Transport Velocity Formulation

Some main principles and consequences of the transport velocity.

Full theoretical description of the transport velocity is beyond the scope of this manual, so some main principles and consequences of the transport velocity will be discussed. For full theoretical derivation and analysis, refer to the work of Adami. ^¹

Accuracy of the SPH method heavily relies on the ability of the code to accurately reconstruct the Shepard coefficient and provide full support to the particles. In reality the value of the Shepard coefficient will be ≈ 1, but rarely exactly 1. If you analyze the SPH method, you can easily understand that the accuracy of the method is actually directly related to the particle distribution. If the particles are uniformly ordered, the reconstruction of the variable fields will be accurate. If there are excesses, such as overly-packed or overly-sparse particle distributions, this will negatively reflect on the accuracy of the solution. It would; therefore, be ideal if you can keep the particles ordered as uniformly as possible without sacrificing computational time.

The transport velocity does precisely this. The numerical formulation of the transport velocity introduces a correction method to otherwise normal SPH velocity computation in a case-independent manner, while preserving the physicality of the solution. The dual correction is applied exclusively through the momentum equation, as opposed to the traditional background pressure approach which is explicitly appearing in the quasi-incompressible equation of state. The correction comes directly from the computation of the transport velocity $\tilde{u}$ , or more precisely the time advancement of the transport velocity:

\tilde{u_{i}} (t + Δ t) = u_{i} (t) + Δ t (\frac{\tilde{d} u_{i}}{d t} - \frac{1}{ρ_{i}} \nabla p_{c})

Where, $p_{c}$ is the corrective pressure field, usually set to be equivalent to the initial pressure of the simulation $p_{0}$ .

These corrections actively maintain particle order which has a number of beneficial influences on the numerical behavior of the code.

The magnitude of the $p_{c}$ pressure directly influences the strength of the correction. The higher the $p_{c}$ value, the more vigorous will be the correction attempt. You should keep this in mind, as specifying the $p_{c}$ value too high, for example, $p_{c}$ = 10 $p_{0}$ , can lead to excessive correction force and in these cases the time step must be appropriately reduced.

The command is actually a coefficient with which you multiply the $p_{0}$ value and therefore determine the $p_{c}$ correction pressure (by default it is set to 1.0).

p_{0} = p_{c}_f a c t o r * p_{0}

In more graphical terms, the transport velocity formulation automatically detects particle vacuum and attempts to populate it with particles. As mentioned, this has profoundly beneficial influence in multiphase simulations, but in single phase simulation where you have intentionally left a large portion of the domain empty (particle vacuum), the use of the transport velocity could actually have detrimental effects. The reason is precisely because transport velocity is seeking for particle vacuum and tries to fill it, which would in single phase cases result in a pop-corn like behavior of the free surface. This is something to be avoided and therefore in single phase cases it is strongly recommended to turn the transport velocity off.

Artificial Particle Displacement (APD)

nanoFluidX has two schemes to address this. The transport velocity is used in combination with the Weighted interaction scheme, whereas with the Riemann interaction scheme, you rely on the Artificial Particle Displacement (APD) method.^²

The APD method relies on adding a small correction vector to the position of the particle:

δ {\vec{r}}_{i} = β \sum_{j = 1}^{N} \frac{{\vec{r}}_{i j}}{r_{i j}^{3}} r_{i, o}^{2} u_{\max} Δ t

Where,

$β$: Numerical coefficient, typically of the value 0.01 to 0.05 for multiphase flows with bounded domain and recommended as 0 for single phase (free surface; open domain) flows
$r_{i, o}$: Average particle distance between a particle and its neighbors
$u_{\max}$: Maximum expected velocity in the domain (reference velocity)
$j$ index: Refers to the standard SPH vector subtraction between the owner particle and its neighbors

Resulting particle fields display better order and provide smoother solutions without noticeable diffusion or detriment to the results. However, it is worth noting that pushing the APD coefficient

β

beyond the recommended values is likely to introduce negative consequences into the results – most notably numerical diffusion.

Important: Be cautious when changing the values of the APD coefficient (

β

¹ S. Adami, Modeling and Simulation of Multiphase Phenomena with Smoothed Particle Hydrodynamics, Chair of Aerodynamics and Fluid Mechanics, Technical University of Munich, 2014

² SPH Modelling of Long-term Sway-Sloshing Motion in a Rectangular Tank, Oezbulut, M.; Tofighi, N.; Yildiz, M.; Goeren Oe.; The Twenty-fifth International Ocean and Polar Engineering Conference, Kona, Hawaii, USA, 2015