This set of Advanced Computational Fluid Dynamics Questions and Answers focuses on “Diffusion Problem – Least-Square Gradient for Cartesian Grids”.
1. Which of these statements is true?
a) The Gauss-Gradient method is a special case of the least-square gradient method
b) The least-square gradient method is a special case of the Gauss-Gradient method
c) The least-square method is not connected with the Gauss-Gradient method
d) The least-square method is suitable only for the Cartesian grids
Explanation: The least-square method is a general method. This can be reduced to the Gauss-Gradient method by altering some values. It is obtained by applying the least-square method to a Cartesian grid system.
2. What is the advantage of the least-square method over the other methods?
a) Computational ease
Explanation: The least-square method offers more flexibility over the other methods to compute gradients for choosing between the order of accuracy used and the stencil on which to work. According to the user’s ease, these two can be chosen.
3. What is the disadvantage of using the least-square method?
b) Less convergence rate
d) Computational cost
Explanation: The cost for the advantage of the least-square gradient methods is its high computational cost. This high computational cost is due to its separate calculation of the weighted average.
4. The weight used in the least-square method is a function of __________
a) twice the distance between the vertex and the centroid of the cells
b) square of the distance between the vertex and the centroid of the cells
c) inverse of the distance between the vertex and the centroid of the cells
d) the distance between the vertex and the centroid of the cells
Explanation: Generally, the weight used is based on the distances. We can choose it to be the inverse of the distance between the vertex and the cell centroids. Mostly, the weight is a function of the distance between the vertex and the cell centroids.
5. In the least-square method, the gradient is computed using ___________
a) trial and error method
b) optimization method
c) weighted average
d) predictor-corrector method
Explanation: The least-square method is based on an optimization procedure. This optimization carried over a function which is a function of the weight used, the gradients and the grid sizes in the x, y and z-directions.
6. The gradient is found in the least-square method by solving ____________
(Note: n is the number of dimensions)
a) a system of n equations
b) a single equation
c) a system of n2 equations
d) a system of 2n equations
Explanation: A set of equations with the number of equations equal to the number of dimensions is formed using some conditions. This is written in the matrix form and the matrix is solved to get the gradients.
7. The least-square method is exact when ____________
a) the system is two-dimensional
b) the system is linear
c) the system is two-dimensional
d) the system is quadratic
Explanation: Unless the field where we solve the problem is linear, the solution using the least square method will not be exact. This is because the number of columns in the coefficient matrix will be more than the number of rows.
8. While using the Cartesian grids, the coefficient matrix becomes ____________
a) a square matrix
b) an upper triangular matrix
c) a diagonal matrix
d) a lower triangular matrix
Explanation: The coefficient matrix in the least-square method gives a square matrix. While substituting the required values in this square matrix for the Cartesian grids, the matrix is reduced into a diagonal one.
9. The least-square method is ____________
a) at least first-order accurate
b) at least second-order accurate
c) first-order accurate
d) second-order accurate
Explanation: The accuracy of the gradient found using this method is at least first-order. This can be proved by expanding the values of the flow variable using the Taylor series. It can be of higher orders also.
10. The diagonal elements of the coefficient matrix obtained by applying the least-square to Cartesian grids are ___________
a) the ratio of the grid sizes and the weights in the x, y, z-directions
b) the product of the grid sizes and the weights in the x, y, z-directions
c) the weights in the x, y, z-directions
d) the grid sizes in the x, y, z-directions
Explanation: The coefficient matrix obtained by applying the least-square to Cartesian grids has non-zero elements only in the diagonal. These non-zero elements are Δx, Δy and Δz (the grid sizes in the x, y, z-directions).
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