Numerical Simulation of Particle Dispersion in the Wake of a Circular Cylinder

Figures & data

FIG. 1 A sketch of the computational domain.

Table

γ 0	α 0	α 1	α 2	β 0	β 1	β 2
11/6	3	−3/2	1/3	3	−3	1

TABLE 1 Numerical results with different grids division at Re = 30

Cases	Elements	X1	X0	Y1	−C pb	C df	C dp	Convergence	h min ∗	h avg ∗
G1	30	24	−7	14	0.65	0.81	1.25	1.7e0-4	0.08	0.52
G2	40	36	−7	14	0.66	0.80	1.25	1.5e0-4	0.03	0.24
G3	50	40	−14	28	0.54	0.70	1.05	2.0e0-4	0.01	0.12
G4	50	40	−7	28	0.64	0.72	1.26	3.4e0-4	0.01	0.12
∗∗	202	—	—	—	0—56	0.64	1.1	−e0-4	—	—
∗ h min and h avg are the minimum and average element size in the region |R| ≤ 2; ∗∗Henderson (1995).

TABLE 2 Numerical results with different grids division at Re = 100

Cases	Elements	X1	X0	Y1	−C pb	C df	C dp	St g	u∗	h min ∗∗	h avg ∗∗
G1	30	24	−7	14	0.79	0.40	1.24	0.192	1.392	0.08	0.52
G2	40	36	−7	14	0.81	0.40	1.24	0.192	1.392	0.03	0.24
G3	50	40	−14	28	0.69	0.35	1.05	0.172	1.255	0.01	0.12
G4	50	40	−7	28	0.79	0.36	1.24	0.192	1.392	0.01	0.12
G5	40	36	−7	28	0.81	0.40	1.24	0.191	1.392	0.05	0.36
G6	60	40	−14	28	0.69	0.35	1.04	0.172	1.251	0.008	0.08
G7	80	40	−14	28	0.68	0.35	1.04	0.172	1.250	0.005	0.05
—	—	—	—	—	0.74^+	0.35^+	1.0^+	0.165^+ +	1.20^+ +	—	—
∗ u-component larger magnitude at x/R = 3.24, y/R = 1.65; ∗∗ h min and h avg are the minimum and average element size in the region |R| ≤ 2; ^+ Henderson (1995), ^++ Williamson (1989), ^++ Braza, Chassaing, and Minh (1986).

FIG. 2 Coefficient versus Reynolds number: (a) C _df ; (b) C _dp ; (c) C _Pb ; (d) C _d .

TABLE 3 Numerical results with different grids division at Re = 300

Cases	Elements	X1	X0	Y1	−C pb	C df	C dp	St g	u∗	h min ∗∗	h avg ∗∗
G3	50	40	−14	28	1.14	0.23	1.29	0.222	0.381	0.01	0.12
G6	60	40	−14	28	1.15	0.23	1.27	0.222	0.379	0.008	0.08
G7	80	40	−14	28	1.15	0.22	1.27	0.222	0.379	0.005	0.05
—	—	—	—	—	1.22^+	0.22^+	1.18^+	0.206^++	0.351^+++	—	—
∗ u-component larger magnitude at x/R = 1.94, y/R = 0; ∗∗ h min and h avg are the minimum and average element size in the region |R| ≤ 2; ^+ Henderson (1995), ^+ + Williamson (1989), ^+ + + Braza, Chassaing, and Minh (1986).

FIG. 3 Time-dependent evolution of: (a) U-component; (b) V-component; (c) drag coefficient; (d) lift coefficient (x/R = 2, y/R = 0.5).

FIG. 4 Distribution of fluid flow (Re = 100) variables at the nondimensional time of t = 100: (a) vorticity; (b) velocity; (c) pressure.

TABLE 4 Test of particle number independence

	The number of particles introduced at each step
	10	20	50	100	200	500
	Averaged particle concentration function (Nrm(x))
St = 4, Re = 40	0.321	0.327	0.339	0.340	0.340	0.340
Re = 100	0.350	0.357	0.368	0.369	0.368	0.369
Re = 300	0.349	0.358	0.370	0.372	0.373	0.372
Re = 100, St = 0.01	0.379	0.381	0.387	0.388	0.388	0.388
St = 1	0.345	0.350	0.359	0.359	0.359	0.359
St = 10	0.341	0.349	0.354	0.354	0.354	0.354

TABLE 5 Total number of particle tracked for each case

St = 4	Re =	30	40	50	60	100	300
Particle number		1091327	87646	663105	57791	29797	10457
Re = 100	St =	0.01	0.1	1	4	10	100
Particle number		28645	29157	27943	29797	35679	46789

FIG. 5 Particle dispersion pattern t = 100 in the flow: Re = 100 (left), 300 (right): (a) St = 0.01, (b) 0.1, (c) 1, (d) 10.

FIG. 6 Distribution of relative velocity vector for different Stokes number at nondimensional time of t = 100 at Re = 100: (a) St = 0.01, (b) St = 0.1, (c) St = 1, (d) St = 10.

TABLE 6 Mean relative slip velocity (t = 100)

		Mean relative slip velocity
Particle St		|u − u p |	|v − v p |
0.01	Re = 100	7.033e-04	9.094e-04
0.1		5.406e-03	6.322e-03
1		2.365e-02	3.122e-02
10		1.175e-01	7.400e-02
0.01	Re = 300	6.220e-04	8.889e-04
0.1		4.868e-03	7.390e-03
1		2.322e-02	3.495e-02
10		1.093e-01	8.460e-01

FIG. 7 Particle concentration function in the flow t = 100 (a), (b) St = 4: (a) Nrms(x); (b) Nrms(y); (c), (d) Re = 100: (c) Nrms(x); (d) Nrms(y).

FIG. 8 Time-dependent particle dispersion in the vertical direction: (a) St = 4 in various flows (Re = 30 ∼ 300); (b) different particles in the flow: Re = 100.

Henderson , R. D. 1995 . Details of the Drag Curve Near the Onset of Vortex Shedding . Phys. Fluids , 7 : 2102 – 2104 .

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Braza , M. , Chassaing , P. and Minh , H. H. 1986 . Numerical Study and Physical Analysis of the Pressure and Velocity Fields in the Near Wake of a Circular Cylinder . J. Fluid Mech. , 165 : 79 – 130 .

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