Numerical Solution of Turbine Cascade Flow by Two Equations, EARSM and Bypass Transition Turbulence Models

Abstract

This work deals with numerical solution of transonic flow in turbine cascades. We compare results achieved by SST, TNT and EARSM turbulence models and we also present results achieved by two equations turbulence model with an algebraic bypass transition model. Mathematical models and numerical methodsWe consider the conservative system of Favre averaged Navier-Stokes equations for compressible flow.Two types of two equation turbulence models have been used -TNT k − ω model of Kok and SST k − ω/k − model of Menter. We implemented also the explicit Reynolds stress model of Wallin [2] which uses proper explicit formulae to evaluate turbulent stress tensor components as a function of k, ω (computed by two equations turbulence model, e.g. TNT or SST) and other flow paramers. Next we consider a two equation SST turbulence model extended by an algebraic model of the bypass transition. It is based on the empirical relation for the intermittency coefficient γ ∈< 0, 1 > proposed by Narasimha, and proper algebraic relations for transition onset and lenght of transition region. The position of the transition onset is described by the Reynolds numberRe xt determined by means of the momentum Reynolds number Re θt = f (T u, λ t ), where T u is the free-stream turbulence level and λ t is the pressure gradient parameter. Both parameters are considered at the location of the transition onset. The length of the transition region is given by parameters describing the spot generation raten and spot propagation parameter σ.The problem is solved by several alternative implicit schemes based on the finite volume method in cell-centered formulation: Method 1. The method is based on AUSM scheme and linear interpolation with van Leer's limiter on multi-block structured grid. Some numerical resultsIn-house developed implicit methods differs in type of used grid, numerical flux, reconstruction on finite volume or limiter. They can be used with different turbulence models. This fact gives us opportunity to study numerically an influence of a numerical scheme, a grid topology, a turbulence model on the flow field in turbine cascade and especially on loss prediction. We present here numerical results for flow through SE1050 cascade at design conditions (inlet velocity angle 19.3deg and isoentropic outlet Mach number M 2is = 1.198. The first two results in Fig.1 shows isolines of Mach number achieved by method 1 and two diferent turbulence models -EARSM and SST. The right figure shows results achieved by method 3 on hybrid grid also with SST turbulence model but together with the algebraic model of boundary layer transition. We can observe similar shape of isolines and very good agreement of computed shock waves structure. Diferences between SST and EARSM models are more visible on isolines of kinetic turbulent energy k ploted in Fig.2. We can observe especialy much stronger production of k along relatively weak shock wave behind the trailing edge of profile in case of SST model. The influence of boundary layer transition model is shown in Fi...