To understand the process of pattern formation in a low-density granular flow, we propose a simple particle model. This model considers spherical particles moving over an inclined flat surface based on three forces: gravity as the driving force, repulsive force due to particle collision, and drag force as the particle-- interaction through the ambient fluid. Numerical simulations of this model are conducted in two different types of two-dimensional planes, i.e., the monolayer was treated. In the horizontal plane parallel to the slope, particles aggregate at the moving front of the granular flow; and subsequently, flow instability occurs as a wavy pattern. This flow pattern is caused by the interparticle interaction arising from the drag force. Additionally, a vortex convection of particles is formed inside the aggregations. Meanwhile, in the vertical plane on the slope, particle aggregation is also found at the moving front of the granular flow. The aggregation resembles a head--tail structure, where the frontal angle against the slope approaches 60 degree from a larger angle as time progresses. Comparing the numerical result by varying the particle size, the qualitative dynamics of the granular flow are independent of size. To elucidate this reason, we perform a nondimensionalization for this model. The result indicates that our model can be simplified to dimensionless equations with one dimensionless parameter that represents the ratio of the gravity term to the excluded volume effect.
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