figure 6.ipynb

Kernel: Python 3 (system-wide)

import numpy as np
import sys, os, h5py
import matplotlib.pyplot as plt
import seaborn as sns
import cmocean

# Aesthetics
sns.set_theme()
sns.set_style('ticks')
sns.set_context('paper')
plt.rc('mathtext', fontset='stix')
plt.rc('font', family='serif')

cspeed = cmocean.tools.crop_by_percent(cmocean.cm.tempo, 30, which='both', N=None)

import pandas as pd

Load sheared VC data

df = pd.read_csv('../data/P-VC_global_response.csv')
df.head()

Load sheared RB data

PRB_df = pd.read_csv('../data/P-RB_global_response.csv')
PRB_df.head()

Add Grashof number to datasets $Gr=Ra/Pr$

# P-VC
df['Gr'] = df.Ra/df.Pr

# P-RB
PRB_df['Gr'] = PRB_df.Rayleigh/PRB_df.Prandtl

Write function to solve Prandtl friction law prediction

from scipy.optimize import newton
def Cfunc(Reb):
    Cf = np.zeros(Reb.size)
    for i in range(Reb.size):
        def f(x):
            # k = 0.41
            # B = 5
            k = 0.384
            B = 4.27
            return (2/x)**0.5 - 1/k*np.log(Reb[i]/2*(x/2)**0.5) - B
        Cf[i] = newton(f, 2.5e-3)
    return Cf

fig, axs = plt.subplots(1,2, figsize=(5.2,2.2), layout='constrained', dpi=200)
cmap = cspeed

ax = axs[0]

# P-RB
sc = ax.scatter(PRB_df['Bulk Reynolds'], PRB_df['Nusselt (stafield)']/PRB_df['Prandtl']**0.5, c=np.log10(PRB_df.Gr), cmap=cmap, s=5, label='Mixed RB')

# P-VC
ax.scatter(df.Re*(df.Pr==1), df.Nu*(df.Pr==1)/df.Pr**0.5, c=np.log10(df.Gr), cmap=cmap, ec='k', s=25, label='Mixed VC')
ax.scatter(df.Re*(df.Pr==4), df.Nu*(df.Pr==4)/df.Pr**0.5, c=np.log10(df.Gr), cmap=cmap, ec='k', s=25, marker='^')
ax.scatter(df.Re*(df.Pr==10), df.Nu*(df.Pr==10)/df.Pr**0.5, c=np.log10(df.Gr), cmap=cmap, ec='k', s=25, marker='v')

fig.colorbar(sc, ax=axs, label='$\log_{10} Gr$')

# Add theoretical comparison
R = 10**np.linspace(1.5,4.5,101)
ax.loglog(R, 0.25*Cfunc(R)*R, 'k--')
ax.annotate('Prandtl', (2e2, 1.25e-2), ha='center', va='center', rotation=-17)

# Labels and aesthetics
ax.grid(True)
ax.set(
    xlabel='$Re$',
    ylabel='$Nu/Pr^{1/2}$',
    xlim = [8e1,2e4],
    ylim=[2,50]
)
ax.annotate('$(a)$', (-0.05, 1.05), xycoords='axes fraction', ha='right', va='bottom')


ax = axs[1]

# P-RB
ax.scatter(PRB_df['Bulk Reynolds']/PRB_df['Re0'], PRB_df['Nusselt (stafield)']/PRB_df['Nu0'], c=np.log10(PRB_df.Gr), cmap=cmap, s=5)

# P-VC
ax.scatter(df.Re*(df.Pr==1)/df.Re0, df.Nu/df.Nu0, c=np.log10(df.Gr), cmap=cmap, ec='k', s=25, label='$Pr=1$')
ax.scatter(df.Re*(df.Pr==4)/df.Re0, df.Nu/df.Nu0, c=np.log10(df.Gr), cmap=cmap, ec='k', s=25, marker='^', label='$Pr=4$')
ax.scatter(df.Re*(df.Pr==10)/df.Re0, df.Nu/df.Nu0, c=np.log10(df.Gr), cmap=cmap, ec='k', s=25, marker='v', label='$Pr=10$')

# Add theoretical comparison
R = 10**np.linspace(-1,1,101)
ax.loglog(R, (1 + R**2)**-0.1, 'k--')

# Labels and aesthetics
ax.grid(True)
ax.legend()
ax.set(
    xlabel='$Re/Re_0$',
    ylabel='$Nu/Nu_0$',
    xlim=[1e-1, 1e2],
    ylim=[0.6,2]
)
ax.annotate('$(b)$', (-0.05, 1.05), xycoords='axes fraction', ha='right', va='bottom')

# fig.savefig('Nusselt_numbers.pdf')

plt.show()