Aim:The code to plot the first order, second and fourth order error for a range of dxclear allclose allclcx=pi/3;dx=linspace(pi/5,pi/5000,20);% f(x)=sin(x)/x^3;% f'(x)=(cos(x)*x^3 - 3*x^2*sin(x))/x^6;analytical_derivative =(cos(x)*x^3 - 3*x^2*sin(x))/x^6;for i=1:length(dx) % first order approximation first_order_approximation(i)=(sin(x+dx(i))/(x+dx(i))^3-sin(x)/x^3)/dx(i);…

Objectives: Simulate a supersonic flow through a duct having a sharp corner to observe the formation of Shock waves. Literature review on Shock waves. Literature review on Shock wave boundary conditions. Observe the effect of SGS parameter on shock wave location. Observe the effect of SGS temperature value on cell count.…

Objective: The main aim of this project is simulating the Engine manifold, we can call this Conjugate Heat Transfer. DO Conjugate heat transfer simulation. Calculating the Temperature at the outlet of the manifold. Calculate the Heat transfer Coefficient. 1. Conjugate Heat Transfer: The term conjugate heat transfer is…

Objective: In this project, we were worked on solving ODE\' s of simple reaction Mechanism. For solving ODE\'s we will be using compact notations method which is converted into the matrix for Computer understanding. We have given four reaction mechanisms below CO + O2 -> CO2 + O O + H2o -> OH + OH CO + OH -> CO2…

Objective: The main aim of this project is working on Conjugate Heat Transfer On a Graphics Card. Perform a steady-state conjugate heat transfer analysis on a model of a graphics card.  1. Find out the maximum temperature attained by the processor. 2. Prove that the simulation has achieved convergence with appropriate…

Objective: For this challenge, you will run both steady and transient flow simulations of flow over a cylinder for Re = 100. In both approaches, show that the flow has converged. That is, what quantities or plots need to be looked at to determine that the flow has converged? How would you calculate Strouhal number in the…

Aim:The code to plot the first order, second and fourth order error for a range of dxclear allclose allclcx=pi/3;dx=linspace(pi/5,pi/5000,20);% f(x)=sin(x)/x^3;% f'(x)=(cos(x)*x^3 - 3*x^2*sin(x))/x^6;analytical_derivative =(cos(x)*x^3 - 3*x^2*sin(x))/x^6;for i=1:length(dx) % first order approximation first_order_approximation(i)=(sin(x+dx(i))/(x+dx(i))^3-sin(x)/x^3)/dx(i);…

Objectives of the Project: (1) Perform a transient simulation for the flow of air over a throttle body in Converge Studio and post-process the results in ParaView. (2) Show the plots for pressure, velocity, and mass flow rate. (3) The throttle valve should rotate by a given angle during the simulation time.   Gemoetry:…

MATLAB SCRIPT close all clear all clc %% Input Values gamma=1.4; t3=2800; t1=800; p1=101325; bore=0.1; stroke=0.1; cr=13; con_rod=0.15; v_s=(pi/4)*bore^2*stroke; v_c=v_s/(cr-1); %% Find out the values of t2,t4,p2,p3,p4,v1,v2 v1=v_s+v_c; v2=v_c; v3=v2; v4=v1; p2=p1*cr^gamma; t2=p2*v2*t1/(p1*v1); p3=p2*t3/t2; p4=p2*(v3/v4)^gamma;…

close allclear allclc% input variablesA=[1,2,3,4,5];B=[1;2;3;4;5];C=A*B;D=B*A;E=A(1:end);F=A(1:2);G=B(1:2,3)H=(length(B)-2);At line number 15 'H' their is error because its defined row vector but above B matrix their is cloumn matrix error: A(I,J): column index out of bounds; value 3 out of bound 1

% matlab program to calculate the stochiometric coefficient cumbution clear all close all clc % for first 20 carbon atoms in hydrocarbon range ar=zeros(1,20); x=linspace(1,20,20); for i=1:length(x) alkane(i)=3/2*i+1/2; alkene(i)=3/2*i; alkyne(i)=3/2*i-1/2; end % ploting grid on hold on plot(x,alkane) plot(x,alkene) plot(x,alkyne)…

Objective: This project we are working on calculate the 1D Flame Speed Analysis for Methane and Hydrogen. You need to use a separate Hydrogen mechanism to perform Flame Speed Analysis for Hydrogen. What is Flame speed?: The flame speed is the measured rate of expansion of the flame front in a combustion reaction.…

% matlab program to calculate the stochiometric coefficient cumbution clear all close all clc % for first 20 carbon atoms in hydrocarbon range ar=zeros(1,20); x=linspace(1,20,20); for i=1:length(x) alkane(i)=3/2*i+1/2; alkene(i)=3/2*i; alkyne(i)=3/2*i-1/2; end % ploting grid on hold on plot(x,alkane) plot(x,alkene) plot(x,alkyne)…

% matlab program to calculate the stochiometric coefficient cumbution clear all close all clc % for first 20 carbon atoms in hydrocarbon range ar=zeros(1,20); x=linspace(1,20,20); for i=1:length(x) alkane(i)=3/2*i+1/2; alkene(i)=3/2*i; alkyne(i)=3/2*i-1/2; end % ploting grid on hold on plot(x,alkane) plot(x,alkene) plot(x,alkyne)…

Aim: To perform the three-dimensional Numerical simulation for a port fuel injected, spark            ignited Gasoline engine  Pre-Analysis-  1) Importing the geometry and checking for surface errors:   The three-dimensional CAD geometry of the engine is imported…

Objective: Perform a combustion simulation on the combustor model and plot the variation of the mass fraction of the different species’ in the simulation using line probes at different locations of the combustor as shown in Fig. You need to plot for CO2, H2O, CH4, N2, O2 and NOx emissions. If you do not obtain the…

Objective: In this project, we are going to calculate the sensitivity Analysis using python and Cantera.  Write a code that takes all the reactions from the GRI mechanism and calculates 10 most sensitive reactions The sensitivity parameter should be with respect to the Temperature (It should find the sensitivity with…

Objectives of the Project: (1) Calculation of Flow Variables. (2) Generation of blockMeshDict using the symmetry boundary condition. Create the blockMeshDict file in MATLAB such that the mesh is automatically created for various wedge angle. (3) Simulating the pipe flow in OpenFOAM using icoFoam solver. (4) Post process…

Objective: The main aim of this project is writing code for 2nd order ODE in python. The way the pendulum moves depends on the Newtons second law. When this law is written down, we get a second order Ordinary Differential Equation that describes the position of the \"ball\" w.r.t time. For each value in time Create an…

Objective: In this project, we are going to learn how species reactions occur in the combustion chamber and solve their ODE\'s using Multivariate Newton Raphson solver but instead of species we are taken y1, y2, y3 and have written Python code for below mention equations. `(dy1)/(dt)=-0.04 y_1 + 10^4 y_2 y_3` `(dy2)/(dt)=0.04…

INTERPOLATION SCHEM: The approximation of surface and volume integrals may require values of the variables at location other than the computational nodes of the cv.values at these locationare obtained using interpolation formula. UPWIND INTERPOLATION:                   The…

1D LINEAR CONVECTION EQUATION: `(du)/(dt)+c*(du)/(dx)=0` The first step to solve the problem is to discretize the equation. The time derivative is discretized using a first-order approximation forward difference scheme, whereas a first-order approximation backward difference scheme is used to discretize the space derivative.…

Introduction : Nozzles are most commonly used in every field in order to study the liquid and air properties, because those two element plays a major role in engineering fields. However we may have many properties and the behaviour of fluids and air but we are still in position to find the exact solutions for the equations…

Aim: To Apply Reynold\'s decomposition to the NS equations and come up with the expression for Reynold\'s stress Methodology:The Reynolds-averaged Navier–Stokes equations (or RANS equations) are time-averaged[1] equations of motion for fluid flow. The idea behind the equations is Reynolds…

Objective: This project is working on the literature review of ODE stability. Solving ODE\'s was very difficult to get an accurate solution because they contain large data, which has been converted into user understanding and computer understanding. In computational we were using a compact notation method for an understanding…

1)Kinematic viscosity is the measure of the inherent resistance of a fluid to flow when no external force is exerted, except gravity.v=dynmic viscosity/densityunits=m^2/s.2)Dynamic Viscosity (also known as absolute viscosity) is the measurement of the fluid’s internal resistance to flow.dynamic viscosity gives…

simple pendulum program clear all close all clc % Solving second order ODE % Let theta = y and angular velocity = v t=linspace(0,20,1050); theta=[0 3]; y=lsode(\"pendulum_fun\",theta,t); figure(1) subplot(3,1,1) plot(t,y(:,1)) xlabel(\'Time\',\'FontWeight\',\'bold\',\'FontSize\',10) ylabel(\'angular displacement\',\'FontWeight\',\'bold\',\'FontSize\',10)…

Objective:. Here are the requirements 1. Your code should create a PV diagram 2. You should output the thermal efficiency of the engine Otto Cycle: An Otto cycle is an idealized thermodynamic cycle that describes the functioning of a typical spark ignition piston engine. It is the thermodynamic…

% matlab program to calculate the stochiometric coefficient cumbution clear all close all clc % for first 20 carbon atoms in hydrocarbon range ar=zeros(1,20); x=linspace(1,20,20); for i=1:length(x) alkane(i)=3/2*i+1/2; alkene(i)=3/2*i; alkyne(i)=3/2*i-1/2; end % ploting grid on hold on plot(x,alkane) plot(x,alkene) plot(x,alkyne)…

Objective: For this challenge, you will have to perform a parametric study on a gate valve model and find out the mass flow rates at each design point. we will have to perform a parametric study on the gate valve simulation by setting 5 design points starting from the initial condition of lift with the least value of mass…

PROBLEM STATEMENT:cofficient matrix:`A=[[5,1, 2],[-3,9,4],[1,2,-7]]`solution matrix`x=[[x1],[x2],[x3]]`right side matrix`B=[[10],[-14],[33]]`we have a 3 gernal linear sytems the equationsAX=B`X=[A^-]B`Eigen values of given matrix=`[A-lamdaI]``A=[[5,1, 2],[-3,9,4],[1,2,-7]]`-`lamda[[1,0,0],[0,1,0],[0,0,1]]`we get the equtions…

Object: In this project, we are writing code for curve fitting. 1. What does popt and pcov mean? 2. What does np.array(temperature) do? 3. What does the * in *popt mean? 4. Write code to fit a linear and cubic polynomial for the Cp data. Explain if your results are good or bad.  5. What needs to be done in order to…

% matlab program to calculate the stochiometric coefficient cumbution clear all close all clc % for first 20 carbon atoms in hydrocarbon range ar=zeros(1,20); x=linspace(1,20,20); for i=1:length(x) alkane(i)=3/2*i+1/2; alkene(i)=3/2*i; alkyne(i)=3/2*i-1/2; end % ploting grid on hold on plot(x,alkane) plot(x,alkene) plot(x,alkyne)…

% matlab program to calculate the stochiometric coefficient cumbution clear all close all clc % for first 20 carbon atoms in hydrocarbon range ar=zeros(1,20); x=linspace(1,20,20); for i=1:length(x) alkane(i)=3/2*i+1/2; alkene(i)=3/2*i; alkyne(i)=3/2*i-1/2; end % ploting grid on hold on plot(x,alkane) plot(x,alkene) plot(x,alkyne)…

Matlab program for three solving approximations Central difference Skewed right sided difference Skewed left sided difference clear all close all clc % function=exp(x)*cos(x) x=pi/3; dx=pi/40; % analytical derivative=-2*exp(x)*sin(x) analytical_derivative=-2*exp(x)*sin(x); % fourth order approximation using central differencing…

MATLAB SCRIPT: close all clear all clc l1=1; l2=0.5; x0=0; y0=0; theta_1=linspace(0,90,10); theta_2=linspace(0,90,10); frame=1 for i=1:length(theta_1) Theta_1=theta_1(i); x1=l1*cosd(Theta_1); y1=l2*sind(Theta_1); for j=1:length(theta_2) Theta_2=theta_2(j); x2=x1+l2*cosd(Theta_2) y2=x2+l2*sind(Theta_2) plot([x1 x2],[y1…

OBJECTIVE: To run the simulation close cycle of CAT3410 Diesel engine with two different pistions W_piston and Omega_based_pistion.And run simulations with both these pistons and characterize the emissions (Soot, Nox and UHC) and showing cut plane animation of these emissions. INTODUCTION: Now-a-days the raising competition…

Objectives of the Project: (1) Perform a steady state simulation for the flow of air over a throttle body in Converge Studio and post-process the results in ParaView. (2) Show the plots for velocity & pressure countur, and mass flow rate. Geometry: The throttle inside the pipe is aligned parallel to the plane of the…

(1) Central difference scheme: `(d^2f)/(dx^2)=(a.f(i-2)+b.f(i-1)+c.(i)+d(i+1)+e.f(i+2))/(Delta*x^2)` The coefficient of 2nd derivative of x is 1 while that of other derivatives is zero and of 6th derivative of x is an unknown finite term (which is considered as the error term).Thus, in the error term, the power of…

Introduction: This project is based on setting up an Backward facing step flow using Converge CFD. The main objective of this project is to study the velocity profiles of a Backward facing block which is meshed by using different kinds of grid points. The changes in these grid points will be helpful to study the velocity…

Stability Analysis: The Stability of a Numerical Scheme depends on how the error propagates in the equation at different time stepsIf the error magnifies during the course of equation solving then it causes the solution to get unstableIf the error remains stable during the equation solving then it remains stable.An…

(1). vanilla option`(delv)/(delt)+1/2sigma^2s^2(del^2v)/(dels^2 )+(r-q)s(delv)/(dels)-rv=0`0<s<+`oo`,t`in`[0,T),`V(S,T)={(S-X)^+,For call,`           {(X-S)^+,For put,(2). Barrier options.`(delv)/(delt)+1/2sigma^2s^2(del^2v)/(dels^2 )+(r-q)s(delv)/(dels)-rv=0,``0<S<H,tin[O,T)``V(S,T)=(S-H)^+``V(H,T)=0,`(3).Arthimatic…

Objective of the project: 1.The main objective of this project is to simulate the laminar flow through pipe. The wedge portion is taken as the whole circular section (cylinder) may take time to simulate or it may be costly if your computer do not have much resources. 2. The other objectives is to write the Matlab code…

Objective: In determining the minimum cushion pressure needed to break a given thickness of the iceusing an air cushion vehicle, Muller (“Ice Breaking with an Air Cushion Vehicle”) inMathematical Modeling: Classroom Notes in Applied Mathematics, SIAM 1987) derivedthe equation where p denotes the cushion pressure,…

Objective: For this challenge, we have created two versions of the mixing tee. One of them is longer than the other. Case 1 (60 points) Short mixing tee with a hot inlet velocity of 3m/s Momentum ratio of 2 Case 2 (60 points) Long mixing tee with a hot inlet velocity of 3m/s Momentum ratio of 2 Case 3 (60 points) Short…

Introduction: Conjugate heat transfer corresponds with the combination of heat transfer in solids and heat transfer in fluids. In solids, conduction often dominates whereas, in fluids, convection usually dominates. Conjugate heat transfer is observed in many situations. For example, heat sinks are optimized to combine…

PROBLEM STATEMENT:cofficient matrix:`A=[[5,1, 2],[-3,9,4],[1,2,-7]]`solution matrix`x=[[x1],[x2],[x3]]`right side matrix`B=[[10],[-14],[33]]`we have a 3 gernal linear sytems the equationsAX=B`X=[A^-]B`Eigen values of given matrix=`[A-lamdaI]``A=[[5,1, 2],[-3,9,4],[1,2,-7]]`-`lamda[[1,0,0],[0,1,0],[0,0,1]]`we get the equtions…

Objective: This project we were working on calculating Adiabatic Flame Temperature that rises from the combustion process using python and Cantera. Two cases were presented to find Adiabatic Flame Temperature, The first case put a volume of methane (CH4) is constant and try different Equivalent Ratios, and the Second case…

Objective: In this project to perform an analysis on the cyclone separator model for different number of injected particles Introduction: Cyclone Separator: Cyclone separators or simply cyclones are separation devices (dry scrubbers) that use the principle of inertia to remove particulate…

Objective : In this challenge, we are going to calculate the combustion efficiency of Recuperator. Calculate, the effect of preheating Temperature on the Adiabatic Flame Temperature. Calculate, the effect of preheating temperature on combustion efficiency. What is Recuperator: A recuperator is a special purpose counter-flow energy…

Governing Equation:  `(du)/(dt)+c((du)/(dx))`we use forward differce scheem for time derivative and use backward difference scheem for time spacematlab code for linear convectionclear allclose allclc% inputs for the linear convectionl=1;n=160;c=1;dt=0.01;%calculationx=linspace(0,l,n);dx=x(2)-x(1);% dx=l/(n-1)%…

% matlab program to calculate the stochiometric coefficient cumbution clear all close all clc % for first 20 carbon atoms in hydrocarbon range ar=zeros(1,20); x=linspace(1,20,20); for i=1:length(x) alkane(i)=3/2*i+1/2; alkene(i)=3/2*i; alkyne(i)=3/2*i-1/2; end % ploting grid on hold on plot(x,alkane) plot(x,alkene) plot(x,alkyne)…

Objective: Q1. What are some practical CFD models that have been based on the mathematical analysis of Rayleigh Taylor waves? In your own words, explain how these mathematical models have been adapted for CFD calculations Q2. Perform the Rayleigh Taylor instability simulation for 3 different mesh sizes with the base mesh…

Formula for 2D Heat Conduction Equation :-     1) For Transient State Analysis    `((del^2T)/(delx^2)+(del^2T)/(dely^2))alpha=(delT)/(delt)`     2) For Steady state analysis    `(del^2T)/(delx^2)+(del^2T)/(dely^2)=0` Some of the constant taken for solving the equation :-…

Object : Handling Mixtures (O2 + N2 +CH4) in Cantera and calculating the Adiabatic Flame Temperature. Calculating Mole Fraction:  The General equation CH4 + 2 (O2 + 3.72 N2)=CO2+2H2O+7.52N2 Calculating the mole fraction of O2 and N2, These were we defined O2 = 0.21 and N2 = 0.79 that equal to 1. Now, A is contain…

Objective: The main aim of this project is Simulating Ahmed Body. Describe Ahmed body and its importance In order to refine the mesh around the area of interest. Create an enclosure around it and refine the mesh locally. Choose appropriate dimensions for the second enclosure and use inflation layers around the body. Perform…

Objective: This project worked on Auto-Ignition and Ignition Delay depending on variation temperature and pressure using Cantera. simulate the Auto Ignition Time for Methane under the following conditions, (simulation time = 10 secs) Plot the variation of Auto Ignition time of Methane with a constant temperature of…

Objective: Setup a trasient shock tube simulation. Plot the pressure and temperature history in the entire domain. plot  the cell count asa function of time - Exolain the result. Introduction: A simple shock tube is a tube, rectangular or circular in cross-section, usually constructed of metal, in which a gas at low…

STEADY STATE SOLUTIONS:The steady state solution does not involve time marching .Temperature Gradient does not change with respect to time .The number of grid points are same i.e nx=ny  Following are the results for steady state solution:jacobi methodNumber of iterations done are 217Time taken for solving equation…