Kelvin-Helmholtz Instability Visualizer
Kelvin-Helmholtz Instability Visualizer computes growth rate of instability at fluid interface using densities, velocity, and wave number.
Kelvin-Helmholtz Instability Overview
The Kelvin-Helmholtz instability occurs at the interface of two fluids with different velocities, leading to wave growth. The growth rate is:
Growth Rate: \\( \sigma = k \sqrt{\frac{\rho_1 \rho_2 (\Delta u)^2}{(\rho_1 + \rho_2)^2}} \\)
Where:
- \\(\sigma\\): Instability growth rate (1/s)
- \\(k\\): Wave number (1/m)
- \\(\rho_1, \rho_2\\): Densities of fluids 1 and 2 (kg/m³)
- \\(\Delta u = |u_1 – u_2|\\): Velocity difference (m/s)
Note: This assumes negligible viscosity and surface tension for simplicity.
Example Calculations
Example 1: Air over Water
\\(\rho_1 = 1.225 \, \text{kg/m}^3\\), \\(\rho_2 = 1000 \, \text{kg/m}^3\\), \\(u_1 = 10 \, \text{m/s}\\), \\(u_2 = 0 \, \text{m/s}\\), \\(k = 1 \, \text{1/m}\\)
Growth Rate: \\( \sigma = 1 \times \sqrt{\frac{1.225 \times 1000 \times (10 – 0)^2}{(1.225 + 1000)^2}} \approx 0.0349 \, \text{1/s} \\)
Example 2: Air over Oil
\\(\rho_1 = 1.225 \, \text{kg/m}^3\\), \\(\rho_2 = 900 \, \text{kg/m}^3\\), \\(u_1 = 15 \, \text{m/s}\\), \\(u_2 = 0 \, \text{m/s}\\), \\(k = 2 \, \text{1/m}\\)
Growth Rate: \\( \sigma = 2 \times \sqrt{\frac{1.225 \times 900 \times (15 – 0)^2}{(1.225 + 900)^2}} \approx 0.1148 \, \text{1/s} \\)
Example 3: Water over Mercury
\\(\rho_1 = 1000 \, \text{kg/m}^3\\), \\(\rho_2 = 13534 \, \text{kg/m}^3\\), \\(u_1 = 5 \, \text{m/s}\\), \\(u_2 = 0 \, \text{m/s}\\), \\(k = 1 \, \text{1/m}\\)
Growth Rate: \\( \sigma = 1 \times \sqrt{\frac{1000 \times 13534 \times (5 – 0)^2}{(1000 + 13534)^2}} \approx 0.0173 \, \text{1/s} \\)
Example 4: Air over Air (Shear Layer)
\\(\rho_1 = 1.225 \, \text{kg/m}^3\\), \\(\rho_2 = 1.225 \, \text{kg/m}^3\\), \\(u_1 = 20 \, \text{m/s}\\), \\(u_2 = 0 \, \text{m/s}\\), \\(k = 0.5 \, \text{1/m}\\)
Growth Rate: \\( \sigma = 0.5 \times \sqrt{\frac{1.225 \times 1.225 \times (20 – 0)^2}{(1.225 + 1.225)^2}} \approx 2.5 \, \text{1/s} \\)