Derive the `4^(th)` order approximation of the second-order derivative using central difference scheme let f be the function whose second derivative is approximated at `i^(th)` point using function values at (i-2), (i-1), i, (i+1), (i+2) points. let h be the distance between the consecutive points.  let `(d^2f)/dx^2=a*f_(i-2)+b*f_(i-1)+c*f_i+d*f_(i+1)+e*f_(i+2)+O(h^n)`…

2-D steady state is solved first using the three methods. The three methods are formulated in MATLAB as follows. close all clear all clc % Solving steady state 2D heat conduction equation % Domain is a square of length equal to one n=20; % Number of nodes in x and y direction % Discretization of domain x=linspace(0,1,n);…

In this assignment, unsteady flow through the 1-D convergent-divergent nozzle is explored using MacCormak's technique. The transient flow is solved using finite difference methods on the non-dimensionalized governing equation's conservative and non-conservative forms. Separate functions are created for conservative and…

The appropriate solver for the given flow condition is icoFoam. From the tutorials folder of the OpenFoam, the cavity tutorial which uses icoFoam solver is copied to the run folder. The blockMeshDict file is edited as below according to the following configuration.    blockMeshDict file /*--------------------------------*-…

Finite Volume Method (FVM) is a numerical technique for solving partial differential equations governing the phenomena. The method involves discretizing the domain into smaller volumes. These control volumes are connected by common faces. The governing equation in integral form is applied for all the control volumes followed…

The following is the simulation of a laminar flow of an incompressible fluid through the pipe in OpenFoam.  The following figure depicts the physical situation of the flow. For a laminar flow, the hydrodynamic length, `L_h=0.06*D*Re_D`, where `D` is pipe diameter, and `Re_D` is Reynolds number…

Unlike in the previous assignment, where the simulation is performed using a transient solver icoFoam, the simulation is carried out using a steady-state solver simpleFoam since the emphasis is on the steady-state flow field.  This assignment aims to compare the boundary conditions of Wedge and Symmetry…

This assignment aims to evaluate the mixing effectiveness of a Tee joint with two different outlet pipe lengths, namely small and long. The required Tee geometry is provided, and the interior volume for CFD analysis is extracted as shown below.    Tee geometry with shorter outlet pipe:  Extracted…