In lattice Boltzmann method, we have a concept, which is called pseudo-compressibility and it is defined based on the fact that LBM simulates incompressible flows by having small Mach number to ensure density fluctuations in time (or pressure fluctuations in time) are not that big. Note, that introduced Mach number here is just a numerical parameter calculated as:

$$Ma = \frac{u_{max}\frac{\Delta t}{\Delta x}}{\hat{c}_{s}}$$

Where $Ma$ is Mach number, $u_{max}$ is temporal maximum of velocity presents in the simulation, $\Delta t$ and $\Delta x$ are temporal and spatial discretization resolutions respectively, and $\hat{c}_{s} = \frac{1}{\sqrt{3}}$ is the lattice parameter. As a result pressure is calculated based on ideal gas equation as:

$$\Delta P = \Delta \rho c_{s}^{2}$$

Where $\Delta P = P - P_{\mathrm{ref}}$, $\Delta \rho = \rho - \rho_{f}$, and $c_{s} = \hat{c}_{s} \frac{\Delta x}{\Delta t}$ is speed of sound in the lattice, and $\rho_{f}$ is known fluid density (e.g. for water it is 1000 $\frac{\mathrm{kg}}{\mathrm{m}^{3}}$), and finally $\rho$ is instantaneous density, where we call $\Delta \rho$ density fluctuations. For small $Ma$ values, this fluctuations are smaller than 5%.

Now, let's define mass flux as for example in an arbitrary cut plane ($\Gamma$) of computational domain ($\Omega$) as:

$$M(t) = \int_{\Gamma} \rho \mathbf{u}(\mathbf{r},t) \cdot d \mathscr{A}$$

Where $M(t)$ is mass flux, $\rho$ is instantaneous density, $\mathbf{u}(\mathbf{r},t)$ is velocity, and $d \mathscr{A}$ is surface differential form of $\Gamma$ cut plane. On the other hand, volumetric mass flux is defined as:

$$Q(t) = \int_{\Gamma} \mathbf{u}(\mathbf{r},t)\cdot d \mathscr{A}$$

Obviously: $M(t) \neq \rho_{f} Q(t)$. I know $M(t)$ and $\rho_{f} Q(t)$ must be close and their difference must be in the order of that 5% fluctuation, but I'm confused that if we want to calculate for example volumetric flux, which number should be used or reported? $Q(t)$ or $\frac{M(t)}{\rho_{f}}$? I appreciate if someone could explain this.


1 Answer 1


I think that both can be considered to be equally valid. In fact, asymptotically (with an infinite number of lattice) you will tend towards the incompressible athermal Navier-Stokes equations. In this limit, both measures of the flow should be equal. If I were you, I would initially report the two. Then I would measure the difference between them as a measure of the impact of your pseudo-compressibility on your flow. Then, I think I would use the velocity-field alone to calculate the mass flux. Don't forget that your density is really there to act as a pseudo pressure in your equation of state and is not a measure of your real density.

I hope that makes sense.


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