# Figure 6 # Fig. 6: frequency ratios of the quasi air-water simulation, as functions of kappa_x*H. # delta = omega_0/omega_t compares intrinsic surface dynamics with turbulent forcing; zeta = gamma/omega_t # compares surface damping with turbulent forcing. # Parameters (simulation units, H = 1): g = 0.36, sigma = 0.036, rho_w = 1, rho_a = 1/80, # mu_w = 1/5000, mu_w/mu_a = 55, dissipation rates eps_w = 0.00645606716 and eps_a = 0.00214390149. # omega_t = eps^(1/3) kappa^(2/3), eps^(4/3) = (rho_w^2 eps_w^(4/3) + rho_a^2 eps_a^(4/3))/(rho_w^2 + rho_a^2). # The grid is the one-sided streamwise grid kappa_x = 2*pi*i/(4*pi), i = 1..1151 (k = 0 excluded). # Columns: # 1: kxH - streamwise wavenumber, kappa_x*H # 2: delta - delta = omega_0/omega_t (left panel); omega_0 gravity-capillary natural frequency # 3: zeta - zeta = gamma/omega_t (right panel); gamma = viscous damping rate of the interface # ------------------------------------------------------------------------------ # kxH delta zeta 5.000000e-01 3.616984e+00 4.286578e-04 1.000000e+00 3.339582e+00 1.080150e-03 1.500000e+00 3.295868e+00 1.854694e-03 2.000000e+00 3.360642e+00 2.721807e-03 2.500000e+00 3.490631e+00 3.664972e-03 3.000000e+00 3.663538e+00 4.673537e-03 3.500000e+00 3.865815e+00 5.739950e-03 4.000000e+00 4.088597e+00 6.858524e-03 4.500000e+00 4.325858e+00 8.024797e-03 5.000000e+00 4.573404e+00 9.235151e-03 5.500000e+00 4.828266e+00 1.048659e-02 6.000000e+00 5.088311e+00 1.177658e-02 6.500000e+00 5.351985e+00 1.310293e-02 7.000000e+00 5.618149e+00 1.446377e-02 7.500000e+00 5.885952e+00 1.585741e-02 8.000000e+00 6.154761e+00 1.728240e-02 8.500000e+00 6.424095e+00 1.873740e-02 9.000000e+00 6.693592e+00 2.022122e-02 9.500000e+00 6.962974e+00 2.173279e-02 1.000000e+01 7.232030e+00 2.327112e-02 1.050000e+01 7.500599e+00 2.483532e-02 1.100000e+01 7.768557e+00 2.642455e-02 1.150000e+01 8.035811e+00 2.803805e-02 1.200000e+01 8.302291e+00 2.967511e-02 1.250000e+01 8.567944e+00 3.133507e-02 1.300000e+01 8.832732e+00 3.301732e-02 1.350000e+01 9.096628e+00 3.472128e-02 1.400000e+01 9.359614e+00 3.644641e-02 1.450000e+01 9.621678e+00 3.819220e-02 1.500000e+01 9.882814e+00 3.995818e-02 1.550000e+01 1.014302e+01 4.174389e-02 1.600000e+01 1.040230e+01 4.354891e-02 1.650000e+01 1.066066e+01 4.537284e-02 1.700000e+01 1.091811e+01 4.721528e-02 1.750000e+01 1.117465e+01 4.907588e-02 1.800000e+01 1.143029e+01 5.095428e-02 1.850000e+01 1.168505e+01 5.285016e-02 1.900000e+01 1.193894e+01 5.476320e-02 1.950000e+01 1.219196e+01 5.669309e-02 2.000000e+01 1.244414e+01 5.863956e-02 2.050000e+01 1.269548e+01 6.060231e-02 2.100000e+01 1.294600e+01 6.258108e-02 2.150000e+01 1.319571e+01 6.457563e-02 2.200000e+01 1.344463e+01 6.658569e-02 2.250000e+01 1.369276e+01 6.861105e-02 2.300000e+01 1.394013e+01 7.065146e-02 2.350000e+01 1.418674e+01 7.270671e-02 2.400000e+01 1.443261e+01 7.477659e-02 2.450000e+01 1.467774e+01 7.686090e-02 2.500000e+01 1.492216e+01 7.895943e-02 2.550000e+01 1.516586e+01 8.107201e-02 2.600000e+01 1.540888e+01 8.319843e-02 2.650000e+01 1.565121e+01 8.533853e-02 2.700000e+01 1.589287e+01 8.749214e-02 2.750000e+01 1.613387e+01 8.965908e-02 2.800000e+01 1.637422e+01 9.183919e-02 2.850000e+01 1.661392e+01 9.403232e-02 2.900000e+01 1.685300e+01 9.623832e-02 2.950000e+01 1.709147e+01 9.845702e-02 3.000000e+01 1.732932e+01 1.006883e-01 3.050000e+01 1.756657e+01 1.029320e-01 3.100000e+01 1.780323e+01 1.051880e-01 3.150000e+01 1.803931e+01 1.074562e-01 3.200000e+01 1.827482e+01 1.097364e-01 3.250000e+01 1.850977e+01 1.120285e-01 3.300000e+01 1.874416e+01 1.143324e-01 3.350000e+01 1.897800e+01 1.166479e-01 3.400000e+01 1.921131e+01 1.189751e-01 3.450000e+01 1.944408e+01 1.213136e-01 3.500000e+01 1.967634e+01 1.236635e-01 3.550000e+01 1.990808e+01 1.260245e-01 3.600000e+01 2.013931e+01 1.283967e-01 3.650000e+01 2.037004e+01 1.307800e-01 3.700000e+01 2.060027e+01 1.331741e-01 3.750000e+01 2.083002e+01 1.355790e-01 3.800000e+01 2.105929e+01 1.379946e-01 3.850000e+01 2.128809e+01 1.404209e-01 3.900000e+01 2.151641e+01 1.428576e-01 3.950000e+01 2.174428e+01 1.453049e-01 4.000000e+01 2.197169e+01 1.477624e-01 4.050000e+01 2.219866e+01 1.502302e-01 4.100000e+01 2.242518e+01 1.527082e-01 4.150000e+01 2.265126e+01 1.551963e-01 4.200000e+01 2.287691e+01 1.576944e-01 4.250000e+01 2.310214e+01 1.602025e-01 4.300000e+01 2.332694e+01 1.627204e-01 4.350000e+01 2.355133e+01 1.652481e-01 4.400000e+01 2.377531e+01 1.677854e-01 4.450000e+01 2.399888e+01 1.703324e-01 4.500000e+01 2.422205e+01 1.728890e-01 4.550000e+01 2.444483e+01 1.754551e-01 4.600000e+01 2.466721e+01 1.780305e-01 4.650000e+01 2.488921e+01 1.806153e-01 4.700000e+01 2.511082e+01 1.832094e-01 4.750000e+01 2.533206e+01 1.858127e-01 4.800000e+01 2.555292e+01 1.884252e-01 4.850000e+01 2.577341e+01 1.910468e-01 4.900000e+01 2.599354e+01 1.936773e-01 4.950000e+01 2.621331e+01 1.963169e-01 5.000000e+01 2.643272e+01 1.989653e-01 5.050000e+01 2.665178e+01 2.016226e-01 5.100000e+01 2.687048e+01 2.042887e-01 5.150000e+01 2.708884e+01 2.069635e-01 5.200000e+01 2.730686e+01 2.096469e-01 5.250000e+01 2.752454e+01 2.123390e-01 5.300000e+01 2.774189e+01 2.150396e-01 5.350000e+01 2.795890e+01 2.177488e-01 5.400000e+01 2.817559e+01 2.204664e-01 5.450000e+01 2.839195e+01 2.231924e-01 5.500000e+01 2.860799e+01 2.259267e-01 5.550000e+01 2.882372e+01 2.286694e-01 5.600000e+01 2.903912e+01 2.314203e-01 5.650000e+01 2.925422e+01 2.341794e-01 5.700000e+01 2.946901e+01 2.369466e-01 5.750000e+01 2.968349e+01 2.397220e-01 5.800000e+01 2.989767e+01 2.425054e-01 5.850000e+01 3.011154e+01 2.452968e-01 5.900000e+01 3.032512e+01 2.480962e-01 5.950000e+01 3.053841e+01 2.509035e-01 6.000000e+01 3.075141e+01 2.537186e-01 6.050000e+01 3.096411e+01 2.565416e-01 6.100000e+01 3.117653e+01 2.593724e-01 6.150000e+01 3.138867e+01 2.622110e-01 6.200000e+01 3.160053e+01 2.650572e-01 6.250000e+01 3.181210e+01 2.679111e-01 6.300000e+01 3.202340e+01 2.707726e-01 6.350000e+01 3.223443e+01 2.736417e-01 6.400000e+01 3.244519e+01 2.765184e-01 6.450000e+01 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5.395000e+02 1.914892e+02 4.743941e+00 5.400000e+02 1.916371e+02 4.749804e+00 5.405000e+02 1.917850e+02 4.755669e+00 5.410000e+02 1.919328e+02 4.761536e+00 5.415000e+02 1.920806e+02 4.767404e+00 5.420000e+02 1.922284e+02 4.773274e+00 5.425000e+02 1.923761e+02 4.779146e+00 5.430000e+02 1.925239e+02 4.785020e+00 5.435000e+02 1.926716e+02 4.790896e+00 5.440000e+02 1.928193e+02 4.796774e+00 5.445000e+02 1.929669e+02 4.802653e+00 5.450000e+02 1.931146e+02 4.808534e+00 5.455000e+02 1.932622e+02 4.814417e+00 5.460000e+02 1.934098e+02 4.820302e+00 5.465000e+02 1.935574e+02 4.826188e+00 5.470000e+02 1.937050e+02 4.832076e+00 5.475000e+02 1.938525e+02 4.837966e+00 5.480000e+02 1.940000e+02 4.843858e+00 5.485000e+02 1.941475e+02 4.849752e+00 5.490000e+02 1.942950e+02 4.855647e+00 5.495000e+02 1.944424e+02 4.861545e+00 5.500000e+02 1.945898e+02 4.867444e+00 5.505000e+02 1.947372e+02 4.873345e+00 5.510000e+02 1.948846e+02 4.879247e+00 5.515000e+02 1.950320e+02 4.885152e+00 5.520000e+02 1.951793e+02 4.891058e+00 5.525000e+02 1.953266e+02 4.896966e+00 5.530000e+02 1.954739e+02 4.902875e+00 5.535000e+02 1.956211e+02 4.908787e+00 5.540000e+02 1.957684e+02 4.914700e+00 5.545000e+02 1.959156e+02 4.920615e+00 5.550000e+02 1.960628e+02 4.926532e+00 5.555000e+02 1.962100e+02 4.932451e+00 5.560000e+02 1.963571e+02 4.938371e+00 5.565000e+02 1.965043e+02 4.944293e+00 5.570000e+02 1.966514e+02 4.950217e+00 5.575000e+02 1.967985e+02 4.956143e+00 5.580000e+02 1.969455e+02 4.962071e+00 5.585000e+02 1.970926e+02 4.968000e+00 5.590000e+02 1.972396e+02 4.973931e+00 5.595000e+02 1.973866e+02 4.979864e+00 5.600000e+02 1.975336e+02 4.985798e+00 5.605000e+02 1.976806e+02 4.991735e+00 5.610000e+02 1.978275e+02 4.997673e+00 5.615000e+02 1.979744e+02 5.003613e+00 5.620000e+02 1.981213e+02 5.009554e+00 5.625000e+02 1.982682e+02 5.015498e+00 5.630000e+02 1.984150e+02 5.021443e+00 5.635000e+02 1.985618e+02 5.027390e+00 5.640000e+02 1.987087e+02 5.033339e+00 5.645000e+02 1.988554e+02 5.039289e+00 5.650000e+02 1.990022e+02 5.045241e+00 5.655000e+02 1.991489e+02 5.051195e+00 5.660000e+02 1.992957e+02 5.057151e+00 5.665000e+02 1.994424e+02 5.063109e+00 5.670000e+02 1.995890e+02 5.069068e+00 5.675000e+02 1.997357e+02 5.075029e+00 5.680000e+02 1.998823e+02 5.080991e+00 5.685000e+02 2.000289e+02 5.086956e+00 5.690000e+02 2.001755e+02 5.092922e+00 5.695000e+02 2.003221e+02 5.098890e+00 5.700000e+02 2.004686e+02 5.104860e+00 5.705000e+02 2.006152e+02 5.110831e+00 5.710000e+02 2.007617e+02 5.116805e+00 5.715000e+02 2.009081e+02 5.122780e+00 5.720000e+02 2.010546e+02 5.128756e+00 5.725000e+02 2.012010e+02 5.134735e+00 5.730000e+02 2.013475e+02 5.140715e+00 5.735000e+02 2.014939e+02 5.146697e+00 5.740000e+02 2.016402e+02 5.152680e+00 5.745000e+02 2.017866e+02 5.158666e+00 5.750000e+02 2.019329e+02 5.164653e+00 5.755000e+02 2.020792e+02 5.170642e+00 # ============================================================================== # Figure 6 # Fig. 6: frequency ratios of the quasi air-water simulation, as functions of kappa_x*H. # delta = omega_0/omega_t compares intrinsic surface dynamics with turbulent forcing; zeta = gamma/omega_t # compares surface damping with turbulent forcing. # Parameters (simulation units, H = 1): g = 0.36, sigma = 0.036, rho_w = 1, rho_a = 1/80, # mu_w = 1/5000, mu_w/mu_a = 55, dissipation rates eps_w = 0.00645606716 and eps_a = 0.00214390149. # omega_t = eps^(1/3) kappa^(2/3), eps^(4/3) = (rho_w^2 eps_w^(4/3) + rho_a^2 eps_a^(4/3))/(rho_w^2 + rho_a^2). # The grid is the one-sided streamwise grid kappa_x = 2*pi*i/(4*pi), i = 1..1151 (k = 0 excluded). # Columns: # 1: kxH - streamwise wavenumber, kappa_x*H # 2: delta - delta = omega_0/omega_t (left panel); omega_0 gravity-capillary natural frequency # 3: zeta - zeta = gamma/omega_t (right panel); gamma = viscous damping rate of the interface # ------------------------------------------------------------------------------ # kxH delta zeta 5.000000e-01 3.616984e+00 4.286578e-04 1.000000e+00 3.339582e+00 1.080150e-03 1.500000e+00 3.295868e+00 1.854694e-03 2.000000e+00 3.360642e+00 2.721807e-03 2.500000e+00 3.490631e+00 3.664972e-03 3.000000e+00 3.663538e+00 4.673537e-03 3.500000e+00 3.865815e+00 5.739950e-03 4.000000e+00 4.088597e+00 6.858524e-03 4.500000e+00 4.325858e+00 8.024797e-03 5.000000e+00 4.573404e+00 9.235151e-03 5.500000e+00 4.828266e+00 1.048659e-02 6.000000e+00 5.088311e+00 1.177658e-02 6.500000e+00 5.351985e+00 1.310293e-02 7.000000e+00 5.618149e+00 1.446377e-02 7.500000e+00 5.885952e+00 1.585741e-02 8.000000e+00 6.154761e+00 1.728240e-02 8.500000e+00 6.424095e+00 1.873740e-02 9.000000e+00 6.693592e+00 2.022122e-02 9.500000e+00 6.962974e+00 2.173279e-02 1.000000e+01 7.232030e+00 2.327112e-02 1.050000e+01 7.500599e+00 2.483532e-02 1.100000e+01 7.768557e+00 2.642455e-02 1.150000e+01 8.035811e+00 2.803805e-02 1.200000e+01 8.302291e+00 2.967511e-02 1.250000e+01 8.567944e+00 3.133507e-02 1.300000e+01 8.832732e+00 3.301732e-02 1.350000e+01 9.096628e+00 3.472128e-02 1.400000e+01 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