Moving least-squares aided finite element method (MLS-FEM): A powerful means to predict pressure discontinuities of multi-phase flow fields and reduce spurious currents
Introduction
Spurious or parasite-like currents are known as unphysical velocity components in the vicinity of the interface in multi-phase flow problems, which are also accompanied by meaningless pressure fluctuations. Spurious currents can destroy the numerical predictions of the morphology development in two-phase flow problems, especially for problems with low values of Capillary number.
This unfavorable and unphysical phenomenon is attributed to the deficiencies of the different numerical methods. Therefore, the attention of many studies is directed to resolve this problem and minimize or remove the spurious currents. In the following, some strategies for resolving the Spurious current problem will be reviewed, then our approach toward this problem is introduced.
Spurious currents were firstly reported by Gunstensen [1] in his Lattice-Boltzmann study of multi-phase flow. He believed that these anomalous currents exist due to the small pressure gradients, which are generated as a result of spatial discretization of the interface. He also stated that since the spurious currents do not exceed the interface neighborhood and are two orders of magnitude smaller than the maximum velocity of the system, they can not have a considerable effect on the bulk flow. He showed that the magnitude of the spurious currents is proportional to the surface tension coefficient (γ) and the inverse viscosity (1/μ). Since the spurious currents can lead to unphysical results in a two-phase flow problem [2], the Lattice-Boltzmann method for two-phase flow problem is then improved to provide a more reasonable prediction of the spurious currents [3], [4], [5], [6].
Lafaurie et al. [7] observed parasite currents in their method, named SURFER. They observed small-amplitude velocity components in the vicinity of a macroscopic static bubble. They attributed the existence of the currents to the locally unbalanced stress components and the pressure in the interfacial region. In their study, a relation between the amplitude of the currents and surface tension coefficient as well as viscosity was reported. As it is summarized in [8], the maximum velocity due to the spurious et al. (us) can be scaled as Eq. (1).
Renardy and Renardy studied the amplitude of the spurious current using the Volume-Of-Fluid method (VOF) [9]. In their study, different methods of surface tension force calculation including their own-developed method (parabolic reconstruction of surface tension or PROST) were compared in an unsteady situation, and it was shown that using a more accurate estimation of the body force due to the surface tension (i.e., PROST) can considerably reduce the magnitude of the unwanted spurious currents.
Francois et al. [10] applied the VOF method to study the flow of two-phase-fluid systems in the presence of the surface tension coefficient and developed two different methods (continuous and sharp) to represent the interface. They show that the sharp method can provide better results for pressure jump due to the surface tension; nevertheless, both methods show the existence of spurious currents of the same magnitude. It is recently shown that by evaluating the flow field parameters using the finite volume method and capturing the interface using the level set method and the ghost fluid method the magnitude of the spurious currents can be reduced [11].
Aulisa et al. [12] also stated that the main source of spurious currents is an inaccurate definition of the surface tension force in the flow problem. Therefore, they suggested a new method for the calculation of the curvature and the surface tension force when using a Laplacian smoothing operator, which removes some inappropriate data. Although they do not show the magnitude of the spurious current in their research explicitly, they provided some data regarding curvature calculation of a stationary droplet after some time evolution, which has a good agreement with analytical data.
Ganesan et al. [13] showed that the unphysical spurious currents also appear in finite element calculations. The authors suggested different possible reasons for the existence of the parasite currents in FEM calculations and demonstrated that even a Lagrangian mesh could not remove the spurious velocity components. A Lagrangian mesh is a meshing or griding system, which is fitted to the boundaries of the flow domain or the interfaces between different phases. This type of mesh is also recalled as the boundary-fitted mesh.
Groß and Reusken [14] stated that the amplitude of the spurious currents could be considerably reduced by employing a more accurate method for the calculation of the surface tension force, in conjunction with the pressure extended finite element method, which is accompanied by mesh refinement in the vicinity of the interface. The enrichment of the pressure shape function is also used by Ausas et al. [15] as an effective method for a more precise prediction of the pressure jump due to the surface tension force.
Popinet [16] employed an adaptive mesh refinement in the interface neighborhood to provide a better variable spatial resolution along with the interface. He also implemented different methods to evaluate the surface tension force. He showed that the exactness of the surface tension force evaluation has a considerable effect on the magnitude of the spurious currents.
Marchandise et al. [17] showed that the calculation of the surface tension force based on the Dirac delta function can develop pressure fluctuations at the interface area. Nevertheless, a smoothened Dirac delta function will prevent the oscillatory behavior of pressure. Cho et al. [18] also obtained similar results when they evaluated the surface tension force using the derivatives of the distance function on the interface.
Zahedi et al. [19] also revealed the fact that the surface tension calculation has a great impact on the amplitudes of the spurious currents. They found that the implementation of a sharp interface force, which is calculated by the surface tension force integration over the interface surface, will increase the spurious currents. However, employing a regularized indicator function instead of the linear sign distance function can alleviate the size of the parasite currents. In the recent case, pressure jump due to the existence of the surface tension cannot be predicted, satisfactorily. They finally suggested that a discontinuous treatment of the pressure is necessary for the FEM to obtain a correct pressure jump and reduce the spurious currents.
A new variational formulation besides a pressure enriched extended FEM was employed by Barrett et al. [20] to reduce the magnitude of the spurious current and obtain a better prediction for the pressure jump across the interface.
In this paper, our main focus is to reduce the spurious currents and provide a better prediction for the pressure jump in two-phase flow problems. For this purpose, the governing equations and the finite element formulation, which is introduced in Mostafaiyan et al. [21] for two-phase flow problems will be briefly reviewed, in Section 2. Then in the third section, the reason for the existence of the spurious current will be discussed, and it will be shown that even by employing a Lagrangian (the boundary-fitted) mesh the spurious currents do not vanish. In Section 4, our new strategy to consider the pressure discontinuity will be presented. Considering the surface tension force as a pressure boundary condition is the subject of Section 5. In Sections 6 and 7, the magnitude of the spurious currents and the streamlines will be studied for the stationary drops and the flow problems with very low Capillary numbers, respectively. In Section 8, the capability of the PMLS method in an unstructured mesh is examined. Finally, in Section 9, the limitation of the introduced method is discussed.
Section snippets
Governing equations and the finite element formulation
The flow domain comprises a drop with a radius Rdrop, which is located between two parallel plates of a two-dimensional channel, as shown in Fig. 1. The parameters L and H are the length and depth of the channel, respectively. The relative velocity of the upper and lower plates is U. We name this flow problem two-phase simple shear flow field, which contains two immiscible major (matrix) and minor (drop) phases with the viscosity values of μ1 and μ2, respectively.
By assuming the flow regime of
The spurious currents
According to the review in the introduction section, the magnitude of the unphysical spurious currents, which are predicted by finite element and finite volume methods, can be reduced by a precise evaluation of the surface tension force. Besides the inaccurate calculation of the surface tension force as a reason for the existence of the spurious currents, we believe that the discontinuity of the pressure (i.e., a pressure jump) due to surface tension is another reason for the misprediction of
Pressure shape function enhancement by MLS interpolating technique
The flow domain, in a multi-phase flow problem, comprises at least two different fluids. One way to numerically evaluate the flow field parameters (the velocity components and the pressure) in such a flow domain, is to employ the classical FEM method, as introduced in section (2). For this purpose, one shall firstly provide a suitable mesh for the whole flow domain. Since the generated mesh or grid is independent of the phase morphology or the microstructural changes during the flow, it is
Surface tension force as a pressure boundary condition
Since the surface tension force is not considered as a source term in the PMLS method, we can cancel the corresponding source term on the right-hand side of Eqs. (6-a), and re-write the equation as follows:
However, the boundary term in Eq. (20-b) will be calculated to consider the effect of the surface tension forces. The boundary term, after assembling of the finite element matrix, on
The pressure jump and the spurious current magnitude
In this section, we compare the pressure jump and the magnitude of the spurious current in a stationary two-phase flow problem for two different approaches: the PMLS method and the classical method.
For this purpose, the two-phase simple shear flow domain (as shown in Fig. 1) with unit dimensions (), a drop diameter and stationary upper and lower plates () is supposed. The surface tension coefficient and the main phase viscosity are assumed to be and ,
Drop deformation at very low Capillary numbers (Ca ≤ 0.01)
The Capillary number is defined as the ratio of the viscous forces to the surface tension forces in a two-phase flow field. In a two-phase simple shear flow problem, where the drop with initial circular shape lays in the middle between the upper and lower plates, as shown in Fig. 1, the Capillary number is defined as stated in Eq. (25):
According to Eq. (25), a simple shear flow field with the property of a low Capillary number comprises high values of surface
Unstructured mesh
Since in many complex geometries, it is not always possible to provide a structured mesh; the PMLS method is adapted to be compatible with any unstructured mesh. To investigate this capability, the evolution and the steady shape of a drop with , , , , and , which is calculated in an unstructured mesh using the PMLS method, are compared with similar results in a structured mesh.
Fig. 17a shows the unstructured mesh in a rectangular channel. The number of the
The limitation of the PMLS method
The PMLS method can predict the deformation extent of a drop when the viscosity ratio is one () because in this case, there is no velocity discontinuity in the system. Figs. 13a and 18a, which show the streamlines and the pressure after equilibrium; respectively, are a typical illustration of the recent case.
When deviating from , a velocity discontinuity will be introduced to the system; which can be ignored up to a limit. Neglecting the discontinuity of the velocity components is also
Conclusion
In this research, the pressure has been introduced as a discontinuous parameter in a two-phase flow system, which cannot be reasonably predicted by the continuous shape functions of the Galerkin finite element method. It has also been shown that the mentioned discontinuity can lead to an unphysical prediction of the pressure values, and non-zero velocity components or spurious currents for a stationary drop problem.
To reduce the magnitude of the spurious currents and have a more accurate
Declaration of Competing Interest
Some novelties and valuable results are included in this paper:
1-The moving least-squares aided finite element method (MLS-FEM) as a new method is introduced.
2-The MLS-FEM uses element splitting (similar to extended FEM or XFEM). Using the MLS-FEM method no new unknown will be introduced to the finite element calculations. (in spite of XFEM)
3-The stiffness matrix of the MLS-FEM retains its symmetric nature. (in spite of XFEM)
4-Since in this research the MLS interpolation functions are used for
Acknowledgment
We would like to thank Dr. Mahdi Salami Hosseini (Sahand university of technology in Tabriz) for his cooperation through helpful discussions in this subject.
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