When multiphase flow is simulated by solving a single set of equations for the whole flow field, it is necessary to account for differences in the material properties of the different fluids and to add appropriate interface terms for interfacial phenomena, such as surface tension. Since these terms are con- centrated at the boundary between the different fluids, they are represented by delta (δ) functions. When the equations are discretized, the δ-functions must be approximated along with the rest of the equations. The material properties and the flow field are, in general, also discontinuous across the in- terface and all variables must therefore be interpreted in terms of generalized functions.
The various fluids can be identified by a step (Heaviside) function H which is 1 where a particular fluid is located and 0 elsewhere. The interface itself is marked by a nonzero value of the gradient of the step function. To find
then for cubic interpolated propagation and most recently for constrained interpolation profile.
Immersed Boundary Methods 39 the gradient it is most convenient to express H in terms of the integral over multidimensional δ-functions. For a two-dimensional field
H(x, y) =
A
δ(x− x)δ(y− y) da, (3.1) where the integral is over an area A bounded by a contour S. H is obviously 1 if the point (x, y) is enclosed by S and 0 otherwise. To find the gradient of H, first note that since the gradient is with respect to the unprimed variables, the gradient operator can be put under the integral sign. Since the gradient of the δ-function is antisymmetric with respect to the primed and unprimed variables, the gradient with respect to the unprimed variables can be replaced by the gradient with respect to the primed variables. The resulting area (or volume, in three dimensions) integral can be transformed into a line (surface) integral by a variation of the divergence theorem for gradients. Symbolically: ∇H = A ∇δ(x− x)δ(y− y)da (3.2) =− A ∇δ(x− x)δ(y− y) da =− S δ(x− x)δ(y− y)nds.
Here the prime on the gradient symbol denotes the gradient with respect to the primed variables and n is the outward unit normal vector to the interface. Although we have used that S is a closed contour, the contribution of most of the integral is zero. We can therefore replace the integral by one over a part of the contour and drop the circle on the integral:
∇H = −
S
δ(x− x)δ(y− y)nds. (3.3) By introducing local coordinates tangent (s) and normal (n) to the front, we can write
δ(x− x)δ(y− y) = δ(s)δ(n), (3.4) and evaluate the integral as
− S δ(x− x)δ(y− y)nds =− S δ(s)δ(n)nds =−δ(n)n. (3.5) This allows us to use a one-dimensional delta function of the normal variable, instead of the two-dimensional one in equation (3.2). Although we have assumed a two-dimensional flow in the discussion above, the same arguments apply to three-dimensional space.
If the density of each phase is assumed to be constant, the density at each point in the domain can be represented by the constant densities and the Heaviside function:
ρ(x, y) = ρ1H(x, y) + ρ0[1− H(x, y)]. (3.6) Here, ρ1 is the density of the fluid in which H = 1 and ρ0 is the density where H = 0. The gradient of the density is given by
∇ρ = ρ1∇H − ρ0∇H = (ρ1− ρ0)∇H (3.7)
= ∆ρ
δ(x− x)δ(y− y)nds = ∆ρδ(n)n,
where ∆ρ = ρ0− ρ1. Similar equations can be derived for other material properties.
The Navier–Stokes equations, as derived in Chapter1, allow for arbitrary changes in the material properties of the fluids. While the differential form does not, strictly speaking, allow discontinuous material properties, numer- ical methods based on the finite-volume method are equivalent to working with the integral form of the governing equations, where no smoothness as- sumption is made. Furthermore, if we work with generalized functions, then we can use the equations in their original form. This is the approach that we take here. The equations as written in Chapter1 do not, however, account for interface effects such as surface tension. Surface tension acts only at the interface and we can add this force to the Navier–Stokes equations as a singular interface term by using a δ-function,
ρ∂u
∂t + ρ∇ · uu = −∇p + ρf + ∇ · µ(∇u + ∇
Tu) + γκδ(n)n. (3.8)
Here, κ is the curvature for two-dimensional flows and twice the mean cur- vature in three-dimensions, and n is a properly oriented unit vector normal to the front. n is a normal coordinate to the interface, with n = 0 at the in- terface. As in Chapter1, u is the velocity, p is the pressure, and f is a body force. With the singular term added, this equation is valid for the whole flow field, including flows with interfaces across which ρ, and the viscosity field,
µ, change discontinuously. This fact justifies the “one-fluid” denomination.
For incompressible flows, the mass conservation equation is the same as for a single-phase flow
Immersed Boundary Methods 41 showing that volume is conserved. For a single-phase flow where the density is constant, there is no need to follow the motion of individual fluid particles. If the density varies from one particle to another, but remains constant for each particle as it moves (as it must do for an incompressible flow), it is necessary to follow the motion of each fluid particle. This can be done by integrating the equation
Dρ
Dt = 0. (3.10)
For multiphase flows with well-defined interfaces, where the density of each phase is a constant, we only need to find H and then construct the den- sity directly from H as discussed above. The same arguments hold for the viscosity and other properties of the fluid.
The “one-fluid” equations are an exact rewrite of the Navier–Stokes equa- tions for the fluid in each phase and the interface boundary conditions. The governing equations as listed above assume that the only complication in multifluid flows is the presence of a moving phase boundary with a con- stant surface tension. While these equations can be used to describe many problems of practical interest, additional complications quickly emerge. The presence of a surfactant or contaminants at the interface between the fluids is perhaps the most common one, but other effects such as phase changes, are common too. We will not address these more complex systems in this book.
There is, however, one issue that needs to be addressed here. While the Navier–Stokes equations, with the appropriate interface conditions, govern the evolution of a system of two fluids separated by a sharp interface, the topology of the interface can change by processes that are not included in the continuum description. Topology changes are common in multiphase flow, such as when drops or bubbles break up or coalesce. These changes can be divided into two broad classes: films that rupture and threads that break. If a large drop approaches another drop or a flat surface, the fluid in between must be “squeezed” out before the drops are sufficiently close so that the film between them becomes unstable to attractive forces and ruptures. A long, thin cylinder of one fluid will, on the other hand, generally break by Rayleigh instability where one part of the cylinder becomes sufficiently thin so that surface tension “pinches” it in two. The exact mechanisms of how threads snap and films break are still being actively investigated. There are, however, good reasons to believe that threads can become infinitely thin in a finite time and that their breaking is “almost” described by the Navier– Stokes equations (Eggers, 1995). Films, on the other hand, are generally
believed to rupture due to short-range attractive forces, once they are a few hundred angstroms thick (Edwards, Brenner, and Wasan,1991). At the moment, most numerical simulations of topological changes treat the process in a very ad hoc manner and simply fuse interfaces together when they come close enough and allow threads to snap when they are thin enough. “Enough” in this context is generally about a grid spacing and it should be clear that this approach can, when the exact time of rupture is important, as it sometimes is for thin films, lead to results that do not converge under grid refinement. The pinching of threads, on the other hand, appears to be much less sensitive to the exact numerical treatment and results that are essentially independent of the grid can be obtained. How to incorporate more exact models for the rupture of films into simulations of multiphase flows is a challenging problem that is, as of this writing, mostly unsolved. Current efforts on multiscale computing, however, hold great promise.
The one-fluid formulation of the Navier–Stokes equations allows multi- phase flow to be treated in more or less the same way as homogeneous flows and any standard algorithm based on fixed grids can, in principle, be used to integrate the discrete Navier–Stokes equations in time. The main difference between a Navier–Stokes solver for multiphase flows and the simple method outlined in Chapter2is that we must allow for variable density and viscos- ity and add the surface tension as a body force. For the simple first-order method of Chapter2, the equation for the predicted velocity (equation 2.3) is modified by explicitly identifying at which time the density is computed and by adding the surface tension:
u∗= un+ ∆t −Ah(un) + 1 ρnDh(u n) + 1 ρnf n b + 1 ρnF n γ . (3.11)
The exact calculation of the surface tension term Fγ generally depends on how the marker function is advected, which is discussed below. The Poisson equation for the pressure is the same, except that the density is no longer a constant and must be explicitly included under the divergence operator
∇h
1
ρn · ∇hp =
1
∆t∇h· ˜u. (3.12)
This simple-looking change has rather profound implications for the solu- tion of the pressure equation, since highly developed methods for separable equations cannot be used. Considerable progress has, however, been made in the development of efficient methods for elliptic equations with variable coefficients like equation (3.12). We refer the reader to Wesseling (2004) for a discussion and additional references. The projected velocity at time step
Immersed Boundary Methods 43 density computed at time n:
un+1 = ˜u−∆t
ρn∇hP. (3.13)
Several variations of this algorithm are possible. Here we have used the momentum equations in the non-conservative form. If the conservative form is used, the density must be available at time n + 1 so the density must be advected at the beginning of the time step. The conservative form can, however, lead to unrealistically large velocities around the interface, so the non-conservative form is generally recommended. For serious simulations, higher order time integration is usually used. The spatial discretization of the advection terms is the same as discussed in Chapter 2, but we must use the full deformation tensor for the viscous terms, since the viscosity is generally not constant. Although the viscous fluxes are usually computed by simply using the linearly interpolated value of viscosity for the boundaries of the control volumes, the resulting approximation does not ensure the proper continuity of the viscous fluxes when the viscosity changes rapidly. Many authors have found that working with the inverse of the viscosities improves the results (Patankar,1980; Ferziger, 2003).