# How do I calculate Reynold's number using artifical viscosity?

Hello!

I've done a 3D simulation of water and I want to know the Reynolds number. Usually we would say something like this:

Say my alpha value in DualSPHysics is 0.02, would I then plug that in the kinematic viscosity, in the last equation or how?

Kind regards

## Comments

Following the work of Colagrossi et al 2011:

The corresponding numerical kinematic viscosity is theoretically given as

nu=alpha xh / 8 in 2D and nu=alpha xh / 10 in 3D

CsxCsxColagrossi et al 2011 in Phys. Rev. E. “Theoretical analysis and numerical verification of the consistency of viscous smoothed-particle-hydrodynamics formulations in simulating free-surface flows”Rather good point, but the answer is no. There is a relationship between the

artificial viscosityand the equivalent natural viscosity, which also depends on the smoothing length, the (numerical) reference speed of sound, the reference speed of sound, and whether the problem is 2/3D. If you change only one of these parameters, you are effectively changing the simulated viscosity of your flow or, in dimensionless terms, the Reynolds number it has. This point has been recently addressed, possibly among others, inA dynamic δ-SPH model: How to get rid of diffusive parameter tuningby Meringolo et al in https://doi.org/10.1016/j.compfluid.2018.11.012Thanks guys! Just to be completely sure:

h is the smoothing length which is stated in the run file as "H"?

If I were to use SPS-LES, I would just do it the normal way? I looked through the paper which @sph_tudelft_nl and

it seems to be what is done there.

Kind regards and thanks for the help

@Asalih3d Concerning the work of Meringolo I guess you could restrict yourself to the considerations of section 3.2 -- at least I had those in mind.

A few points for consideration may be worth sharing.

On the one hand, one could derive a Reynolds number for "any" fluid flow problem from dimensional analysis without knowing much details of the physics in the interior. In this case, the Re number is a determinant of the fluid problem by construction, for you have put the variables that make it in the list of important items from the start. Buckingham’s theorem indeed. In this case the Re number is an explaining parameter, a useful tag and uses overall parameters of the flow.

On the other hand, the Re number can come from the scaling of the flow variables and from non-dimensionalizing the governing equations. In that case, the Re number expresses the ratio of convective to diffusive forces. The little surprise is that in the Lagrangean framework in which SPH lives, convection is not even properly defined. It is incorporated in the Lagrangean derivative.

This sound a bit of paradox: Meringolo et al effectively talk about `cell Reynolds numbers` for a meshless method. Probably, cell is a bit of misnomer there. Effectively they work with interparticle distances Dx and have assumed that the ratio smoothing length to interparticle distance is 2. There is no cell, clearly, there are smoothing kernels. If I understood it correctly, their reasoning aims to address the issue: which local scales of flow oscillations can we resolve upon holding to a certain set of SPH parameters and changing it?

So we should be careful in distinguishing global and local Reynolds numbers, in the first place. The gap is closed by knowing the relationship between the global and local Reynolds number that applies to a certain flow type – if we know it. Plus let’s recall that, when people use the eddy viscosity to represent turbulence, a turbulent Reynolds number is then used for comparison. So there is a Reynolds number for anything you do :-)

The sole substitution of the natural viscosity with the artificial viscosity is always possible, but may not have enough explaining power in and of itself. I do not know whether a tweaked global Reynolds number explains the effective dynamics at that tweaked Re number. I guess that in SPH we’d better think of local Reynolds numbers. I presume that’s why people have developed the laminar + SPS modelling of viscosity; they may be able to say more on this.