Week 7 - Simulation of a 1D Super-sonic nozzle flow simulation using Macormack Method

Simulation of a 1D Super-sonic nozzle flow simulation using Macormack Method.

Objective:

To Simulate the isentropic flow through a Quasi 1D subsonic nozzle by deriving both the conservation and non-conservation forms of the governing equations and solve them by implementing the MacCormack's technique using MATLAB.

Problem Statement:

Determine the steady-state temperature distribution for the flow-field variables and to investigate the difference between the two forms of governing equations by comparing their solutions along with the plots stated below.

1. Steady-State distribution of primitive variable inside the nozzle.
2. Time-wise variation of the primitive variable.
3. Variation of Mass rate distribution inside the nozzle at different time steps during the time-marching process.
4. Comparision of Normalized mass flow rate distributions of both forms.                                                                                                                                                                                                                                                                                                                                         

   Theory:

   Any numerical solution of isentropic flow though a quasi 1D subsonic-supersonic nozzle is unreasonable, while analytic solution exists for these kind of problems. However we are trying here to establish the broadened capability of CFD in various streams rather in representing the approximated form of anayltic solution. Here, we implement the numerical solution with Conservative and Non-Conservative forms.

   Governing Equations are marched in terms of PDE for both forms in order to use Time Marching solution for the problem. These forms represent in numerical aspect only, if we analytic all the methods are same.

   Non Conservative Form:

 

Conservative Form:

 

 

Non Dimensional Quantities:

Also as part of the problem setup we represent quantites in terms of non-dimensional quantities.

 

Nozzle diagram with sections:

Since the problem setup is 1D, we consider only distribution in x-axis. Here Nozzle can be split into two sections

• Convergent Section 
• Divergent Section

 

 

MacCarmack method:

To discretize the numerical solution we use the technique called MacCarmack method. The MacCormack method is a variation of the two-step Lax-Wendroff Scheme but is much simpler in application.

The application of MacCormack method proceeds in two steps; a predictor step which is followed by a corrector step.

• Predictor step: In the predictor step, a "provisional" value of    at time level    (denoted by   ) is estimated obtained by replacing the spatial and temporal derivatives in the previous first order equation using Forward Difference Scheme.  is used for Timestep.
• Corrector step: In the corrector step, the predicted value    is corrected according to the equation. Note that the corrector step uses Backward Difference Scheme for spatial derivative. The time-step used in the corrector step is    
• in contrast to the    used in the predictor step.

 

 

MAIN PROGRAM:

In order for the code to be modular, program is split into three sections. They are 

• Main Program of Quasi 1D subsonic-supersonic nozzle
  
  Two functions namely
  
• Non Conservative form 
• Conservative form.

 

SAMPLE CODE FOR MAIN PROGRAM

%Simulation of a 1D Super-sonic nozzle flow simulation using Macormack Method
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','c')
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','r')
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','b')
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','m')
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','r')
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','m')
ylabel('Temperature')
legend({'Temperature'});
axis([0 3 0 1])
grid minor;

%Pressure
subplot(4,1,3);
plot(x,Pressure_NC,'color','c')
ylabel('Pressure')
legend({'Pressure'});
axis([0 3 0 1])
grid minor;

%Mach number
subplot(4,1,4);
plot(x,Mach_number_NC,'color','g')
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','r')
ylabel('Temperature')
legend({'Temperature at throat'});
axis([0 1400 0.6 1]);
grid minor;

%Pressure
subplot(4,1,3)
plot(Pressure_throat_C,'color','b')
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','g')
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','k')
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','r')
ylabel('Temperature')
legend({'Temperature'});
axis([0 3 0 1])
grid minor;

%Pressure
subplot(4,1,3);
plot(x,Pressure_C,'color','c')
ylabel('Pressure')
legend({'Pressure'});
axis([0 3 0 1])
grid minor;

%Mach number
subplot(4,1,4);
plot(x,Mach_number_C,'color','m')
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,'c','linewidth',1.25)
hold on;
plot(x,Mass_flow_rate_C,'r','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:

• Code is started with clear all, close all and clc to enable a fresh start without the manipulation of previous values.
• Input parameters are assigned for the variables x, dx, n, gamma, nt and C.
• Non Conservative and Conservative forms are created as separate functions with the neccessary inputs which is detailed below this section.
• tic toc command is used for calculating the time elasped between the functions.
• Plotting is done for the following solutions
  1)Timestep variation at throat area of both forms
  2)Variation of Flow Rate Distribution of both forms
  3)Comparison of Normalized Mass Flow Rate of both forms
• Each plot is computed as subplots of four primitive variables such as Density, Temperature, Pressure and Mach Number at throat as well as Convergent/Divergent section.

 

 

NON CONSERVATIVE 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, 'y', 'linewidth', 1.25);
hold on
elseif k == 100
plot(x, Mass_flow_rate_NC, 'm', 'linewidth', 1.25);
hold on
elseif k == 200
plot(x, Mass_flow_rate_NC, 'c', 'linewidth', 1.25);
hold on
elseif k == 400
plot(x, Mass_flow_rate_NC, 'r', '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

end

 

Explanation:

• The purpose of the function is to calculate the values of
  Mass flow rate, Pressure, Mach number, Density, velocity, Temperature, Density at throat, velocity at throat, Temperature at throat, Mass flow rate at throat, Pressure at throat, Mach number at throat by the known equations for all the grid points.
  
  
• For this initial boundary conditions are established for Density, Temperature & Velocity and profiles for calculating area & throat.
• Now Outer Time loop is computed for for loop to calculate all the grid points for various formulas inside the loop.
• MacCarmack technique is used to solve the numerical solution of governing equations such as Continuity, Momentum & Energy Equation.
• The technique implies two methods for solving namely Predictor method followed by Corrector method.
• Predictor & Corrector method are assigned within the for loop individually for calculating values.
• The calculated values are averaged in the next step and updating the solution with old values.
• Boundary Conditions are applied within the for loop and subsequently used for calculating the Mass Flow Rate, Mach Number & Pressure and also calculating the values throat section.
• Finally plotting is done for Distance(x) & Mass Flow Rate for differrent Timesteps.
• Program is ended

 

 

 

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, 'y', 'linewidth', 1.25);
hold on
elseif k == 100
plot(x, Mass_flow_rate_C, 'm', 'linewidth', 1.25);
hold on
elseif k == 200
plot(x, Mass_flow_rate_C, 'c', 'linewidth', 1.25);
hold on
elseif k == 400
plot(x, Mass_flow_rate_C, 'r', '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

end

 

Explanation:

• The purpose of the function is to calculate the values of
  Mass flow rate, Pressure, Mach number, Density, velocity, Temperature, Density at throat, velocity at throat, Temperature at throat, Mass flow rate at throat, Pressure at throat, Mach number at throat by the known equations for all the grid points.
  To determine the initial profile of Density & Temperature, for loop is used inputing the known conditions within an else-if command.
• Initial values of area, throat, velocity & Pressure are calculated using formulas.
• In order to simplify and avoid errors, equation is splitted into Solution Vector(U), Flux Vector(F) and Source Term(J).
• Now Outer Time loop is computed for for loop to calculate all the grid points for various formulas inside the loop.
• MacCarmack technique is used to solve the numerical solution of governing equations such as Continuity, Momentum & Energy Equation.
• The technique implies two methods for solving namely Predictor method followed by Corrector method.
• Predictor & Corrector method are assigned within the for loop individually for calculating values.
• The calculated values are averaged in the next step and updating the solution with old values.
• Boundary Conditions are applied within the for loop and subsequently used for calculating the Mass Flow Rate, Mach Number & Pressure and also calculating the values throat section.
• Finally plotting is done for Distance(x) & Mass Flow Rate for differrent Timesteps.
• Program is ended.

 

 

 

Outputs:

Time steps variation of primitive variable at Throat area

Non-Conservative Form

 

Conservative form

Observation:

From the above plots we can conclude that initial variations observed in Non Conservative form for a certain period of time makes it consume higher amount of time to acheive stable solution when compared to Conservative form.

 

Variation of Flow Rate Distribution

Non Conservative form

Conservative form

Observation:

Here by observing the above plots we can 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.

 

COMPARISON PLOT of Normalized Mass Flow Rate of both NON CONSERVATIVE & CONSERVATIVE FORMS:

Observation:

From the above representation we can colclude that  both the forms are almost similar, the Non Conservative form oscillates much when compared to Conservative form. The latter also gives almost stable solution along the length. By this we can conclude that Conservative form attains stability in lesser number of iterations with earliest time.

 

GRID DEPENDENCE TEST:

 

GRID DEPENDENCE TEST Comparison Of Solutions After 1400 Steps
Types Of Forms	Grid Number (n)	Primitive Variables					Time Elasped (in seconds)
		Density	Pressure	Temperature	Mach Number
Non Conservative Form	n=31	0.638690887	0.534239917	0.836460842		0.999372038	155.4
	n=61	0.637558551	0.532609261	0.835388788		0.999699114	140.7
Conservative Form	n=31	0.651619331	0.547765088	0.840621298		0.979517537	130.7
	n=61	0.638305711	0.533075405	0.835141214		0.999245249	122.2
Expected Solution		0.639	0.534	0.836		0.99

 

 

Observation:

Grid dependency is an important factor to determine whether the problem can be solved faster. So here we increased the Grid points from 31 to 61 and populated them in a sheet.

It is clear from the above table that the values are almost same along all the Primitive Variables for both the forms even when the Grid points(n) is changed from 31 to 61 showcasing that it doesn't get affected by the increase of Grid points.

So we can safely conclude that the above problem is Grid Dependant.

 

CONCLUSION:

Accumulating all the above results we can ostensively conclude that Conservative form converges slightly faster than Non Conservative form offering stable solution along the simulation. Yet both the forms are dependable in calculating the solutions which is almost precise.