Deriving SPH - Question about Dissipation


In the DualSPHyiscs wiki I find that;

Where GAMMA is the dissipation term. When I start from the original Navier-Stokes equation I reach this point;

My question is then, I suppose that this GAMMA term in the wiki corresponds to that which I have marked with yellow? (Of course I just need to divide by density as well)

If this is the case, then can anyone explain to me why do we need to use something as artificial viscosity, can we not just evaluate this term marked with yellow directly?

I have had a hard time finding papers starting from the Navier-Stokes eqn., so if somebody knows of a detailed derivation I would be happy to read it.

Kind regards


  • i am not sure i'm right, so just for discussion

    in SPH, most only care about inviscid incompressable flow, then gamma canbe ignore. But the artificial viscosity is another thing, not viscosity in physics.

    Have you learn FDM before? or FVM.... SPH, something like a kind of central difference in FDM, which is unstability. With artificial viscosity, to fix this problem, just like Lax-wendroff (maybe i forget the name)

    by the way, you can find lots of similarities in FDM,FVM

  • @JOJO I liked your explanation so I digged a bit deeper into it.

    In Monaghans paper from 1992 he writes;

    So from this one would think that your explanation is wrong. So what I did was to find the paper from 1985 and in here it says;

    So this supports your explanation that the equations were originally derived with an assumption of no viscosity (inviscid fluid), and then later the artifical viscosity term was added as a stabilizing mechanic. This stabilization then mimics the viscosity terms shown in the original equations according to Monaghan 1985, Particle Methods for Hydrodynamics, "5.2. Artificiul viscosity for SPH "(

    So my current conclusion is that in fact that the artifical viscosity scheme is not an inviscid scheme, but in fact a modelling of the viscous terms. Why this modelling has to be done and why it cannot just be calculated directly as shown in Navier-Stokes equation is still unclear to me.

    Kind regards

  • Indeed, it modeling a viscous terms. But as fas as i know, to stablize the field, it is necessary for both inviscid and viscosity, rather than an option only for viscous flow, and i think that is why we call it 'artifical'.

    and it can be equivalented to a physical ones.

  • and why we can not directly modelling the real one. i think it is not a question only for SPH, but for all numerical methods.

    as you know, NS contains momentum quation and continuity equation. In this system, pressure and velocity is coupled, which we can not solve them together. so, the frist thing we should do is decouple it. Inteartion, prediction and revision, very common in mesh-based method.

    But in SPH, we just simplify it. you know, sph is quite, slow

  • Thank you both of you. Will have to spend some more time on it before I can add more questions etc.

    Kind regards

  • Okay, I have finally understood why artificial viscosity has to be included. It is because when particles move in a simulation, different shock fronts might occur, and if this is not smoothened out, the simulation will become unstable.

    Thanks for all of the help

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