An equilibrium wall model for reacting turbulent flows with heat transfer Daiki Muto a,⇑ , Yu Daimon b , Taro Shimizu a , Hideyo Negishi b a Research Unit III, Japan Aerospace Exploration Agency, 3-1-1 Yoshinodai, Chuo-ku, Sagamihara, Kanagawa, Japan b Research Unit III, Japan Aerospace Exploration Agency, 2-1-1 Sengen, Tsukuba, Ibaraki, Japan article info Article history: Received 27 September 2018 Received in revised form 27 May 2019 Accepted 28 May 2019 Keywords: Wall model Heat transfer Reacting flow Boundary layer equation abstract A wall model for reacting turbulent flows for predicting heat transfer in rocket engine combustion cham- bers is presented. The wall model is developed based on boundary layer equations and includes the effects of chemical reactions and variable properties. Assuming an equilibrium state near the wall, a sim- ple set of momentum and enthalpy equation is formulated. A table look-up procedure is employed to cal- culate the contributions of chemical effects and mixture properties. For turbulence modeling of the inner layer, a modified mixing length model based on semi-local scaling is applied. The proposed equilibrium wall model is validated in two hydrogen/oxygen reacting cases. In the first validation case of a reacting turbulent channel flow, the equilibrium wall model accurately predicts the near-wall velocity and tem- perature fields. In contrast, the wall model when assuming frozen chemistry shows a discrepancy in tem- perature gradient and mixture properties. The heat flux balance in the equilibrium wall model highlights the important contributions of chemical effects. The second test case of a rocket combustion chamber shows that the equilibrium wall model is superior to the frozen wall model and wall function models in predicting wall shear stress and heat flux. Ó 2019 Elsevier Ltd. All rights reserved. 1. Introduction Combustion chamber walls in liquid rocket engines represent severe thermal environments. The burnt gas temperature is typi- cally around 3500 K, while the chamber wall is cooled to about 400–800 K. Thus a drastic temperature gradient of about 3000 K is developed in a boundary layer less than 1 mm thick. Accordingly, the chamber wall is exposed to high heat flux that locally reaches values greater than 100 MW/m 2 . In rocket chamber design, unex- pected high heat flux locally imposed on the wall could lead to wall damage and melting, thereby making an accurate prediction of wall heat flux essential for preventing engine failure and estimat- ing chamber lifetime [1,2]. Numerical simulations are promising tools for estimating the wall heat flux in combustion chambers, and Reynolds-Averaged Navier-Stokes (RANS) approaches have been widely applied in the chamber design process. While several RANS studies of the combustion chambers have been conducted [3–5], the quantitative prediction of wall heat flux, particularly its spatial distribution and local maxima, is still difficult. This is due to intrinsic limitations of conventional RANS approaches in capturing complex turbulent flowfields including a recirculation and attachment of combustion gas and subsequent heat transfer on the wall. Meanwhile, Large eddy simulation (LES) of combustion chambers has actively been performed and shown its validity on simulating turbulent mixing and combustion [6–9]. For estimating wall heat flux, however, LES of combustion chambers has often used a too coarse computa- tional grid due to the severe requirements in grid resolutions for resolving boundary layers: the viscous length scale at the chamber wall approaches O 1 lm ð Þ. Consequently, LES of near-wall flows requires a huge computational cost, and an accurate prediction of the wall heat flux in combustion chambers by using LES remains a challenging problem. One approach to overcoming the difficulty of the near-wall grid requirements is a wall-modeled LES [10,11]. The concept of wall- modeled LES is resolving the outer layer while modeling the inner layer, as inner layer models are used to estimate wall shear stress and heat flux. One such inner layer model uses an algebraic wall function based on the law of the wall [12,13]. Another numerically solves boundary layer equations [14–16] in a grid with fine resolu- tion that is embedded in an outer flow calculation domain. Kawai and Larsson [16] showed the use of equilibrium boundary layer equations as an inner layer model. This approach was successfully demonstrated regarding aerodynamic problems [17,18] and heat transfers [19–21], and applied to supersonic combustion [22]. https://doi.org/10.1016/j.ijheatmasstransfer.2019.05.101 0017-9310/Ó 2019 Elsevier Ltd. All rights reserved. ⇑ Corresponding author. E-mail address: muto.daiki@jaxa.jp (D. Muto). International Journal of Heat and Mass Transfer 141 (2019) 1187–1195 Contents lists available at ScienceDirect International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt