Solving the steady and transient state 2D heat conduction problem using discretization methods and iterative solvers- Jacobi Gauss Siedal and SOR method for both implicit and explicit time solutions

Perform Gradient Descent in Python with any loss function Let us consider the below dataset for performing gradient descent: x = [1,2,3,4,5] y = [2,4,6,8,10] The loss function to be used is the mean square error which is defined as: `MSE = sum_(i=1)^n ( y_a - y_p)^2/n` n = total number of data points `y_p` = predicted…

1)Calculate all 4 business moments using pen and paper for the below data set? First business moment The first business moment is the measure of central tendency Mean or expected value, `mu = sumx.P(x)` For the data, we calculate the data as follows: `mu = sumx.P(x) = 1.4` Second business moment The second business…

Why there is a difference in the formula of variance for population and sample? The formula used for determining variance for population is `sigma^2 = sum(x_i - mu_x)^2/N` Whereas, the variance for a sample is calculated using `s^2 = sum(x_i - mu_x)^2/(N-1)` It can be noted that the denominator for sample variance is reduced…

Aim To setup a full hydrodynamic case of PFI and perform simulation to determine the engine characteristics Introduction Using the boundary flagging and mesh motion obtained from the no-hydro case, this project involves spray and combustion modeling to perform a full hydrodynamic case of Port Fuel Injection system. The…

Reynolds- Averaged Navier Stokes(RANS) equation RANS equations are time-averaged equations governing the motion of fluid flow. The methodology of reynold's decomposition, wherein an instantaneous property is decomposed into mean or time-avergaged and fluctuating components, is used to approximate the NS equations. RANS…

Aim To simulate a case of shock tube using converge CFD and study the pressure and temperature characteristics Introduction A shock tube is an instrument wherein shock waves are generated. The shock wave is produced inside a shock tube when the diaphragm separating a high pressure region (driver medium) and a lower region…

Aim To perform a conjugate heat transfer analysis for a case of air flow inside a hollow cylinder Introduction Conjugate Heat Transfer is the heat transfer occurring at the interface of solid and fluid medium. This heat transfer is a virtue of temperature difference and finds application in various automotive and aerospace…

Aim To simulate the Prandtl Meyer Shock Problem using converge CFD Introduction and Theory 1.1 Shock wave A shock is a type of propagating disturbance that moves faster than the local speed of sound in the medium. There is abrupt, nearly discontinuous, change in pressure, temperature, and density of…

Aim To simulate a transient state case of fluid flow over a throttle body Objective To set up and run the transient state simulation for flow over throttle body and post process for the following: Mesh profile (i.e surface with edges) Plots for pressure, velocity, mass flow rate and total cell count Animation capturing…

Aim To simulate a steady state case of fluid flow over a throttle body Introduction Throttle body finds its application in mechanical devices where the fluid flow is to be controlled by constriction or obstruction. The most common application is in the internal combustion engines where a butterfly valve is used to control…

Aim To simulate the case of backward facing step (BFS) flow using converge CFD Introduction The backward step flow is a representative model for separation flows and is widely studied for its applications in aerodynamic flows, engine flows, and vehicles and also extended to flow over boats and buildings. The accuracy of…

Aim To set up an channel flow case in Converge CFD and simulate the flow for different mesh grid sizes and plot the output parameters using Paraview Introduction An open channel flow is a type of fluid flow with a free surface. Free surface can be defined as the surface with zero shear stress and constant normal stress.…

Aim To write a code to import all the reactions from the GRI mechanism and calculate the 10 most sensitive reactions with respect to temperature parameter Introduction Sensitivity analysis is the study performed to determine the influence of a reaction parameter such as species concentration, pressure, temperature etc.…

Aim To simulate the case of auto ignition of methane fuel to determine the ignition delay and study the influence of temperature and pressure on the ignition delay Introduction The auto ignition temperature is the lowest temperature at which the fuel mixture spontaneously ignites without an external source of ignition. …

Aim To solve a set of coupled ordinary differential equations using multivariate Newton Rhapson solver Introduction For an ordinary differential equation with independent variable t and dependent variable y, using Newton Rhapson method, the new value can be calculated as ynew = yguess – `f(y_(guess),t)/f^'(y_(guess),t)`…

When does a ODE become unstable? An ordinary differential equation (ODE) is a differential equation with an independent variable and the derivatives of that variable. In computational mechanics, ODE is solved using a numerical integration method and the solution is said to converge when the absolute error between the iterations…

Derivation of reaction rate of species Reaction mechanism: `CO + O_2 => CO_2 + O` `O + H_2O => 2 OH` `CO + OH => CO_2 + H` `H + O_2 => OH + O`   The first step is to identify the species and elementary reactions with index j and i respectively j Species 1 CO 2 `O_2` 3 `CO_2` 4 O 5 `H_2O` 6 OH 7 H The elementary…

Aim To study the effect of preheating air on combustion of methane fuel and plot the variations in the adiabatic flame temperature and combustion efficiency using Cantera Introduction A recuperator is a counter-flow energy recovery heat exchanger device positioned within the air inlet and outlet systems. The exhaust gases…

Aim To determine the adiabatic flame temperature for combustion of hydrocarbon fuels and study the trend for variation in equivalence ratio, number of carbon atoms, type of carbon bond using Python and Cantera Introduction In a chemical reaction, the conversion of reactants to products involves heat transfer in the system.…

Aim The project aims at simulating the flow characteristics of lubricant substance in a 2D gear box to study the sloshing effect. The study is extended by using two different lubricants, namely, engine oil and n-heptane and by varying the immersion volume to study the effects on fluid flow.     Introduction…

Aim To simulate axisymmetric 2D combustion of methane fuel in presence of air and study the characteristics of species. The study is further extended to investigate the influence of water content in the fuel on emissions of CO2, NOx and soot formation. Introduction Combustion of a fuel in presence of air results in release…

Aim To perform a parametric study on a gate valve assembly and to simulate the flow in the pipe for different valve openings Introduction A gate valve is a mechanical component assembly used to control the flow in a pipe having pressure difference across the inlet and outlet. The components of gate valve include a gate…

Aim To perform analysis on cyclone separator and calculate the separation efficiency and pressure drop.   Introduction Cyclone separators, also known as cyclones, are used to separate particles from a carrier gas. It employs a centrifugal force by spinning gas stream inside the separator geometry. It is widely…

Aim To perform CHT analysis on a basic graphic card to study the temperature characteristics of components and determine the maximum temperature attained by the processor in ANSYS Fluent Introduction A graphic card or a video card is a specialized electronic circuit designed to accelerate the creation of images displayed…

  Aim To simulate Rayleigh-Taylor instability using VOF method in ANSYS Fluent Introduction Rayleigh-Taylor instability is a type of instability occurring at the interface of two fluids with difference in densities in the presence of gravitational field. The fluid with higher density is located above the less dense…

1. Introduction Conjugate heat transfer (CHT) analysis is the term used for studying the heat transfer in solids and fluids where the convective heat transfer takes place at the interface of the solid and fluid flow. The heat transfer is dependent on the temperature difference of the two mediums. CHT analysis can be extended…

Introduction External car aerodynamics plays a significant role in reduction of fuel consumption, noise and driver stability. With manufacturers subjected to stringent limitations of carbon dioxide, NOx emissions and market demand to continually improve the efficiency and mileage, immense resources are expended in studying…

Objective The project involves solving the heat conduction equation in 2D for transient and steady state condition. The differential equation is solved using iterative solvers by incorporating Jacobian, Gauss Siedal and Successive over relaxation (SOR) method Assumptions Consider a system defined by the below parameter:…

Objective To solve the system of equations for quasi steady convergent-divergent 1D nozzle using Maccormack method applying both conservative and non-conservative form of equations   Flow field 1. Consider an steady,  isentropic flow 2. Since the flow is quasi 1D flow, the properties vary only along the x-axis…

4th order approximation of the second order derivative using Taylor table method 1. Central difference scheme For second order derivative(p=2) and fourth order approximation (q=4), the number of nodes reqd = p+q-1= 5 The equation for the second order derivative,say, dx^2*f \'\'(x) = a*f(i-2)+b*f(i-1)+c*f(i)+d*f(i+1)+e*f(i+2) …

Interpolation schemes Unlike the finite difference method, in finite volume method (FVM), the integral form of the governing equations are solved considering an arbitrary volume with central point P (about which the properties in the control volume are averaged). Now, in integrating the governing equation about the control…

Objective To study the icoFoam solver and simulate flow through backward-facing step     Specification The number of cells along the direction of flow = 200 The number of cells along the y-direction for each block = 10 Inlet Velocity is 1 m/sec.    Geometry Define the geometry using the vertices as…

Objective Using the program algorithm derived for wedge Boundary Condition, simulate the flow through pipe using symmetric and compare the results for three different angles- 10, 25 and 45 deg. Also, to plot comparison for the results of  HP vs wedge BC vs symmetric BC using Paraview.   Matlab code for blockMeshDict…

Objective To simulate the velocity profile for a laminar flow of water through a pipe using icoFoam solver. In addition:  To write a program in Matlab that can generate the computational mesh automatically for any wedge angle and grading schemes To show that the velocity profile is fully developed To post process…

2-D steady and unsteady state flow simulation over a cylinder to demonstrate vortex shedding and study the relationship of force coefficients with Reynolds number Abstract One of the most complex phenomena in the field of fluid dynamics is the Von Karman vortex shedding, the periodic oscillation developed at the wake region.…

Abstract The project involves studying the flow characteristics- velocity and temperature in T junction where cold and hot air mixing occurs. The study compares the simulation results of a particular case using two turbulence models, K-Ɛ realizable and K-ω SST, in order to select the most appropriate solver for the…