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462159 A Dynamic Drag Model Using Sub-Grid Scalar Variance of Solid Volume Fraction for Gas-Solid Suspensions

In the present study, we have performed highly resolved Computational Fluid Dynamics (CFD) – Discrete Element Method (DEM) simulations of gas-fluidization of mono-disperse particles with three different diameters (75, 150, 300 μm) in periodic domains at various solid volume fractions. Simulations of this kind are being done by several research groups to learn more about meso-scale structures in gas-particle flows; for example, see [6,7].

We first mapped the Lagrangian results onto the Eulerian field and then filtered them by volume averaging in order to evaluate the sub-filter contribution of Eulerian drag force. We found that the sub-filter contribution of drag force can be captured via a model relating the filtered drag coefficient to the filtered particle volume fraction, the sub-filter scalar variance of solid volume fraction, and the particle Froude number. The sub-filter scalar variance is a measure of the degree of local inhomogeneity of the solid volume fraction within the filter. The Froude number is based on particle diameter, terminal settling velocity, and gravitational acceleration.

As the sub-filter scalar variance of solid volume fraction cannot be obtained from the resolved field, one must develop a closure model or an additional transport equation for it. In this study, we have formulated an algebraic closure for this scalar variance in terms of the filter size and filtered solid volume fraction. Towards this end, we have analyzed the CFD-DEM simulation results and extracted the functional dependence of the sub-filter scalar variance of the solid volume fraction on the filtered volume fraction and filter size to within an unspecified multiplicative constant. It is then proposed that this constant be determined dynamically in coarse simulations by using a scale similarity assumption [8], and a test filter following the approach proposed by Germano et al. [9].

We assessed the accuracy of the model by computing correlation coefficients between model predictions and exact values calculated from mapped results. The correlation coefficients are around 0.7 even for large filter sizes, indicating that the sub-filter contribution is well captured by the model.

As a further study, we plan to implement the proposed model into a two-fluid model in order to assess a posteriori performance of the model.

**References**

[1] Igci, Y. and Sundaresan, S., 2011. Constitutive models for filtered two-fluid models of fluidized gas–particle flows. *Industrial & Engineering Chemistry Research*, *50*(23), pp.13190-13201.

[2] Schneiderbauer, S., Puttinger, S. and Pirker, S., 2013. Comparative analysis of subgrid drag modifications for dense gas‐particle flows in bubbling fluidized beds. *AIChE Journal*, *59*(11), pp.4077-4099.

[3] Parmentier, J.F., Simonin, O. and Delsart, O., 2012. A functional subgrid drift velocity model for filtered drag prediction in dense fluidized bed. *AIChE Journal*, *58*(4), pp.1084-1098.

[4] Milioli, C.C., Milioli, F.E., Holloway, W., Agrawal, K. and Sundaresan, S., 2013. Filtered two‐fluid models of fluidized gas‐particle flows: New constitutive relations. *AIChE Journal*, *59*(9), pp.3265-3275.

[5] Ozel, A., Fede, P. and Simonin, O., 2013. Development of filtered Euler–Euler two-phase model for circulating fluidised bed: high resolution simulation, formulation and a priori analyses. *International Journal of Multiphase Flow*, *55*, pp.43-63.

[6] Capecelatro, J. and Desjardins, O., 2013. An Euler–Lagrange strategy for simulating particle-laden flows. *Journal of Computational Physics*, *238*, pp.1-31.

[7] Capecelatro, J., Desjardins, O. and Fox, R.O., 2014. Numerical study of collisional particle dynamics in cluster-induced turbulence. *Journal of Fluid Mechanics*, *747*, p.R2.

[8] Bardina, J., Ferziger, J.H. and Reynolds, W.C., 1983. Improved turbulence models based on large eddy simulation of homogeneous, incompressible, turbulent flows. Stanford University.

[9] Germano, M., Piomelli, U., Moin, P. and Cabot, W.H., 1991. A dynamic subgrid‐scale eddy viscosity model. *Physics of Fluids A: Fluid Dynamics (1989-1993)*, *3*(7), pp.1760-1765.

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