Diffuse interfaces appear with any Eulerian discontinuity capturing compressible flow solver. When dealing with multifluid and multimaterial computations, interfaces smearing results in serious difficulties to fulfil contact conditions, as spurious oscillations appear. To circumvent these difficulties, several approaches have been proposed. One of them relies on multiphase flow modelling of the numerically diffused zone and is based on extended hyperbolic systems with stiff mechanical relaxation (Saurel and Abgrall, 1999 [4], Saurel et al., 2009 [6]). This approach is very robust, accurate and flexible in the sense that many physical effects can be included: surface tension, phase transition, elastic-plastic materials, detonations, granular effects etc. It is also able to deal with dynamic appearance of interfaces. However it suffers from an important drawback when long time evolution is under interest as the interface becomes more and more diffused. The present paper addresses this issue and provides an efficient way to sharpen interfaces. A sharpening flow model is used to correct the solution after each time step. The sharpening process is based on a hyperbolic equation that produces a steady shock in finite time at the interface location. This equation is embedded in a “sharpening multiphase model” redistributing volume fractions, masses, momentum and energy in a consistent way. The method is conservative with respect to the masses, mixture momentum and mixture energy. It results in diffused interfaces sharpened in one or two mesh points. The method is validated on test problems having exact solutions.
Diffuse interfaces appear with any Eulerian discontinuity capturing compressible flow solver. When dealing with multifluid and multimaterial computations, interfaces smearing results in serious difficulties to fulfil contact conditions, as spurious oscillations appear. To circumvent these difficulties, several approaches have been proposed. One of them relies on multiphase flow modelling of the numerically diffused zone and is based on extended hyperbolic systems with stiff mechanical relaxation (Saurel and Abgrall, 1999 [
Keywords:Material Interfaces; Multifluid; Multiphase; Multimaterial; Shocks; Non-Conservative; Hyperbolic Equations
Numerical smearing of interfaces (or contact discontinuities) appears with any Eulerian discontinuity capturing compressible flow solver. In the context of the Euler equations, it results in at least four points in the capturing zone and the number of points increases during time evolution with most methods. When dealing with multifluid and multimaterial computations, diffuse interfaces result in serious difficulties to match interface conditions, as spurious oscillations appear. To circumvent these difficulties, several approaches have been proposed.
Lagrangian methods and Front Tracking schemes are aimed to remove artificial smearing but result in other difficulties.
Interface reconstruction methods (Hirt and Nichols, 1981 [
Level-set methods (Fedkiw et al., 1999 [
Another option relies on multiphase flow modelling of the numerically diffused zone and is based on extended hyperbolic systems with stiff relaxation (Saurel and Abgrall, 1999 [
• surface tension (Perigaud and Saurel, 2005 [
This method is also able to deal with dynamic interfaces appearance. However it suffers of an important drawback when long time evolutions are under interest as the interface becomes more and more diffused. This has been clearly understood by Kokh and Lagoutière (2010) [
The present paper addresses this issue with another way to sharpen interfaces. The artificial compression method of Harten (1977 [
We continue this effort in the present paper by addressing:
• A more general diffuse interfaces flow model, where the physical effects previously mentioned can be considered, in particular dynamic interfaces creation.
• Mass, momentum and energy redistribution across the interface during the sharpening process. This issue has never been considered prior to the present work.
The present method uses an extra hyperbolic equation for a function that is equal to zero on one side of the interface and to one on the other side. Contrarily to Volume of Fluid, Level-set and multiphase flow models, this equation is not a transport one. It corresponds to a hyperbolic conservation law that admits shocks. More precisely, steady shocks appear at the location where this function is equal to 0.5. Thus, starting from a diffused solution, after for example one or several steps of the diffuse interface method, the volume fraction, density, energy fields are diffused. The volume fraction field is then sharpened with the help of the new function during a correction step. The main difficulty is to synchronise mass, momentum and energy sharpening with the volume fraction one. Also, this correction has to be conservative. Synchronisation of the various sharpening processes while maintaining conservation are the main goals and difficulties of the present work.
The resulting scheme is able to handle material interfaces in one or two mesh points. The single phase limit of the multiphase model and method corresponds to a scheme that improves the results of conventional capturing schemes used for the Euler equations.
The paper is organised as follows. In Section 2, the sharpening function is introduced with its evolution equation. Its ability to sharpen diffused profiles by the means of shock formation is demonstrated. In Section 3, the diffuse interface model of Saurel et al. (2009) [
Let’s consider two phases separated by an interface. The sharpening function
Let’s consider a particular case of a mixture zone separating phase 1 on the right and phase 2 on the left, as shown in
In this particular situation where
where a is a positive arbitrary parameter that controls the rate at which sharpening occurs. This parameter is not describing any physical effects and this equation will be used only to sharpen interfaces. Since the sharpening parameter a is arbitrary, in the following we will pose
As
The characteristic slope is thus,
This slope is positive if
Equation (2.1) admits shocks appearing in finite time. Consider a left state (L) and a right state (R) separated by a shock. The shock propagation speed
If
The Riemann problem solution is schematized in the
For any point of the initial data corresponding to
The Godunov method for equation (2.1) reads:
This method is stable under the following CFL condition:
i.e.
Let’s examine the solution given by this method for the initial condition shown in the
Equation (2.1) is however a simplified form of a more general equation, valid only if
In one-dimension, we suppose here that the volume fraction regular enough i.e. two interfaces are not too close the general form of the sharpening equation is,
This is equivalent to (in the case of monotonic behaviour of
In multi-dimensional case, it generalizes as:
The sharpening equation (2.6) or (2.7) is going to play a central role in the interface sharpening process. Indeed, the sharpening function
Before going on with the sharpening method, it is important to recall and summarize the basis of the diffuse interface method given in Saurel et al. (2009) [
The diffuse interface model under consideration is the one of Kapila et al. (2001) [
The Kapila et al. (2001) model in the context of two fluids reads:
where
The mixture internal energy is defined as,
and the mass fractions are given by
The mixture density is defined by
Each fluid is governed by its own equation of state (EOS),
The mixture pressure p is determined by solving Equation (3.2). In the particular case of fluids governed by the stiffened gas EOS,
the resulting mixture EOS reads,
The mixture sound speed corresponds to the Wood (1930) [
Stiffened gas EOS parameters are determined by the method given in Le Metayer et al. (2004).
It is straightforward to obtain the entropy equations:
As System (3.1) is non-conservative, specific relations for its closure in the presence of shocks are needed. In the weak shocks limit, appropriate shock relations have been determined in Saurel et al. (2007) [
where
Even equipped with these relations, this apparently simple model involves many difficulties:
• With the help of relations (3.6), it is possible to solve exactly or approximately the Riemann problem. When dealing with shock propagation in multiphase mixtures, the convergence of the numerical scheme to the exact solution is extremely difficult as the system is non-conservative: The cell average of non-conservative variables has no physical sense. This difficulty appears even with exact Riemann solvers. To reach convergence for shock propagating in multiphase mixtures, a special correction has been developed in Petitpas et al. (2009) [
• Another issue is related to the volume fraction positivity that is difficult to preserve due to the nonconservative term present in the right hand side of the first equation of System (3.1).
In spite of these difficulties, System (3.1) presents very nice features. It is able to create dynamically interfaces as a consequence of the mechanical relaxation process. It involves two entropy equations as well as two temperatures. These properties are important for the extensions to extra physics, as mentioned in the introduction.
To solve System (3.1) for interfaces separating fluids governed by different equations of state, an augmented system is preferred.
The augmented system with pressure disequilibrium has better properties for numerical resolution than the target system:
• Positivity of the volume fraction is easily preserved.
• The mixture sound speed has a monotonic behavior during the hyperbolic step.
These two properties are key points for the building of a simple, robust and accurate hyperbolic solver. Moreover, with proper treatment of relaxation terms, solutions of the target system (Kapila et al., 2001) [
The augmented model reads:
The interfacial pressure reads (Saurel et al., 2003 [
The combination of the two internal energy equations with mass and momentum equations results in the additional mixture energy equation:
This extra equation will be important during numerical resolution, in order to correct inaccuracies due to the numerical approximation of the two non-conservative internal energy equations.
The phasic entropy equations are readily obtained,
This model exhibits a nice feature with respect to the mixture sound speed. The mixture (frozen) sound speed,
has a monotonic behavior versus volume and mass fractions.
The model is thus strictly hyperbolic with waves speeds:
The numerical resolution of this model is summarized in Section 5, details being available in Saurel et al. (2009). As this model is solved with an Eulerian method, interface capturing results in numerical smearing. In the following section an interface sharpening model is built to remove excessive numerical diffusion.
The aim is to build a mathematical model able to sharpen interfaces with the help of Equation (2.6). The subsequent model is not a physical one but a “sharpening model” to reduce artificial diffusion at interfaces. During the sharpening process, mass, momentum and energy will have to be redistributed. To do this, a list of conditions has to be fulfilled:
1) When the initial conditions correspond to a mechanical equilibrium state (uniform velocity and uniform pressure) this state has to remain invariant during the sharpening process. Such initial conditions correspond to those resulting of the hyperbolic flow solver, as interface conditions of equal normal velocities and equal pressures have to be fulfilled.
2) Mass, momentum and energy are redistributed with the help of two contact waves. They must be linearly degenerate in order that the mass, momentum and energy of each phase redistribution be synchronised. The first contact wave will be used for conservative variables redistribution of phase 1 across the artificial shock, the second contact wave will be used similarly for phase 2.
3) The sharpening system must admit a conservative formulation.
4) Mixture mass, momentum and energy must be conserved in a weak sense that will be defined latter.
5) The model must create a single artificial shock, at the interface location.
Before using these various conditions for the model building, a first issue has to be addressed, regarding the link between the sharpening function
In a sake of clarity we performed the following analysis considering 1-D evolution with
The sharpening function
The volume fraction
It is important to have non-zero volume fractions for three reasons:
• First, System (3.1) cannot be used at any mesh point if
• Second, it is useful to have non-zero volume fractions in order to allow dynamic interfaces creation from nearly pure fluids. This is possible with the right hand side of the preceding equation and is important for example in cavitating flow modelling.
• Last, the interface can separate a pure fluid and a mixture of materials, like for example of compacted powder. In this case, the fluid volume fraction in the materials mixture can be far from 0 and 1. An example of interface separating multiphase mixtures will be considered in Section 5.5.
Let us denote by
Here
Consider a linear relation between the volume fraction
The variation of
Equipped with Relations (4.1) and (4.2) it is now possible to derive the ‘sharpening multiphase model’.
Let us consider again the particular case of 1D evolution when
The sharpening multiphase model reads:
This 8-equation model expresses in non-conservative form as:
The two internal energy equations can be expressed in terms of pressure equations as,
With the help of the mass equation of phase 1 it becomes:
Considering the two pressure equations and the two velocity equations of System (4.4), it appears that for mechanical equilibrium conditions
, (4.5)
Therefore, the model clearly respects the most important criterion of the sharpening method, corresponding to the first one of the list given previously: Interface conditions must be unchanged during the sharpening process. In other words, no spurious pressure and velocity oscillations must appear. System (4.3) respects this requirement.
Moreover, the sharpening System (4.4) is hyperbolic with the characteristics:
However, it is not strictly hyperbolic. The fields
The waves
The flux structure of System (4.3) guarantees synchronised evolutions of the sharpening function, volume fraction, mass, momentum and energy of the various phases.
System (4.3) admits the following set of jump conditions across a discontinuity propagating at speed D:
where, for any function f we denote
As shown previously, the first equation of this system admits shocks propagating at the velocity:
The second jump condition becomes:
Using (4.2), the volume fraction jump is related to the sharpening function jump by,
Thus the second jump condition reduces to the first one.
The mass jump condition now becomes:
Or,
Under expanded form it reads:
It thus reduces to:
Across shocks propagating at speed
These jump conditions express that across the sharpening shock, the densities, velocities and internal energies are prolonged. This is a very nice feature already observed with diffuse interface methods. It also means that the pressures are prolonged too. Thus, System (4.7) means that weak solutions of System (4.3) respect interface conditions of equal pressures and equal normal velocities, corresponding to the main criterion the method has to fulfil.
We now consider contact discontinuities of the sharpening model propagating at speeds
For example, for
As
It results in:
Consequently,
System (4.3) thus admits a single shock and two contact discontinuities, corresponding to the one genuinely non-linear field and the two linearly degenerate fields. These various jump conditions will be useful for the Riemann problem (RP) resolution. Before that, let us examine conservation properties.
Summing phase’s mass equations of System (4.3), the mixture mass conservation equation appears:
Denoting the mixture density by
This equation expresses that the mixture density is going to sharpen during the resolution of System (4.3). However, as the mass transport has already been solved during the diffuse interface model resolution (3.1 or 3.7) no extra mass flux must appear. Equation (4.9) violates the local mass conservation. However, when integrated between the left and right side of the diffuse interface, mass is preserved. Indeed,
The same remark holds for phase’s mass, momentum and energy:
and consequently for the phase’s total energies,
The method is thus conservative in the weak sense.
As the sharpening function gradient
Jump conditions are the same as previously with obvious modifications related to the
Flows computations proceed in two steps. First the flow model (System 3.7) is solved with Saurel et al. (2009) [
Numerical resolution of the 6-equation augmented model (3.7) in the limit of stiff pressure relaxation has been addressed in Saurel et al. (2009) [
The system to consider during numerical resolution thus involves 7 equations: System (3.7) and Equation (3.8).
Many approximate Riemann solvers can be considered to determine the cell boundary states, but the HLLC solver of Toro et al. (1994) [
Saurel et al. (2009) [
• At each cell boundary solve the Riemann problem of System (3.7) with HLLC solver.
• Evolve all flow variables with a Godunov type scheme (or its higher order variants).
• Determine the relaxed pressure and especially the volume fraction.
• Compute the mixture pressure with Equation (3.4).
• Reset the internal energies with the computed pressure with the help of their respective EOS.
• Go to the first item for the next time step.
The main drawback of this method is related to the excessive numerical diffusion of interfaces, especially for long time evolutions. The sharpening System (4.11) is addressed in this aim. Its numerical resolution is detailed hereafter.
Consider a cell boundary separating a left state (L) and a right state (R). At each cell boundary an initial discontinuity is present and three waves are emerging:
• Two contact discontinuities with velocities:
• A shock wave propagating with speed:
For the sake of simplicity we still assume that
It is important to consider
For a given
*
Let us examine the first situation, the others being similar. The corresponding configuration is depicted in the
The shock speed is determined immediately from the initial data,
The sharpening function solution is unchanged compared to the scalar case of Section 2:
Consequently, the volume fractions are determined as:
System (4.4) can be cast under compact form as:
with
The approximate Riemann problem solution for variables
, (5.4)
It is worth to mention that in this approximate solution, the solution for
Therefore, when
Compute
If
Compute
If
The fluxes of System (4.11) are thus easily deduced. They are used in the Godunov (1959) method that reads:
with,
The fluxes
The method is stable under conventional CFL restriction based on the fastest wave speed:
Theoretically, the interface is ideally sharpened when the steady state solution of (4.11) is obtained. For practical computations, a CFL number of 0.9 is used and a single time step is done.
In Section 4.5 we have shown that the sharpening method was not creating spurious pressure and velocity oscillations when the sharpening process was starting from initial data corresponding to mechanical equilibrium state. However, when a shock or an expansion wave interacts with an interface, the velocities and pressures are different on both sides of the interface at the discrete level. Thus, the mass redistribution achieved with the sharpening method is associated to momentum and energy redistribution that are going to produce non-equilibrium states on both sides of the interface. In order to restore mechanical equilibrium conditions, which consequence will be to fulfill the interface conditions of equal normal velocities and equal pressures, the following correction is done:
• The center of mass velocity is computed at each mesh point as,
• A correction is also needed for the internal energies and volume fractions. Indeed, as the mixture becomes out of pressure equilibrium, a pressure relaxation step is done, exactly in the same way as in Saurel et al. (2009) [
It means that System (4.11) has been complemented by velocity and pressure relaxation terms, in the same way as System (3.7) and that the limit solution when velocities and pressures are relaxed is used to reset the variables. Such correction is obviously conservative with respect to the masses, mixture momentum and mixture total energy. See for example Saurel and Abgrall (1999) [
We now consider method validation on various test problems of increasing difficulty. A single pseudo time step with CFL = 0.9 is used. With such a choice the convergence is reached at each time step and the interface is computed in one point.
A volume fraction discontinuity associated to a mixture density discontinuity is moving in a uniform pressure and velocity flow at 100 m/s. Initially the discontinuity is located at
• Liquid water on the left defined by
• Air on the right defined by
In the left chamber, the water volume fraction is set to
A mesh involving 100 cells is used. The results are shown in the
The sharpening algorithm clearly improves the results as the interface is handled in two points instead of 4 - 5 points. The pressure and velocity fields are maintained invariant.
The aim of this example is to show that the diffuse interface model with the sharpening algorithm is able to improve solutions of the Euler equations. Let’s consider a shock tube filled with air in both chambers. At initial time a pressure discontinuity is present at
In the
involving 100 cells.
A small oscillation due to the Superbee limiter is present on the velocity graph at shock front when the sharpening algorithm is used. These oscillations are absent with other limiters. Except regarding this discrepancy, the sharpening method improves the results.
A 1 m long shock tube containing two chambers separated by an interface at the location
A single time step with CFL = 0.9 is done with the sharpening solver after each hyperbolic time step of the diffuse interface model. The interface separating liquid and gas is clearly sharpened, as shown on the mixture density and volume fractions graphs.
The same shock tube problem is solved, but initially, the left chamber pressure is set to 1 TPa (10 Mbars). The initial discontinuity is located at
Again, the sharpening algorithm clearly improves the solution. An oscillation appears on the mixture density graph with the sharpening method, with similar behavior as Lagrangian and antidiffusion schemes (Kokh and Lagoutière, 2010) [
This test addresses method ability to deal with interfaces separating mixtures of materials. It highlights the need for an evolution equation for the sharpening variable
present in System (4.11).
A 1 m long shock tube containing two chambers separated by an interface at the location
In this example, the volume fraction and the sharpening function are not anymore linearly dependent. Their dependency is varying with time and space, showing the importance of Relation (5.1). The interface smearing is lowered with the sharp interface method.
The multiphase sharpening system can be rewritten as follows:
These systems are solved independently for an arbitrary number of fluids. The Godunov method presented in Section 5 results in the fulfilment of the constraints
An interface sharpening method has been presented. It is based on Harten (1977 [
It preserves spurious pressure and velocity oscillations and handles interfaces in about two mesh points. It is also able to handle interfaces separating mixtures of materials.
Multidimensional extension is under examination.
This work was partially supported by A * MIDEX and ANR, the grant ANR-11-LABX-0092 and ANR-11- IDEX-0001-02.