A Diffuse Interface Model with Immiscibility Preservation

Abstract

A new, simple, and computationally efficient interface capturing scheme based on a diffuse interface approach is presented for simulation of compressible multiphase flows. Multi-fluid interfaces are represented using field variables (interface functions) with associated transport equations that are augmented, with respect to an established formulation, to enforce a selected interface thickness. The resulting interface region can be set just thick enough to be resolved by the underlying mesh and numerical method, yet thin enough to provide an efficient model for dynamics of well-resolved scales. A key advance in the present method is that the interface regularization is asymptotically compatible with the thermodynamic mixture laws of the mixture model upon which it is constructed. It incorporates first-order pressure and velocity non-equilibrium effects while preserving interface conditions for equilibrium flows, even within the thin diffused mixture region. We first quantify the improved convergence of this formulation in some widely used one-dimensional configurations, then show that it enables fundamentally better simulations of bubble dynamics. Demonstrations include both a spherical bubble collapse, which is shown to maintain excellent symmetry despite the Cartesian mesh, and a jetting bubble collapse adjacent a wall. Comparisons show that without the new formulation the jet is suppressed by numerical diffusion leading to qualitatively incorrect results.

1. Introduction

Aspherical bubble dynamics has been studied extensively using boundary integral numerical method [1-9]. Since this formulation is for incompressible fluids, it must therefore be supplemented, when feasible, with additional models whenever compressibility effects are important. For example, in violent collapses, artificial energy extraction is needed to account for the substantial energy lost to acoustic emission [9]. In bubble-cluster collapse, it is known that these same pressure pulses affect the dynamics of nearby bubbles [10, 11]. This limitation motivates the development of methods that explicitly include compressibility. Unfortunately, it is often difficult for compressible flow formulations to ensure that the interface between the liquid and the gas remains realistically sharp, especially given the dissipation inherent in shock-capturing schemes. A challenge in any multiphase compressible numerical method is the simultaneous and faithful representation of both shocks and interfaces.

One class of compressible multiphase flow approaches is Lagrangian, which includes arbitrary Lagrangian–Eulerian [12], free-Lagrange [13, 14], and front tracking methods [15-18]. These maintain sharp interfaces via explicit representation, but they introduce geometric complexity for large deformations and topological changes. A level-set [19, 20] based Eulerian approach coupled with the ghost fluid technique mitigates topological difficulties [21, 22]. However, the method requires special thermodynamic management at the interface for flows with high density or pressure ratios [23-31]. Another way to cope with the geometric challenges of Lagrangian tracking is to allow the interface to artificially diffuse into thin zones where multiple fluids overlap, forming in effect a mixture region. Multiphase flow theory is invoked to define thermodynamic variables in these mixture zones [32-42]. To be consistent with the nominally discontinuous interface of an actual immiscible fluid, these zones should be thinner than the features of the flow. In the well-known “single-fluid” model, this description is implemented by augmenting the Euler equations with equations that describe the evolution of smooth functions that mark different fluids [43-49].

There are two main challenges specific to the diffuse interface approach: (1) obtaining consistent thermodynamic laws for the mixture, and (2) preventing artificial spatial distortions of the interface functions. These two challenges are coupled as we demonstrate for the bubble collapse configuration in § 2. The second of these challenges is the more obvious. Without correction, distortions make the interface either too sharp to be represented on the mesh or so diffuse that it is no longer thin relative to other relevant flow features. Furthermore, it is also widely recognized that inconsistent interface thermodynamics can lead to mechanical incompatibilities. Mixture models obtained by asymptotic reduction of the Baer–Nunziato non-equilibrium multiphase model [32, 33] under stiff mechanical relaxation [38, 40, 41] provide transport of interface functions consistent with first-order pressure and velocity non-equilibrium effects. Incorporation of first-order non-equilibrium effects constitutes a significant improvement, but the interfaces can still become so diffused and distorted without proper numerical regularization that key flow features can be completely lost. We will show in § 2 and § 5 that this level of thermodynamic consistency alone is insufficient, particularly when strain smears the interface, as for a near-wall bubble collapse. Such distortions can be countered with specially designed terms in the governing equations that cause the interface function transition to span an approximately constant-thickness zone, as proposed recently by Shukla et al.[50] for compressible flows. Unfortunately, these source terms are not necessarily compatible with the thermodynamic mixture models in the interface zone, which can lead to accumulation of errors in space and time, even far from the interface. So et al.[51] have recently developed an anti-diffusive method to address this problem. This technique, however, is intimately tied to the underlying numerical scheme and it is therefore difficult to generalize to different discretizations, such as an increase in the order of accuracy. There is also the risk that anti-diffusive fluxes can over-sharpen the interface in flow regions already drawn thin by the resolved strain field (discussed in § 3 and § 5.4). Alternatively, Kokh and Lagoutière [52] have shown encouraging results with an anti-diffusive Lagrange-Remap scheme that controls numerical diffusion via limited downwind fluxes in the remap stage. The method, however, requires additional steps to compute, and remap from, Lagrangian variables, and its extension to more general interfacial mixture laws has not been explored yet.

In this paper, we present a new mixture-consistent interface regularization approach that addresses the limitations identified in the previous discussion. It maintains the integrity of the thin interface between immiscible fluids with a sharpening term, but this term is now crafted in such as way that it remains consistent with the first-order non-equilibrium mixture model. This regularization operator is incorporated into the continuum model, making the overall model independent of the numerical scheme employed. For example, it is easily demonstrated on a standard shock-tube problem in one dimension, and is also directly compatible with a multi-resolution adaptive mesh refinement (AMR) implementation and a fifth-order WENO scheme we use to demonstrate its properties for three-dimensional bubble collapse.

2. Motivation: mixture-zone models and discretization

Faced with the limited availability of exact solutions that can be used for comparison and analysis of our method, we frame our initial discussion on a free-space spherical-bubble collapse in a compressible liquid. In the weak compressibility limit, this configuration has a semi-analytic solution, obtained upon solving an ordinary differential equation [53, 54], which we treat as nominally “exact”. Though one dimensional in the radial coordinate, the configuration is advantageous in that it can be used to evaluate simulations on a three-dimensional mesh, and thereby illustrate key features of methods that will be important for complex-geometry applications. If a method is unable to do this, there is little hope that more complex flows can be well approximated by such a computational model. We therefore use it to motivate the development of the proposed method, which we present in detail in § 3.

In all the simulations, water is modeled with a stiffened-gas equation of state (see (10) in § 3 where we present the formulation in detail) using γ = 4.4 and p^∞ = 600 MPa and air is modeled as an ideal gas with γ = 1.4 (p^∞ = 0 MPa). The simulation domain is a 40 × 40 × 40 mm³ cube with a bubble of initial radius R₀ = 1 mm at its center. The bubble is discretized with a mesh of minimum spacing Δx_min such that R̄_min/Δx_min ≈ 13.9, where R̄_min is its minimum radius over the entire simulated time. Details of the initial conditions are provided in § 5.2, where we present one-dimensional (radial coordinate) simulation results for this problem. Here, we compare six approaches on the same three-dimensional mesh:

• Ã)Interface capturing based upon an established equilibrium mixture model without any interface regularization [39, 46-49];
• A)The model of Shukla et al. [50], which employs the two-fluid equilibrium model of Allaire et al. [39] and the interface regularization first proposed by Olsson & Kreiss [55] for incompressible flows;
• B)The method of So et al. [51], which employs anti-diffusive fluxes to sharpen the interface;
• B̃)Method B without the anti-diffusive fluxes;
• C̃)The asymptotically reduced model of Kapila et al. [38, 40], which includes first-order non-equilibrium effects without any interface regularization; and
• C)The new model, presented in detail in § 3, which incorporates an interface regularization in a similar spirit to that we used previously [50], but is now consistent with first-order non-equilibrium effects [38, 40] in the mixture zone. This demonstration will show how advantageous such consistency can be;

The numerical method used for solving models Ã, A, C̃ and C is detailed in § 4. Since the anti-diffusion approach depends on the diffusion of the underlying solver, the numerical solver used for methods B̃ and B is the same as that employed by So et al. [51]. (Method B̃ is included in the analysis only to highlight the effect of anti-diffusion.) Figure 1a compares the radius of the bubble from three-dimensional simulations with the Keller–Miksis (KM) [53] solution. The equivalent radius R̄(t) is defined in terms of the gas volume fraction α₂, according to

$$\frac{4}{3}\pi {\overline{R}}^3(t)=\mathit{\iiint }{\alpha }_2(\mathit{x},t)d\mathit{x},$$ (1)

where x = {x, y, z}.

Figure 1.

Evolution of (a) radius R̄(t) defined in (1) and (b) thickness of the two-fluid interface d̄(t) defined in (2) relative to the initial thickness using models Ã(◇), A(Δ), B̃(▷), B(◁), C̃(□), and C(○) for isolated single-bubble collapse under a pressure ratio of 10. The results are compared with the semi-analytic KM [53] solution (—) in (a), and the desired constant interface thickness with exact immiscibility (—) in (b).

The equilibrium model à and our previous mixture-inconsistent model A both lead to rebound significantly earlier compared to the reference solution. In both cases, the assumption of total thermodynamic equilibrium at the interface increases the internal bubble pressure at an erroneously faster rate. Models C̃ and C are far more accurate, providing a fundamentally better minimum radius and collapse time predictions on the same mesh as Ã, A, B̃ and B.

To assess the interface preservation property of different models, we compute the thickness

$$\overline{d}(t)=\frac{1}{\overline{S}(t)}\mathit{\iiint }{\alpha }_2(\mathit{x},t)[1-{\alpha }_2(\mathit{x},t)]d\mathit{x},$$ (2)

where S̄(t) is the bubble surface area

$$\overline{S}(t)=\mathit{\iiint }|\nabla {\alpha }_2(\mathit{x},t)|d\mathit{x}.$$ (3)

This definition yields the exact thickness d if the volume fraction obeys

$${\alpha }_2=\frac{1}{2}\left[1-\tanh \left(\frac{r-R}{2d}\right)\right],$$ (4)

where R is the radial coordinate at which α₂ = 0.5. As can be seen in figure 1b, method B̃ is the most diffusive; the anti-diffusive fluxes in method B reduce this. The regularization operator in our model C has a user-defined mesh-independent length-scale that governs the thickness of the interface. By construction, such a length-scale does not exist in the anti-diffusion approach. The thickness of the sharpened interface using B, therefore, does not remain fixed as it is determined by the dynamics of the flow.

Compared to A and C, models à and C̃ significantly thicken the interface during collapse, with peak thickness smeared by more than a factor of two around the time of minimum volume. To maintain a thin interface relative to the corresponding interface features in a complex three-dimensional configuration, this would necessitate 8 times the mesh, and this situation would degrade further as the interface diffuses further throughout the course of a simulation. Models A and C, which use regularization to ensure the thickness is approximately constant, do not show this behavior. Model A, which does not respect self-consistent mixture rules, is modestly better than C in preserving fluid immiscibility, but is far inferior for the generally more important bubble radius history R̄(t) in figure 1a. We also note in figure 1a that the radius history obtained using model C agrees well with that obtained using C̃ upon which it is built. Therefore, our consistent interface regularization preserves large-scale features (associated with the bubble radius R̄) while better resolving small-scale features (associated with the interface thickness d̄). Such an agreement is clearly not observed for the interface sharpening methods A and B. It is thus clear that the proposed model C is advantageous for spherical bubble collapse on a three-dimensional Cartesian grid, a model for any challenging complex simulation on a non-interface-conforming mesh. In making these comparisons for bubble collapse and rebound, we recognize that different methods are better suited to meet the diverse challenges presented by different classes of problems. There is no expectation that any method will be universally superior.

Since the new model C, which is formulated in the following section, is a continuum model, it is compatible with different underlying discretizations. Figure 2 shows how the better resolution properties of a fifth-order WENO scheme, as an example, is advantageous despite its nominally equivalent low-order behavior at interfaces. We see in figures 2a and 2b that it is far more effective at maintaining the spherical symmetry of the bubble, which is confirmed quantitatively in figure 2c, which shows the behavior of spherical harmonics. For spherical harmonics $Y_l^m$ which satisfy

$$\frac{1}{4\pi }\underset{0}{\overset{2\pi }{\mathit{\int }}}\underset{0}{\overset{\pi }{\mathit{\int }}}{Y}_l^m(\theta ,\varphi )Y_{l^{\prime }}^{m^{\prime }}(\theta ,\varphi )\sin \theta \mathrm{d}\theta \mathrm{d}\varphi ={\delta }_{{\mathit{ll}}^{\prime }}{\delta }_{{\mathit{mm}}^{\prime }},$$ (5)

a Parseval equality provides a convenient measure of the spectrum,

$${\zeta }_l^2=\sum _{m=-l}^l{r}_{\mathit{lm}}^2,$$ (6)

where l and m represents degree and order of the spherical harmonic function, respectively, ζ_l denotes the power in mode l, and r_lm represents the coefficients corresponding to $Y_l^m$. To compute ζ_l, we first obtain the bubble radius R̂(θ ϕ) defined by α₂ = 0.5 from the three-dimensional simulations, then compute the coefficients r_lm using the SPHEREPACK library [56]. The clear superiority of fifth-order WENO will also benefit it in more complex configurations. The main advantage of the high-order method is that it has a much lower grid anisotropy error and therefore lowers the unavoidable directionality introduced by the Cartesian mesh. The more accurate (spherically symmetric) flow fields around the bubble yield a more accurate radius history, as seen in the inset of figure 2d. Though these errors are small compared to the large errors seen for the different formulations in figure 1a, the WENO scheme provides a significantly more accurate representation of the bubble radius.

Figure 2.

Bubble shapes obtained from model D using (a) second-order minmod and (b) fifth-order WENO at t = 110 μs for isolated single-bubble collapse under a pressure ratio of 10; (c) evolution of spherical harmonics power-spectrum for l = 1 (Δ), l = 2 (∇), l = 3 (▷), l = 4 (□), l = 5 (◁) and l = 6 (○); and (d) radius history using second-order minmod (○) and fifth-order WENO (□), compared with the Keller–Miksis [53] solution (—).

3. Model formulation

The two-fluid model is developed starting with a variant [33] of the Baer–Nunziato model [32]. It assumes neither pressure nor velocity equilibrium at the interface, with separate conservation of mass, momentum, and energy equations for each fluid, along with an equation for the evolution of volume fraction (interface function) of one of the two fluids. We present the seven-equation model [33, 40], in terms of entropy (s_k), velocity (u_k), pressure (p_k), and volume fraction (α_k) of fluids k = 1 and 2,

$$\begin{array}{lll}{\alpha }_1{\rho }_1{T}_1\frac{{\mathrm{D}}_1{s}_1}{\mathrm{D}t}& =& (p_{\mathrm{I}}-p_1)({\mathbf{u}}_{\mathrm{I}}-{\mathbf{u}}_1)·\nabla {\alpha }_1+\mu (p_{\mathrm{I}}-p_1)(p_2-p_1)\\ & & +\mathrm{\lambda }({\mathbf{u}}_{\mathrm{I}}-{\mathbf{u}}_1)·({\mathbf{u}}_2-{\mathbf{u}}_1),\end{array}$$ (7a)

$$\begin{array}{lll}{\alpha }_2{\rho }_2{T}_2\frac{{\mathrm{D}}_2{s}_2}{\mathrm{D}t}& =& (p_{\mathrm{I}}-p_2)({\mathbf{u}}_{\mathrm{I}}-{\mathbf{u}}_2)·\nabla {\alpha }_2+\mu (p_{\mathrm{I}}-p_2)(p_1-p_2)\\ & & +\mathrm{\lambda }({\mathbf{u}}_{\mathrm{I}}-{\mathbf{u}}_2)·({\mathbf{u}}_1-{\mathbf{u}}_2),\end{array}$$ (7b)

$${\alpha }_1{\rho }_1\frac{{\mathrm{D}}_1{\mathbf{u}}_1}{\mathrm{D}t}+\nabla ({\alpha }_1{p}_1)=p_{\mathrm{I}}\nabla {\alpha }_1+\mathrm{\lambda }({\mathbf{u}}_2-{\mathbf{u}}_1),$$ (7c)

$${\alpha }_2{\rho }_2\frac{{\mathrm{D}}_2{\mathbf{u}}_2}{\mathrm{D}t}+\nabla ({\alpha }_2{p}_2)=p_{\mathrm{I}}\nabla {\alpha }_2+\mathrm{\lambda }({\mathbf{u}}_1-{\mathbf{u}}_2),$$ (7d)

$$\begin{array}{lll}{\alpha }_1\frac{{\mathrm{D}}_1{p}_1}{\mathrm{D}t}+{\alpha }_1{\rho }_1{a}_1^2\nabla ·{\mathbf{u}}_1& =& {\rho }_1{a}_{1\mathrm{I}}^2({\mathbf{u}}_{\mathrm{I}}-{\mathbf{u}}_1)·\nabla {\alpha }_1+\mu {\rho }_1{a}_{1\mathrm{I}}^2(p_2-p_1)\\ & & +\mathrm{\lambda }{\sigma }_1({\mathbf{u}}_{\mathrm{I}}-{\mathbf{u}}_1)·({\mathbf{u}}_2-{\mathbf{u}}_1),\end{array}$$ (7e)

$$\begin{array}{lll}{\alpha }_2\frac{{\mathrm{D}}_2{p}_2}{\mathrm{D}t}+{\alpha }_2{\rho }_2{a}_2^2\nabla ·{\mathbf{u}}_2& =& {\rho }_2{a}_{2\mathrm{I}}^2({\mathbf{u}}_{\mathrm{I}}-{\mathbf{u}}_2)·\nabla {\alpha }_2+\mu {\rho }_2{a}_{2\mathrm{I}}^2(p_1-p_2)\\ & & +\mathrm{\lambda }{\sigma }_2({\mathbf{u}}_{\mathrm{I}}-{\mathbf{u}}_2)·({\mathbf{u}}_1-{\mathbf{u}}_2),\end{array}$$ (7f)

$$\frac{{\mathrm{D}}_{\mathrm{I}}{\alpha }_2}{\mathrm{D}t}=\mu (p_2-p_1)+R({\alpha }_2).$$ (7g)

The new term R added to (7g) is a user specified regularization operator that sets the interface thickness; our specific R is defined in (20). The subscript I, appearing in the right-hand side of several of the equations, indicates an interface quantity. At this point, the interface velocity u_I and pressure p_I are retained as distinct variables. Material derivatives for each fluid and the interface are

$$\frac{{\mathrm{D}}_l}{\mathrm{D}t}=\frac{\partial }{\partial t}+{\mathbf{u}}_l·\nabla ,$$ (8)

for l = 1, 2 and I. The fluid densities, temperatures and sound speeds are ρ_k, T_k and a_k, respectively. The mass-specific total energy is E_k = e_k +|u_k|²/2, where e_k is the mass-specific internal energy. The interfacial variables (u_I and p_I) are, in general, obtained by weighted averaging of the respective quantities on either side of the interface [33], but the choice of averaging is not explicit in our formulation after we reduce the model to equilibrium velocity and pressure at the interface. The parameters λ and μ are the rates at which velocity and pressure relax to thermodynamic equilibrium, respectively, and

$${\sigma }_k=\frac{1}{{\rho }_k}{\left(\frac{\partial p_k}{\partial e_k}\right)}_{{\rho }_k}.$$ (9)

The modified seven-equation non-equilibrium model (7), including the new R in (7g), is reduced consistently to the five-equation model under stiff mechanical relaxation. The assumption underlying this procedure is that the relaxation times for the velocities and pressures of the two fluids to reach equilibrium are small compared to λ and μ → ∞. The procedure is lengthy, but straightforward; details are included in Appendix A. The reduced system (A.16) incorporates first-order mechanical non-equilibrium effects as well as the new, now thermodynamically self-consistent, interface regularization (Appendix A).

The analysis in Appendix A provides a consistent way of regularizing all the conservative variables by incorporating R in the continuum formulation, which distinguishes our model from the interface sharpening approaches of Shukla et al. [50] and So et al. [51]. In the approach of Shukla et al. [50], the density and interface function are sharpened first, then momentum and energy are discretely updated such that velocity and pressure in each cell remain unchanged. In the anti-diffusion method of So et al. [51], discrete sharpening fluxes for the conservative variables are obtained using the discrete anti-diffusive fluxes for volume fraction such that pressure and velocity equilibrium is maintained in each cell. Thermodynamic self-consistency guarantees that our continuum formulation (A.16), by construction, conserves specific entropies (s_k) of each phase under isentropic conditions along with preserving velocity and pressure equilibrium. The consistency of this regularization is shown in § 2.

The asymptotic analysis that unifies this regularization is carried out for fluids whose states are fixed by two thermodynamic variables and is therefore broadly applicable. The stiffened gas equation of state [57] provides a good model for our bubble dynamics application:

$${\rho }_k{e}_k={\mathrm{\Gamma }}_k{p}_k+{\mathrm{\Pi }}_k,$$ (10)

with

$${\mathrm{\Pi }}_k=\frac{{\gamma }_k{p}_k^{\infty }}{{\gamma }_k-1},$$ (11)

$${\mathrm{\Gamma }}_k=\frac{1}{{\gamma }_k-1},$$ (12)

where γ_k and $p_k^{\infty }$ are the usual stiffened gas parameters [57]. For this equation of state, the reduced system with consistent regularization (A.16) becomes

$$\frac{\partial {\alpha }_1{\rho }_1}{\partial t}+\nabla ·({\alpha }_1{\rho }_1\mathbf{u})=-{\rho }_{1}R({\alpha }_2),$$ (13a)

$$\frac{\partial {\alpha }_2{\rho }_2}{\partial t}+\nabla ·({\alpha }_2{\rho }_2\mathbf{u})={\rho }_{2}R({\alpha }_2),$$ (13b)

$$\frac{\partial \rho \mathbf{u}}{\partial t}+\nabla ·(\rho \mathbf{u}\otimes \mathbf{u})+\nabla p=({\rho }_2-{\rho }_1)\mathbf{u}R({\alpha }_2),$$ (13c)

$$\frac{\partial \rho E}{\partial t}+\nabla ·((\rho E+p)\mathbf{u})=(({\rho }_2-{\rho }_1)\kappa +p({\mathrm{\Gamma }}_2-{\mathrm{\Gamma }}_1)+{\mathrm{\Pi }}_2-{\mathrm{\Pi }}_1)R({\alpha }_2),$$ (13d)

$$\frac{\partial {\alpha }_2}{\partial t}+\mathbf{u}·\nabla {\alpha }_2={\alpha }_1{\alpha }_2\frac{{\rho }_1{a}_1^2-{\rho }_2{a}_2^2}{{\alpha }_1{\rho }_2{a}_2^2+{\alpha }_2{\rho }_1{a}_1^2}\nabla ·\mathbf{u}+R({\alpha }_2),$$ (13e)

where u and p are the equilibrium velocity and pressure (Appendix A),

$${\alpha }_1+{\alpha }_2=1,$$ (14)

$$\rho ={\alpha }_1{\rho }_1+{\alpha }_2{\rho }_2,$$ (15)

$$\mathit{\rho E}={\alpha }_1{\rho }_1{E}_1+{\alpha }_2{\rho }_2{E}_2,$$ (16)

$$\kappa =\frac{1}{2}|\mathbf{u}{|}^2.$$ (17)

The mixture speed of sound a is obtained using

$$\frac{1}{\rho a^2}=\frac{{\alpha }_1}{{\rho }_1{a}_1^2}+\frac{{\alpha }_2}{{\rho }_2{a}_2^2},$$ (18)

which reflects the eigenvalues of the hyperbolic part of the reduced system, (A.16). Any other averaging to obtain the mixture sound speed (frozen sound speed, for example) would be inconsistent with (A.16). For stiffened gases, the phasic sound speeds (a_k) are given by

$$a_k=\sqrt{{\gamma }_k\frac{p+p_k^{\infty }}{{\rho }_k}}.$$ (19)

The specification of R completes the system. The goal is to represent the interface with a smeared Heaviside function of constant and uniform thickness throughout the simulation. For incompressible multifluid flows, Olsson et al. [55, 58] used the hyperbolic tangent as the smeared Heaviside function. For compressible flows, Shukla et al. [50] proposed a modified form that has been shown to work well for the diffuse interface methods based on the advection model [39]. We use a similar operator in this study for the asymptotically reduced model,

$$R({\alpha }_2)=L({\alpha }_2)U_0\mathbf{n}·\nabla (\epsilon |\nabla {\alpha }_2|-{\alpha }_2(1-{\alpha }_2)),$$ (20)

where the balance of interface sharpening (α₂(1 − α₂)) and diffusion (ε|∇α₂|) fluxes maintains the interface thickness. The parameter ε, which is of the order of the grid spacing but controlled explicitly by the user, defines the thickness of the interface, and U₀ represents the characteristic regularization rate. The interface normal,

$$\mathbf{n}=\nabla {\alpha }_2/|\nabla {\alpha }_2|,$$ (21)

directs the operation across the interface, and

$$L({\alpha }_2)=\{\begin{array}{ll}1& \mathrm{for}{10}^{-6}<{\alpha }_2<1-{10}^{-6}\\ 0& \mathrm{otherwise}\end{array}.$$ (22)

Since the distortions of the interface function field are largely driven by the velocity field near the interface, it is convenient to choose U₀ = ∥u_I∥_max. We obtain an estimate of ∥u_I∥_max using:

$${\Vert {\mathbf{u}}_{\mathrm{I}}\Vert }_{\max }=\frac{{({\alpha }_2(1-{\alpha }_2)|\mathbf{u}|)}_{\max }}{{({\alpha }_2(1-{\alpha }_2))}_{\max }}=4{({\alpha }_2(1-{\alpha }_2)|\mathbf{u}|)}_{\max }.$$ (23)

Having ε as an independent length scale is advantageous. The terms u·∇α₂ and β∇·u in (A.16e) both can thicken or thin the interface. While regularization schemes are usually designed with interface sharpening in mind to counter the smearing of numerical diffusion, a purely sharpening term does not prevent over-thinning the interface when it works in concert with an interface-thinning strain. In the absence of such a length scale, especially if there is complex flow, strain can disrupt the balance between the numerical diffusion and the artificial anti-diffusion. Relying on the mesh size rather than ε is also ill-suited for AMR-WENO, because the usual refinement criteria are predicated on the mesh-independence of the underlying solution.

It should also be noted that computation of phase densities for the right-hand side of (13) using the standard formula ρ_k = α_kρ_k/α_k could lead to spurious oscillatory behavior near the interface where both α_kρ_k and α_k have large gradients. To make the formulation more robust, we consider the assumption that phase densities ρ_k, though not of course the mixture density ρ, vary slowly across the interface in the computation of regularization terms

$$\frac{\partial {\rho }_k}{\partial n}\approx 0.$$ (24)

With this approximation, we avoid the explicit computation of ρ_k in the right-hand side of (13). To this end, we first take the gradient of ρ_kα_k,

$$\nabla ({\rho }_k{\alpha }_k)={\rho }_k\nabla {\alpha }_k+{\alpha }_k\nabla {\rho }_k,$$ (25)

then, using (14) and (24), we obtain

$$\mathbf{n}·\nabla ({\rho }_2{\alpha }_2)\approx {\rho }_2\mathbf{n}·\nabla {\alpha }_2,$$ (26a)

$$\mathbf{n}·\nabla ({\rho }_1{\alpha }_1)\approx -{\rho }_1\mathbf{n}·\nabla {\alpha }_2.$$ (26b)

Subsequently, we use (21) and (26) to develop an approximation for the first term of ρ_kR(α₂) in (13a) and (13b)

$${\rho }_2|\nabla {\alpha }_2|\approx \mathbf{n}·\nabla ({\rho }_2{\alpha }_2),$$ (27a)

$$-{\rho }_1|\nabla {\alpha }_2|\approx \mathbf{n}·\nabla ({\rho }_1{\alpha }_1),$$ (27b)

and for the second term

$${\rho }_k\mathbf{n}·\nabla ({\alpha }_k(1-{\alpha }_k))\approx (1-2{\alpha }_k)\mathbf{n}·\nabla ({\rho }_k{\alpha }_k).$$ (28)

From (27) and (28), we get

$$\begin{array}{lll}{\rho }_{2}R({\alpha }_2)& \approx & L({\alpha }_2)U_0\mathbf{n}·(\nabla (\epsilon \mathbf{n}·\nabla ({\rho }_2{\alpha }_2))-(1-2{\alpha }_2)\nabla ({\rho }_2{\alpha }_2))\\ & \equiv & {\widehat{R}}_2({\rho }_2{\alpha }_2,{\alpha }_2),\end{array}$$ (29a)

$$\begin{array}{lll}-{\rho }_{1}R({\alpha }_2)& \approx & L({\alpha }_2)U_0\mathbf{n}·(\nabla (\epsilon \mathbf{n}·\nabla ({\rho }_1{\alpha }_1))-(1-2{\alpha }_2)\nabla ({\rho }_1{\alpha }_1))\\ & \equiv & {\widehat{R}}_1({\rho }_1{\alpha }_1,{\alpha }_2).\end{array}$$ (29b)

Substituting these R̂_k for ρ_kR in (13) yields the specific quasi-conservative system we use:

$$\frac{\partial {\rho }_1{\alpha }_1}{\partial t}+\nabla ·({\rho }_1{\alpha }_1\mathbf{u})={\widehat{R}}_1,$$ (30a)

$$\frac{\partial {\rho }_2{\alpha }_2}{\partial t}+\nabla ·({\rho }_2{\alpha }_2\mathbf{u})={\widehat{R}}_2,$$ (30b)

$$\frac{\partial \rho \mathbf{u}}{\partial t}+\nabla ·(\rho \mathbf{u}\otimes \mathbf{u})+\nabla p=\mathbf{u}\widehat{R},$$ (30c)

$$\frac{\partial \mathit{\rho E}}{\partial t}+\nabla ·((\mathit{\rho E}+p)\mathbf{u})=\kappa \widehat{R}+(p({\mathrm{\Gamma }}_2-{\mathrm{\Gamma }}_1)+{\mathrm{\Pi }}_2-{\mathrm{\Pi }}_1)R,$$ (30d)

$$\frac{\partial {\alpha }_2}{\partial t}+\mathbf{u}·\nabla {\alpha }_2={\alpha }_1{\alpha }_2\frac{{\rho }_1{a}_1^2-{\rho }_2{a}_2^2}{{\alpha }_1{\rho }_2{a}_2^2+{\alpha }_2{\rho }_1{a}_1^2}\nabla ·\mathbf{u}+R,$$ (30e)

where R̂ = R̂₁ + R̂₂.

4. Numerical method

The governing equations are discretized on a fixed Cartesian mesh with an established block-structured Adaptive Mesh Refinement (AMR) [59, 60]. To achieve maximum resolution near the interface, refinement is based on the scaled gradient

$$\mathrm{\Delta }|\nabla {\alpha }_2|>\delta ,$$ (31)

where Δ is the geometrically averaged grid spacing. We take δ = 10⁻⁶ in the present simulations. When (31) is satisfied, the grid is refined an additional level and the solution is interpolated to the new grid.

The compressible flow equations are solved using the finite-volume method with the well-known reconstruct-evolve-average (REA) technique [61, 62]. A total variation diminishing third-order Runge–Kutta method [63] is used for time integration. The reduced system (30) is not exactly conservative in divergence form due to the advective form of the volume fraction equation and the regularization terms. In the presence of discontinuities, the numerical solution of a non-conservative hyperbolic system poses computational difficulties owing to the lack of generalized weak solutions that accurately define the shock relations. Even with approximate jump conditions [64], it is difficult to develop a path-conservative numerical scheme that converges to the correct solution [65-68]. A first step toward developing a numerical method to solve (30e) is to recast it as

$$\frac{\partial {\alpha }_2}{\partial t}+\nabla ·({\alpha }_2\mathbf{u})=K\nabla ·\mathbf{u}+R({\alpha }_2),$$ (32)

$$K=\frac{{\alpha }_2{\rho }_1{a}_1^2}{{\alpha }_1{\rho }_2{a}_2^2+{\alpha }_2{\rho }_1{a}_1^2}.$$ (33)

The next step involves volume averaging of the governing system of equations, in which, we use the following approximation to numerically compute the right-hand side of (32),

$$\frac{1}{V_{\mathit{ijk}}}{\mathit{\int }}_{V_{\mathit{ijk}}}(K\nabla ·\mathbf{u})\mathrm{d}V\approx \frac{1}{V_{\mathit{ijk}}}{K}_{\mathit{ijk}}{\mathit{\int }}_{S_{\mathit{ijk}}}\mathbf{u}·\mathrm{d}\mathbf{S},$$ (34)

where subscript ijk refers to the average values at the corresponding computational cell; V and S denote volume and surface of the cell, respectively. It should be noted that (34) is at best second-order accurate near the interface. However, it does not alter the overall spatial accuracy of our method because the shock-capturing scheme will invariably reduce the method to sub-first-order accurate near sharp flow field transitions or discontinuities [61].

We use a centered second-order approximation to evaluate R in (30). The Riemann problem is solved approximately using the HLLC method [62, 69, 70], though this necessitates special care in the discrete evolution of the interface function to ensure compatibility with the Euler system [49]. Johnsen and Colonius [49] have proposed an adaptation of the HLLC Riemann solver to the volume fraction evolution for the diffuse interface method based on the advection model [39]. We use the equivalent adaptation for the asymptotically reduced model. For spatial reconstruction, we use a WENO method [71-73], but we observed only marginal improvement in results upon implementing the two-point (fourth-order accurate) Gaussian integration proposed by Titarev and Toro [74] for finite-volume WENO methods. Therefore, for computational efficiency, we use a fifth-order accurate WENO reconstruction without the Gaussian integration in this study. Reconstruction is performed on primitive variables to avoid spurious pressure oscillations at the interface [49]. We show in Appendix B that appropriate interface conditions for equilibrium flows are preserved with our method, despite the regularization.

5. Tests and validations

We first demonstrate convergence of our method on one-dimensional problems on one-dimensional meshes. This is followed by simulations of free-space and near-wall bubble collapses. A uniform Cartesian mesh is taken in all the simulations, with the regularization parameter ε = 0.75 Δx_min in (20), where Δx_min denotes the finest AMR mesh spacing. Characteristic based non-reflecting boundary conditions are applied at farfield boundaries [75]. The one-dimensional simulations are carried out without AMR, with extrapolation boundary conditions, and employing second-order spatial reconstruction using the minmod limiter. Three-dimensional simulations are performed with fifth-order WENO spatial reconstruction.

5.1. Air-helium shock-tube problem

We consider the initial condition [48]

$$(\rho ,u,p,{\alpha }_2)=\{\begin{array}{ll}(1,0,1,1)& \mathrm{for}0\le x\le 1,\\ (0.125,0,0.1,0)& \mathrm{otherwise},\end{array}$$

where α₂ denotes the air volume fraction. Air and helium are treated as ideal gases with γ = 1.4 and 1.6, respectively. The interface regularization is activated after the interface thickness as defined in (2) becomes greater than the grid spacing. Simulation results using 800 grid cells at t = 0.4 with and without R are compared with the exact solution in figure 3. The resolution study in Table 1 demonstrates better convergence of the new method.

Figure 3.

Simulation results for the air-helium shock-tube problem without (□) and with (○) R compared with the exact solution (—).

Table 1.

Convergence of air volume fraction (α₂) for the air-helium shock-tube problem. The error for any solution variable ϕ is defined by ${\sum }_i^n|{\varphi }_{i,\mathrm{simulation}}-{\varphi }_{i,\mathrm{exact}}|/n$, where n is the number of grid cells.

Grid cells	R = 0		With R
	Error	Order	Error	Order
200	1.10 × 10−2		6.80 × 10−3
400	7.14 × 10−3	0.62	3.80 × 10−3	0.84
800	4.48 × 10−3	0.67	1.70 × 10−3	1.16
1600	2.85 × 10−3	0.65	8.47 × 10−4	1.01

It should be noted that the volume fraction transport equation in the underlying model of Kapila et al. [38] is not in conservation form, which leads to non-conservative products on the right-hand side of the entire system in (A.16). Shukla et al. [50] showed that if the numerical implementation without R is conservative, then the density regularization is asymptotically mass conservative with mesh refinement provided it is done according to (A.16a) and (A.16b). The present model has analogous terms in the right hand side of the entire system, yielding asymptotic conservation to mixture mass, momentum and energy. This is demonstrated in Table 2, which shows better mass and energy conservation with R upon mesh refinement. We further show in Tables 3 and 4 that even though mixture mass and energy have better conservation without regularization, the numerical smearing hampers convergence to the correct solution of these quantities as well.

Table 2.

Conservation of mass (M ≡ ∫ ρ dV) and energy (ε ≡ ∫ ρE dV) for the air-helium shock-tube problem with Δ(·)≡ |(·)_initial − (·)_final|/(·)_initial.

Grid cells	R = 0		With R
	ΔM	Δε	ΔM	Δε
200	5.05 × 10−15	1.13 × 10−14	2.54 × 10−4	4.67 × 10−5
400	8.84 × 10−15	8.31 × 10−15	2.20 × 10−4	3.97 × 10−5
800	1.54 × 10−14	2.81 × 10−14	1.12 × 10−4	2.01 × 10−5
1600	2.54 × 10−14	2.39 × 10−14	5.67 × 10−5	1.02 × 10−5

Table 3.

Convergence of mixture density (ρ) for the air-helium shock-tube problem. (Error is defined in Table 1.)

Grid cells	R = 0		With R
	Error	Order	Error	Order
200	4.43 × 10−3		3.65 × 10−3
400	2.63 × 10−3	0.75	2.00 × 10−3	0.87
800	1.45 × 10−3	0.87	9.17 × 10−4	1.12
1600	8.54 × 10−4	0.76	4.70 × 10−4	0.96

Table 4.

Convergence of volume-specific total energy (ρE) for the air-helium shock-tube problem. (Error is defined in Table 1.)

Grid cells	R = 0		With R
	Error	Order	Error	Order
200	1.08 × 10−2		9.40 × 10−3
400	6.17 × 10−3	0.81	5.04 × 10−3	0.90
800	3.29 × 10−3	0.90	2.36 × 10−3	1.10
1600	1.88 × 10−3	0.81	1.20 × 10−3	0.97

5.2. Isolated single-bubble collapse: one-dimensional simulation

We consider the collapse of an air bubble in water under an initial pressure ratio p_∞/p₀ = 10, with the bubble at its maximum radius R₀ = 1 mm at t = 0. The simulation domain is large enough to avoid boundary effects: 0 ≤ r ≤ 160 × R₀. The initial pressure profile in water p(r) is obtained from the Rayleigh–Plesset model [10]:

$$\begin{array}{lll}\rho & =& {\rho }_1{\alpha }_1+{\rho }_2{\alpha }_2,\\ u& =& 0,\\ p& =& \{\begin{array}{ll}{p}_0& \mathrm{for}0\le r\le R_0,\\ p(r)=p_{\infty }+\frac{R_0}{r}(p_0-p_{\infty })& \mathrm{otherwise},\end{array}\\ {\alpha }_2& =& \frac{1}{2}\left[1-\tanh \left(\frac{r-R_0}{2\epsilon }\right)\right],\end{array}$$

where α₂ denotes the air volume fraction, ρ₁ = 1000 kg/m³, and ρ₂ = 0.19 kg/m³. Three-dimensional simulations for the same problem were discussed in § 2. Table 5 demonstrates improvement in convergence properties of the numerical scheme upon incorporating R.

Table 5.

Convergence of air volume fraction (α₂) for the isolated single bubble collapse at t = 100 μs. (Error is defined in Table 1.)

R0/Δr	R = 0		With R
	Error	Order	Error	Order
12.5	7.65 × 10−4		6.82 × 10−4
25	5.11 × 10−4	0.58	3.77 × 10−4	0.85
50	2.98 × 10−4	0.78	1.60 × 10−4	1.23
100	1.94 × 10−4	0.62	7.74 × 10−5	1.05

5.3. Isolated single-bubble collapse: three-dimensional simulation

We next consider the collapse of an air bubble in water under a pressure ratio p_∞/p₀ = 25, with R̄_min/Δx_min ≈ 7.3. Without AMR, the total mesh size required to maintain the finest resolution used in this simulation would have been 2048³ ≈ 86 × 10⁸, while the number of grid cells we required with AMR at t = 0 is about 7 × 10⁶. Figure 4 compares the radial profiles obtained from the simulation with the KM solution. The bubble shapes obtained from the overall numerical scheme with regularization are also shown in figure 4. Higher pressure ratios are, of course, more challenging both because the bubble becomes much smaller and the radiated acoustic energy is more important. Figure 5 shows a similar collapse, but with p_∞/p₀ = 100, using R̄_min/Δx_min ≈ 5.8. Both the average bubble-interior pressure and the radius history agree well with theoretical predictions despite a factor of 1.7 × 10⁵ change in the interior pressure. It is also clear that even the rebound is accurately predicted.

Figure 4.

(a) Radius profile comparison with the Keller–Miksis [53] solution (—), for isolated single-bubble collapse under a pressure ratio of 25. (b) Bubble shapes at various times during collapse (top) and re-expansion (bottom) obtained from our overall numerical scheme. The innermost and outermost surfaces correspond to the minimum and maximum volumes, respectively. Arrows indicate increasing time.

Figure 5.

Radius (a) and pressure (b) profiles obtained from the new model compared with the Keller–Miksis [53] solution (—), for isolated single-bubble collapse under a pressure ratio of 100.

5.4. Near-wall single-bubble collapse

Finally, we consider the expansion and collapse of a gas bubble whose initial center is d = 1.8 mm away from a perfectly rigid wall (figure 6). At t = 0, the bubble is at its minimum radius (R₀ = 0.5 mm) under an initial pressure of 50 MPa, while the ambient water is initially at a uniform pressure (p_∞) of 10 MPa. The wall breaks the spherical symmetry, and an equivalent radius R̄ = (3V/4π)^1/3 is used to discuss the results. For this test case, R₀/Δx_min ≈ 25.6. Figure 7 shows radial profile and volume fraction contours (at three successive minima) obtained from the simulations, showing the benefits of including R in the analysis. While the domain integrals of gas volume fraction agree, consistent with the spherical results, visualization shows that without the regularization term the interface is both diffused to the same scale as the bubble shape and, in places, thinner than its target thickness. This spurious behavior increases obviously during the course of the simulation.

Figure 6.

Schematic of the arrangement for the near-wall collapse.

Figure 7.

Radius history for R = 0 and the new model for the near-wall single-bubble expansion and collapse with an initial pressure ratio of 5. The insets show corresponding volume fraction contours (ranging from 0.1 to 0.9) at three successive minima.

The smearing of the volume fraction field suppresses the kinetic energy of the jet that penetrates the bubble, which can be anticipated to be important for predictions of damage [10, 11] (in marine applications) or injury [76] (in bio-medical ultrasound). To investigate this, we consider a test case with more asymmetry due to closer wall proximity. The bubble, in this case, has a smaller initial radius (0.25 mm), and is placed closer to the wall (1.1 mm), with a higher initial interior pressure (500 MPa). Figure 8 shows good equivalent radius agreement between the two models in terms of total gas volume fraction. However, without R, the distal surface is obviously smeared, with the consequence of a significantly weaker liquid jet (figure 9). This prevents the jet from fully penetrating the bubble. To quantify this, the spatial profile of normalized volumespecific kinetic energy (ρ|u|²/2p_∞) along the symmetry axis is plotted versus time in figure 10. It is also clear that increasing resolution shows apparent global convergence for the new method, whereas no convergence yet (though it might be expected to be the correct trend) for R = 0. It would be prohibitively expensive to converge the diffuse interface model.

Figure 8.

Radius history for R = 0 (- -) and the new model (—) for the near-wall single-bubble expansion and collapse for an initial pressure ratio of 50 with R₀/Δx_min ≈ 6.4, 12.8, 25.6, and 51.2.

Figure 9.

Colors of normalized specific kinetic energy, and volume fraction contours (lines). Volume fraction contours range from 0.05 to 0.95; kinetic energy contours range from 1 to 13 in (a) and (c), and 1 to 9 in (b) and (d).

Figure 10.

Normalized specific kinetic energy along the axis of symmetry (the x-axis) at t = 22.2 μs for R = 0 (dashed) and the new model (solid) with R₀/Δx_min ≈ 6.4, 12.8, 25.6, and 51.2. (x₀ is the x-coordinate of the bubble center at t = 0.)

The stronger and steeper jet, obtained from the new model with regularization, penetrates though the bubble, and upon impact on the proximal bubble surface emits a shock wave. To visualize this phenomenon, figure 11 shows contours of ψ [45]:

$$\psi =\exp \left(-50\frac{|\nabla \rho |}{|\nabla \rho {|}_{\max }}\right).$$ (36)

Figure 11.

Contours of ψ in (36) [45] for R = 0 and the new model using R₀/Δx_min ≈ 51.2.

In figure 11, from left to right, the jet hits the liquid layer between the bubble and the wall in the first contour plot, the emitted shock wave reaches the wall in the second one, and the reflected wave interacts with the bubble in the third subfigure. The model without regularization fails to capture this phenomenon.

It is clear that a sharper and steeper jet kinetic energy will lead to a more focussed impact with higher wall pressure p_w. To examine this further, in figure 12, we plot the normalized impulse

$$I_w={\mathit{\int }}_{t_1}^{t_2}\frac{p_w-p_{\infty }}{p_{\infty }}\mathrm{d}t,$$ (37)

for t₁ = 21.0 μs and t₂ = 24.5 μs along the wall y-axis. I_w also appears converged globally with the new model and at a significantly higher value than the corresponding simulations with R = 0.

Figure 12.

Variation of I_w in (37) along the wall for R = 0 (dashed) and the new model (solid) with R₀/Δx_min ≈ 6.4, 12.8, 25.6, and 51.2.

6. Conclusion

An interface regularization scheme consistent with the first-order non-equilibrium mixture model is developed for bubble dynamics and other high-speed, multi-fluid flows. Smearing of the interface in the diffuse interface or mixture approach renders the solutions incorrect at long times. The interface sharpening approach of Shukla et al. [50] keeps the interface thickness fixed, but leads to qualitatively wrongs results due to thermodynamic inconsistencies. We have re-crafted the asymptotic multi-fluid formulation of [38, 40] to incorporate the immiscibility condition into the model in a thermodynamically consistent fashion. This reduces a general seven-equation two-fluid compressible flow model to a five equation model including interface regularization terms. The analysis is performed for general fluids whose states are fixed by two thermodynamic variables. The regularization operator is incorporated into the continuum equations, and is consequently independent of the underlying numerical scheme.

Upon establishing consistency and convergence of the model, we tested it on three-dimensional bubble collapses employing an AMR framework built on a fifth-order WENO method. We obtained excellent agreement of the simulations results with the spherically symmetric semi-analytic models for the free-space bubble collapses. The new model consistently preserves fluid immiscibility with superior convergence properties. The utility of the regularization scheme is further realized in the near-wall bubble collapses, for which, the new model prevents numerical smearing of the jet kinetic energy leading to stronger and converged jet impacts on the wall.

Appendix A

Asymptotic analysis

To reduce (7) in the limit of zero relaxation time, we seek an asymptotic solution of the form

$$(s_1,s_2,{\mathbf{u}}_1,{\mathbf{u}}_2,p_1,p_2,{\alpha }_2)={\mathbf{q}}^{(0)}+\delta {\mathbf{q}}^{(1)}+\dots ,$$ (A.1)

where δ is such that

$$\delta ~\frac{1}{\mathrm{\lambda }}~\frac{1}{\mu }.$$ (A.2)

Assuming

$$R\left({\alpha }_2^{(0)}\right)\mathrm{to\; be}O(1),$$

retaining terms to O(1/δ), and dropping the superscript notation, we arrive at

$$\mu (p_{\mathrm{I}}-p_1)(p_2-p_1)+\mathrm{\lambda }({\mathbf{u}}_{\mathrm{I}}-{\mathbf{u}}_1)·({\mathbf{u}}_2-{\mathbf{u}}_1)=0,$$ (A.3a)

$$\mu (p_{\mathrm{I}}-p_2)(p_1-p_2)+\mathrm{\lambda }({\mathbf{u}}_{\mathrm{I}}-{\mathbf{u}}_2)·({\mathbf{u}}_1-{\mathbf{u}}_2)=0,$$ (A.3b)

$$\mathrm{\lambda }({\mathbf{u}}_2-{\mathbf{u}}_1)=0,$$ (A.3c)

$$\mathrm{\lambda }({\mathbf{u}}_1-{\mathbf{u}}_2)=0,$$ (A.3d)

$$\mu {\rho }_1{a}_{1\mathrm{I}}^2(p_2-p_1)+\mathrm{\lambda }{\sigma }_1({\mathbf{u}}_{\mathrm{I}}-{\mathbf{u}}_1)·({\mathbf{u}}_2-{\mathbf{u}}_1)=0,$$ (A.3e)

$$\mu {\rho }_2{a}_{2\mathrm{I}}^2(p_1-p_2)+\mathrm{\lambda }{\sigma }_2({\mathbf{u}}_{\mathrm{I}}-{\mathbf{u}}_2)·({\mathbf{u}}_1-{\mathbf{u}}_2)=0,$$ (A.3f)

$$\mu (p_2-p_1)=0,$$ (A.3g)

which gives,

$${\mathbf{u}}_{\mathrm{I}}={\mathbf{u}}_1={\mathbf{u}}_2=\mathbf{u}(\mathrm{equilibrium\; velocity}),$$ (A.4)

$$p_{\mathrm{I}}=p_1=p_2=p(\mathrm{equilibrium\; pressure}),$$ (A.5)

To O(1), using (A.4) and (A.5), we obtain

$$\frac{\mathrm{D}{s}_1}{\mathrm{D}t}=0,$$ (A.6a)

$$\frac{\mathrm{D}{s}_2}{\mathrm{D}t}=0,$$ (A.6b)

$${\alpha }_1{\rho }_1\frac{\mathrm{D}\mathbf{u}}{\mathrm{D}t}+{\alpha }_1\nabla p=\mathrm{\lambda }\delta ({\mathbf{u}}_2^{(1)}-{\mathbf{u}}_1^{(1)}),$$ (A.6c)

$${\alpha }_2{\rho }_2\frac{\mathrm{D}\mathbf{u}}{\mathrm{D}t}+{\alpha }_2\nabla p=\mathrm{\lambda }\delta ({\mathbf{u}}_1^{(1)}-{\mathbf{u}}_2^{(1)}),$$ (A.6d)

$${\alpha }_1\frac{\mathrm{D}p}{\mathrm{D}t}+{\alpha }_1{\rho }_1{a}_1^2\nabla ·\mathbf{u}=\mu \delta {\rho }_1{a}_1^2(p_2^{(1)}-p_1^{(1)}),$$ (A.6e)

$${\alpha }_2\frac{\mathrm{D}p}{\mathrm{D}t}+{\alpha }_2{\rho }_2{a}_2^2\nabla ·\mathbf{u}=\mu \delta {\rho }_1{a}_2^2(p_1^{(1)}-p_2^{(1)}),$$ (A.6f)

$$\frac{\mathrm{D}{\alpha }_2}{\mathrm{D}t}=\mu \delta (p_2^{(1)}-p_1^{(1)})+R({\alpha }_2).$$ (A.6g)

Elimination of

$$\mathrm{\lambda }\delta ({\mathbf{u}}_2^{(1)}-{\mathbf{u}}_1^{(1)})\mathrm{and}\mu \delta (p_2^{(1)}-p_1^{(1)}),$$

from (A.6), gives

$$\frac{\mathrm{D}{s}_1}{\mathrm{D}t}=0,$$ (A.7a)

$$\frac{\mathrm{D}{s}_2}{\mathrm{D}t}=0,$$ (A.7b)

$$\frac{\mathrm{D}\mathbf{u}}{\mathrm{D}t}+\frac{1}{\rho }\nabla p=0,$$ (A.7c)

$$\frac{\mathrm{D}p}{\mathrm{D}t}+\rho a^2\nabla ·\mathbf{u}=0,$$ (A.7d)

$$\frac{\mathrm{D}{\alpha }_2}{\mathrm{D}t}={\alpha }_1{\alpha }_2\frac{{\rho }_1{a}_1^2-{\rho }_2{a}_2^2}{{\alpha }_1{\rho }_2{a}_2^2+{\alpha }_2{\rho }_1{a}_1^2}\nabla ·\mathbf{u}+R({\alpha }_2),$$ (A.7e)

where ρ and a are as defined in (15) and (18), respectively.

We now have the system in terms of P = (s₁, s₂, u, v, w, p, α₂)^t:

$$\frac{\partial \mathbf{P}}{\partial t}+\mathbf{F}\frac{\partial \mathbf{P}}{\partial x}+\mathbf{G}\frac{\partial \mathbf{P}}{\partial y}+\mathbf{H}\frac{\partial \mathbf{P}}{\partial z}=\mathbf{R},$$ (A.8)

where

$$\mathbf{F}=\left(\begin{array}{ccccccc}u& 0& 0& 0& 0& 0& 0\\ 0& u& 0& 0& 0& 0& 0\\ 0& 0& u& 0& 0& \frac{1}{\rho }& 0\\ 0& 0& 0& u& 0& 0& 0\\ 0& 0& 0& 0& u& 0& 0\\ 0& 0& \rho a^2& 0& 0& u& 0\\ 0& 0& -\beta & 0& 0& 0& u\end{array}\right),$$ (A.9)

with

$$\beta ={\alpha }_1{\alpha }_2\frac{{\rho }_1{a}_1^2-{\rho }_2{a}_2^2}{{\alpha }_1{\rho }_2{a}_2^2+{\alpha }_2{\rho }_1{a}_1^2},$$ (A.10)

and

$$\mathbf{R}={(\begin{array}{ccccccc}0,& 0,& 0,& 0,& 0,& 0,& R({\alpha }_2)\end{array})}^{\mathrm{t}},$$ (A.11)

where G and H are defined similarly. Note that we focus only on F in this analysis. The objective at this stage is to write the system in terms of the quasi-conservative variables, Q=(ρ₁α₁, ρ₂α₂, ρu, ρv, ρw, ρE, α₂)^t, where ρE is as defined in (16). To this end, we first construct the transformation matrix B = ∂Q/∂P, so that (A.8) can be equivalently written as

$$\frac{\partial \mathbf{Q}}{\partial t}+{\mathbf{BFB}}^{-1}\frac{\partial \mathbf{Q}}{\partial x}+{\mathbf{BGB}}^{-1}\frac{\partial \mathbf{Q}}{\partial y}+{\mathbf{BHB}}^{-1}\frac{\partial \mathbf{Q}}{\partial z}=\mathbf{BR}.$$ (A.12)

For fluids whose states are fixed by two thermodynamic variables, we first invoke Gibb’s relation to obtain the partial derivates needed to construct B:

$$\begin{array}{cccc}{\left(\frac{\partial {\rho }_k}{\partial s_k}\right)}_p& =& -\frac{T_k{\rho }_k{\sigma }_k}{a_k^2},& {\left(\frac{\partial {\rho }_k}{\partial p}\right)}_{s_k}=\frac{1}{a_k^2},\\ {\left(\frac{\partial e_k}{\partial s_k}\right)}_p& =& -\frac{T_k{\rho }_k{\sigma }_k{\xi }_k}{a_k^2},& {\left(\frac{\partial e_k}{\partial p}\right)}_{s_k}=\frac{p_k}{a_k^2{\rho }_k^2},\end{array}$$ (A.13)

where

$${\xi }_k={\left(\frac{\partial e_k}{\partial {\rho }_k}\right)}_p=\frac{1}{{\rho }_k}\left(\frac{p}{{\rho }_k}-\frac{a_k^2}{{\sigma }_k}\right),$$ (A.14)

from which, we obtain

$$\begin{array}{lll}\frac{\partial {\alpha }_2}{\partial \mathbf{P}}& =& (0,0,0,0,0,0,1),\\ \frac{\partial {\rho }_1}{\partial \mathbf{P}}& =& \left(-\frac{{\rho }_1{\sigma }_1{T}_1}{a_1^2},0,0,0,0,\frac{1}{a_1^2},0\right),\\ \frac{\partial {\rho }_2}{\partial \mathbf{P}}& =& \left(0,-\frac{{\rho }_2{\sigma }_2{T}_2}{a_2^2},0,0,0,\frac{1}{a_2^2},0\right),\\ \frac{\partial {\alpha }_1{\rho }_1}{\partial \mathbf{P}}& =& {\alpha }_1\frac{\partial {\rho }_1}{\partial \mathbf{P}}+(0,0,0,0,0,0,-{\rho }_1),\\ \frac{\partial {\alpha }_2{\rho }_2}{\partial \mathbf{P}}& =& {\alpha }_2\frac{\partial {\rho }_2}{\partial \mathbf{P}}+(0,0,0,0,0,0,{\rho }_2),\\ \frac{\partial \rho }{\partial \mathbf{P}}& =& \frac{\partial {\alpha }_1{\rho }_1}{\partial \mathbf{P}}+\frac{\partial {\alpha }_2{\rho }_2}{\partial \mathbf{P}},\\ \frac{\partial \rho u}{\partial \mathbf{P}}& =& u\frac{\partial \rho }{\partial \mathbf{P}}+\rho (0,0,1,0,0,0,0),\\ \frac{\partial \rho \nu }{\partial \mathbf{P}}& =& \nu \frac{\partial \rho }{\partial \mathbf{P}}+\rho (0,0,0,1,0,0,0),\\ \frac{\partial \rho w}{\partial \mathbf{P}}& =& w\frac{\partial \rho }{\partial \mathbf{P}}+\rho (0,0,0,0,1,0,0),\\ \frac{\partial \rho \kappa }{\partial \mathbf{P}}& =& \kappa \frac{\partial \rho }{\partial \mathbf{P}}+\rho (0,0,u,v,w,0,0),\\ \frac{\partial {\rho }_1{e}_1}{\partial \mathbf{P}}& =& \left(-\frac{({\rho }_1{\xi }_1+e_1)T_1{\rho }_1{\sigma }_1}{a_1^2},0,0,0,0,\frac{e_1{\rho }_1+p}{{\rho }_1{a}_1^2},0\right),\\ \frac{\partial {\rho }_2{e}_2}{\partial \mathbf{P}}& =& \left(0,-\frac{\left({\rho }_2{\xi }_2+e_2\right)T_2{\rho }_2{\sigma }_2}{a_2^2},0,0,0,\frac{e_2{\rho }_2+p}{{\rho }_2{a}_2^2},0\right),\\ \frac{\partial \rho e}{\partial \mathbf{P}}& =& {\alpha }_1\frac{\partial {\rho }_1{e}_1}{\partial \mathbf{P}}+{\alpha }_2\frac{\partial {\rho }_2{e}_2}{\partial \mathbf{P}}+(0,0,0,0,0,0,{\rho }_2{e}_2-{\rho }_1{e}_1).\end{array}$$ (A.15)

Substituting B in (A.12) yields the final quasi-conservative system for interfacial flows involving any two immiscible simple fluids:

$$\frac{\partial {\alpha }_1{\rho }_1}{\partial t}+\nabla ·({\alpha }_1{\rho }_1\mathbf{u})=-{\rho }_{1}R({\alpha }_2),$$ (A.16a)

$$\frac{\partial {\alpha }_2{\rho }_2}{\partial t}+\nabla ·({\alpha }_2{\rho }_2\mathbf{u})={\rho }_{2}R({\alpha }_2),$$ (A.16b)

$$\frac{\partial \rho \mathbf{u}}{\partial t}+\nabla ·(\rho \mathbf{u}\otimes \mathbf{u})+\nabla p=({\rho }_2-{\rho }_1)\mathbf{u}R({\alpha }_2),$$ (A.16c)

$$\frac{\partial \rho E}{\partial t}+\nabla ·((\rho E+p)\mathbf{u})=(({\rho }_2-{\rho }_1)\kappa +\eta )R({\alpha }_2),$$ (A.16d)

$$\frac{\partial {\alpha }_2}{\partial t}+\mathbf{u}·\nabla {\alpha }_2=\beta \nabla ·\mathbf{u}+R({\alpha }_2),$$ (A.16e)

where κ and β are as defined in (17) and (A.10), respectively, and η = ρ₂e₂ − ρ₁ e₁.

Appendix B

Preservation of interface conditions for equilibrium flows

The utility of our approach is predicated upon maintaining mechanical equilibrium at interfaces. We confirm that this is exact for uniform conditions using a test problem proposed by Abgrall [43]. We check that uniform velocity, pressure and density at time tⁿ implies the same uniformity at time tⁿ⁺¹, despite the interface regularization. To do this, we start with the quasi-conservative system (A.16) in discrete one-dimensional form for uniform pressure (p₀) and velocity (u₀),

$${\left({\alpha }_1{\rho }_1\right)}_i^{n+1}={\left({\alpha }_1{\rho }_1\right)}_i^n-\frac{\mathrm{\Delta }t}{\mathrm{\Delta }x}\mathrm{\Delta }({\alpha }_1{\rho }_{1}u)-\mathrm{\Delta }t{\rho }_{1,i}^{n}R,$$ (B.1a)

$${\left({\alpha }_2{\rho }_2\right)}_i^{n+1}={\left({\alpha }_2{\rho }_2\right)}_i^n-\frac{\mathrm{\Delta }t}{\mathrm{\Delta }x}\mathrm{\Delta }({\alpha }_2{\rho }_{2}u)+\mathrm{\Delta }t{\rho }_{2,i}^{n}R,$$ (B.1b)

$${(\mathit{\rho u})}_i^{n+1}={(\mathit{\rho u})}_i^n-\frac{\mathrm{\Delta }t}{\mathrm{\Delta }x}\mathrm{\Delta }(\rho u^2)+\mathrm{\Delta }t{u}_0({\rho }_{2,i}^n-{\rho }_{1,i}^n)R,$$ (B.1c)

$${(\mathit{\rho E})}_i^{n+1}={(\mathit{\rho E})}_i^n-\frac{\mathrm{\Delta }t}{\mathrm{\Delta }x}\mathrm{\Delta }(\mathit{\rho Eu})+\mathrm{\Delta }t({\kappa }_0({\rho }_{2,i}^n-{\rho }_{1,i}^n)+{\eta }_i^n)R,$$ (B.1d)

$${({\alpha }_2)}_i^{n+1}={({\alpha }_2)}_i^n-\frac{\mathrm{\Delta }t}{\mathrm{\Delta }x}{u}_0\mathrm{\Delta }({\alpha }_2)+\mathrm{\Delta }\mathit{tR},$$ (B.1e)

where Δ(·) denotes discrete flux for the hyperbolic part, and R denotes discrete regularization operator. Without loss of generality, we assume u₀ > 0 and choose simple upwind flux of the form

$$\mathrm{\Delta }(\varphi )={\varphi }_i^n-{\varphi }_{i-1}^n.$$ (B.2)

(Extensions are available for more general Δ(·) involving spatial reconstruction and Riemann solver [39, 40, 43, 48, 49].) Using (15), u₀((B.1a)+(B.1b))-(B.1c) yields

$$u_i^{n+1}=u_0,$$ (B.3)

and κ₀((B.1a)+(B.1b))-(B.1d) yields

$${\left(\mathit{\rho e}\right)}_i^{n+1}={\left(\mathit{\rho e}\right)}_i^n-\frac{\mathrm{\Delta }t}{\mathrm{\Delta }x}\mathrm{\Delta }(\mathit{\rho eu})+\mathrm{\Delta }t{\eta }_i^{n}R,$$ (B.4)

where ρe = ρE − ρκ. From (16) and (10),

$$\mathit{\rho e}=p({\mathrm{\Gamma }}_1{\alpha }_1+{\mathrm{\Gamma }}_2{\alpha }_2)+{\prod }_1{\alpha }_1+{\prod }_2{\alpha }_2,$$ (B.5)

and

$$\eta =p({\mathrm{\Gamma }}_2-{\mathrm{\Gamma }}_1)+{\prod }_2-{\prod }_1.$$ (B.6)

Using (14) and (B.6), (B.5) can be written as

$$\mathit{\rho e}=p{\mathrm{\Gamma }}_1+{\prod }_1+\eta {\alpha }_2,$$ (B.7)

which upon substitution in (B.4) yields

$${(p{\mathrm{\Gamma }}_1+{\prod }_1+\eta {\alpha }_2)}_i^{n+1}={(p{\mathrm{\Gamma }}_1+{\prod }_1+\eta {\alpha }_2)}_i^n-\frac{\mathrm{\Delta }t}{\mathrm{\Delta }x}\mathrm{\Delta }((p{\mathrm{\Gamma }}_1+{\prod }_1+\eta {\alpha }_2)u)+\mathrm{\Delta }t{\eta }_i^{n}R.$$ (B.8)

Since ${\eta }_i^n=p_0({\mathrm{\Gamma }}_2-{\mathrm{\Gamma }}_1)+{\mathrm{\Pi }}_2-{\mathrm{\Pi }}_1$, ${\eta }_i^n$ (B.1e)-(B.8) shows preserved uniformity:

$$p_i^{n+1}=p_0.$$ (B.9)

If density is also uniform (ρ₁ (x) = ρ₂(x) = ρ₀), then using (14), ρ₀(B.1e)+(B.1a) and ρ₀(B.1e)-(B.1b) yield

$${\rho }_{1,i}^{n+1}={\rho }_{2,i}^{n+1}={\rho }_0.$$ (B.10)