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Correct answer is (a) a LINE integral

For EXPLANATION: For a one-dimensional flow, the volume of the control volume (cell) is in a SINGLE dimension which is the length of that cell. So, there will be no need for a volume, area or surface integrals. It is enough to INTEGRATE over the length of that cell.

The correct answer is (b) PRODUCT of the integrand at the control volume CENTRE and the volume of the control volume

The explanation is: Using the midpoint rule, the volume integral is APPROXIMATED as the product of the integrand at the centre of the cell (the control volume) and the volume of the cell. REPRESENTING it mathematically,

∫VqdV = qP×ΔV.

Right ANSWER is (c) Triangle

Easy explanation: A tetrahedron has four TRIANGULAR faces. Each face is formed by connecting three vertices of the tetrahedron. This is one of the most common shapes of element USED in UNSTRUCTURED GRIDS.

The correct choice is (b) need less computational cost

Easy explanation: Accuracy wise, both the grids result in the same level of accuracy as they do not involve any approximation. But, the unstructured grid algorithm needs less computational cost as it is written in TERMS of the GLOBAL INDICES. This can be ADOPTED for structured grids ALSO.

Correct option is (d) ∇Φk = \(\frac{1}{V_k}(- \Sigma_{n\leftarrow f<K}\Phi_n\vec{S_n}+\Sigma_{n\leftarrow f>k} \Phi_n\vec{S_n})\)

For explanation: The formula for the GRADIENT, in general, is the summation of the product of the flow variables and the area of the faces. As the direction matters here, the summation will be NEGATIVE till ‘n’ reaches ‘k’. Therefore,

∇Φk = \(\frac{1}{V_k}(- \Sigma_{n\leftarrow fk} \Phi_n\vec{S_n})\).

The correct answer is (a) Pre-defined shape functions

For 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.

Correct choice is (a) fnSn

The explanation is: For surface INTEGRAL, the value of the integrand at the face centre is USED in the MIDPOINT approximation. The NEIGHBOURING values are not used. So,

∫Snfd\(\vec{S}\)=fnS

The correct option is (d) its GEOMETRIC centre

The EXPLANATION: First, the geometric centre of the POLYHEDRON is located. This is the apex of the pyramid. The distance of this apex from the FACE of the element gives the HEIGHT of the pyramid. From the base and height of the pyramid, its volume is calculated.

The CORRECT CHOICE is (b) Pyramids

The EXPLANATION: The first step of calculating the volume and CENTROID of any three-dimensional element is to pyramids. The summation of the volumes of these pyramids is the volume of the whole polyhedron.

The correct answer is (c) Shape functions or interpolation profiles

For explanation I would say: 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.

Correct option is (c) Second-order

Easiest EXPLANATION: The given diagram REPRESENTS the Trapezoidal rule for the approximation of an integral. It uses the VALUES at the starting and the ending of the function. The order of ACCURACY of the Trapezoidal rule is TWO.

Right ANSWER is (d) 4 VERTICES

Best EXPLANATION: At least four points are needed to make an element. A TETRAHEDRON is a shape which has the least number of vertices and the least number of faces. It has 4 faces and 4 vertices. A 2-D element can be formed USING three vertices.

The CORRECT option is (d) COMPLEXITY

Easy explanation: All these properties such as consistency, stability and ACCURACY depends upon the method used for discretization. They do not depend on whether the grid is structured or unstructured. The cost of increased FLEXIBILITY in unstructured GRIDS is its complexity.

The correct option is (b) Φe, Φw

The BEST I can explain: Discretization indexing is the type of indexing which is used locally while mentioning the NEIGHBOURS of an element. Here, Φe is the EASTERN node; Φw is the western node; Φn is the northern node; Φs is the SOUTHERN node.

Correct answer is (c) indices

Best explanation: The topological (spatial relation) INFORMATION is embedded in the MESH structure through the INDEXING system. This also LEADS to greater efficiency in coding, cache UTILIZATION and vectorization.

The correct CHOICE is (a) face CENTRE

To ELABORATE: The convective and diffusive fluxes undergo a SURFACE integral. To get the surface integral, they are integrated over the area of the face considered. While approximating USING the mean value approximation, the central value is used. So, the value at the face centre is used.

The correct answer is (a) \(\Sigma_{k=1}^4 \int_{S_k} f d\vec{S}\)

Explanation: In a two-dimensional flow, the number of faces BOUNDING a control volume is four. So, the summation of the integrals along these four faces will be equal to the total flux of the control volume. Here, f may be convective or DIFFUSIVE flux.

The correct OPTION is (c) SOURCE TERM

The BEST explanation: The convection and diffusion terms need surface integrals and not volume integrals. The source and rate of change terms need volume integral. Since the flow taken is a steady flow, the rate of change term will be equal to zero. So, in this case, only the source term needs a volume integral.

Correct answer is (b) Trapezoidal RULE

Easy explanation: The trapezoidal rule ASSUMES that the dependent VARIABLE varies LINEARLY with the independent variable. So, the mean of the values at the endpoints is taken to calculate the integral value.

Correct answer is (d) Dot product of the surface VECTOR and the vector joining the element’s centroid with the FACE’s centroid

Explanation: The vector joining the centroid of the element with the centroid of the face always POINTS outwards. So, the dot product of this vector with the surface vector is FOUND. If the sign of the dot product is POSITIVE, the surface vector points outwards. Otherwise, it points inwards.

Correct answer is (b) FINITE CONTROL volume fixed in space

Best explanation: Finite volume method USES the conservation equation in the INTEGRAL form without any substantial derivative. This can be given by a finite control volume fixed in space. This directly results in the conservative integral form of the governing equations.

Right option is (d) convection TERM, SOURCE term

The 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).

The CORRECT choice is (a) Cell-centred and Vertex-centred

To explain: 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.

The correct ANSWER is (b) cell centre

To explain I would SAY: SOURCE term integral is a volume integral. APPROXIMATING the volume integral, the value at the centre of the cell and the volume of the cell are multiplied. So, the value at the cell centre is used.

Correct CHOICE is (c) \(\frac{S_w}{6}\)(fnw+4fw+fsw)

To elaborate: The Simpson’s rule uses values of the integrand at THREE points – centre of the FACE and the two vertices in the same face. It is given by

\(\frac{S_w}{6}\)(fnw+4fw+fsw).

Correct ANSWER is (d) Triangles

To elaborate: The faces of a 3-D element are 2-D polygons. With the intention of finding the AREA of these faces, they are divided into triangles of DIFFERENT type and SIZES. Triangles give the advantage of having any length in all three of its sides.

The correct answer is (a) Quadrilateral, 4

To EXPLAIN: For a two-dimensional grid, there are four faces and four vertices for every element. And, each element is surrounded by four elements. With this, we cannot SAY if they are cubes or CUBOIDS. In general, we can say they will definitely be quadrilaterals.

Right CHOICE is (d) Overlapping elements

Easy 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.

Correct CHOICE is (a) Using spatial variation

The explanation: The mean VALUE theorem APPROXIMATES the function to be linear. If the real spatial variation of the function is found and the TWO are SUBTRACTED, the order of accuracy can be obtained.

Right choice is (d) three terms

For explanation: For a two-dimensional flow and the surface integral, the midpoint rule NEEDS only one TERM (the value of the integrand at the FACE CENTRE). The trapezoidal rule needs two terms (the value of the integrand at the nodes above and below the face centre). The Simpson’s rule needs three terms (the value of the integrand at the face centre and at the nodes above and below the face centre).

The correct choice is (C) Polyhedron

Explanation: The number of SIDES in an ELEMENT is not RESTRICTED. The GENERAL shape of a two-dimensional element can be named polygon which can have any number of sides. The same way, the general shape of a three-dimensional element can be named polygon which can have any number of faces.

Right choice is (d) vectorization

The best EXPLANATION: As it is EASY to index the elements in structured grids, the values are stored USING global indexing which does not need three-dimensional indexing. So, it can be vectorized and saved as a single array of elements. This SAVES computer MEMORY.

The correct answer is (b) Global index has a single index and LOCAL index has multiple indices

Easiest explanation: Local indexing is the indexing of elements AROUND a PARTICULAR cell. It has to mention the direction ALSO. So, it involves multiple indexing. Global indexing is just a single number representing a cell or ELEMENT.

The correct choice is (b) easy access to elements, less memory for storage

To explain: Structured grids have properly ORDERED cells. So, it is easy to access their elements or GRID points. As access is easy, it is easy to STORE the values ALSO in memory. So, less memory is REQUIRED.

The correct CHOICE is (d) vertices

For explanation I would say: 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.

Correct choice is (C) Second-order

Explanation: The midpoint rule is the method which uses the value of the function only at its midpoint to APPROXIMATE the INTEGRATION. It is second order accurate. This is the simplest method of APPROXIMATION.

Right CHOICE is (c) \(\frac{1}{2}\)(fne+fse)

For explanation I would say: The trapezoidal rule is a second-order accurate approximation. It needs the values of the integrand at two points. Here, as we need the SURFACE integral in the eastern face, the VALUE is approximated using the northern and the southern nodes of the eastern face.

∫Sefd\(\vec{S}=\frac{1}{2}\) (fne+fse).

The correct choice is (c) Area-weighted average of the CENTROID of the sub-elements

Easiest EXPLANATION: The centroid of each of the sub-elements are first LOCATED. The weighted average of these centroids is the centroid of the whole face. Areas of the TRIANGLES are USED as the weight for this average.

The CORRECT CHOICE is (b) HALF of the vector product of two sides

The explanation is: Sub-elements are in triangular shape. Half of the base times height is the formula generally used to find the area of a TRIANGLE. But, CFD uses the vector-based formula to make it algorithmically easier, which is GIVEN by half of the vector product of two sides.

The correct option is (C) Geometric CENTRE of the face

Easiest explanation: The sub-elements are all triangular in shape. The apex of these TRIANGLES are all the same POINT. The geometric centre of the face is chosen as the apex of the triangles. The geometric centre is not the centroid of the POLYGON.