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

Aim:Simulation of 1D Super-sonic nozzle flow simulation using the Macormack Method.

Objective:

Simulate isentropic flow through a Quasi 1D subsonic nozzle by deriving conservative and non-conservative forms of governing equations and solving them by implementing the MacCormack technique using MATLAB.

Problem Statement:

Determine the steady-state temperature distribution for the flow-field variables and explore the difference between the two forms of the governing equations by comparing their solutions to the graphs below.

Steady state distribution of the primitive variable inside the nozzle.
1 Time variation of a primitive variable.
2 Variation of the mass velocity distribution inside the nozzle at different time steps during the time marching process.
3 Comparison of normalized mass flow distributions of both forms.

Theory:

Any numerical solution of isentropic flow using a quasi 1D subsonic-supersonic nozzle is unreasonable, while an analytical solution exists for this kind of problem. However, we are trying here to introduce the extended capability of CFD in different flows rather in the representation of an approximate form of the analytical solution. Here we implement a numerical solution with conservative and non-conservative forms.

The governing equations are marched in PDE terms for both forms to apply a Time Marching solution to the problem. These forms represent only numerically if we analyze all methods the 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. in thi Nozzle can be split into two sections which are

• Convergent Section
• Divergent Section

The MacCarmack Method:

We use a technique called the MacCarmack method to discretize the numerical solution. MacCormack's method is a variation of the two-step Lax-Wendroff scheme, but is much simpler to apply.

The application of the MacCormack method takes place in two steps;

a predictor step followed by a corrector step.

• Predictor step: In the predictor step, a "provisional" value of u at time level 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 splited into three sections.

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

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:

• The code is run with the clear all, close all, and clc commands to allow a new run without manipulating the previous values.
• The variables x, dx, n, gamma, nt and C are assigned input parameters.
• The non-conservative and conservative forms are built as separate functions with the necessary inputs detailed below in this section.
• The tic toc command is used to calculate the elapsed time between functions.
  Rendering is done for the following solution
  1) Variation of the time step in the throat area of ​​both forms
  2) Variation of flow distribution of both forms
  3) Comparison of the normalized mass flow rate of both forms
  Each plot is calculated as sub-plots of four primitive variables such as density, temperature, pressure and throat Mach number, as well as convergent/divergent sections.

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, '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

end

Explanation:

• The purpose of the function is to calculate values
  Mass Flow, Pressure, Mach Number, Density, Velocity, Temperature, Throat Density, Throat Velocity, Throat Temperature, Throat Mass Flow, Throat Pressure, Throat Mach Number according to known equations for all grid points.
• For this initial boundary conditions are established for density, temperature and velocity and profiles for area and throat calculation.
• Now the outer while loop is calculated for the for loop to calculate all the grid points for the various formulas inside the loop.
• The MacCarmack technique is used to numerically solve governing equations such as the continuity, momentum, and energy equations.
• The technique includes two solution methods, a predictor method followed by a corrector method.
• The predictor and corrector methods are assigned within the for loop individually to calculate the values.
• The calculated values ​​are averaged in the next step and the solution is updated with the old values.
• The boundary conditions are applied within the for cycle and subsequently used to calculate the mass flow rate, Mach number and pressure, as well as to calculate the throat part of the values.
• Finally, a plot is made for distance (x) and mass flow for different time steps.
  The program is closed.

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

end

Explanation:

• The purpose of the function is to calculate values
  Mass Flow, Pressure, Mach Number, Density, Velocity, Temperature, Throat Density, Throat Velocity, Throat Temperature, Throat Mass Flow, Throat Pressure, Throat Mach Number according to known equations for all grid points.
• A for loop is used to specify the known conditions in an else-if statement to determine the initial density and temperature profile.
• The initial values ​​of area, throat, velocity and pressure are calculated using formulas.
• To simplify and eliminate errors, the equation is divided into Solution Vector(U), Flux Vector(F) and Source Term(J).
• Now the outer while loop is calculated for the for loop to calculate all the grid points for the various formulas inside the loop.
• The MacCarmack technique is used to numerically solve governing equations such as the continuity, momentum, and energy equations.
• The technique includes two solution methods, a predictor method followed by a corrector method.
• The predictor and corrector methods are assigned within the for loop individually to calculate the values.
• The calculated values ​​are averaged in the next step and the solution is updated with the old values.
• The boundary conditions are applied within the for cycle and subsequently used to calculate the mass flow rate, Mach number and pressure, as well as to calculate the throat part of the values.
• Finally, a plot is made for distance (x) and mass flow for different time steps.
  The program is closed.

Outputs:

Time steps variation of primitive variable at Throat area

Non-Conservative Form

Conservative Form

Observation:

From the above graphs, we can conclude that the initial deviations observed in the non-conservative form over a period of time require a higher amount of time to reach a stable solution compared to the conservative form.

Variation of Flow Rate Distribution

Non Conservative form

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.

Observation:

Although the final solutions of the two forms are almost similar, the non-conservative form oscillates considerably compared to the conservative form. The latter also provides an almost stable solution along the entire length. From this we can conclude that the conservative form reaches stability in fewer iterations with the earliest possible time.

GRID DEPENDENCE TEST:

GRID DEPENDENCE TEST(comparision of solution after 1400 steps)

Types of forms	Grid numbers(n)	Density	pressure	temperature	mach number	Time elapsed(in sec)
Non-conservative form	n=31 n=61	0.638690887 0.637558551	0.534239917 0.532609261	0.836460842 0.83538878	0.9999372 0.9996991	155.4 140.7
Conservative form	n=31 n=61	0.651619331 0.63830571	0.547765088 0.533075405	0.8406221298 0.835141214	0.979517537 0.999245249	130.7 122.
Solution to be expected		0.639	0.534	0.836	0.99

Observation:

• Grid dependency is an important factor in determining whether a problem can be solved faster. So here we increased the grid points from 31 to 61 and filled them into the sheet.
• From the table above, it is clear that the values ​​are almost the same for all primitive variables for both forms, even if the grid points (n) change from 31 to 61, this shows that it does not affect the increase in grid points. .
• So we can safely conclude that the above problem is grid dependent.

CONCLUSION:

Summarizing all the above results, we can obviously conclude that the conservative form converges slightly faster than the non-conservative form, which offers a stable solution along the simulation. Nevertheless, both forms are reliable in computing a solution that is nearly exact.