Numerical solution of the heat transfer problem in a short channel with backward-facing step

A. A. Fomin   ; L. N. Fomina

Vestnik Udmurtskogo universiteta. Matematika, mehanika, kompʹûternye nauki, Tome 27 (2017) no. 3, pp. 431-449

Résumé

A test problem of the laminar steady incompressible flow and heat transfer over backward-facing step in a $\mathrm{2D}$ short channel is presented. The focus of the study is on the changes in heat transfer characteristics of the flow field inside the channel due to different boundary conditions for heat flux at the outflow border of the domain. The Navier–Stokes equations in a velocity-pressure formulation and energy equation are numerically solved using a uniform grid of $6001\times301$ points. The control-volume technique for the second-order difference approximation for spatial derivatives is used. The solutions were validated for a wide range of Reynolds numbers $(100 \leq \text{Re} \leq 1000)$ and Prandtl number $\text{Pr} = 0.71$, comparing them to experimental and numerical results found in the literature. The isotherm patterns and behaviors of Nusselt number along the heated bottom wall of the channel are examined. The study results showed that a condition for the heat flow (temperature) at the outlet border can influence the heat transfer in the whole domain. The nonlinear boundary condition for temperature at the outflow border is claimed as the best.

Détail

Keywords: incompressible fluid flow, heat transfer, outflow boundary condition.

Bibliographie

[1] Roache P. J., Computational fluid dynamics, Hermosa Publs., Albuquerque, 1976, 446 pp. ; Rouch P., Vychislitelnaya gidrodinamika, Mir, M., 1980, 616 pp. | MR

[2] Orlanski I., “A simple boundary condition for unbounded hyperbolic flows”, J. Comput. Phys., 21:3 (1976), 251–269 | DOI | Zbl

[3] Camerlengo A. L., O'Brien J. J., “Open boundary conditions in rotating fluids”, J. Comput. Phys., 35:1 (1980), 12–35 | DOI | MR | Zbl

[4] Han T. Y., Meng J. C. S., Innis G. E., “An open boundary condition for incompressible stratified flows”, J. Comput. Phys., 49:2 (1983), 276–297 | DOI | Zbl

[5] Sani R. L., Gresho P. M., “Resume and remarks on the open boundary condition minisymposium”, Internat. J. Numer. Methods Fluids, 18:10 (1994), 983–1008 | DOI | MR | Zbl

[6] Ol'shanskii M. A., Staroverov V. M., “On simulation of outflow boundary conditions in finite difference calculations for incompressible fluid”, Internat. J. Numer. Methods Fluids, 33:4 (2000), 499–534 | 3.0.CO;2-7 class='badge bg-secondary rounded-pill ref-badge extid-badge'>DOI | Zbl

[7] Blayo E., Debreu L., “Revisiting open boundary conditions from the point of view of characteristic variables”, Ocean Modelling, 9:3 (2005), 231–252 | DOI

[8] Dong S., Karniadakis G. E., Chryssostomidis C., “A robust and accurate outflow boundary condition for incompressible flow simulations on severely-truncated unbounded domains”, J. Comput. Phys., 261 (2014), 83–105 | DOI | MR | Zbl

[9] Kawamura T., Tanaka S., Mabuchi I., Kumada M., “Temporal and spatial characteristics of heat transfer at the reattachment region of a backward-facing step”, Experimental Heat Transfer, 1:4 (1987), 299–313 | DOI

[10] Sparrow E. M., Kang S. S., Chuck W., “Relation between the points of flow reattachment and maximum heat transfer for regions of flow separation”, International Journal of Heat and Mass Transfer, 30:7 (1987), 1237–1246 | DOI

[11] Nakamura H., Takaki S., Yamada S., “Spatio-temporal characteristics of heat transfer in separated and reattaching flows”, Proc. ASME-JSME-KSME 2011 Joint Fluids Engineering Conference, Symposia – Parts A, B, C, and D, v. 1, ASME, 2011, 3977–3988 | DOI

[12] Nakamura H., Yamada S., “Quantitative evaluation of spatio-temporal heat transfer to a turbulent air flow using a heated thin-foil”, International Journal of Heat and Mass Transfer, 64 (2013), 892–902 | DOI

[13] Aung W., Baron A., Tsou F.-K., “Wall independency and effect of initial shear-layer thickness in separated flow and heat transfer”, International Journal of Heat and Mass Transfer, 28:9 (1985), 1757–1771 | DOI

[14] Kondoh T., Nagano Y., Tsuji T., “Computational study of laminar heat transfer downstream of a backward-facing step”, International Journal of Heat and Mass Transfer, 36:3 (1993), 577–591 | DOI | Zbl

[15] Valencia A., Hinojosa L., “Numerical solutions of pulsating flow and heat transfer characteristics in a channel with a backward-facing step”, Heat Mass Transf., 32:3 (1997), 143–148 | DOI | MR

[16] Kanna P. R., Das M. K., “Conjugate heat transfer study of a two-dimensional laminar incompressible wall jet over a backward-facing step”, Journal of Heat Transfer, 129:2 (2006), 220–231 | DOI | MR

[17] Tihon J., Pěnkavová V., Havlica J., Šimčík M., “The transitional backward-facing step flow in a water channel with variable expansion geometry”, Experimental Thermal and Fluid Science, 40 (2012), 112–125 | DOI

[18] Yang Y. T., Huang M. L., “Numerical studies of heat transfer characteristics by using jet discharge at downstream of a backward-facing step”, Acta Mech., 128:1–2 (1998), 29–37 | DOI | Zbl

[19] Tsay Y.-L., Chang T. S., Cheng J. C., “Heat tranfer enhancement of backward-facing step flow in a channel by using baffle installation on the channel wall”, Acta Mech., 174:1–2 (2005), 63–76 | DOI | Zbl

[20] Hong B., Armaly B. F., Chen T. S., “Laminar mixed convection in a duct with a backward-facing step: the effects of inclination angle and Prandtl number”, International Journal of Heat and Mass Transfer, 36:12 (1993), 3059–3067 | DOI | Zbl

[21] Abu-Hijleh B., “Convection heat transfer from a laminar flow over a 2-D backward facing step with asymmetric and orthotropic porous floor segments”, Numerical Heat Transfer. Part A: Applications, 31:3 (1997), 325–335 | DOI

[22] Batenko S. R., Terekhov V. I., “Friction and heat transfer in a laminar separated flow behind a rectangular step with porous injection or suction”, J. Appl. Mech. Tech. Phys., 47:1 (2006), 12–21 | DOI | MR | Zbl

[23] Abu-Nada E., Al-Sarkhi A., Akash B., Al-Hinti I., “Heat transfer and fluid flow characteristics of separated flows encountered in a backward-facing step under the effect of suction and blowing”, Journal of Heat Transfer, 129:11 (2007), 1517–1528 | DOI

[24] Saha S., Mamun A. H., Hossain Z., Islam S., “Mixed convection in an enclosure with different inlet and exit configurations”, Journal of Applied Fluid Mechanics, 1:1 (2008), 78–93

[25] Teruel F. E., Fogliatto E., “Numerical simulations of flow, heat transfer and conjugate heat transfer in the backward-facing step geometry”, Mecánica Computacional, 32:39 (2013), 3265–3278

[26] Gada V. H., Datta D., Sharma A., “Analytical and numerical study for two-phase stratified-flow in a plane channel subjected to different thermal boundary conditions”, International Journal of Thermal Sciences, 71 (2013), 88–102 | DOI

[27] Mitsoulis E., Malamataris N. A., “Free (open) boundary condition: some experiences with viscous flow simulation”, Internat. J. Numer. Methods Fluids, 68:10 (2012), 1299–1323 | DOI | MR | Zbl

[28] Papanastasiou T. C., Malamataris N., Ellwood K., “A new outflow boundary condition”, Internat. J. Numer. Methods Fluids, 14:5 (1992), 587–608 | DOI | MR | Zbl

[29] Dimakopoulos Y., Karapetsas G., Malamataris N. A., Mitsoulis E., “The free (open) boundary condition at inflow boundaries”, J. Non-Newton. Fluid Mech., 187–188 (2012), 16–31 | DOI | MR

[30] Patankar S. V., Numerical heat transfer and fluid flow, Hemisphere Publishing Corporation, New York, 1980, 197 pp. ; Patankar S., Chislennye metody resheniya zadach teploobmena i dinamiki zhidkosti, Energoatomizdat, M., 1984, 152 pp. | Zbl

[31] Belotserkovskii O. M., Gushchin V. A., Shchennikov V. V., “Use of the splitting method to solve problems of the dynamics of a viscous incompressible fluid”, USSR Computational Mathematics and Mathematical Physics, 15:1 (1975), 190–200 | DOI | MR

[32] Fomin A. A., Fomina L. N., “Numerical simulation of viscous incompressible fluid in a short plane channel with backward-facing step”, Matem. Mod., 28:5 (2016), 32–46 (in Russian) | MR | Zbl

[33] Fomin A. A., Fomina L. N., “Acceleration of the line-by-line recurrent method in Krylov subspaces”, Vestn. Tomsk. Gos. Univ. Mat. Mekh., 2011, no. 2 (14), 45–54 (in Russian)

[34] Armaly B. F., Durst F., Pereira J. C. F., Schönung B., “Experimental and theoretical investigation of backward-facing step flow”, J. Fluid Mech., 127 (1983), 473–496 | DOI

[35] Chiang T. P., Sheu T. W. H., Fang C. C., “Numerical investigation of vortical evolution in a backward-facing step expansion flow”, Appl. Math. Model., 23:12 (1999), 915–932 | DOI | Zbl

[36] Erturk E., “Numerical solutions of 2-D steady incompressible flow over a backward-facing step, Part I: High Reynolds number solutions”, Comput. Fluids, 37 (2008), 6 633–655 | DOI | Zbl

[37] Rogers S. E., Kwak D., “An upwind differencing scheme for the incompressible Navier–Stokes equations”, Appl. Numer. Math., 8:1 (1991), 43–64 | DOI | MR | Zbl

[38] Lewis R. W., Nithiarasu P., Seetharamu K. N., Fundamentals of the finite element method for heat and fluid flow, John Wiley Sons, 2004, xiv+341 pp. | DOI

[39] Cruchaga M. A., “A study of the backward-facing step problem using a generalized streamline formulation”, Communications in Numerical Methods in Engineering, 14:8 (1998), 697–708 | 3.0.CO;2-0 class='badge bg-secondary rounded-pill ref-badge extid-badge'>DOI | Zbl

[40] Røed L. P., Smedstad O. M., “Open boundary conditions for forced waves in a rotating fluid”, SIAM Journal on Scientific and Statistical Computing, 5:2 (1984), 414–426 | DOI | MR | Zbl