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23.5.7 Solids Shear Stresses

The solids stress tensor contains shear and bulk viscosities arising from particle momentum exchange due to translation and collision. A frictional component of viscosity can also be included to account for the viscous-plastic transition that occurs when particles of a solid phase reach the maximum solid volume fraction.

The collisional and kinetic parts, and the optional frictional part, are added to give the solids shear viscosity:


 \mu_{s} = \mu_{s,{\rm col}} + \mu_{s,{\rm kin}} + \mu_{s,{\rm fr}} (23.5-62)



Collisional Viscosity


The collisional part of the shear viscosity is modeled as [ 121, 364]


 \mu_{s,{\rm col}} = \frac{4}{5} \alpha_{s} \rho_{s} d_{s} g_{0,ss}(1 + e_{ss}) \left(\frac{\Theta_{s}}{\pi} \right)^{1/2} (23.5-63)



Kinetic Viscosity


FLUENT provides two expressions for the kinetic part.

The default expression is from Syamlal et al. [ 364]:


 \mu_{s,{\rm kin}} = \frac{ \alpha_{s} d_{s} \rho_{s} \sqrt{\... ... \right) \left(3e_{ss} - 1 \right) \alpha_{s} g_{0,ss} \right] (23.5-64)

The following optional expression from Gidaspow et al. [ 121] is also available:


 \mu_{s,{\rm kin}} = \frac{ 10 \rho_{s} d_{s} \sqrt{\Theta_{s... ...{4}{5} g_{0,ss} \alpha_{s} \left( 1 + e_{ss} \right) \right]^2 (23.5-65)



Bulk Viscosity


The solids bulk viscosity accounts for the resistance of the granular particles to compression and expansion. It has the following form from Lun et al. [ 222]:


 \lambda_{s} = \frac{4}{3} \alpha_{s} \rho_{s} d_{s} g_{0,ss} (1 + e_{ss}) \left(\frac{\Theta_{s}}{\pi} \right)^{1/2} (23.5-66)

Note that the bulk viscosity is set to a constant value of zero, by default. It is also possible to select the Lun et al. expression or use a user-defined function.



Frictional Viscosity


In dense flow at low shear, where the secondary volume fraction for a solid phase nears the packing limit, the generation of stress is mainly due to friction between particles. The solids shear viscosity computed by FLUENT does not, by default, account for the friction between particles.

If the frictional viscosity is included in the calculation, FLUENT uses Schaeffer's [ 319] expression:


 \mu_{s,{\rm fr}}= \frac{p_s \sin \phi}{2\sqrt{I_{2D}} } (23.5-67)

where $p_s$ is the solids pressure, $\phi$ is the angle of internal friction, and $I_{2D}$ is the second invariant of the deviatoric stress tensor. It is also possible to specify a constant or user-defined frictional viscosity.

In granular flows with high solids volume fraction, instantaneous collisions are less important. The application of kinetic theory to granular flows is no longer relevant since particles are in contact and the resulting frictional stresses need to be taken into account. FLUENT extends the formulation of the frictional viscosity and employs other models, as well as providing new hooks for UDFs. See the separate UDF Manual for details.

The frictional stresses are usually written in Newtonian form:


 \tau_{friction} = -P_{friction}\vec{\vec{I}} + \mu_{friction} (\nabla{\vec{u}}_s + (\nabla{\vec{u}}_s)^T) (23.5-68)

The frictional stress is added to the stress predicted by the kinetic theory when the solids volume fraction exceeds a critical value. This value is normally set to 0.5 when the flow is three-dimensional and the maximum packing limit is about 0.63. Then


 P_S = P_{kinetic} + P_{friction} (23.5-69)


 \mu_S = \mu_{kinetic} + \mu_{friction} (23.5-70)

The derivation of the frictional pressure is mainly semi-empirical, while the frictional viscosity can be derived from the first principles. The application of the modified Coulomb law leads to an expression of the form


 \mu_{friction} = \frac{P_{friction} \sin\phi}{2 \sqrt{I_{2D}}} (23.5-71)

Where $\phi$ is the angle of internal friction and $I_{2D}$ is the second invariant of the deviatoric stress tensor.

Two additional models are available in FLUENT: the Johnson and Jackson [ 166] model for frictional pressure and Syamlal et al [ 364].

The Johnson and Jackson [ 166] model for frictional pressure is defined as


 P_{friction} = Fr \frac{(\alpha_s - \alpha_{s,min})^n}{(\alpha_{s,max} - \alpha_s)^p} (23.5-72)

With coefficient Fr = 0.05, n=2 and p = 3 [ 267]. The critical value for the solids volume fraction is 0.5. The coefficient Fr was modified to make it a function of the volume fraction:


 Fr = 0.1\alpha_s (23.5-73)

The frictional viscosity for this model is of the form


 \mu_{friction} = P_{friction} \sin\phi (23.5-74)

The second model that is employed is Syamlal et al [ 364], described in Equation  23.5-64. Comparing the two models results in the frictional normal stress differing by orders of magnitude.

The radial distribution function is an important parameter in the description of the solids pressure resulting from granular kinetic theory. If we use the models of Lun et al. [ 222] or Gidaspow [ 120] the radial function tends to infinity as the volume fraction tends to the packing limit. It would then be possible to use this pressure directly in the calculation of the frictional viscosity, as it has the desired effect. This approach is also available in FLUENT by default.

figure   

The introduction of the frictional viscosity helps in the description of frictional flows, however a complete description would require the introduction of more physics to capture the elastic regime with the calculation of the yield stress and the use of the flow-rule. These effects can be added by the user via UDFs to model static regime. Small time steps are required to get good convergence behavior.


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