zbMATH — the first resource for mathematics

Numerical investigation of double-diffusive convection in rectangular cavities with different aspect ratio. I: High-accuracy numerical method. (English) Zbl 07351739
Summary: In this paper, being based on the idea of dispersion-relation-preserving optimization, a class of high-order high-resolution upwind compact schemes with a free parameter and their consistent boundary scheme are proposed for the solution of the governing equations of the 2D double-diffusive convection. The first derivative and the second derivative, in the vorticity equation, the temperature equation and the concentration equation, are discretized by using the optimal upwind compact scheme on a uniform mesh and the fourth-order symmetrical Padé compact scheme, respectively. The pressure Poisson equation is approximated using a fourth-order compact difference scheme constructed currently on the nine-point two dimensional stencil. The fourth-order Runge-Kutta method is utilized for the temporal discretization. To assess numerical capability of the proposed algorithm, particularly its spatial behavior, the problems about the convection diffusion equation, the nonlinear Burgers equation and the nature convection flows in the square cavity with adiabatic horizontal walls and differentially heated vertical walls are computed by using the newly proposed algorithm. The numerical results are in excellent agreement with the benchmark solutions and some of the accurate results available in the literature, and show that effectiveness, accuracy and the advantage of better resolution of the present method. After that, steady and unsteady solutions for the double-diffusive convection in a rectangular cavity are also used to assess the efficiency of the present algorithm. The period of oscillation and flow field profiles are in great agreement with the data in the literature for buoyancy ratio, \(\lambda=1\). The typical separation and secondary vortices at the bottom corner of the cavity as well as the top corner can be captured well for various buoyancy ratio. These indicate that the present method is suitable for simulating effectively the unsteady double-diffusive convection.
76-XX Fluid mechanics
65-XX Numerical analysis
Full Text: DOI
[1] Stern, M. E., The salt-fountain and thermohaline convection, Tellus, 12, 172-175 (1960)
[2] Herbert Huppert, E.; Daniel, R. M., Nonlinear double-diffusive convection, J. Fluid Mech., 78, 821-854 (1976) · Zbl 0353.76028
[3] Spiegel, E. A., Convection in stars II: special effects, Annu. Rev. Astron. Astrophys., 10, 261-304 (1972)
[4] Turner, J. S., Double-diffusive phenomena, Annu. Rev. Fluid Mech., 6, 37-54 (1974) · Zbl 0312.76028
[5] Graham Hughes, O.; Ross, W. G., Horizontal convection, Annu. Rev. Fluid Mech., 40, 185-208 (2008) · Zbl 1136.76046
[6] Arpino, F.; Massarotti, N.; Mauro, A., Artificial compressibility based CBS solutions for double diffusive natural convection in cavities, Int. J. Numer. Methods Heat, 23, 205-225 (2013) · Zbl 1356.76307
[7] Fakher Brahim, O.; Taieb, L., Double-diffusive natural convection and entropy generation in an enclosure of aspect ratio 4 with partial vertical heating and salting sources, Alex. Eng. J., 52, 605-625 (2013)
[8] Yang, Y. T.; Roberto, V.; Detlef, L., From convection rolls to finger convection in double-diffusive turbulence, J. Comput. Phys., 113, 69-73 (2016)
[9] Rodolfo, O. M.; Yang, Y. T.; Erwin, P. V.D. P., A multiple-resolution strategy for direct numerical simulation of scalar turbulence, J. Comput. Phys., 301, 308-321 (2015) · Zbl 1349.76136
[10] Yang, Y. T.; Erwin, P. V.D. P.; Rodolfo, O. M., Salinity transfer in bounded double diffusive convection, J. Fluid Mech., 768, 476-491 (2015)
[11] Ababaei, A.; Abbaszadeh, M.; Arefmanesh, A., Numerical simulation of double-diffusive mixed convection and entropy generation in a lid-driven trapezoidal enclosure with a heat source, Numer. Heat Transf., Part A, Appl., 1-19 (2018)
[12] Moolya, S.; Satheesh, A., Role of magnetic field and cavity inclination on double diffusive mixed convection in rectangular enclosed domain, Int. Commun. Heat Mass, 118, Article 104814 pp. (2020)
[13] Bihiche, K.; Lamsaadi, M.; Hasnaoui, M., Multiple steady state solutions for double-diffusive convection in a shallow horizontal rectangular cavity uniformly heated and salted from the side and filled with non-Newtonian power-law fluids, J. Non-Newton. Fluid, 283, Article 104349 pp. (2020)
[14] Chen, C. K.; Zhang, X. H.; Liu, Z., A high-order compact finite difference scheme and precise integration method based on modified Hopf-Cole transformation for numerical simulation of n-dimensional Burgers’ system, Appl. Math. Comput., 372, 1-28 (2020) · Zbl 1433.65156
[15] Pradip, R.; Goura, V. M.K. P., A new higher order compact finite difference method for generalised blackcscholes partial differential equation: European call option, J. Comput. Appl. Math., 363, 464-484 (2020) · Zbl 1418.91602
[16] Sun, Y. X.; Tian, Z. F., High-order upwind compact finite-difference lattice Boltzmann method for viscous incompressible flows, Comput. Math. Appl., 80, 1858-1872 (2020) · Zbl 1451.76086
[17] Yu, P. X.; Tian, Z. F., An upwind compact difference scheme for solving the streamfunction-velocity formulation of the unsteady incompressible Navier-Stokes equation, Comput. Math. Appl., 75, 3224-3243 (2018) · Zbl 1409.76093
[18] Yang, X. J.; Ge, Y. B.; Zhang, L., A class of high-order compact difference schemes for solving the Burgers’ equations, Appl. Math. Comput., 358, 394-417 (2019) · Zbl 1429.65204
[19] Tian, Z. F.; Yu, P. X., A high-order exponential scheme for solving 1D unsteady convection-diffusion equations, J. Comput. Appl. Math., 235, 2477-2491 (2011) · Zbl 1209.65090
[20] Lele, S. K., Compact finite difference schemes with spectral-like resolution, J. Comput. Phys., 103, 16-42 (1992) · Zbl 0759.65006
[21] Peter, C. C.; Fan, C. W., A three-point combined compact difference scheme, J. Comput. Phys., 6, 370-399 (1998) · Zbl 0923.65071
[22] Rao, S. Chandra Sekhara; Kumar, M., An almost fourth order parameter robust numerical method for a linear system of \((M \geq 2)\) coupled singularly perturbed reaction-diffusion problems, Int. J. Numer. Anal. Model., 10, 603-621 (2013) · Zbl 1281.65102
[23] Liu, X. L.; Zhang, S. H.; Zhang, H. X., A new class of central compact schemes with spectral-like resolution i: linear schemes, J. Comput. Phys., 248, 235-256 (2013) · Zbl 1349.76504
[24] Wang, Z. K.; Li, J. Y.; Wang, B. F., A new class of central compact finite-difference scheme with high spectral resolution for acoustic wave equation, J. Comput. Phys., 366, 191-206 (2018) · Zbl 1406.65066
[25] Liu, X. L.; Zhang, S. H.; Zhang, H. X., A new class of central compact schemes with spectral-like resolution ii: hybrid weighted nonlinear schemes, J. Comput. Phys., 284, 133-154 (2015) · Zbl 1351.76170
[26] Roul, P.; Gouraa, V. P.; Agarwal, R., A compact finite difference method for a general class of nonlinear singular boundary value problems with Neumann and Robin boundary conditions, Appl. Math. Comput., 350, 283-304 (2019) · Zbl 1429.65165
[27] Fu, D. X.; Ma, Y. W., A high order accurate difference scheme for complex flow fields, J. Comput. Phys., 134, 1-15 (1997) · Zbl 0882.76054
[28] Tian, Z. F.; Li, Y. A., Numerical solution of the incompressible Navier-Stokes equations with a three-point fourth-order upwind compact difference schemes, (Proceedings of Fourth International Conference Nonlinear Mechanic (2002)), 942-948
[29] Chiu, P. H.; Sheu, T. W.H., On the development of a dispersion-relation-preserving dual-compact upwind scheme for convection-diffusion equation, J. Comput. Phys., 228, 3640-3655 (2009) · Zbl 1166.65391
[30] Yu, C.; Bhumkar, Y. G.; Sheu, T. W., Dispersion relation preserving combined compact difference schemes for flow problems, J. Sci. Comput., 62, 482-516 (2015) · Zbl 1326.76081
[31] Zhao, L.; Yu, C. H.; He, Z., Numerical modeling of lock-exchange gravity/turbidity currents by a high-order upwinding combined compact difference scheme, Int. J. Sediment Res., 34, 240-250 (2019)
[32] Qin, Q.; Xia, Z. A.; Tian, Z. F., High accuracy numerical investigation of double-diffusive convection in a rectangular enclosure with horizontal temperature and concentration gradients, Int. J. Heat Mass Transf., 71, 405-423 (2014)
[33] Weinan, E.; Liu, J. G., Vorticity boundary condition and related issues for finite difference schemes, J. Comput. Phys., 124, 368-382 (1996) · Zbl 0847.76050
[34] Wang, T.; Liu, T. G., A consistent fourth-order compact finite difference scheme for solving vorticity-stream function form of incompressible Navier-Stokes equations, Numer. Math., Theory Methods, 12, 312-330 (2019) · Zbl 1438.65273
[35] Shu, C. W.; Osher, S., Efficient implementation of essentially non-oscillatory shock capturing schemes, J. Comput. Phys., 2, 439-471 (1988) · Zbl 0653.65072
[36] Spotz, W. F., Accuracy and performance of numerical wall boundary conditions for steady, 2D, incompressible streamfunction vorticity, Int. J. Numer. Methods Fluids, 28, 737-757 (1998) · Zbl 0930.76059
[37] Zhao, B. X.; Tian, Z. F., High-resolution high-order upwind compact scheme-based numerical computation of natural convection flows in a square cavity, Int. J. Heat Mass Transf., 98, 313-328 (2016)
[38] De, A. K.; Eswaran, V., Analysis of a new high resolution upwind compact scheme, J. Comput. Phys., 218, 398-416 (2006) · Zbl 1103.65092
[39] Kim, J. W.; Lee, D. J., Optimized compact finite difference schemes with maximum resolution, AIAA J., 34, 887-893 (1996) · Zbl 0900.76317
[40] Gottlieb, S.; Shu, C. W., Total variation diminishing Runge-Kutta schemes, Math. Comput., 67, 73-85 (1998) · Zbl 0897.65058
[41] Davis, G. D.V., Natural convection of air in a square cavity, a benchmark numerical solution, Int. J. Numer. Methods Fluids, 3, 249-264 (1983) · Zbl 0538.76075
[42] Tian, Z. F.; Ge, Y. B., A fourth-order compact finite difference scheme for the steady stream function-vorticity formulation of the Navier-Stokes/Boussinesq equations, Int. J. Numer. Methods Fluids, 41, 495-518 (2003) · Zbl 1038.76029
[43] Yu, P. X.; Tian, Z. F., Compact computations based on a stream-function-velocity formulation of two-dimensional steady laminar natural convection in a square cavity, Phys. Rev. E, 85, Article 36703 pp. (2012)
[44] Tian, Z. F.; Liang, X.; Yu, P. X., A higher order compact finite difference algorithm for solving the incompressible Navier-Stokes equations, Int. J. Numer. Methods Eng., 88, 6, 511-532 (2011) · Zbl 1242.76216
[45] Quéré, P. L., Accurate solutions to the square thermally driven cavity at high Rayleigh number, Comput. Fluids, 20, 29-41 (1991) · Zbl 0731.76054
[46] Arpino, F.; Massarotti, N.; Mauro, A., High Rayleigh number laminar-free convection in cavities: new benchmark solutions, Numer. Heat Transf., Part B, Fundam., 58, 73-97 (2010)
[47] Nonino, C.; Croce, G., An equal-order velocity-pressure algorithm for incompressible thermal flows, part 2: validation, Numer. Heat Transf., Part B, Fundam., 32, 17-35 (1997)
[48] Morega, M. A.; Tatsuo, Double diffusive convection by a Chebyshev collocation method, Technol. Rep. Yamaguchi Univ., 5, 259-276 (1996)
[49] Nishimura, T.; Wakamatsu, M.; Morega, A. M., Oscillatory double-diffusive convection in a rectangular enclosure with combined horizontal temperature and concentration gradients, Int. J. Heat Mass Transf., 41, 1601-1611 (1998) · Zbl 0962.76616
[50] Liang, X.; Li, X. L.; Fu, D. X., Complex transition of double-diffusive convection in a rectangular enclosure with height-to-length ratio equal to 4: part I, Commun. Comput. Phys., 6, 247-268 (2009)
[51] Chen, S.; Tölke, J.; Krafczyk, M., Numerical investigation of double-diffusive (natural) convection in vertical annuluses with opposing temperature and concentration gradients, Int. J. Heat Fluid Flow, 31, 217-226 (2010)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.