This set of Tough Computational Fluid Dynamics Questions and Answers focuses on “Variable Arrangement in FVM”.
1. What are the two possible variable arrangements for the finite volume method?
a) Cell-centred and Vertex-centred
b) Cell-centred and Face-centred
c) Face-centred and Vertex-centred
d) Face-centred and Boundary-centred
Explanation: Unlike FDM, in FVM, the grid points are taken inside the elements. There are two ways of arranging these elements for a finite volume method. They are cell-centred and Vertex centred arrangements.
2. The variables are calculated at the __________ in the vertex-centred arrangements.
Explanation: In the vertex-centred arrangements, all the flow variables and their related quantities are calculated and stored in the vertices. Elements are constructed around these vertices using different methods.
3. Dual cell or dual mesh method is used to __________
a) create boundary elements
b) create elements around a vertex
c) create cell-centred arrangement
d) create boundary faces
Explanation: In the vertex-centred arrangements, elements are created around the vertices by joining either their centroids or their centroids with the face-centroids. This method of forming the vertex-centred arrangement is called dual cell or dual mesh method.
4. Consider a 2-D finite volume problem. A vertex-centred arrangement is created by connecting the centroids of the elements sharing the vertex. What is the problem that may arise because of this arrangement?
a) Unstructured elements
b) Conjunctional elements
c) Orthogonal elements
d) Overlapping elements
Explanation: The above-mentioned method is a method of creating the vertex-centred arrangement. As mentioned, elements are created by joining the centroids of the surrounding elements. This way, the lines connecting the centroids may overlap and result in overlapping elements.
5. Consider a 3-D problem. Non-overlapping elements can be created by joining __________
a) face-centroids and edge-centroids
b) cell-centroids and edge -centroids
c) cell-centroids, face-centroids and edge-centroids
d) cell-centroids and face-centroids
Explanation: To overcome the problem of overlapping elements, in two-dimensional problem, the cell-centroids and the face-centroids are connected. Extending this to a three-dimensional problem, cell-centroids, face-centroids and edge-centroids are connected. Therefore, ensuring that the connecting lines do not overlap.
6. In vertex-centred arrangements, the variables at the vertices are only known. How is the variation of variables in these elements calculated?
a) Interpolation profiles or Taylor series expansion
b) Shape functions or Taylor series expansion
c) Shape functions or interpolation profiles
d) Shape functions or Fourier series expansion
Explanation: In vertex-centred arrangements, the flow variables are calculated and stored at the vertices only. The variation of flow properties through the element can be obtained by using either shape functions or interpolation profiles.
7. Vertex-centred approach gives accurate solution for ___________ but not for _________
a) diffusion term, convection term
b) source term, convection term
c) convection term, diffusion term
d) convection term, source term
Explanation: Vertex-centred arrangements give an accurate resolution of face fluxes (surface integrals) such as convection and diffusion terms. But, they yield a lower order of accuracy for element based integrations (source terms).
8. Cell-centred FVM is __________ accurate.
Explanation: Cell-centred arrangements have predefined elements and their centroids are the grid points. Here, the elements are identical to discretization elements and they are second-order accurate.
9. Which of these is needed for a vertex-centred arrangement and not for cell-centred elements?
a) Pre-defined shape functions
Explanation: Cell-centroids are must for cell-centred arrangements as they are the grid points here. Face-centroids and edge-centroids are also needed here during numerical integration. Pre-defined shape functions are not needed as they use general polygonal elements.
10. The cell-centred arrangements are disadvantageous when we have ___________
a) Non-conjunctional and non-orthogonal elements
b) Conjunctional and orthogonal elements
c) Unstructured and non-orthogonal elements
d) Structured and orthogonal elements
Explanation: The treatment of non-conjunctional elements and the manner the diffusion term is discretized for non-orthogonal grids are disadvantages of cell-centred arrangements. They cannot produce good results in these cases.
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