|
Nodal Integral Methods for Heat and Mass Transfer in Curvilinear Coordinates
ABSTRACT: The main advantage of using the Nodal integral methods (NIM) in solving partial differential equations (PDEs) is that they provide an accurate solution over relatively coarse meshes compared to other numerical methods. This is because of the use of approximate analytical solutions within cells that more accurately capture the physical processes modeled by the governing differential equations than the polynomials often used in other schemes. In the NIM, the PDEs are reduced to a set of ordinary differential equations by applying the transverse integration procedure. This step requires the discretized cells to be rectangles or cuboids. However, it is difficult to capture the desired geometries using such elements in most practical applications. Very fine elements near the boundaries are required, which negatively impacts the efficiency of nodal integral methods. More complicated shapes, such as quadrilateral and hexahedral shapes, are often needed to represent complex geometries. Therefore, in order to exploit the NIM for complex domains, the applicability of the NIM should be extended to irregularly shaped elements.
This work aims to extend the applicability of the NIM to arbitrary geometries while keeping the same order of accuracy as in the traditional NIM for regularly shaped elements. This is achieved by deriving the NIM in curvilinear coordinates and using it to solve PDEs in domains discretized using quadrilateral elements in 2D and hexahedral elements in 3D. The quadrilateral and hexahedral elements in the Cartesian system are mapped into rectangular and cuboidal elements in curvilinear coordinates. The mapping functions are derived using Lagrangian interpolation functions. All mathematical operators essential for the development of the NIM, such as the transverse-integration operators and continuity conditions, are derived in curvilinear coordinates. A second-order approximation for the transverse-integration operators is developed in curvilinear coordinates.
The approach developed in this dissertation is used to solve the 2D and 3D, time-dependent convection-diffusion and Navier-Stokes equations. The numerical scheme developed for the Navier-Stokes equations is based on the Poisson pressure equation (PPE) instead of the continuity equation. The governing PDEs are transformed into curvilinear coordinates, where the standard NIM procedure is applied. Transforming the governing PDE to the new coordinate system converts the isotropic diffusion to anisotropic diffusion, introducing cross-derivative terms, and thus making the steps in the derivation of the NIM more complicated. In this dissertation, a new approximation for the cross-derivative terms based on the discrete unknowns of the NIM is developed. The order of accuracy of the new approximation is consistent with the order of accuracy of the traditional NIM. Moreover, Neumann boundary condition for the PPE is developed based on the momentum equation normal to the boundary. Padé approximation is used to approximate the second derivative of the normal velocity in arbitrary geometries using only four points stencil in 2D and five points stencil in 3D.
Several numerical test problems are solved to assess the accuracy and efficiency of the scheme. The results of all test cases show that the NIM can provide accurate solutions over coarse meshes even for highly distorted quadrilateral and hexahedral elements. The scheme is second-order accurate in space and time for all deformation levels and flow conditions studied, which is the same as the order of accuracy of the NIM for regularly shaped elements. Compared with other second-order finite-volume schemes, NIM is found to have a better convergence rate and be more accurate than the other schemes considered in this work.
In summary, highly efficient, second-order accurate nodal schemes are developed and tested in curvilinear coordinates for the convection-diffusion and Navier-Stokes equations.
|
