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An Arbitrary High-Order Conservative Level Set Runge-Kutta Discontinuous Galerkin Method for Capturing Interfaces
| Content Provider | Semantic Scholar |
|---|---|
| Author | Jibben, Z. Herrmann, Marcus |
| Copyright Year | 2013 |
| Abstract | We present an arbitrary high-order, quadrature-free, Runge-Kutta discontinuous Galerkin (RKDG) method for the solution of the conservative level set equation (Olsson et al., 2007), used for capturing phase interfaces in atomizing multiphase flows. Special care is taken to maintain high-order accuracy in the reinitialization equation, using appropriate slope limiters when necessary and a shared basis across cell interfaces for the diffusive flux. This further develops the ideas of Czajkowski and Desjardins (2011) by implementing a quadrature-free approach and allowing for arbitrary polynomial degree. For efficiency, we implement the method in the context of the dual narrow band overset mesh approach of the Refined Level Set Grid method (Herrmann, 2008). The accuracy, consistency, and convergence of the resulting method is demonstrated using the method of manufactured solutions (MMS) and several standard test cases, including Zalesak’s disk and columns and spheres in prescribed deformation fields. Using MMS, we demonstrate k + 1 order spatial convergence for k-th order orthonormal Legendre polynomial basis functions. We furthermore show several orders of magnitude improvement in shape and volume errors over traditional WENO based distance function level set methods, and k− 1 order spatial convergence of interfacial curvature using direct neighbor cells only. ∗Corresponding Author: zjibben@asu.edu Introduction The level set method is a popular approach to follow the motion of interfaces in numerical simulations [1] and has been widely used in simulations of multiphase flows involving phase interfaces. While exhibiting some advantages over alternative numerical approaches to capture the interface, level set methods have the distinct disadvantage that they are not locally volume conserving for divergence free velocity fields. That is, there is no built in discrete constraint that conserves the volume enclosed by the iso-surface of the level set scalar that defines the position of the interface. Numerous numerical methods have been devised to overcome this issue by coupling the level set method to other, better volume conserving interface capturing or tracking methods, see for example [2, 3]. The approach proposed by Olsson and Kreiss [4], Olsson et al. [5], on the other hand, reformulates the level set scalar as a conserved quantity itself by using the divergence free velocity constraint of low Mach number flows. As such, the level set scalar, in essence, becomes a smeared out Heaviside function. While this Conservative Level Set (CLS) method strictly speaking still does not guarantee discrete conservation of the level set iso-scalar enclosed volume if the thickness of the smeared out Heaviside function is non-zero, the method exhibits drastically improved volume conservation properties compared to other popular level set methods that are based, for example, on a distance function formulation. The CLS method has, for example, been successfully applied to atomizing flows by [6, 7, 8]. The discrete volume conservation quality of the CLS method is directly linked to the imposed thickness of the smeared out Heaviside function. Numerical methods that are able to solve the linear level set advection equation with minimum numerical dissipation and dispersion for a nearly discontinuous solution variable are thus ideal candidates for the CLS method. Discontinuous Galerkin methods potentially fall into this category. They have the added benefit that they are easy to parallelize and thus applicable to many modern massively parallel supercomputer platforms. In this paper, we present a Runge-Kutta Discontinous Galerkin method in quadrature-free form for solving the advection and reinitialization equations of the CLS method. We follow to a significant extent the work done by [9], however, we introduce some key modifications and developments that allow the method to be formally of order k + 1 for k-th order Legendre polynomial basis functions and provide a method to calculate interface curvature with order k − 1. Method To capture the location of the phase interface, we follow the Conservative Level Set (CLS) method, originally introduced by [4, 5]. Identifying the isosurface of the level set scalar G = 0.5 to coincide with the location of the phase interface, the advection equation for the level set scalar in the incompressible limit, i.e. ∇ · u = 0, is ∂G ∂t +∇ · (Gu) = 0 . (1) Following a standard interpretation of a level set scalar, i.e. G has meaning only at the phase interface, one is theoretically free to choose any definition of G away from the G = 0.5 iso-surface. Following Olsson et al. [5], we propose to define G as G(x, t) = 1 2 ( tanh ( φ(x, t) 2ε ) + 1 ) (2) away from the interface, where φ is the signed minimum distance to the fluid interface and ε defines a spreading width of the level set scalar G. To ensure G remains close to Eq. (2), one needs to periodically reinitialize the level set scalar by solving the following conservative PDE [5] ∂G ∂τ +∇ · (G (1−G)n) = ∇ · (ε (∇G · n)n) , (3) where τ is a pseudo time and n is the fluid interface normal. To solve Eq. (1) we will follow in essence the approach introduced by [9] and use a Runge-Kutta Discontinuous Galerkin (RKDG) method in quadrature free form, see Cockburn and Shu [10] and references therein. With it, one expresses G as a linear combination of basis functions bn within each control volume, |
| File Format | PDF HTM / HTML |
| Alternate Webpage(s) | http://www.ilass.org/2/conferencepapers/47_2013.pdf |
| Language | English |
| Access Restriction | Open |
| Content Type | Text |
| Resource Type | Article |
