Week 7 - 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 MaCormack'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.
- Steady-State distribution of primitive variable inside the nozzle.
- Time-wise variation of the primitive variable.
- Variation of Mass rate distribution inside the nozzle at different time steps during the time-marching process.
- 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.
Conservative Form:
Non Conservative Form:
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
MaCarmack 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.
MAIN PROGRAM,
clear all
close all
clc
%SOLVING QUASI 1D SUPERSONIC NOZZLE FLOW EQUATIONS
%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')
%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
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 ')
%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 ')
%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
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 o')
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
- 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.
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);
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('MFR - NON CONSERVATIVE )')
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
- 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.
O/P
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 Mass flow rate
Mass flow rates:
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 of Normalized mass flow rate distributions of both forms:
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