Abstract
Some observations on numerical predictions of temporally periodic fluid flow and heat transfer in spatially periodic geometries, in both spatially developing and fully developed regions, are presented and discussed in this chapter. Special attention is given to several issues that have not been fully resolved in earlier publications. The key points and ideas are demonstrated in the context of computationally convenient finite volume solutions of the mathematical models of two-dimensional, laminar, constant-property Newtonian fluid flow and forced convection heat transfer in uniform arrays of staggered rectangular plates. A dimensionless plate length of 1, dimensionless plate thicknesses of 1/4, 1/8, 1/12, and 1/16, time-mean Reynolds number values ranging from 100 to 1000, and a Prandtl number of 0.7 were considered. The simulations of developing fluid flow and heat transfer were conducted with calculation domains consisting of one row of ten consecutive geometric modules, followed by a plate-free exit zone of suitable length. Calculation domains consisting of single and multiple geometric modules were considered in simulations of fluid flow and heat transfer in the temporally and spatially periodic region. Findings of particular interest include the following: (1) multiple-module simulations of temporally and spatially periodic fluid flow and heat transfer yielded multiple solutions, but the absolute percentage differences in the corresponding values of time-mean modular friction and Colburn factors were all less than 6.4% and 5.1%, respectively; (2) in simulations of unsteady temporally periodic flows, the values of fully developed time-mean modular friction factor obtained from the predictions of the developing flow and the flow in a single module in the spatially periodic region differed by up to 22%; and (3) the instantaneous spatial periodicity conditions imposed in simulations of temporally periodic flows in single or multiple modules in the spatially periodic region are less restrictive than the boundary conditions employed in the corresponding simulations of developing flows, so the former yielded unsteady fluid flow over a wider range of Reynolds number.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Patankar, S. V., & Spalding, D. B. (1967). A finite-difference procedure for solving the equations of the two-dimensional boundary layer. International Journal of Heat and Mass Transfer, 10, 1389–1411.
Patankar, S. V., & Spalding, D. B. (1970). Heat and mass transfer in boundary layers (2nd ed.). London, UK: Intertext Books.
Runchal, A. K. (1972). Convergence and accuracy of three finite difference schemes for a two-dimensional conduction and convection problem. International Journal for Numerical Methods in Engineering, 4, 541–550.
Spalding, D. B. (1972). A novel finite difference formulation for differential expressions involving both first and second derivatives. International Journal for Numerical Methods in Engineering, 4, 551–559.
Patankar, S. V., & Spalding, D. B. (1972). A calculation procedure for heat, mass and momentum transfer in three-dimensional parabolic flows. International Journal of Heat and Mass Transfer, 15, 1787–1806.
Launder, B. E., & Spalding, D. B. (1974). The numerical computation of turbulent flows. Computer Methods in Applied Mechanics and Engineering, 3, 269–289.
Pratap, V. S., & Spalding, D. B. (1976). Fluid flow and heat transfer in three-dimensional duct flows. International Journal of Heat and Mass Transfer, 19, 1183–1188.
Pollard, A., & Spalding, D. B. (1978). The prediction of the three-dimensional turbulent flow field in a flow-splitting tee-junction. Computer Methods in Applied Mechanics and Engineering, 13, 293–306.
Patankar, S. V. (1980). Numerical heat transfer and fluid flow. New York: McGraw-Hill.
Patankar, S. V., Pollard, A., Singhal, A. K., & Vanka, S. P. (1983). Numerical prediction of flow, heat transfer, turbulence, and combustion—Selected works of professor D. Brian Spalding. New York: Pergamon Press.
Artemov, V., Beale, S. B., de Vahl Davis, G., Escudier, M. P., Fueyo, N., Launder, B. E., et al. (2009). A tribute to D.B. Spalding and his contributions in science and engineering. International Journal of Heat and Mass Transfer, 52, 3884–3905.
Runchal, A. K. (2009). Brian Spalding: CFD and reality—A personal recollection. International Journal of Heat and Mass Transfer, 52, 4063–4073.
Kakac, S., Bergles, A. E., & Mayinger, F. (1981). Heat exchangers—Thermal-hydraulic fundamentals and design. New York: McGraw-Hill.
Fraas, A. P. (1989). Heat exchanger design (2nd ed.). New York: Wiley.
Kays, W. M., & London, A. L. (1984). Compact heat exchangers (3rd ed.). Malabar, Florida: Kreiger Publishing Company.
Shah, R. K. (2006). Advances in science and technology of compact heat exchangers. Heat Transfer Engineering, 27, 3–22.
Han, J. C., Glicksman, L. R., & Rohsenow, W. M. (1978). An investigation of heat transfer and friction for rib-roughened surfaces. International Journal of Heat and Mass Transfer, 21, 1143–1156.
Acharya, S., Myrum, T., Qiu, X., & Sinha, S. (1997). Developing and periodically developed flow, temperature and heat transfer in a ribbed duct. International Journal of Heat and Mass Transfer, 40, 461–479.
Wang, L., & Sunden, B. (2007). Experimental investigation of local heat transfer in a square duct with various-shaped ribs. Heat Mass Transfer, 43, 759–766.
Liu, J., Xie, G., & Simon, T. W. (2015). Turbulent flow and heat transfer enhancement in rectangular channels with novel cylindrical grooves. International Journal of Heat and Mass Transfer, 81, 563–577.
Metwally, H. M., & Manglik, R. M. (2004). Enhanced heat transfer due to curvature-induced lateral vortices in laminar flows in sinusoidal corrugated-plate channels. International Journal of Heat and Mass Transfer, 47, 2283–2292.
Pedras, M. H. J., & de Lemos, M. J. S. (2001). Simulation of turbulent flow in porous media using a spatially periodic array and a low Re two-equation closure. Numerical Heat Transfer; Part A, 39, 35–59.
Sparrow, E. M., Baliga, B. R., & Patankar, S. V. (1977). Heat transfer and fluid flow analysis of interrupted-wall channels, with application to heat exchangers. ASME Journal of Heat Transfer, 99, 4–11.
Patankar, S. V., Liu, C. H., & Sparrow, E. M. (1977). Fully developed flow and heat transfer in ducts having streamwise-periodic variations of cross-sectional area. ASME Journal of Heat Transfer, 99, 180–186.
Acharya, S., Dutta, S., Myrum, T., & Baker, R. S. (1993). Periodically developed flow and heat transfer in a ribbed duct. International Journal of Heat and Mass Transfer, 36, 2069–2082.
Wang, G. V., & Vanka, S. (1995). Convective heat transfer in periodic wavy passages. International Journal of Heat and Mass Transfer, 38, 3219–3230.
Murthy, J. Y., & Mathur, S. (1997). Periodic flow and heat transfer using unstructured meshes. International Journal for Numerical Methods in Fluids, 25, 659–677.
Beale, S. B., & Spalding, D. B. (1998). Numerical study of fluid flow and heat transfer in tube banks with stream-wise periodic boundary conditions. Transactions CSME, 22, 397–416.
Beale, S. B., & Spalding, D. B. (1999). A numerical study of unsteady fluid flow in in-line and staggered tube banks. Journal of Fluids and Structures, 13, 723–754.
Bahaidarah, H. M. S., Anand, N. K., & Chen, H. C. (2005). A numerical study of fluid flow and heat transfer over a bank of flat tubes. Numerical Heat Transfer, Part A, 48, 359–385.
Bahaidarah, H. M. S., Ijaz, M., & Anand, N. K. (2006). Numerical study of fluid flow and heat transfer over a series of in-line noncircular tubes confined in a parallel-plate channel. Numerical Heat Transfer; Part B, 50, 97–119.
Fullerton, T. L., & Anand, N. K. (2010). Periodically fully-developed flow and heat transfer over flat and oval tubes using a control-volume finite element method. Numerical Heat Transfer; Part A, 57, 642–665.
Fullerton, T. L., & Anand, N. K. (2010). An alternative approach to study periodically fully-developed flow and heat transfer problems subject to isothermal heating conditions. International Journal of Engineering Science, 48, 1253–1262.
Hutter, C., Zenklusen, A., Kuhn, S., & von Rohr, P. R. (2011). Large eddy simulation of flow through a streamwise-periodic structure. Chemical Engineering Science, 66, 519–529.
Labbé, O. (2013). Large-eddy-simulation of flow and heat transfer in a ribbed duct. Computers & Fluids, 76, 23–32.
Iacovides, H., Launder, B., & West, A. (2014). A comparison and assessment of approaches for modelling flow over in-line tube banks. International Journal of Heat and Fluid Flow, 49, 69–79.
Ruck, S., & Arbeiter, F. (2018). Detached eddy simulation of turbulent flow and heat transfer in cooling channels roughened by variously shaped ribs on one wall. International Journal of Heat and Mass Transfer, 118, 388–401.
Hasis, F. B. A., Krishna, P. M., Aravind, G. P., Deepu, M., & Shine, S. R. (2018). Thermo hydraulic performance analysis of twisted sinusoidal wavy microchannels. International Journal of Thermal Sciences, 128, 124–136.
Dezan, D. J., Yanagihara, J. I., Jenovencio, G., & Salviano, L. O. (2019). Parametric investigation of heat transfer enhancement and pressure loss in louvered fins with longitudinal vortex generators. International Journal of Thermal Sciences, 135, 533–545.
London, A. L., & Shah, R. K. (1968). Offset rectangular plate-fin surfaces—Heat transfer and flow friction characteristics. ASME Journal of Engineering for Gas Turbines and Power, 90, 218–228.
Wieting, A. R. (1975). Empirical correlations for heat transfer and flow friction characteristics of rectangular offset-fin plate-fin heat exchangers. ASME Journal of Heat Transfer, 97, 488–490.
Cur, N., & Sparrow, E. M. (1979). Measurements of developing and fully developed heat transfer coefficients along a periodically interrupted surface. ASME Journal of Heat Transfer, 101, 211–216.
Patankar, S. V., & Prakash, C. (1981). An analysis of the effect of plate thickness on laminar flow and heat transfer in interrupted-plate passages. International Journal of Heat and Mass Transfer, 24, 1801–1810.
Mullisen, R. S., & Loehrke, R. I. (1986). A study of the flow mechanisms responsible for heat transfer enhancements in interrupted-plate heat exchangers. ASME Journal of Heat Transfer, 108, 377–385.
Joshi, H. M., & Webb, R. L. (1987). Heat transfer and friction in the offset strip-fin heat exchanger. International Journal of Heat and Mass Transfer, 30, 69–84.
McBrien, R. K., & Baliga, B. R. (1988). Module friction factors and intramodular pressure distributions for periodic fully developed turbulent flow in rectangular interrupted-plate ducts. ASME Journal of Fluids Engineering, 110, 147–154.
Kelkar, K. M., & Patankar, S. V. (1989). Numerical prediction of heat transfer and fluid flow in rectangular offset-fin arrays. Numerical Heat Transfer, Part A, 15, 149–164.
Amon, C. H., Majumdar, D., Herman, C. V., Mayinger, F., Mikic, B. B., & Sekulic, D. P. (1992). Numerical and experimental studies of self-sustained oscillatory flows in communicating channels. International Journal of Heat and Mass Transfer, 35, 3115–3129.
Suzuki, K., Xi, G. N., Inaoka, K., & Hagiwara, Y. (1994). Mechanism of heat transfer enhancement due to self-sustained oscillation for an in-line fin array. International Journal of Heat and Mass Transfer, 37, 83–96.
Grosse-Gorgemann, A., Weber, D., & Fiebig, M. (1995). Experimental and numerical investigation of self-sustained oscillations in channels with periodic structures. Experimental Thermal and Fluid Science, 11, 226–233.
Manglik, R. M., & Bergles, A. E. (1995). Heat transfer and pressure drop correlations for the rectangular offset strip fin compact heat exchanger. Experimental Thermal and Fluid Science, 10, 171–180.
Zhang, L. W., Balachandar, S., Tafti, D. K., & Najjar, F. M. (1997). Heat transfer enhancement mechanisms in inline and staggered parallel-plate fin heat exchangers. International Journal of Heat and Mass Transfer, 40, 2307–2325.
Tafti, D. K., Zhang, L. W., & Wang, G. (1999). Time-dependent calculation procedure for fully developed and developing flow and heat transfer in louvered fin geometries. Numerical Heat Transfer; Part A, 35, 225–249.
Shah, R. K., Heikal, M. R., Thonon, B., & Tochon, P. (2001). Progress in numerical analysis of compact heat exchanger surfaces. Advances in Heat Transfer, 34, 363–443.
Saidi, A., & Sundén, B. (2001). A numerical investigation of heat transfer enhancement in offset strip fin heat exchangers in self-sustained oscillatory flows. International Journal of Numerical Methods for Heat & Fluid Flow, 11, 699–717.
Muzychka, Y. S., & Yovanovich, M. M. (2001). Modeling the f and j characteristics for transverse flow through an offset strip fin at low Reynolds number. Enhanced Heat Transfer, 8, 243–259.
Candanedo, J. A., Aboumansour, E., & Baliga, B. R. (2003). Time-mean wall static pressure distributions and module friction factors for fully developed flows in a rectangular duct with spatially periodic interrupted-plate inserts. In Proceedings of the ASME International Mechanical Engineering Congress and Expo (IMECE 2003), Washington, D.C., Nov. 16–21 (pp. 1–10).
Lamoureux, A., Camargo, L., & Baliga, B. R. (2005). Strouhal numbers and power spectrums for turbulent fully-developed flows in rectangular ducts with spatially-periodic interrupted-plate inserts. In Proceedings of the Fifth International Conference on Enhanced, Compact and Ultra-Compact Heat Exchangers (CHE 2005), Whistler, B.C. (pp. 73–80).
Ismail, L. S., Ranganayakulu, C., & Shah, R. K. (2009). Numerical study of flow patterns of compact plate-fin heat exchangers and generation of design data for offset and wavy fins. International Journal of Heat and Mass Transfer, 52, 3972–3983.
Lamoureux, A., & Baliga, B. R. (2007). Temporally- and spatially-periodic laminar flow and heat transfer in staggered-plate arrays. In Proceedings of the ASME-JSME Thermal Engineering Heat Transfer Summer Conference (HT2007), Vancouver, B.C., Canada, July 8–12 (pp. 1–10).
Lamoureux, A., & Baliga, B. R. (2007). Temporally-periodic developing laminar flow and heat transfer in staggered-plate arrays. In Proceedings of the Sixth International Conference on Enhanced, Compact and Ultra-Compact Heat Exchangers (CHE 2007), Potsdam, Germany (pp. 306–313).
Benarji, N., Balaji, C., & Venkateshan, S. P. (2008). Unsteady fluid flow and heat transfer over a bank of flat tubes. Heat Mass Transfer, 44, 445–461.
Korichi, A., Oufer, L., & Polidori, G. (2009). Heat transfer enhancement in self-sustained oscillatory flow in a grooved channel with oblique plates. International Journal of Heat and Mass Transfer, 52, 1138–1148.
Sui, Y., Teo, C. J., & Lee, P. S. (2012). Direct numerical simulation of fluid flow and heat transfer in periodic wavy channels with rectangular cross-sections. International Journal of Heat and Mass Transfer, 55, 73–88.
Zheng, Z., Fletcher, D. F., & Haynes, B. S. (2014). Transient laminar heat transfer simulations in periodic zigzag channels. International Journal of Heat and Mass Transfer, 71, 758–768.
Ramgadia, A. G., & Saha, A. K. (2016). Numerical study of fully developed unsteady flow and heat transfer in asymmetric wavy channels. International Journal of Heat and Mass Transfer, 102, 98–112.
Sebben, S., & Baliga, B. R. (1996). A benchmark numerical solution involving steady, spatially-periodic, fully-developed laminar flow and heat transfer. In Current Topics in Computational Heat Transfer, ASME National Heat Transfer Conference, Houston, Texas, August 3–6.
Ghaddar, N. K., Korczak, K. Z., Mikic, B. B., & Patera, A. T. (1986). Numerical investigation of incompressible flow in grooved channels. Part 1. Stability and self-sustained oscillations. Journal of Fluid Mechanics, 163, 99–127.
Ghaddar, N. K., Magen, M., Mikic, B. B., & Patera, A. T. (1986). Numerical investigation of incompressible flow in grooved channels. Part 2. Resonance and oscillatory heat-transfer enhancement. Journal of Fluid Mechanics, 168, 541–567.
Sebben, S. (1996). Temporally and spatially periodic flows in interrupted-plate rectangular ducts. Ph.D. Thesis, Dept. Mech. Eng., McGill University, Montreal, Canada.
Stone, K., & Vanka, S. P. (1999). Numerical study of developing flow and heat transfer in a wavy passage. ASME Journal of Fluids Engineering, 121, 713–719.
Croce, G., & D’Agaro, P. (2006). Two-dimensional and three-dimensional self-sustained flow oscillations in interrupted fin heat exchangers. Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, 220, 297–307.
Schlichting, H. (1979). Boundary layer theory (3rd ed.). New York: McGraw-Hill.
Tritton, D. J. (1988). Physical fluid dynamics (2nd ed.). Oxford: Oxford Univ. Press.
Ferziger, J. H., & Perić, M. (2002). Computational methods for fluid dynamics (3rd ed.). New York: Springer.
Baliga, B. R., & Atabaki, N. (2006). Control-volume-based finite-difference and finite-element methods. In W. J. Minkowycz, E. M. Sparrow, & J. Y. Murthy (Eds.), Handbook of numerical heat transfer, Chapter 6 (2nd ed., pp. 191–224). New York: Wiley.
Leonard, B. P. (1979). A stable and accurate convective modelling procedure based on quadratic upstream interpolation. Computer Methods in Applied Mechanics and Engineering, 19, 59–98.
Rhie, C. M., & Chow, W. L. (1983). Numerical study of the turbulent flow past an airfoil with trailing edge separation. AIAA Journal, 21, 1525–1532.
Saabas, H. J., & Baliga, B. R. (1994). A co-located equal-order control-volume finite element method for multidimensional, incompressible fluid flow, Part I: Formulation. Numerical Heat Transfer, Part B, 26, 381–407.
Sebben, S., & Baliga, B. R. (1995). Some extensions of tridiagonal and pentadiagonal matrix algorithms. Numerical Heat Transfer, Part B, 28, 323–351.
Saad, Y. (2003). Iterative methods for sparse linear systems (2nd ed.). Philadelphia: Society for Industrial and Applied Mathematics (SIAM).
Richardson, L. F. (1910). The approximate arithmetical solution by finite differences of physical problems involving differential equations with application to a masonry dam. Transactions of the Royal Society of London, Series A, 210, 307–357.
Cotta, R. M. (1994). Benchmark results in computational heat and fluid flow: The integral transform method. International Journal of Heat and Mass Transfer, 37, 381–393.
Roache, P. J. (2002). Code verification by the method of manufactured solutions. ASME Journal of Fluids Engineering, 124, 4–10.
Cotta, R. M., & Mikhailov, M. D. (2006). Hybrid methods and symbolic computations. In W. J. Minkowycz, E. M. Sparrow, & J. Y. Murthy (Eds.), Handbook of numerical heat transfer, Chapter 16 (2nd ed., pp. 493–522). New York: Wiley.
Oberkampf, W. L., & Roy, C. J. (2010). Verification and validation in scientific computing. Cambridge, U.K: Cambridge University Press.
Baliga, B. R., & Lokhmanets, I. (2016). Generalized Richardson extrapolation procedures for estimating grid-independent numerical solutions. International Journal of Numerical Methods for Heat & Fluid Flow, 26, 1121–1144.
Franke, R., Rodi, W., & Schönung, B. (1990). Numerical calculation of laminar vortex-shedding flow past cylinders. Journal of Wind Engineering and Industrial Aerodynamics, 35, 237–257.
Davis, R. W., & Moore, E. F. (1982). A numerical study of vortex shedding from rectangles. Journal of Fluid Mechanics, 116, 475–506.
Lamoureux, A. (2006). Oscillatory flows in periodically interrupted rectangular passages in heat exchangers. M. Eng. thesis, Dept. Mech. Eng., McGill University, Montreal, Canada.
Acknowledgements
Financial support of this work by the Natural Sciences and Engineering Research Council of Canada (NSERC) and the Fonds québécois de la recherche sur la nature et les technologies (FQRNT) is gratefully acknowledged by both authors. The second author (on behalf of all his former students, current students, and himself) would also like to express his deep gratitude and admiration for Professor D. B. Spalding for his numerous inspiring, lasting, and peerless contributions to the subject of computational fluid dynamics and heat transfer.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2020 Springer Nature Singapore Pte Ltd.
About this chapter
Cite this chapter
Lamoureux, A., (Rabi) Baliga, B.R. (2020). Numerical Predictions of Temporally Periodic Fluid Flow and Heat Transfer in Spatially Periodic Geometries. In: Runchal, A. (eds) 50 Years of CFD in Engineering Sciences. Springer, Singapore. https://doi.org/10.1007/978-981-15-2670-1_6
Download citation
DOI: https://doi.org/10.1007/978-981-15-2670-1_6
Published:
Publisher Name: Springer, Singapore
Print ISBN: 978-981-15-2669-5
Online ISBN: 978-981-15-2670-1
eBook Packages: EngineeringEngineering (R0)