Towards front-tracking based on conservation in two space dimensions III, tracking interfaces

Abstract

This is the third paper in the series of our conservative front-tracking method. In this paper, we describe how our method tracks fluid interfaces in multi-fluid flows. Two important ingredients in our conservative front-tracking method in tracking fluid interfaces are: (1) the velocities and pressures of the left and right cell-averages in a discontinuity cell are respectively maintained to be equal to each other, and in doing so the physics that the normal velocity and pressure are continuous cross the interface is simulated but that the tangential velocity may be discontinuous is ignored, and (2) a so-called numerical surface dissipation is designed on the tracked interface to eliminate possible numerical instability there, and we believe that this numerical surface dissipation is a good substitute for the missing physical dissipation acting on the interface. We then present numerical simulation of Haas–Sturtevant’s two shock–bubble interaction experiments [J.F. Haas, B. Sturtevant, Interaction of weak shock waves with cylindrical and spherical gas inhomogeneities, J. Fluid Mech. 181 (1987) 41–76] using this method. Our numerical results are in good agreement with the experimental and other numerical results in the early times of the flow. Moreover, our numerical results are also in good agreement with the experimental results in the later times of the flow and give clear pictures of the bubble deformation then, which show that the right boundaries of the bubble behave just as Rychtmyer–Meshkov instabilities with the shock coming from either heavy or light gases.

Introduction

This is the third paper in the series of our 2D conservative front-tracking method beginning with [37], and the topic discussed in this paper is the tracking of interfaces in compressible multi-fluid flows. As was shown in the previous two papers [36], [37], one of the major features of our method that distinguishes it from other tracking methods is that the evolution of a 2D discontinuity curve is locally described by 1D conservation laws, and the tracking of the curve is accomplished by numerically solving these 1D conservation laws in capturing fashion on 1D subgrids. In the following we are going to use a simple example to explain the idea.

Consider a curve driven by a solenoidal velocity field $(u(x\text{,}y)\text{,}v(x\text{,}y))$, i.e. $u$ and $v$ satisfy

$$\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}=0\text{.}$$

It is known that the solenoidal velocity field conserves the spatial volume. We are now going to see how the traditional front-tracking methods, see [7], [8], [9], [15], [16], [17], [32], [52], [56] and the papers cited therein, VOF methods, see [4], [5], [6], [20], [45], [46], [50] and the papers cited therein, level-set methods, see [2], [3], [11], [12], [40] and the papers cited therein, and our conservative front-tracking method track this moving curve.

Conceptionally speaking, the traditional front-tracking methods track the curve by dragging selected points on the curve, called markers, with the given velocity field to new positions, and then connect the new markers with line segments to form the curve at the new time level, see Fig. 1.1. Logical connectivity between neighboring markers is required in the description of the curve. The algorithms are usually complex, the volume conservation is not considered in the approach, and therefore it will be very difficult to preserve the volume conservation in the designed numerical schemes.

The VOF methods use a volume fraction or “color” function $\mathcal{C}(x\text{,}y\text{,}t)$ to describe the moving curve, which takes 1 on one side of the curve and 0 on the other side, see Fig. 1.1. The volume fraction $\mathcal{C}$ then satisfies the following volume conservation equation,

$$\frac{\partial \mathcal{C}}{\partial t}+\frac{\partial (u\mathcal{C})}{\partial x}+\frac{\partial (v\mathcal{C})}{\partial y}=0\text{,}$$

where the derivatives are in distribution sense. The VOF methods track the moving curve by numerically and conservatively solving (1.2) to compute the volume fraction $\mathcal{C}$, and then use the volume fraction to reconstruct the curve segments in the grid cells. In the approach, the volume conservation is preserved; actually, it is part of the tracking mechanism. Moreover, the logical connectivity of neighboring interface cells is not required. Nevertheless, the reconstruction of the curve is cumbersome, and the methods will have difficulty in resolving thin structures of the curve forming “bobbly” filaments.

The level-set methods also solve Eq. (1.2); however, $\mathcal{C}(x\text{,}y\text{,}t)$ is now a smooth function, called level-set function, with the curve represented by the isocontour $C=0$. Usually, the level-set function $\mathcal{C}$ is taken as the signed distance to the curve so that $|\nabla \mathcal{C}|=1$. The level-set method is probably the simplest among all the sharp interface methods discussed in this paper since they are purely front-capturing. Nevertheless, they do not maintain the volume conservation and also have difficulty in resolving thin structures losing (or gaining) volume.

It should be pointed out that in applications of all the above methods to track interfaces in fluid dynamics the external velocity field is provided as the fluid velocity; therefore, the interfaces are passively driven by fluid flows in the simulations.

Our conservative front-tracking method still adopts (1.2) as the fundamental description of the moving curve. However, rather than directly solving (1.2), our method solves some 1D conservation laws derived from (1.2). If in a local area the curve can be represented as a function $y=s(x\text{,}t)$, see Fig. 1.1, we integrate (1.2) in the y-direction from $y=a$ to $y=b$, where the points with $y=a$ and $y=b$ are on the two sides of the curve, and obtain, with the notation that derivatives are in distribution sense,

$$\frac{\partial s}{\partial t}+\frac{\partial F(s\text{,}x)}{\partial x}=v(x\text{,}a)\text{,}$$

where

$$F(s\text{,}x)={\int }_a^{s}u(x\text{,}y)\mathit{dy}\text{.}$$

Eq. (1.3) is not something new. If we take $\mathfrak{F}(x\text{,}y\text{,}t)=s(x\text{,}t)-y=0$ as the description of the moving curve in the $(x\text{,}y\text{,}t)$-space, then the normal of the surface $(\frac{\partial \mathfrak{F}}{\partial t}\text{,}\frac{\partial \mathfrak{F}}{\partial x}\text{,}\frac{\partial \mathfrak{F}}{\partial y})$ is orthogonal to the tangential of the surface $(1\text{,}u\text{,}v)$, i.e.

$$\left(\frac{\partial \mathfrak{F}}{\partial t}\text{,}\frac{\partial \mathfrak{F}}{\partial x}\text{,}\frac{\partial \mathfrak{F}}{\partial y}\right)·(1\text{,}u\text{,}v)=\frac{\partial s}{\partial t}+u(x\text{,}s)\frac{\partial s}{\partial x}-v(x\text{,}s)=0\text{.}$$

It is easy to verify that (1.3) with (1.4) is equivalent to (1.5). However, Eq. (1.3) with (1.4) is in a volume-conservative form.

Likewise, if in a local area the curve can be represented as a function $x=w(y\text{,}t)$, see also Fig. 1.1, we then integrate (1.2) in the x-direction from the points with $x=c$ to the points with $x=d$, and obtain

$$\frac{\partial w}{\partial t}+\frac{\partial G(w\text{,}y)}{\partial y}=u(c\text{,}y)\text{,}$$

with

$$G(w\text{,}y\text{,}t)={\int }_a^{w}v(x\text{,}y)\mathit{dx}\text{.}$$

Again, Eq. (1.6) with (1.7) is in a volume-conservative form and it is easy to see that it is equivalent to

$$\left(\frac{\partial \mathfrak{F}}{\partial t}\text{,}\frac{\partial \mathfrak{F}}{\partial x}\text{,}\frac{\partial \mathfrak{F}}{\partial y}\right)·(1\text{,}u\text{,}v)=\frac{\partial w}{\partial t}+v(w\text{,}y)\frac{\partial w}{\partial y}-u(w\text{,}y)=0\text{.}$$

It is obvious that a continuous curve can always be represented either as $y=s(x\text{,}t)$ or $x=w(y\text{,}t)$ in a local area; therefore, its evolution can always be described by either (1.3) with (1.4) or (1.6) with (1.7). Our conservative front-tracking method then numerically solves either (1.3) on x-sub-grid, with $y=a$ and $y=b$ properly chosen as the horizontal grid lines, or (1.6) on y-sub-grid, with $x=c$ and $x=d$ properly chosen as the vertical grid lines, to track the moving curve, see the discussion in Section 3.

As was seen above, Eqs. (1.3), (1.6) are both 1D conservation laws with source terms, with $s(x\text{,}t)$ and $w(y\text{,}t)$, respectively, the unknowns. There are plenty of well developed finite-volume schemes for numerically solving these equations. Moreover, since the curve is continuous, $s(x\text{,}t)$ and $w(y\text{,}t)$ are also continuous and they satisfy (1.3), (1.6) in classical sense; therefore, their numerical solutions do not involve jump discontinuities. In tracking the curve in this way, the volume conservation is preserved, which is actually also part of the tracking mechanism. Unlike the VOF methods, the reconstruction of curve and computation of volume fraction are now well organized in a unified 1D conservative numerical scheme for solving either (1.3), (1.6); the reconstruction of curve is then never cumbersome.

Like the traditional front-tracking methods, our conservative front-tracking method also requires logical connectivity between neighboring discontinuity cells in the description of the tracked curve, which makes the method very powerful, especially in resolving thin structures of the tracked curve with the help of some “ghost” techniques, and also showing a good possibility in resolving and tracking the interactions of discontinuities, the so-called “triple points”, see [36], [37], [38].

Conceptionally speaking, a 2D interface in a compressible multi-fluid flow is also a curve moving with the velocity field, though the velocity field is usually not solenoidal. However, our conservative front-tracking method does not use an advection equation of the volume fraction $\mathcal{C}$ to describe the interface evolution as the VOF or level-set methods do. Our method applies the previously described approach directly to the conservation laws of mass, momentums and total energy to obtain 1D conservation laws that locally describe the evolution of the interface. In doing so, the interface is tracked by the physical conservation of the fluid rather than passively driven by the fluid velocity, which is probably the basic character that distinguishes it from the other tracking methods. These 1D conservation laws are then numerically solved by 1D finite-volumes schemes using extrapolation data from the smooth region to track the interface. In doing so, all the physics on and across the interface can be well simulated; especially, the conservations of mass, momentum and total energy are all well maintained. Also in doing so, the 2D front-tracking becomes a 2D capturing plus a 1D capturing; the 2D capturing computes the numerical solution in smooth region, and the 1D capturing tracks the interface. The algorithm is thus much simpler than the traditional front-tracking methods.

To make the description of the method relatively complete and independent, we will review the method in some details in the following discussion to avoid many citations of the previously published works. The rest of the paper is organized as the follows:

In the second section, we briefly recall the Euler system for compressible fluids and the interface, and we then derive the 1D conservation laws with source terms that locally describe the evolution of the interface.

The third section is the main part of the paper in which we describe our conservative front-tracking method tracking interface in details. The section consists of subsections of “data structure of numerical solution”, “computation of solution in smooth region” and “front tracking”. The first and third subsections also consist of, respectively, several sub-subsections to describe the different data types and different phases of the tracking algorithm. There are two important features of the tracking algorithm: (1) The velocities and pressures of the left and right cell-averages in a discontinuity cell, a cell crossed by the interface, will respectively be maintained to be equal to each other. In doing so, the physics that the normal velocity and pressure are continuous cross the interface is simulated, but that the tangential velocity may be discontinuous is ignored. (2) A so-called “numerical surface dissipation” is designed on the tracked interface to eliminate possible numerical instability on the interface. This numerical surface dissipation is a good substitute of the missing physical dissipation on the tracked interface to smooth and stabilize it. Our numerical simulations show that the numerical surface dissipation eliminates the numerical artifacts that may be presented in other numerical simulations.

We have done quite a lot of numerical experiments for the verification and validation of our conservative front-tracking method, see the previously referenced papers and also some more recent publications [54], [55]. To further verify and validate the method, and also to show its efficiency and effectiveness, we present in the fourth section our numerical simulations of Haas–Sturtevant’s two shock–bubble interaction experiments using this method. Our numerical results are in good agreement with the experimental results and other numerical results in the early times of the flows. Moreover, our numerical results are also in good agreement with the flow in the later times of the flow and give clear pictures of the bubble deformation then, which show that the right boundaries of the bubble behave just as Rychtmyer–Meshkov (RM) instabilities with the shock coming from either heavy or light gases. Finally, the fifth section is the conclusion.