# Gas Turbine nozzle cascade

Application Challenge 6-06 © copyright ERCOFTAC 2004

## Overview of CFD Simulations

Numerical simulations have been performed using Two 3D CFD solvers for internal flow applications for the investigation of heat transfer problems in gas turbine components. The numerical approaches considered are respectively based on a structured (XFLOS) and on an unstructured (HybFlow) methods. Both codes are also implemented for the porting on parallel architectures for the improvement of the computational efficiency in the application to complex 3D stage investigations. The main objective of the computation performed is devoted to the turbulent flow modelling which assumes great concern for heat transfer predictions on high loaded transonic turbine blades. According this observation, the two CFD approaches have been applied to the heat transfer investigation of some of the test cases presented in the previous section. The model equation and the main features of the solver will be briefly described in the following sections. Firstly a brief overview of the model equations will be provided and special attention will be devoted to the transition modelling which plays a crucial role in the heat transfer computation.

3.1.1 The Governing Equations for Turbulent Flows

The gasdynamic model considered is based on the classical Navier-Stokes systems of governing equations. In order to simulate high Reynolds viscous conditions a RANS (Reynolds Averaged Navier-Stokes) approach is applied for turbulent industrial flows. A Favre mass averaging is performed on the whole system while the family of two-equation eddy-viscosity models (eg. k-ω) is considered for the closure of turbulent stresses in the averaged equations. The resulting set of governing equations considered for internal viscous flows can be cast in strong conservative form as follows:

${\displaystyle {\frac {\partial Q}{\partial t}}+{\frac {\partial F_{i}}{\partial x_{i}}}={\frac {\partial F^{v_{i}}}{\partial x_{i}}}+S}$

${\displaystyle F_{i}=\left\{{\begin{array}{cc}\rho u_{i}\\\rho u_{1}u_{i}+p\delta _{1i}\\\rho u_{2}u_{i}+p\delta _{2i}\\\rho u_{3}u_{i}+p\delta _{3i}\\\rho Eu_{i}+pu_{i}\\\rho u_{i}k\\\rho u_{i}\omega \end{array}}\right\}}$

${\displaystyle F^{v_{i}}=\left\{{\begin{array}{c}0\\{\hat {t}}_{1i}\\{\hat {t}}_{2i}\\{\hat {t}}_{3i}\\{\hat {t}}_{ij}u_{j}-{\hat {q}}_{i}\\{(\mu +\mu _{t}/\sigma _{k})}{\frac {\partial k}{\partial x_{i}}}\\{(\mu +\mu _{t}/\sigma _{\omega })}{\frac {\partial \omega }{\partial x_{i}}}\end{array}}\right\}}$ (1)

Here Q stands for the vector of the averaged variables, which characterise the flow field solution in terms of ρ, ρui, ρE and the turbulent quantities ρk - ρω. All the governing equations within the flow field satisfy the same conservation balances obtained accounting for the accumulation, convection, diffusion, and production phenomena. The flux ${\displaystyle F_{i}}$ represents the convective transport while ${\displaystyle F^{v_{i}}}$ is the diffusive counterpart. The tensor ${\displaystyle {\hat {t}}_{ij}}$ introduces the total turbulent-laminar stresses, while ${\displaystyle {\hat {q}}_{i}}$ accounts for the effective heat transferred by conduction. Thanks to the Boussinesq assumption the total stresses and heat conduction terms are expressed using the effective viscosity ${\displaystyle {\hat {\mu }}=\mu _{lam}+\mu _{T}}$ and the effective Fourier coefficient ${\displaystyle {\hat {\lambda }}=\lambda _{lam}+c_{p}\mu _{T}/Pr_{T}}$ where ${\displaystyle Pr_{T}}$ is a constant turbulent Prandtl number. The contribution to the source vector comes from the turbulence production and dissipation terms of the k-ω model.

3.1.2 Transition Modelling

Two equation models represent traditionally a good compromise between accuracy and computational efficiency for prediction of turbulent flows. In order to verify the performances of two equation closures, several turbulence models of the two-equation eddy viscosity family have been implemented. The accurate boundary layer definition assumes a relevant importance in heat transfer load simulation especially on the blade suction side where laminar/turbulent transition generally occurs in the adverse pressure gradient region before the trailing edge. The comparison of the heat transfer load on the blade against experiments allows therefore a valuable analysis of the turbulence closure performances for laminar/turbulent boundary layer computation. Except for the first algebraic approach, two equations models have been considered for such a purpose in the present work:

• Baldwin-Lomax

• k-ω (low Re and High Re; Wilcox)

• k-ε (Launder-Sharma)

• cubic k-ω (Sofialidis-Prinos)

• k-ω (Kalitzin and Gould)

Despite the success of two-equation models for fully turbulent flows, several limitations arise when facing with the investigation of the transition problem, such as wrong location of the transition onset, under-prediction of transition length, too high sensitivity from initial conditions and weak dependency from the Mach number. In order to face transition modelling inaccuracies, the commonly used two-equations approaches are combined with integral methods or intermittency transport models. The integral methods attempt to characterise the boundary layer state by correlations based on the displacement or momentum thickness. Several correlations have been successfully proposed and tested in literature for turbomachinery applications such as the Abu-Ghannam and Shaw, 1980 and the Mayle, 1991. The Abu-Ghannam and Shaw correlation takes into account both the effect of turbulence level and pressure gradient. Conversely, focusing on turbulence levels above 3%, Mayle suggests a formula which accounts only for external turbulence intensity:

${\displaystyle Re_{\text{strans}}=400\cdot Tu^{-5/8}}$

Integral methods determine the transition onset accurately and predict transition for attached flows, but require a considerable degree of empiricism for practical application to three-dimensional flows. These models still need an extra correlation for the intermittency function distribution downstream the transition onset. According the definition the intermittency γ assumes values between 0 and 1 from laminar to fully turbulent flow. The literature offers a wide choice of solutions such as that proposed by Simon and Stephens 1991:

${\displaystyle \gamma =1-exp{(f{(Re_{{\text{strans}}^{5}})})}}$

being s the streamwise abscissa downstream the transition onset. An approach has also been suggested and tested by Michelassi and Rodi, 1997 using:

${\displaystyle \gamma ={\left({\frac {A_{r}^{+}}{A_{r}^{+}+\left(300-A_{r}^{+}\right)\left(1-\sin {\frac {\pi }{2}}\left({\frac {Re_{\theta }-Re_{\text{tr}}}{Re_{\text{tr}}}}\right)\right)}}\right)}^{\alpha }}$

The above expression suggests the shape of the function γ that reaches unity when ${\displaystyle Re_{\theta }=2\cdot Re_{tr}}$ and indicates also the transition length of the boundary layer downstream the onset point. The parameter a in equation controls the rise of g in the transition region (the larger the value of a, the longer is the transition length). The eddy viscosity computed from the turbulence model is therefore expressed as follows:

${\displaystyle \mu _{t}=y\cdot C_{\mu }\cdot {\frac {\rho k}{\omega }}}$

The concept of intermittency function is based on the existence of a transitional boundary-layer, which is between a laminar and turbulent state. Despite experimental results indicate that transitional profiles are not a simple combination of laminar/turbulent solutions, the simplified assumption behind g proves important success for practical applications (see Steelant and Dick, 1999). A simple application of the intermittency function approach can be defined coupling a two-equation closure to the intermittency function which is defined by the correlations defined above. A different approach suggested in literature is based on the evaluation of g from its own transport equation which is solved throughout the computational domain. In this case the lack of history effects of zero equation correlation is overcome thanks not only to the kinetic energy and turbulent dissipation transport equations but also from the solution of the evolution equation for the intermittency function which is meant to improve the model physical basis. According the suggestion of Vicedo et al. the transport equation for γ is defined as follows:

3.1.2 The Structured Approach

The structured solver XFLOS is an implicit code based on the scalar form of the approximate factorisation method as proposed by Pulliam in order to reduce the computational costs of the implicit approximate factorisation method by Beam and Warming. The implicit matrix to be inverted comes out to be scalar penta-diagonal with a considerable savings of memory and CPU requirements for its inversion. Several variants of the basic scalar penta-diagonal solver are included ranging from the scalar tri-diagonal to the tri and penta diagonal block solvers using different versions of LU decompositions coupled with a diagonal dominance enhance (d3adi). The spatial discretisation is performed in the rectangular computational image of the physical domain by means of centred finite differences with second order accuracy in space. The artificial dissipation scheme is based on guidelines given by Jameson, 1981 and consists of a non-linear blend of implicit-explicit second order plus fourth order differences of the solution. A parallel version of XFLOS is also available under the MPI environment.

3.1.3 The Unstructured Approach

The solver HybFlow performs a numerical discretization for the spatial gradients of the governing equation (1) using a cell centred finite volume scheme. The spatial discretization scheme is grid transparent and does not require any specific information on the local cell topology or element type, allowing a method which is suitable for generic unstructured hybrid grids. The computational domain is subdivided into an arbitrary set of 3D elements selected among exahedra, prisms, pyramids and tetrahedra covering without overlapping the whole computational domain. The numerical fluxes are computed integrating the flux function in equation (1) over all cell faces dividing every element from its neighbours. To this aim a simple midpoint quadrature formula is considered. The inviscid fluxes approximation is based on a reconstruction evolution approach. With the reconstruction phase the solution stored on the cell centres is interpolated onto faces mid-point considering a linear law. The solution monotonicity is enforced thanks to a slope limiting of the gradients using the solpe limiter defined by Barth, 1991 (Adami, 1998). The reconstructed values are considered to interact assuming an analogy with the evolution of a 1D and the resulting convective fluxes on the face are computed from solution of the Riemann problem with the Roe approximated method(Roe, 1986 and Barth, 1991). The viscous terms of governing equations are numerically integrated using a centred scheme based on the same FVM approach. The viscous stresses are computed on every element face using centred differences of the solution stored in neighbouring elements. As far as spatial discretization no difference arise between the flow equations and these of the turbulence model.

## References

Martelli F. - "Numerical Methodology for Turbomachinery " Invited Lecture at ERCOFTAC Seminar and Workshop on 3D Turbomachinery Flow Prediction II 10-13 January 1994 Val d'Iser , France

C.H. Sieverding, T. Arts, R. Denos, F. Martelli - "Investigation of the Flow Field Downstream of a Transonic Trailing Edge Cooled Nozzle Guide Vane" -ASME Trans Journal of Turbomachinery -Vol 118 , n.2. April 1996- Pg 291-301

Michelassi V., Martelli F., Adami P.- "An Implicit Algorithm for stator-rotor interaction Analysis" - presented ASME Turbo Expo (Gas Turbine Conference) June 10-13, 1996 Birmingham,UK

F. Martelli, V. Michelassi, (invited lecturer) “Numerical Simulation of Stator-Rotor Interaction in BRITE Turbine stage”, Lecture Notes on “Blade Row Interference Effects in Axial Turbomachinery Stages”, Von Karman Institute for Fluid Dynamics, 9-12 February 1998.

V. Michelassi, F. Martelli R. Denos, T. Arts, C.H. Sieverding “ Unsteady heat Transfer in Rotor-Stator Interaction Study By Numerical and experimental Approaches “ to be presented at ASME-IGTC June 1998-Stockholm, ASME Journal of Turbomachinery, July 1999, vol. 121, pp. 436-447.

Denos R., Arts T., Paniaagua G., Martelli F. Michelassi V. “ Investigation on the Unsteady Rotor Aerodynamics in a Transonic Turbine Stage”” ASME TURBO 2000 8-11 May 2000 Munich , Germany, ASME 2000 GT-634 Accepted for publication on J.of Turboamchinery

Adami P., Belardini E., Martelli F. “Development of an Unsteady Parallel Approach for 3D Stator-Rotor Interaction” 4th European Conference on Turbomachinery, Fluid Dynamics and Thermodynamics- Firenze 20-23 March 2001, Accepted for publication on ImechI Journal of Power and Energy

Michelassi, V., Giangiacomo, P., Martelli, F., “On the Choice of Variables and Matching Criteria for the Steady Simulation of Transonic Axial Turbine Stages”” 4th European Conference on Turbomachinery, Fluid Dynamics and Thermodynamics- Firenze 20-23 March 2001, Accepted for publication on ImechI Journal of Power and Energy