An Effective Model for the Simulation of Transpiration Cooling

Abstract

Transpiration cooling is numerically investigated, where a cooling gas is injected through a carbon composite material into a hot gas channel. To account for microscale effects at the injection interface, an effective problem is derived. Here, effects induced by microscale structures on macroscale variables, e.g., cooling efficiency, are taken into account without resolving the microscale structures. For this purpose, effective boundary conditions at the interface between hot gas and porous medium flow are derived using an upscaling strategy. Numerical simulations in 2D with effective boundary conditions are compared to uniform and non-uniform injection. The computations confirm that the effective model provides a more efficient and accurate approximation of the cooling efficiency than the uniform injection.

Keywords

Transpiration cooling / Darcy-Forchheimer flow / Multiscale modeling / Effective boundary conditions

References

[1.] Achdou Y, Pironneau O, Valentin F. Effective boundary conditions for laminar flows over periodic rough boundaries. J. Comput. Phys., 1998, 147: 187-218,

[2.] Bangerth W, Hartmann R, Kanschat G. Deal.II - a general-purpose object-oriented finite element library. ACM Trans. Math. Softw., 2007, 33(4): 24-12427,

[3.] Bramkamp F, Lamby P, Müller S. An adaptive multiscale finite volume solver for unsteady and steady state flow computations. J. Comput. Phys., 2004, 197(2): 460-490,

[4.] Carraro T, Goll C, Marciniak-Czochra A, Mikelic A. Effective interface conditions for the forced infiltration of a viscous fluid into a porous medium using homogenization. Comput. Methods Appl. Mech. Eng., 2015, 292: 195-220,

[5.] Dahmen, W., Gerber, V., Gotzen, T., Müller, S., Rom, M., Windisch, C.: Numerical simulation of transpiration cooling with a mixture of thermally perfect gases. In: Proceedings of the Jointly Organized WCCM XI - ECCM V - ECFD VI 2014 Congress, Barcelona, pp. 3012–3023 (2014)

[6.] Dahmen W, Gotzen T, Müller S, Rom M. Numerical simulation of transpiration cooling through porous material. Int. J. Numer. Methods Fluids, 2014, 76(6): 331-365,

[7.] Dahmen W, Müller S, Rom M, Schweikert S, Selzer M, von Wolfersdorf J. Numerical boundary layer investigations of transpiration-cooled turbulent channel flow. Int. J. Heat Mass Transf., 2015, 86: 90-100,

[8.] Deolmi G, Dahmen W, Müller S. Effective boundary conditions for compressible flows over rough boundaries. Math. Model. Methods Appl. Sci., 2015, 25(7): 1257-1297,

[9.] Deolmi G, Dahmen W, Müller S. Effective boundary conditions: a general strategy and application to compressible flows over rough boundaries. Commun. Comput. Phys., 2017, 21(2): 358-400,

[10.] Deolmi G, Müller S. A two-step model order reduction method to simulate a compressible flow over an extended rough surface. Math. Comput. Simul., 2018, 150: 49-65,

[11.] Gotzen, T.: Numerical Investigation of Film and Transpiration Cooling. PhD thesis, RWTH Aachen University (2013)

[12.] Herbertz A, Selzer M. Analysis of coolant mass flow requirements for transpiration cooled ceramic thrust chambers. Trans. JSASS Aerosp. Tech. Japan, 2014, 12(29): 31-39

[13.] Jäger W, Mikelic A. Modeling effective interface laws for transport phenomena between an unconfined fluid and a porous medium using homogenization. Transp. Porous Media, 2009, 78(3): 489-508,

[14.] Jiang P, Yu L, Sun J, Wang Y. Experimental and numerical investigation of convection heat transfer in transpiration cooling. Appl. Therm. Eng., 2004, 24: 1271-1289,

[15.] König, V.: Effective Boundary Conditions for Transpiration Cooling Applications. PhD thesis, RWTH Aachen University (2023)

[16.] König V, Rom M, Müller S. Adams NA, Schröder W, Radespiel R, Haidn OJ, Sattelmayer T, Stemmer C, Weigand B. A coupled two-domain approach for transpiration cooling. Future Space-Transport-System Components Under High Thermal and Mechanical Loads: Results from the DFG Collaborative Research Center TRR40, 2021 Cham Springer 33-49,

[17.] König V, Rom M, Müller S, Schweikert S, Selzer M, von Wolfersdorf J. Numerical and experimental investigation of transpiration cooling with Carbon/Carbon characteristic outflow distributions. J. Thermophys. Heat Transf., 2019, 33(2): 449-461,

[18.] Lacis U, Bagheri S. A framework for computing effective boundary conditions at the interface between free fluid and a porous medium. J. Fluid Mech., 2017, 812: 866-889,

[19.] Langener T, von Wolfersdorf J, Selzer M, Hald H. Experimental investigations of transpiration cooling applied to C/C material. Int. J. Therm. Sci., 2012, 54: 70-81,

[20.] Linn, J., Keller, M., Kloker, M.J.: Effects of Inclined Blowing on Effusion Cooling in a Mach-2.67 Boundary Layer. Annual Report SFB TRR40 2010. Munich (2010)

[21.] Linn J, Kloker MJ. Gülhan A. Numerical investigations of film cooling and its influence on the hypersonic boundary-layer flow. RESPACE-Key Technologies for Reusable Space Systems, NNFM, 2008 Springer 151-169,

[22.] Linn J, Kloker MJ. Effects of wall-temperature conditions on effusion cooling in a Mach-2.67 boundary layer. AIAA J., 2011, 49(2): 299-307,

[23.] Liu Y, Jiang P, Xiong Y, Wang Y. Experimental and numerical investigation of transpiration cooling for sintered porous flat plates. Appl. Therm. Eng., 2013, 50: 997-1007,

[24.] Nield DA, Bejan A. . Convection in Porous Media, 2013 4 Cham Springer,

[25.] Ortelt, M., Hald, H., Herbertz, A., Müller, I.: Advanced design concepts for ceramic thrust chamber components of rocket engines. In: 5th European Conference for Aeronautics and Space Sciences (EUCASS), Munich (2013)

[26.] Rom M, Müller S. Derivation and analysis of a 1D porous medium flow solver embedded in a two-domain model for 2D and 3D transpiration cooling. Int. J. Heat Mass Transf., 2022, 195,

[27.] Schweikert, S., von Wolfersdorf, J., Selzer, M., Hald, H.: Experimental investigation on velocity and temperature distributions of turbulent cross flows over transpiration cooled C/C wall segments. In: 5th European Conference for Aeronautics and Space Sciences (EUCASS), Munich (2013)

[28.] Selzer, M., Langener, T., Hald, H., von Wolfersdorf, J.: Production and characterization of porous C/C material. Annual Report SFB TRR40. Munich (2009)

[29.] Steins E, Bui-Thanh T, Herty M, Müller S. Probabilistic constrained Bayesian inversion for transpiration cooling. Int. J. Numer. Methods Fluids, 2022, 94(12): 2020-2039,

[30.] Yang G, Coltman E, Weishaupt K, Terzis A, Helmig R, Weigand B. On the Beavers-Joseph interface condition for non-parallel coupled channel flow over a porous structure at high Reynolds numbers. Transp. Porous Media, 2019, 128(2): 431-457,

[31.] Yang L, Min Z, Yue T, Rao Y, Chyu MK. High resolution cooling effectiveness reconstruction of transpiration cooling using convolution modeling method. Int. J. Heat Mass Transf., 2019, 133: 1134-1144,

Funding

Deutsche Forschungsgemeinschaft; RWTH Aachen University (3131)