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Heat transfer over rotor surface in a pre-swirl rotating-disc system.

Abstract

Pre-swirl is often used in the internal cooling-air systems of gas turbines to reduce the temperature of the cooling air relative to the rotating turbine blades. In a "direct-transfer" system, the air passes axially across the wheel-space from stationary pre-swirl nozzles to receiver holes in the rotating turbine disc. This paper investigates heat transfer over rotating disc of such a system, using a 3D steady, incompressible turbulent flow in rotating frame of reference. Computed results are compared with measurements of static and total pressure coefficients and heat transfer coefficients obtained by [4]. There is mainly good agreement between computed and measured values of pressure coefficients. There are less agreement between measured and computed local hest transfer coefficients. The computed as well as measured local hest transfer coefficients shows axisymmetric distribution on rotor except near receiver cooling holes in which a small region of high heat transfer is observable.

Keywords: Numerical methods, pre-swirl systems, gas turbine cooling system.

Introduction

Fig. 1 shows the flow of the cooling air due to confined discs of typical internal-air system of gas-turbines. The exit of the compressor is the source for the turbine blade-cooling air. The blade-cooling air is usually supplied to the rotating high-pressure blades by stationary pre-swirl nozzles. The cooling air is swirled, which reduces the work done by the rotating turbine disc in accelerating the air to the disc speed. This in turn reduces the total temperature of the air entering the receiver holes in the disc. There is a need to understand the heat transfer between the cooling air and the turbine disc, particularly the possible creation of local non-uniform temperatures in the metal that could lead to large thermal stresses.

A rotor-stator system, as shown in Fig. 2, provides a simplified model for the flow and heat transfer that occurs in the wheel-space between an air-cooled turbine disc and adjacent stationary casing. This is known as "direct-transfer" pre-swirl system and such arrangement is used in some gas turbine engines. The pre-swirl nozzles are located at a low radius on the stator and the cooling air flows radially outward to the receiver holes through the rotating cavity between the stationary disc and rotating disc.

Enim et al. [1] studied "direct transfer" pre swirl systems using the commercial multi-purpose CFD code Fluent. The computations were carried out using a steady-state 3D method and a so-called "frozen rotor" approach for treating the interface between the stationary and rotating domains.

[FIGURE 1 OMITTED]

Geis et al. [2] measured the cooling efficiency of a pre-swirl rotor-stator system equipped with a small number of pre-swirl nozzles of circular shape, located on a radius equal to that of the receiver holes. They compared their experimental data with a simple theoretical model, which predicted air temperatures in an "ideal" pre-swirl system. It was found that the pre-swirl system performed worse, in terms of cooling air temperature reduction, than was expected for isentropic flow.

The flow and heat transfer in a "direct-transfer" pre-swirl model of pre-swirl rotor-stator system in which pre-swirl nozzles and blade cooling holes were located at same radius were studied by Wilson et al. [3], who found that that the pre-swirl flow mixed fully with a superposed radial outflow of disc-cooling air before entering the receiver holes. Greater losses in total pressure are expected for direct transfer system compared with the free-vortex flow found in other arrangement of rotor-stator system, known as "cover-plate" arrangement, owing to strong mixing between the pre-swirl flow and the re-circulating rotor-stator flow in the chamber.

Yan et al. [4] carried out measurements and three dimensional computations for the flow structure in an idealised pre-swirl rotor-stator system, and Farzaneh et al. [5] described an investigation of the heat transfer in the same system where blade cooling holes were modeled as a ring. The results obtained show that the flow in the pre-swirl system has some similarities with that found in classical rotor-stator systems. The measurements and computations showed that significant losses in total pressure occurred between the inlet nozzles and the mid-axial plane between the rotor and stator (where pitot-tube measurements were made). These mixing losses, which were caused by a momentum exchange between the primary pre-swirl flow and the recirculation secondary flow, increased as the inlet pre-swirl ratio increased.

[FIGURE 2 OMITTED]

Karabay [6] carried out a combined experimental and computational study of flow in a "cover-plate" pre-swirl system. The cooling air from the stationary pre-swirl nozzles flowed radially outward (to the receiver holes) in a rotating cavity formed by the rotating disc and a cover-plate attached to it. Free vortex flow was found to occur for this system, and a theoretical analysis was used to show that there was an optimal value of the pre-swirl ratio, for which the average Nusselt number, Nu, for a heated rotating disc would be a minimum.

Earlier research into direct transfer systems, and rotor-stator systems in general, was described by Owen and Rogers [7] and Owen and Wilson [8] gave a brief review of more recent heat transfer research.

Lewis et al. [9] have compared heat transfer measurements from a pre-swirl rotor-stator experiment with 3D steady state results obtained from a commercial CFD code. The computed and measured contours of Nu showed that there is a small region of high heat transfer close to the receiver holes. They concluded this is due to the two routes by which flow enters the holes: a "direct" route from the pre-swirl nozzles and an "indirect" route from the core.

In this paper, an in-house 3 dimensional code has been developed to study heat transfer over a rotating disc. The regions of high heat transfer are of importance for designers as they may result in thermal stresses. As 3D steady code has been developed in rotating frame of reference, in order to model blade cooling holes, and the pre-swirl nozzles were modeled as an annular inlet. This allows computing suspected high local heat transfer coefficients around the blade cooling holes. For validating the numerical method, the measured data and computed values of [4] and [5] are also compared with computation values

Governing equations

The three dimensional, steady state, incompressible Reynolds-averaged flow equations in a cylindrical polar coordinate system r, [phi], z and in rotating frame of reference with velocity components [V.sub.r], [V.sub.[phi]], [V.sub.z] can be written in the common form:

1 / r [partial derivative] / [partial derivative]r ([rho]r[V.sub.r][PHI]) + 1 / r [partial derivative] / [partial derivative][phi] ([rho][V.sub.[phi]][PHI]) + [partial derivative] / [partial derivative]z ([rho][V.sub.z][PHI])

= 1 / r [partial derivative] / [partial derivative]r (r[[GAMMA].sub.r] [partial derivative][PHI] / [partial derivative]r) + 1 / [r.sup.2] [partial derivative] / [partial derivative][phi] ([[GAMMA].sub.[phi]] [partial derivative][PHI] / [partial derivative]z) + [S.sub.[PHI]] (1)

where [PHI] represents the generalized momentum variable and the net source [S.sub.[PHI]] is different for each component of momentum. [[GAMMA].sub.r], [[GAMMA].sub.[PHI]], and [[GAMMA].sub.z] are the effective diffusivities for the radial, circumferential and axial directions comprising both laminar and turbulent components. In the momentum equation, P denotes reduced pressure which is defined as:

P = [P.sub.s] - [rho][r.sup.2][[OMEGA].sup.2] (2)

where [P.sub.s] is the pressure in fixed frame of reference and [OMEGA] is the angular speed.

In this paper, turbulent flow computations have been made using the low-Reynolds number k - [epsilon] turbulence model proposed by Launder-Sharma [11] with turbulent heat transfer represented using a turbulent Prandtl number, [Pr.sub.t], equal to 0.9.

The k - [epsilon] turbulence model equations can be represented in the same common form of equation (1). The relevant expressions are given in Table (1), in which [[mu].sub.eff] = [mu] + [[mu].sub.t] is the effective viscosity.

In the k - [epsilon] equations, [P.sub.e] denotes the rate of production of turbulent kinetic energy and is given as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)

Other terms appearing in the k-[epsilon] equations are given in Table 2. The wall damping function, [f.sub.[mu]], is associated with low Reynolds number models, [R.sub.t] is a local turbulence Reynolds number and [y.sup.+] is the non-dimensional distance from the solid surface:

[R.sub.t] = [rho][[kappa].sup.2] / [mu][epsilon] (4)

and

[y.sup.+] = [y.sub.min] [square root of [[tau].sub.w] / [rho]] / v (5)

In equation (5), [y.sub.min] is taken as the minimum distance between the wall and the mesh point and [[tau].sub.w] is the average wall shear stress.

Yap empirical correction, YC, is added to source term of 0 equation intending to reduce unrealistically large levels of near-wall turbulence that are returned by the Launder-Sharma model in regions of flow separation. Craft et al [14] showed that the unrealistic large peak in Nusselt number prediction for impinging flows can be improved with the use of the Yap correction term as:

YC = max[ 0.83 (l / [l.sub.[theta]] - 1)(l / [l.sub.[theta]]) 2[[epsilon].sup.2] / k, 0)] (6)

where

[l.sub.[theta]] = [k.sup.1.5] / [epsilon] (7)

and l = 2.55y with y the normal wall distance.

Numerical Method

Computational procedure

The governing equations were discretised using a finite-volume method with hybrid-differencing for convection terms. The SIMPLE approach is adopted within staggered grid arrangement, and the algebraic equations were solved using a tri-diagonal matrix algorithm (TDMA) iteratively in an ADI fashion. A Gosman [11] type damping function, GSF, (as in equation 8) was used to improve convergence for radial momentum equation.

GSF = [[alpha].sub.G] [rho][absolute value of [V.sub.[phi]]] / r ([V.sup.old.sub.r] - [V.sup.new.sub.r]) (8)

where [[alpha].sub.G] is an empirical constant.

Geometry and grid distribution

A schematic diagram of the geometry modeled is shown in Fig. 3. The red shaded areas being stationary whilst the blue shaded areas rotate with angular velocity [OMEGA]. Axial and circumferential velocities are prescribed at the inlet to give the flow angle of 20[degrees] to the tangential direction.

[FIGURE 3 OMITTED]

It is based on an experimental rig of [4] in which there were 24 pre-swirl nozzles, angled at 20[degrees] to the tangential direction, in the stator, and 60 axial receiver holes in the rotor. The radial location of the nozzles, for which [x.sub.p] = [r.sub.p]/b = 0.74, was less than the centerline radius of the blade cooling holes at [x.sub.b] = [r.sub.b]/b = 0.93.

[FIGURE 4 OMITTED]

The 3D steady incompressible-flow model involves one of the blade-cooling holes on the rotor in rotating frame of reference with cyclic symmetry conditions applied at the tangential faces of the domain. In order to permit steady flow computations, an annular inlet is used on the stator (see Fig. 3).

This inlet matches the centreline radius and total flow area of the 24 pre-swirl inlets. In order to satisfy the Low-Reynolds k - [epsilon] model requirements ([y.sup.+] < 0.5), a large number grid points was packed near the wall and to model the inlet and outlet, fine grid used in these area in an attempt to keep the expansion/contraction parameter lower than 1.2, The grid distribution tests showed that 140 x 297 x 12 grid points (axial x radial x tangential) was required. The mesh is illustrated in Fig. 4.

Boundary conditions

No-slip boundary condition was used for the velocity components on the solid surface. The uniform axial velocity was used from the specified mass flow rate. At the outlet, the uniform axial velocity was used to ensure continuity, and tangential velocities were computed from a zero normal derivative condition. The radial velocity component was set to zero at inlet and outlet.

For heat transfer computation, adiabatic thermal boundary conditions were used for solid surfaces other than for the rotating disc for which was kept constant at 20[degrees]C. It should be also pointed out that a fitted value to measured adiabatic wall temperature for a typical case does not change the computed value of heat transfer coefficients. The inlet flow total temperature was kept constant at 55[degrees]C, and a zero normal-derivative condition was used for the outlet. Fluid properties were calculated at 20[degrees]C, as in the experiments [4].

In this paper, cyclic boundary conditions were applied in tangential direction. For N receiver cooling holes, this enables modeling only 360[degrees]/N angular sector to represent the whole system.

Referring to Fig. 4, a summary of boundary condition which were used, are given in Table 3.

Range of flow parameters

According to [7] the most effective dimensionless parameters that control flow inside a rotor-stator system are inlet swirl ratio, [beta]p, non-dimensional mass flow rate, [C.sub.w], rotational Reynolds number, [Re.sub.[phi]], and turbulent flow parameter, [[lambda].sub.T]. It is also shown by [8] that rotational Reynolds number is less effective parameter in cover plate pre-swirl arrangement. Yan et al. [4] are also concluded that the flow structure in the pre-swirl chamber is controlled principally by the pre-swirl ratio and the turbulent flow parameter.

The parameter ranges covered numerically are same as in [4] & [5] which were as follows:

0.78 x [10.sup.6] < [Re.sub.[phi]] < 1.2 x [10.sup.6]

0.6 x [10.sup.4] < [C.sub.w] < 2.8 x [10.sup.4] (0.11 < [[lambda].sub.T] < 0.36)

0.5 <[[beta].sub.p] < 3.0

The ranges of [[lambda].sub.T] and [[beta].sub.p] are typical of those used in practice. Maximum [Re.sub.[phi]] value is an order of magnitude smaller than that found in gas turbines. Although a comprehensive computational work has been carried out, only results for a case which [Re.sub.[phi]] = 0.78x[10.sup.6], [C.sub.w] = 6600, [[lambda].sub.T] = 0.127 and [[beta].sub.p] = 0.52 are presented. The same results also obtained for the other cases.

Results and Discussion

Radial variation of static and total pressure

The static pressure coefficient is defined as

[C.sub.p] = P - [P.sub.p] / 1/2 [rho][OMEGA][r.sup.2.sub.p] (9)

where P is the static pressure on the stator and [P.sub.p] is the computed static pressure at the pre-swirl inlet. Measured values for [C.sub.p] are calculated in the same way (see [4]). It should be noted that the datum for P was set by matching the computed and measured values at x = 0.89, consequently the measured value of [C.sub.p] at inlet is not equal zero.

Total pressure coefficient is defined as:

[C.sub.o,p] = [P.sub.o,[infinity]] - [P.sub.o,p] / 1/2 [rho][OMEGA][r.sup.2.sub.p] (10)

where

[P.sub.o,[infinity]] = [P.sub.[infinity]] + 1/2 [rho][V.sup.2.sub.[phi],[infinity]] (11)

For the dynamic term, [V.sub.[PHI]] is the most significant velocity and, for comparison with measured data, the pitot tubes used in experiment were oriented tangentially.

[FIGURE 4 OMITTED]

The radial variation of static and total pressure coefficients in mid rz plane are shown in Figure 4a and 4b. The measured and computed results of [4] are also compared. As it can be seen static pressure coefficients show good agreement with measured values. The computation values of total pressure coefficients over predict slightly measured values.

Radial and distributed variation of local heat transfer coefficient

The local heat transfer coefficient number, h, is defined as:

h = q / ([T.sub.w] - [T.sub.aw]) (12)

where

[T.sub.aw] = [T.sub.a] + 1 / 2[c.sub.p] {R[([OMEGA]r - [V.sub.[phi],[infinity]]).sup.2] - [V.sup.2.sub.[phi],[infinity]]} (13)

In Eq. (13) [T.sub.aw] is based on the theoretical adiabatic-wall temperature given by [6], although making use of the measured total temperature of the inlet air, [T.sub.a]. The transient heat transfer experiments give rise to a time-varying distribution of temperature [T.sub.w] over the unheated rotor; for the steady-state computations, the initial air temperature measured prior to the start of each transient test was used as the uniform disc temperature.

Fig. 4 shows computed radial variation of local heat transfer coefficients between holes. Measured values of [4] are also compared with computed results. The computed results show a large peak around the pre-swirl nozzle radius, [x.sub.p] = 0.74. This peak results from the use of a low-Reynolds-number, isotropic turbulence model in a region where anisotropic impinging-flow effects occur near the rotating disc surface (due to the axial component of the pre-swirl flow). Pilbrow et al [13] found similar behaviour for a different pre-swirl system, and tested an alternative to the Launder-Sharma low-Reynolds-number model used here. They show, in general, a lower peak in the corresponding impingement region. However, this model did not improve the results in the current study.

[FIGURE 5 OMITTED]

Fig. 5 shows distributions of computed local heat transfer coefficients over the rotating disc surface, As it can be seen, it shows nearly axisymmetric behaviour except in the regions around and between blade cooling holes in which a small region of high heat transfer is observable. (In an engine, high heat transfer in this region could result in thermal stresses within the rotor.)

The same result also obtained by Lock et al [12] using liquid crystal measurement. Their results also show generally good repeatability of heat transfer coefficients obtained between different pairs of holes

Conclusions

A computational study has been carried out to investigate the effects of blade cooling holes on heat transfer coefficients over rotating disc in a pre-swirl rotating-disc system, representative of those found in gas-turbine cooling systems. Measured and computed values of [4] and [5] are also compared with computed results. Computed values of static and total pressure coefficients show a good agreement with computed values.

Computation values of heat transfer coefficients mainly underpredicted measured values, except in the region around the pre-swirl nozzle radius where inertial "impingement" effects are predicted poorly by the low-Reynolds-number k-[epsilon] turbulence model used.

Distributed heat transfer coefficients over rotating disc show nearly axisymmetric behaviour except in the regions around and between blade cooling holes in which a small region of high heat transfer is observable. The regions of high heat transfer are of importance for designers as they may result in thermal stresses around the receiver holes in turbine discs.

Acknowledgments

The author would like to thank the reviewers for their constructive feedback during the review process.
Nomenclature

a, b inner, outer radius of disc

[c.sub.p] specific heat of air at constant pressure

[C.sub.w] nondimensional mass flow rate (=[??] /[mu]b)

d pre-swirl nozzle diameter

G gap ratio (= s/b)

k turbulence kinetic energy, thermal conductivity of air

[??] mass flow rate

N number of pre-swirl nozzles (or blade cooling holes)

Pr Prandtl number (=mcp /k)

Q heat flux (from air to disc)

r, [phi], z radial, tangential and axial coordinates

R recovery factory (=Pr1/3)

[Re.sub.[phi]] Rotational Reynolds number

s axial spacing between stator (z=0) and rotor (z=s)

T temperature

[T.sub.a] total temperature of pre-swirl air at inlet

[T.sub.w] temperature at rotor disc surface (wall)

[U.sub.t] friction velocity (= [square root of [[tau].sub.w]/
 [rho])]

[V.sub.r], time-averaged radial, circumferential, axial
[V.sub.[phi]], components of velocity in stationary frame
[V.sub.z]

x nondimensional radius (= r/b)

y distance normal to the wall

[y.sup.+] nondimensional distance (= [rho]y[U.sub.[tau]]/[mu])

z axial distance

[beta] swirl ratio (= [V.sub.[phi]] /[[OMEGA].sub.r])

[[beta].sub.p] pre-swirl ratio (= [V.sub.[phi],p] /[OMEGA][r.sub.p])

[epsilon] turbulence energy dissipation rate

[[lambda].sub.T] turbulent flow parameter (= [C.sub.w] /
 [Re.sub.[phi].sup.0.8])

[mu] dynamic viscosity

[rho] density

[[tau].sub.w] wall shear stress

[OMEGA] angular speed of disc

Subscripts

Aw adiabatic wall value

b blade-cooling air, at receiver hole radius

p pre-swirl air, at pre-swirl nozzle radius

s sealing air

[infinity] mid-plane z/s=0.5 (outside of boundary layers)


References

[1] Eenim, A.C., Brillert D. and Cagan M., 2004, Investigation into the computational analysis of direct-transfer pre-swirl systems for gas turbine cooling' ASME paper GT2004-54151

[2] Geis T., Dittmann M. and Dullenkopf K., 2003, Cooling air temperature reduction in a direct transfer preswirl system, ASME paper GT2003-38231

[3] Wilson, M, Pilbrow, R. and Owen, J. M. (1997) Flow and heat transfer in a pre-swirl rotor-stator system, J. Turbomachinery, v. 119, pp. 364-373.

[4] Yan, Y., Farzaneh-Gord, M., Lock, G. D., Wilson, M. and Owen, J. M, 2003, Fluid dynamics of a pre-swirl rotor-stator system, Journal of Turbomachinery, vol. 125, pp. 641-647.

[5] Farzaneh-Gord, M., Wilson, M. and Owen, J. M., 2003, Effects of swirl and flow rate on the flow and heat transfer in a pre-swirl rotating-disc system International Gas Turbine Congress, Tokyo, November, paper TS-064

[6] Karabay, H., Wilson, M. and Owen, J. M., 2001, Predictions of effect of swirl on flow and heat transfer in a rotating cavity, Int. J. Heat Fluid Flow, v. 22, pp. 143-155.

[7] Owen, J. M. and Rogers, R. H. (1989) Flow and heat transfer in rotating disc systems: Vol. 1, Rotor-stator systems, Research Studies Press, Taunton, UK and John Wiley, NY

[8] Owen, J. M. and Wilson, M., 2000, Some current research in rotating-disc systems, Turbine 2000 Int. Symp. on Heat Transfer in Gas Turbine Systems, Turkey, August 13-18, in Heat Transfer in Gas Turbine Systems, Annals of the New York Academy of Sciences, v 934, pp. 206-221.

[9] Lewis P., Wilson M., Lock G., Owen J. M., Physical interpretation of flow and heat transfer in pre-swirl systems, ASME Turbo Expo 2006, May 8-11, 2006, Barcelona, Spain.

[10] Launder, B.E., Sharma, B.L. 1974, Applications of the energy dissipation model of turbulence to the calculation of flow near a spinning disc. Letters in Heat and Mass Transfer, 1, pp. 131-138.

[11] Gosman, A.D. and Iferiah F.J.K., 1976, Tech-T: A general computer program for two dimensional turbulent recirculating flows in calculations of recirculating flows, Mech. Eng. Dept., Imperial college, Univ. of London.

[12] Lock, G. D., Yan, Y., Newton, P. J., Wilson, M. and Owen, J._M., 2003, Heat transfer measurements using liquid crystal in a pre-swirl rotating-disc system, ASME paper GT-2003-38123

[13] Pilbrow, R., Karabay, H., Wilson, M. and Owen, J. M., 1999, Heat transfer in a "cover-plate" pre-swirl rotating-disc system, J. Turbomachinery, v. 121, pp. 249-256.

[14] Craft, T.J., Graham, L.J.W. and Launder, B.E. 1993 Impinging jet studies for turbulence model assessment-II. An examination of the performance of four turbulence models. Int. J. Heat Mass Transfer, 36, pp. 2685-2697.

Mahmood Farzaneh-Gord

Faculty of Mechanical Engineering,

Shahrood university of Technology,

Shahrood, Iran

E-mail: mgord@shahroodut.ac.ir
Table 1: The components of the transport equations.

[phi] [[GAMMA].sub.Z] [[GAMMA].sub.r] [[GAMMA].sub.[phi]]

1 0 0 0

[V.sub.r] 2[[mu].sub.t] [[mu].sub.t] [[mu].sub.t]
 + [mu] + [mu] + [mu]

[V.sub.[phi]] [[mu].sub.t+] 2 [[mu].sub.t] [[mu].sub.t] + [mu]
 [mu] + [mu]

[V.sub.Z]] 2[[mu].sub.t] + [[mu].sub.t] + [[mu].sub.t] + [mu]
 [mu] [mu]

T [mu]/Pr + [mu]/ [mu]/Pr + [mu]/Pr +
 P[r.sub.t] [[mu].sub.t]/ [[mu].sub.t]/
 P[r.sub.t] P[r.sub.t]

k [mu] + [mu] + [mu] +
 [[mu].sub.t]/ [[mu].sub.t]/ [[mu].sub.t]/
 [[sigma].sub.k] [[sigma].sub.k] [[sigma].sub.k]

[epsilon] [mu] + [mu] + [mu] + [[mu].sub.t]
 [[mu].sub.t]/ [[mu].sub.t]/ /[[sigma].sub.
 [[sigma].sub. [[sigma].sub. [epsilon]]
 [epsilon]] [epsilon]]

[phi] [S.sub.[phi]]

1 0

[V.sub.r] - [partial derivative]/[[partial derivative].sub.r] +
 [rho][V.sup.2.sub.[phi]]/r - 2[[mu].sub.eff]/[r.sup.2]
 [partial derivative][V.sub.[phi]]/[partial derivative]
 [phi] -

 (2[[mu].sub.t] + [mu]) [V.sub.r]/[r.sup.2] -
 [partial derivative]([rho]k)/[partial derivative]r +
 [partial derivative]/[partial derivative]z
 ([[mu].sub.t] [partial derivative][V.sub.t]/[partial
 derivative]r) + [[mu].sub.t] [partial derivative]/
 [partial derivative][phi] ([partial derivative]/
 [partial derivative]r([V.sub.[phi]]/r)) + 2[rho][OMEGA]
 [V.sub.[phi]]

[V.sub.[phi]] 1/r [partial derivative]/[partial derivative]r
 ([[mu].sub.t] [partial derivative][V.sub.t]/[partial
 derivative][phi]) + [V.sub.[phi]]/r [partial
 derivative] [[mu].sub.t]/[partial derivative]r +
 3[[mu].sub.t] + 2[mu]/[r.sup.2] [partial derivative]
 [V.sub.r]/([partial derivative][phi] 2[V.sub.t]/
 [r.sup.2] [partial derivative][[mu].sub.t]/[partial
 derivative][phi] -

 1/r [partial derivative]/[partial derivative] [phi]
 (P + [rho]k) - [[mu].sub.eff] [V.sub.[phi]]/[r.sup.2]
 + 1/r [partial derivative]/[partial derivative]Z
 ([[mu].sub.t] [partial derivative] [V.sub.t]/
 [partial derivative][phi]) - [rho][V.sub.t]
 [V.sub.[phi]]/r - 2[rho][OMEGA][V.sub.t]

[V.sub.Z]] - [partial derivative]/[partial derivative]Z (P +
 [rho]k) + 1/r [partial derivative]/[partial derivative]
 r (r[[mu].sub.t] [partial derivative][V.sub.r]/[partial
 derivative]Z) 1/r [partial derivative]/[partial
 derivative][phi] ([[mu].sub.t] [partial derivative]
 [V.sub.[phi]]/[partial derivative]Z)

T 2[[mu].sub.eff]/[c.sub.p] [[([partial derivative]
 [V.sub.r]/ [partial derivative]r).sup.2] + [([partial
 derivative][V.sub.[phi]]/r[partial derivative][phi] +
 [v.sub.t]/r).sup.2] + [([partial derivative]
 [V.sub.t]/z).sup.2]] + 2[[mu].sub.eff]/[c.sub.p]

 [[r [partial derivative]/[partial derivative]r
 ([V.sub.[phi]]/r) + [partial derivative][V.sub.t]/r
 [partial derivative][phi]].sup.2] + [[mu].sub.eff]/
 [c.sub.p] [[([partial derivative][V.sub.t]/r[partial
 derivative][phi] + [partial derivative][V.sub.[phi]]/
 [partial derivative]Z).sup.2] + [([partial derivative]
 [V.sub.t]/[partial derivative]Z + [partial derivative]
 [V.sub.t]/[partial derivative]r).sup.2]]

k [P.sub.0] - [rho][epsilon] - D

[epsilon] [epsilon]/k ([C.sub.[epsilon]1][P.sub.0] -
 [C.sub.[epsilon]2][rho][epsilon] + E

Table 2: Terms appearing in the turbulence model.

Term Launder and Sharma Morse model

[C.sub.[mu]] 0.09

[C.sub.[epsilon]1] 1.44

[C.sub.[epsilon]2] 1.92 [f.sub.1]

D 2[mu] [[([partial derivative][square root of k]/
 [partial derivative]z).sup.2] + [([partial
 derivative][square root of k]/[partial
 derivative]r).sup.2] + ([partial derivative]
 [square root of k]/r[partial derivative]
 [phi]).sup.2]]

E 2[mu][[mu].sub.t]/[rho] [[([[partial
 derivative].sup.2][V.sub.r]/[partial derivative]
 [z.sup.2]).sup.2] + [([[partial
 derivative].sup.2][V.sub.[phi]]/[partial
 derivative][z.sup.2]).sup.2] + [([[partial
 derivative].sup.2][V.sub.z]/[partial
 derivative][r.sup.2]).sup.2] + [([[partial
 derivative].sup.2][V.sub.[phi]]/[partial
 derivative][r.sup.2]).sup.2] + (1/[r.sup.2]
 [[partial derivative].sup.2][V.sub.r]/[partial
 derivative][[phi].sup.2]).sup.2] + (1/[r.sup.2]
 [[partial derivative].sup.2][V.sub.z]/[partial
 derivative][[phi].sup.2]).sup.2]]

F 0

[f.sub.1] 1 - 0.3 exp(- [R.sup.2.sub.t)

[f.sub.[mu]] exp [- 3.4/[(1 + [R.sub.t]/50).sup.2]]

[[sigma].sub.k] 1

[[sigma].sub.e] 1.3

Table 3: A summary of boundary conditions.

 [V.sub.r] [V.sub.[phi]] [V.sub.z]

Inlet 0 Fixed Fixed

Outlet 0 [partial derivative][V.sub.[phi]]/ Fixed
 [partial derivative]z = 0

stator 0 0 0

rotor 0 r[OMEGA] 0

 T

Inlet Fixed

Outlet [partial derivative]T/
 [partial derivative]z = 0

stator [partial derivative]T/
 [partial derivative]z = 0

rotor Fixed

 k

Inlet [V.sup.2.sub.[phi]] + [V.sup.2.sub.z]/[100.sup.4]

Outlet [partial derivative]k/[partial derivative]z = 0

stator 0

rotor 0

 [epsilon]

Inlet [[kappa].sup.2][rho]/100[mu]

Outlet [partial derivative][epsilon]/[partial derivative]z = 0

stator 0

rotor 0
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Article Details
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Author:Farzaneh-Gord, Mahmood
Publication:International Journal of Dynamics of Fluids
Article Type:Technical report
Geographic Code:7IRAN
Date:Jun 1, 2007
Words:4661
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