Reynold Averaged Navier Stokes (RANS)
In some RANS models like κ-ω, κ-ε, and κ-ω SST, the iteration process can sometimes become sensitive to the input of boundary conditions or initial conditions for turbulent variables such as κ, ε, and ω. This is why differences in the initialization process can lead to distinct solutions.
Although in several commercial CFD software, these boundary conditions are usually computed automatically with inputs like velocity, characteristic length (hydraulic diameter), or turbulence intensity, it’s important to understand how these parameters are calculated.
For internal flows, these boundary conditions differ from external flows, thus requiring separate equations.
Internal Flow Equation
For internal flow scenarios (e.g., flow in pipes, ducting, turbomachinery, etc.), the following equations are utilized:
External Flow Equation
Then, for external flows (e.g., aircraft aerodynamics, race car aerodynamics, wind around buildings, etc.), the κ equation takes the same form as in internal flows. However, for other variables, the relationships are as follows:
Meanwhile, the value of I can be calculated using the approximation:
With U being the reference velocity (usually the inlet velocity), C,miu is the coefficient in the turbulence model typically set to 0.09, L is the length scale usually set to 0.038 times the pipe diameter (for fully developed turbulent flow), and β has a value depending on the Reynolds number as follows:
| Re | I | β |
| less than 3000 | 0-1% | 0.1-0.2 |
| 3000-5000 | 1% | 11.6%-16.5% |
| 5000-15000 | 1-5% | 16.5-26.7 |
| 15000–20000 | 5-20% | 26.7-34 |
| more than 20000 | 5-20% | 100 |
Reference Values
Disclaimer: These reference values are just coarse estimations to help you quickly estimate the input. For more scientific projects, please use your own values!
| Fluid | Dynamic Viscosity (Pa.s) | Kinematic viscosity (m^2/s) |
| Air | 1.79e-5 | 1.48e-5 |
| Water | 8.9e-4 | 1e-6 |
It’s important to note that the calculations above serve as initial approximations to facilitate convergence, and eventually, they will be “washed out” as iterations progress. Therefore, extreme precision in these calculations isn’t necessary.
Some Unit conversions:
Density:
- 1 g/cm³ = 1,000 kg/m³
- 1 g/mL = 1,000 kg/m³
- 1 lb/ft³ ≈ 16.0185 kg/m³
- 1 lb/gal ≈ 8.3454 kg/m³
- 1 oz/in³ ≈ 27,679.9 kg/m³
- 1 kg/L = 1,000 kg/m³
Velocity:
- 1 mph≈0.44704 m/s
- 1 ft/s≈0.3048 m/s
- 1 in/s≈0.0254 m/s
- 1 knot (nautical miles/hr) ≈0.51444 m/s
- 1 Mach≈343 m/s
Length:
- 1 mm=0.001 m
- 1 cm=0.01 m
- 1 km=1,000 m
- 1 in≈0.0254 m
- 1 ft≈0.3048 m
- 1 yd≈0.9144 m
- 1 mi≈1,609.344 m
- 1 nmi≈1,852 m
Viscosity:
- 1 Poise = 0.1 Pa.s
- 1 cPoise = 0.001 Pa.s
- 1 Stokes = 0.0001 m^2/s
- 1 cSt=0.000001 m^2/s