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A composite grid solver for conjugate heat transfer in fluid-structure systems. (English) Zbl 1396.80006

Summary: We describe a numerical method for modeling temperature-dependent fluid flow coupled to heat transfer in solids. This approach to conjugate heat transfer can be used to compute transient and steady state solutions to a wide range of fluid-solid systems in complex two- and three-dimensional geometry. Fluids are modeled with the temperature-dependent incompressible Navier-Stokes equations using the Boussinesq approximation. Solids with heat transfer are modeled with the heat equation. Appropriate interface equations are applied to couple the solutions across different domains. The computational region is divided into a number of sub-domains corresponding to fluid domains and solid domains. There may be multiple fluid domains and multiple solid domains. Each fluid or solid sub-domain is discretized with an overlapping grid. The entire region is associated with a composite grid which is the union of the overlapping grids for the sub-domains. A different physics solver (fluid solver or solid solver) is associated with each sub-domain. A higher-level multi-domain solver manages the entire solution process.
We propose and analyze some centered discrete approximations to the interface equations that have some desirable stability properties. The coupled interface equations may be solved directly when using explicit time-stepping methods in the sub-domains, resulting in a strongly coupled approach. The stability of the interface treatment in this case is independent of the relative sizes of the material properties in the two domains with the time-step only depending on the usual von Neumann conditions for each sub-domain. For implicit time-stepping methods we solve the interface equations in a weakly coupled fashion to avoid forming a coupled implicit system across all sub-domains. The convergence of this approach does depend on the relative sizes of the thermal conductivities and diffusivities. We analyze different iteration strategies for solving these implicit equations including the use of mixed (Robin) approximations at the interface.
Numerical results are presented to illustrate the method. The accuracy of the technique is verified using the method of analytic solutions and by computing the solution to some heat exchanger problems where the exact solution is known. The technique is also applied to the modeling of an inertial-confinement-fusion hohlraum target and the flow of coolant past an hexagonal array of heated fuel rods. The multi-domain solver runs in parallel on distributed memory computers and some parallel results are provided.

MSC:

80M25 Other numerical methods (thermodynamics) (MSC2010)
76M25 Other numerical methods (fluid mechanics) (MSC2010)
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
65M08 Finite volume methods for initial value and initial-boundary value problems involving PDEs
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs

Software:

Ogen; CMPGRD; OVERFLOW; PETSc
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Full Text: DOI

References:

[1] Giles, M. B., Stability analysis of numerical interface conditions in fluid-structure thermal analysis, Int. J. Numer. Method Eng., 25, 421-436 (1997) · Zbl 0891.76058
[2] Chesshire, G.; Henshaw, W., Composite overlapping meshes for the solution of partial differential equations, J. Comput. Phys., 90, 1, 1-64 (1990) · Zbl 0709.65090
[3] Chesshire, G.; Henshaw, W., A scheme for conservative interpolation on overlapping grids, SIAM J. Sci. Comput., 15, 4, 819-845 (1994) · Zbl 0805.65086
[4] Henshaw, W. D.; Schwendeman, D. W., An adaptive numerical scheme for high-speed reactive flow on overlapping grids, J. Comput. Phys., 191, 420-447 (2003) · Zbl 1134.76427
[5] Henshaw, W. D.; Schwendeman, D. W., Moving overlapping grids with adaptive mesh refinement for high-speed reactive and non-reactive flow, J. Comput. Phys., 216, 2, 744-779 (2006) · Zbl 1220.76052
[6] Volkov, E. A., A finite difference method for finite and infinite regions with piecewise smooth boundaries, Doklady, 168, 5, 744-747 (1966) · Zbl 0227.65050
[7] Volkov, E. A., The method of composite meshes for finite and infinite regions with piecewise smooth boundaries, Proc. Steklov Inst. Math., 96, 145-185 (1968) · Zbl 0207.09502
[8] Starius, G., Composite mesh difference methods for elliptic and boundary value problems, Numer. Math., 28, 243-258 (1977) · Zbl 0363.65078
[9] Starius, G., On composite mesh difference methods for hyperbolic differential equations, Numer. Math., 35, 241-255 (1980) · Zbl 0475.65059
[10] Starius, G., Constructing orthogonal curvilinear meshes by solving initial value problems, Numer. Math., 28, 25-48 (1977) · Zbl 0363.65072
[11] Kreiss, B., Construction of a curvilinear grid, SIAM J. Sci. Stat. Comput., 4, 2, 270-279 (1983) · Zbl 0536.65086
[12] Steger, J. L.; Benek, J. A., On the use of composite grid schemes in computational aerodynamics, Comput. Method Appl. Mech. Eng., 64, 301-320 (1987) · Zbl 0607.76061
[13] Kapila, A.; Schwendeman, D.; Bdzil, J.; Henshaw, W., A study of detonation diffraction in the ignition-and-growth model, Combust. Theor. Model., 11, 5, 781-822 (2007) · Zbl 1145.80011
[14] Henshaw, W. D.; Schwendeman, D. W., Parallel computation of three-dimensional flows using overlapping grids with adaptive mesh refinement, J. Comput. Phys., 227, 16, 7469-7502 (2008) · Zbl 1213.76138
[15] Banks, J. W.; Schwendeman, D. W.; Kapila, A. K.; Henshaw, W. D., A high-resolution Godunov method for compressible multi-material flow on overlapping grids, J. Comput. Phys., 223, 262-297 (2007) · Zbl 1163.76032
[16] Banks, J. W.; Schwendeman, D. W.; Kapila, A. K.; Henshaw, W. D., A study of detonation propagation and diffraction with compliant confinement, Combust. Theor. Model., 12, 4, 769-808 (2008) · Zbl 1144.80379
[17] Tu, J. Y.; Fuchs, L., Calculation of flows using three-dimensional overlapping grids and multigrid methods, Int. J. Numer. Method Eng., 38, 259-282 (1995) · Zbl 0823.76059
[18] P.G. Buning, I.T. Chiu, S. Obayashi, Y.M. Rizk, J.L. Steger, Numerical simulation of the integrated space shuttle vehicle in ascent, Paper 88-4359-CP, AIAA, 1988.; P.G. Buning, I.T. Chiu, S. Obayashi, Y.M. Rizk, J.L. Steger, Numerical simulation of the integrated space shuttle vehicle in ascent, Paper 88-4359-CP, AIAA, 1988.
[19] Hinatsu, M.; Ferziger, J., Numerical computation of unsteady incompressible flow in complex geometry using a composite multigrid technique, Int. J. Numer. Method Fluid, 13, 971-997 (1991) · Zbl 0741.76044
[20] R. Meakin, Moving body overset grid methods for complete aircraft tiltrotor simulations, Paper 93-3350, AIAA, 1993.; R. Meakin, Moving body overset grid methods for complete aircraft tiltrotor simulations, Paper 93-3350, AIAA, 1993.
[21] D. Pearce, S. Stanley, F. Martin, R. Gomez, G.L. Beau, P. Buning, W. Chan, T. Chui, A. Wulf, V. Akdag, Development of a large scale Chimera grid system for the space shuttle launch vehicle, Paper 93-0533, AIAA, 1993.; D. Pearce, S. Stanley, F. Martin, R. Gomez, G.L. Beau, P. Buning, W. Chan, T. Chui, A. Wulf, V. Akdag, Development of a large scale Chimera grid system for the space shuttle launch vehicle, Paper 93-0533, AIAA, 1993.
[22] Maple, R.; Belk, D., A new approach to domain decomposition, the Beggar code, (Weatherill, N., Numerical Grid Generation in Computational Fluid Dynamics and Related Fields (1994), Pineridge Press Limited), 305-314
[23] D. Jespersen, T. Pulliam, P. Buning, Recent enhancements to OVERFLOW, Paper 97-0644, AIAA, 1997.; D. Jespersen, T. Pulliam, P. Buning, Recent enhancements to OVERFLOW, Paper 97-0644, AIAA, 1997.
[24] Meakin, R. L., Composite overset structured grids, (Thompson, J. F.; Soni, B. K.; Weatherill, N. P., Handbook of Grid Generation (1999), CRC Press), 1-20, (Ch. 11)
[25] Kiris, C.; Kwak, D.; Rogers, S.; Chang, I., Computational approach for probing the flow through artificial heart devices, J. Biomech. Eng., 119, 4, 452-460 (1997)
[26] Henshaw, W. D., A high-order accurate parallel solver for Maxwell’s equations on overlapping grids, SIAM J. Sci. Comput., 28, 5, 1730-1765 (2006), <http://link.aip.org/link/?SCE/28/1730/1> · Zbl 1127.78011
[27] Tahara, Y.; Wilson, R.; Carrica, P.; Stern, F., RANS simulation of a container ship using a single-phase level-set method with overset grids and the prognosis for extension to a self-propulsion simulator, J. Mar. Sci. Technol., 11, 4, 209-228 (2006)
[28] Olsson, F.; Yström, J., Some properties of the upper convected Maxwell model for viscoelastic fluid flow, J. Non-Newtonian Fluid Mech., 48, 125-145 (1993) · Zbl 0779.76003
[29] Petersson, N. A., A numerical method to calculate the two-dimensional flow around an underwater obstacle, SIAM J. Numer. Anal., 29, 20-31 (1992) · Zbl 0742.76010
[30] P. Fast, Dynamics of interfaces in non-Newtonian Hele-Shaw flow, Ph.D. Thesis, New York University, Courant Institute of Mathematical Sciences, 1999.; P. Fast, Dynamics of interfaces in non-Newtonian Hele-Shaw flow, Ph.D. Thesis, New York University, Courant Institute of Mathematical Sciences, 1999.
[31] Fast, P.; Shelley, M. J., A moving overset grid method for interface dynamics applied to non-Newtonian Hele-Shaw flow, J. Comput. Phys., 195, 117-142 (2004) · Zbl 1087.76084
[32] Kao, K.-H.; Liou, M.-S., Application of Chimera/unstructured hybrid grids to conjugate heat transfer, AIAA J., 35, 9, 1472-1478 (1997) · Zbl 0900.76337
[33] Patankar, S., Numerical Heat Transfer and Fluid Flow (1980), McGraw-Hill · Zbl 0521.76003
[34] Desrayaud, G.; Fichera, A.; Lauriat, G., Natural convection air-cooling of a substrate-mounted protruding heat source in a stack of parallel boards, Int. J. Heat Fluid Flow, 28, 469-482 (2007)
[35] Schäfer, M.; Teschauer, I., Numerical simulation of coupled fluid-solid problems, Comput. Method Appl. Mech. Eng., 190, 3645-3667 (2001) · Zbl 0982.74074
[36] Wansophark, N.; Malatip, A.; Dechaumphai, P., Streamline upwind finite element method for conjugate heat transfer problems, Acta Mech. Sin., 21, 436-443 (2005) · Zbl 1200.76123
[37] Chen, X.; Han, P., A note on the solution of conjugate heat transfer problems using SIMPLE-like algorithms, Int. J. Heat Fluid Flow, 21, 463-467 (2000)
[38] Liaqat, A.; Baytas, A., Numerical comparison of conjugate and non-conjugate natural convection for internally heated semi-circular pools, Int. J. Heat Fluid Flow, 22, 650-656 (2001)
[39] N.B. Sambamurthy, A. Shaija, G.S.V.L. Narasimham, M.V. Krishna Murthy, Laminar conjugate natural convection in horizontal annuli, Int. J. Heat Fluid Flow 29 (5) (2008) 1347-1359.; N.B. Sambamurthy, A. Shaija, G.S.V.L. Narasimham, M.V. Krishna Murthy, Laminar conjugate natural convection in horizontal annuli, Int. J. Heat Fluid Flow 29 (5) (2008) 1347-1359.
[40] Roe, B.; Jaiman, R.; Haselbacher, A.; Geubelle, P. H., Combined interface boundary condition method for coupled thermal simulations, Int. J. Numer. Method Fluid, 57, 329-354 (2008) · Zbl 1241.80009
[41] Funaro, D.; Quarteroni, A.; Zanolli, P., An iterative procedure with interface relaxation for domain decomposition methods, SIAM J. Numer. Anal., 25, 6, 1213-11236 (1998)
[42] Rice, J. R.; Tsompanopoulou, P.; Vavalis, E., Fine tuning interface relaxation methods for elliptic differential equations, Appl. Numer. Math., 43, 4, 459-481 (2002) · Zbl 1017.65098
[43] Garbey, M., Acceleration of the Schwarz method for elliptic problems, SIAM J. Sci. Comput., 26, 6, 1871-1893 (2005) · Zbl 1081.65116
[44] Henshaw, W., A fourth-order accurate method for the incompressible Navier-Stokes equations on overlapping grids, J. Comput. Phys., 113, 1, 13-25 (1994) · Zbl 0808.76059
[45] Henshaw, W.; Kreiss, H.-O.; Reyna, L., A fourth-order accurate difference approximation for the incompressible Navier-Stokes equations, Comput. Fluid, 23, 4, 575-593 (1994) · Zbl 0801.76055
[46] Henshaw, W. D.; Petersson, N. A., A split-step scheme for the incompressible Navier-Stokes equations, (Hafez, M., Numerical Simulation of Incompressible Flows (2003), World Scientific), 108-125 · Zbl 1074.76036
[47] Henshaw, W. D., On multigrid for overlapping grids, SIAM J. Sci. Comput., 26, 5, 1547-1572 (2005) · Zbl 1076.65113
[48] W. Henshaw, Ogen: an overlapping grid generator for Overture, Research Report UCRL-MA-132237, Lawrence Livermore National Laboratory, 1998.; W. Henshaw, Ogen: an overlapping grid generator for Overture, Research Report UCRL-MA-132237, Lawrence Livermore National Laboratory, 1998.
[49] Petersson, N. A., Stability of pressure boundary conditions for Stokes and Navier-Stokes equations, J. Comput. Phys., 172, 1, 40-70 (2001) · Zbl 1014.76064
[50] Gustafsson, B.; Kreiss, H.-O.; Oliger, J., Time Dependent Problems and Difference Methods (1995), John Wiley and Sons Inc.
[51] Roache, P., Code verification by the method of manufactured solutions, ASME J. Fluid Eng., 124, 1, 4-10 (2002)
[52] S. Balay, W.D. Gropp, L.C. McInnes, B.F. Smith, The portable extensible toolkit for scientific computation, Technical Report, Argonne National Laboratory, 1999. <http://www.mcs.anl.gov/petsc/petsc.html>; S. Balay, W.D. Gropp, L.C. McInnes, B.F. Smith, The portable extensible toolkit for scientific computation, Technical Report, Argonne National Laboratory, 1999. <http://www.mcs.anl.gov/petsc/petsc.html>
[53] Sanchez, J. J.; Giedt, W. H., Thin films for reducing tamping gas convection heat transfer effects in a National Ignition Facility hohlraum, Fus. Sci. Technol., 44, 811-819 (2003)
[54] London, R. A.; McEachern, R. L.; Kozioziemski, B. J.; Bittner, D. N., Computational design of infrared enhanced layering of ICF capsules, Fus. Sci. Technol., 45, 2, 245-252 (2004)
[55] Collins, G.; Souers, P.; Fearon, E.; Mapoles, E.; Tsugawa, R.; Gaines, J., Thermal conductivity of condensed \(D\)−\(T\) and \(T_2\), Phys. Rev. B, 41, 4, 1816-1823 (1990)
[56] Lamarsch, J.; Baratta, A., Introduction to Nuclear Engineering (2001), Prentice Hall: Prentice Hall New Jersey
[57] W. Henshaw, Mappings for Overture, a description of the Mapping class and documentation for many useful Mappings, Research Report UCRL-MA-132239, Lawrence Livermore National Laboratory, 1998.; W. Henshaw, Mappings for Overture, a description of the Mapping class and documentation for many useful Mappings, Research Report UCRL-MA-132239, Lawrence Livermore National Laboratory, 1998.
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