... chamber through the wall into the cooling fluid of up to 85 MW/m2 in Ariane 5's Vulcain ... turbulence model [2] with the standard Pope correction [6] is used.
PAMM · Proc. Appl. Math. Mech. 2, 360–361 (2003) / DOI 10.1002/pamm.200310164
Haarmann, T. M.; Koschel, W. W.
Numerical Simulation of Heat Loads in a Cryogenic H2 /O2 Rocket Combustion Chamber A Finite Element solver for a coupled simulation of fluid and structure in an axisymmetric domain is presented. The method employs an explicit solution of the flow and structure variables. The computational domain of the fluid is discretised by unstructured triangles and rectangles while the sturcture domain is discretised by unstructured triangles only. For the purpose of code validation the solution of in total three test cases are shown. One test case deals with the structure only while the other two simulate heat transfer problems with a defined temperature distribution along a boundary wall and coupled conditions. Finally the code is used to simulate the heat load in a cryogenic H2 /O2 rocket combustion chamber. 1. Introduction In order to design and develop the cooling systems of modern cryogenic H2 /O2 rocket combustion chambers and nozzles a detailed and accurate information of the heat fluxes into the axial chamber wall are crucial. To determine these heat fluxes from the chamber through the wall into the cooling fluid of up to 85 MW/m2 in Ariane 5’s Vulcain main engine nozzle [1] numerical tools are necessary. Therefore a code is required which does not only simulate the flow regime with its physical problems but solves the heat conduction in the wall and the heat transfer into the cooling fluid. 2. Numerical Method To simulate heat loads in numerical domains the Navier-Stokes equations with the additional Low-Reynolds q − ω turbulence model [2] with the standard Pope correction [6] is used. These models are in a Finite Element Solver for axisymmetric calculations using unstructured grids. Two Taylor-Series expansions (2nd order accurate) in time with the weighted residual method in space are applied. This scheme was developed by L¨ohner[3] and validated for supersonic laminar flows by Schulte[4]. 3. Results The first validation case is performed for the structure solver. In figure 1a the numerical domain with an analytic heat source/sink distribution is shown. By means of these sources and sinks an analytic temperature distribution can be computed represented by: T(x,r) = 300 + x(0.5)(1.5-r)(1-x) arctan β. In figure 1b the values along the diagonal from (0,0.5) to (1,1.5) show very a good agreement with the analytic distribution. The numerical result has a relative error of 2.5 10−5 %. For validation purposes of the fluid solver and the coupling algorithm the two cases of the flow in an intake duct with an isothermal wall boundary condition and the flow along a lance with a defined wall temperature distribution were performed. In figure 2 the resulting Nusselt-numbers of the isothermal and coupled simulation are compared to two Nusselt-laws formulated by Eckert and from VDI-W¨armeatlas. Even though all curves do not match each other beyond one duct diameter length they all seem to fade to the same value after one diameter of length. In figure 3 the heat flux along the cylindrical part of a lance is displayed. On its surface a linear temperature distribution has been established causing the heat flux rise over the length. The numerical result shows a good agreement with the measurements published by Eichhorn et al [5]. As a final test case the code is applied for a subscale test engine to perform calorimetric investigations on cryogenic H2 /O2 fuel combinations[7]. The selected results were performed with a total pressure at the nozzle throat of 60 bar and a Oxidizer/Fuel ratio O/F = 6. The viscosity was computed by the Sutherland law and the heat conductivity with the assumption of constant Prandtl-number of 0.72. At the symmetry line slip boundary condition were taken while at the nozzle wall no-slip condition with either isothermal or coupled conditions were used. In figure 4 the wall heat fluxes of the computation, the wall surface temperature of the coupled simulation and the measured data are presented. While the isothermal case is showing an acceptable agreement with the measured data the coupled simulation does not show the expected distribution. This result can be explained by the surface temperature distribution showing a temperature peak in the throat. It seems the used Nusselt-correlation for helically coiled
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tubes does not represent the heat transfer into the cooling fluid properly, 4. Acknowledgements The authors acknowledge the support of this work by providing the measurement data from Astrium GmbH, Space Infrastructure Division, gained from the General Support Technolgy Program 2 comissioned by ESA/ESTEC. These measurements were performed on P8 at the rocket engine test site of the German Aerospace Center (DLR) at Lampoldshausen. 5. References 1 Kuhl D., Riccius J., Haidn O.: Thermo-mechanical Analysis and Optimization of Cryogenic Liquid Rocket Engines. AIAA Journal of Propulsion and Power, to be published 2002 2 Coakley T.J.: Turbulence Modeling for High Speed Flows, AIAA-92-0436, 1992. 3 Lohner R., Baum J., Loth E., Ramamurti E.R.: A Finite Element Solver for Axisymmetric Compressible Flows, AIAA 89-1794, 1989 4 Schulte D.: Beeinflussung viskoser Str¨ omungseffekte in Hyperschall-Einl¨ aufen, Dissertation RWTH Aachen 2001. 5 Eichhorn R., Eckert E.R.G., Anderson A.D.: An Experimental Study of the Effects of Nonuniform Wall Temperature on Heat Transfer in Laminar and Turbulent Axisymmetric Flow along a Cylinder, Journal of Heat Transfer, 11/1960, P.349-359, 1960 6 Pope S.B.: An Explanation of the Turbulent Round-Jet/Plane Jet Anomaly, AIAA Journal, 16(3), p. 279-281, 1978 7 Preclik D., Wiedmann D., Oechslein W. Kretschmer J.: Cryogenic Rocket Calorimetric Chamber Experiments and Heat Transfer Simulations, AIAA-98-3440, 1998.
Dipl.-Ing. T. M. Haarmann, Lehr- und Forschungsgebiet Betriebsverhalten der Strahlantriebe, RWTH Aachen, Templergraben 55, D-52062 Aachen Univ.-Prof. Dr.-Ing. W. W. Koschel, Institut f¨ ur Raumfahrtantriebe, Deutsches Zentrum f¨ ur Luft- und Raumfahrt e.V., Lampoldshausen, D-74239 Hardthausen
Fig.1: Heat source/sink distribution (left) and the resulting temperature distribution along the diagonal(right)
Fig.3: Computed wall heat flux along a lance compared to measured values
Fig.2: Nusselt-numbers along a intake duct flow
Fig.4: Computed and measured wall heat fluxes and wall surface temperature of a cryogenic rocket combustion chamber
