Week 7 - Simulation of a 1D Super-sonic nozzle flow simulation using Macormack Method
% witting program for simulation of 1D NOZZEL SUPERSONIC FLO USING MACORMAC
% METHOD
clear all
close all
clc
%defing the inputs
%number of grids points
n=31;
x=linspace(0,3,n);
dx=x(2)-x(1);
gamma=1.4;
%numbers of time steps
nt=1400;
CFL=0.5;
%defing the function for non conservation form of simulation
tic;
[mass_flow_rate_nc, presure_nc, mach_num_nc, rho_nc, v_nc, T_nc, rho_trottle_nc, v_trottle_nc, T_trottle_nc, mass_flow_rate_trottle_nc, presure_trottle_nc, mach_num_trottle_nc]=non_conservative_form_original(x,dx,n,gamma,nt,CFL)
time_elapsed=toc;
fprintf('time elapesed for non coservation form:%f',time_elapsed)
hold on
%ploting the graphs
figure(3)
%for density
subplot(4,1,1)
plot(linspace(1,nt,1400),rho_trottle_nc,"Color","m");
ylabel("density")
legend({'density at trottle'})
title('timestep variation at trottle area-non conservation form')
%for tempreture
subplot(4,1,2)
plot(linspace(1,nt,1400),T_trottle_nc,"Color","b");
ylabel("tempreture")
legend({'tempreture at trottle'})
title('timestep variation at trottle area-non conservation form')
% for presure
subplot(4,1,3)
plot(linspace(1,nt,1400),presure_trottle_nc,"Color","r");
ylabel("presure")
legend({'presure at trottle'})
title('timestep variation at trottle area-non conservation form')
%for mach number
subplot(4,1,4)
plot(linspace(1,nt,1400),mach_num_trottle_nc,"Color","m");
xlabel('time step')
ylabel("mach nuber")
legend({'mach num at trottle'})
title('timestep variation at trottle area-non conservation form')
% steady sTATE simulation for primitive value for nom conservation form
figure(4)
%for density
subplot(4,1,1)
plot(x,rho_nc,"Color","m");
ylabel("density")
legend({'density'})
title('timestep variation at trottle area-non conservation form')
%for tempreture
subplot(4,1,2)
plot(x,T_nc,"Color","b");
ylabel("tempreture")
legend({'tempreture'})
title('timestep variation at trottle area-non conservation form')
% for presure
subplot(4,1,3)
plot(x,presure_nc,"Color","r");
ylabel("presure")
legend({'presure'})
title('timestep variation at trottle area-non conservation form')
%for mach number
subplot(4,1,4)
plot(x,rho_nc,"Color","m");
xlabel('nozzel lenth')
ylabel("mach nuber")
legend({'mach num at trottle'})
title('timestep variation at trottle area-non conservation form')
% %defing the function for conservation form of simulation
tic;
[mass_flow_rate_c, presure_c, mach_num_c, rho_c,v_c, T_c, rho_trottle_c, v_trottle_c, T_trottle_c, mass_flow_rate_trottle_c, presure_trottle_c, mach_num_trottle_c]=conservation_form(x,dx,n,gamma,nt,CFL);
time_elapsed_c=toc;
fprintf('time elapesed for coservation form:%f',time_elapsed_c)
hold on
%plotting
figure(5)
%for density
subplot(4,1,1)
plot(linspace(1,nt,1400),rho_trottle_c,'Color','m');
ylabel("density")
legend({'density at trottel'})
title('timestep variation at trottle area- conservation form')
%for tempreture
subplot(4,1,2)
plot(linspace(1,nt,1400),T_trottle_c,'Color',"b");
ylabel("tempreture")
legend({'tempreture at trottle'})
title('timestep variation at trottle area- conservation form')
% for presure
subplot(4,1,3)
plot(linspace(1,nt,1400),presure_trottle_c,"Color","r");
ylabel("presure")
legend({'presure at trottle'})
title('timestep variation at trottle area- conservation form')
%for mach number
subplot(4,1,4)
plot(linspace(1,nt,1400),mach_num_trottle_c,"Color","m");
xlabel('time step')
ylabel("mach nuber")
legend({'mach num at trottle'})
title('timestep variation at trottle area- conservation form')
% steady sTATE simulation for primitive value for nom conservation form
figure(6)
%for density
subplot(4,1,1)
plot(x,rho_c,"Color","m");
ylabel("density")
legend({'density'})
title('timestep variation at trottle area- conservation form')
%for tempreture
subplot(4,1,2)
plot(x,T_c,"Color","b");
ylabel("tempreture")
legend({'tempreture'})
title('timestep variation at trottle area-conservation form')
% for presure
subplot(4,1,3)
plot(x,presure_c,"Color","r");
ylabel("presure")
legend({'presure'})
title('timestep variation at trottle area- conservation form')
%for mach number
subplot(4,1,4)
plot(x,mach_num_c,"Color","m");
xlabel('nozzel lenth')
ylabel("mach nuber")
legend({'mach num at trottle'})
title('timestep variation at trottle area- conservation form')
%camparing the plots of narmalization mass flow rte of both forms
figure(7)
hold on
plot(x,mass_flow_rate_nc,'r','linewidth',2)
hold on
plot(x,mass_flow_rate_c,'b','linewidth',2)
hold on
grid on
legend('non conversion','conversion')
xlabel('lenght of the domain')
ylabel('mass flow rate')
title('camparison of narmalization mass flow rate of both forms')
Function for consevation form:
%writting the function for non conservation form using macormac method
function [mass_flow_rate_c, presure_c, mach_num_c, rho_c, v_c, T_c, rho_trottle_c, v_trottle_c, T_trottle_c, mass_flow_rate_trottle_c, presure_trottle_c, mach_num_trottle_c]=conservation_form(x,dx,n,gamma,nt,CFL)
%[mass_flow_rate_c, presure_c, mach_num_c, rho_c, v_c, T_c, rho_trottle_c, v_trottle_c, T_trottle_c, mass_flow_rate_trottle_c, presure_trottle_c, mach_num_trottle_c]=conservation_form(x,dx,n,gamma,nt,CFL);
%defining the intial boundry condition for density and tempreture
for m =1:n
if(x(m) >=0 && x(m) <=0.5)
rho_c(m)=1;
T_c(m) =1;
elseif (x(m) >=0.5 && x(m) <=1.5)
rho_c(m)=1-0.366*(x(m)-0.5);
T_c(m) =1-0.167*(x(m)-0.5);
elseif (x(m) >=1.5 && x(m) <=3.5)
rho_c(m)=0.634-0.3879*(x(m)-1.5);
T_c(m) =0.833-0.3507*(x(m)-1.5);
end
end
%defining for velocity
a=1+2.2*(x-1.5).^2;
v_c =0.59./(rho_c.*a);
presure_c=rho_c.*T_c;
% defining the trottle
trottle = find(a==1);
% defing the initiel solution variabels
U1 =rho_c.*a;
U2 =rho_c.*a.*v_c;
U3 =rho_c.*a.*(T_c./(gamma-1))+((gamma/2).*v_c.^2);
%defining the time loop
for k =1:nt
%defining the time step
dt =min(CFL.*dx./(sqrt(T_c)+v_c));
%defing the old values
U1_old=U1;
U2_old=U2;
U3_old=U3;
%calculating the flux
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)));
% calculatinf the predector method
for j = 2:n-1
% calculating the souse term
J2(j)=(1/gamma)*rho_c(j)*T_c(j)*((a(j+1)-a(j))/dx);
% continuty eqution
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 the soluton vector
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 the flux
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)));
% know calculating for currector method
for j=2:n-1
% calculating the souse term
J2(j)=(1/gamma)*rho_c(j)*T_c(j)*((a(j)-a(j-1))/dx);
% continuty eqution
dU1_dt_c(j)=-((F1(j)-F1(j-1))/dx);
% momentum equation
dU2_dt_c(j)=-((F2(j)-F2(j-1))/dx)+J2(j);
%energy equation
dU3_dt_c(j)=-((F3(j)-F3(j-1))/dx);
end
%computing the avarege value
dU1_dt_av=0.5*(dU1_dt_p+dU1_dt_c);
dU2_dt_av=0.5*(dU2_dt_p+dU2_dt_c);
dU3_dt_av=0.5*(dU3_dt_p+dU3_dt_c);
% calculating the final solution
for m=2:n-1
U1(m)=U1_old(m)+dU1_dt_av(m)*dt;
U2(m)=U2_old(m)+dU2_dt_av(m)*dt;
U3(m)=U3_old(m)+dU3_dt_av(m)*dt;
end
%appliying the boundry conditions
% at inlets
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);
% defing the mass flow rate
%defing the flow rate
mass_flow_rate_c = rho_c.*a.*v_c;
presure_c = rho_c.*T_c;
mach_num_c = (v_c./sqrt(T_c));
% calculating the variables at trottle
rho_trottle_c(k) = rho_c(trottle);
T_trottle_c(k) = T_c(trottle);
v_trottle_c(k) = v_c(trottle);
mass_flow_rate_trottle_c(k) = mass_flow_rate_c(trottle);
presure_trottle_c(k) = presure_c(trottle);
mach_num_trottle_c(k) = mach_num_c(trottle);
%comparing the non dimensional mass flow at different time step
%plotting
figure(2);
if k == 50
plot(x,mass_flow_rate_c,'b','linewidth',1)
hold on
elseif k == 150
plot(x,mass_flow_rate_c,'c','linewidth',1)
hold on
elseif k == 200
plot(x,mass_flow_rate_c,'k','linewidth',1)
hold on
elseif k == 400
plot(x,mass_flow_rate_c,'g','linewidth',1)
hold on
elseif k == 800
plot(x,mass_flow_rate_c,'m','linewidth',1)
hold on
end
end
%labeling
xlabel('domain')
ylabel('mass flow rate conservation')
title('camparing the non dimensinal mass flow rate at diffrent time steps-conservation form ')
legend({'50th time step','150th time step','200th time step','400th time step','800th time step'})
grid on
Function for non conservation form:
%writting the function for non conservation form using macormac method
function [mass_flow_rate_nc, presure_nc, mach_num_nc, rho_nc, v_nc, T_nc, rho_trottle_nc, v_trottle_nc, T_trottle_nc, mass_flow_rate_trottle_nc, presure_trottle_nc, mach_num_trottle_nc]=non_conservative_form_original(x,dx,n,gamma,nt,CFL)
%defining the intial boundry condition
rho_nc=1 - 0.3146*x;
T_nc =1 - 0.2341*x;
v_nc=(0.1+1.09*x).*T_nc.^0.5;
%area of the nozzel
a = 1+2.2*(x-1.5).^2;
% defing trottle
trottle = find(a==1);
%defing the time loop
%defing the nodes
for k =1:nt
dt =min(CFL.*dx./(sqrt(T_nc)+v_nc));
%defing the old values
rho_old=rho_nc;
T_old=T_nc;
v_old =v_nc;
%know we have to solve for pridictor method
for j =2:n-1
% simplifying the speciel terms there fore we have equation for continity momentum energy in that equation we are descritizing using forword scheme
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;
% continuty equation
drho_dt_p(j)=-rho_nc(j)*dv_dx-rho_nc(j)*v_nc(j)*dlog_a_dx-v_nc(j)*drho_dx;
%momemtum 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);
% updating the solution
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
%know we have to define the equations fo currector methods
% simplifying the speciel terms there fore we have equation for continity momentum energy in that equation we are descritizing using backword scheme
for j=2:n-1
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;
% continuty equation
drho_dt_c(j)=-rho_nc(j)*dv_dx-rho_nc(j)*v_nc(j)*dlog_a_dx-v_nc(j)*drho_dx;
%momemtum 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
% computing the averege vtime deravetion or averege solution
drho_dt_av =0.5.*(drho_dt_p+drho_dt_c);
dv_dt_av =0.5.*(dv_dt_p+dv_dt_c);
dT_dt_av =0.5.*(dT_dt_p+dT_dt_c);
% final solution
for m=2:n-1
rho_nc(m)=rho_old(m)+(drho_dt_av(m))*dt;
v_nc(m)=v_old(m)+(dv_dt_av(m))*dt;
T_nc(m)=T_old(m)+(dT_dt_av(m))*dt;
end
% appling the baoundry conditions
%AT INLET
rho_nc(1)=1;
v_nc(1)=2*v_nc(2)-v_nc(3);
T_nc(1)=1;
% at oulet
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);
%defing the flow rate
mass_flow_rate_nc=rho_nc.*a.*v_nc;
presure_nc=rho_nc.*T_nc;
mach_num_nc=(v_nc./sqrt(T_nc));
% calculating the variables at trottle
%trottle=find(a==1);
rho_trottle_nc(k) = rho_nc(trottle);
T_trottle_nc(k) = T_nc(trottle);
v_trottle_nc(k) = v_nc(trottle);
mass_flow_rate_trottle_nc(k) = mass_flow_rate_nc(trottle);
presure_trottle_nc(k) = presure_nc(trottle);
mach_num_trottle_nc(k) = mach_num_nc(trottle);
%comparing the non dimensional mass flow at different time step
%plotting
figure(1);
if k == 50
plot(x,mass_flow_rate_nc,'b','linewidth',1)
hold on
elseif k == 150
plot(x,mass_flow_rate_nc,'c','linewidth',1)
elseif k == 200
plot(x,mass_flow_rate_nc,'m','linewidth',1)
elseif k == 400
plot(x,mass_flow_rate_nc,'r','linewidth',1)
elseif k == 800
plot(x,mass_flow_rate_nc,'k','linewidth',1)
end
end
%labeling
xlabel('domain')
ylabel('mass flow rate non conservation')
title('camparing the non dimensinal mass flow rate at diffrent time steps-non conservation form ')
legend({'50th time step','150th time step','200th time step','400th time step','800th time step'})
grid onPROGRAM EXPLANATIONS:
* First we have to start the program by writting by writting the clear all,close all,clc
* Then we have so solve the equations by using the maccormack mathed for both conservation and non consevation method
* Then we have to plot the graph for all the defferent variabels
OUTPUTS:
Figure 1:
* Plotting the mass flow rate againts the different time step and variation of this values are exactly same
Figure 2:
* The variation of non consarvation mass flow rate againt the diffrent time steps and the variation is same
Figure 3:
* The variation of density,time,velocity and mach number at trottle with respect the given time step.
Figure 4:
* the varion of variables with respect to the nozzel lenth of the non conservation form.
Figure 5:
* The varion of variables with respect to the nozzel lenth of the conservation form.
Figure 6:
* The varion of variables with respect to the nozzel lenth of the conservation form.
Figure 7:
* The mass flow rate comparition of conservation and non conservation form.
