Simulation of transitional flows through a turbine blade cascade with heat transfer for various flow conditions

. The contribution deals with the simulation of the transitional flows with heat transfer by means the EARSM turbulence model of Hellsten [1] completed by the algebraic transition model of Straka and P(cid:2)íhoda [2] and by the three-equation model of Walters and Cokjlat [3]. The both mathematical models were tested for the flat plate flow on a heated wall measured by Sohn and Reshotko [16] and then applied to the simulation of compressible flow through the VKI turbine blade cascade according to measurements of Arts et al. [4]. The simulations were carried out for subsonic and transonic regimes at various free-stream turbulence levels. The best agreement of numerical results with experimental data was achieved by the URANS approach applied for the EARSM model with the algebraic transition model giving good results for both subsonic and transonic regimes as well.


Introduction
The numerical simulation of compressible flows in turbomachinery by means of the RANS approach should be based not only on the advanced turbulence model but also on adequate models of the laminar/turbulent transition and the turbulent heat transfer. The bypass transition models come out mostly from the intermittency concept using the algebraic and/or transport equation for the intermittency coefficient, see e.g. Thermann and Niehuis [5], Straka and PĜíhoda [2] and/or Langtry and Menter [6], Kubacki et al. [7], and Menter et al. [8]. Besides, the three-equation k-kL-Z model by Walters and Cokljat [3] with the equation for the energy of non-turbulent fluctuations can be used for transition modelling as well. Nevertheless both types of transition models need empirical relations describing the onset and the length of the transition region. The turbulent heat transfer is mainly modelled by the simplest assumption of the constant turbulent Prandtl number.
The contribution deals with the numerical simulation of the compressible flow through a turbine blade cascade with heat transfer by means two different turbulence models. The EARSM turbulence model of Hellsten [1] was completed by the algebraic bypass transition model of Straka and PĜíhoda [2] and by the model of turbulent heat transfer based on the generalized gradient hypothesis of Launder [9]. Futher the k-kL-Z model of Walters and Cokjlat [3] where the relation for the turbulent thermal diffusivity corresponds to the variable turbulent Prandtl number was used. Both models were tested by means of the flow over a heated flat plate and applied for simulation of flow through a turbine blade cascade.

Mathematical model
The governing equations for compressible flow are given by the Favre-averaged Navier-Stokes equations and by constitutive relations. The system of equations was closed partly by the EARSM model with the algebraic transition model implemented into the in-house numerical code and partly by the three-equation k-kL-Z model implemented into the open-source code OpenFOAM.
The EARSM model is used in the form corresponding to turbulent-viscosity models with the turbulent stress given by the relation where the extra-anisotropy tensor a ij (ex) is dependent on strain-rate and vorticity tensors and their invariants. Transport equations for the turbulent energy k and the specific dissipation rate Z are given by the SST model according to Menter [11] in the form Turbulent viscosity is given by the relation with the turbulent time scale where the Kolmogorov viscous time scale is used near the wall and E * = 0.09 and C W = 6. The coefficient C P is obtained from the equation where coefficients E 1 and E 6 depend on strain-rate and vorticity tensors and their invariants II S and II : . The detailed description of the EARSM model is given by Hellsten [1]. The production term in the turbulent energy equation was modified for the reduction of the undesirable overproduction of the turbulent energy in the stagnation region.
The turbulent heat transfer is modelled according to Launder [9] by the generalized gradient hypothesis in the form with the constant C t = 0.3. For the prediction of the transitional flows, the production and destruction terms in where Tu (%) is the free-stream turbulence level and Ȝ t is the pressure-gradient parameter. The length of the transition region is expressed using the parameter N = nV Re Tt 3 where n is the spot generation rate and ı is the spot propagation rate introduced by Narasimha [12]. The effect of the free-stream turbulence and the pressure gradient on the parameter N is correlated by an empirical relation proposed for the attached flow by Solomon et al. [13]. The onset of transition in separated flow is given by the correlation proposed by Mayle [14] in the form Re xt = f (Re Ts , Re xs ) where Re Ts is the momentum Reynolds number at the separation and Re xs is the Reynolds number related to the distance of the separation from the leading edge. The application of local variables is necessary for prediction of transitional flows in complex geometries. Therefore the momentum Reynolds number is replaced by the maximum of the vorticity Reynolds number according to Langtry and Menter [6]. The vorticity Reynolds number is given by the relation 2 / Re y : : Q where y is the distance from the wall and : is the absolute value of the vorticity tensor. The link between both Reynolds numbers is expressed by the relation Re T = Re :max /C where the parameter C depends on the pressure gradient expressed by the parameter The three-equation model of Walters and Cokljat [2] is based on the assumption that the transition process can be expressed by the transfer from the energy of nonturbulent velocity fluctuations k L to the turbulent energy k T of three-dimensional turbulent velocity fluctuations.
The transport equations for the turbulent energy k T , laminar energy k L and the specific dissipation rate Z are given by equations The terms P kT and P kL express the production of turbulent and non-turbulent velocity fluctuations respectively. Similarly, terms D T and D L express the diffusion. The transfer of energy between non-turbulent and turbulent vortices is modelled by terms R BP and R NAT expressing the effect of the decay of laminar fluctuations during natural and bypass transitions. The turbulent heat transfer in the energy equation is expressed in the form where h is the enthalpy and Į ș is the turbulent thermal diffusivity used in the form corresponding to the variable turbulent Prandtl number. The both mathematical models are described in detail by Fürst et al. [15].

Results
The mathematical models were tested using the transitional boundary layer on a heated flat plate according to Sohn and Reshotko [16].   The skin friction distribution on the heated flat plate is compared in Fig. 1 with experimental data of Sohn and Reshotko [16] and with results of numerical simulation by means of the DNS method completed by Madavan [17]. For comparison, well-known skin friction laws for the flat-plate laminar and turbulent boundary layers are shown as well.
It is surprising that the algebraic transition model gives the skin friction distribution with a rather early onset of the transition. The k-kL-Z model and the DNS method agree well with experimental data, but the predicted length of the transition region is rather shorter.
The heat transfer at the wall is described by the Stanton number determined by the relation Pr Re x (16) and for the turbulent flat-plate boundary layer 0 111 0 9 0 4 0 2 0 028 1 Pr Re x (17) where x o is the entrance unheated length and the Prandtl number Pr = 0.708. The predicted distribution of the Stanton number on the heated flat plate is compared with experimental data of Sohn and Reshotko [16] in Fig. 2. The transition onset predicted by the algebraic model by means of the Stanton number distribution is again slightly upstream similarly as for the skin friction distribution.
The k-kL-Z model gives a good agreement with experimental data comparable with DNS results. The both models were further applied for the numerical simulation of the compressible flow through the plane VKI high pressure turbine blade cascade with the relative spacing t/c = 0.85 and the zero inlet flow angle according to measurements of Arts et al. [5]. Experiments were carried out in the test facility enabling an independent selection of both Mach and Reynolds numbers. Besides measurements of the blade velocity distribution, the loss coefficient and exit flow angle, experiments were focused on measurements of the blade convective heat transfer.
Predictions were carried out for two subsonic (MUR 132 and MUR 218) and two transonic regimes (MUR 210 and MUR 213) at the isentropic outlet Reynolds number Re 2is = 1u10 6 , free-stream turbulence Tu = 1 and 4 %, and the wall temperature T w = 300 K. Other boundary conditions were prescribed according to individual test cases, i.e. total inlet pressure and total inlet temperature, and the outlet static pressure. The overview of free-stream conditions for selected VKI test cases is given in Table 1. In the case of the k-kL-Z model, the computational grid containing about 62000 cells is formed by rectangular cells in the boundary layer and triangles and rectangles in the free stream. The nearest node was in the distance from the wall y + | 0.2. The onset of the grid is in the distance 0.9c upstream of the leading edge. The detail of the computational grid near the trailing edge is shown in Fig. 3. The distribution of the wall heat flux along the blade obtained by the k-kL-Z model for the free-stream turbulence about Tu = 4% is shown in Fig. 4. Numerical results are very similar for subsonic and transonic regimes. The agreement with experimental data is quite good on the suction side (s ! 0) where the transition onset is near upstream of the trailing edge. However, the k-kL-Z model overestimates the heat flux on the pressure side at higher free-stream turbulence. The Mach number isolines for the case MUR 213 obtained by the k-kL-Z model are shown in Fig. 5 where the shock wave is apparent at the suction side near upstream of the trailing edge.
The distribution of the wall heat flux obtained by the k-kL-Z model for the free-stream turbulence about Tu | 1% is shown in Fig. 6. For the lower level of free-stream turbulence, the agreement of numerical results with experiment is quite good on the pressure and suction side, but the transition onset on the suction side in the case MUR 132 is predicted as early as at s | 0.06 m. The difference between the prediction and experimental data can be caused by the bypass-transition threshold function which is not dependent on the pressure gradient.  The distribution of the wall heat flux obtained by the EARSM model with the algebraic transition model for the free-stream turbulence Tu = 4% and 1% is shown in Fig. 8 and 9. The distribution of the wall heat flux is given for the averaged URANS solution. A very good agreement with experimental data was obtained on the pressure and suction sides for both subsonic regimes where the RANS and URANS approaches give the practically same results.  The URANS approach gives a better agreement with experimental data even though the transition onset on the suction side at s | 0.083 m is probably caused by the interaction with the exit shock wave. Besides, the instantaneous field of the Mach number isolines obtained by the URANS method shows the noticeable vortex line behind the trailing edge of the blade (see Fig. 10c). Similar flow structures including vortex shedding were found out by the RANS and LES simulations of flow with heat transfer in the VKI blade cascade accomplished by Gourdain et al. [18].

Conclusions
The transitional flow with heat transfer was simulated by the k-kL-Z model of Walters and Cokljat [3] and by the EARSM model of Hellsten [1] completed by the algebraic transition model of Straka and PĜíhoda [2] and the turbulent heat transfer model based on the generalized gradient hypothesis. The both mathematical models were tested by experimental data of Sohn and Reshotko [16] for the boundary-layer flow on a heated plat plate and then applied to the simulation of compressible flow through the VKI turbine blade cascade according to measurements of Arts et al. [4].
The simulations were carried out for subsonic and transonic regimes at various values of free-stream turbulence. Due to finite thickness of the blade trailing edge, the RANS and URANS approach was applied for the EARSM model with the algebraic transition model. A good agreement was obtained by both approaches for subsonic regimes but only the URANS approach gives adequate results of numerical simulations for transonic regimes where the RANS approach gives a pseudo shock wave whose interaction with the boundary layer leads to the premature transition on the suction side.