Area Mach Number Relation

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Steven Castrillon 2019-9-30

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此问题的直接链接

https://ww2.mathworks.cn/matlabcentral/answers/482773-area-mach-number-relation

评论： Steven Castrillon 2019-9-30

I need to plot Mach number (M) as a function of Area ratios (A/A*) for subsonic and supersonic cases. I am trying to use newton-raphson method to iterate and find a converging solution for Mach number (M) at specific area ratios (A/A*), however i would like to be able to call in an array of A/A* inputs. The values for this array would from 0.1 to 10 (or 0.1:0.1:10) .

The function in question is

@(M) (1/M^2)*(((2+gm1*M^2)/gp1)^(gp1/gm1))-ARatio^2;

the variables for this function are;

g = 1.4;

gm1 = g-1;

gp1 = g+1;

Any help would be greatly appreciated!!!

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Answer 1

darova 2019-9-30

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此回答的直接链接

https://ww2.mathworks.cn/matlabcentral/answers/482773-area-mach-number-relation#answer_394115

在 MATLAB Online 中打开

Here is what i reached using polyxpoly

g   = 1.4; 
gm1 = g-1; 
gp1 = g+1;
F = @(M) 1./M.^2.*((2+gm1*M.^2)/gp1).^(gp1/gm1);%-ARatio^2;
M = linspace(0.1,3.5);                    % Mach number
A = sqrt( F(M) );                         % A ratio
plot(M,A)                                 % draw function
hold on
    % find Mach number of each A 
for a = linspace(0.1,5,10)
    mm = [0 4];                           % just horizontal line
    aa = [a a];
    [xm,ya] = polyxpoly(mm,aa,M,A);
    plot(xm,ya,'.-r')
end
hold off

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Steven Castrillon 2019-9-30

在 MATLAB Online 中打开

The following code shows me how to get solutions for both sub and supersonic, however, how can i input a range of ARatio instead of just one value?
clear;
clc;
%% INPUTS
% Define some paramters
g   = 1.4;
gm1 = g-1;
gp1 = g+1;
% Define anonymous function with two inputs (M and ARatio)
% - Will be used in the methods below
% - Pass M and ARatio as arguments to AM_EQN to get function value
% -     funVal = AM_EQN(M,ARatio)
AM_EQN = @(M,ARatio) sqrt((1/M^2)*(((2+gm1*M^2)/gp1)^...
                            (gp1/gm1)))-ARatio;
% Solve for Msub and Msup using this area ratio (A/A*)
ARatio = 1.5;
% Error tolerance
errTol = 1e-4;
% Flags for printing iterations to screen
verboseBisection   = 0;
verboseIncremental = 0;
%% SUBSONIC INCREMENTAL SEARCH
% Initial values
dM       = 0.1;                                                             % Initial M step size
M        = 1e-6;                                                            % Initial M value
iConvSub = 0;                                                               % Initial converge index
if (verboseIncremental == 1)
    fprintf('Incremental Search Method: Subsonic\n');
    fprintf('-----------------------------------\n');
end
% Iterate to solve for root
iterMax = 100;                                                              % Maximum iterations
stepMax = 100;                                                              % Maximum step iterations
for i = 1:1:iterMax
    for j = 1:1:stepMax
        
        % Evaluate function at j and j+1
        fj   = AM_EQN(M,ARatio);
        fjp1 = AM_EQN(M+dM,ARatio);
        
        % Print iterations to command window
        if (verboseIncremental == 1)
            fprintf('fj | fjp1: %3.4f\t%3.4f\n',fj,fjp1);
        end
        
        % Update M depending on sign change or not
        % - If no sign change, then we are not bounding root yet
        % - If sign change, then we are bounding the root, update dM
        if (fj*fjp1 > 0)
            M = M + dM;                                                     % Update M
        elseif (fj*fjp1 < 0)
            dM = dM*0.1;                                                    % Refine the M increment
            break;                                                          % Break out of j loop
        end
        
    end % END: j Loop
    
    % Check for convergence
    if (abs(fj-fjp1) <= errTol)                                             % If converged
        iConvSub = i;                                                       % Set converged index
        break;                                                              % Exit loop
    end
    
end % END: i Loop
% Set subsonic Mach number to final M from iterations
Msub = M;
%% SUPERSONIC INCREMENTAL SEARCH
% Initial values
dM       = 1;                                                               % Initial M step size
M        = 1+1e-6;                                                          % Initial M value
iConvSup = 0;                                                               % Initial converge index
if (verboseIncremental == 1)
    fprintf('\nIncremental Search Method: Supersonic\n');
    fprintf('-------------------------------------\n');
end
% Iterate to solve for root
iterMax = 100;                                                              % Maximum iterations
stepMax = 100;                                                              % Maximum step iterations
for i = 1:1:iterMax
    for j = 1:1:stepMax
        
        % Evaluate function at j and j+1
        fj   = AM_EQN(M,ARatio);
        fjp1 = AM_EQN(M+dM,ARatio);
        
        % Print iterations to command window
        if (verboseIncremental == 1)
            fprintf('fj | fjp1: %3.4f\t%3.4f\n',fj,fjp1);
        end
        
        % Update M depending on sign change or not
        % - If no sign change, then we are not bounding root yet
        % - If sign change, then we are bounding the root, update dM
        if (fj*fjp1 > 0)
            M = M + dM;                                                     % Update M
        elseif (fj*fjp1 < 0)
            dM = dM*0.1;                                                    % Refine the M increment
            break;                                                          % Break out of j loop
        end
        
    end % END: j Loop
    
    % Check for convergence
    if (abs(fj-fjp1) <= errTol)                                             % If converged
        iConvSup = i;                                                       % Set converged index
        break;                                                              % Exit loop
    end
    
end % END: i Loop
% Set supersonic Mach number to final M from iterations
Msup = M;
% Print solutions to command window
fprintf('==== INCREMENTAL SEARCH SOLVER ====\n');
fprintf('Msub: %3.4f after %i iterations\n',Msub,iConvSub);
fprintf('Msup: %3.4f after %i iterations\n',Msup,iConvSup);
fprintf('===================================\n\n');

darova 2019-9-30

I like it

Steven Castrillon 2019-9-30

yes but can you please help me to introduce an array of values for ARatio?

in the code i provided, ARatio is set as : ARatio = 1.5

when i set it as: ARatio = [0.1:0.1:10] i get an error at

Error in AREAMACH2 (line 62)

if (fj*fjp1 > 0)

Please help

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Area Mach Number Relation

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