# Python Program to Implement Steady State Equations Assignment Solution

June 28, 2024
Dr. Olivia
🇺🇸 United States
Python
Dr. Olivia Campbell holds a Ph.D. in Computer Science from the University of Cambridge. With over 800 completed assignments, she specializes in developing complex Python applications, including fitness trackers and exercise planners. Dr. Campbell's expertise lies in algorithm design and data analysis, ensuring optimal performance and accuracy in every project she undertakes.
Key Topics
• Instructions
• Objective
• Requirements and Specifications
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## Instructions

### Objective

Write a python assignment program to implement steady state equations.

## Requirements and Specifications

Source Code

import numpy as np import matplotlib.pyplot as plt M = 5.0 gam = 5./3. # ========================================================================== # ========================================================================== def ddx(x, F): Ns = F.size - 1 dFdx = np.zeros(Ns + 1, dtype='double') dFdx[0] = (F[1] - F[0]) / (x[1] - x[0]) for i in range(1, Ns): dFdx[i] = (F[i + 1] - F[i - 1]) / (x[i + 1] - x[i - 1]) dFdx[Ns] = (F[Ns] - F[Ns - 1]) / (x[Ns] - x[Ns - 1]) return dFdx # ========================================================================== # ========================================================================== if __name__ == '__main__': electron = np.loadtxt('./electron_quantites.txt') ion = np.loadtxt('./ion_quantites.txt') x = electron[:,0] # positions rho = ion[:,1] # this is rho/rho0. ke = electron[:,2] Te = electron[:,1] Ti = ion[:,2] ki = ion[:,3] eta = ion[:,4] u = ion[:,5] p = ion[:,6] E = ion[:,7] # First, calculate du/dx dudx = ddx(x, u) # Calculate dTi/dx dTidx = ddx(x, Ti) # Calculate dTe/dx dTedx = ddx(x, Te) # From equation (1b), calculate p0u0^2 + p0 eq1b = p + gam*M**2.*rho*u**2. - eta*dudx eq1b_right = gam*M**2 + p[0] # Equation (1c) eq1c = E*u + p*u - ki*dTidx - ke*dTedx - eta*u*dudx eq1c_right = E[0]/rho[0] + p[0] # Eq. 1a eq1a = rho*u #========================================================================== #========================================================================== # create plot plt.figure(figsize=(7,7)) # Plot dTidx plt.plot(x, rho, label = r'$\rho$') plt.plot(x, ke, label = '$k_{e}$') plt.plot(x, Te, label = '$T_{e}$') plt.plot(x, Ti, label = '$T_{i}$') plt.plot(x, ki, label = r'$k_{i}$') plt.plot(x, eta, label = r'$\eta$') plt.grid(True) plt.xlabel('Position') plt.legend() plt.show() plt.figure(figsize=(7,7)) plt.plot(x, u, label = 'u') plt.plot(x, p, label = 'p') plt.plot(x, E, label = 'E') plt.plot(x, dudx, label = r'$\frac{du}{dx}$') plt.plot(x, dTidx, label = r'$\frac{dT_{i}}{dx}$') plt.plot(x, dTedx, label = r'$\frac{dT_{e}}{dx}$') plt.xlabel('Position') plt.grid(True) plt.legend() plt.show() """ Now, plot each terms of each equation """ # Terms for Equation (1a) plt.figure() plt.plot(x, rho*u, label = r'$\rho u$') plt.xlabel('Position') plt.ylabel(r'$\rho u$') plt.set_title('Equation (1a)') plt.legend() plt.grid(True) plt.show() # Terms for question (1b). All terms in the same Figure # p + gam*M**2.*rho*u**2. - eta*dudx fig, axes = plt.subplots(nrows = 1, ncols = 2, figsize=(7,7)) axes[0].plot(x, p, label = r'$p$') axes[0].plot(x, gam*(M**2)*rho*(u**2), label = r'$\gamma M^{2}\rho u^{2}$') axes[0].plot(x, -eta*dudx, label = r'$-\eta \frac{du}{dx}$') axes[0].plot(x, eq1b, 'r--', label = r'$p + \gamma M^{2}\rho u^{2} - \eta \frac{du}{dx}$') axes[0].grid(True) axes[0].set_title('Left-side terms of Equation (1b)') axes[0].legend() axes[1].plot(x, gam*(M**2)*np.ones(len(x)), label = r'$\gamma M^{2}$') axes[1].plot(x, p[0]*np.ones(len(x)), label = r'$p_{0}$') axes[1].plot(x, eq1b_right*np.ones(len(x)), 'r--', label = r'$\gamma M^{2} + p_{0}$') axes[1].grid(True) axes[1].set_title('Right-side terms of Equation (1b)') axes[1].legend() plt.show() # Terms for Equation (1c). # E*u + p*u - ki*dTidx - ke*dTedx - eta*u*dudx = E[0]/rho[0] + p[0] fig, axes = plt.subplots(nrows = 1, ncols = 2, figsize=(7,7)) axes[0].plot(x, E*u, label = r'$Eu$') axes[0].plot(x, p*u, label = r'$pu$') axes[0].plot(x, -ki*dTidx, label = r'$-k_{i} \frac{dT_{i}}{dx}$') axes[0].plot(x, -ke*dTedx, label = r'$-k_{e} \frac{dT_{e}}{dx}$') axes[0].plot(x, -eta*u*dudx, label = r'$-\eta u \frac{du}{dx}$') axes[0].plot(x, eq1c, 'r--', label = r'$Eu + pu - k_{i} \frac{dT_{i}}{dx} - k_{e} \frac{dT_{e}}{dx} - \eta u \frac{du}{dx}$') axes[0].grid(True) axes[0].set_title('Left-side terms of Equation (1c)') axes[0].legend() axes[1].plot(x, E[0]/rho[0] *np.ones(len(x)), label = r'$\frac{E_{0}}{\rho_{0}}$') axes[1].plot(x, p[0]*np.ones(len(x)), label = r'$\rho_{0}$') axes[1].plot(x, eq1c_right*np.ones(len(x)), 'r--', label = r'$\frac{E_{0}}{\rho_{0}} + \rho_{0}$') axes[1].grid(True) axes[1].set_title('Right-side terms of Equation (1c)') axes[1].legend() plt.show() fig = plt.figure(figsize=(7, 5), dpi=120) plot = plt.plot(x, (eq1b-eq1b_right)/eq1b_right*100., 'royalblue', linestyle='--', label='Eq. (1b) percent error') plot = plt.plot(x, (eq1c-eq1c_right)/eq1c_right*100., 'tomato', linestyle='-', label='Eq. (1c) percent error') plot = plt.plot(x, (eq1a - 1.0)/1.0 *100, 'purple', linestyle='-', label = 'Eq. (1a) percent error') plt.ylabel(r'Percent error [%]') plt.xlabel(r'Position $\hat{x}$') #plt.xlim([25.0, 32.5]) plt.xlim([min(x), max(x)]) legend = plt.legend(loc='best', shadow=False, fontsize='small') #plt.savefig('equation_error.eps', format='eps', dpi=1000) plt.grid(True) plt.show() plt.close() #========================================================================== #========================================================================== 

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