Aerodynamic Entropy Generation Rate in a Boundary Layer With High Free Stream Turbulence (2004)

Contributed by the Fluids Engineering Division for publication in the JOURNAL OF FLUIDS ENGINEERING. Manuscript received by the Fluids Engineering Division June 27, 2002; revised manuscript received March 5, 2004. Associate Editor: T. B. Gatski. The loss mechanism under consideration in this study is that fraction of turbomachinery blade profile loss 1 due to viscous dissipation and turbulent phenomena in the blade boundary layer. In the consideration of profile loss it is customary to neglect any end wall effects and as such the flow is approximately two-dimensional. Hence loss measurements can be based on cascade tests and boundary layer calculations. It has been proposed in 1 that the concept of entropy generation be used as the key measure of lost performance in turbomachinery, as opposed to the more traditional loss calculation methods based on stagnation pressure and kinetic energy losses. Assessment of loss in terms of the entropy generation rate is not dependant on whether it is examined from the perspective of a stationary or rotating blade row, thereby enabling direct comparison between measurements from cascades and rotating facilities. Previous researchers 2345 at the author’s university have measured transitional and turbulent boundary layer losses. This paper, however, is mainly focused on loss in the laminar boundary layer and how it is influenced by FST. This is important in turbomachinery because laminar boundary layers are subjected to extremes of FST of varying intensity. As the flow under consideration is assumed to be adiabatic, the only contributory factor to loss is that due to viscous dissipation and Reynolds stresses. The entropy generation rate is therefore obtained directly from the boundary layer velocity profile; a first integration through the boundary layer thickness gives the entropy generation rate at a chordwise point, a second integration in the streamwise direction yields the entropy generation rate over the entire suction surface length. The procedures for acquiring such have been described in 2, 4 and 5. In short, the contributions to the entropy generation rate S˙gen″ per unit area in an incompressible, two-dimensional boundary layer are given by the sum of the viscous dissipation and turbulence production terms:(1)For the case of increased FST it will be shown that it is necessary to include the Reynolds stress term in the calculation of laminar boundary layer loss. However, the very thin boundary layer present in this investigation precluded the use of multi-sensor hot-wire probes. As it was not possible to directly measure, the Reynolds stress in the turbulent boundary layer, an approximation was required. This approximation is based on the assumption that in the near-wall region of a turbulent boundary layer, the total shear stress is constant and equal to the wall value. Therefore, the required element of the Reynolds stress tensor is approximated as: (2)μ ∂u¯∂yy=0−μ ∂u¯∂y≅τxy′=−ρu′v′¯.This relationship is only valid in the absence of large streamwise pressure gradients and holds approximately true for the first 10% of a turbulent boundary layer. Fortunately this constitutes the most loss-affected region of the turbulent boundary layer 4. Additionally, studies on flat-plate boundary layers by 6 have confirmed that increased FST only alters the velocity defect region of the turbulent boundary layer. Their investigation illustrated that at FST levels of up to 20%, the near-wall region agrees very closely with the classical universal velocity profile for a turbulent boundary layer. By combining Eqs. (1) and (2), the entropy generation rate becomes: (3)S˙gen″=μT ∂u¯∂yy=0∫0δ ∂u¯∂y dy.However, as the constant stress approximation is not valid in the presence of strong pressure gradients (adverse or favorable), calculation of the entropy generation rate in laminar and transitional boundary layers only involved integration of the first term in Eq. (1). Therefore, all laminar and transitional loss measurements presented here are slightly underestimated as they do not explicitly account for the Reynolds shear stress. The turbine cascade and associated instrumentation have previously been described in detail in Refs. 2345. An illustration is given in Fig. 1. The cascade consists of seven aluminum blades, each of 92.6 mm chord, 300 mm span and spaced 70.44 mm apart. Boundary layer velocity data was acquired at a sampling frequency of 20 kHz (over a 0.2 sec duration) by traversing a single sensor hot-wire probe (Dantec type P11) operating at 250°C overheat normal to the surface. Traversing was initially carried out in 10 μm increments as very fine resolution is required within the first 0.5 mm from the wall as the majority of entropy is generated here. Hence motion of the sensor normal to the surface was achieved using a precision traversing mechanism synchronized with the data acquisition system for automatic measurement. In addition, data was also acquired from an array of 28 hot-film sensors (spaced in 3.5% increments of suction surface length) located at mid-span of the central blade in the cascade. A perforated plate was used to raise the turbulence intensity at the inlet to the blade passage from 0.8 to 5.0% 3. The standard error analysis technique of Kline and McClintock 7 was used to find the uncertainty in S˙gen″. Firstly, based on all possible sources of error in the hot-wire measurement technique the error in the boundary layer velocity gradient and subsequent wall shear stress estimate was estimated as ±10% 4. The primary source of error comes from the low velocity hot-wire calibration needed in the near-wall region. The error shown is believed to be greatly over estimated, because the wire calibration at higher velocities is extendible to low velocities, provided that the flow velocity is not so low that natural convection effects are significant. Repeat experiments with different calibrated hot-wire sensors have shown a lower uncertainty than that estimated. Based on the estimated error of ±10% in the measurement of the velocity gradient, an estimate of the error of S˙gen‴‴‴ was found to be ±20% using the method described in 7. Overall, the calculated maximum error in the S˙gen″ values is of the order of ±10%, half that of S˙gen‴‴‴ as the integration step averages out the errors. Figure 2 illustrates a distribution of the pressure coefficient along the blade suction surface for both turbulence levels at Rec=76,000, in which peak suction occurs at 35% of suction surface length. Figure 3 shows a typical distribution of the mean and turbulent velocity components for the high turbulence case in the laminar boundary layer. Clearly visible are increased fluid strain rates under conditions of high FST as well as appreciable velocity fluctuations across the boundary layer. These effects are not due to an earlier transition as the skew distribution shown in Fig. 4(b) illustrates an unambiguous transition for the high turbulence case at 54% suction surface length. As this laminar boundary layer possesses both laminar and turbulent characteristics we have named it a hybrid boundary layer. In a classical turbulent boundary layer the fluctuating velocities are at a maximum very close to the wall, however in a perturbed laminar boundary layer maxima are encountered further out from the wall (η&gt;0.1). These findings have previously been observed in similar boundary layers by 8 and 9. The measured local FST in the laminar region was found to equal that convected from the grid at the boundary layer edge, according to: Tuδ=Tu∞ U∞Uδ.There is therefore very little damping of the FST. Using Eq. (3), velocity data is transformed into loss data. Often it is more convenient to express the measured values of S˙gen″ in terms of a non-dimensional dissipation coefficient defined as 1: (4)Cd=TS˙gen″ρUδ3.The distribution of Cd is plotted in Fig. 4(a) for both high and low FST levels at Rec=76,000. Significant losses are observed in the laminar and transitional regions with increased FST. Losses in the pre-transitional boundary layer are substantially increased in the presence of FST primarily due to changes in the mean velocity profile. This loss is even greater if the induced Reynolds stresses are considered. If a CFD code is to accurately model flows under these conditions it must account fully for the affects of a turbulent free stream. These phenomena are neglected in the conventional modelling of laminar boundary layers which would likely require the use of different turbulence models for the laminar and turbulent boundary layers. In contrast, the turbulent boundary layer is largely unaffected by FST with a near constant value Cd≈0.003 (inclusive of Reynolds stress effects) for both FST levels at the same Reynolds number. This view is supported in 10, noting that FST only effects the low loss outer boundary layer. The overall loss generated in the entire suction surface boundary layer can best be viewed when Eq. (3) is integrated in the streamwise direction: (5)S˙gen′=∫0SSLS˙gen″dx.Table 1 summarizes the relative distribution of the blade suction surface loss between the laminar, transitional and turbulent regions. The total loss for the 5.0% FST test case is almost 45% higher than its low turbulence counterpart due to increased losses in the laminar and transitional regions for the former. Greater losses are observed in the turbulent boundary layer of the high FST case, due to an earlier transition and hence larger coverage of the suction surface by a turbulent boundary layer. Interestingly, the relative distribution of loss between the laminar, transitional and turbulent regions remains unchanged. The fact that the laminar boundary layer accounts for the majority of the suction surface loss may appear contrary to common perception. However, as noted by 4, the very thin laminar boundary layer coupled with a high freestream velocity due to acceleration of the flow over the fore section of the blade, generates very large velocity gradients. Furthermore, the validity of the results has been verified by 11 in which a comparison is made between the integrated boundary layer entropy generation rate and that estimated by Eq. (6) from stagnation pressure losses in the turbine blade wake (inclusive of trailing edge loss and turbulent phenomena upstream of the turbulent boundary layer): (6)S˙gen′=−ρR∫0δu ¯lnp02p01dy.Figure 5 shows frequency spectra at different wall-normal positions in the laminar boundary layer for both FST levels including those of the free stream (η&gt;1). Evident in all spectra is the similarity to the classical Kolmogorov −5/3 power slope characterizing the inertial subrange 1213. This represents the intermediate scales of turbulence; those lying between the larger energy containing and smaller dissipatative scales. Adherence to the Kolmogorov −5/3 power-law slope indicates a flow with an equilibrium of production and dissipation of turbulent kinetic energy 12. It is also apparent that the high free stream turbulence spectra contain greater spectral energy over the same range of frequencies than their low turbulence counterparts. For both cases the highest spectral power is observed moving into the boundary layer, with the highest encountered in the region corresponding to the maximum value of u′2/Uδ at η=0.15 in Fig. 3. In addition, the boundary layer spectra exhibit greater spectral power in the lower frequency range, which increases towards the wall. This has been previously observed by 9 and is due to the receptivity of the boundary layer to disturbances in the free stream 14. The boundary layer can be considered as a ‘noise amplifier’, which selectively amplifies low frequency fluctuations from the free stream. The maximum amplification was not observed in the near-wall region (i.e. at η&lt;0.05, where Tollmien-Schlichting instabilities would prevail), but further out. An increase in the free-stream turbulence level from 0.8% to 5.0% resulted in almost a 45% increase in suction surface boundary layer loss for a comparable Reynolds number. Laminar and transitional losses were most influenced by the freestream turbulence. As the perturbed laminar suction surface boundary layer exhibits significant turbulent characteristics it has been termed a hybrid boundary layer. In contrast, the local turbulent boundary layer loss was unaffected by an increase in freestream turbulence, as a constant value of Cd≈0.003 was observed for both freestream turbulence levels. The measured entropy generation rate compares favourably with that determined from the pressure loss in the turbine blade wake. The discrepancy at high FST arises due to the Reynolds stresses in the laminar and transitional boundary layers which were not included in the hot-wire determination. Therefore, if CFD codes are to accurately model boundary layers under these conditions, they must account for the turbulent phenomena. It is not clear, which, if any of the existing models will work. Cd= Dissipation coefficient (Eq. 4) − Cp= Pressure coefficient =p−p∞/0.5ρU∞2 − Rec= Reynolds number based on blade chord and inlet free-stream velocity − S˙gen″= Entropy generation rate per unit area Wm−2 K S˙gen′= Entropy generation rate per unit length Wm−1 K p01= Stagnation pressure at cascade inlet Nm−2p02= Stagnation pressure at cascade outlet Nm−2R= Gas constant J Kg−1 K−1Tuδ= Local percentage turbulence intensity % Tu∞= Inlet percentage turbulence intensity % T= Bulk fluid temperature K u= Instantaneous x-direction velocity ms−1u¯= Mean velocity in x-direction ms−1u′2¯= RMS average of x-direction fluctuating velocity component ms−1Uδ= Local free-stream velocity ms−1U∞= Inlet free-stream velocity ms−1y= Wall normal coordinate m δ = Boundary layer thickness m η = Dimensionless wall-normal coordinate =y/δ − μ = Dynamic viscosity Nsm−2τxy′= Turbulent shearing stress Nm−2