CSF表面张力模型：一种连续方法处理两相流

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"本文介绍了一种新的方法，即CSF（Color-Tracking Surface Force）表面张力模型，用于模拟两相流体界面的表面张力效应。该模型由J. U. Brackbill、D. B. Kothe和C. Zemach在1992年发表在《计算物理》期刊上，它提供了一个连续的方法来处理不同性质流体之间的界面，这些界面被表示为具有有限厚度的过渡区域。" 在计算机模拟和流体力学中，处理两相流体的相互作用，特别是涉及表面张力的现象，一直是一项挑战。CSF模型通过引入一种连续变化的"颜色"变量，成功地解决了这个问题。这里的"颜色"并不是指视觉上的色彩，而是用来区分不同流体的抽象概念。在两个流体的交界处，这个"颜色"变量会连续变化，形成一个过渡区。关键在于，CSF模型定义了一种力密度，该力密度与该点处颜色常数表面的曲率成比例。当局部过渡区域的厚度与局部曲率半径之比趋近于零时，这个力密度就恢复了传统表面张力的描述。这意味着，即使在存在复杂形状和动态变化的界面时，模型也能准确地捕捉到表面张力的影响。传统的方法通常需要重建流体界面，这既复杂又容易出错。CSF模型消除了这种需求，简化了表面张力的计算，并且能够处理由表面力驱动的二维和三维流体流动问题。这种模型对于模拟如泡沫、液滴、气泡等两相流体系统的运动尤其有用，特别是在工业应用和科学研究中，如燃烧、喷雾、沸腾等现象。 CSF模型的优势还在于其灵活性，它可以轻松地集成到现有的数值求解器中，而不需要对流体界面进行特殊的追踪或重建。此外，由于它避免了界面追踪的难题，因此在处理快速变化的流体动力学问题时，CSF模型能够提供更稳定和准确的解决方案。 CSF表面张力模型是一种创新的数值技术，它在处理流体动力学中的表面张力效应时，不仅提高了计算效率，也提高了模拟精度。对于研究者和工程师来说，这一方法是理解和模拟复杂两相流体系统的重要工具。

338

BRACKBILL, KOTHE, AND ZEMACH

direction to A, fi(x,,,), passes through x. Then xSO is the

B. Properties of F,,(x)

surface point closest to x. The integral in (21) is, approxi-

mately,

In summary, the volume force, F,,(x), has the following

properties:

ri(x,)Y(x-x,)dA

The volume force in the transition region, where the

color varies smoothly from c1 to c2, is designed to simulate

= 2 i(%o)

h 2

Y(x-x,)dA+Q z

(( >>

3 (22)

the surface pressure on the interface between the fluids.

Thus, the line integral of F,,(x) across the transition region,

e.g., from P, to P, in Fig. 1, is approximately equal to the

where R is the radius of curvature of the surface at xSO. One

conventional surface pressure (x, is the interface point on

can bound the integral in (22) by

the line P, P,):

Wx -x,) dA <9(x -x,,,).

As h + 0, 9(x - x,,,) is zero everywhere but x = xSO, and

d?(x)

the corresponding limit of the integral of V?(x) across the

Mx) fi(x) ccl

interface is given by

= mc(x,) fi(x,) for h>O.

(29)

lim

h-0

6(x,,) -W(x) dx= [cl.

(24)

2. In the limit that the width of the transition region in

a direction normal to the interface goes to zero (h + 0), the

Thus, the limit h -+ 0 of V?(x) can be written

volume force becomes

lim VC(x)=A[c] S[ri.(x-x,)] =Vc(x).

(25)

lim F,,(x) = F,,(x) X%x,). (x -x,)1,

(30)

h-0

This delta function can be used to rewrite F,,(x,) as a

which yields the conventional surface pressure at x, given

volume integral for h = 0:

by (9).

F,,(x) b[fi(x,) . (x - xS)] d3x

m(x) ii(x) cS[iz(x,). (x - x,)] d3x. (26)

The delta function converts the integral of F,,(x) over a

volume V containing the interface A to an integral over A of

F,,(x,) evaluated at that surface. The integral relation in

(26), an identity for discontinuous interfaces (h = 0), can be

used to approximate interfaces having a finite thickness h

when (25) is substituted for the delta function. Upon

substituting (25) into (26), we find that

F,,(x,) dA = lim

Wx) d3X.

4x) ccl

(27)

h-0 y

By comparing (27) with (13), we identify the volume force,

F,,(x), for finite h as

V?(x)

F,,(x)= 04x1 ccl . (28)

III. NUMERICAL IMPLEMENTATION

A. Choice of the Color Function

For cases of incompressible flow, there is a natural

alternative to the definition of Z(a) by (16) in terms of an

interpolation function. Instead we set

S(x) = P(X)

(31)

at grid points, with p(x) derived from the evolution

equation (15). The volume force is still given by (28).

The transition region thickness is then of the order of the

grid spacing and, at points outside the transition region,

F(x) has the values p, , p2 in fluids 1, 2, respectively. The

interface between the fluids is given by the surface

P(xs)=th +Pz)‘<P).

One can multiply the integrand on the right side of (27)

by a function g(x) = Z(x)/(c) without changing the value

of the integral in the limit h + 0, since at the interface

x =x, and g(xS) = 1. If, for incompressible flow, we use

E(x) = p(x), then g(x) is given by

g(x) = P(X)/(P),

(32)

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