[maysmech]

Hi all,
Based on Hertzian Theory as discussed in Tsuji (1992, 1993) papers, the spring stiffness is a function of Young's modulus and Poison's ratio. It says we can consider the elastic part of particle collision as (k_spring*delat_n) or (K_n*delta_n^3/2).
In the first Tsuji paper (1992) the relation between k_spring and K_n is: k_spring=K_n*(delta_n)^0.5

I see in the second paper (1993) of Tsuji uses k_spring=800 N/s2 but doesn't say about the process of calculation. I don't know how delta_n can be calculated.

In DPMFoam which is based on Tsuji model what are applied as collision properties are: Young's modulus (E) and Poisson's ratio (sigma)

According to Tsuji, K_n calculates by: K_n=((D^0.5)*E)/(3*(1-sigma^2))

How can we understand k_spring? We need the displacement (delta_n). Is it constant in different collisions? If no it means we have different spring constant in different collisions? I think it should be a constant because in all DEM papers they use spring stiffness with constant magnitude(for example 400 or 800).

If anyone know about the method of calculation of delta_n and spring constant in DPMFoam, please share here.
Regards,

[kmooney]

Hi Maysam,

As far as I know DPMFoam uses the same lagrangian classes as everything else, the DPM particulars come in higher up on the solver solver with Uc, volFraction sources etc...

I'm not sure where the actually K is calculated from but I'm guessing its somewhere in the only pairCollision model in OpenFOAM-2.3, 'pairSpringSliderDashot', which can be found here:

/src/lagrangian/submodels/Kinematic/CollisionModel/PairCollision/PairModel/PairSpringSliderDashpot/

Cheers,
Kyle

[ansubru]

Hi Guys!!

You are quite right Kyle, the stiffness K is calculated in the code for pair collision as follows -

Kn = (4/3)*sqrt(R)*Estar

Now clearly the stiffness Kn is just dependent on the effective radius R --> ((da*db)/(2*(da+db))

and Estar which is nothing but a function of young's modulus.

Hence we could say that Kn would be a constant, provided that the system contains symmetrically sized and shaped particles. Additionally, Kn is independent of deltaN as it is a material property. Kn*delta^x is the form that would give the total a normal component of hertzian contact.

BR
ansubru

[maysmech]

Thanks Kyle and Ananda,
If I understand correctly we have one spring stiffness for both of linear and Hertzian models. The difference is in "b" parameter (1 or 1.5) which is the power of delta_n . I thought these models (linear and Hertzian) have different stiffness.
Now It is true that it is based on Equation 17 of Tsuji (1992) but why it is not same as Equation 16 which is stated that suitable for same properties particle collision? It is a sqrt(2) difference between Eq. 16 and 17. Is it a mistake in Tsuji paper?


The next question is the process of particle time step calculation. in the stated C file:

template<class CloudType>
Foam::label Foam::PairSpringSliderDashpot<CloudType>::nSubCycles() const
{
    if (!(this->owner().size()))
    {
        return 1;
    }

    scalar RMin;
    scalar rhoMax;
    scalar UMagMax;

    findMinMaxProperties(RMin, rhoMax, UMagMax);

    // Note:  pi^(7/5)*(5/4)^(2/5) = 5.429675
    scalar minCollisionDeltaT =
        5.429675
       *RMin
       *pow(rhoMax/(Estar_*sqrt(UMagMax) + VSMALL), 0.4)
       /collisionResolutionSteps_;

    return ceil(this->owner().time().deltaTValue()/minCollisionDeltaT);
}

What is the criteria for calculating dt_particle or number of subcycle? (It is different with Tsuji 1993 criteria for time step)

[maysmech]

It is a sqrt(2) difference between Eq. 16 and 17. Is it a mistake in Tsuji paper?

What is the criteria for calculating dt_particle or number of subcycle?

If anyone has any Idea will be appreciated to share.
Regards,

[maysmech]

If definition of Kn and kn are same in Fn=Kn*delta_n^1.5 and Fn=kn*delta_n^1, it will cause many difference in Fn if we change "b" magnitude and switch between linear (b=1) and hertzian (1.5) because of very small order of magnitude of normalOverlapMag (delta_n).

[ansubru]

Hi Maysam!!

I think I have an idea as to how the number of nsubcycles are being estimated. From my understanding, the nsubcycles represents a ratio of fluid resolution time scale (based on particle rayleigh criterion) to the minimum collision time scale for the bi-particle system estimated from fundamental hertzian contact laws. The reference for 'mincollisionDeltaT'; is here :

https://www.dropbox.com/s/6lojt1s49h...epted.pdf?dl=0

refer pg 12, eq.3.19

Additionally, ur observations on stiffness are valid. OF calculates an individual Kn for every scenario/collision (through young's modulus and Poisson ratio specifications). However, we use linear spring slider (or b = 1) when we know that the system can be approximated by a linear model (which is applicable in most cases actually).. But, when you have the time and resources its best if you use the hertzian law as this would be representative of the actual elastic collisional behavior.

Regards
ansubru

[bjnieuwboer]

Dear Maysam,

I am also looking into the collision model of OpenFOAM. As it is my understanding, it uses the contact model with the wall from the Tjuji 1992 paper. However, I think since then the concept of a equivalent particle radius has been defined.

When one accepts that difference, there is still one discrepancy between the paper and the model. This is the tangential damping coefficient eta_t. In the OpenFOAM code this is set equal to eta_n. However, in all the papers I can find on the subject, it is computed from k_t. Do you know why it is implemented in this manner?

Best regards,

Bas

[bjnieuwboer]

I overlooked a little section in the Tjuji 1992 paper, which states that ete_t = eta_n.

[arch.mr]

Hello Ananda Kannan
Can you let us know the reference for 'mincollisionDeltaT' calculation?
The dropbox link is already outdated..
I tried to find the reference but cannot...
If you let us know the title of the reference, it should be great help for the people who study particle collision model.
Thank you for your help in advance.

[bjnieuwboer]

Dear Arch,

I too was not able to find a reference for my dissertation. Eventually, I found two other references, one predicting a higher value and one a lower. This gave me confidence that the value of openfoam is in the right order. I used the two reference:

Timoshenko, S.P., & Goodier, J.N. 1970. Theory of Elasticity. 3rd edn. McGraw-Hill, inc

Maw, N, Barber, J R, & Fawcett, J N. 1976. The oblique impact of elastic spheres. Wear,
38(1), 101–114.

You do need to rewerite these equations to get 3 similar forms.


Bas

[arch.mr]

Dear Bas Nieuwboer

Your help gave me deeper understand about the term in the opnefoam code.
Really appreciated
Thank you for your help!