Particles Trajectories in Uniform Flows

Let us now consider the Flow Simulation capability to predict particles trajectories in a gas flow (i.e. two-phase flow of fluid + liquid droplets or solid particles).

In accordance with the particles motion model accepted in Flow Simulation, particle trajectories are calculated after completing a fluid flow calculation (which can be either steady or time-dependent). That is, the particles mass and volume flow rates are assumed substantially lower than those of the fluid stream, so that the influence of particles’ motions and temperatures on the fluid flow parameters is negligible, and motion of the particles obeys the following equation:



where m is the particle mass, t is time, Vp and Vf are the particle and fluid velocities (vectors), accordingly, ρf is the fluid density, Cd is the particle drag coefficient, A is the particle frontal surface area, and Fg is the gravitational force.

Particles are treated as non-rotating spheres of constant mass and specified (solid or liquid) material, whose drag coefficient is determined from Henderson’s semi-empirical formula (Ref. 11). At very low velocity of particles with respect to carrier fluid (i.e., at the relative velocity’s Mach number M → 0) this formula becomes:



where Reynolds number is defined as:



d is the diameter of particles, and μ is the fluid dynamic viscosity.

To validate Flow Simulation, let us consider three cases of injecting a particle perpendicularly into an incoming uniform flow, Figure 1. Since both the fluid flow and the particle motion in these cases are 2D (planar), we will solve a 2D (i.e. in the XY-plane) flow problem.

Figure 1. Injection of a particle into a uniform fluid flow.

Due to the same reason as in the previous validation examples with flow over flat plates, we will solve this validation as an internal problem. The corresponding geometry model is shown in Figure 2. Both of the walls are ideal, the channel has length of 0.233 m and height of 0.12 m, all the walls have thickness of 0.01 m. We specify the uniform fluid velocity Vin, the fluid temperature of 293.2 K, and the default values of turbulent flow parameters with the laminar boundary layer at the channel inlet, and the static pressure of 1 atm at the channel outlet. All the fluid flow calculations are performed at a result resolution level of 5.

Figure 2. The model geometry: (1) inlet, (2) origin and particle injection point, (3) outlet, (4) ideal wall.

To validate calculations of particles trajectories by comparing them with available analytical solutions of the particle motion equation, we consider the following three cases:

  1. the low maximum Reynolds number of Remax = 0.1 (air flow with Vin = 0.002 m/s, gold particles of d = 0.5 mm, injected at the velocity of Vp = 0.002 m/s perpendicularly to the wall),

  2. the high maximum Reynolds number of Remax = 105 (water flow with Vin = 10 m/s, iron particles of d = 1 cm, injected at the velocities of Vp = 1, 2, 3 m/s perpendicularly to the wall),

  3. a particle trajectory in the Y-directed gravitational field (gravitational acceleration gy = -9.8 m/s2, air flow with Vin = 0.6 m/s, an iron particle of d = 1 cm, injected at the velocity of Vp = 1.34 m/s at the angle of 63.44o with the wall).

In the first case, due to small Re values, the particle drag coefficient is close to Cd = 24/Re (i.e., obeys the Stokes law). Then, neglecting gravity, we obtain the following analytical solution for the particle trajectory:





where Vfx, Vpx, Vfy, Vpy are the X- and Y-components of the fluid and particle velocities, accordingly, ρp is the particle material density. The Flow Simulation calculation and the analytical solution are shown in Figure 3. It is seen that they are very close to one another. Special calculations have shown that the difference is due to the CD assumptions only.

Figure 3. Particle trajectories in a uniform fluid flow at Remax = 0.1, predicted by Flow Simulation and obtained from the analytical solution.

In the second case, due to high Re values, the particle drag coefficient is close to Cd = 0.38. Then, neglecting the gravity, we obtain the following analytical solution for the particle trajectory:





The Flow Simulation calculations and the analytical solutions for three particle injection velocities, Vpy(t = 0) = 1, 2, 3 m/s, are shown in Figure 4. It is seen that the Flow Simulation calculations coincide with the analytical solutions. Special calculations have shown that the difference is due to the CD assumptions only.

In the third case, the particle trajectory is governed by the action of the gravitational force only, the particle drag coefficient is very close to zero, so the analytical solution is:



The Flow Simulation calculation and the analytical solution for this case are presented in Figure 5. It is seen that the Flow Simulation calculation coincides with the analytical solution.

Figure 4. Particle trajectories in a uniform fluid flow at Remax = 105, predicted by Flow Simulation and obtained from the analytical solution.

Figure 5. Particle trajectories in the Y-directed gravity, predicted by Flow Simulation and obtained from the analytical solution.I