Handbook of Thermal Science and Engineering pp 527-602 | Cite as

# Free Convection: External Surface

## Abstract

Some representative recent basic studies involving external free convective heat transfer in situations that are of practical interest are discussed in this chapter. Most of the results considered have been obtained numerically, and a very brief discussion of the methodology used to generate these results is presented. Attention has here first been given to the heat transfer rate from narrow vertical plane surfaces with the effect of the width-to-height ratio of the surface on the heat transfer rate in particular being discussed. Heat transfer from horizontally and vertically spaced pairs of narrow plates has also been considered. Consideration has then been given to the heat transfer rate from horizontal heated surfaces of complex shape, to adjacent pairs of horizontal heated surfaces, and to heat transfer from two-sided circular horizontal surfaces. The effect of a covering surface on the heat transfer from a horizontal heated surface has also been considered. The heat transfer rate from bodies with wavy surfaces has been considered next with attention being given to situations involving two-dimensional flow over vertical and horizontal surfaces and to the effect of surface waviness on the heat transfer rate from cylindrical bodies. The heat transfer from relatively short vertical circular and square cylinders with exposed top surfaces is also considered. Lastly, brief attention is given to external free convective heat transfer to nanofluids.

## Nomenclature

*A*Total surface area

*A*_{bottom}Area of bottom surface

*A*_{c}Area of cylindrical outer surface of cylinder

*A*_{top}Area of top surface

*A*_{t}Area of top surface of cylinder

*A*_{total}Total surface area

*AR*Aspect ratio

*a*Mean side length

*c*_{pf}Specific heat of fluid in which nanoparticles are placed

*c*_{pnf}Specific heat of nanofluid

*c*_{ps}Specific heat of nanoparticles

*D*_{i}Dimensionless diameter of inner adiabatic section of surface

*D*Diameter of circular surface

*d*Heated surface diameter

*d*_{i}Diameter of inner adiabatic section of surface

*G*Gap between heated surfaces

*Gap*Gap between vertically spaced heated surfaces

*g*Gravitational acceleration

*H*_{Gap}Dimensionless size of gap between surfaces

*H*Height of surface and dimensionless recess depth,

*h*/*d**h*Recess depth

*k*Thermal conductivity

*h*Recess depth

*L*l/d

*L*_{out}Dimensionless side length of rectangular adiabatic covering surface,

*L*_{out}*= l*_{out}*/w**l*Reference length and cylinder height

*l*_{out}Side lengths of rectangular adiabatic covering surface

*m*Characteristic length scale of surface

*Nu*Nusselt number

*Nu*_{0}Reference Nusselt number

*Nu*_{a}Mean Nusselt number based on the mean side length,

*a*, of rectangular heated surface*Nu*_{bottom}Mean Nusselt number averaged over bottom surface

*Nu*_{c}Mean Nusselt number averaged over vertical side surface of cylinder

*Nu*_{m}Mean Nusselt number based on

*m**Nu*_{r}Mean Nusselt number based on

*r**Nu*_{rbottom}Mean Nusselt number based on

*r*averaged over bottom surface*Nu*_{rtop}Mean Nusselt number based on

*r*averaged over top surface*Nu*_{t}Mean Nusselt number averaged over horizontal top surface of cylinder

*Nu*_{top}Mean Nusselt number averaged over top surface

*Nu*_{total}Mean Nusselt number averaged over entire surface

*n*Coordinate normal to surface or number of surface waves

*P*Total perimeter of heated surface

*Pr*Prandtl number

- \( {\overline{Q}}^{\prime } \)
Total mean heat transfer rate

- \( {{\overline{Q}}^{\prime}}_{\mathrm{bottom}} \)
Total mean heat transfer rate from bottom surface

- \( {{\overline{Q}}^{\prime}}_{\mathrm{top}} \)
Total mean heat transfer rate from top surface

*q*^{′}Local heat transfer rate per unit area

- \( {\overline{q}}^{\prime } \)
Mean heat transfer rate per unit area from entire cylinder

- \( \overline{q_c^{\prime }} \)
Mean heat transfer rate per unit area from cylindrical outer surface of cylinder

- \( \overline{q_s^{\prime }} \)
Mean heat transfer rate per unit area from vertical side surface of square cylinder

- \( \overline{q_t^{\prime }} \)
Mean heat transfer rate per unit area from top surface of cylinder

- \( \overline{q_w^{\prime }} \)
Mean heat transfer rate per unit area

*R*Dimensionless cylinder radius,

*r*/*l**Ra*Rayleigh number

*Ra*_{a}Rayleigh number based on the mean side length,

*a*, of rectangular heated surface*Ra*_{m}Rayleigh number based on

*m**Ra*_{r}Rayleigh number based on

*r**Ra*^{*}Heat Flux Rayleigh number

*r*Characteristic length scale of surface and the radius of cylinder

*r*_{bottom}Characteristic length scale of bottom surface

*r*_{top}Characteristic length scale of top surface

*s*Arm size of +- shaped and I-shaped heated surfaces

*S*Dimensionless distance between sides of square heated surfaces

*T*Temperature

*T*_{f}Fluid temperature far from surface

*T*_{w}Wall temperature

*T*_{0}Reference fluid temperature

*V*_{Gap}Dimensionless size of gap between surfaces,

*Gap*/*h**W*Dimensionless surface width,

*w*/*h**W*_{out}The dimensionless side lengths of the rectangular surrounding adiabatic surface

*W*_{out}=*w*_{out}/*w**w*Surface width

*w*_{out}Side lengths of the rectangular surrounding adiabatic surface

## Greek Symbols

*α*Thermal diffusivity

*β*Bulk coefficient of thermal expansion

*δ*Measure of the boundary layer thickness

*ϕ*Nanoparticle volumetric fraction

*ν*Kinematic viscosity

*ξ*1/(

*R Ra*^{0.25}) or 1/(*W Ra*^{0.25})*φ*Angle of inclination

*μ*Viscosity

*ρ*Density

*ρ*_{0}Reference density value

## References

- Ahmed SE, Abd El-Aziz MM (2013) Effect of local thermal non-equilibrium on unsteady heat transfer by natural convection of a nanofluid over a vertical wavy surface. Meccanica 48(1):33–43MathSciNetCrossRefGoogle Scholar
- Albets-Chico X, Oliva A, Perez-Segarra CD (2008) Numerical experiments in turbulent natural convection using two-equation eddy-viscosity models. J Heat Tran 130(7):072501–072501-11CrossRefGoogle Scholar
- Bejan A (1995) Convection heat transfer, 2nd edn. Wiley, New YorkMATHGoogle Scholar
- Bhavnani SH, Bergles AE (1990) Effect of surface geometry and orientation on laminar natural convection heat transfer from a vertical flat plate with transverse roughness elements. Int J Heat Mass Transf 33(5):965–981CrossRefGoogle Scholar
- Bhavnani SH, Bergles AE (1991) Natural convection heat transfer from sinusoidal wavy surfaces. Wärme-Stoffübertragung 26(6):341–349CrossRefGoogle Scholar
- Bredmose H, Madsen PA, Schäffer HA (2001) On the accuracy of Boussinesq evolution equations. In: Proceedings of the 27th international conference on coastal engineering (ICCE 2000) 276. p. 162–175, Sydney, Australia, 16–21 July 2000. https://doi.org/10.1061/40549(276)13
- Burmeister LC (1993) Convective heat transfer, 2nd edn. Wiley, New YorkGoogle Scholar
- Churchill SW, Chu HHS (1975) Correlating equations for laminar and turbulent free convection from a vertical plate. Int J Heat Mass Transf 18(11):1323–1329CrossRefGoogle Scholar
- Clausing AM, Berton JJ (1989) An experimental investigation of natural convection from an isothermal horizontal plate. J Heat Transf 111(4):904–908CrossRefGoogle Scholar
- Das SK, Choi SUS, Patel HE (2006) Heat transfer in nanofluids – a review. Heat Tran Eng 27(10):3–19CrossRefGoogle Scholar
- Ede AJ (1968) Advances in free convection. Adv Heat Tran 4:1–64Google Scholar
- Gebhart B (1973) Natural convection flows and stability. Adv Heat Tran 9:273–348CrossRefGoogle Scholar
- Gebhart B, Jaluria Y, Mahajan RL, Sammakia B (1988) Buoyancy-induced flows and transport. Hemisphere Publishing, Washington, DCMATHGoogle Scholar
- Gray DD, Giorgini A (1976) The validity of the Boussinesq approximation for liquids and gases. Int J Heat Mass Transf 19(5):545–551. https://doi.org/10.1016/0017-9310(76)90168-XCrossRefMATHGoogle Scholar
- Gryzagoridis J (1971) Natural convection from a vertical flat plate in the low Grashof number range. Int J Heat Mass Transf 14:162–164CrossRefGoogle Scholar
- Haddad Z, Oztop HF, Abu-Nada E, Mataoui A (2012) A review on natural convective heat transfer of nanofluids. Renew Sust Energ Rev 16(7):5363–5378CrossRefGoogle Scholar
- Husar RB, Sparrow EM (1968) Patterns of free convection flow adjacent to horizontal heated surfaces. Int J Heat Mass Transf 11(7):1206–1208CrossRefGoogle Scholar
- Jaluria Y (1980) Natural convection: heat and mass transfer. Pergamon Press, New YorkGoogle Scholar
- Kakaç S, Pramuanjaroenkij A (2009) Review of convective heat transfer enhancement with nanofluids. Int J Heat Mass Transf 52(13–14):3187–3196CrossRefGoogle Scholar
- Kakaç S, Yener Y (1995) Convective heat transfer, 2nd edn. CRC Press, Boca RatonGoogle Scholar
- Kalendar AY (2011) Numerical and experimental studies of natural convective heat transfer from vertical and inclined flat plates and short cylinders. PhD thesis, Queen’s UniversityGoogle Scholar
- Kalendar AY, Kalendar A, Karar S, Oosthuizen PH (2015) Correlation equations for natural convective heat transfer from two inclined vertically spaced narrow isothermal flat plates. J Heat Transf. https://doi.org/10.1115/1.4029594
- Kalendar AY, Oosthuizen PH (2008) Natural convective heat transfer from an inclined narrow isothermal flat plate. In: Proceedings of the ASME National heat transfer conference, heat transfer: Volume 1, Jacksonville, 10–14 Aug 2008Google Scholar
- Kalendar AY, Oosthuizen PH (2009) Natural convective heat transfer from two vertically spaced narrow isothermal flat plates. In: Proceedings of the 17th annual conference of the CFD society of Canada, 3–5 May 2009Google Scholar
- Kalendar AY, Oosthuizen PH (2010a) Experimental study of natural convective heat transfer from an inclined isothermal square cylinder with an exposed top surface mounted on a flat adiabatic base. In: Proceedings of the 14th international heat transfer conference, Washington, DC, 7–13 Aug 2010Google Scholar
- Kalendar AY, Oosthuizen PH (2010b) Natural convective heat transfer from two adjacent isothermal narrow vertical and inclined flat plates. JP J Heat Mass Transf 4(1):61–80MATHGoogle Scholar
- Kalendar AY, Oosthuizen PH (2010c) A numerical study of natural convective heat transfer from vertical and inclined narrow isothermal flat plates in the transition and turbulent flow regions. In: Proceedings of the 2010 CSME Forum, Victoria, 7–9 June 2010Google Scholar
- Kalendar AY, Oosthuizen PH (2011) Numerical and experimental studies of natural convective heat transfer from vertical and inclined narrow isothermal flat plates. Heat Mass Transf 47(9):1181–1195CrossRefGoogle Scholar
- Kalendar AY, Oosthuizen PH (2013) A numerical and experimental study of natural convective heat transfer from an inclined isothermal square cylinder with an exposed top surface. Heat Mass Transf 49(5):601–616. https://doi.org/10.1007/s00231-012-1106-7CrossRefGoogle Scholar
- Kalendar AY, Oosthuizen PH, Alhadhrami A (2011) Experimental study of natural convective heat transfer from an inclined isothermal cylinder with an exposed top surface mounted on a flat adiabatic base. In: Proceedings of the 8th international conference on heat transfer, fluid mechanics and thermodynamics (HEFAT2011), Pointe Aux Piments, Mauritius, 26 June–1 July 2011Google Scholar
- Kalendar AY, Oosthuizen PH, Kalandar B (2009) A numerical study of natural convective heat transfer from two adjacent narrow isothermal inclined flat plates. In: Proceedings of the ASME 2009 heat transfer summer conference (HT2009), San Francisco, 19–23 July 2009Google Scholar
- Kaviany M (1994) Principles of convective heat transfer. Springer, New YorkMATHGoogle Scholar
- Kobus CJ, Wedekind GL (1995) An experimental investigation into forced, natural and combined forced and natural convective heat transfer from stationary isothermal circular disks. Int J Heat Mass Transf 38(18):3329–3339CrossRefGoogle Scholar
- Kobus CJ, Wedekind GL (2001) An experimental investigation into natural convection heat transfer from horizontal isothermal circular disks. Int J Heat Mass Transf 44:3381–3384CrossRefGoogle Scholar
- Kobus CJ, Wedekind GL (2002) An empirical correlation for natural convection heat transfer from thin isothermal circular disks at arbitrary angles of inclination. Int J Heat Mass Transf 45:1159–1163CrossRefGoogle Scholar
- Lienhard VJH, Lloyd JR, Moran WR (1974) Natural convection adjacent to horizontal surface of various planforms. J Heat Transf 96:443–447CrossRefGoogle Scholar
- Lin S-J, Churchill SW (1978) Turbulent free convection from a vertical, isothermal plate. Num Heat Tran 1:129–145Google Scholar
- Mahajan RL, Gebhart B (1979) An experimental determination of transition limits in a vertical natural convection flow adjacent to a surface. J Fluid Mech 91(1):131–154CrossRefGoogle Scholar
- Mickle F, Marient ST (2009) Convective heat transfer. Wiley and ISTE, LondonGoogle Scholar
- Molla M, Hossain A, Yao L (2007) Natural-convection flow along a vertical complex wavy surface with uniform heat flux. ASME J Heat Transfer 129(10):1403–1407CrossRefGoogle Scholar
- Moulic SG, Yao LS (1989) Natural convection along a vertical wavy surface with uniform heat flux. ASME J Heat Transfer 111(4):1106–1108CrossRefGoogle Scholar
- Nikitin LV, Ryzhak EI (1981) Accuracy of the Boussinesq approximation for an incompressible fluid. Fluid Dyn 16(2):174–180. https://doi.org/10.1007/BF01090344MathSciNetCrossRefMATHGoogle Scholar
- Noto K, Matsumoto R (1975) Turbulent heat transfer by natural convection along an isothermal vertical plate surface. J Heat Transf 97(4):621–624CrossRefGoogle Scholar
- Oleg GM, Pavel PK (2005) Free-convective heat transfer. Springer, BerlinGoogle Scholar
- Oosthuizen PH (1964) A note on the transition point in a free convective boundary layer on an isothermal vertical plane surface. JSA Inst Mech Eng 13(10):265–268Google Scholar
- Oosthuizen PH (2007) Natural convective heat transfer from an isothermal vertical cylinder with an exposed upper surface mounted on a flat adiabatic base. In: Proceedings of the 2007 ASME international mechanical engineering congress and exposition, Seattle, 11–15 Nov 2007Google Scholar
- Oosthuizen PH (2008) Natural convective heat transfer from an isothermal vertical square cylinder mounted on a flat adiabatic base. In: Proceedings of the ASME national heat transfer conference, Jacksonville, 10–14 Aug 2008. Paper HT2008-56025, pp 499–505. https://doi.org/10.1115/HT2008-56025
- Oosthuizen PH (2010) A numerical study of laminar and turbulent natural convective heat transfer from an isothermal vertical plate with a wavy surface. In: Proceedings of the ASME 2010 international mechanical engineering congress and exposition, volume 7: fluid flow, heat transfer and thermal systems, Parts A and B, Vancouver, 12–18 Nov 2010Google Scholar
- Oosthuizen PH (2011) Natural convective heat transfer from an inclined isothermal plate with a wavy surface. In: Proceedings of the 42nd AIAA thermophysics conference, Honolulu, 27–30 June 2011Google Scholar
- Oosthuizen PH (2013a) A numerical study of natural convective heat transfer from isothermal high aspect ratio rectangular cylinders. In: Proceedings of the ASME summer heat transfer conference (HT 2013), Minnesota, Minneapolis, 14–19 July 2013Google Scholar
- Oosthuizen PH (2013b) Numerical study of natural convective heat transfer from a very short isothermal cylinder mounted on a flat adiabatic base. In: Proceedings of the 2013 ASME international mechanical engineering congress and exposition, San Diego, 15–21 Nov 2013Google Scholar
- Oosthuizen PH (2013c) Natural convective heat transfer from a short isothermal square cylinder mounted on a flat adiabatic base. In: Proceedings of the 21st annual conference of the CFD society of Canada, Sherbrooke, 6–9 May 2013Google Scholar
- Oosthuizen PH (2014a) Natural convective heat transfer from a horizontal rectangular isothermal element imbedded in a plane adiabatic surface with a parallel adiabatic covering surface. In: Proceedings of the 2014 ASME international mechanical engineering congress and exposition, Montreal, 14–20 Nov 2014Google Scholar
- Oosthuizen PH (2014b) Natural convective heat transfer from a horizontal isothermal circular element imbedded in a flat adiabatic surface with a parallel adiabatic covering surface In: Proceedings of the 11th AIAA/ASME joint thermophysics and heat transfer conference, Atlanta, 16–20 June 2014Google Scholar
- Oosthuizen PH (2014c) A numerical study of natural convective heat transfer from a horizontal isothermal square element imbedded in an adiabatic surface with a parallel adiabatic covering surface. In: Proceedings of the 10th international conference on heat transfer, fluid mechanics and thermodynamics (HEFAT 2014), Orlando, 14–16 July 2014Google Scholar
- Oosthuizen PH (2014d) Natural convective heat transfer from an inclined isothermal square flat element mounted in a flat adiabatic surrounding surface. In: Proceedings of the 15th international heat transfer conference (IHTC-15), Kyoto, 10–15 Aug 2014Google Scholar
- Oosthuizen PH (2015a) Laminar, transitional and turbulent natural convective heat transfer from a horizontal rectangular isothermal element imbedded in a flat adiabatic surrounding surface. In: Proceedings of the 6th International Symposium on Advances in Computational Heat Transfer (CHT-15), Rutgers University, Piscataway, 25–29 May 2015Google Scholar
- Oosthuizen PH (2015b) A numerical study of natural convective heat transfer from a horizontal isothermal square element with an unheated adiabatic inner section. In: Proceedings of the 11th international conference on heat transfer, fluid mechanics and thermodynamics, Skukuza, 20–23 July 2015Google Scholar
- Oosthuizen PH (2015c) A numerical study of natural convective heat transfer from horizontal isothermal heated elements of complex shape. In: Proceedings of the 1st thermal & fluids engineering summer conference (ASTFE), New York City, 9–12, Aug 2015Google Scholar
- Oosthuizen PH (2016a) A numerical study of the effect of a plane horizontal covering surface on natural convective heat transfer from a circular horizontal isothermal element that has an inner adiabatic circular section. In: Proceedings of the 24th annual conference of the CFD society of Canada, Kelowna, 26–29 June 2016Google Scholar
- Oosthuizen PH (2016b) Numerical study of natural convective heat transfer from horizontal heated elements of relatively complex shape that have a uniform surface heat flux. In: Proceedings of the 12th international conference on heat transfer, fluid mechanics and thermodynamics (HEFAT 2016), Malaga, 11–13 July 2016Google Scholar
- Oosthuizen PH (2016c) A numerical study of natural convective heat transfer from upward facing recessed and protruding heated horizontal isothermal circular surfaces. In: Proceedings of the 12th international conference on heat transfer, fluid mechanics and thermodynamics (HEFAT 2016), Malaga, 11–13 July 2016Google Scholar
- Oosthuizen PH (2016d) A numerical study of natural convective heat transfer from a horizontal isothermal surface with rectangular surface roughness elements. In: Proceedings of the 1st Pacific rim thermal engineering conference (PRTEC 2016), Hawaii, 13–17 Mar 2016Google Scholar
- Oosthuizen PH (2016e) A numerical study of the effect of triangular roughness elements on natural convective heat transfer from an upward facing heated horizontal isothermal surface. In: Proceedings of the ASME 2016 international mechanical engineering congress and exposition, 11–17 Nov 2016Google Scholar
- Oosthuizen PH (2016f) External natural convective heat transfer from bodies having a wavy surface for conditions under which laminar, transitional and turbulent flow can exist. In: Sparrow EM, Abraham JP, Gorman JM (eds) Advances in heat transfer. Elsevier, Oxford, pp 261–317Google Scholar
- Oosthuizen PH, Chow K (1986) An experimental study of free convective heat transfer from short cylinders with ‘wavy’ surfaces. In: Proceedings of the 8th international heat transfer conference, San Francisco, 17–22, Aug 1986Google Scholar
- Oosthuizen PH, Garrett M (2001a) A numerical study of natural convective heat transfer from an inclined plate with a ‘wavy surface’. In: Proceedings of the 2001 ASME national heat transfer conference, Anaheim, 10–12 June 2001Google Scholar
- Oosthuizen PH, M. Garrett (2001b) A numerical study of three-dimensional natural convective heat transfer from a plate with a ‘wavy’ surface. In: Proceedings of the 2001 ASME international. mechanical engineering congress & exposition, New York City, 11–16 Nov 2001Google Scholar
- Oosthuizen PH, Kalendar AY (2013) Natural convective heat transfer from narrow plates. In: Kulacki FA (ed) Springer briefs in applied sciences and technology, thermal engineering and applied science. Springer, New York. https://doi.org/10.1007/978-1-4614-5158-7CrossRefGoogle Scholar
- Oosthuizen PH, Kalendar AY (2014) Natural convective heat transfer from short inclined cylinders. In: Kulacki FA (ed) Springer briefs in applied sciences and technology, thermal engineering and applied science. Springer, New York. https://doi.org/10.1007/978-3-319-02459-2CrossRefGoogle Scholar
- Oosthuizen PH, Kalendar AY (2015a) A numerical study of natural convective heat transfer from a pair of adjacent horizontal isothermal square elements embedded in an adiabatic surface-effect of element spacing on heat transfer rate. In: Proceedings of the 11th international conference on heat transfer, fluid mechanics and thermodynamics (HEFAT 2015), Skukuza, 20–23 July 2015Google Scholar
- Oosthuizen PH, Kalendar AY (2015b) Laminar and turbulent natural convective heat transfer from a horizontal isothermal circular element with an unheated inner circular section. In: Proceedings of the CFD Society of Canada 23rd annual conference, Waterloo, 7–10 June 2015Google Scholar
- Oosthuizen PH, Kalendar AY (2016) A numerical study of the simultaneous natural convective heat transfer from the upper and lower surfaces of a thin isothermal horizontal circular plate. In: Proceedings of the 2016 ASME international mechanical engineering congress and exposition (IMECE2016), Phoenix, 11–17 Nov 2016Google Scholar
- Oosthuizen PH, Naylor D (1999) Introduction to convective heat transfer analysis. McGraw-Hill, New YorkGoogle Scholar
- Oosthuizen PH, Paul JT (2006) Natural convective heat transfer from a narrow isothermal vertical flat plate. In: Proceedings of the 9th AIAA/ASME joint thermophysics and heat transfer, San Francisco, 5–8 June 2006Google Scholar
- Oosthuizen PH, Paul JT (2007a) Natural convective heat transfer from a recessed narrow vertical flat plate with a uniform heat flux at the surface. In: Proceedings of the 5th international conference on heat transfer, fluid mechanics and thermodynamics (HEFAT2007), Sun City, South Africa, 1–4 July 2007Google Scholar
- Oosthuizen PH, Paul JT (2007b) Effect of edge conditions on natural convective heat transfer from a narrow vertical flat plate with a uniform surface heat flux. In: Proceedings of the ASME international mechanical engineering congress and exposition (IMECE2007) volume 8: heat transfer, fluid flows, and thermal systems, Parts A and B, Seattle, 11–15 Nov 2007Google Scholar
- Oosthuizen PH, Paul JT (2007c) Natural convective heat transfer from a narrow vertical isothermal flat plate with different edge conditions. In: Proceedings of the 15th annual meeting of the computational fluid dynamics Society of Canada, Toronto, 27–31 May 2007Google Scholar
- Oosthuizen PH, Paul JT (2007d) Natural convective heat transfer from a narrow isothermal vertical flat plate with a uniform heat flux at the surface. In: Proceedings of the 2007 ASME/JSME thermal engineering summer heat transfer conference (HT2007) Vancouver, 8–12 July 2007Google Scholar
- Oosthuizen PH, Paul JT (2010) Natural convective heat transfer from a narrow vertical flat plate with a uniform surface heat flux and with different plate edge conditions. Front Heat Mass Transf. https://doi.org/10.5098/hmt.v1.1.3006
- Oosthuizen PH, Paul JT (2011) A numerical study of natural convective heat transfer from an inclined isothermal plate having a square wave surface. In: Proceedings of the ASME 2011 international mechanical engineering congress and exposition, volume 10: heat and mass transport processes, parts A and B, Denver, 11–17 Nov 2011Google Scholar
- Oosthuizen PH, Paul JT (2012) A numerical study of natural convective heat transfer from an inclined isothermal plate with a ‘sinusoidally wavy’ surface. In: Proceedings of the 12th ICHMT international symposium on advances in computational heat transfer (CHT-12), Bath, 1–6 July 2012Google Scholar
- Pakravan HA, Yaghoubi M (2013) Analysis of nanoparticles migration on natural convective heat transfer of nanofluids. Int J Therm Sci 68:79–93CrossRefGoogle Scholar
- Plumb OA, Kennedy LA (1977) Application of a
*k-e*turbulence model to natural convection from a vertical isothermal surface. J Heat Transf 99(1):79–85CrossRefGoogle Scholar - Polidori G, Fohanno S, Nguyen CT (2007) A note on heat transfer modelling of Newtonian nanofluids in laminar free convection. Int J Therm Sci 46(8):739–744CrossRefGoogle Scholar
- Prétot S, Miriel J, Bailly Y, Zeghmati B (2003) Visualization and simulation of the natural-convection flow above horizontal wavy plates. Num Heat Transf Part A: Appl: Int J Comput Methodol 43(3):307–325CrossRefGoogle Scholar
- Prétot S, Zeghmati B, Caminat P (2000) Influence of surface roughness on natural convection above a horizontal plate. Adv Eng Softw 31(10):793–801CrossRefGoogle Scholar
- Rahman SU (2001) Natural convection along vertical wavy surfaces: an experimental study. Chem Eng J 84(3):587–591CrossRefGoogle Scholar
- Raithby GD, Hollands KGT (1985) Natural convection, Chapter 6. In: Rohsenow WM, Hartnett JP, Ganić EN (eds) Handbook of heat transfer fundamentals, 2nd edn. McGraw-Hill, New YorkGoogle Scholar
- Ravipati D (2008) Free convection along a vertical wavy surface in a nanofluid. MSc (Mechanical Engineering) thesis, Cleveland State UniversityGoogle Scholar
- Sahraoui M, Kaviany M, Marshall H (1990) Natural convection from horizontal disks and rings. J Heat Transf 112:110–116CrossRefGoogle Scholar
- Sarkar J (2011) A critical review on convective heat transfer correlations of nanofluids. Renew Sust Energ Rev 15(6):3271–3277CrossRefGoogle Scholar
- Savill AM (1993) Evaluating turbulence model predictions of transition. An ERCOFTAC special interest group project. Appl Sci Res 51(1–2):555–562CrossRefGoogle Scholar
- Schmidt RC, Patankar SV (1991) Simulating boundary layer transition with low-Reynolds-number
*k-ε*turbulence models: part 1- an evaluation of prediction characteristics. ASME J Turbomach 113(1):10–17CrossRefGoogle Scholar - Scott DA, Oosthuizen PH (2000) An experimental study of three-dimensional mixed convective heat transfer from short vertical cylinders in a horizontal forced flow. In: Proceedings of the IDMME’2000 and Canadian society for mechanical engineering (CSME) Forum 2000, Montreal, 6–19 May 2000Google Scholar
- Sergis A, Hardalupas Y (2011) Anomalous heat transfer modes of nanofluids: a review based on statistical analysis. Nanoscale Res Lett 6(1):1–37CrossRefGoogle Scholar
- Siddiqa S, Hossain MA (2013) Natural convection flow over wavy horizontal surface. Adv Mech Eng. https://doi.org/10.1155/2013/743034
- Siddiqa S, Hossain MA, Gorla RSR (2015) Natural convection flow of viscous fluid over triangular wavy horizontal surface. Comput Fluids 106(5):130–134MathSciNetCrossRefGoogle Scholar
- Sparrow EM (1955) Laminar free convection on a vertical plate with prescribed nonuniform wall heat flux or prescribed nonuniform wall temperature. National Advisory Committee for Aeronautics, Lewis Flight Propulsion Lab. NACA-TN-3508. Available via NASA Technical Notes Reports ServerGoogle Scholar
- Sparrow EM, Gregg JL (1956) Laminar free convection from a vertical flat plate with uniform surface heat flux. Trans ASME 78:435–440Google Scholar
- Sparrow EM, Gregg (1958) Similar solutions for free convection from a nonisothermal vertical plate. Trans ASME 80:379–386Google Scholar
- Szewczyk AA (1962) Stability and transition of the free convection layer along a vertical flat plate. Int J Heat Mass Transf 5:903–914CrossRefGoogle Scholar
- Terekhov VI, Kalinina SV, Lemanov VV (2010a) The mechanism of heat transfer in nanofluids: state of the art (review). Part 1. Synthesis and properties of nanofluids. Thermophys Aeromech 17(1):1–14CrossRefGoogle Scholar
- Terekhov VI, Kalinina SV, Lemanov VV (2010b) The mechanism of heat transfer in nanofluids: state of the art (review). Part 2. Convective heat transfer. Thermophys Aeromech 17(2):157–171CrossRefGoogle Scholar
- Tetsu F, Motoo F, Masanori T (1973) Influence of various surface roughness on the natural convection. Int J Heat Mass Transf 16(3:629–636CrossRefGoogle Scholar
- Vliet GC, Ross DC (1975) Turbulent natural convection on upward and downward facing inclined constant heat flux surfaces. J Heat Transf 97:549–554CrossRefGoogle Scholar
- Warner CY, Arpaci VS (1968) An experimental investigation of turbulent natural convection in air at low pressure along a vertical heated plate. Int J Heat Mass Transf 11:397–406CrossRefGoogle Scholar
- Wen DS, Ding YL (2005) Formulation of nanofluids for natural convective heat transfer applications. Int J Heat Fluid Flow 26(6):855–864CrossRefGoogle Scholar
- Wen DS, Ding YL (2006) Natural convective heat transfer of suspensions of titanium dioxide nanoparticles (nanofluids). IEEE Trans Nanotechnol 5(3):220–227CrossRefGoogle Scholar
- Yao LS (1983) Natural convection along a vertical wavy surface. ASME J Heat Transfer 105(3):465–468CrossRefGoogle Scholar
- Yao L-S (2006) Natural convection along a vertical complex wavy surface. Int J Heat Mass Transf 49(1–2):281–286CrossRefGoogle Scholar
- Yousef WW, Tarasuk JD, McKeen WJ (1982) Free convection heat transfer from upwardfacing isothermal horizontal surfaces. J Heat Transf 104:493–500CrossRefGoogle Scholar
- Zaraki A, Ghalambaz M, Chamkha AJ, Ghalambaz M, De Rossi D (2015) Theoretical analysis of natural convection boundary layer heat and mass transfer of nanofluids: effects of size, shape and type of nanoparticles, type of base fluid and working temperature. Adv Powder Technol 26(3):935–946CrossRefGoogle Scholar
- Zheng X, Liu C, Liu F, Yang C-I (1998) Turbulent transition simulation using the k-ω model. Int J Numer Methods Eng 42(5):907–926CrossRefGoogle Scholar