Simulation of 1D Supersonic Nozzle Flow using MacCormack Method in MATLAB

Governing Equations for Non-Conservative form:

(A) Continuity Equation:

`(delrho)/(delt) = -rho(delV)/(delx)-rhoV((del(lnA))/(delx)) - V((delrho)/(delx))`

(B) Momentum Equation:

`(delV)/(delt) = -V(delV)/(delx) - 1/gamma((delT)/(delx) + T/rho(delrho)/(delx))`

(C) Energy Equation:

`(delT)/(delt) = -V(delT)/(delx) - (gamma-1)T[(delV)/(delx) + V(del(lnA))/(delx)]`

Governing Equations for Conservative form:

`(delU_1)/(delt) = - (delF_1)/(delx)`

`(delU_2)/(delt) = - (delF_2)/(delx) + J_2`

`(delU_3)/(delt) = - (delF_3)/(delx)`

Here,

`U_1 = rhoA, U_2 = rhoAV, U_3 = rhoA(e/(gamma-1)+gamma/2V^2)`

`F_1 = rhoAV,F_2 = rhoAV^2 + 1/gammapA, F_3 = rho(e/(gamma-1)+gamma/2V^2 )VA+ pAV`

`J_2 = 1/gamma*p*(delA)/(delx)`

Again,

`V=(U_2)/(U_1), e=T and p=rhoT`

Simulation is computed at 1) Grid Points = 31, 61 and 91

2) Various CFL Number

3) Error Tolerance = 1e-05

PLOTS shown for Conservative form at CFL = 0.5 and Grid Points = 31:

PLOTS shown for Non-Conservative form at CFL = 0.5 and Grid Points = 31:

RESULT Discussion:

The minimum Number of Cycles and Time required for the convergence of simulation ( at CFL = 0.5 ):

Convergence Rate:

We noticed that if `CFL <1`, the Non-Conservative form converges at faster rate than the Conservative form for the same Number of Grid Points. But for `CFL =1`, the Conservative form is faster compared to the other one.

Here CFL No. is incresaed for the same No. of Grid Points to converge the simulation fast.

Mass Flow Rate Variation:

Conservative form --> In this case the Mass Flow Rate remains fairly constant before and after the throat (x = 1.5). The plot will be smoother if we increase the No. of Grid Points.

For CFL = 0.8 and Grid Points = 31,

For CFL = 0.8 and Grid Points = 91,

Non-Conservative form --> For a fixed CFL No. increasing the Grid Points to 61 and 91 decreases the deviation of Mass Flow Rate before and after the throat (x = 1.5).

For CFL = 0.8 and Grid Points = 31,

For CFL = 0.8 and Grid Points = 91,

## Again between these Conservative and Non-conservative Mass Flow Rate, deviation in the first one is less compared to the second one. Hence the Conservative form is better for Mass Flow Rate ##

At last we can conclude that to minimize the deviation or error, finer grid should be used.

NB:  In the book, John D Anderson; dt = 0.0267 is taken for the Conservative simulation.That 
     value is a TYPO. Because of that value, 'Simulation Time' and 'No. of Time Steps' may 
     get differ from the present results stated above when comparing with the Non- 
     Conservative form.