# Analysis of laminar flow across a triangular periodic array of heated cylinders

## Abstract

The laminar flow and heat transfer across a triangular periodic array of heated cylinders are simulated computationally and analyzed. The study has been carried out at Reynolds number 10–100 for fluid volume fraction ranging from 0.7 to 0.99 and Prandtl number ranging from 0.7 to 50. The size of the wake region increases continuously with an increase in the Reynolds number for all values of fluid volume fraction. The recirculation bubble from the rear of a cylinder is reaching the front of the next cylinder in the same column of the periodic array for low values of free volume fraction, but this is not the case with the highest free volume fraction, i.e., 0.99. At high Reynolds number, the flow is separating early on the cylinder surfaces. The wake size at higher Reynolds number 75 and 100 for the lowest free volume fraction 0.7 is more in comparison with the wake size at free volume fraction 0.99, which is explained by plotting the location of flow separation against Reynolds number for both the extreme values of free volume fractions, i.e., 0.7 and 0.99. The isovorticity contours are concentrated in the vicinity of the cylinders on increasing the Reynolds number irrespective of free volume fraction and then convected downstream. On increasing free volume fraction, the friction and pressure drags in the array decrease. The increase in Reynolds number also results in the decrease in the values of the individual (friction and pressure drag coefficients) as well as total drag coefficients for all values of free volume fraction. At high values of Reynolds number, the emergence of carbuncle or thermal spike on isotherm near the cylinder’s surface is observed where the value of the local Nusselt number is observed low. The heat transfer improves and the Nusselt number increases as the Reynolds number and/or Prandtl number increases. On the contrary, heat transfer decreases as free volume fraction increases.

## Keywords

Triangular array Cylinders Drag coefficient Free volume fraction Periodicity Flow separation Isotherms Nusselt number## List of symbols

*C*_{D}Overall mean drag coefficient

*C*_{DF}Friction/viscous drag coefficient

*C*_{DP}Pressure drag coefficient

*C*_{DR}Ratio of pressure and friction drag coefficients

*C*_{P}Coefficient of pressure

*c*_{p}Specific heat of the fluid (J kg

^{−1}K^{−1})*D*Cylinder diameter (m)

*F*Dimensionless drag force (

*F*_{D}/*µU*)*F*_{D}Drag force (N m

^{−1})*F*_{DF}Friction drag force (N m

^{−1})*F*_{DP}Pressure drag force (N m

^{−1})*G*Grid size

*h*Convective heat transfer coefficient (W m

^{−2}K^{−1})*k*Thermal conductivity of fluid (W m

^{−1}K^{−1})*L*Center to center distance between cylinders (m)

*Nu*Average Nusselt number

*p*Pressure

*p*_{f}Dimensionless free stream pressure

*p*_{θ}Dimensionless pressure on the surface of a cylinder

*Pr*Prandtl number

*r*Radius of the cylinder (m)

*Re*Reynolds number

*T*Dimensionless temperature

*T*_{d}Dimensional temperature (K)

*T*_{w}Temperature at the cylinders surface (K)

*T*_{∞}Free stream temperature (K)

*U*Volume-averaged fluid velocity (m s

^{−1})- u
Cross-stream velocity (m s

^{−1})*u*Dimensionless cross-stream velocity

- v
Stream-wise velocity (m s

^{−1})*v*Dimensionless stream-wise velocity

- x, y
Dimensional Cartesian coordinates (m)

*x, y*Dimensionless Cartesian coordinates

## Greek symbols

*α*_{sep}Angle of separation (°)

*ϕ*_{f}Porosity/free volume fraction

*µ*Viscosity of fluid (Pa s)

*ρ*Density of fluid (kg m

^{−3})*ψ*Stream function

*θ*Surface angle (°)

*ω*_{z}Dimensionless vorticity

## Notes

### Acknowledgements

The authors would like to thank the reviewers for their valuable suggestions and helpful comments, which have enriched the present work for the wider readership.

## References

- 1.Wang C (1999) Longitudinal flow past cylinders arranged in a triangular array. Appl Math Model 23:219–230CrossRefMATHGoogle Scholar
- 2.Eidsath A, Carbonell R, Whitaker S, Herrmann L (1983) Dispersion in pulsed systems—III: comparison between theory and experiments for packed beds. Chem Eng Sci 38:1803–1816CrossRefGoogle Scholar
- 3.Astrom BT, Pipes RB, Advani SG (1992) On flow through aligned fiber beds and its application to composites processing. J Compos Mater 26:1351–1373CrossRefGoogle Scholar
- 4.Mandhani V, Chhabra R, Eswaran V (2002) Forced convection heat transfer in tube banks in cross flow. Chem Eng Sci 57:379–391CrossRefGoogle Scholar
- 5.Kirsch AA, Fuchs N (1967) Studies on fibrous aerosol filters—II. Pressure drops in systems of parallel cylinders. Ann Occup Hyg 10:23–30Google Scholar
- 6.Fisher TS, Torrance KE, Sikka KK (1997) Analysis and optimization of a natural draft heat sink system. IEEE Trans Compon Packag Manuf Technol A 20:111–119CrossRefGoogle Scholar
- 7.Kiljański T, Dziubiński M (1996) Resistance to flow of molten polymers through filtration screens. Chem Eng Sci 51:4533–4536CrossRefGoogle Scholar
- 8.Sparrow E, Loeffler A (1959) Longitudinal laminar flow between cylinders arranged in regular array. AIChE J 5:325–330CrossRefGoogle Scholar
- 9.Hasimoto H (1959) On the periodic fundamental solutions of the Stokes equations and their application to viscous flow past a cubic array of spheres. J Fluid Mech 5:317–328MathSciNetCrossRefMATHGoogle Scholar
- 10.Happel J (1959) Viscous flow relative to arrays of cylinders. AIChE J 5:174–177CrossRefGoogle Scholar
- 11.Launder B, Massey T (1978) The numerical prediction of viscous flow and heat transfer in tube banks. J Heat Transf 100:565–571CrossRefGoogle Scholar
- 12.Drummond J, Tahir M (1984) Laminar viscous flow through regular arrays of parallel solid cylinders. Int J Multiph Flow 10:515–540CrossRefMATHGoogle Scholar
- 13.Sangani A, Acrivos A (1982) Slow flow past periodic arrays of cylinders with application to heat transfer. Int J Multiph Flow 8:193–206CrossRefMATHGoogle Scholar
- 14.Sangani A, Acrivos A (1982) Slow flow through a periodic array of spheres. Int J Multiph Flow 8:343–360CrossRefMATHGoogle Scholar
- 15.Bruschke MV, Advani S (1993) Flow of generalized Newtonian fluids across a periodic array of cylinders. J Rheol 37:479–498CrossRefGoogle Scholar
- 16.Sangani AS, Mo G (1994) Inclusion of lubrication forces in dynamic simulations. Phys Fluids 6:1653–1662CrossRefMATHGoogle Scholar
- 17.Mo G, Sangani AS (1994) A method for computing Stokes flow interactions among spherical objects and its application to suspensions of drops and porous particles. Phys Fluids 6:1637–1652CrossRefMATHGoogle Scholar
- 18.Koch DL, Ladd AJ (1997) Moderate Reynolds number flows through periodic and random arrays of aligned cylinders. J Fluid Mech 349:31–66MathSciNetCrossRefMATHGoogle Scholar
- 19.Wung T-S, Chen CJ (1989) Finite analytic solution of convective heat transfer for tube arrays in crossflow: part I—flow field analysis. J Heat Transf 111:633–640CrossRefGoogle Scholar
- 20.Chen CJ, Wung T-S (1989) Finite analytic solution of convective heat transfer for tube arrays in crossflow: part II—heat transfer analysis. J Heat Transf 111:641–648CrossRefGoogle Scholar
- 21.Zukauskas AA, Ulinskas RV, Bubelis ES (1978) Average heat transfer and pressure drop in cross flow of viscous fluid over a tube bundle at low Reynolds number. Heat Transf Sov Res 10:90–101Google Scholar
- 22.Dhaubhadel M, Reddy J, Telionis D (1987) Finite-element analysis of fluid flow and heat transfer for staggered bundles of cylinders in cross flow. Int J Numer Methods Fluids 7:1325–1342CrossRefMATHGoogle Scholar
- 23.Fowler A, Bejan A (1994) Forced convection in banks of inclined cylinders at low Reynolds numbers. Int J Heat Fluid Flow 15:90–99CrossRefGoogle Scholar
- 24.Edwards D, Shapiro M, Bar-Yoseph P, Shapira M (1990) The influence of Reynolds number upon the apparent permeability of spatially periodic arrays of cylinders. Phys Fluids A 2:45–55CrossRefMATHGoogle Scholar
- 25.Ghaddar CK (1995) On the permeability of unidirectional fibrous media: a parallel computational approach. Phys Fluids 7:2563–2586CrossRefMATHGoogle Scholar
- 26.Wang W, Sangani AS (1997) Nusselt number for flow perpendicular to arrays of cylinders in the limit of small Reynolds and large Peclet numbers. Phys Fluids 9:1529–1539CrossRefGoogle Scholar
- 27.Martin A, Saltiel C, Shyy W (1998) Frictional losses and convective heat transfer in sparse, periodic cylinder arrays in cross flow. Int J Heat Mass Transf 41:2383–2397CrossRefMATHGoogle Scholar
- 28.Satheesh V, Chhabra RP, Eswaran V (1999) Steady incompressible fluid flow over a bundle of cylinders at moderate Reynolds numbers. Can J Chem Eng 77:978–987CrossRefGoogle Scholar
- 29.Beale S, Spalding D (1999) A numerical study of unsteady fluid flow in in-line and staggered tube banks. J Fluids Struct 13:723–754CrossRefGoogle Scholar
- 30.Wilson AS, Bassiouny MK (2000) Modeling of heat transfer for flow across tube banks. Chem Eng Process 39:1–14CrossRefGoogle Scholar
- 31.Kuwahara F, Shirota M, Nakayama A (2001) A numerical study of interfacial convective heat transfer coefficient in two-energy equation model for convection in porous media. Int J Heat Mass Transf 44:1153–1159CrossRefMATHGoogle Scholar
- 32.Wakao N, Kaguei S (1982) Heat and mass transfer in packed beds. Gorden and Breach, New York, pp 243–295Google Scholar
- 33.Gamrat G, Favre-Marinet M, Le Person S (2008) Numerical study of heat transfer over banks of rods in small Reynolds number cross-flow. Int J Heat Mass Transf 51:853–864CrossRefMATHGoogle Scholar
- 34.Roychowdhury DG, Das SK, Sundararajan T (2002) Numerical simulation of laminar flow and heat transfer over banks of staggered cylinders. Int J Numer Meth Fluids 39:23–40CrossRefMATHGoogle Scholar
- 35.Ram RP, Bharti RP, Dhiman AK (2016) Forced convection flow and heat transfer across an in-line bank of circular cylinders. Can J Chem Eng 94:1381–1395CrossRefGoogle Scholar
- 36.Sadeghifar H, Djilali N, Bahrami M (2016) A compact closed-form Nusselt formula for laminar longitudinal flow between rectangular/square arrays of parallel cylinders with unequal row temperatures. Int J Therm Sci 100:248–254CrossRefGoogle Scholar
- 37.Crowdy DG (2016) Uniform flow past a periodic array of cylinders. Eur J Mech B/Fluids 56:120–129MathSciNetCrossRefGoogle Scholar
- 38.Fornarelli F, Lippolis A, Oresta P (2017) Buoyancy effect on the flow pattern and the thermal performance of an array of circular cylinders. J Heat Transf 139:022501–022510CrossRefGoogle Scholar
- 39.Cho HW, Seo YM, Mun GS, Ha MY, Park YG (2017) The effect of instability flow for two-dimensional natural convection in a square enclosure with different arrays of two inner cylinders. Int J Heat Mass Transf 114:307–317CrossRefGoogle Scholar
- 40.Maleki M, Martinuzzi RJ, Herzog W, Federico S (2017) Orthotropic hydraulic permeability of arrays of parallel cylinders. Phys Rev E 96:033112-11CrossRefGoogle Scholar
- 41.ANSYS user manual. Ansys, Inc., Canonsburg, 2009Google Scholar
- 42.Verzicco R, Camussi R (1997) Transitional regimes of low-Prandtl thermal convection in a cylindrical cell. Phys Fluids 9:1287–1295CrossRefGoogle Scholar
- 43.Dennis SCR, Hudson J, Smith N (1968) Steady laminar forced convection from a circular cylinder at low Reynolds numbers. Phys Fluids 11:933–940CrossRefMATHGoogle Scholar
- 44.Soares A, Ferreira J, Chhabra R (2005) Flow and forced convection heat transfer in crossflow of non-Newtonian fluids over a circular cylinder. Ind Eng Chem Res 44:5815–5827CrossRefGoogle Scholar
- 45.Bharti RP, Chhabra R, Eswaran V (2007) A numerical study of the steady forced convection heat transfer from an unconfined circular cylinder. Heat Mass Transf 43:639–648CrossRefGoogle Scholar
- 46.Sparrow EM, Abraham JP, Tong JC (2004) Archival correlations for average heat transfer coefficients for non-circular and circular cylinders and for spheres in cross-flow. Int J Heat Mass Transf 47:5285–5296CrossRefGoogle Scholar
- 47.Sanyal A, Dhiman A (2017) Wake interactions in a fluid flow past a pair of side-by-side square cylinders in presence of mixed convection. Phys Fluids 29:103602CrossRefGoogle Scholar