Drag models in OpenFOAM

OpenFOAM is a great tool, however the public documentation is not exhaustive and there is a lot to find out on your own if you are trying to master the tool.¹

So today, I am trying to give an overview of the drag models available in OpenFOAM v11. In the forums out there you can find information about the models but it is quite scattered. I hope this will save some time.

The information provided below is mainly extracted using the command foamInfo -k <name of model>. I have also read some of the articles cited in the brief explanations provided by that command.

The table below shows the currently summarized models:

The section TODO lists the models for which I have not yet summarized. Feel free to send me your summary!

Drag models are implemented by calculating a drag coefficient, returned by the method CdRe()in the C++ class definition. This coefficient is then used to compute the drag force using the drag equation.

The notation in OpenFOAM is such that the sub-index _\(_C\) refers to the continuous phase, and _\(_D\) to the discrete or dispersed phase. This means, for example that \(\alpha_C\) stands for the fraction of the continuous phase.

Ergun, Gibilaro, WenYu

The Ergun and Gibilaro are correlations based on pressure gradient (relative velocity is the interstitial relative velocity: \(U_C - U_D\)). WenYu is a particle based model (relative velocity is the superficial relative velocity, aka apparent relative velocity: \(\alpha_C (U_C - U_D)\))

Ergun

Implements a drag coefficient given by

$$ C_d = \frac{4}{3}\left(150\, \frac{\alpha_D}{\alpha_C} + 1.75\, \text{Re} \right) $$

where \(\alpha_D\) and \(\alpha_C\) are the fractions of the dispersed and continuous phase (a phase is considered dispersed into the other). Many researchers, e.g. Gidaspow (1994)², assume the Ergun equation to be valid under fluidized conditions, for \(\alpha_C < 0.8\). It corresponds to Eq. 104 in the reference Enwald et al.³.

Gibilaro

Implements a drag coefficient given by⁴

$$ C_d = \frac{4}{3}\alpha_C^{-1.8}\left(\frac{17.3}{\alpha_C} + 0.336\, \text{Re} \right) $$

It corresponds to Eq. 106 in the reference Enwald et al.³:

The correlation given by Ergun shows quite good agreement with the correlations based on the drag coefficient, even though it was originally developed for a fixed bed, while the correlation of Gibilaro et al. gives the lowest values of all the drag function correlations, at least for the flow conditions chosen in the comparison. It should be noted that, of all the drag correlations included here, the Ergun correlation is the only one to take into account the particle sphericity.

WenYu

$$ R_C = \alpha_C\, \text{Re}\\ C_d = \alpha_C^{-2.65} \begin{cases} 24 \left(1.0 + 0.15\, R_C^{0.687}\right) & R_C < 1000 \\ 0.44\, R_C & R_C \geq 1000 \end{cases} $$

It corresponds to Eq. 85-87 in the reference Enwald et al.³. In that article the model has the exponent for alpha of -4.65, but here -2.65 is used. I do not know of there is an explanation for the deviation from the article.

GidaspowSchillerNaumann

Seems to be exactly the same as WenYu² for \(R_C < 1000\)

$$ R_C = \alpha_C\, \text{Re}\\ C_d = \alpha_C^{-1.65} \begin{cases} \frac{24}{\alpha_C} \left(1.0 + 0.15\, R_C^{0.687}\right) & R_C < 1000 \\ 0.44\, R_C & R_C \geq 1000 \end{cases} $$

It corresponds to Eq. 85-87 in the reference Enwald et al. below³.

GidaspowErgunWenYu

Provides is a piece-wise combination of Ergun and WenYu model:

$$ C_d = \begin{cases} C_\text{WengYu} & \alpha_C \geq 0.8 \\ C_\text{Ergun} & \alpha_C < 0.8 \end{cases} $$

SchillerNaumann

Schiller and Naumann drag model for dispersed bubbly flows.

$$ \text{Re}_i = \vert \Delta U\vert \frac{d_D}{\nu_C} \;: \text{interface Re}\\ C_d = \begin{cases} 24 \left(1.0 + 0.15\, \text{Re}_i^{0.687}\right) & \text{Re}_i < 1000 \\ 0.44\, \text{Re}_i & \text{Re}_i \geq 1000 \end{cases} $$

Compare with GidaspowSchillerNaumann and WenYu.

aerosolDrag

Stokes drag with Cunningham slip correction⁵. The drag coefficient calculated with standard correlations is divided by the Cunningham slip correction factor, \(C_c\) , given below:

$$ C_c = 1 + \frac{\lambda}{d} [A_1 + A_2 \exp(-A_3 \frac{d}{\lambda})] $$

The coefficients \(A_i\) default to the values proposed by Davis (1945)⁶. The mean free path is computed by

$$ \lambda = \frac{kT}{\sqrt{2} \pi p \sigma^{2}}. $$

where \(\sigma\) is the Lennard-Jones parameter in meters.

AttouFerschneider

Attou and Ferschneider’s Drag model for film flow through packed beds. The implementation follows the description of Gunjal and Ranade⁷, who, in the reference below, formulate the model in more convenient terms.

Beetstra

Drag model of Beetstra et al.⁸ for mono-disperse gas-particle flows obtained with direct numerical simulations with the Lattice-Boltzmann method and accounting for the effect of particle ensembles.

IshiiZuber

Ishii and Zuber (1979)⁹ drag model for dense dispersed bubbly flows. It considers the Eötvös number and the shape of the bubbles.

Lain

Drag model of Lain et al.^{10 11}

$$ \text{Re}_i = \vert \Delta U\vert \frac{d_D}{\nu_C} \;: \text{interface Re}\\ C_d = \begin{cases} 16 & \text{Re}_i < 1.5 \\ 14.9\, \text{Re}_i^{0.22} & 1.5 \geq \text{Re}_i < 80 \\ 48 \left(1 - 2.21 \, \text{Re}_i^{-0.5}\right) & 80 \geq \text{Re}_i < 1500 \\ 2.61\, \text{Re}_i & \text{Re}_i \geq 1500 \end{cases} $$ where \(d_D\) is the diameter associated to the dispersed phase and \(\Delta U\) is the relative velocity between the phases.

SyamlalOBrien

¹²

$$ A = \alpha_C^{4.14}\\ B = \begin{cases} 0.8 \, \alpha_C^{1.28} & \alpha_C < 0.85\\ \alpha_C^{2.65} & \alpha_C \geq 0.85 \end{cases}\\ V_r = 0.5 \left(A - 0.06 \text{Re}_i + \sqrt{(0.06\text{Re}_i)^2 + 0.12\text{Re}_i (2B - A) + A^2}\right)\\ C_d = \alpha_C V_r^{-2} (0.63\sqrt{\text{Re}_i} + 4.8\sqrt{V_r}); $$

TODO

• segregated
• Tenneti
• timeScaleFilteredDrag
• TomiyamaAnalytic
• TomiyamaCorrelated
• TomiyamaKataokaZunSakaguchi