Two-dimensional Flow of Power-law Fluids over a Pair of Cylinders in a Side-by-Side Arrangement in the Laminar Regime

Panda, Saroj K.

doi:10.1590/0104-6632.20170342s20150504

Saroj K. Panda ^**To whom correspondence should be addressed. Email: sarojdeep@gmail.com Tel.: +91 11 2659 6282, fax: +91 11 2658 1120

Department of Chemical Engineering, Indian Institute of Technology Delhi New Delhi 110016, India

*To whom correspondence should be addressed. Email: sarojdeep@gmail.com Tel.: +91 11 2659 6282, fax: +91 11 2658 1120

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Figure 1
(a) Schematic representation of the physical model (b) Computational domain (c) Close up view of the grid in the vicinity of cylinders.

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Figure 2
Vorticity contours for different wake patterns (a) single bluff-body pattern (Re=100, G=1.2); (b) flip-flopping pattern (Re=100, G=1.7); (c) in-phase-synchronized pattern (Re=100, G=2.5); (d) anti-phase- synchronized pattern (Re=100, G=4); (e) steady pattern (Re=40, G=2)

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Figure 3
(a) Streamline contours for Re=0.1 at different G and n

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Figure 3
(b) Streamline contours for Re=10 at different G and n

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Figure 3
(b) Streamline contours for Re=10 at different G and n

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Figure 3
(d) Streamline contours for Re=100 at different G and n

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Figure 3
(e) Streamline contours for Re = 0.1, 40 at G = 1.2, 4 and n = 1.2, 1.6 and 1.8

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Figure 4
(a). Representative pressure profiles on the surface of the upper cylinder

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Figure 4
(b). Representative pressure profiles on the surface of the lower cylinder

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Figure 5
Variation of (a) pressure drag coefficient (C _DP ) and (b) friction drag coefficient (C _DF ) with Reynolds number (Re)

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Figure 6
Variation of total lift coefficient(C_DP/ C_DF ) with Reynolds number (Re)

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Figure 7
Variation of total drag coefficient (C _D ) with (a) Reynolds number (Re) and (b) gap ratio (G)

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Figure 8
Variation of total drag coefficient (C _D ) with power-law index for Reynolds number 0.1 and 40

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Figure 9
Variation of total lift coefficient (C _L ) with Reynolds number (Re)

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Figure 10
Variation of total lift coefficient (C _L ) with power-law index for different Reynolds number 0.1 and 40

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Table 1
Physical properties of the fluid used

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Table 2
Selection of optimum domain at Re = 0.1

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Table 3
Selection of optimum domain at Re = 5

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Table 4
Details of grids used for the grid independence study

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Table 5
Effect of grid details on the results at Re = 0.1 (H/D = 240)

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Table 6
Effect of grid details on the results at Re = 5 (H/D = 120)

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Table 7
Comparison between the present and literature values for Re = 100 and 40; G = 2 and 4; n = 1

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Table 8
Comparison of drag coefficient values between a single cylinder and the present case for Re = 0.1, 1 and 40 (G = 4)

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Re	ρ (Kg/m³ )	m (Kg/m-s)
Re	ρ (Kg/m³ )			n = 0.2	n = 0.4	n = 0.6	n = 0.8	n = 1	n = 1.2
0.1	0.1	0.01584	0.02511	0.03981	0.06309	0.1	0.15848
0.2	0.2	0.01584	0.02511	0.03981	0.06309	0.1	0.15848
0.5	0.5	0.01584	0.02511	0.03981	0.06309	0.1	0.15848
1	1	0.01584	0.02511	0.03981	0.06309	0.1	0.15848
2	2	0.01584	0.02511	0.03981	0.06309	0.1	0.15848
5	5	0.01584	0.02511	0.03981	0.06309	0.1	0.15848
10	10	0.01584	0.02511	0.03981	0.06309	0.1	0.15848
20	20	0.01584	0.02511	0.03981	0.06309	0.1	0.15848
30	30	0.01584	0.02511	0.03981	0.06309	0.1	0.15848
40	40	0.01584	0.02511	0.03981	0.06309	0.1	0.15848
100	100	0.01584	0.02511	0.03981	0.06309	0.1	0.15848

H/D	Re = 0.1, n = 1, G = 4
	Upper Cylinder				Lower Cylinder
		C_DP	C_D	C_LF	C_L	C_DP	C_D	C_LP	C_L
220	21.6440	43.7499	1.9171	3.8090	21.6429	43.7506	-1.9194	-3.8081
240	21.3989	43.2549	1.8931	3.7618	21.3979	43.2556	-1.8954	-3.7610
260	21.3984	43.2546	1.8924	3.7612	21.3981	43.2558	-1.8955	-3.7625
H/D	Re = 0.1, n = 0.2, G = 4
	Upper Cylinder				Lower Cylinder
	C_DP	C_D	C_LP	C_L	C_DP	C_D	C_LP	C_L
220	183.551	262.335	0.6632	-0.8828	183.541	262.325	0.6312	0.8464
240	183.581	262.378	-0.6704	-0.8927	183.571	262.369	0.6375	0.8553
260	183.542	262.338	-0.6724	-0.8962	183.568	262.364	0.6324	0.8592

H/D	Re = 5, n = 1, G = 4
	Upper Cylinder				Lower Cylinder
		C_DP	C_D	C_LP	C_L	C_DP	C_D	C_LP	C_L
100	2.2058	4.2072	0.3475	0.6494	2.2058	4.2072	-0.3477	-0.6491
120	2.1853	4.1696	0.3464	0.6475	2.1853	4.1697	-0.3466	-0.6472
180	2.1838	4.1706	0.3458	0.6470	2.1855	4.1710	-0.3465	-0.6485
H/D	Re = 5, n = 0.2, G = 4
	Upper Cylinder				Lower Cylinder
	C_DP	C_D	C_LP	C_L	C_DP	C_D	C_LP	C_L
100	4.2432	5.8273	-0.021	-0.032	4.2433	5.8275	0.0201	0.0308
120	4.2447	5.8290	-0.021	-0.032	4.2448	5.8293	0.0200	0.0308
180	4.2454	5.8297	-0.021	-0.032	4.2454	5.8299	0.0201	0.0309

Grid	H/D = 240
Grid		N _i	N	M	N _cells
G1	100	5	16	48550
G2	200	10	33	196254
G3	400	20	66	782224

Grid	H/D = 120
Grid	N _i	N	M	N _cells
G1	100	5	16	37780
G2	200	10	33	153054
G3	400	20	66	609424

Grid	Re = 0.1, n = 1, G = 4
	Upper Cylinder				Lower Cylinder
		C_DF	C_D	C_LF	C_L	C_DF	C_D	C_LF	C_L
G1	20.0215	42.2026	1.8351	3.3564	20.0225	42.2046	-1.8342	-3.3586
G2	21.3989	43.2549	1.8931	3.7618	21.3979	43.2556	-1.8954	-3.7610
G3	21.4007	43.2452	1.9020	3.7681	21.3906	43.2451	-1.8920	-3.7583
Grid	Re = 0.1, n = 0.2, G = 4
	Upper Cylinder				Lower Cylinder
	C_DF	C_D	C_LF	C_L	C_DF	C_D	C_LF	C_L
G1	179.338	255.226	-0.8358	-1.0313	179.312	255.182	0.8298	1.0223
G2	183.581	262.378	-0.6704	-0.8927	183.571	262.369	0.6375	0.8553
G3	183.301	261.990	-0.6722	-0.8955	183.423	262.112	0.6339	0.8504

Grid	Re = 5, n = 1, G = 4
	Upper Cylinder				Lower Cylinder
		C_DF	C_D	C_LF	C_L	C_DF	C_D	C_LF	C_L
G1	2.1033	4.1561	0.3235	0.6356	2.1014	4.1512	-0.3227	-0.6438
G2	2.1853	4.1696	0.3464	0.6475	2.1853	4.1697	-0.3466	-0.6472
G3	2.1844	4.1633	0.3467	0.6488	2.1849	4.1658	-0.3459	-0.6483
Grid	Re = 5, n = 0.2, G = 4
	Upper Cylinder				Lower Cylinder
	C_DF	C_D	C_LF	C_L	C_DF	C_D	C_LF	C_L
G1	4.1291	5.8207	-0.035	-0.043	4.1321	5.8224	0.0355	0.0435
G2	4.2447	5.8290	-0.021	-0.032	4.2448	5.8293	0.0200	0.0308
G3	4.2431	5.8278	-0.020	-0.031	4.2444	5.8294	-0.0200	-0.0310

Upper Cylinder
Source	C _D	C _L
Re = 40, n = 1, G = 2
Present	1.7064	0.3665
Kang, 2003	1.70	0.3651
Re = 100, n = 1, G = 4
Present	1.4686	--
Chaitanya and Dhiman, 2012	1.4727	--
Ding et al., 2007	1.514	--
Lower Cylinder
Source	C _D	C _L
Re = 40, n = 1, G = 2
Present	1.7066	-0.3658
Kang, 2003	1.70	-0.3706
Re = 100, n = 1, G = 4
Present	1.4702	--
Chaitanya and Dhiman, 2012	1.4726	--
Ding et al., 2007	1.514	--

C_D Single Cylinder (Panda and Chhabra, 2010)			C_D Upper Cylinder (Present work at G = 4)
Re	n = 0.2	n = 1	Re	n = 0.2	n = 1
0.1	270.318	61.127	0.1	262.567	43.254
1	27.063	10.566	1	26.526	9.086
40	1.136	1.513	40	1.176	1.64

\frac{\partial U_{x}}{\partial t} + \frac{\partial (U_{x} U_{x})}{\partial x} + \frac{\partial (U_{x} U_{y})}{\partial y} = - \frac{\partial P}{\partial x} + \frac{1}{Re} (\frac{\partial τ_{x x}}{\partial x} + \frac{\partial τ_{y x}}{\partial y})

\frac{\partial U_{y}}{\partial t} + \frac{\partial (U_{x} U_{y})}{\partial x} + \frac{\partial (U_{y} U_{y})}{\partial y} = - \frac{\partial P}{\partial y} + \frac{1}{Re} (\frac{\partial τ_{x y}}{\partial x} + \frac{\partial τ_{y y}}{\partial y})

C_{D} = \frac{2 F_{D}}{ρ U_{0}^{2} D}, C_{D P} = \frac{2 F_{D P}}{ρ U_{0}^{2} D} = \int_{S} C_{P} n_{x} d S and C_{D F} = \frac{2 F_{D F}}{ρ U_{0}^{2} D} = \frac{2^{n + 1}}{Re} \int_{S} (τ \cdot n_{s}) d S

[1] *To whom correspondence should be addressed. Email: sarojdeep@gmail.com Tel.: +91 11 2659 6282, fax: +91 11 2658 1120

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Two-dimensional Flow of Power-law Fluids over a Pair of Cylinders in a Side-by-Side Arrangement in the Laminar Regime

Abstract