Abstract
We present a finite volume method formulated on a mixed Eulerian-Lagrangian mesh for highly advective 1D hyperbolic systems altogether with its application to plug-flow heat exchanger modeling/simulation. Advection of sharp moving fronts is an important problem in fluid dynamics, and even a simple transport equation cannot be solved precisely by having a finite number of nodes/elements/volumes. Finite volume methods are known to introduce numerical diffusion, and there exist a wide variety of schemes to minimize its occurrence; the most recent being adaptive grid methods such as moving mesh methods or adaptive mesh refinement methods. We present a solution method for a class of hyperbolic systems with one nonzero time-dependent characteristic velocity. This property allows us to rigorously define a finite volume method on a grid that is continuously moving by the characteristic velocity (Lagrangian grid) along a static Eulerian grid. The advective flux of the flowing field is, by this approach, removed from cell-to-cell interactions, and the ability to advect sharp fronts is therefore enhanced. The price to pay is a fixed velocity-dependent time sampling and a time delay in the solution. For these reasons, the method is best suited for systems with a dominating advection component. We illustrate the method’s properties on an illustrative advection-decay equation example and a 1D plug flow heat exchanger. Such heat exchanger model can then serve as a convection-accurate dynamic model in estimation and control algorithms for which it was developed.
Highlights
Numerical computation is a fundamental tool for the study of complex phenomena described by sets of partial differential equations (PDE)
Finite volume method (FVM) is one of the numerical solution techniques used for solving fluid dynamics problems
For increasing N, mixed-mesh finite volume method (MMFVM) approximates ever so closely the continuum of the governing PDE; the solution delay of Theorem 2 diminishes and so does the numerical diffusion introduced in Remark 4
Summary
Numerical computation is a fundamental tool for the study of complex phenomena described by sets of partial differential equations (PDE). Picking the FVM, which differs from the latter by engaging the divergence theorem, the main ingredients of the solutions are a spatial reconstruction scheme, numerical flux selection, grid (adaptation) selection, and a time integration method. A breakthrough paper by Godunov [2] presents a first-order upwind method and discusses the monotonicity of the solution schemes. Regarding spatial accuracy, his method is of the first order. FVM may add significant artificial numerical diffusion at the low number of states/nodes; the presented mixed-mesh method eliminates this drawback. The mixed-mesh finite volume method (MMFVM) is defined An application to a simple advection equation, together with a heat exchanger application, is presented at the end of the article
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