%%
%Isentropic, Normal shock Relations and Linear Interpolation Formulas
%Isentropic Relations
%Calculate Total Pressure Over Local Pressure (Pt/Px)
FindPtOverPLoc = @(G,MAtPLoc) (1+.5*(G-1)*(MAtPLoc^2))^(G/(G-1));
%Formula to determine Pt/Ploc
%Calculate Area/Area_Star
FindAOverAStar = @(G,MAtALoc) (1/MAtALoc)*(((2/(G+1))*(1+.5*(G1)*(MAtALoc^2)))^((.5*(G+1))/(G-1))); %Formula to detmerine A/AStar
%Calculate Total Temperature Over Local Temperature
FindTtOverTLoc = @(G,MAtT1)(1+.5*(G-1)*(MAtT1^2)); %Formula to
determine Tt/Tloc
%Shock Relations
%Calculate M2
FindM2 = @(G,M1) ((1+.5*(G-1)*(M1^2))/(G*(M1^2)-.5*(G-1)))^.5;
%Formula to detmerine M2 after the shock
%Calculate P2/P1 (pressure relation before and after shock)
FindP2OverP1 = @(G,M1) 1+((2*G)/(G+1))*((M1^2)-1); %Formula to
detmerine P2/P1(Pressure After Shock/Pressure Before Shock)
%Linear Interpolation Scheme
%y2 = y1 + (y3-y1)*((x2-x1)/(x3-x1))
LinearInterpolate = @(y1,y3,x1,x2,x3) y1 + (y3-y1)*((x2-x1)/(x3-x1));
%%
% inputing the ratios necessary for the problem
prompt = 'What is the ratio of exit static pressure to reservoir total
pressure? ';
PeOverPt1Initial = input(prompt);
prompt = 'What is the ratio of the exit area to the throat area? ';
AeOverA1Star = input(prompt);
%% Properties of air
G = 1.4; %Specific heat ratio of air, gamma
%%
%The throat area is found by dividing Ae by Ae/A1*. Ae = 1
A1Star = 1/AeOverA1Star;
% A guess at the LOCATION of the shockwave. Iteration will begin with this
% value
XShockGuess = 0.01;
% Use an iteration scheme to converge onto a final value of Pe/Pt1
% Using the steps below.
%%
repeat = 1; %Initial condition for Pressure Loop
%Initializing arrays for specific outputs
Machs = zeros;
x = zeros;
StaticPressure = zeros;
Temperature = zeros;
%Initializing variable for Interpolation
PeOverPt1Upper = 0;
PeOverPt1Lower = 0;
ShockLocationUpper = 0;
ShockLocationLower = 0;
i = 1; %initial i counter
%While loop to find Pressure ratio for changing shock locations
while repeat == 1
AShock = A1Star + (1-A1Star)*XShockGuess; %Area of shock
AoA1star = AShock / A1Star; %Area of Shock / Area of First Throat
% First use iteration to find M1 at the guessed shock location, then use the
% formula for p1/pt1
%%
%Initial values for Secant Method to Find Mach number
Mtol = 1;
M1a = 1;
M1b = 3;
%Area Ratio for Lower Mach Number Guess
Y1 = FindAOverAStar(G,M1a)- AoA1star;
%Secant Method to Find Mach Number
while Mtol > .01
%Area Ratio for upper Mach Number Guess
Y2 = FindAOverAStar(G,M1b)- AoA1star;
%Estimated Mach Number
M1next = M1b-(((Y2)*(M1b-M1a))/(Y2-Y1));
%Exchanging of Values
M1a = M1b;
M1b = M1next;
%New Value becomes Old Value
Y1 = Y2;
%Set Tolerance
Mtol = abs(M1b-M1a)/M1a;
end
%Mach Number Value
M1 = M1b;
Pt1OverP1 = FindPtOverPLoc(G,M1);
%Finds P1/Pt1 at M1
M2 = FindM2(G,M1); %Formula to detmerine M2 after the shock
Pt2OverP2 = FindPtOverPLoc(G,M2);
%Finds P2/Pt2 at M2
P2OverP1 = FindP2OverP1(G,M1); %Formula to detmerine P2/P1(Pressure After
Shock/Pressure Before Shock)
Pt2OverPt1 = (Pt2OverP2 / Pt1OverP1) * P2OverP1; %Finds Pt2/Pt1 using
pressure ratios
% Use formula to calculate A2/A2* from isentropic relations using M2
A2OverA2Star = FindAOverAStar(G,M2); %Formula to detmerine A2/A2Star
% Ae/A2* = Ae/A2 * A2/A2* (Ae is 1 and A2 is the same as the shock Area.)
AeOverA2Star = 1/(A1Star + ( 1 - A1Star ) * XShockGuess)*A2OverA2Star;
%Use Secant Method to Calculate Me using Ae/A2*
%%
%Initialize Tolerance Value
Mtol = 1;
%Bounds: Change as needed
Mea = .3;
Meb = .6;
%Area Ratio for Lower Mach Number Guess
Y1 = FindAOverAStar(G,Mea)- AeOverA2Star;
%Secant Method to Find Mach Number
while Mtol > .01
%Area Ratio for upper Mach Number Guess
Y2 = FindAOverAStar(G,Meb)- AeOverA2Star;
%Estimated Mach Number
Menext = Meb-(((Y2)*(Meb-Mea))/(Y2-Y1));
%Exchanging of Values
Mea = Meb;
Meb = Menext;
%New Value becomes Old Value
Y1 = Y2;
%Set Tolerance
Mtol = abs(Meb-Mea)/Mea;
end
%Mach Number at the Exit
Me = Meb;
% Use Me to find Pe/Pt2 with isentropic formulas.
PeOverPt2 = 1/FindPtOverPLoc(G,Me); %Formula to detmerine Pe/Pt2
%PeOverPt1new = PeOverPt2 * Pt2OverPt1
if (PeOverPt2 * Pt2OverPt1) < PeOverPt1Initial
PeOverPt1Upper = PeOverPt2 * Pt2OverPt1;
ShockLocationUpper = XShockGuess;
repeat = 0;
else
PeOverPt1Lower = PeOverPt2 * Pt2OverPt1;
ShockLocationLower = XShockGuess;
repeat = 1;
end
%Fill Arrays for Specified Variables
Machs(i) = M1;
x(i) = XShockGuess;
StaticPressure(i)=1/FindPtOverPLoc(G,M1);
Temperature(i) = 1/FindTtOverTLoc(G,M1);
%Increment Shock Guess Location
XShockGuess = XShockGuess + 0.001;
%Increment i counter
i = i+1;
end
%%
% To find M between shock and exit.
A2Star = AShock/A2OverA2Star; %AShock = A2
X = XShockGuess; %Location after the shock
%Loop to find Specified Values after Shock based on Mach Number
while X <= 1
%Area of Linearly Expanding Nozzle
A = A1Star + ( 1 - A1Star ) * X;
AOverA2Star = A/A2Star; %Area Ratio after shock
%% Secant Method Again
%Uses different bounds
Mtol = 1;
Ma = .2;
Mb = .7;
Y1 = FindAOverAStar(G,Ma)- AOverA2Star;
while Mtol > .01
Y2 = FindAOverAStar(G,Mb)- AOverA2Star;
Mnext = Mb-(((Y2)*(Mb-Ma))/(Y2-Y1));
Ma = Mb;
Mb = Mnext;
Y1 = Y2;
Mtol = abs(Mb-Ma)/Ma;
end
%% Continue to Fill Arrays for Specified Variables
Machs(i) = Mb;
x(i) = X;
StaticPressure(i) = Pt2OverPt1/FindPtOverPLoc(G,Mb);
Temperature(i) = 1/FindTtOverTLoc(G,Mb);
%
i
%
X
i counter
= i + 1;
Increment Location in Nozzle
= X + .001;
end
%%
%Linear Interpolation Check to find shock location
ShockLocation =
LinearInterpolate(ShockLocationLower,ShockLocationUpper,PeOverPt1Upper,PeOver
Pt1Initial,PeOverPt1Lower);
%display shock location
disp('Shock Location = '); disp(ShockLocation);
%%
%Plots
figure (1)
subplot(3,1,1)
plot(x,Machs)
title('Mach Number vs Distance Downstream');
xlabel('Distance Downstream, x');
ylabel('Mach Number');
subplot(3,1,2)
plot(x,StaticPressure)
title('Local Static Pressure/Reservoir Total Pressure vs Distance
Downstream');
xlabel('Distance Downstream, x');
ylabel('P Static/Pt1');
subplot(3,1,3)
plot(x,Temperature)
title('Local Temperature/ Total Temperature vs Distance Downstream');
xlabel('Distance Downstream, x');
ylabel('T local/ T total');