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heat transfer measurements in a linear turbine cascade
Applied Thermal Engineering 27 (2007) 771–778 www.elsevier.com/locate/apthermeng
Numerical heat transfer predictions and mass/heat transfer measurements in a linear turbine cascade M. Papa a, R.J. Goldstein a
a,*
, F. Gori
b
Department of Mechanical Engineering, University of Minnesota, 111 Church Street, Minneapolis, MN 55455, USA b University of Rome ‘‘Tor Vergata’’, Rome, Italy
Abstract The effect of secondary flows on mass transfer from a simulated gas turbine blade and hubwall is investigated. Measurements performed using naphthalene sublimation provide non-dimensional mass transfer coefficients, in the form of Sherwood numbers, that can be converted to heat transfer coefficients through the use of an analogy. Tests are conducted in a linear cascade composed of five blades having the profile of a first stage rotor blade of a high-pressure turbine aircraft engine. Detailed mass transfer maps on the airfoil and endwall surfaces allow the identification of significant flow features that are in good agreement with existing secondary flow models. These results are well suited for validation of numerical codes, as they are obtained with an accurate technique that does not suffer from conduction or radiation errors and allows the imposition of precise boundary conditions. The performance of a RANS (Reynolds-Averaged Navier–Stokes) numerical code that simulates the flow and heat/mass transfer in the cascade using the SST (Shear Stress Transport) k–x model is evaluated through a comparison with the experimental results. Ó 2006 Published by Elsevier Ltd. Keywords: Heat/mass transfer analogy; Turbulence model; Turbine
1. Introduction In modern gas turbine engines the temperature of the combustion gases exiting the combustor often reach values that exceed the failure temperatures of the turbine blade materials. The engine can operate only thanks to the adoption of extensive cooling schemes that use air bled from the compressor. Knowledge of the local thermal loads is therefore essential to design cooling systems that are effective in protecting the blades using the least amount of cooling air. In recent years designers have been increasingly relying on numerical simulations to predict the temperature and flow fields within the turbine. While computationally intensive technique such as LES (large eddy simulation) and DES (detached eddy simulation) are becoming more attractive, RANS (Reynolds-Averaged Navier–Stokes) simulations still constitute the most diffused industrial *
Corresponding author. Tel.: +1 612 625 5552; fax: +1 612 625 3434. E-mail address: [email protected] (R.J. Goldstein).
1359-4311/$ - see front matter Ó 2006 Published by Elsevier Ltd. doi:10.1016/j.applthermaleng.2006.10.017
practice thanks to their low computational cost. In the development of such codes, validation with experimental results remains one of the most important and delicate tasks. In the present study, the flow and heat transfer in a linear gas turbine cascade are investigated both experimentally and numerically. The heat transfer measurements are performed using an indirect technique that makes use of the analogy between heat and mass transfer [1]. The simulated blade and endwall surfaces are coated with naphthalene and exposed to the wind tunnel flow. The profiles of the two surfaces are measured before and after the wind tunnel test to determine the sublimation of the naphthalene layer. The data is then reduced to derive non-dimensional mass transfer coefficients in the form of Sherwood numbers Sh. The mass (heat) transfer data are analyzed and compared to the secondary flow model proposed by Wang et al. [2], who performed flow visualization experiments in the same facility used for the present study. They identified
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a horseshoe vortex system, composed of a pressure side leg (Vph) and a suction side leg (Vsh), which forms around the blade leading edge. The pressure side leg develops into a passage vortex system (Vp), which induces a wall vortex (Vwip) and a corner vortex (Vsc). Mass transfer measurements with naphthalene sublimation are free from conduction and radiation errors typical of direct heat transfer measurements and allow the imposition of well-defined boundary conditions. In an equivalent heat transfer experiment, naphthalene coated surfaces would in fact correspond to isothermal boundaries while the non-coated surfaces would correspond to adiabatic boundaries. These features, in addition to the high spatial resolution that can be achieved with this technique, make the data obtained using naphthalene sublimation particularly attractive for validation of numerical codes. 2. Experimental procedure Experiments are conducted in a blowing type wind tunnel with a five-blade linear cascade. The blades are largescale models of a first stage rotor blade of a high-pressure
turbine. Details of the facility, located in the Heat Transfer Laboratory at the University of Minnesota, can be found in a study by Jin and Goldstein [3]. The choice of coordinate systems and the nomenclature used for the cascade parameters are illustrated in Fig. 2. The coordinates ss and sp are the curvilinear distances along the blade pressure and suction sides measured from the leading edge. In the present study, the chord C is 184 mm and the axial chord Cax 130 mm. A casting technique is used to coat the central blade and the two central endwall passages with naphthalene. The coating on the endwall is applied starting from a distance of 43 mm upstream of the cascade in the incoming flow direction (x/Cax = 0.26). The coating on the blade is applied for z P 2.5 mm (z/C P 0.014) on the blade. An LVDT (linear variable differential transformer) measures the profiles before and after exposure to the flow in the wind tunnel. A correction is applied to account for the natural sublimation losses that occur during the profile measurements and the mounting operations in the wind tunnel. The uncertainty in the value of Sherwood number is 7% at a 95% confidence level. Goldstein and Cho [1]
Fig. 1. Secondary flow model [2].
M. Papa et al. / Applied Thermal Engineering 27 (2007) 771–778
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Fig. 2. Cascade coordinate system and parameters.
present a detailed analysis of the experimental uncertainties of naphthalene sublimation measurements. The measurement is performed at an exit Reynolds number Reex = 600,000 based on cascade exit velocity and chord C. The boundary layer on the endwall is tripped with a 1 mm wire placed 828 mm upstream of the leading edge of the central blade of the cascade, producing a turbulent boundary layer with close to zero free-stream turbulence (<0.2%). Hot wire measurements performed at four locations upstream of the cascade show that the boundary layer profile follows a power law with exponent 1/6.45. The projected boundary layer thickness and momentum thickness at the location of the central blade leading edge are 22 mm and 2.3 mm, respectively. 3. Numerical procedure The analogy between heat and mass transfer indicates that for a constant property flow situation, if equivalent boundary conditions are imposed and the non-dimensional groups in the energy and diffusion equation have equal values, the non-dimensional heat transfer coefficients expressed as Nusselt numbers (Nu) are equal to the nondimensional mass transfer coefficients expressed as Sherwood numbers (Sh). The Sherwood number is defined as Sh = hmC/D, where hm is the convective mass transfer coefficient, C is the blade chord and D is the binary diffusion coefficient of naphthalene vapour in air. The Schmidt number (Sc) that appears in the diffusion equation is 2.28 for naphthalene vapour diffusing in air, while the Prandtl number (Pr) that appears in the energy equation is approximately 0.7 for air at room temperature and atmospheric pressure. When the goal of the simulation is that of obtaining heat transfer data for Pr = 0.7, two approaches come to mind to use the mass transfer data for validation. A first possibility
is that of converting the experimental mass transfer data to heat transfer data, using correlations available in the literature on the effect of Pr on heat transfer. Eckert et al. [4] give indications on how to convert the data for boundary layers on airfoil surfaces. It is clear, however, that this procedure would introduce a new source of uncertainty in the experimental data. A second method is preferred in this study. The heat transfer numerical simulations are carried out for a fluid having Pr = 2.28. The value of thermal conductivity of air is adjusted to satisfy this condition. In this way the validation process can take advantage of the high accuracy of the mass transfer data. In a constant property flow, for a given geometry the flow physics is solely dependent on the Reynolds number, i.e. transition, separation, reattachment and secondary flows processes will be the same for Pr = 0.7 and Pr = 2.28. Comparisons to mass/heat transfer data constitute a severe test for the code’s ability to capture the flow physics and for the performance of the turbulence model. With this in mind, it is expected that a code that can accurately match the results obtained for Pr = 2.28 would be able to accurately predict the heat transfer at Pr = 0.7, also considering the vicinity of these two values. As a further check, data computed from the simulations at Pr = 2.28 and Pr = 0.7 could be used to calculate values of the analogy factor on all the surfaces of interest, characterized by different flow regimes. These values can be compared with the ones proposed in the literature, based on a wide database of experimental results. The numerical part of this study is carried out using the software ‘‘Fluent’’, performing a RANS simulation that uses the SST k–x model as a turbulence closure [5]. The computational domain extends for 0.72Cax upstream of the leading edge and for Cax downstream of the trailing edge in the nominal direction of the inlet and outlet flow, respectively. An unstructured mesh having approximately
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1.2 million nodes, clustered near the endwall and blade surfaces, was used to obtain the results presented in Section 4. The mesh near the solid boundaries is sufficiently refined to resolve the viscous sublayer, i.e. no wall-functions are used. Simulations were also performed with grids having 900,000 and 2.0 million nodes, constructed following the same procedure, to assess mesh independence. Profiles of velocity, turbulent kinetic energy and specific turbulent dissipation are assigned at the domain inlet. The profiles are derived from a separate 2D simulation that provides a turbulent boundary layer profile with boundary layer and momentum thicknesses that match those measured in the experiment. Periodic conditions are assigned in the pitchwise direction, a symmetry condition is assigned at the blade midspan and an ‘‘outflow’’ condition is assigned at the domain outlet. The imposition of constant properties, justified by the low-speed character of the problem, allows the decoupling of the momentum and the energy equation that are solved independently by the solver. The simulation is performed using the default values provided by the software for all model constants to test the predictive capabilities of the code, i.e. the code was not ‘‘tuned’’ to improve the agreement with the experimental results. 4. Results and discussion Numerical convective coefficients were obtained performing a heat transfer simulation for Pr = 2.28, obtaining local values of Nu. As explained in Section 3, the heat/mass transfer analogy shows that Nu = Sh for this condition, provided all other requirements are satisfied. Simulations results are therefore expressed in terms of Sh for consistency with the mass transfer data used for their validation. Figs. 3 and 4 compare the experimental and numerical results on the endwall using contour plots and line plots
at lines of constant x, respectively. Figs. 5 and 6 provide qualitative comparison of measurements and predictions using Sh contours on the blade pressure and suction surfaces. In these last two figures, LE indicates the blade leading edge and TE indicates the blade trailing edge. Mass transfer coefficients in the two-dimensional region (z/C > 0.5) are compared in Fig. 7 using line plots at a location of constant span on the blade pressure and suction sides. 4.1. Measurements on the endwall A contour plot of local mass transfer coefficients on the hub-endwall surface is reported in Fig. 4. The horseshoe vortex system originates upstream of the blade leading edge, locally enhancing mass transfer. The suction side leg of the vortex moves into the passage remaining close the blade surface, causing the local increase of the convective coefficients. A trace of the movement of the pressure side leg of the horseshoe vortex is very visible, starting from the leading edge and terminating near the suction side surface of the neighboring blade. The two vortex systems meet at approximately x/Cax = 0.45. The effect of the corner vortex in the second half of the passage is clearly noticeable, producing a streak of high mass transfer rates near the suction side of the blade which increases in size and intensity moving toward the trailing edge. Vortex shedding from the trailing edges of the blades dominates the mass transfer downstream of the passage, causing a strong increase of the convective coefficients which spreads rapidly in the cross-stream direction. 4.2. Measurements on the blade pressure side Fig. 5a shows high mass transfer rates at the leading edge, where the incoming flow stagnates. The mass transfer coefficients drop rapidly moving downstream, as the
Fig. 3. Sh contours on the endwall surface at Reex = 600,000, Tu = 0.2% (a) measured and (b) computed.
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Fig. 4. Sh on the endwall at locations of constant x/Cax, Reex = 600,000, Tu = 0.2%.
Fig. 5. Sh contours on the blade pressure surface at Reex = 600,000, Tu = 0.2% (a) measured and (b) computed.
boundary layer gets thicker. A local minimum is observed at approximately sp/C = 0.15 and is related to the local curvature and pressure gradients along the blade surface. It is well predicted by a laminar wedge flow solution [6] and is captured by the numerical simulation. An interesting feature shown by the measurements is the appearance of ‘‘waviness’’ in the Sherwood number distribution starting from approximately sp/C = 0.4 and increasing moving downstream. This phenomenon was observed by Wang et al. [7] who performed extensive mass transfer and flow visualization experiments in the two-dimensional region
of the blade, using the same facility of the present study. They related the ‘waviness’ to the formation and development of Taylor–Go¨rtler vortices on the concave pressure blade surface. Their action is believed to cause the enhanced mass transfer streaks observed in the results presented in this study. As the objective of this work was the investigation of near-wall effects, the spatial resolution of the measurement is much higher near the endwall than in the two-dimensional region. The distance between the streaks is therefore not indicative of the periodicity of the vortices in the spanwise direction.
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Fig. 6. Sh contours on the blade suction surface at Reex = 600,000, Tu = 0.2% (a) measured and (b) computed.
Fig. 7. Sh on the blade surface at z/C = 0.88, Reex = 600,000, Tu = 0.2% (a) pressure side and (b) suction side.
4.3. Measurements on the blade suction side Results reported in Fig. 6a show high mass transfer rates by the leading edge (ss/C = 0) where the approaching flow stagnates. The Sherwood number decreases sharply as the boundary layer develops in the streamwise direction on the blade surface. In the two-dimensional region (z/C > 0.5), transition to turbulence can be observed at approximately ss/C = 1.1. Until that location the mass transfer results match those that can be calculated through a local laminar wedge-flow solution [6], indicating that the flow is indeed laminar. The mass transfer coefficients then increase abruptly, clearly indicating a transition to a turbulent boundary layer. The absence of free-stream turbulence and the smoothness of the naphthalene surface cause the laminar to turbulent transition to occur late in the cascade test. In a real engine situation, the high free-stream turbulence intensity and the unsteady wakes are known to cause a much earlier transition acting through different physical mechanisms (bypass transition). Near the endwall, a large triangular region of high mass transfer is evident starting at ss/C 0.35. The mass transfer results can be interpreted using the secondary flow model
proposed by Wang et al. [2] and reported in Fig. 1. Two separate high mass transfer regions are identified within the triangular region. A large region occupying most of the enhanced mass transfer area is believed to be caused by the action of the passage vortex that has been drawn toward the suction side of the blade by the pressure difference existing across the passage. Above this region, a high mass transfer streak is caused by the action of the induced wall vortex described in the model. These vortices tend to bring portions of naphthalene-free mainstream fluid in contact with the blade surface, yielding elevated mass transfer rates. 4.4. Numerical simulation on the endwall Fig. 3b shows that high heat transfer rates on the endwall are predicted in front of the blade leading edge, in good agreement with the experimental measurements. This is the location where the horseshoe vortex system forms. The suction side leg of the horseshoe vortex moves along the suction side of the blade, enhancing the local heat transfer rates. The pressure side leg of the horseshoe vortex system moves across the passage and reaches the suction side surface of the neighboring blade leaving a clear trace
M. Papa et al. / Applied Thermal Engineering 27 (2007) 771–778
of enhanced mass transfer in its path, once again in good agreement with the experimental measurements. Moving further downstream within the passage, the numerical simulation sensibly over-predicts the heat transfer rates and fails to predict the heat transfer enhancement caused by the corner vortex described in the model reported in Fig. 1 and obtained in the streak in Fig. 3a. A quantitative comparison between the numerical and experimental results can be made referring to the line plots reported in Fig. 4 which show the measured and simulated Sherwood numbers at locations of constant x on the endwall surface. The rectangular bars in the figure indicate the locations of the blades. At x/Cax = 0.015 the high heat transfer peak in the stagnation region in front of the blade leading edge due to the horseshoe vortex is generally well predicted. At x/Cax = 0.299 the pressure side leg of the horseshoe vortex system produces a local heat transfer enhancement near the center of the passage. The simulation predicts that this vortex produces two distinct heat transfer peaks while only a single peak is measured. It is interesting to notice that the flow visualization experiments performed by Wang et al. [2] suggest that the horseshoe vortex system is characterized by a system at times containing a single horseshoe vortex and other times containing two horseshoe vortices that alternate in a periodic way, as shown in their model (Fig. 1). Capturing such a phenomenon goes beyond the capabilities of a RANS simulation; nevertheless the time-averaged heat transfer levels and the drift of the passage vortex system across the blade passage are well predicted. Starting from x/Cax 0.57, shortly after the passage vortex reaches the blade suction side surface, the simulation begins sensibly over-predicting the heat transfer rate in the middle of the passage. This trend continues all the way to the trailing edge and is consistent with the results obtained by Pasinato et al. [8], who carried out simulations on a turbine blade cascade using the RNG k–e, the Spalart– Allmaras model and a Reynolds-Stress model. It is important to notice that while the measurements clearly indicate the presence of a strong corner vortex by the suction side surface, this feature is barely noticeable in the simulations that do not predict high heat transfer rates near the blade surface. It can be imagined that the presence of this vortex acts to reduce the cross-flow within the passage near the endwall. From the measurements it can be inferred that, moving from the pressure side to the suction side in the aft portion of the passage, the fluid coming from the pressure side is initially accelerated, accounting for an increase of mass transfer rates. Once the presence of the corner vortex is felt, the fluid decelerates, causing the boundary layer to grow and the mass transfer to be reduced, and then lifts off the surface when it meets the corner vortex. Closer to the blade suction side, high mass transfer rates are caused directly by the presence of the corner vortex. The absence or very limited size of the corner vortex in the simulations causes a stronger acceleration of the cross-flow fluid, producing higher heat transfer rates. In the numerical analysis
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the fluid then decelerates due to the presence of the suction side wall, producing a decrease in mass transfer rates that continues almost all the way to the blade. Downstream of the trailing edge, vortex shedding phenomena cause the measured mass transfer rates to increase sharply. Numerical results in this region are reported for completeness, even though the objective of the simulations was the study of heat transfer within the blade passage, as a RANS simulation is not expected to capture the effects of unsteady vortex shedding. 4.5. Numerical simulation on the blade pressure side A contour plot of the numerical heat/mass transfer results on the blade pressure surface is reported in Fig. 5b. High heat transfer rates are predicted at the blade leading edge and decrease sharply moving downstream on the pressure side. The results on the pressure surface appear to be uniform, showing no trace of Taylor–Go¨rtler vortices. Aside from this, the agreement with the experimental measurements is very good, as documented in Fig. 7a. The numerical predictions deviate from the measurements only very near the wall, for z/C < 0.05. In both the experimental and numerical results there is no evidence of transition on the blade pressure side. 4.6. Numerical simulation on the blade suction side The numerical heat/mass transfer results on the blade suction surface are reported in Fig. 6b. The most noticeable feature is the triangular region of enhanced heat transfer in the aft part of the blade by the bottom wall, which corresponds to that observed in the experimental measurements and is due to the action of the passage vortex system. A close examination of the triangular region in the trailing portion of the blade shows two separate regions of high heat transfer, separated by a streak of lower heat transfer. This agrees well with the measurements and with the model that suggest that the passage vortex system is directly causing the enhancement in the lower portion of the triangular region, and that an induced wall vortex is causing the enhancement in the top portion of the triangular region. In the two-dimensional region of the blade suction surface (z/C > 0.5), the laminar to turbulent transition is very evident. This can be well observed in the line plot reported in Fig. 7b which shows a sharp increase in Sh at ss/C 0.9. Though the agreement with the measurements is excellent until that location, the transition inferred from the measurements occurs at ss/C 1.1. Moving to lower spanwise locations, the transition point shifts forward in both simulations and measurements, with the numerical code still predicting an early transition. 5. Conclusions Detailed heat/mass transfer measurements have been performed on the airfoil and hubwall surfaces of a
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simulated gas turbine blade using the naphthalene sublimation technique. Results show that the high spatial resolution of the measurements allows a clear identification of the effects that the secondary flows have on the convective coefficients. An interpretation of the convective heat/mass transfer distribution is proposed and supported by comparisons with a secondary flow model derived from flow visualization experiments previously performed in the same facility. A Reynolds-Averaged Navier–Stokes computation was carried out with the software ‘‘Fluent’’ to simulate the flow and heat/mass transfer in the linear cascade used in the experiments. The SST k–x turbulence model was used to close the turbulence problem. The flow and thermal boundary conditions were carefully matched to the ones measured in the experiments. Results show that the code generally performs well in predicting the main features of the secondary flows in the near-wall region. The numerical simulation captures the formation of the horseshoe vortex system upstream on the blade leading edge, and the migration of the pressure side leg of the horseshoe vortex toward the suction side of the neighboring blade, where it meets the suction side leg of neighboring horseshoe vortex system. The heat (mass) transfer rates on the endwall are well predicted in the first portion of the passage, until the two legs of the horseshoe vortex system meet. Further downstream the code underpredicts the effect of the corner vortex and overpredicts the heat (mass) transfer rates associated with the cross-flow within the passage. On the airfoil surface, the simulation captures the enhancement
of the heat transfer coefficients on the suction side of the blade due to the action of passage vortex system. The agreement in the two-dimensional region of the airfoil is excellent, but the code predicts an earlier transition to turbulence on the blade suction side than is found in the experiment. References [1] R.J. Goldstein, H.H. Cho, A review of mass transfer measurements using naphthalene sublimation, Experimental Thermal and Fluid Science 10 (1995) 416–434. [2] H.P. Wang, S.J. Olson, R.J. Goldstein, E.R.G. Eckert, Flow visualization in a linear turbine cascade of high performance turbine blades, Journal of Turbomachinery 119 (1997) 1–8. [3] P. Jin, R.J. Goldstein, Local mass/heat transfer on a turbine blade tip, International Journal of Rotating Machinery 9 (2) (2003) 81–95. [4] E.R.G. Eckert, H. Sakamoto, T.W. Simon, The heat/mass transfer analogy factor, Nu/Sh, for boundary layers on turbine blade profiles, International Journal of Heat and Mass Transfer 44 (2001) 1223–1233. [5] F.R. Menter, Two-equation eddy-viscosity turbulence models for engineering applications, AIAA Journal 32 (1994) 1598–1605. [6] H.P. Wang, R.J. Goldstein, S.J. Olson, Effect of high free-stream turbulence with large length scale on blade heat/mass transfer, Journal of Turbomachinery 121 (1999) 217–224. [7] H.P. Wang, S.J. Olson, R.J. Goldstein, Development of Taylor– Go¨rtler vortices over the pressure surface of a turbine blade, Journal of Heat Transfer 127 (2005) 540–543. [8] H.D. Pasinato, K.D. Squires, R.P. Rov, Assessment of reynoldsaveraged turbulence models for prediction of the flow and heat transfer in an inlet vane-endwall passage, Journal of Fluids Engineering Transactions of the ASME 126 (3) (2004) 305–315.
Numerical heat transfer predictions and mass/heat transfer measurements in a linear turbine cascade M. Papa a, R.J. Goldstein a
a,*
, F. Gori
b
Department of Mechanical Engineering, University of Minnesota, 111 Church Street, Minneapolis, MN 55455, USA b University of Rome ‘‘Tor Vergata’’, Rome, Italy
Abstract The effect of secondary flows on mass transfer from a simulated gas turbine blade and hubwall is investigated. Measurements performed using naphthalene sublimation provide non-dimensional mass transfer coefficients, in the form of Sherwood numbers, that can be converted to heat transfer coefficients through the use of an analogy. Tests are conducted in a linear cascade composed of five blades having the profile of a first stage rotor blade of a high-pressure turbine aircraft engine. Detailed mass transfer maps on the airfoil and endwall surfaces allow the identification of significant flow features that are in good agreement with existing secondary flow models. These results are well suited for validation of numerical codes, as they are obtained with an accurate technique that does not suffer from conduction or radiation errors and allows the imposition of precise boundary conditions. The performance of a RANS (Reynolds-Averaged Navier–Stokes) numerical code that simulates the flow and heat/mass transfer in the cascade using the SST (Shear Stress Transport) k–x model is evaluated through a comparison with the experimental results. Ó 2006 Published by Elsevier Ltd. Keywords: Heat/mass transfer analogy; Turbulence model; Turbine
1. Introduction In modern gas turbine engines the temperature of the combustion gases exiting the combustor often reach values that exceed the failure temperatures of the turbine blade materials. The engine can operate only thanks to the adoption of extensive cooling schemes that use air bled from the compressor. Knowledge of the local thermal loads is therefore essential to design cooling systems that are effective in protecting the blades using the least amount of cooling air. In recent years designers have been increasingly relying on numerical simulations to predict the temperature and flow fields within the turbine. While computationally intensive technique such as LES (large eddy simulation) and DES (detached eddy simulation) are becoming more attractive, RANS (Reynolds-Averaged Navier–Stokes) simulations still constitute the most diffused industrial *
Corresponding author. Tel.: +1 612 625 5552; fax: +1 612 625 3434. E-mail address: [email protected] (R.J. Goldstein).
1359-4311/$ - see front matter Ó 2006 Published by Elsevier Ltd. doi:10.1016/j.applthermaleng.2006.10.017
practice thanks to their low computational cost. In the development of such codes, validation with experimental results remains one of the most important and delicate tasks. In the present study, the flow and heat transfer in a linear gas turbine cascade are investigated both experimentally and numerically. The heat transfer measurements are performed using an indirect technique that makes use of the analogy between heat and mass transfer [1]. The simulated blade and endwall surfaces are coated with naphthalene and exposed to the wind tunnel flow. The profiles of the two surfaces are measured before and after the wind tunnel test to determine the sublimation of the naphthalene layer. The data is then reduced to derive non-dimensional mass transfer coefficients in the form of Sherwood numbers Sh. The mass (heat) transfer data are analyzed and compared to the secondary flow model proposed by Wang et al. [2], who performed flow visualization experiments in the same facility used for the present study. They identified
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a horseshoe vortex system, composed of a pressure side leg (Vph) and a suction side leg (Vsh), which forms around the blade leading edge. The pressure side leg develops into a passage vortex system (Vp), which induces a wall vortex (Vwip) and a corner vortex (Vsc). Mass transfer measurements with naphthalene sublimation are free from conduction and radiation errors typical of direct heat transfer measurements and allow the imposition of well-defined boundary conditions. In an equivalent heat transfer experiment, naphthalene coated surfaces would in fact correspond to isothermal boundaries while the non-coated surfaces would correspond to adiabatic boundaries. These features, in addition to the high spatial resolution that can be achieved with this technique, make the data obtained using naphthalene sublimation particularly attractive for validation of numerical codes. 2. Experimental procedure Experiments are conducted in a blowing type wind tunnel with a five-blade linear cascade. The blades are largescale models of a first stage rotor blade of a high-pressure
turbine. Details of the facility, located in the Heat Transfer Laboratory at the University of Minnesota, can be found in a study by Jin and Goldstein [3]. The choice of coordinate systems and the nomenclature used for the cascade parameters are illustrated in Fig. 2. The coordinates ss and sp are the curvilinear distances along the blade pressure and suction sides measured from the leading edge. In the present study, the chord C is 184 mm and the axial chord Cax 130 mm. A casting technique is used to coat the central blade and the two central endwall passages with naphthalene. The coating on the endwall is applied starting from a distance of 43 mm upstream of the cascade in the incoming flow direction (x/Cax = 0.26). The coating on the blade is applied for z P 2.5 mm (z/C P 0.014) on the blade. An LVDT (linear variable differential transformer) measures the profiles before and after exposure to the flow in the wind tunnel. A correction is applied to account for the natural sublimation losses that occur during the profile measurements and the mounting operations in the wind tunnel. The uncertainty in the value of Sherwood number is 7% at a 95% confidence level. Goldstein and Cho [1]
Fig. 1. Secondary flow model [2].
M. Papa et al. / Applied Thermal Engineering 27 (2007) 771–778
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Fig. 2. Cascade coordinate system and parameters.
present a detailed analysis of the experimental uncertainties of naphthalene sublimation measurements. The measurement is performed at an exit Reynolds number Reex = 600,000 based on cascade exit velocity and chord C. The boundary layer on the endwall is tripped with a 1 mm wire placed 828 mm upstream of the leading edge of the central blade of the cascade, producing a turbulent boundary layer with close to zero free-stream turbulence (<0.2%). Hot wire measurements performed at four locations upstream of the cascade show that the boundary layer profile follows a power law with exponent 1/6.45. The projected boundary layer thickness and momentum thickness at the location of the central blade leading edge are 22 mm and 2.3 mm, respectively. 3. Numerical procedure The analogy between heat and mass transfer indicates that for a constant property flow situation, if equivalent boundary conditions are imposed and the non-dimensional groups in the energy and diffusion equation have equal values, the non-dimensional heat transfer coefficients expressed as Nusselt numbers (Nu) are equal to the nondimensional mass transfer coefficients expressed as Sherwood numbers (Sh). The Sherwood number is defined as Sh = hmC/D, where hm is the convective mass transfer coefficient, C is the blade chord and D is the binary diffusion coefficient of naphthalene vapour in air. The Schmidt number (Sc) that appears in the diffusion equation is 2.28 for naphthalene vapour diffusing in air, while the Prandtl number (Pr) that appears in the energy equation is approximately 0.7 for air at room temperature and atmospheric pressure. When the goal of the simulation is that of obtaining heat transfer data for Pr = 0.7, two approaches come to mind to use the mass transfer data for validation. A first possibility
is that of converting the experimental mass transfer data to heat transfer data, using correlations available in the literature on the effect of Pr on heat transfer. Eckert et al. [4] give indications on how to convert the data for boundary layers on airfoil surfaces. It is clear, however, that this procedure would introduce a new source of uncertainty in the experimental data. A second method is preferred in this study. The heat transfer numerical simulations are carried out for a fluid having Pr = 2.28. The value of thermal conductivity of air is adjusted to satisfy this condition. In this way the validation process can take advantage of the high accuracy of the mass transfer data. In a constant property flow, for a given geometry the flow physics is solely dependent on the Reynolds number, i.e. transition, separation, reattachment and secondary flows processes will be the same for Pr = 0.7 and Pr = 2.28. Comparisons to mass/heat transfer data constitute a severe test for the code’s ability to capture the flow physics and for the performance of the turbulence model. With this in mind, it is expected that a code that can accurately match the results obtained for Pr = 2.28 would be able to accurately predict the heat transfer at Pr = 0.7, also considering the vicinity of these two values. As a further check, data computed from the simulations at Pr = 2.28 and Pr = 0.7 could be used to calculate values of the analogy factor on all the surfaces of interest, characterized by different flow regimes. These values can be compared with the ones proposed in the literature, based on a wide database of experimental results. The numerical part of this study is carried out using the software ‘‘Fluent’’, performing a RANS simulation that uses the SST k–x model as a turbulence closure [5]. The computational domain extends for 0.72Cax upstream of the leading edge and for Cax downstream of the trailing edge in the nominal direction of the inlet and outlet flow, respectively. An unstructured mesh having approximately
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1.2 million nodes, clustered near the endwall and blade surfaces, was used to obtain the results presented in Section 4. The mesh near the solid boundaries is sufficiently refined to resolve the viscous sublayer, i.e. no wall-functions are used. Simulations were also performed with grids having 900,000 and 2.0 million nodes, constructed following the same procedure, to assess mesh independence. Profiles of velocity, turbulent kinetic energy and specific turbulent dissipation are assigned at the domain inlet. The profiles are derived from a separate 2D simulation that provides a turbulent boundary layer profile with boundary layer and momentum thicknesses that match those measured in the experiment. Periodic conditions are assigned in the pitchwise direction, a symmetry condition is assigned at the blade midspan and an ‘‘outflow’’ condition is assigned at the domain outlet. The imposition of constant properties, justified by the low-speed character of the problem, allows the decoupling of the momentum and the energy equation that are solved independently by the solver. The simulation is performed using the default values provided by the software for all model constants to test the predictive capabilities of the code, i.e. the code was not ‘‘tuned’’ to improve the agreement with the experimental results. 4. Results and discussion Numerical convective coefficients were obtained performing a heat transfer simulation for Pr = 2.28, obtaining local values of Nu. As explained in Section 3, the heat/mass transfer analogy shows that Nu = Sh for this condition, provided all other requirements are satisfied. Simulations results are therefore expressed in terms of Sh for consistency with the mass transfer data used for their validation. Figs. 3 and 4 compare the experimental and numerical results on the endwall using contour plots and line plots
at lines of constant x, respectively. Figs. 5 and 6 provide qualitative comparison of measurements and predictions using Sh contours on the blade pressure and suction surfaces. In these last two figures, LE indicates the blade leading edge and TE indicates the blade trailing edge. Mass transfer coefficients in the two-dimensional region (z/C > 0.5) are compared in Fig. 7 using line plots at a location of constant span on the blade pressure and suction sides. 4.1. Measurements on the endwall A contour plot of local mass transfer coefficients on the hub-endwall surface is reported in Fig. 4. The horseshoe vortex system originates upstream of the blade leading edge, locally enhancing mass transfer. The suction side leg of the vortex moves into the passage remaining close the blade surface, causing the local increase of the convective coefficients. A trace of the movement of the pressure side leg of the horseshoe vortex is very visible, starting from the leading edge and terminating near the suction side surface of the neighboring blade. The two vortex systems meet at approximately x/Cax = 0.45. The effect of the corner vortex in the second half of the passage is clearly noticeable, producing a streak of high mass transfer rates near the suction side of the blade which increases in size and intensity moving toward the trailing edge. Vortex shedding from the trailing edges of the blades dominates the mass transfer downstream of the passage, causing a strong increase of the convective coefficients which spreads rapidly in the cross-stream direction. 4.2. Measurements on the blade pressure side Fig. 5a shows high mass transfer rates at the leading edge, where the incoming flow stagnates. The mass transfer coefficients drop rapidly moving downstream, as the
Fig. 3. Sh contours on the endwall surface at Reex = 600,000, Tu = 0.2% (a) measured and (b) computed.
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Fig. 4. Sh on the endwall at locations of constant x/Cax, Reex = 600,000, Tu = 0.2%.
Fig. 5. Sh contours on the blade pressure surface at Reex = 600,000, Tu = 0.2% (a) measured and (b) computed.
boundary layer gets thicker. A local minimum is observed at approximately sp/C = 0.15 and is related to the local curvature and pressure gradients along the blade surface. It is well predicted by a laminar wedge flow solution [6] and is captured by the numerical simulation. An interesting feature shown by the measurements is the appearance of ‘‘waviness’’ in the Sherwood number distribution starting from approximately sp/C = 0.4 and increasing moving downstream. This phenomenon was observed by Wang et al. [7] who performed extensive mass transfer and flow visualization experiments in the two-dimensional region
of the blade, using the same facility of the present study. They related the ‘waviness’ to the formation and development of Taylor–Go¨rtler vortices on the concave pressure blade surface. Their action is believed to cause the enhanced mass transfer streaks observed in the results presented in this study. As the objective of this work was the investigation of near-wall effects, the spatial resolution of the measurement is much higher near the endwall than in the two-dimensional region. The distance between the streaks is therefore not indicative of the periodicity of the vortices in the spanwise direction.
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Fig. 6. Sh contours on the blade suction surface at Reex = 600,000, Tu = 0.2% (a) measured and (b) computed.
Fig. 7. Sh on the blade surface at z/C = 0.88, Reex = 600,000, Tu = 0.2% (a) pressure side and (b) suction side.
4.3. Measurements on the blade suction side Results reported in Fig. 6a show high mass transfer rates by the leading edge (ss/C = 0) where the approaching flow stagnates. The Sherwood number decreases sharply as the boundary layer develops in the streamwise direction on the blade surface. In the two-dimensional region (z/C > 0.5), transition to turbulence can be observed at approximately ss/C = 1.1. Until that location the mass transfer results match those that can be calculated through a local laminar wedge-flow solution [6], indicating that the flow is indeed laminar. The mass transfer coefficients then increase abruptly, clearly indicating a transition to a turbulent boundary layer. The absence of free-stream turbulence and the smoothness of the naphthalene surface cause the laminar to turbulent transition to occur late in the cascade test. In a real engine situation, the high free-stream turbulence intensity and the unsteady wakes are known to cause a much earlier transition acting through different physical mechanisms (bypass transition). Near the endwall, a large triangular region of high mass transfer is evident starting at ss/C 0.35. The mass transfer results can be interpreted using the secondary flow model
proposed by Wang et al. [2] and reported in Fig. 1. Two separate high mass transfer regions are identified within the triangular region. A large region occupying most of the enhanced mass transfer area is believed to be caused by the action of the passage vortex that has been drawn toward the suction side of the blade by the pressure difference existing across the passage. Above this region, a high mass transfer streak is caused by the action of the induced wall vortex described in the model. These vortices tend to bring portions of naphthalene-free mainstream fluid in contact with the blade surface, yielding elevated mass transfer rates. 4.4. Numerical simulation on the endwall Fig. 3b shows that high heat transfer rates on the endwall are predicted in front of the blade leading edge, in good agreement with the experimental measurements. This is the location where the horseshoe vortex system forms. The suction side leg of the horseshoe vortex moves along the suction side of the blade, enhancing the local heat transfer rates. The pressure side leg of the horseshoe vortex system moves across the passage and reaches the suction side surface of the neighboring blade leaving a clear trace
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of enhanced mass transfer in its path, once again in good agreement with the experimental measurements. Moving further downstream within the passage, the numerical simulation sensibly over-predicts the heat transfer rates and fails to predict the heat transfer enhancement caused by the corner vortex described in the model reported in Fig. 1 and obtained in the streak in Fig. 3a. A quantitative comparison between the numerical and experimental results can be made referring to the line plots reported in Fig. 4 which show the measured and simulated Sherwood numbers at locations of constant x on the endwall surface. The rectangular bars in the figure indicate the locations of the blades. At x/Cax = 0.015 the high heat transfer peak in the stagnation region in front of the blade leading edge due to the horseshoe vortex is generally well predicted. At x/Cax = 0.299 the pressure side leg of the horseshoe vortex system produces a local heat transfer enhancement near the center of the passage. The simulation predicts that this vortex produces two distinct heat transfer peaks while only a single peak is measured. It is interesting to notice that the flow visualization experiments performed by Wang et al. [2] suggest that the horseshoe vortex system is characterized by a system at times containing a single horseshoe vortex and other times containing two horseshoe vortices that alternate in a periodic way, as shown in their model (Fig. 1). Capturing such a phenomenon goes beyond the capabilities of a RANS simulation; nevertheless the time-averaged heat transfer levels and the drift of the passage vortex system across the blade passage are well predicted. Starting from x/Cax 0.57, shortly after the passage vortex reaches the blade suction side surface, the simulation begins sensibly over-predicting the heat transfer rate in the middle of the passage. This trend continues all the way to the trailing edge and is consistent with the results obtained by Pasinato et al. [8], who carried out simulations on a turbine blade cascade using the RNG k–e, the Spalart– Allmaras model and a Reynolds-Stress model. It is important to notice that while the measurements clearly indicate the presence of a strong corner vortex by the suction side surface, this feature is barely noticeable in the simulations that do not predict high heat transfer rates near the blade surface. It can be imagined that the presence of this vortex acts to reduce the cross-flow within the passage near the endwall. From the measurements it can be inferred that, moving from the pressure side to the suction side in the aft portion of the passage, the fluid coming from the pressure side is initially accelerated, accounting for an increase of mass transfer rates. Once the presence of the corner vortex is felt, the fluid decelerates, causing the boundary layer to grow and the mass transfer to be reduced, and then lifts off the surface when it meets the corner vortex. Closer to the blade suction side, high mass transfer rates are caused directly by the presence of the corner vortex. The absence or very limited size of the corner vortex in the simulations causes a stronger acceleration of the cross-flow fluid, producing higher heat transfer rates. In the numerical analysis
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the fluid then decelerates due to the presence of the suction side wall, producing a decrease in mass transfer rates that continues almost all the way to the blade. Downstream of the trailing edge, vortex shedding phenomena cause the measured mass transfer rates to increase sharply. Numerical results in this region are reported for completeness, even though the objective of the simulations was the study of heat transfer within the blade passage, as a RANS simulation is not expected to capture the effects of unsteady vortex shedding. 4.5. Numerical simulation on the blade pressure side A contour plot of the numerical heat/mass transfer results on the blade pressure surface is reported in Fig. 5b. High heat transfer rates are predicted at the blade leading edge and decrease sharply moving downstream on the pressure side. The results on the pressure surface appear to be uniform, showing no trace of Taylor–Go¨rtler vortices. Aside from this, the agreement with the experimental measurements is very good, as documented in Fig. 7a. The numerical predictions deviate from the measurements only very near the wall, for z/C < 0.05. In both the experimental and numerical results there is no evidence of transition on the blade pressure side. 4.6. Numerical simulation on the blade suction side The numerical heat/mass transfer results on the blade suction surface are reported in Fig. 6b. The most noticeable feature is the triangular region of enhanced heat transfer in the aft part of the blade by the bottom wall, which corresponds to that observed in the experimental measurements and is due to the action of the passage vortex system. A close examination of the triangular region in the trailing portion of the blade shows two separate regions of high heat transfer, separated by a streak of lower heat transfer. This agrees well with the measurements and with the model that suggest that the passage vortex system is directly causing the enhancement in the lower portion of the triangular region, and that an induced wall vortex is causing the enhancement in the top portion of the triangular region. In the two-dimensional region of the blade suction surface (z/C > 0.5), the laminar to turbulent transition is very evident. This can be well observed in the line plot reported in Fig. 7b which shows a sharp increase in Sh at ss/C 0.9. Though the agreement with the measurements is excellent until that location, the transition inferred from the measurements occurs at ss/C 1.1. Moving to lower spanwise locations, the transition point shifts forward in both simulations and measurements, with the numerical code still predicting an early transition. 5. Conclusions Detailed heat/mass transfer measurements have been performed on the airfoil and hubwall surfaces of a
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simulated gas turbine blade using the naphthalene sublimation technique. Results show that the high spatial resolution of the measurements allows a clear identification of the effects that the secondary flows have on the convective coefficients. An interpretation of the convective heat/mass transfer distribution is proposed and supported by comparisons with a secondary flow model derived from flow visualization experiments previously performed in the same facility. A Reynolds-Averaged Navier–Stokes computation was carried out with the software ‘‘Fluent’’ to simulate the flow and heat/mass transfer in the linear cascade used in the experiments. The SST k–x turbulence model was used to close the turbulence problem. The flow and thermal boundary conditions were carefully matched to the ones measured in the experiments. Results show that the code generally performs well in predicting the main features of the secondary flows in the near-wall region. The numerical simulation captures the formation of the horseshoe vortex system upstream on the blade leading edge, and the migration of the pressure side leg of the horseshoe vortex toward the suction side of the neighboring blade, where it meets the suction side leg of neighboring horseshoe vortex system. The heat (mass) transfer rates on the endwall are well predicted in the first portion of the passage, until the two legs of the horseshoe vortex system meet. Further downstream the code underpredicts the effect of the corner vortex and overpredicts the heat (mass) transfer rates associated with the cross-flow within the passage. On the airfoil surface, the simulation captures the enhancement
of the heat transfer coefficients on the suction side of the blade due to the action of passage vortex system. The agreement in the two-dimensional region of the airfoil is excellent, but the code predicts an earlier transition to turbulence on the blade suction side than is found in the experiment. References [1] R.J. Goldstein, H.H. Cho, A review of mass transfer measurements using naphthalene sublimation, Experimental Thermal and Fluid Science 10 (1995) 416–434. [2] H.P. Wang, S.J. Olson, R.J. Goldstein, E.R.G. Eckert, Flow visualization in a linear turbine cascade of high performance turbine blades, Journal of Turbomachinery 119 (1997) 1–8. [3] P. Jin, R.J. Goldstein, Local mass/heat transfer on a turbine blade tip, International Journal of Rotating Machinery 9 (2) (2003) 81–95. [4] E.R.G. Eckert, H. Sakamoto, T.W. Simon, The heat/mass transfer analogy factor, Nu/Sh, for boundary layers on turbine blade profiles, International Journal of Heat and Mass Transfer 44 (2001) 1223–1233. [5] F.R. Menter, Two-equation eddy-viscosity turbulence models for engineering applications, AIAA Journal 32 (1994) 1598–1605. [6] H.P. Wang, R.J. Goldstein, S.J. Olson, Effect of high free-stream turbulence with large length scale on blade heat/mass transfer, Journal of Turbomachinery 121 (1999) 217–224. [7] H.P. Wang, S.J. Olson, R.J. Goldstein, Development of Taylor– Go¨rtler vortices over the pressure surface of a turbine blade, Journal of Heat Transfer 127 (2005) 540–543. [8] H.D. Pasinato, K.D. Squires, R.P. Rov, Assessment of reynoldsaveraged turbulence models for prediction of the flow and heat transfer in an inlet vane-endwall passage, Journal of Fluids Engineering Transactions of the ASME 126 (3) (2004) 305–315.