Two-phase continuum theory for windblown sand

We outline the derivation of a two-phase continuum theory for grains, jumping above a bed of sand, while accelerated by a turbulent shearing flow, colliding with the bed, rebounding, and, perhaps, generating other grains. Relations between the shear and normal stresses and vertical derivatives of components of the average particle velocity are determined by averaging the dynamical equations for the particle trajectories. This provides the closure for the system of differential equations that govern the behavior of the wind and particles above the bed. Boundary conditions are obtained by averaging the results of experiments on rebound and ejection of particles from a particle bed. We solve the resulting system of equations subject to the derived boundary conditions for steady, uniform flows over both particle and rigid beds, and obtain unsteady, uniform solutions and steady, nonuniform solutions that provide information regarding saturation times and lengths, respectively.

Figure 1

Logarithm of the concentration vs dimensionless height for $d=0.025\mathrm{cm}$. Spherical grains of sand in air at $S^{*}=0.035$, 0.059, 0.065, and 0.080. Line weights increase with increasing values of $S^{*}$.

Figure 2

Average dimensionless particle and gas velocities vs dimensionless height for $d=0.025\mathrm{cm}$. Spherical grains of sand in air at $S^{*}=0.035$, 0.059, 0.065, and 0.080. Line weights increase with increasing values of $S^{*}$.

Figure 3

Dimensionless continuum particle shear stress (dashed lines) and measured simulated particle shear stress (solid lines) vs dimensionless height for $d=0.025\mathrm{cm}$. Spherical grains of sand in air for $u_0=23.06,T=25.14,S^{*}=0.050$, and $\alpha =20$.

Figure 4

Concentration vs dimensionless height for slip velocities of 15, 33, 48, and 60, with the line weights increasing with slip velocity. The corresponding values of the particle flux and holdup are 0.7 and 0.04, 1.8 and 0.04, 2.1 and 0.03, and 1.8 and 0.02, respectively.

Figure 5

Dimensionless particle and gas velocities vs dimensionless height for slip velocities of 20, 39, 59, and 80, with the line weights increasing with slip velocity. The corresponding values of the particle flux and holdup are 1.8 and 0.072, 3.5 and 0.065, 4.3 and 0.052, and 3.5 and 0.032, respectively.

Figure 6

The relationship between dimensionless particle flux and dimensionless particle slip velocity for $S^{*}=0.05$.

Figure 7

The relationship between dimensionless particle flux and dimensionless particle holdup for $S^{*}=0.05$.

Figure 8

Evolution of the dimensionless particle vertical velocity in space at three equal intervals between 0 and 12000 in units of dimensionless time with $\delta =10$, for $d=0.025\mathrm{cm}$. Spherical grains of sand in air at ${S^{*}}^{\prime }=0.0025,S^{*}=0.05$, and $u_0^0=23.06$. Line weights increase with dimensionless time.

Figure 9

Evolution of the dimensionless temperature in space at three equal intervals between 0 and 12000 in units of dimensionless time with $\delta =10$, for $d=0.025\mathrm{cm}$. Spherical grains of sand in air at ${S^{*}}^{\prime }=0.0025,S_0^{*}=0.05$, and $u_0^0=23.06$. Line weights increase with dimensionless time.

Figure 10

Dimensionless particle flux vs time in seconds for spherical grains of sand in air showing the evolution with time on a particle bed for $S^{*}=0.05$, from initial states that are steady states on a rigid bed with slip velocities of 19.0, 21.6, 23.1, 24.4, and 26.1, indicated by increasing line weights. Here, $d=0.25\mathrm{mm},\varepsilon =0.01$, and $\delta =5$.

Figure 11

Dimensionless particle flux vs time in seconds showing the evolution with time on a particle bed from a very small initial value $(Q=0.005$ at $S^{*}=0.009)$ for increases in Shields numbers of 0.002, 0.003, 0.004, 0.006, and 0.008, indicated by increasing line weights. Here, $d=0.25\mathrm{mm}$, with $\varepsilon =0.01$ and $\delta =1$.

Figure 12

Evolution of dimensionless total particle flux with distance over a rigid base for $S^{*}=0.05$. The slip velocities of the three stable solutions (solid) are 59, 78, and 83, while those of the unstable solution are 37 (dotted), 40 (dotted dashed), and 45 (dashed).

Figure 13

Concentration vs height for four equal intervals of dimensionless distance between 0 and ${10}^6$, with $S^{*}=0.05,u_0=59$, and $\delta =5$. Line weights increase with distance.

Figure 14

Horizontal velocities vs height for four equal intervals of dimensionless distance between 0 and ${10}^6$, with $S^{*}=0.05,u_0=59$, and $\delta =5$. Line weights increase with distance.

Figure 15

Vertical velocity vs height for four equal intervals of dimensionless distance between 0 and ${10}^6$, with $S^{*}=0.05,u_0=59$, and $\delta =5$. Line weights increase with distance.

Figure 16

Granular temperature vs height for four equal intervals of dimensionless distance between 0 and ${10}^6$, with $S^{*}=0.05,u_0=59$, and $\delta =5$. Line weights increase with distance.

Figure 17

Total particle flux vs dimensionless downstream distance, with $S^{*}=0.05,u_0=59$, and $\delta =5$.

Figure 18

Initial variation of the total particle flux vs dimensionless downstream distance, with $S^{*}=0.05,u_0=59$, and $\delta =5$.