Week 7 - Simulation of a 1D Super-sonic nozzle flow simulation using Macormack Method
Aim:
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.
Objective:
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.
Explanation:
We consider the study, isentropic flow through a convergent divergent nozzle. The flow at the inlet to the nozzle comes from reservoir. The cross section area of the reservoir is very large and hence velocity is very small. The flow is locally subsonic the convergent section of the nozzle, Sonic at the throat and Supersonic at the divergent section.
A nozzle is a type of device designed to control the direction or characteristics of fluid flow at the exit of a pipe (or such system). A nozzle is often a pipe or tube of varying cross-sectional area, thus can be used to direct or modify flow of a fluid.
A convergent-divergent nozzle (also known as CD nozzle or de Laval nozzle) is a device used to accelerate a hot, pressurized gas passing through it to a higher supersonic speed in the axial direction by converting heat energy of the flow to kinetic energy. Hence, such nozzles are widely used in steam turbines, supersnic jet engines and rocket engine nozzles. Its operation relies on the different properties of gases flowing at subsonic, sonic and supersonic speeds.
In the field of CFD, MacCormack method is a prominent discretization scheme for obtaining numerical solutions of hyperbolic partial differential equations. (based on second-order finite difference method). It is the variation of two-step Lax-Wendroff scheme but much simple in application.
Nozzle diagram with sections:
Non Conservative Form:
Conservative Form:
These equations are solved by Macormack method using time marching process
Mach number:
Mach number is a dimensionless quantity in fluid dynamics representing the ratio of flow velocity past a boundry to the local speed of sound.
where:
- M is the local Mach number,
- u is the local flow velocity with respect to the boundaries (either internal, such as an object immersed in the flow, or external, like a channel), and
- c is the speed of sound in the medium, which in air varies with the square root of the thermodynamics temeprature.
For air at standard conditions, γ=1.4">γ=1.4
γ=1.4">where γ">γ is the ratio of specific heat
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.
1.inital profile:
Before the calculation, the initial profile for all the flow field variables which are velocity, density and temperature has to be determined. These are guess values but should be specified with more awareness on the problem.
2.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.
3.solution update:
fp=">fp=Flow field variables calculated after the predictor step
fInitial=">fInitial=flow field variable in the initial step
Δt=">Δt=time step size
4.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.
5. Final solution update:
Main Code:
close all
clear all
clc
%Input values
L=3; %length of nozzle
n=31; %number of grid points
x=linspace(0,L,n);
dx = x(2)-x(1);
CFL = 0.5; %number of time steps
nt = 1400; %number of time steps
gamma = 1.4;
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, CFL);
Time_Period_NC = toc;
fprintf('Time Period for Non Conservative form: %0.4f', Time_Period_NC);
hold on
% PLOTTING
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','b')
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','b')
xlabel('Nozzle Length')
ylabel('Mach number')
legend({'Mach number'});
axis([0 3 0 4])
grid minor;
%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
% Inlet
rho_NC(1) = 1;
v_NC(1) = 2*v_NC(2) - v_NC(3);
T_NC(1) = 1;
% 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
CONSERVATIVE FORM:
close all
clear all
clc
%Input values
L=3; %length of nozzle
n=31; %number of grid points
x=linspace(0,L,n);
dx = x(2)-x(1);
CFL = 0.5; %number of time steps
nt = 1400; %number of time steps
gamma = 1.4;
%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, CFL);
Time_Period_C = toc;
fprintf('Time Elasped for Conservative form: %0.4f', Time_Period_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;
%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
% 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));
% 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
Outputs:
Time steps variation of primitive variable at Throat area Non-Conservative Form:
Time steps variation of primitive variable at Throat area Conservative Form:
Variation of Flow Rate Distribution Non-Conservative form:
Variation of Flow Rate Distribution Conservative form:
Mass flow rates at different Timesteps Non-Conservative form:
Mass flow rates at different Timesteps Conservative form:
CONCLUSION:
Cumulating all the above results we can evidently 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.
- For comparing the conservative and non-conservative form, the values of the flow variables at the throat of the nozzle are plotted at each time step. It can be observed that at smller time-steps, the behaviour of non-conservative form shows unstable results. (it shows oscillations) but with increase in time step, the difference in conservative and non-conservative approach diminshes.
- If we notice the residuals (average time derivatives of variables) of both the approaches the conservative form converges to a steady state faster than non-conservative form.
- There is an appreciable difference in how the mass flow rate varies with time step, but it can be observed that at the time step of 700Δt
- , the mass flow becomes constant. Moreover, the conservative form produces an almost constant mass flow rate but the non-conservative approach gave a highly fluctuating mass flow rate.
- On varying the grid size, it is noted that the conservative form almost maintains its consistency but the non conservative form tends to become stable (on increasing grid size).
- Thus MaCormack method provides quite stable and close to exact solutions, hence, it is widely implemented to solve various flow problems.