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Transport Models

Viscosity models available in flow_state. All models compute dynamic viscosity \(\mu\) as a function of temperature, as well as the temperature derivative \(d\mu/dT\).

Overview

Model Class Gases Reference
Sutherland's Law Sutherland Air, N2, Custom 1
Sutherland's Law with LTC SutherlandLowTemp Air 2
Sutherland's Law with LTC (Blended) SutherlandBlended Air -
Keyes Keyes Air, N2 3
Power Law PowerLaw Air 4

Sutherland's Law

The standard Sutherland formula 1:

\[ \mu(T) = \mu_{\text{ref}} \left( \frac{T}{T_{\text{ref}}} \right)^{3/2} \frac{T_{\text{ref}} + S}{T + S} \]

The temperature derivative is:

\[ \frac{d\mu}{dT} = \frac{\mu_{\text{ref}} (T_{\text{ref}} + S)}{T_{\text{ref}}^{3/2}} \cdot \frac{\sqrt{T} \left( \frac{1}{2} T + \frac{3}{2} S \right)}{(T + S)^2} \]

Valid for moderate temperatures (~100 K to ~1900 K for air).

Gas \(\mu_{\text{ref}} \, [Pa \cdot s]\) \(T_{\text{ref}} \, [K]\) \(S \, [K]\)
Air \(1.716 \times 10^{-5}\) 273.15 110.4
Nitrogen \(1.663 \times 10^{-5}\) 273.15 106.7

Sutherland's Law with Low-Temperature Correction (LTC)

Prevents unphysical viscosity at very low temperatures where Sutherland's law breaks down 2:

\[ \mu(T) = \begin{cases} C_0 T_1 & T < T_1 \\ C_0 T & T_1 \leq T \leq S \\ \mu_{\text{ref}} \left( \frac{T}{T_{\text{ref}}} \right)^{3/2} \frac{T_{\text{ref}} + S}{T + S} & T > S \end{cases} \]

where:

  • \(\mu_{\text{ref}} = 1.716 \times 10^{-5}\) kg/(m·s)
  • \(T_{\text{ref}} = 273.15\) K
  • \(S = 110.4\) K
  • \(C_0 = 6.93873 \times 10^{-8}\) kg/(m·s·K)
  • \(T_1 = 40\) K

The temperature derivative is:

\[ \frac{d\mu}{dT} = \begin{cases} 0 & T < T_1 \\ C_0 & T_1 \leq T \leq S \\ \text{(Sutherland derivative)} & T > S \end{cases} \]

Sutherland's Law with Low-Temperature Correction Blended (LTC-blended)

Smoother transition using 8th-order polynomial blending. This avoids discontinuities in the derivative that can cause numerical issues:

\[ \mu(T) = \begin{cases} C_0 T & T < T_1 \\ a_0 \sum_{i=1}^{8} a_i \frac{T^{8-i}}{S^{8-i}} & T_1 \leq T \leq T_2 \\ \mu_{\text{ref}} \left( \frac{T}{T_{\text{ref}}} \right)^{3/2} \frac{T_{\text{ref}} + S}{T + S} & T > T_2 \end{cases} \]

where:

  • \(\mu_{\text{ref}} = 1.716 \times 10^{-5}\) kg/(m·s)
  • \(T_{\text{ref}} = 273.15\) K
  • \(S = 110.4\) K
  • \(C_0 = 6.93873 \times 10^{-8}\) kg/(m·s·K)
  • \(T_1 = 100\) K
  • \(T_2 = 130\) K

Polynomial coefficients:

\(i\) \(a_i\)
0 \(7.659704848 \times 10^{-6}\) kg/(m·s)
1 \(-44.79148053679334\)
2 \(319.5188079744342\)
3 \(-971.6235566382709\)
4 \(1632.645086771892\)
5 \(-1637.375578884298\)
6 \(980.2775658900685\)
7 \(-323.4667180557399\)
8 \(45.8157988617632\)

The temperature derivative in the polynomial region (\(T_1 \leq T \leq T_2\)) is:

\[ \frac{d\mu}{dT} = \frac{a_0}{S} \sum_{i=1}^{7} (8-i) \, a_i \left( \frac{T}{S} \right)^{7-i} \]

Keyes Model

High-temperature viscosity law 3 5 6:

\[ \mu(T) = \frac{a_0 \sqrt{T}}{1 + a_1 \cdot 10^{-a_2/T} / T} \]

Often used for hypersonic flows where temperatures can be very high.

Gas \(a_0 [kg/(m \cdot s \cdot \sqrt{K})]\) \(a_1 [K]\) \(a_2 [-]\)
Air \(1.488 \times 10^{-6}\) 122.1 5.0
Nitrogen \(1.418 \times 10^{-6}\) 116.4 5.0

The temperature derivative is:

\[ \frac{d\mu}{dT} = a_0 \left( \frac{1}{2\sqrt{T} \cdot D} - \frac{\sqrt{T} \cdot D'}{D^2} \right) \]

where \(D = 1 + a_1 \cdot 10^{-a_2/T} / T\) and \(D' = a_1 \cdot 10^{-a_2/T} (a_2 \ln 10 - T) / T^3\).

Power Law

Simple power-law model 4:

\[ \mu(T) = \mu_{\text{ref}} \left( \frac{T}{T_{\text{ref}}} \right)^m \]

The temperature derivative is:

\[ \frac{d\mu}{dT} = \frac{m \, \mu_{\text{ref}}}{T_{\text{ref}}} \left( \frac{T}{T_{\text{ref}}} \right)^{m-1} \]

Useful for simplified analyses or specific temperature ranges. Typical exponent \(m\) ranges from 0.5 to 1.0.

References

  1. William Sutherland. The viscosity of gases and molecular force. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 36(223):507–531, 1893. URL: https://doi.org/10.1080/14786449308620508, doi:10.1080/14786449308620508. ↩↩

  2. Leslie M. Mack. Boundary-layer stability theory. Technical Report, Jet Propulsion Laboratory, 1969. Document ID 19900029782. URL: https://ntrs.nasa.gov/citations/19900029782. ↩↩

  3. Frederick G. Keyes. A summary of viscosity and heat conduction data for he, a, h2, o2, n2, co, co2, h2o and air. Transactions of the American Society of Mechanical Engineers, 73:589–596, 1951. URL: https://doi.org/10.1115/1.4016335, doi:10.1115/1.4016335. ↩↩

  4. Frank M. White. Viscous Fluid Flow. McGraw-Hill, 3rd edition, 2006. ISBN 978-0-07-240231-5. ↩↩

  5. Stephan Priebe and M. Pino Martin. Low-frequency unsteadiness in shock wave-turbulent boundary layer interaction. Journal of Fluid Mechanics, 699:1–49, 2012. URL: https://doi.org/10.1017/jfm.2011.560, doi:10.1017/jfm.2011.560. ↩

  6. Christopher J. Roy and Frederick G. Blottner. Review and assessment of turbulence models for hypersonic flows. Progress in Aerospace Sciences, 42(7-8):469–530, 2006. URL: https://doi.org/10.1016/j.paerosci.2006.12.002, doi:10.1016/j.paerosci.2006.12.002. ↩