Numerical modeling of two-phase flows using the two-fluid two-pressure approach
Summary (3 min read)
1 Introduction
- For instance the transport of Liquefied Natural Gas (LNG) in LNG carriers.
- The numerical modeling of two-phase flow involves dealing with a multitude of jumps of fluid properties across the interface separating the fluids.
- Numerically such an underresolved interface boundary layer effectively results in a velocity field which has a discontinuity in the tangential direction.
- Concluding remarks are made in Section 5.
2.1 Primary equations
- The authors consider incompressible flow for which the mass conservation equations result in a volume constraint on the evolution of the interface d dt |ωl|+.
- The influence of surface tension is of interest in their application and the authors therefore include it via Laplace’s law −JpK = σκ. (4) Here κ denotes the interface mean curvature and JϕK denotes jump of some flow variable ϕ JϕK := ϕg−ϕl.
- Note that this model does not impose any smoothness on the tangential velocity component uτ = τ ·u, where τ denotes the interface tangent.
2.2 The pressure Poisson problem
- Addition of the mass conservation equations, when divided by their respective densities, yields ∫ ∂ω uη dS = ∫ ∂ωl\I ulη dS+ ∫ ∂ωg\I ugη dS = 0, (7) thus showing that the mixture velocity field is divergence free.
- The latter condition (9) is necessary for having a well-posed coupled Poisson problem.
3.2 Momentum equations
- In the interior of each of the phases the momentum equations are discretized using the symmetry-preserving finite-volume discretization by Verstappen and Veldman [8].
- Near the interface the authors choose to discretize the momentum equations in strong form, thereby sacrificing exact momentum conservation at the interface but alleviating difficulties faced with having arbitrarily small cells |cπ|(t) and non-smooth in time face areas | f π|(t).
- At the interface the authors use a first-order upwind convection scheme per phase, which relies on constant extrapolation of velocities.
3.3 Pressure Poisson problem
- The solution of the Poisson problem plays a central role in the numerical model, since this is where the two phases are implicitly coupled.
- The authors will now precisely define the divergence D̄ and the gradient G.
3.3.1 Divergence operator
- At the interface their velocity field is discontinuous, and therefore the divergence operator needs to be modified.
- A finite difference approach such as the Ghost Fluid Method (GFM) [4] will result in an incompatible discretization of the Poisson problem ∃u ∈ F̂h0 s.t. ∑ c∈C D̄(u)c 6= 0, (12) where F̂h0 is the set of velocity fields which vanish at the boundary.
- A consequence of this incompatibility is that the resulting linear system of equations, resulting from the pressure Poisson equation, has no solution.
- A finite-volume approach however naturally preserves the flux cancellation property which a divergence operator should satisfy, and therefore the authors propose to use the cut-cell method [6] for discretization of the divergence operator.
- This divergence operator satisfies the discrete equivalent to Gauss’s theorem exactly, and therefore the term in (12) vanishes exactly for all u ∈ F̂h0 .
3.3.2 The gradient operator
- Near the interface the gradient needs modification to sharply capture the imposed jumps.
- Hence the authors know the liquid pressure plc on one side of the face and the gas pressure p g c′ on the other side of the face.
- From the discussion in Section 2.2 the authors know that they should not impose a jump on the full pressure gradient, but rather only on the component normal to the interface.
- The discretization of the gradient operator for an interface normal face is as follows.
- For faces f ∈ FIη for which this ratio exceeds 1 the authors interpret the face as an interface tangential face instead.
4 Validation
- Here the authors consider the validation of the proposed discretization.
- The authors first assess the accuracy of the discretized Poisson problem and then consider the more exciting dam-break problem.
4.1 Poisson problem
- The authors compare their proposed method to the Immersed Interface Method (IIM) [3] which sharply imposes jump conditions directly on the Laplacian.
- The 1d-GFM [4], implemented as described in Section 3.3.2, is also included in the comparison.
- Whenever the methods 1d-GFM or MdGFM are referred to in the context of a Laplace operator, the composition with the CCM divergence operator (13) is implied.
4.1.1 Mesh refinement
- As expected, the 1d-GFM is firstorder accurate, whereas the Md-GFM is second-order accurate, and of comparable accuracy to the IIM.
- The main advantage of using the Md-GFM is that the Laplace operator itself follows from the composition of a divergence operator and a gradient operator which is required in the context of solving incompressible two-phase problems.
4.1.2 Varying the density ratio
- To asses the dependence of the errors on the density ratio the authors fix the mesh-width h = 2/80, and vary the density ratio.
- The resulting gradient errors are shown in Table 2.
- Hence the proposed method can be used to accurately simulate near the one-phase limit ρg→ 0.
4.2 A dam-break problem
- The proposed discretization has been implemented in their in-house free-surface NavierStokes solver ComFLOW.
- Local and adaptive mesh refinement is used, as detailed in Van der Plas [7].
- The authors validate their model using a smooth version of the classic dam-break problem.
4.2.1 Problem description
- The domain is a rectangle of size 20×12m with an elliptic bathymetry of half lengths 18 and 2.8m whose center lies in the left-hand side bottom corner.
- Slip boundary conditions are imposed on the velocity field.
- The liquid density is given by ρl = 103kg/m3, the gas density varies and will always be indicated.
- Both fluids are initially at rest and separated by the following interface profile y(x) = 7.6+3.6tanh(0.36(x−12.5)) , (25) which will result in a flip-through impact (FTI) [1] in which the wave trough and crest reach the wall at the same time instance, resulting a violent impact.
- The authors consider several levels of mesh refinement, where they refine the mesh near the interface using blocks of size 16× 16.
4.2.2 Comparison to standard two-phase method
- Here the authors demonstrate the efficacy of the proposed model when compared to the standard two-phase model2.
- The authors also show the initial interface profile and at a later time t = 1.47 .
- Note that the standard two-phase model has a thin region at the interface in which the velocity transitions from gas to liquid.
- The proposed model captures this transition in a discontinuity, which allows the breaking wave to develop properly, as seen by the interface profile at t = 1.47.
4.2.3 Mesh refinement
- Results by the CADYF code3[1] are included.
- Except for the highest level of refinement, l = 4, the authors observe convergence towards the reference solution.
- For the highest refinement level, the authors observe a small fragmentation of the interface at the location where the velocity discontinuity is largest.
- The reference solution clearly does not exhibit this behavior, but this could be attributed to numerical damping.
4.2.4 Varying the density ratio
- Furthermore the authors consider the dependence of the solution on the gas density.
- For the 2The numerical scheme was identical except for the condition JuτK = 0 and the use of first-order upwind throughout the entire domain.
- The authors find fairly good agreement in terms of the interface profile.
- Moreover, for the larger gas densities, oscillations in the tangential velocity jump can be observed just before impact.
- This suggests that free surface instabilities are about to develop.
5 Conclusion
- The authors presented a discretization approach for capturing contact discontinuities in two-phase flow.
- The discretization of the pressure Poisson problem plays a central role since it implicitly couples the phases using a gradient jump condition which corresponds to imposing smoothness of the velocity field only in the interface normal direction.
- A novel combination of their proposed Md-GFM and the CCM was used to achieve this.
- Using the dam-break problem the authors then demonstrated that this approach is able to capture contact discontinuities sharply and accurately, even at high density ratios (demonstrated up to 10−6) close to the one-phase limit.
- The model will be extended with gas compressibility and the effects of viscosity will be included.
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...[20], and at last the Godunov scheme of Schwend-...
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...[20] (see also Guillemaud [24]), the approximation of the convective terms of the system is based on the Rusanov scheme [36] and the so-called VFRoe-ncv scheme [8], these strategies being adapted to the nonconservative framework....
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...3) (u2 remains continuous even if α1 presents a discontinuity, see also [20])....
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...Rusanov denotes a non conservative version of the Rusanov scheme which is explained in detail in [20]....
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...4) which corresponds to a particular choice of the closures proposed for instance in [20,25]....
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...[20] for a comprehensive study of general closure laws concerning uI and pI ....
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...[20] (see also Guillemaud [25]), the approximation of the convective terms of the system is based on the Rusanov scheme (Rusanov [34]) and the so-called VFRoe-ncv scheme (Buffard et al....
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References
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Frequently Asked Questions (2)
Q2. What are the future works mentioned in the paper "Numerical modeling of contact discontinuities in two-phase flow" ?
Eventually the authors want to use this model to study the effects of free surface instabilities in sloshing of LNG and its vapor.