Unsteady Temperature Variation

The results presented in this paper were computed using three Newton sub­iterations per time-step and 2700 time-steps per cycle. Here, a cycle is defined as the time required for a rotor to travel a distance equal to the pitch length at midspan. To ensure time-periodicity, each simulation was run in excess of 80 cycles.

The variation of total enthalpy for the three in situ reheat cases and for the no combustion case is shown in Fig. 2. The abscissa indicates the axial location. S1 denotes stator 1, R1 denotes rotor 1, etc. The total enthalpy is calculated at inlet and outlet of each row. Depending on the row type, that is, stator or rotor, the total enthalpy is calculated using either the absolute or the relative velocity. The switch between using absolute or relative velocities generates discontinuities between rows. As shown in Fig. 2, for all fuel injection cases the total enthalpy increases compared to the no combustion case. The largest enthalpy increase is located on the first rotor, where most of the combustion takes place. The combustion and heat release continue throughout the second stator and rotor, as indicated by the total enthalpy variation shown in Fig. 2.

Figure 2. Variation of averaged total en – Figure 3. Variation of stagnation temper-

thalpy (absolute or relative) ature along first row of rotors for the case

without combustion and case 1 of in situ re­heat

The stagnation temperature variation along the first row of rotors is strongly inflienced by the in situ reheat, as shown in Fig. 3. Figure 3 shows the av­eraged, minimum and maximum stagnation temperature for the ft>w without combustion and for case 1 of fbw with combustion. On the pressure side, the averaged temperature of case 1 is approximately 180 K larger than the no combustion case temperature. At the leading edge, however, the averaged tem­perature of case 1 is approximately 70 K lower than in the no combustion case. On the suction side, the averaged temperature of case 1 is slightly higher than
in the no combustion case. On most of the suction side, the averaged tempera­ture of case 1 is approximately 15 to 20 K larger than the no combustion case temperature.

The averaged temperature indicates that combustion takes place on the pres­sure side of the rotor airfoil. The existence of small regions where the averaged temperature of the case with combustion is lower than the average temperature of the case without combustion indicates that combustion is not completed. Consequently, the low enthalpy of the fuel injected reduces locally the airfoil temperature. The maximum temperature of the case with combustion is larger than the maximum temperature of the no combustion case over the entire air­foil. On the pressure side, the minimum temperature of the case with combus­tion is larger than the minimum temperature of the case without combustion. On most of the suction side, however, the minimum temperature of the case with combustion is smaller than the minimum temperature of the case without combustion, indicating that the unburned, cold fuel injected is affecting this region.

Accuracy of Numerical Results

Currently there are no experimental data available for the validation of nu­merical simulation of transport phenomena in a turbine-combustor. To validate the accuracy of the numerical results corresponding to the governing equations used, it was necessary to show that the results were independent of the grid which discretizes the computational domain. The verification of grid indepen­dence results was presented in [Isvoranu and Cizmas, 2002], where a one-stage turbine-combustor was simulated. Note that the grids were generated such that, for the given fbw conditions, the y + number was less than 1. Approximately 20 grid points were used to discretize the boundary layer regions.

Following the conclusions of accuracy investigation presented in [Isvoranu and Cizmas, 2002], the medium grid was used herein, since it provides the best compromise between accuracy and computational cost. This grid has 53 grid points normal to the airfoil and 225 grid points along the airfoil in the O-grid, and 75 grid points in the axial direction and 75 grid points in the circumferen­tial direction in the H-grid. The stator airfoils and rotor airfoils have the same number of grid points. The inlet and outlet H-grids have each 36 grid points in the axial direction and 75 grid points in the circumferential direction. The grid is shown in Fig. 1, where for clarity every other grid point in each direction is shown.

Figure 1. Detail of the medium grid (every other grid point in each direction shown)

Geometry and Flow Conditions

The blade count of the four-stage turbine-combustor requires a full-annulus simulation for a dimensionally accurate computation. To reduce the compu­tational effort, it was assumed that there were an equal number of airfoils in each turbine row. As a result, all airfoils except for the inlet guide vane airfoils were rescaled by factors equal to the number of airfoils per row i divided by the number of airfoils per row one. An investigation of the influence of air­foil count on the turbine fbw showed that the unsteady effects were amplified when a simplified airfoil count 1:1 was used [Cizmas, 1999]. Consequently, the results obtained using the simplified airfoil count represent an upper limit for the unsteady effects.

The inlet temperature in the turbine-combustor exceeds 1800 K and the inlet Mach number is 0.155. The inlet fbw angle is 0 degrees and the inlet Reynolds number is 194,000 per inch, based on the axial chord of the first – stage stator. The values of the species concentrations at inlet in the turbine – burner are: yco2 = 0.0775, ун2о = 0.068, yco = 5.98 ■ 10-06, ун2 =

2.53 ■ 10-07, yO2 = 0.1131, yN2 = 0.7288 and yAr = 0.0125. The rotational speed of the test turbine-burner is 3600 RPM.

The unsteady effects of in situ reheat will be investigated by comparing the turbine without combustion against three cases of fuel injection in the turbine – combustor. Pure methane will be injected at the trailing edge of the first vane in all the cases of in situ reheat presented herein. The injection velocity and pressure, methane temperature and injection slot dimension are presented in Table 1.

Table 1. Parameters of fuel injection


Case 1

Case 2

Case 3

Injection velocity [m/s]




Pressure [bar]




Temperature [K]




Injection slot size [mm]




Boundary Conditions

Two classes of boundary conditions must be enforced on the grid bound­aries: (1) natural boundary conditions, and (2) zonal boundary conditions. The natural boundaries include inlet, outlet, periodic and the airfoil surfaces. The zonal boundaries include the patched and overlaid boundaries.

The inlet boundary conditions include the specification of the flow angle, average total pressure and downstream propagating Riemann invariant. The upstream propagating Riemann invariant is extrapolated from the interior of the domain. At the outlet, the average static pressure is specified, while the downstream propagating Riemann invariant, circumferential velocity, and en­tropy are extrapolated from the interior of the domain. Periodicity is enforced by matching ft>w conditions between the lower surface of the lowest H-grid of a row and the upper surface of the top most H-grid of the same row. At the airfoil surface, the following boundary conditions are enforced: the "no slip”
condition, the adiabatic wall condition, and the zero normal pressure gradient condition.

For the zonal boundary conditions of the overlaid boundaries, data are trans­ferred from the H-grid to the O-grid along the O-grid’s outermost grid line. Data are then transferred back to the H-grid along its inner boundary. At the end of each iteration, an explicit, corrective, interpolation procedure is per­formed. The patch boundaries are treated similarly, using linear interpolation to update data between adjoining grids [Rai, 1985].

2. Results

This section presents selected the results of the numerical simulation of un­steady transport phenomena inside a four-stage turbine-combustor. The sec­tion begins with a description of the geometry and ft>w conditions, followed by a brief discussion of the accuracy of numerical results. The last part of this section presents the effects of in situ reheat on the unsteady flow and blade loading.

Discretization of Governing Equations

The transport of chemical species is modeled by the mass, momentum, energy and species balance equations. These gas-dynamics and chemistry
governing equations are solved herein using a fully decoupled implicit algo­rithm. Further discussions on the coupled vs. decoupled algorithms for com­bustion problems can be found in [Eberhardt and Brown, 1986, Yee, 1987, Balakrishnan, 1987, Li, 1987]. A correction technique has been developed to enforce the balance of mass fractions. The governing equations are dis­cretized using an implicit, approximate-factorization, finite difference scheme in delta form [Warming and Beam, 1978]. The discretized operational form of both the Reynolds-averaged Navier-Stokes (RANS) and species conservation equations, combined in a Newton-Raphson algorithm [Rai and Chakravarthy,

1986], is:

where A and B are the fhx Jacobian matrices A = dF/dQ, B = dG/dQ. The Y and C matrices are Y = dS/dQ and C = dSch/dQ. Note that the flix Jacobian matrices are split into A = A+ + A-, where A± = PA±P-1. Л is the spectral matrix of A and P is the modal matrix of A. The spectral matrix Л is split into Л = Л+ + Л-, where the components of Л+ and Л – are A- = 0.5(Aj — |AiI) and A+ = 0.5(Ai + |AjI), respectively [Steger and Warming, 1981]. The same flux vector splitting approach is applied to the matrix B. In equation (5), A, V and 5 are forward, backward and central differences operators, respectively. Qp is an approximation of Q”+1. At any time step n, the value of Qp varies from Qn at first internal iteration when p = 0, to Qn+1 when integration of equation (5) has converged. Additional details on the implementation of the inter-cell numerical flixes and on the Roe’s approximate Riemann solver are presented in [Cizmas et al., 2003].

Numerical Model

The numerical model used herein is based on an existing algorithm devel­oped for unsteady fbws in turbomachinery [Cizmas and Subramanya, 1997]. The Reynolds-averaged Navier-Stokes equations and the species equations are written in the strong conservation form. The fully implicit, finite-difference approximation is solved iteratively at each time level, using an approximate factorization method. Three Newton-Raphson sub-iterations are used to re­duce the linearization and factorization errors at each time step. The con­vective terms are evaluated using a third-order accurate upwind-biased Roe scheme. The viscous terms are evaluated using second-order accurate central differences. The scheme is second-order accurate in time.

Grid Generation

The computational domain used to simulate the flow inside the turbine – combustor is reduced by taking into account flow periodicity. Two types of grids are used to discretize the flow field surrounding the rotating and station­ary airfoils, as shown in Fig. 1. An O-grid is used to resolve the governing equations near the airfoil, where the viscous effects are important. An H-grid is used to discretize the governing equations away from the airfoil. The O-grid is generated using an elliptical method. The H-grid is algebraically gener­ated. The O – and H-grids are overlaid. The fbw variables are communicated between the O – and H-grids through bilinear interpolation. The H-grids corre­sponding to consecutive rotor and stator airfoils are allowed to slip past each other to simulate the relative motion.

Chemistry Model

The chemistry model used herein to simulate the in situ reheat is a two-step, global, finite rate combustion model for methane and combustion gases [West­brook and Dryer, 1981, Hautman et al., 1981]

CH4 + 1.5O2 ^ CO + 2H2O

CO + 0.502 ^ CO2. (2)

The rate of progress (or Arrhenius-like reaction rate) for methane oxidation is given by:

qi = Ai exp (Ei/Rm/T) [CHJ-0’3 [O2j1’3 , (3)

where A1 = 2.8 ■ 109 s-1, E1/Rm = 24360 K. The reaction rate for the CO/CO2 equilibrium is:

q2 = A2 exp (E2/Rm/T) [COj [O2]0’25 [H2Oj0’5

with A2 = 2.249 ■ 1012 (m3/kmol)°’75 s-1 and E2/RM = 20130 K. The symbols in the square brackets represent local molar concentrations of various species. The net formation/destruction rate of each species due to all reactions is:


Wi = ^2 MiVikqk, k=i

where vik are the generalized stoichiometric coefficients. Note that the gen­eralized stoichiometric coefficient is vik = vik — vik where v’ik and v"k are stoichiometric coefficients for species i in reaction k appearing as reactant or as a product. Additional details on the implementation of the chemistry model can be found in [Isvoranu and Cizmas, 2002].

Physical Model

The flow and combustion through a multi-row turbine-burner with arbitrary blade counts is modeled by the Reynolds-averaged Navier-Stokes equations and the species conservation equations. To reduce the computational time, the flow and combustion are modeled as quasi-three-dimensional. This section will present the details of the governing equations and the chemistry model.

Governing Equations

The unsteady, compressible flow through the turbine-combustor is modeled by the Reynolds-averaged Navier-Stokes equations. The ft>w is assumed to be fully turbulent and the kinematic viscosity is computed using Sutherland’s law. The Reynolds-averaged Navier-Stokes equations and species conserva­
tion equations are simplified by using the thin-layer assumption [Isvoranu and Cizmas, 2002].

In the hypothesis of unity Lewis number, both the Reynolds-averaged Navier – Stokes and species equations can be written as [Balakrishnan, 1987]:

dQ dF_ dG Vt^Mqq dS

дт dij Reoo dij ch

Note that equation (1) is written in the body-fitted curvilinear coordinate sys­tem (f, n, t ).

The state and flix vectors of the Reynolds-averaged Navier-Stokes equa­tions in the Cartesian coordinates are






f ns ___

pu2 + p puv

nns ___

, П



pv2 + p


(e + p) u

( e + p) v

The state and flix vectors of the species conservation equations in the Cartesian coordinates are

py 1




, fsp =


,nsp =


_ pyN _




Further details on the description of the viscous terms and chemical source terms are presented in [Cizmas et al., 2003].


Horia C. Flitan and Paul G. A. Cizmas

Department of Aerospace Engineering, Texas A&M University College Station, Texas 77843-3141 horiaf@plano. tamu. edu and cizmas@tamu. edu

Thomas Lippert and Dennis Bachovchin

Siemens Westinghouse Power Corporation Pittsburgh, Pennsylvania

Dave Little

Siemens Westinghouse Power Corporation Orlando, Florida

Abstract This paper presents a numerical investigation of the unsteady transport phe­nomena in a turbine-combustor. The fbw and combustion are modeled by the Reynolds-averaged Navier-Stokes equations coupled with the species conserva­tion equations. The chemistry model used herein is a two-step, global, finite rate combustion model for methane and combustion gases. The governing equations are written in the strong conservation form and solved using a fully implicit, finite difference approximation. This numerical algorithm has been used to in­vestigate the airfoil temperature variation and the unsteady blade loading in a four-stage turbine-combustor. The numerical simulations indicated that in situ reheat increased the turbine power by up to 5.1%. The turbine combustion also increased blade temperature and unsteady blade loading. Neither the tempera­ture increase nor the blade loading increase exceeded acceptable values for the turbine investigated.

Keywords: Turbine-combustor, in situ reheat, unsteady flow, turbine flow

1. Introduction

In the quest to increase the thrust-to-weight ratio and decrease the thrust specific fuel consumption, turbomachinery designers are facing the fact that


K. C. Hall et al. (eds.),

Unsteady Aerodynamics, Aeroacoustics and Aeroelasticity of Turbomachines, 551-566. © 2006 Springer. Printed in the Netherlands.

the combustor residence time can become shorter than the time required to complete combustion. As a result, combustion would continue in the turbine, which up to recently has been considered to be undesirable. A thermody­namic cycle analysis demonstrates performance gains for turbojet engines with turbine-burner [Sirignano and Liu, 1999]. Even better performance gains for specific power and thermal efficiency are predicted for power generation gas – turbine engines when the turbine is coupled with a heat regenerator.

The process of combustion in the turbine is called in situ reheat and the turbine in which combustion takes place is called turbine-burner. The fuel is commonly injected in the turbine-burner through the vanes. Several challenges are, however, associated with the combustion in the turbine-burner: mixed sub­sonic and supersonic fews; fews with large unsteadiness due to the rotating blades; hydrodynamic instabilities and large straining of the flow due to the very large three-dimensional acceleration and stratified mixtures [Sirignano and Liu, 1999]. The obvious drawback associated with the strained fbws in the turbine-burner is that widely varying velocities can result in widely vary­ing residence time for different flow paths and as a result there are flammabil­ity difficulties for regions with shorter residence times. In addition, transverse variation in velocity and kinetic energy can cause variations in entropy and stagnation entropy that impact heat transfer. The heat transfer and mixing may be enhanced by increasing interface area due to strained flows.

Turbine aerodynamics might be drastically modified by strong exothermic combustion processes in a turbine-burner. Thermal expansion due to combus­tion could significantly change the pressure variation and the shock strength and location. As a result, the blade loading would be modified. There is evidence on a low pressure turbine without in situ reheat, that the tempera­ture non-uniformities can generate strong entropic and vortical waves. These waves produced excitations large enough to generate unsteady loadings and stresses on the 5th stage of alow pressure turbine, sufficient to cause high-cycle fatigue failures of a disk/blade/tip-shroud system mode crossing [Manwaring and Kirkeng, 1997]. The danger of high-cycle fatigue is even more imminent for a turbine-burner because larger temperature non-uniformities are likely to produce stronger entropic and vortical waves.

Experimental data for conventional (i. e., without in situ reheat) gas-turbines have shown the existence of large radial and circumferential temperature gra­dients downstream of the combustor [Dills and Follansbee, 1979, Elmore et al., 1983]. These temperature non-uniformities, called hot streaks, have a signifi­cant impact on the secondary flow and wall temperature of the entire turbine. Since the combustor exit few may contain regions where the temperature ex­ceeds the allowable metal temperature by 260-520°C [Butler et al., 1989], un­derstanding the effects of temperature non-uniformities on the flow and heat transfer in the turbine is essential for increasing vane and blade durability. It is estimated that an error of 55oC in predicting the time-averaged tempera­ture on a turbine rotor can result in an order of magnitude change in the blade life [Graham, 1980, Kirtley et al., 1993].

Temperature non-uniformities generated by the upstream combustor can be amplified in a turbine-burner. Consequently, it is expected that not only the secondary ft>w and wall temperature be affected but also the blade loading due to the modified pressure distribution. Temperature non-uniformities in a turbine-burner can also affect the location of hot spots on airfoils and as a result can impact on the internal and film cooling schemes.

There are extensive experimental [Whitney et al., 1980, Schwab et al., 1983, Stabe et al., 1984, Butler et al., 1989, Sharma et al., 1992, Shang et al., 1995] and numerical [Rai and Dring, 1990, Krouthen and Giles, 1988, Taka – hashi and Ni, 1991, Shang and Epstein, 1996, Dorney et al., 2000, Dorney et al., 1999] results for the infhence of temperature non-uniformities on the flow and heat transfer in a conventional turbine. To the best knowledge of the authors, however, there are no data available in the open literature for the effect of in situ reheat on turbine-burners. The objective of this paper is to investigate the effects of in situ reheat on the unsteady aerothermodynamics in a multi-stage turbine-combustor. This numerical simulation is crucial for the development of turbine-burners which, in spite of their challenges, can pro­vide significant performance gains for turbojet engines and power generation gas-turbine engines.

The next section presents the physical model used for the simulation of flow and combustion in the turbine-combustor. The governing equations and the chemistry model are presented. The third section describes the numerical model. This section includes information about the grid generation, boundary conditions, numerical method and parallel algorithm. The results are presented in the fourth section.

Flow Details

In contrast to a less labour – and CPU-intensive set-up with source terms (Kluge et. al. (2003)) the meshing of every single cooling hole, including the plenum provides a much higher level of detailed information. Since the local

Steady, Mixing Plane

o-o Unsteady, Time Averaged

— Unsteady, Time Dependent

Normalized Axia Distance

Figure 9. Heat Transfer Coefficient in NGV

fbw conditions at the cooling hole exits are not longer a fixed boundary con­dition as in the source term approach, the local blowing rate has the freedom to adapt itself according to the local few conditions. Consequently, the local blowing rate varies from hole to hole, obvious in the distribution of the heat transfer coefficient in Figure 9.

The cooling flow enters the blade passage in a typical flow pattern (Fig. 10). In dependence on the inclination angle, the local blowing rate and the shape of the cooling hole, the emerging jet acts much like a solid obstacle. The incoming boundary layer of the main flow rolls up into a horseshoe vortex, causing a counter-rotating kidney vortex behind the jet. (Hildebrandt et. al. (2002), Wilfert (1994)). This vortex configuration is responsible for the hot gas entrainment beneath the cool air, a distinct and undesired feature of cylindrical cooling holes.


Unsteady calculations of a transonic film cooled turbine stage where the cooling holes and the cold air plenum is discretized represent a high level

of very detailed information from the fbw. Clearly, on the downside of this approach are the high CPU requirements and the quite labour intense pre­processing. Both limitations prohibit the use of such a method in the frame of the daily design work in industry, which is characterized by short turn around times. The source term approach, presented in Kluge et. al. (2003) is more suitable in such an environment, but suffers not only from a lack of detailed fbw information, but more important from an uncertainty in the specification of the correct boundary condition for the source terms. Here, a full discretiza­tion offers the advantage that no boundary conditions are necessary on the exit surface of the cooling holes as long as the plenum is taken into account. How­ever, the boundary conditions for the plenum are relatively straightforward to obtain. An option is proposed to combine these two approaches. First, a set of fully discretized simulations are conducted for typical configurations and operating conditions. From these results, boundary conditions for the source term approach can be derived in order to calibrate the source term boundary conditions. But even then, the immediate vicinity of the cooling holes will be better captured using a full discretization of holes and plenum.


The reported work was carried out under the contract of the European Com­mission as part of the BRITE EURAM project contract number BRPR-CT-97- 0519, Project number BE97-4440 (TATEF).The authors wish to acknowledge the financial support as well as the contributions from ALSTOM POWER,



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