Rayleigh-Taylor Instability Simulator
Rayleigh-Taylor Instability Simulator calculates growth rate of interface perturbations using densities difference, gravity, and wavelength, with plots.
Rayleigh-Taylor Instability Overview
Rayleigh-Taylor instability occurs when a heavier fluid lies above a lighter fluid, leading to interface perturbations. The growth rate is given by:
Growth Rate: \\( n = \sqrt{\frac{g k (\rho_h – \rho_l)}{\rho_h + \rho_l}} \\)
Where: \\( k = \frac{2\pi}{\lambda} \\) (wavenumber)
Variables:
- \\(n\\): Growth rate (s⁻¹)
- \\(g\\): Acceleration due to gravity (m/s²)
- \\(\rho_h\\): Density of heavy fluid (kg/m³)
- \\(\rho_l\\): Density of light fluid (kg/m³)
- \\(\lambda\\): Wavelength of perturbation (m)
- \\(k\\): Wavenumber (m⁻¹)
Example Calculations
Example 1: Water over Air
\\(\rho_h = 1000 \, \text{kg/m}^3\\), \\(\rho_l = 1.225 \, \text{kg/m}^3\\), \\(g = 9.81 \, \text{m/s}^2\\), \\(\lambda = 0.1 \, \text{m}\\)
Wavenumber: \\( k = \frac{2\pi}{0.1} \approx 62.832 \, \text{m}^{-1} \\)
Growth Rate: \\( n = \sqrt{\frac{9.81 \times 62.832 \times (1000 – 1.225)}{1000 + 1.225}} \approx 24.786 \, \text{s}^{-1} \\)
Example 2: Oil over Water
\\(\rho_h = 900 \, \text{kg/m}^3\\), \\(\rho_l = 1000 \, \text{kg/m}^3\\), \\(g = 9.81 \, \text{m/s}^2\\), \\(\lambda = 0.05 \, \text{m}\\)
Wavenumber: \\( k = \frac{2\pi}{0.05} \approx 125.664 \, \text{m}^{-1} \\)
Growth Rate: \\( n = \sqrt{\frac{9.81 \times 125.664 \times (900 – 1000)}{900 + 1000}} \approx 8.064i \, \text{s}^{-1} \\) (stable, negative \\(\rho_h – \rho_l\\))
Example 3: Mercury over Oil
\\(\rho_h = 13534 \, \text{kg/m}^3\\), \\(\rho_l = 900 \, \text{kg/m}^3\\), \\(g = 9.81 \, \text{m/s}^2\\), \\(\lambda = 0.2 \, \text{m}\\)
Wavenumber: \\( k = \frac{2\pi}{0.2} \approx 31.416 \, \text{m}^{-1} \\)
Growth Rate: \\( n = \sqrt{\frac{9.81 \times 31.416 \times (13534 – 900)}{13534 + 900}} \approx 17.676 \, \text{s}^{-1} \\)
Example 4: Water over Air (Small Wavelength)
\\(\rho_h = 1000 \, \text{kg/m}^3\\), \\(\rho_l = 1.225 \, \text{kg/m}^3\\), \\(g = 9.81 \, \text{m/s}^2\\), \\(\lambda = 0.01 \, \text{m}\\)
Wavenumber: \\( k = \frac{2\pi}{0.01} \approx 628.319 \, \text{m}^{-1} \\)
Growth Rate: \\( n = \sqrt{\frac{9.81 \times 628.319 \times (1000 – 1.225)}{1000 + 1.225}} \approx 78.339 \, \text{s}^{-1} \\)