Derive the following 4th order approximations of the second-order derivative.  1. Central difference Based on hint (i), central difference scheme for second order derivative will need 5 nodes in the numerical stencil to get an order of accuracy of 4. `P = N - Q + 1` P = 4, Q = 2. Therefore N = 5 Formulating discretized…

1. Solve steady 2D heat conduction equation by using the point iterative techniques `(del^2T)/(delx^2) + (del^2T)/(dely^2) = 0` Discretizing the partial differential terms using truncated Taylor series - `i` and `j` indices represent x and y node position respectively `( del^2T)/(delx^2) = ( T_((i-1)(j))…

Isentropic flow through a quasi 1D subsonic-supersonic nozzle: The nozzle is assumed to be have flow variations in one dimension only - x. Hence the primitive variables such as -> x-velocity(`V`), pressure(`p`), density(`rho`) and temperature(`T`) vary along x but stays constant within the cross-section area at a particular…

Transient Incompressible-laminar-viscous flow simulation through a backward facing step: 1. Uniform Mesh: IcoFoam solver is used to solve transient incompressible laminar flow problems in openFoam. Hence the same is chosen for our simulaiton here. Couple of tutorials are available for icoFoam solver in the package ->…

1. Finite Volume Method(FVM): FVM is a method of discretizing partial differential governing equation of fluid flow into numerical algebraic equations. The domain, where the solution of partial differential equations are to be obtained, is split into multiple finite volumes. The finite volumes could be of any arbitrary…

Incompressible laminar flow through a pipe: Hagen-Poiseuille's equation provides analytical solution for pressure drop due to fluid viscosity in a pipe of constant cross-section having incompressible laminar flow. The goal of this exercise is to simulate Hagen-Poiseuille's flow with axisymmetric approximation in OpenFoam…

Laminar pipe flow in OpenFOAM using symmetry and wedge BC: In last exercise, the pipe flow was simulated using wedge BC on a pipe sector with `2^0` wedge section, 11.55 [m]  length and 0.01 [m] diameter. The same pipe dimensions are used for this exercise as well with varying sector angles of `10^0, 25^0 and…

Solving scalar transport equation in Openfoam: A scalar transport equation (given below) is to be solved in a computational domain along with the momentum and continuity equation for unsteady incompressible flow. Icofoam solver is customized to include the solution for scalar transport equation for this purpose. …

Using function object to calculate and write total pressure field: Function objects are defined in system/controlDict file in the root directory. They are used to perform additional post-postprocessing of user defined variables on the mesh using the solution data during runtime or at the end of simulation. Different function…

Adding temperature equation to icoFoam solver: Solver setup: 1. IcoFoam solver is copied from $FOAM_SOLVERS/incompressible folder to $FOAM_RUN. The folder is renamed to icoTempFoam. icoFoam.C file is renamed to icoTempFoam.cpp and Make/files is updated for this change as shown below to create the executable named icoTempFoam…

1. Adiabatic flame temperature variation with Equivalence ratio (In constant volume-chamber): Equivalence ration, `phi` is defined as ratio of actual fuel-air ratio to stoichiometric fuel-air ratio `phi = (F//A)/(F//A)_(st)` Combustion of methane with air in stoichiometric conditions is represented by following…

Handling Mixtures with Cantera: The python code to simulate a stoichiometric combustion of Methane and air at constant pressure using cantera quantity object is shown below ''' Stoichiometric combusion of methane using cantera quantity class ''' import cantera as ct # instantiate solution object gas = ct.Solution("gri30.cti")…

Fuel Combustion using recuperator for preheating air: Often times, useful enthalpy in products of combustion is not completely utilized by harvesting devices due to the small time-scales products are present in the system before being replaced by more high energy products from continuous combustion. These enthalpies go…

Compact notation to represent reaction mechanism: In real combusiton applicaitons, there are a lot of species involved including multiple number of reactions. The list of reactions that could occur based on various parameters can be listed out in a mechanism file and solved using computer. But these mechanisms should be…

Stability of ODE: An ordinary differential equation has one independent variable and one or more orders of differential terms of the dependent variable with respect to the independent variable. In general, mathematical stability is defined such that when small pertubations are added to the independant variable, the function…

The following set of ODEs are solved using Euler backward differencing method for time discretization and multivariate Newton Raphson technique to solve each time-step in iterative scheme.  `(dy_1)/(dt) = -0.04*y_1 + 10^4*y_2*y_3``(dy_2)/(dt) = 0.04*y_1 - 10^4*y_2*y_3 - 3*10^7*y_2^2``(dy_3)/(dt) = 3*10^7*y_2^2` Initial…

Auto-ignition time delay with varying temperature and pressure: The auto-ignition time delay is the time taken for the mixture to reach a tempeture of 400 [K] greater then the initial temperature. This parameter is very important fuel property since it determines the ability to start the combusiton of a fuel rapidly. Such…

Sensitivity analysis: In a combustion process of fuel such as methane, there are lot of intermediatory reactions involved. In order to tune reaction mechanisms to match experimental conditions, the effect of each reaction on reactors dependent variables must be known. This is given in terms of sensitivity. First order…

Aim: To simulate flow of air through a square duct in converge. Introduction:The flow is induced in the square duct of 0.1 m length with a pressure difference of 1 Pa. Given the minor change in pressure, the velocity and hence reynolds number would also be insignificant. A laminar flow assumption is valid for…

Aim: To simulate flow through a backward facing step internal flow geometry using Converge with different mesh resolutions. Theory: Backward facing step geometry is a basic geometry that is preferred in the validation of turbulence models or their modifications. Various experimental results are available for this geometry…