Week 7 - Simulation of a 1D Supersonic nozzle flow simulation using Macormack Method
AIM:
To write a Matlab program to simulate the isentropic flow through a quasi 1D subsonic-supersonic nozzle by both the conservation and non-conservation forms of the governing equations and to solve them using the MacCormack's technique.
OBJECTIVE:
- To write a Matlab program to simulate the isentropic flow through a quasi 1D subsonic-supersonic nozzle.
- To solve the problem by MacCormack's technique considering both the conservation and non-conservation forms of the governing equations.
- To plot steady state distribution and time wise variation of primitive variables inside the nozzle.
- To compare the mass flow rate at steady state for both conservation form and non-conservation form.
- To plot the variation of Mass flow rate distribution inside the nozzle at different time steps during the time-marching process.
- To perform a grid independence test on the solution.
Theory:
Nozzle:
A nozzle is a tool used to change the properties of a fluid flow as it leaves (or enters) a closed chamber or conduit, particularly to enhance velocity.
A nozzle can be used to guide or change the flow of a fluid (liquid or gas), and it is frequently a pipe or tube with a variable cross sectional area. Nozzles are widely used to regulate the stream that emerges from them in terms of flow rate, speed, direction, mass, form, and/or pressure. The fluid's pressure energy is lost as its velocity rises in a nozzle.
The Bernoulli Principle is what causes the pressure in a convergent nozzle to decrease. The Bernoulli principle asserts that an increase in a fluid's speed and a reduction in the pressure the fluid exerts happen at the same time.
Convergent Nozzle:
A converging nozzle is a nozzle whose cross-sectional area decreases as it grows smaller from a large initial size. Due to mass conservation, a fluid must speed up when it reaches the smaller cross-section. The fluid must travel more quickly in order to keep the amount of fluid passing through the nozzle's constrained area constant.
This fluid needs energy to accelerate, therefore it must originate from someplace. The random movement of molecules, which we experience as pressure, contains some energy. This random motion's energy is transformed into stream flow, a quicker forward motion. The pressure decreases due to this modification.
If all else is constant, pressure has an inverse relationship to area. In this instance, the energy that creates pressure is changed into a different kind of energy, resulting in a reduction in both pressure and area.
As indicated in the accompanying figure, we will now investigate the steady, isentropic flow via a convergent diverging nozzle.
isentropic flow via a convergent diverging nozzle.
MacCormack's Technique:
The MacCormack method is a second order accurate explicit finite-difference methodology that works in both space and time. This approach is well-liked because it works well in many situations involving fluid movement.A variation of the Lax-Wendroff approach, the MacCormack method requires less work to apply because there is no need to analyse the second order time derivatives as there was with the Lax-Wendroff method.The MacCormack technique is a second order accurate method that uses a two-step predictor-corrector sequence with forward differences on the predictor and with rearward differences on the corrector.
Step-1 - Governing equations
for Non conservetive :
CONTINUTY EQUATION
dp/dt'=p'dv'/dx'-pv' d(lnA')/dx'-v'dp'/dx'
MOMENTUM ERQUATION
dv'/dt'=-v'dv'/dx'-1/y(dt'/dx'+t'/p' dp'/dx')
ENERGY EQUATION
dT'/dt'=-v'dt'/dx'-(y-1)t'[dv'/dx'+v'd(lnA')/dx'
for conservative equation
CONTINUTY EQUATION
d(p'A')/dt'+d(P'A'V')/dt'=0
MOMENTUM EQUATION
d(p'A'V')/dt'+d[p'A'V'2+(1/Y)P'A'/dx'=1/y p'dA'/dx'
ENERGY EQUATION
d[(e'/y-1+y/2 v'2)A']/dt'+d[p'(e'/y-1+y/2v'2)v'A'+p'A'V']/dx'=0
PREDICTOR STEP
(df/dt)p=f(i+1)-f(i)/dx
SOLUTION STEP
fp=f initial+(df/dt)p.dx
CPRRECTOR STEP
(dfp/dt)c=f(i)-f(i-1)/dx
AVERAGE DERIVATIVE
(dfp/dt)avg=((df/dt)p+(dfp/dt)c)
FINAL SOLUTION UPDATE
fsol=finitial+(df/dt)avg.dt
INITIAL CONDITION
{P=1,T=1}FOR 0
p=1-0.366(x-0.5) for 0.5
t=1-0.167(x-0.5)
p=0.634-0.3879(x-1.5) for 1.5<3.5
v=o.59/pA
nozzle sgape A1+2.2(X-1.5
BOUNDARY CONDITION
INLET
U2(i=1)2u2(i=2)-u2(i=3)
u3is also updated by using the value of updated V
OUTLET
all the three varaible are float
(U1)=2(U1)N-1(U1)N-2
(U2)N=2(U2)N-1-(U2)N-2
(U3)N=2(U3)N-1-(U3)N-3
clear all
close all
clc
%SOLVING QUASI 1D SUPERSONIC NOZZLE FLOW EQUATIONS
%INPUT PARAMETERS
%No. of grid points
n = 31;
gamma = 1.4;
x = linspace(0,3,n);
dx = x(2) - x(1);
%No. of time step
nt = 1400;
%Courant Number
C = 0.5;
%FUNCTION FOR NON CONSERVATIVE FORM
tic;
[Mass_flow_rate_NC, Pressure_NC, Mach_number_NC, rho_NC, v_NC, T_NC, rho_throat_NC, v_throat_NC, T_throat_NC, Mass_flow_rate_throat_NC, Pressure_throat_NC, Mach_number_throat_NC] = Non_conservative_form(x, dx, n, gamma, nt, C);
Time_elapsed_NC = toc;
fprintf('Time Elasped for Non Conservative form: %0.4f', Time_elapsed_NC);
hold on
%PLOTTING
%Timestep variation of properties at throat area in Non Conservative form
figure(2)
%Density
subplot(4,1,1)
plot(linspace(1,nt,nt),rho_throat_NC,'color','m')
ylabel('Density')
legend({'Density at throat'});
axis([0 1400 0 1.3]);
grid minor;
title('Timestep variation at throat area - NON CONSERVATIVE FORM')
%Temperature
subplot(4,1,2)
plot(linspace(1,nt,nt),T_throat_NC,'color','c')
ylabel('Temperature')
legend({'Temperature at throat'});
axis([0 1400 0.6 1]);
grid minor;
%Pressure
subplot(4,1,3)
plot(linspace(1,nt,nt),Pressure_throat_NC,'color','g')
ylabel('Pressure')
legend({'Pressure at throat'});
axis([0 1400 0.4 1.1]);
grid minor
%Mach Number
subplot(4,1,4)
plot(linspace(1,nt,nt),Mach_number_throat_NC,'color','y')
xlabel('Timesteps')
ylabel('Mach Number')
legend({'Mach Number at throat'});
axis([0 1400 0.6 1.4]);
grid minor;
%Steady state simulation for primitve values for Non Conservative form
figure(3);
%Density
subplot(4,1,1);
plot(x,rho_NC,'color','m')
ylabel('Density')
legend({'Density'});
axis([0 3 0 1])
grid minor;
title('Variation of Flow Rate Distribution - NON CONSERVATIVE')
%Temperature
subplot(4,1,2);
plot(x,T_NC,'color','c')
ylabel('Temperature')
legend({'Temperature'});
axis([0 3 0 1])
grid minor;
%Pressure
subplot(4,1,3);
plot(x,Pressure_NC,'color','g')
ylabel('Pressure')
legend({'Pressure'});
axis([0 3 0 1])
grid minor;
%Mach number
subplot(4,1,4);
plot(x,Mach_number_NC,'color','y')
xlabel('Nozzle Length')
ylabel('Mach number')
legend({'Mach number'});
axis([0 3 0 4])
grid minor;
%FUNCTION FOR CONSERVATIVE FORM
tic;
[Mass_flow_rate_C, Pressure_C, Mach_number_C, rho_C, v_C, T_C, rho_throat_C, v_throat_C, T_throat_C, Mass_flow_rate_throat_C, Pressure_throat_C, Mach_number_throat_C] = Conservative_form(x, dx, n, gamma, nt, C);
Time_elasped_C = toc;
fprintf('Time Elasped for Conservative form: %0.4f', Time_elasped_C);
%PLOTTING
%Timestep variation of properties at throat area in Conservative form
figure(5)
%Density
subplot(4,1,1)
plot(rho_throat_C,'color','m')
ylabel('Density')
legend({'Density at throat'});
axis([0 1400 0 1]);
grid minor;
title('Timestep variation at throat area - CONSERVATIVE FORM')
%Temperature
subplot(4,1,2)
plot(T_throat_C,'color','c')
ylabel('Temperature')
legend({'Temperature at throat'});
axis([0 1400 0.6 1]);
grid minor;
%Pressure
subplot(4,1,3)
plot(Pressure_throat_C,'color','g')
ylabel('Pressure')
legend({'Pressure at throat'});
axis([0 1400 0.2 0.8]);
grid minor
%Mach Number
subplot(4,1,4)
plot(Mach_number_throat_C,'color','y')
xlabel('Timesteps')
ylabel('Mach Number')
legend({'Mach Number at throat'});
axis([0 1400 0.6 1.4]);
grid minor;
%Steady state simulation for primitve values for Non Conservative form
figure(6);
%Density
subplot(4,1,1);
plot(x,rho_C,'color','m')
ylabel('Density')
legend({'Density'});
axis([0 3 0 1])
grid minor;
title('Variation of Flow Rate Distribution - CONSERVATIVE FORM')
%Temperature
subplot(4,1,2);
plot(x,T_C,'color','c')
ylabel('Temperature')
legend({'Temperature'});
axis([0 3 0 1])
grid minor;
%Pressure
subplot(4,1,3);
plot(x,Pressure_C,'color','g')
ylabel('Pressure')
legend({'Pressure'});
axis([0 3 0 1])
grid minor;
%Mach number
subplot(4,1,4);
plot(x,Mach_number_C,'color','y')
xlabel('Nozzle Length')
ylabel('Mach number')
legend({'Mach number'});
axis([0 3 0 4])
grid minor;
%COMPARISON PLOT of Normalized Mass Flow Rate of both forms
figure(7)
hold on
plot(x,Mass_flow_rate_NC,'r','linewidth',1.25)
hold on;
plot(x,Mass_flow_rate_C,'b','linewidth',1.25)
hold on;
grid on
legend('Non Conservative','Conservative');
xlabel('Length of the Domain')
ylabel('Mass Flow Rate')
title('Comparison of Normalized Mass Flow Rate of both forms')
Explanation:
• To enable a fresh run without changing the previously used values, the code is executed using the clear all, close all, and clc commands.
The input parameters for the variables x, dx, n, gamma, nt, and C are given.
• Using the essential inputs, which are listed below in this section, the non-conservative and conservative forms are created as distinct functions.
• To determine the amount of time that has passed between functions, use the tic toc command.
For the ensuing resolution, rendering is completed
1) Changes in the time step in both types' throat regions
2) Changes in both kinds' flow distribution
3) Evaluation of both types' normalised mass flow rates
Each graphic is computed using four simple variables, such as density, temperature, pressure, and throat Mach number, as well as convergent/divergent sections.
NON COSERVETIVE FORM
%Function for Non-Conservative form
function [Mass_flow_rate_NC, Pressure_NC, Mach_number_NC, rho_NC, v_NC, T_NC, rho_throat_NC, v_throat_NC, T_throat_NC, Mass_flow_rate_throat_NC, Pressure_throat_NC, Mach_number_throat_NC] = Non_conservative_form(x, dx, n, gamma, nt, C)
%Intial boundary conditions
rho_NC = 1 - 0.3146*x;
T_NC = 1 - 0.2341*x;
v_NC = (0.1 + 1.09*x).*T_NC.^0.5;
%Area & Throat profiles
a = 1 + 2.2*(x - 1.5).^2;
throat = find(a==1);
%Outer Time Loop
for k = 1:nt
%Time-step control
dt = min(C.*dx./(sqrt(T_NC) + v_NC));
%Saving a copy of Old Values
rho_old = rho_NC;
T_old = T_NC;
v_old = v_NC;
%Predictor Method
for j = 2:n-1
%Simplifying the Equation's spatial terms
dv_dx = (v_NC(j + 1) - v_NC(j))/dx;
drho_dx = (rho_NC(j + 1) - rho_NC(j))/dx;
dT_dx = (T_NC(j + 1) - T_NC(j))/dx;
dlog_a_dx = (log(a(j + 1)) - log(a(j)))/dx;
%Continuity Equation
drho_dt_P(j) = -rho_NC(j)*dv_dx - rho_NC(j)*v_NC(j)*dlog_a_dx - v_NC(j)*drho_dx;
%Momentum Equation
dv_dt_P(j) = -v_NC(j)*dv_dx - (1/gamma)*(dT_dx + (T_NC(j)*drho_dx/rho_NC(j)));
%Energy Equation
dT_dt_P(j) = -v_NC(j)*dT_dx - (gamma - 1)*T_NC(j)*(dv_dx + v_NC(j)*dlog_a_dx);
%Solution Update
rho_NC(j) = rho_NC(j) + drho_dt_P(j)*dt;
v_NC(j) = v_NC(j) + dv_dt_P(j)*dt;
T_NC(j) = T_NC(j) + dT_dt_P(j)*dt;
end
%Corrector Method
for j = 2:n-1
%Simplifying the Equation's spatial terms
dv_dx = (v_NC(j) - v_NC(j - 1))/dx;
drho_dx = (rho_NC(j) - rho_NC(j - 1))/dx;
dT_dx = (T_NC(j) - T_NC(j - 1))/dx;
dlog_a_dx = (log(a(j)) - log(a(j - 1)))/dx;
%Continuity Equation
drho_dt_C(j) = - rho_NC(j)*dv_dx - rho_NC(j)*v_NC(j)*dlog_a_dx - v_NC(j)*drho_dx;
%Momentum Equation
dv_dt_C(j) = - v_NC(j)*dv_dx - (1/gamma)*(dT_dx + (T_NC(j)*drho_dx/rho_NC(j)));
%Energy Equation
dT_dt_C(j) = - v_NC(j)*dT_dx - (gamma - 1)*T_NC(j)*(dv_dx + v_NC(j)*dlog_a_dx);
end
%Compute Average Time derivative
drho_dt = 0.5.*(drho_dt_P + drho_dt_C);
dv_dt = 0.5.*(dv_dt_P + dv_dt_C);
dT_dt = 0.5.*(dT_dt_P + dT_dt_C);
%Final Solution Update
for i = 2:n-1
rho_NC(i) = rho_old(i) + drho_dt(i)*dt;
v_NC(i) = v_old(i) + dv_dt(i)*dt;
T_NC(i) = T_old(i) + dT_dt(i)*dt;
end
%Applying Boundary Conditions
%At Inlet
rho_NC(1) = 1;
v_NC(1) = 2*v_NC(2) - v_NC(3);
T_NC(1) = 1;
%At Outlet
rho_NC(n) = 2*rho_NC(n-1) - rho_NC(n - 2);
v_NC(n) = 2*v_NC(n - 1) - v_NC(n - 2);
T_NC(n) = 2*T_NC(n - 1) - T_NC(n - 2);
%Defining Mass flow rate, Pressure and Mach number
Mass_flow_rate_NC = rho_NC.*a.*v_NC;
Pressure_NC = rho_NC.*T_NC;
Mach_number_NC = (v_NC./sqrt(T_NC));
%Calculating Variables at throat section
rho_throat_NC(k) = rho_NC(throat);
T_throat_NC(k) = T_NC(throat);
v_throat_NC(k) = v_NC(throat);
Mass_flow_rate_throat_NC(k) = Mass_flow_rate_NC(throat);
Pressure_throat_NC(k) = Pressure_NC(throat);
Mach_number_throat_NC(k) = Mach_number_NC(throat);
%PLOTTING
%Comparison of non-dimensional Mass flow rates at different time steps
figure(1);
if k == 50
plot(x, Mass_flow_rate_NC, 'm', 'linewidth', 1.25);
hold on
elseif k == 100
plot(x, Mass_flow_rate_NC, 'c', 'linewidth', 1.25);
hold on
elseif k == 200
plot(x, Mass_flow_rate_NC, 'g', 'linewidth', 1.25);
hold on
elseif k == 400
plot(x, Mass_flow_rate_NC, 'y', 'linewidth', 1.25);
hold on
elseif k == 800
plot(x, Mass_flow_rate_NC, 'k', 'linewidth', 1.25);
hold on
end
%Labelling
title('Mass flow rates at different Timesteps - NON CONSERVATIVE FORM)')
xlabel('Distance')
ylabel('Mass flow rate')
legend({'50^t^h Timestep','100^t^h Timestep','200^t^h Timestep','400^t^h Timestep','800^t^h Timestep'})
axis([0 3 0 2])
grid on
end
OUTPUT
Non-Conservative Form
Mass flow rates at different Timesteps:
Non Conservative form:
Explanation:
• The function's goal is to compute values for all grid points for the following quantities: mass flow, pressure, mach number, density, velocity, temperature, throat density, throat velocity, throat temperature, throat mass flow, throat pressure, and throat m/s ratio.
• Profiles for area and throat computation, as well as conditions for density, temperature, and velocity, are constructed for this initial boundary.
• All grid points for the different formulae inside the for loop are now computed using the outer while loop.
• Using the MacCarmack method, governing equations such the continuity, momentum, and energy equations may be solved numerically.
• The strategy uses two ways for finding solutions: a prediction method and a corrector method.
• The for loop assigns the predictor and corrector techniques separately to compute the values.
• The for loop assigns the predictor and corrector techniques separately to compute the values.
Conservative Form
%Function for Conservative form
function [Mass_flow_rate_C, Pressure_C, Mach_number_C, rho_C, v_C, T_C, rho_throat_C, v_throat_C, T_throat_C, Mass_flow_rate_throat_C, Pressure_throat_C, Mach_number_throat_C] = Conservative_form(x, dx, n, gamma, nt, C)
%Determining Initial profile of Density and Temperature
for i = 1:n
if (x(i) >= 0 && x(i) <= 0.5)
rho_C(i) = 1;
T_C(i) = 1;
elseif (x(i) >= 0.5 && x(i) <= 1.5)
rho_C(i) = 1 - 0.366*(x(i) - 0.5);
T_C(i) = 1 - 0.167*(x(i) - 0.5);
elseif (x(i) >= 1.5 && x(i) <= 3)
rho_C(i) = 0.634 - 0.3879*(x(i) - 1.5);
T_C(i) = 0.833 - 0.3507*(x(i) - 1.5);
end
end
%Area & Throat profiles
a = 1 + 2.2*(x - 1.5).^2;
throat = find(a==1);
%Initial profile fot Velocity and Pressure
v_C = 0.59./(rho_C.*a);
Pressure_C = rho_C.*T_C;
%Calculate the Initial Solution Vectors(U)
U1 = rho_C.*a;
U2 = rho_C.*a.*v_C;
U3 = rho_C.*a.*(T_C./(gamma - 1)) + ((gamma/2).*v_C.^2);
%Outer Time Loop
for k =1:nt
%Time-step control
dt = min(C.*dx./(sqrt(T_C) + v_C));
%Saving a copy of Old Values
U1_old = U1;
U2_old = U2;
U3_old = U3;
%Calculating the Flux vector(F)
F1 = U2;
F2 = ((U2.^2)./U1) + ((gamma - 1)/gamma)*(U3 - (gamma/2)*((U2.^2)./U1));
F3 = ((gamma*U2.*U3)./U1) - (gamma*(gamma - 1)*((U2.^3)./(2*U1.^2)));
%Predictor Method
for j = 2:n-1
%Calculating the Source term(J)
J2(j) = (1/gamma)*rho_C(j)*T_C(j)*((a(j + 1) - a(j))/dx);
%Continuity Equation
dU1_dt_P(j) = -((F1(j + 1) - F1(j))/dx);
%Momentum Equation
dU2_dt_P(j) = -((F2(j + 1) - F2(j))/dx) + J2(j);
%Energy Equation
dU3_dt_P(j) = -((F3(j + 1) - F3(j))/dx);
%Updating New values
U1(j) = U1(j) + dU1_dt_P(j)*dt;
U2(j) = U2(j) + dU2_dt_P(j)*dt;
U3(j) = U3(j) + dU3_dt_P(j)*dt;
end
%Updating Flux vectors
F1 = U2;
F2 = ((U2.^2)./U1) + ((gamma - 1)/gamma)*(U3 - ((gamma*0.5*(U2.^2))./(U1)));
F3 = ((gamma*U2.*U3)./U1) - ((gamma*0.5*(gamma - 1)*(U2.^3))./(U1.^2));
%Corrector Method
for j = 2:n-1
%Updating the Source term(J)
J2(j) = (1/gamma)*rho_C(j)*T_C(j)*((a(j) - a(j - 1))/dx);
%Simplifying the Equation's spatial terms
dU1_dt_C(j) = -((F1(j) - F1(j - 1))/dx);
dU2_dt_C(j) = -((F2(j) - F2(j - 1))/dx) + J2(j);
dU3_dt_C(j) = -((F3(j) - F3(j - 1))/dx);
end
%Compute Average Time derivative
dU1_dt = 0.5*(dU1_dt_P + dU1_dt_C);
dU2_dt = 0.5*(dU2_dt_P + dU2_dt_C);
dU3_dt = 0.5*(dU3_dt_P + dU3_dt_C);
%Final Solution Update
for i = 2:n-1
U1(i) = U1_old(i) + dU1_dt(i)*dt;
U2(i) = U2_old(i) + dU2_dt(i)*dt;
U3(i) = U3_old(i) + dU3_dt(i)*dt;
end
%Applying Boundary Conditions
%At Inlet
U1(1) = rho_C(1)*a(1);
U2(1) = 2*U2(2) - U2(3);
v_C(1) = U2(1)./U1(1);
U3(1) = U1(1)*((T_C(1)/(gamma - 1)) + ((gamma/2)*(v_C(1)).^2));
%At Outlet
U1(n) = 2*U1(n - 1) - U1(n - 2);
U2(n) = 2*U2(n - 1) - U2(n - 2);
U3(n) = 2*U3(n - 1) - U3(n - 2);
%Final Primitive value update
rho_C = U1./a;
v_C = U2./U1;
T_C = (gamma - 1)*((U3./U1 - (gamma/2)*(v_C).^2));
%Defining Mass flow rate, Pressure and Mach number
Mass_flow_rate_C = rho_C.*a.*v_C;
Pressure_C = rho_C.*T_C;
Mach_number_C = (v_C./sqrt(T_C));
%Calculating Variables at throat section
rho_throat_C(k) = rho_C(throat);
T_throat_C(k) = T_C(throat);
v_throat_C(k) = v_C(throat);
Mass_flow_rate_throat_C(k) = Mass_flow_rate_C(throat);
Pressure_throat_C(k) = Pressure_C(throat);
Mach_number_throat_C(k) = Mach_number_C(throat);
%PLOTTING
%Comparison of non-dimensional Mass flow rates at different time steps
figure(4)
if k == 50
plot(x, Mass_flow_rate_C, 'm', 'linewidth', 1.25);
hold on
elseif k == 100
plot(x, Mass_flow_rate_C, 'c', 'linewidth', 1.25);
hold on
elseif k == 200
plot(x, Mass_flow_rate_C, 'g', 'linewidth', 1.25);
hold on
elseif k == 400
plot(x, Mass_flow_rate_C, 'y', 'linewidth', 1.25);
hold on
elseif k == 800
plot(x, Mass_flow_rate_C, 'k', 'linewidth', 1.25);
hold on
end
%Labelling
title('Mass flow rates at different Timesteps - CONSERVATIVE FORM')
xlabel('Distance')
ylabel('Mass flow rate')
legend({'50^t^h Timestep','100^t^h Timestep','200^t^h Timestep','400^t^h Timestep','800^t^h Timestep'})
axis([0 3 0.4 1])
grid on
end
Conservative Form
Observation:
by observing the above plots wecan see that both the forms are in same phase giving the exact same solution for all the values.
Mass flow rates at different Timesteps:
Non Conservative form:
Conservative form:
Observation:
Above plots shows that Mass Flow Rate at different Time steps along the Distance oscillates initially, however the solution becomes stable after the 400th Timestep for both the forms attaining a constant value.
GRID DEPENDENCE TEST:
GRID DEPENDENCE TEST(comparision of solution after 1400 steps)
TYPES OF GRID POINT DENSITY PRESSURE TEMPRATURE MACH TIME ELAPSEDFORM NUMBER
NON CON n=31 0.63869 0.53423 0.83646 0.999993 155.4
SEVATIVE
CONSERVATIV n=61 0.63755 0.53260 0.83538 0.9999969 140.7
n=31 0.65161 0.547765 0.840622 0.97951 130.7
SLUTION TO n=61 0.63830 0.53330 0.8351 0.99924 122
BE EXPECTED 0.6339 0.534 0.836 0.99
Observation:
• Grid reliance has a significant role in deciding how quickly a problem may be handled. Therefore, we filled the page with 61 grid points instead of the previous 31.
It is evident from the table above that even if the grid points (n) change from 31 to 61, the values are nearly same for all basic variables for both forms, demonstrating that the increase in grid points is unaffected.
• Therefore, it is acceptable to say that the aforementioned issue is grid-dependent.
CONCLUSION:
When all of the aforementioned findings are put together, it is clear that the conservative form converges a little bit quicker than the non-conservative form, which provides a stable solution throughout the simulation. However, both methods are trustworthy for calculating a result that is almost accurate.