ ) (xs ) :
(7)
The rst term in (6) is a force acting normal to the interface, proportional to the curvature . The second term is a force acting tangential to the interface toward regions of higher surface tension coecient (). The normal force tends to smooth and propagate regions of high curvature, whereas the tangential force tends to force uid along the interface toward regions of higher . Since interfaces having surface tension are tracked with the methods mentioned in section III., their topology will not in general align with logical mesh coordinates. Discontinuous interfaces are therefore represented in the computational domain as nite thickness transition regions within which uid volume fractions vary smoothly from zero to one over a distance of O(h). The volume force, nonzero only within these transition regions, is given in the CSF model by [21]:
Fs(x) = fs(x) jrc[~c(]x)j ;
(8)
where c~ is the characteristic (\color") function, taken to be the VOF function f in this work, and [c] is the jump in c, which is unity when c is equal to f . The term in (8) above involving c represents a molli ed delta function [21]. This form is not unique; see, for example, reference [52] for other forms. In the current work, only the normal force in (6) is modeled for the volume force in (8), i.e., the surface tension coecient is taken to be constant. Details on modeling the tangential force using the CSF model can be found in reference [33]. The interested reader should also consult reference [63], where the volume force in (8) (for constant ) is approximated as the divergence of a \stress" tensor. The advantages of approximating this force as the divergence of a tensor are not yet clear. A simple yet accurate discretization of this tensor (especially in three dimensions) is an outstanding issue that will determine the usefulness of this approach relative to the CSF method in its original form. 7
AIAA{95{0699 2. Compute the time- and edge-centered velocity eld via the Godunov procedure based solely on data at time n.
steps smaller than a maximum allowable value time step ts necessary to resolve the propagation of capillary waves [21]. As pointed out in [21], this constraint is often restrictive, especially when uid interfaces are undergoing topology changes (e.g., pinch-o ) since ts is roughly proportional to a higher power ( 3=2) of the mesh spacing. This restriction can be alleviated with an implicit treatment of , whereby the advanced-time (n+1) value of is estimated with a Taylor-series of taken at the current time (n): n+1
n
+ t d dt
n
3. Solve the \MAC" projection and correct the timeand edge-centered velocity eld so that 1 Dun+ 2 = 0.
4. Using un+ 12 update the volume fraction data from n to n + 1. 5. Using f n+1 compute n+1 and n+1 .
:
6. Compute term using ? the CSF source 1 f n+ 2 = f n + f n+1 =2. 7. Update un to u;n+1 with advection, di usion and CSF sources. 8. Solve the projection L n+ 21 = Du;n+1 and correct the velocities to un+1 .
An evolution equation for follows from the above expansion which must be solved implicitly. Initial studies by J. U. Brackbill indicates this approach can be successful in alleviating the explicit time step constraint, and will be the focus of future work. Wall adhesion is the surface force acting on uid interfaces at points of contact with \walls," which are static, rigid boundaries. Wall adhesion forces are calculated in the same manner as volume forces due to surface tension are calculated, except that a boundary condition is applied to n^ prior to evaluating (IV.). The condition is applied only to those normals lying on a rigid boundary. Those forces Fs attributed to wall adhesion are therefore only in cells along a boundary. For further details see references [11] and [12]. In summary, surface tension modeled with the CSF method places no restrictions on the number, complexity, or dynamic evolution of interfaces having surface tension. Direct comparisons made in modeling surface tension with the CSF model and a popular interface reconstruction model [64] show that the CSF model makes more accurate use of volume fraction data [21]. The normal surface tension force tends to drive interface topologies toward a minimum surface energy con guration. Reconstruction models for surface tension, on the other hand, can sometimes induce numerical noise from computed graininess in the surface pressures, often leading to unphysical disruptions at the interface (termed \ otsam"). The CSF model is also easy to implement, i.e., surface tension is modeled simply by calculating and applying the volume force given by (8) in the momentum equation. A small fraction of the total CPU time (few percent) is expended in modeling surface tension e ects.
9. Filter the advanced time velocities. 10. Begin the next cycle. Several comments are in order regarding the method we use. The stability restriction on the method is the sum of the usual CFL restriction and the restriction based on the capillary wave speed. Linear algebra is computed using a multigrid preconditioned conjugate gradient method. One for the single pressure equations and a second method for the coupled velocity di usion equations. Our advection method is based on limited fourth-order slopes [39, 38, 65]. Volume tracking is computed using an unsplit methods using a modi ed version of Pilliod and Puckett's fast least squares algorithm.
VI. Numerical Examples We illustrate the properties of our method with four numerical examples: a drop falling into a pool and the subsequent splash; a Rayleigh-Taylor instability; the rise of two bubbles of di erent size; and an idealized simulation of \cream" poured into a cup of \co ee". For each calculation, prototypical results are shown for four di erent interface tracking methods: interface capturing, split and unsplit PLIC methods, and a particlebased approach. The computational domain for the drop splash problem is a unit square partitioned with a 64 64 grid. The drop and pool density are 832.5 times higher than the rest of the medium. The drop, at rest initially, has a radius of 0.15 and is initialized at (0:50; 0:75). The domain boundaries are taken to be solid walls. The pool covers the bottom of the domain and has a depth of 0.25. The drop and pool viscosities are 55 times higher than the surrounding medium. Gravity is equal to ?1 in the
V. Algorithm Summary Algorithm 1 [Godunov-Projection Method] 1. Start with data un , f n, n+ 12 . 8
AIAA{95{0699 Problem Setup − Rayleigh−Taylor Instability Density Ratio = 7.25 Viscosity Ratio = 1.06
Problem Setup − Drop Splash Density Ratio = 832.5
Viscosity Ratio = 55
Water
xo = 0.50 yo = 0.75 ro = 0.15
Air
1 (64 cells) u=0 v=0 g = −1
Air
4 (256 cells)
h=2
Water
Helium 1 (64 cells)
u=0 v=0 g = −1
Figure 3: Problem setup for the drop splash problem.
1 (64 cells)
vertical direction. Surface tension acts on all uid interfaces, and wall adhesion is active with an equilibrium contact angle of 15 degrees on all solid boundaries. Four snapshots of this calculation are shown in Figure 4, all taken at the same simulation time (t = 3:0). In general, all four methods display the same gross physical behavior. The notable exception is the interface capturing method, which displays structure that is not seen in the other methods (including the earliest calculations of similar problems in [8, 9]). This is perhaps one of the strongest arguments regarding the inadequacy of interface capturing. Upon closer examination, the two PLIC methods are quite similar, with the unsplit method exhibiting a slight symmetry break near the peak of the rising water in the center of the pool (at present believed to be due to an implementation problem). Upon comparing the di erence between PLIC and the colored particle method, it is apparent that the particle method retains entrained air below the drop boundary and the drop remnant has a distinctly less circular shape. This is a direct result of the PLIC methods' tendency to possess more inherent \numerical surface tension" relative to the colored particle method. The colored particle method also exhibits noticeably ner structure for the water along along the box wall. Figure 5 shows the actual interfaces and particles corresponding to the plots of volume fractions in Figure 4. These further demonstrate the level of ne scale detail available with the colored particle method. Also evident is the ringing instability that causes spurious pattern formation in PIC-type methods [66]. We have found this to be a annoying property of the method, but its detriment is small when weighed against the weaknesses of the other methods.
Figure 6: Problem setup for the Rayleigh-Taylor instability. The Rayleigh-Taylor instability simulates air above helium, where the density ratio is 7.25:1 (an Atwood number of 0.76). The viscosity of air is 1.06 times larger than the helium (with a value of 0.004 in the air). The computational grid is 64 256. The domain is periodic in the horizontal direction. Gravity is equal to -1 in the vertical direction. A sinusoidal perturbation is applied to the position of the interface with a maximum size of three cell widths. Surface tension is neglected at uid interfaces. In Figure 7 are the Rayleigh-Taylor results generated with the four di erent interface tracking methods. The methods all exhibit the same general ow structure, but again there are signi cant di erences at smaller scales. The captured solution is quite di used and the arms of the roll-up do not show the same structure as the tracked interface methods. As expected, the two PLIC methods are quite similar, with both methods showing the e ects of numerical surface tension when the thin laments in the arms of the roll-up have formed \droplets" rather than to remain attached to the bulk
uid. The colored particle solution shows none of the breakup exhibited by the PLIC methods and the arms bow outward. At the edge of the plot and on the periodic boundary a secondary instability has begun to form. We believe that this is caused by the low dissipation inherent in the particle method (whereas the other methods tend to stabilize this mode). 9
AIAA{95{0699
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Figure 4: Water drop on a water pool in air. Volume fractions of water are shown for four di erent interface tracking methods. The simulation has progressed to the point where the crater has collapsed and water is being ejected upward out of the pool. Volume fractions of water are given with contours at 0:05, 0:5, and 0:95.
10
AIAA{95{0699 Interfaces
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Figure 5: Water drop on a water pool in air. For the PLIC method, reconstructed interfaces denote the computed front. For the colored particle method, lighter particles denote the pool and darker particles denote the drop. In our third problem, two gas bubbles are allowed to rise in water, with the two bubbles being a smaller bubble initialized directly below a larger bubble. The density and viscosity ratios are 8:71 10?3 and 1:8 10?2 , respectively. The surface tension coecient and viscosity are both chosen to be 1 10?3 . The domain measures 1 2, partitioned by 64 cells. The lower bubble (radius 0:10) is placed at (0:5; 0:25) and the upper bubble (radius 0:20) is placed at (0:5; 0:75). The Morton number is 8:8 10?6 and the Eotvos number is 4560 and 1140 for the upper and lower bubbles, respectively. As shown in Figure 9, the colored particle method again appears to generate the higher delity results, exhibiting ner structure. The PLIC-based methods show the e ects of numerical tension, which has led to distinctly separate uid blobs. As with the previous problems, the interface capturing method loses most of the ner scale structure of the ow eld. Beyond the point displayed in Figure 9, the methods continue to separate in quality. Qualitative characteristics of the ow eld are present in each method's solution, but only the particle method maintains the ne scale structure of the
ow. Our last test problem shows this characteristic even more strongly. Our last problem is a \co ee and creamer" test, motivated by curiosity about the process of pouring creamer into co ee. It has been observed that cream rises to the top of a co ee cup after disappearing from view.
Problem Setup − Double Bubble Rise Density Ratio = 0.0087 Viscosity Ratio = 0.018 u=0 v=0 g = −1
Water
2 (128 cells)
xo = 0.5 yo = 0.75 r = 0.2 Gas xo = 0.5 yo = 0.25 r = 0.10
1 (64 cells)
Figure 8: Problem setup for the two bubble rise.
11
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Figure 7: Evolution of the Rayleigh-Taylor instability. Volume fractions of air (0:05, 0:5, and 0:95) are shown for each method at t = 2:5.
12
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Figure 9: Small gas bubble being captured by a larger gas bubble as computed by four di erent methods. All are shown at t = 2:0. Volume fractions of gas are given with contours at 0:05, 0:5, and 0:95.
13
AIAA{95{0699 cream is poured into the co ee and the wall e ects cause the original vortices to de ect toward the opposite wall and push a small vortex toward the top of the cup near the point of the cream's entry. This vortex then acts to block the vortices as they again inpinge on the opposite wall. One of the large original vortices then acts as a pump for material that is located below the main vortical ow. It draws the material to the top of the cup to form the secondary plume often seen opposite of the pouring location in a cup. As might be expected, all of this is much clearer in the video we have prepared. We also hope to place these video segments on WWW in the near future.
Problem Setup − Coffee and Cream Density Ratio = 1.12 Viscosity Ratio = 7 xo = 0.7 yo = 1.25 h = 0.5 w = 0.2
Cream
1.5 (96 cells) Coffee (Black)
Velocity field initialized with a vortex pair in the cream and adjacent coffee.
VII. Future Directions Based on our results we will continue to investigate the promise of the colored particle method. Our other e orts are to move these algorithms to both cylindrical coordinates and three dimensions. In both cases the methods can be extended seamlessly with the exception of volume tracking, where again the colored particle method appears to o er an advantage.
1 (64 cells)
Figure 10: Problem setup for the co ee and creamer test. We seek to determine the mechanism by which this occurs. Our problem setup is idealized and awed (this is a inherently three-dimensional phenomena, and our grid under resolves the physics), but should provide some of the fundamental character of the problem. This problem is also a stringent test of interface tracking methods because of the vigorous mixing generated when the creamer \enters" the co ee. The problem setup for this calculation is shown in Figure 10. Our initial conditions are de ned to give the creamer a velocity downward that would result from the creamer being poured from a distance equal to the width of the computational domain. This is accomplished by giving the cream and surrounding co ee a vortex pair that is de ned so that the centerline of the cream blob is traveling downward at the desired velocity. Furthermore, we set the blob of cream closer to one of the side walls of the domain because the act of pouring cream into co ee is rarely symmetric. In Figure 11 we show the results midway through our simulation. It is obvious that interface capturing method has smeared the creamer a great deal more than the other methods. The PLIC methods have also formed a great number of blobs from laments due to the numerical surface tension. This is particularly evident in Figure 12a. Figure 12b shows how the particle method maintains the ne scale structure of the ow. While the PLIC methods again maintain gross aspects of the ow, more intricate aspects are maintained by the colored particle method. Through the use of the colored particle method, a plausible mechanism for the behavior seen in a co ee cup can be described. The vortices induced when the
VIII. Acknowledgments We would like to thank numerous individuals whose assisted us during the course of this work. Discussions with Jerry Brackbill, Phil Colella, Gerry Puckett, Ann Almgren, Tom Adams, Je Saltzman, Marc Sussman, John Turner, Mike Hall, David Brown, and Bill Henshaw all proved invaluable. We would like to especially thank Mike Krogh for his help with the production of the video displaying this work and Richard Lesar for providing invaluable computing resources.
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ow of uid with a free surface. Physics of Fluids, 8:2182{2189, 1965. [2] A. A. Amsden and F. H. Harlow. The smac method: A numerical technique for calculating incompressible uid ows. Technical Report LA{ 4370, Los Alamos National Laboratory, 1970. [3] F. H. Harlow and A. A. Amsden. A simpli ed MAC technique for incompressible uid ow calculations. Journal of Computational Physics, 6:322{ 325, 1970. [4] F. H. Harlow and J. E. Welch. Numerical study of large-amplitude free-surface motions. Physics of Fluids, 9:842{851, 1966. 14
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(d) Colored Particles
Figure 11: \Co ee and creamer" test as computed by four di erent interface tracking methods. Volume fractions of creamer are given with contours at 0:05, 0:5, and 0:95. All are shown at t = 1:0.
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Interfaces
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(b) Colored Particles
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