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 6, 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 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 suer from cell Reynolds numberbased stability restrictions. The linear systems of equations arising from the projection and diusion 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 instabilAll correspondence should be sent to William J. Rider, Mail Stop B265, 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 of volume tracking with this method. In conjunction with approximate projection methods 15, 16, 17], low error formulations 18] and the ltering of nonsolenoidal modes 19, 16, 20], the method remains robust for dicult problems. This method also merges quite naturally with physical models such as the CSF model for surface tension 21]. A similar eort based on the approximate projection method described in 15] is being made by E. G. Puckett et. al 22]. Capturing uid interfaces with volume tracking techniques remains quite eective, but only when subsequent improvements to the original method (developed 15 years ago) are taken into consideration. An example is the use of piecewise linear or planar approximations to the interface geometry 23]. Volume tracking allows interfaces to be captured and maintained compactly in one cell without imposing restrictions on the topological complexity or the number of interfaces that can be represented. We will also present particle based approaches based on the work of Brackbill and others 24] extending the PIC approach developed at Los Alamos forty years ago by Harlow 25, 26]. Physical processes specic to and localized at uid interfaces (e.g, surface tension, phase change) are modeled by applying the process to uid elements everywhere within the interface transition regions. Surface processes are thereby replaced with volume processes whose integral eect properly reproduces the desired interface physics. This methodology is the underlying feature in the CSF method for surface tension, which has proven successful in a variety of studies
27, 28, 11, 12, 29, 30, 31, 32, 33]. The CSF method lifts all topological restrictions (typically inherent in models for surface tension) without sacricing accuracy, robustness, or reliability. It has been extensively veried and tested in two-dimensional ows through its implementation in a classical algorithm for free surface ows, where complex interface phenomena such as breakup and coalescence have been predicted. This success led to our current focus on extensions and enhancements to the CSF method, namely three-dimensionality, an implicit formulation, and the addition of variable surface tension eects. Computational examples are shown of its capabilities.
with density and viscosity transport and
r
;
ru + ru
T
u = 0
(1c)
+u
rf = 0
(1e)
where f is a volume fraction of a uid having certain properties. This volume fraction is then used to determine the average cell-centered values for density and viscosity (as well as dening the location of a front). Density is recovered from f as follows (for two uids a and b. (f ) = f + (1 f ) and similarly for viscosity (f ) = f + (1 f ) : These equations will be then solved via a projection method described next. We will also discuss several important aspects of the methodology: lters, unsplit advection and the solution of linear systems of equations. a
;
b
a
;
b
II.1 Projection
Our basic goal with projection methods is to advance a velocity eld, V = (V V ) by some convenient means disregarding the solenoidal nature of V, then recover the proper solenoidal velocity eld, V ( V = 0). The means to this end is a projection, , which has the eect V = (V) : The projection accomplishes this through the decomposition of the velocity eld into parts that are divergencefree and curl-free. This is known as a Hodge or Helmholtz decomposition 34]. The curl-free portion will be denoted by the gradient of a potential, r'. This decomposition can be written V = V + r' (2) where = 1=. Taking the divergence of (2) gives V = V + r' V = r': Once ' has been computed, then V can be found through V = V r': (3) y T
x
d
r
d
P
d
P
d
+ F (1a)
r
and the conservation of mass is r
r = 0
@ @t
@f @t
We solve the equations of incompressible multiphase ow. The equations of motion is (uu)+ 1 r = 1
+u
+ u r = 0: (1d) In these equations, u is the velocity vector, is the incompressible pressure, the uid density, the viscosity and F is the momentum (velocity) source term. F can include the eects of gravity or surface tension for instance. In our calculations, the transport of density and viscosity is solved with one equation,
II. Projection/Godunov Methods @u +r @t
@ @t
r
d
r
! r
r
d
(1b)
d
2
;
AIAA{95{0699 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] and 19]). Projection methods can be dened 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 dening 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 and r) will be chosen (not quite true, but nearly). 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 1 r = 1 u +1 1 t +2 r
r
where is the pressure and u locity dened by
u
n
+1 =
u
n
+tF
+ 12
n
;
I 2t 1+ 12 ;
t
r
n
(uu)
+ 21 +t1 +
n t 1 n+ 2
2
r
n
+ 12
n
1 +2
t
;
1
r
1 +2 n
r+r
T
r ; n
ru + ru
;
T
;1
1 2
n
1 2:
In terms of a pressure Poisson equation, the form for the right hand side would be 1 + 12 1 ; + 2 ru + ru r
1 r +2
n
T
n
;
r
(uu)
+ 21
n
+F
+ 12
n
n
u + t n
i-1,j-1
i,j-1
the solution's quality. Because the discrete divergence still has a central dierence form, it does not detect all non-solenoidal modes in the solution thus leaving some divergence error largely unaected 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 nonphysical modes from the solution working only on the velocity eld. These lters are always applied at the end of a computational cycle. A number of lters have been devised to remove or damp spurious non-divergence-free modes from the solution. Unobstructed these modes can seriously damage the accuracy and stability of calculations. This is especially true in several cases: large density jumps, local source terms (such as surface tension) and for longterm integrations. These lters are described more fully in 17, 20] (also see 19] and 36] for a discussion of lters in the context of combustion) or more generally in 16].
is the predicted ve-
n
r
n+ 2
r ;
+1
n
i,j
Figure 1: This diagram shows a geometric representation of the corner-transported upwind method that is the basis of our advection method. The method algebraically computes the area swept out by the characteristics as shown above.
n
r
n
i-1,j
:
This option controls the presence of the discrete divergence without sacricing the accuracy of the method. II.3 Advection A full exposition on this subject is given in 17, 18]. The use of non-operator split high-order Godunov advection is also important to the method's success. This II.2 Filters method avoids the use of operator splitting and allows Because the discrete divergence is nonzero it is impor- the consistent use of high-order monotone advection tant to control the growth of the discrete divergence in in incompressible ows. The method also has excelthe solution. The divergence errors are a measure of lent phase error properties and compares favorably with 3
AIAA{95{0699 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 several sources 37, 13, 14, 38, 15, 17]. 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 eect of the governing equations is used to approximate the time derivatives (source terms, pressure gradients and the incompressibility constraint). The use of the incompressibility constraint leads to the \MAC" projection where time-centered normal velocities at cell edges are used to compute a potential eld that corrects them to be divergence-free 38, 19, 17]. This allows the method to use a divergence-free velocity eld for the purpose of advection (even with an approximate projection) and maintain second-order accuracy. Several points are important to emphasize. This method is not bound by a stability restriction related to the Reynolds number. Thus this method can be used to compute both viscous and inviscid ows. Results show that the method can resolve ows up to a grid Reynolds number of 40. Beyond this the solver will degrade is a graceful fashion and produce physical, but not necessarily accurate results (in the classical sense, see Brown and Minion 39]). The advection method is also robust in cases where the ow contains discontinuities (such as a shear).
We have a symmetric system of equations thus enabling the use of a standard preconditioned conjugate gradient (PCG) method 40]. For a preconditioner we have experimented with the usual incomplete Cholesky method, but we have found a weighted Jacobi (multipass) to be more cost ecient and eective. Our other (and more favored) approach is a multigrid method 41]. As opposed to the PCG method (which scales at best N 5 4 ), the multigrid method's operation count scales with N . We use a multigrid method based upon an approach nearly identical to that described in 42] or 36]. Unfortunately this method has not proved itself to be ecient in all cases. Certain ow topologies cause the method to converge at a slow rate thus destroying the advantages of the multigrid algorithm. In order to maintain the eciency of the multigrid algorithm and integrate the robustness of the PCG method we have combined them into a multigrid preconditioned conjugate gradient (MGCG) algorithm 43]. Care must be taken to assure that the multigrid algorithm produces a symmetric iteration matrix, but this is simple in practice. We have found this to be an ecient and robust method even in our most challenging problems. The method is faster than the pure multigrid method on most problems (where the multigrid worked well before) and seems to scale like a multigrid method with problem size. =
III. Interface Kinematics
We wish to follow the motion of an interface embedded in the overall motion of the ow eld. This can be The most computationally expensive portion of the al- accomplished through the solution of the equation gorithm is the solution of the linear system of equa@f tions arising from the discretization of the projec+ u rf = 0 (4a) @t tion/pressure equation. For periodic or solid wall boundaries this system of equations is semi-positive de- where f is some scalar carrying interface or \color" innite, but solvable. It is vital to solve this problem as e- formation. An equivalent equation for incompressible ciently as possible. Furthermore, this problem has been ows is @f rendered more dicult by solving for the entire ow + (uf ) = 0 (4b) @t eld (thus producing a non-constant coecient Poisson problem). Again, the ante is raised by allowing these since u = 0. coecients to typically vary in a discontinuous manner. A particle method is perhaps the most straightDirect solution (i.e. Gaussian elimination) is too ex- forward, drawing upon recent advances in PIC algopensive for further consideration. Classical iterative ap- rithms 44]. Particles are assigned a mass according to proaches are equally expensive in terms of operation the density of the uid in which they reside and a volume count although they are cheaper in terms of storage. (hence size) according to the interpolation function choFor banded Gaussian elimination and standard itera- sen to interpolate quantities to and from the computative methods like SOR, the operation count scales with tional grid. While this method provides a superior multhe number of equations, N , squared. Because of the tidimensional grid-independent advection scheme, there non-constant coecients of the operators, FFT or cyclic are as a result some practical diculties, namely the reduction methods are also ruled out. We are left with cost and accuracy associated with interpolating the partwo choices: preconditioned conjugate gradient methods ticle information to an Eulerian grid 45, 46]. This problem often exhibits itself as a noisy density eld. Later, and multigrid methods.
II.4 Numerical Linear Algebra
r
r
4
AIAA{95{0699 we will describe a new approach that makes a particle method quite feasible for interface tracking. Simply discretising (4) with a high resolution nite dierence 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, 47]. 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 48, 49]. 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 dierence techniques already well developed for advection, but without allowing the interface to smear. The interface is dened as the zero level set of a distance function, , from that interface. Instead of (4), the following equation is solved @ @t
+u
Interfaces 0.7756
Y
0.7
0.6
0.5533 0.5333
r = 0:
0.6
0.7
0.7644
X
Improvements for incompressible ows are suggested by Sussman, Smereka and Osher 50]. VOF methods have been used for several decades starting at the national laboratories (Livermore 51] and Los Alamos 52] and later Sandia 53]). The earlier work is typied by the SLIC 54] algorithm and the original method with the moniker VOF 10]. In each of these methods the interface is designated as a straight line in a cell dened 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 55]. Recently, Pilliod and Puckett have improved the accuracy 56, 57, 58]. 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 oer excellent, numerical diusion-free results, but have two problems: cost and the problem of mapping the particles to a grid in a manner free of oscillatory modes 59]. 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.
Figure 2: This diagram shows the reconstruction of a portion of a cirle by a PLIC method.
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 signicant vorticity can appear quite compressible in one direction thus leading to sgnicant 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 37, 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 59, 56]. In addition we have implemented an algorithm that allows the method to conserve volume to a high degree 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.
III.2 Colored Particle Method Another particularly promising approach for interface tracking has been named the \colored particle" method. The aspect of this method that makes it similar to the earlier MAC method is that the particles do not carry ei5
AIAA{95{0699 ther volume or mass, but rather identity. The entire ow domain is lled with particles each carrying an identity. These identities or colors are then used to produce a weighted average for a physical property on the grid. This allows the density and viscosity to be smoothly reconstructed on the Eulerian grid. With mass or volume carrying particles, the smoothness of the transfer to the Eulerian grid is a diculty in maintaining quality solutions without a large number of particles (with the problem still present, but lessened in severity). In our calculation, the particles are transported using a second-order predictor-corrector method. First, the position of the particles is estimated at a time centered position x + 12 = x + 2t u (x ) then the advanced time position is computed
x
+1
n
n
n
n
= x + tu n
+ 12
n
to the accurate formulation of the necessary discrete operators. We also outline the CSF enhancements and improvements that are currently being studied. These developments should allow a more ecient modeling of a wider variety of interfacial ows. The central theme of the CSF model is reformulation of interface dynamics into a localized volumetric force, which is quite dierent from earlier numerical models of interfacial phenomena. The basic premise of the CSF model is to replace interfacial surface phenomena (normally applied via a discrete boundary condition) as smoothly varying volumetric forces derived from a product of the appropriate interfacial physics per unit area and a numerical approximation to the surface (interface) delta function. The CSF formulation makes use of the fact that numerical models of discontinuities in nite volume and nite dierence schemes are really continuous transitions within which the uid properties vary smoothly from one uid to another (over a distance of (h), where h is a length comparable to the resolution aorded by the computational mesh). It is not appropriate, therefore, to apply in these schemes a boundary condition at an interface \discontinuity", which in the case of surface tension is a pressure jump across the interface. Surface tension should instead act on uid elements everywhere within the transition region. In the case of surface tension, the relevant surface physics is a force per unit area arising from local interface curvature and local (tangential) variations in the surface tension coecient. Application of interfacial physics using the CSF model then reduces to application of a localized force in the momentum equation, irregardless of interface topology. The CSF model is therefore ideally suited for dynamical interfaces of arbitrary topology. The resulting simplicity, accuracy, and robustness has led to its widespread and popular use 27, 28, 11, 12, 29, 30, 31, 32, 33] in modeling complex interfacial ows that were in many cases previously intractable. In the CSF model, surface tension is reformulated as a volume force Fs satisfying
n
x
n
+ 12 :
O
Velocities are computed at the particle location via bilinear interpolation. With the particles advanced in time, bilinear interpolation is used to transfer the particle identities to the grid as densities and viscosities. This is done in a particle-weighted fashion. According to the bilinear interpolation, the particles each have a weight in a grid cell of ! . For instance, to get a density in a cell P ! = P : k i j
k i j
k
i j
k
k
!
k i j
This method will not conserve mass in the usual fashion, but because particles are used, information about the interface is not lost. Unlike other approaches (capturing, level sets, or VOF) there is no diusion or dispersion to destroy information about the interface as the particles are advected as Lagrangian markers. In each calculation we use 16 particles per cell.
IV. Interface Dynamics Algorithms for interface dynamics must model physics specic to and localized at interfaces, such as 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 than surface tension can be encapsulated easily within the CSF model. Typical examples are phase change and momentum exchange 31], where the surface physics are mass and momentum ux, respectively, transferred across the interface. In this section, we briey review the theory of the CSF model, then discuss some important issues pertaining
lim !0
h
Z
V
Fs(x)d3 x =
Z
S
fs(xs )dS
(5)
where xs is a point on the interface, h is a length comparable to the resolution aorded by a computational mesh with spacing x, 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]: fs (xs ) = (xs )^n(xs ) + s (xs ) (6) where is the surface tension coecient, s is the surface gradient 21], n^ is the interface normal, and is the mean interfacial curvature 60]: (xs ) = ( n ^ ) (xs ) : (7) r
r
;
6
r
AIAA{95{0699 The rst term in equation 6 is a force acting normal to the interface and in proportion to the curvature . The second term is a force acting tangential to the interface toward regions of higher surface tension (). 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 (h). The volume force, nonzero only within these transition regions, is given in the CSF model by 21]: Fs(x) = fs(x) c ~c(]x) (8) where c~ is the characteristic (or \color") function, taken to be the VOF function f in this work, and c] is the jump in c, which is unity for c = f . The term in equation 8 above involving c is essentially a mollied delta function. This form is not unique see, for example, reference 50] for other forms. The unit normal,
Its line integral directed normally through the interface transition region is approximately equal to the pressure jump . The force Fs resides at cell centers our scheme, collocated with all other uid quantities. In practice we nd that the use of more interface information (a larger stencil) leads to a better estimate of curvature. This appears to be the case for Fs at cell centers, which requires a 6 cell stencil when both components of interface normal n are collocated at faces. In contrast to the discretizations presented in 21], we have chosen a conservative discretization for the curvature and the force Fs . This is because we wish to preserve an important physical property of surface tension that is conservative is the lack of any net normal component of the surface tension force on a closed surface, i.e., the net normal force (and also the curvature ) should vanish over any closed surface. A cell-centered force is then obtained by summing over cell faces, which brings the eective stencil to 9 cells. This important fact, regretfully omitted in BKZ, implies that discretization of must also be conservative, expressed as a sum of n^ over faces of each control volume:
O
jr
n n
j
=;
1
Z r
V
Z n^ dV = V1 n^ dA ;
;
1
V
X
f
n^ f dAf
(9) where V is the control volume, dAf the area of face f (pointing outward) on V , and n^ f the unit interface noris derived from a normal vector n, mal on face f of V . Without a conservative discretization of , one might observe, for example, numericallyn= f induced horizontal motion of bubbles that should othwhich is the gradient of VOF data. In this work, only erwise rise vertically due to buoyant forces. A stress the normal force in equation 6 is modeled, i.e., the sur- tensor reformulation of face tension coecient is taken to be constant. Details A smoothed value of the curvature can be computed on modeling the tangential force using the CSF model from equation ?? by using a smoothed VOF function can be found in reference 33]. fe to derive the normals needed in equation IV.. The The interested reader should also consult refer- smoothed VOF function is computed with an expression ence 61], where the volume force in equation 8 is ap- of the form proximated as the divergence of a tensor (for constant ). The attractive features gained from a further apX ( ) (x0 x h) ( ) ;y 0 y h fe = f proximation of this force as the divergence of a stress =1 tensor are not yet clear. A simple yet accurate dis(10) cretization of this tensor (especially in three dimensions) ( ) is a B-spline ?] of degree `, having nite where is an outstanding issue that could determine the usefulness of this approach relative to the CSF method in its support, ( ) (x0 x h) = 0 only for x0 x < (` + 1)h=2. This smoothing tends to mitigate the high wavenumber original form. The normal component of the CSF volume force in contributions to , which may or may not be a result of equation 6 is easily calculated by taking rst and second discretization errors. It should, therefore, be used with order spatial derivatives of the uid volume fractions. caution because the real free surface geometry might At each point within the free surface transition region, be unphysically mollied. Examples of the smoothing a cell-centered value of the force Fs is estimated. This eects can be found in the CSF paper. 21] force is proportional to the curvature of the constant The representation of surface tension in the CSF VOF surface at that point. As shown in equation 5, model as a time tn body force is linearly stable only the force is normalized to recover the conventional de- for time steps smaller than a certain maximum allowscription of surface tension as the local product h 0. able value ts necessary to resolve the propagation of n ^=
j
j
r
n
i0 j 0 S
i j
`
i
;
i
S
`
j
i0 j 0
S
`
S
!
7
`
;
6
;
;
j
AIAA{95{0699 force (equation 8) in the momentum equation. Only a small fraction of the total CPU time (few percent) is expended in modeling surface tension eects.
capillary waves, 21] evaluated as "
t < ts
=
3=2 Jmin 4
#1=2
(11)
V. Algorithm Summary
where Jmin is the minimum mesh cell Jacobian. 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 the free surface unit normal n^ prior to evaluating equation 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. The wall adhesion boundary condition becomes an expression for the unit free surface normal n^ at points of contact ~xw along the wall: n ^ = n^w cos eq + t^w sin eq (12)
Algorithm 1 Godunov/Projection Method] 1. Start with data u , f , + 12 . 2. Compute the time- and edge-centered velocity eld via the Godunov procedure based solely on data at time n. 3. Solve the \MAC" projection and correct the timeand edge-centered velocity eld so that 1 Du + 2 = 0. 4. Using u + 12 update the volume fraction data from n to n + 1. 5. Using f +1 compute +1 and +1 . 6. Compute term using ; the CSF source 1 f + 2 = f + f +1 =2. 7. Update u to u +1 with advection, diusion and CSF sources. 8. Solve the projection L + 12 = Du +1 and correct the velocities to u +1 . 9. Filter the advanced time velocities. 10. Begin the next cycle. n
n
n
n
n
where eq is the static contact angle between the uid and the wall, n^ w is the unit wall normal directed into the wall, and t^w is tangent to the wall, normal to the contact line between the free surface and the wall at ~xw . The equation uses the geometric identity that eq , dened as the angle between the tangent to the uid and t^w , is also the angle between n ^ w and n^. The unit tangent t^w is directed into the uid, and is computed from equation IV. with the VOF function f reected at the wall. The angle eq is not a uid material property, but a system property, depending also on properties of the wall itself. The value of eq is measured experimentally when the uid is at rest. We emphasize that the wall adhesion boundary condition above is applied regardless of whether or not the angle the uid interface with the wall is actually equal to eq . Surface tension modeled with the CSF method eliminates the need for interface reconstruction, so restrictions on the number, complexity, or dynamic evolution of interfaces having surface tension are not imposed. Direct comparisons between modeling surface tension with the CSF model and with a popular interface reconstruction model 62] 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 conguration. 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 (often called \otsam"). In addition to providing a more accurate nite dierence representation of surface tension without the topological restrictions, the CSF model is easy to implement computationally. Surface tension is easily included by calculating and applying the extra body
n
n
n
n
n
n
n
n
n
n
n
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 preconditioined conjugate gradient method. One for the single pressure equations and a second method for the coupled velocity diusion equations. Our advection method is based on limited fourth-order slopes 37, 63]. Volume tracking is computed using an unsplit methods using a modied version of Pilliod and Puckett's fast least squares algorithm.
VI. Numerical Examples Here we will present four numerical examples of the methods we have described. The rst example is a drop falling into a pool and the subsequent splash. Our second problem is a Rayleigh-Taylor instability. Next, we show a Kelvin-Helmholtz instability as an idealized ow of wind over water. Finally, we show a simulation of 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) Air
u=0 v=0 g = −1
4 (256 cells)
h=2
Water
Helium 1 (64 cells)
u=0 v=0 g = −1
Figure 3: Setup for the drop and pool problem is shown.
1 (64 cells)
cream being poured into a cup of coee. We will show prototypical results using four dierent interface methods: interface capturing, split and unsplit PLIC methods and a particle based approach. The drop and splash is set in a 1 1 box with the drop and pool density being 832.5 times higher than the rest of the medium. The drop has a radius of 0.15 and is set at (0:50 0:75) Each boundary is a solid wall. The computational grid is 64 64. and is at rest initially. The pool covers the bottom of the domain and has a depth of 0.25 units. The heavy uid has a viscosity that is 55 times higher than the surrounding medium. The high density uid viscosity is 0.001 and the Bond number, 4gR2=, is 1000. Gravity is set to 1. The wall adhesion model is used with and equilibrium angle of 15 degrees. For this problem the physical units give a Morton number of 1 and a E!otv!os number of 2:25 10;3 . We show four snapshots from the calculation in Figure 4. This pictures are all 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 (or the historical references of similar problems 8, 9]). This is perhaps one of the strongest arguments regarding the inadequecy of interface capturing. Upon closer examination, the two PLIC methods are quite similar, with unsplit method exhibiting a slight symmetry break near the peak of the rising water in the center of the pool (we believe do to coding error). When comparing the dierence between PLIC and the colored particle method, one notices that the particle method still contains entrained air below the drop boundary and the drop remanant has a dis-
Figure 6: Setup for the Rayleigh-Taylor instability is shown.
tinctly less circular shape. The PLIC methods may exhibit this characteristic becasue of the numerical surface tension that is inherent in these methods. With the colored particle method, the water along the box wall has considerably greater structure. 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 64]. We have found this to be a annoying property of the method, but its detriment is small when comparing it with of technique for interface tracking. The Rayleigh-Taylor instability simulates air above helium giving a density ratio of 7.25:1. The air's viscosity is 1.06 times larger than the helium. Its value is set to 0.004 in the air. The computational grid is 64 256. In the horizontal direction the domain is periodic. Gravity is set to -1. A sinusoidal perturbation is applied to the position of the interface with a maximum size of three cell widths. The calculation uses no surface tension. The Atwood number for this problem is 0:76. In Figure ?? we again show the results using our four dierent methods. The dierent methods all show the same general ow structure, but again there are signicant dierences at smaller scales. The captured solution is quite diused and the arms of the rollup do not show
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9
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Figure 4: Water drop on a water pool in air. Volume fractions of water are shown for four dierent 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.
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Figure 5: Water drop on a water pool in air. For volume tracking reconstructed interfaces are used to draw the computed front. The colored particles are displayed with the lighter particles denoting the pool and the darker particles showing the drop. the same structure as any of the tracked interface methods. As might be expected, the two PLIC methods are quite similar with both methods showing the eects of Problem Setup − Double Bubble Rise numerical surface tension as the thin laments in the Density Ratio = 0.0087 Viscosity Ratio = 0.018 arms of the rollup have formed \droplets" rather than u=0 stay in on single structure. The colored particle soluv=0 g = −1 tion 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 instabilWater ity has begun to form. We believe that this is caused by the low dissipation inherent in the particle method (and the other methods thus stabilize this mode). 2 (128 cells) xo = 0.5 yo = 0.75 The third problem has two bubbles (gas in water) r = 0.2 with a smaller drop being below the larger bubble. The ; 3 density ratio is 8:71 10 with a viscosity ratio of 1:8 Gas 10;2 . The surface tension constant and viscosity are xo = 0.5 chosen to each be 1 10;3 . The domain measures 1 2 yo = 0.25 discretised with 64 cells. The lower bubble has a radius r = 0.10 of 0:10 and is placed at (0:5 0:25) and the upper bubble has a radius of 0:20 and is placed at (0:5 0:75). For this problem the physical units give a Morton number 1 (64 cells) of 8:8 10;6 and a E!otv!os number of 4560 for the upper bubble and 1140 for the lower bubble. As shown in Figure ??, the colored particle method 8: Setup for the small and large bubble rise is again shines. The PLIC-based methods again show the Figure shown. eects of numerical tension as lament coagulates. As with the earlier tests, interface capturing loses most of
11
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Figure 7: Rayleigh-Taylor instability evolution. The volume fractions of air are shown for each method at t = 2:5. Volume fractions of creamer are given with contours at 0:05, 0:5, and 0:95.
12
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Figure 9: Small bubble being captured by a larger bubble as computed by four dierent methods. All are shown at t = 2:0. Volume fractions of creamer are given with contours at 0:05, 0:5, and 0:95.
13
AIAA{95{0699 providing invaluable computing resources.
Problem Setup − Coffee and Cream Density Ratio = 1.12 Viscosity Ratio = 7 xo = 0.7 yo = 1.25 h = 0.5 w = 0.2
References
Cream
1] F. H. Harlow and J. E. Welch. Numerical calculation of time-dependent viscous incompressible ow of uid with a free surface. Physics of Fluids, 8:2182{2189, 1965. 1.5 (96 cells) Coffee (Black)
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. Velocity field initialized with a vortex pair in the
3] F. H. Harlow and A. A. Amsden. A simplied cream and adjacent coffee. MAC technique for incompressible uid ow calcu1 (64 cells) lations. Journal of Computational Physics, 6:322{ 325, 1970. Figure 10: Setup for the coee and creamer problem is 4] F. H. Harlow and J. E. Welch. Numerical study of large-amplitude free-surface motions. Physics of shown. Fluids, 9:842{851, 1966.
5] B. J. Daly. Numerical study of two uid rayleighthe ner scale structure of the ow eld. taylor instability. Physics of Fluids, 10:297{307, Our last problem is a coee and creamer test. It 1967. is driven by curiousity about the process of pouring creamer into coee. It has been observed that cream 6] B. J. Daly and W. E. Pracht. Numerical study of rises to the top of a coee cup after disappearing from density-current surges. Physics of Fluids, 11:15{30, view. We sought to attempt to determine the mecha1968. nism by which this occurs. Our problem setup is idealized and awed, but hopefully provides some of the 7] F. H. Harlow, J. P. Shannon, and J. E. Welch. Liqfundamental character of the problem. Furthermore the uid waves by computer. Science, 149:1092{1093, vigorous mixing in this system will provide a stringent 1965. test of interface tracking. Our setup of this calculation
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9] F. H. Harlow and J. P. Shannon. The splash of a VII. Future Directions liquid drop. Journal of Applied Physics, 38:3855{ 3866, 1967. Based on our results we will continue to investigate the promise of the colored particle method. Our other 10] C. W. Hirt and B. D. Nichols. Volume of uid eorts are to move these algorithms to both cylindri(VOF) method for the dynamics of free boundcal coordinates and three dimensions. In both cases the aries. Journal of Computational Physics, 39:201{ methods can be extended seemlessly with the excep225, 1981. tion of volume tracking where again the colored particle method oers an advantage.
11] D. B. Kothe, R. C. Mjolsness, and M. D. Torrey. Ripple: A computer program for incompressible ows with free surfaces. Technical Report LA{ VIII. Acknowledgements 12007{MS, Los Alamos National Laboratory, 1991. We would like to thank numerous individuals whose 12] D. B. Kothe and R. C. Mjolsness. Ripple: A new assisted us during the course of this work. Discussions model for incompressible ows with free surfaces. with Jerry Brackbill, Phil Colella, Gerry Puckett, Ann AIAA Journal, 30:2694{2700, 1992. Almgren, Tom Adams, Je Saltzman, Marc Sussman, John Turner, Mike Hall, David Brown, and Bill Hen- 13] J. B. Bell, P. Colella, and H. M. Glaz. A second-order projection method of the incompressshaw all proved invaluable. We would like to especially ible Navier-Stokes equations. Journal of Computathank Mike Krogh for his help with the production of tional Physics, 85:257{283, 1989. the video displaying this work and Richard Lesar for 14
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1.5
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0 0
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1
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(b) Operator Split PLIC
F
F
1
1
Y
1.5
Y
1.5
0
0 0
1
0
X
1
X
(c) Unsplit PLIC
(d) Colored Particles
Figure 11: Creamer in a cup of coee are shown as computed by four dierent methods. Volume fractions of creamer are given with contours at 0:05, 0:5, and 0:95. All are shown at t = 1:0.
15
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Interfaces
Particles
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1
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1.5
Y
1.5
0
0 0
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0
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1
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(a) Unsplit PLIC Interfaces
(b) Colored Particles
Figure 12: Creamer in a cup of coee are shown for unsplit PLIC and the colored particle methods. Interfaces are used to shade in the coee and particles positions are displayed.
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