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Effect of freestream turbulence on recovery factor of a thermocouple probe and its consequences
International Journal of Heat and Mass Transfer 152 (2020) 119498
Contents lists available at ScienceDirect
International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/hmt
Effect of freestream turbulence on recovery factor of a thermocouple probe and its consequences K.S. Kulkarni, U. Madanan∗, R.J. Goldstein Department of Mechanical Engineering, University of Minnesota, 111 Church St, MN-55455, USA
a r t i c l e
i n f o
Article history: Received 29 September 2019 Revised 28 January 2020 Accepted 11 February 2020
Keywords: Thermocouple Temperature measurement Recovery factor Freestream turbulence Energy separation
a b s t r a c t The temperature recovery factor of a thermocouple probe is important when being used to measure fluid temperature of a high-speed flow. Typically, the recovery factor is assumed to be dependent on the probe geometry and Prandtl number of the fluid, and invariant with freestream turbulence. In the present study, recovery factor of a butt-joint thermocouple probe is determined for two freestream turbulence conditions (0.25% and 7%). Results show that the recovery factor decreases with an increase in the freestream turbulence, which can lead to significant errors if not taken into consideration. The consequences of not taking into account the effect of freestream turbulence on the probe recovery factor is demonstrated with an application to energy separation in the wake of a circular cylinder.
1. Introduction When a stationary thermocouple is placed inside a fluid flowing at high speed, it measures an equilibrium temperature which is neither the local static temperature nor the total temperature of the flow, but a value called the recovery temperature [1]. Due to the irreversible behavior of the fluid, caused by its viscosity and thermal conductivity, this recovery temperature is usually lower than the total temperature (or, the ideal value that would exist if the fluid is stagnated adiabatically). Thus, only a fraction of the kinetic energy is recovered by the thermocouple probe, called the recovery factor of the probe. Significant research has been carried out to determine the recovery factor for a flat plate (or, bodies of revolution) immersed in high-speed flows due to its far-reaching real-world applications such as understanding the excessive heating during re-entry of space vehicles [2–4]. The recovery factor is postulated to be a function of probe geometry and fluid Prandtl number (Pr). Many other researchers have also investigated into the effect of various other factors (such as probe design, probe orientation, Reynolds number (Re), and Mach number (Ma)) that can affect the probe recovery factor [5–7]. They have often also proposed correlating equations in terms of either Mach number and Reynolds number [6] or Reynolds number and Prandtl number [7]. However, studies taking into account the effect of freestream turbulence on the probe recovery factor are scarce. Zukauskas and ∗
Corresponding author. E-mail address: [email protected] (U. Madanan).
https://doi.org/10.1016/j.ijheatmasstransfer.2020.119498 0017-9310/© 2020 Elsevier Ltd. All rights reserved.
© 2020 Elsevier Ltd. All rights reserved.
Ziugzda [8] have studied the effect of freestream turbulence and blockage factor on the flow structure around a cylinder in crossflow. Since some thermocouple probes can be approximated as a cylinder in cross-flow, their findings can have a direct consequence on the cylinder (probe) recovery factor. Stinson and Goldstein [9] have recently reported that the probe recovery factor decreases slightly as the freestream turbulence is increased. They have performed experiments with thermocouple probes of varied wire gauges and covering a wide range of freestream turbulence intensities (1 − 6%) for a fixed Mach number of 0.2. They have proposed a set of correlations to estimate the probe recovery factor for the investigated range freestream turbulence intensities. The present study aims to validate this decreasing trend of recovery factor with freestream turbulence observed in the literature and illustrate the magnitude of potential errors that can surface in an application as a consequence. These objectives are achieved by: (i) performing experiments with a butt-joint thermocouple probe at two different freestream turbulence levels (0.25% and a higher turbulence of 7%) and varied Mach numbers (0.2 ≤ Ma ≤ 0.3) to assess the effect of freestream turbulence intensity on the probe recovery factor and (ii) analyzing the implications of improperly correcting for this effect when determining the total temperature from the probe-measured temperature in a highly turbulent flow situation, i.e., energy separation in the wake of a circular cylinder. 2. Experimental setup and procedures The present study uses a butt-joint K-type thermocouple probe made of 30 gauge ( ≈ 0.25 mm) shielded wires (see Fig. 1). Two
2
K.S. Kulkarni, U. Madanan and R.J. Goldstein / International Journal of Heat and Mass Transfer 152 (2020) 119498
Nomenclature Cp D L Ma Pr Re r r1 rp S S1 STu Td
Specific heat at constant pressure for the fluid [J/Kg · K] Diameter of circular cylinder [m] Wire mesh spacing [m] Mach number Prandtl number Reynolds number Recovery factor of temperature probe for negligible turbulence (0.25%) Recovery factor of temperature probe for 7% turbulence Recovery factor of temperature probe at an arbitrary freestream turbulence Energy separation factor (Eq. (4)) Modified energy separation factor (Eq. (5)) Modified energy separation factor at a higher arbitrary freestream turbulence, rTu (Eq. (6)) Local Dynamic temperature in the wake of cylinder [K] 2 U∞ 2C p ) 2 U00 2C p ) U12 2C p )
Td,∞
Dynamic temperature (=
Td,00
Dynamic temperature (=
Td,1
Dynamic temperature (=
Tr
Recovery temperature measured by the probe in the wake of cylinder [K] Recovery temperature measured by the probe in the absence of cylinder or wire mesh [K] Recovery temperature measured by the probe downstream of wire mesh [K] Static temperature [K] Total (stagnation) temperature [K] Total (stagnation) temperature at wind tunnel inlet [K] Turbulence intensity Instantaneous stream-wise velocity [m/s] Stream-wise velocity in the absence of cylinder or wire mesh [m/s] Spatially and temporally averaged stream-wise velocity upstream of cylinder [m/s] Spatially and temporally averaged stream-wise velocity upstream of wire mesh [m/s] random fluctuations in stream-wise (x) velocity [m/s] random fluctuations in cross stream-wise (y) velocity [m/s] Distance in stream-wise direction [m] Distance in cross stream-wise direction [m]
Tr,00 Tr,1 Ts,00 Tt Tt,∞ Tu U U00 U∞ U1 u v x y
[K] [K] [K]
circular steel tubes support the thermocouple junction and also carry the wires into a long steel tube. This design is based on the suggestions of Moffat [1] and Hottel and Kalitinsky [5] and the details of the probe construction can be found in Goldstein and Kulkarni [10]. A subsonic suction type wind tunnel is used to generate high-speed air flows necessary for the present study. A maximum speed of 95 m/s can be achieved with a low freestream turbulence of ~ 0.25% inside a test section of 203 mm × 120 mm cross section. The total temperature of the flow (Tt,∞ ) is measured with the help of five K-type thermocouples placed at the entrance of the wind tunnel (609 mm × 609 mm cross section), where the dynamic temperature is negligible. The thermocouple probe is then introduced into the test section to measure the recovery temperature (Tr,00 ). The freestream velocity (U00 ) is measured using a
Fig. 1. Schematic diagram of recovery temperature probe (all dimensions in mm) [10].
pitot tube placed close to the thermocouple probe during calibration. The recovery factor of this temperature probe can then be calculated using Eq. (1). Tr,00 − Ts,00 Tr,00 − Ts,00 Tr,00 − Tt,∞ r= = =1+ (1) Tt,∞ − Ts,00 Td,00 Td,00 The calibration is repeated with a uniform freestream turbulence of 7%. A wire mesh of 1 mm diameter steel wire, with square mesh spacing (L) of 6 mm, is used to generate a uniform turbulent flow inside the test section. The wire mesh is fixed on a steel frame and covers the entire cross section of the test section resulting in a blockage ratio of 15%. The distance from the wire mesh where the freestream turbulence is uniform and constant in the cross stream and span wise directions is calculated using theoretical arguments [11] and trial experimental observations. All measurements are carried out at this distance (x), which is determined to be 5.5L for a uniform freestream turbulence of 7%. A TSI 1218-T 1.5 standard boundary layer hot-wire probe is used to measure the average velocity and turbulence intensity at various locations along the cross streamwise direction. Once the hot-wire probe is placed at a desired stream-wise location (x/L), an automated traverse (refer to [10] for details) is used to capture the various cross stream-wise (y/L) measurements. A large number of samples (typically ~ 20 0 0 0) are recorded for the hot-wire voltage data, with the help of a Keithley 194A high-speed voltmeter, to compute the instantaneous velocity. This instantaneous velocity is used to estimate the corresponding time-averaged velocity and random fluctuations. Turbulence intensity is then calculated from the random fluctuations and the timeaveraged velocity. Fig. 2 shows the average velocity and turbulence intensity profiles at xL = 5.5 for an upstream uniform velocity (U1 )
K.S. Kulkarni, U. Madanan and R.J. Goldstein / International Journal of Heat and Mass Transfer 152 (2020) 119498
3
Fig. 2. Normalized velocity and turbulence intensity profile in the wake of cylinder (x/L = 5.5 , U00 = 60 m/s ).
1. The recovery factor of the probe is a constant for constant and uniform turbulence conditions. 2. Under turbulent conditions, there is no significant dependence observed for the recovery factor on the freestream velocity for the investigated range. 3. The recovery factor estimated under turbulent conditions is found to be lower than that measured under negligible turbulence conditions. The present study, thus, highlights the importance of accounting for the freestream turbulence while determining the total temperature of high-speed gas flows with a recovery temperature probe. The significance of the error introduced when measuring the total temperature can be clearly seen in highly turbulent flow applications such as energy separation in the wake of a circular cylinder, which is discussed in the following section. 4. Application to energy separation in the wake of a cylinder Fig. 3. Calibration of recovery temperature probe (circles 0.25% Tu, squares 7% Tu).
of 60 m/s. It is evident from this figure that the turbulence intensity at this downstream location is uniform. The recovery temperature probe is placed at this location and the recovery factor is calculated using the dynamic temperature of the upstream uniform flow, as given by Eq. (2).
r1 = 1 +
Tr,1 − Tt,∞ Td,1
(2)
3. Discussion of results The estimated probe recovery factor values are observed to follow a decreasing trend with the freestream turbulence. Although the freestream turbulence intensities fall slightly outside the range used for their study, these estimated probe recovery factor values are also found to be within 10% of the predictions of Stinson and Goldstein [9] (see Eq. (3)).
rp =
√ √ P r + (1 − P r )(−1.25 − 6.41T u )
(3)
The results of the two calibrations are plotted in Fig. 3. A few noteworthy observations from the comparison of recovery factors for these two cases are:
Energy separation is a spontaneous redistribution of total energy (enthalpy) in a fluid without external work or heat transfer, which results in some portions of fluid having higher total energy (enthalpy) and other portions having lower total energy (enthalpy) than the surrounding fluid. Eckert and Weise [12] are the first to observe this phenomenon in the case of a circular cylinder. Further studies on the energy separation phenomenon have been conducted by many other researchers for different flows such as laminar and turbulent boundary layers [13], impinging jets [14], shear layers [15], and flow behind bluff bodies [10,16]. Most of the literature agree that, in general, the cause of energy separation is convection of unsteady vortices. Kurosaka et al. [16] and Eckert [17] further attribute the reason behind the energy separation in the wake of a circular cylinder to Kármán vortex street that forms behind the cylinder at high Reynolds numbers. For circular cylinders, time-averaged total temperature measurements can be utilized to verify the extent of this energy separation. Time-averaged total temperature measurements show the existence of a central wake region, where the total temperature is found to be lower than that of the freestream, and two symmetrical regions near the outer wake, where the total temperature is higher than the freestream total temperature [10,18]. In general, this total temperature is determined using a recovery temperature probe and expressed in terms of an energy separation factor (S), as given in Eq. (4). Fig. 4 shows the profile of this energy sep-
4
K.S. Kulkarni, U. Madanan and R.J. Goldstein / International Journal of Heat and Mass Transfer 152 (2020) 119498
To resolve this discrepancy, a modified energy separation factor (S1 ) is introduced based on the present experimental work. An equation (Eq. (5)) is proposed to calculate this modified energy separation factor (S1 ) using the recovery factor under significantly turbulent conditions (or, r1 ; see Eq. (2)). For lower turbulence intensities, i.e., Tu < 7%, the general expression (Eq. (4)) may be utilized. Thus, a comparison of Eqs. (4) and (5) will help us appreciate the discrepancy in the estimated energy separation factor that would prevail under turbulent situations when using a probe calibrated at a near-zero turbulence intensity.
S1 =
Fig. 4. Effect of recovery factor on energy separation factor profile in the wake of cylinder (Re = 920 0 0 , x/D = 1 ).
aration factor at Dx = 1 in the cross stream wise direction, nondimensionalized with the cylinder diameter (i.e., Dy ). Note that the present study utilizes an adiabatic hollow cylinder, made of phenolic, with 19 mm outside diameter and 10.2 mm inside diameter. This yields a Reynolds number (Re) of ≈ 920 0 0 at which, for the flow past a circular cylinder, formation of a turbulent wake is evident. Additional information about the experimental setup used to study the energy separation in the wake of a cylinder can be found in Goldstein and Kulkarni [10].
S=1+
Tt − Tt,∞ Tr + (1 − r )Td − Tt,∞ = Td,∞ Td,∞
(4)
The recovery factor of the probe used to calculate S is typically calibrated at a low turbulence intensity. But, the energy separation problem in the wake of a circular cylinder is a highly turbulent flow situation. As is apparent from Fig. 5, showing the measured mean velocity and turbulence intensity at a location Dx = 1, the turbulence intensity is much higher (∼ 40 − 50%) than the value at which the recovery factor probe is typically calibrated. This can introduce significant variation in the energy separation factor.
Tr + (1 − r1 )Td − Tt,∞ Td,∞
(T u ≥ 7% )
(5)
In this analysis, the recovery factor (r1 ) is assumed to be constant for turbulence intensities equal to or higher than 7%. Although this equation (Eq. (5)) is a general expression, for turbulence intensities much higher than 7%, using the same r1 value as that for the 7% Tu case may yield lower predictions for the energy separation factor. Thus, it is recommended that, for accurate predictions for the energy separation factor, the value of r1 (for the 7% Tu case in the present analysis) needs to be replaced with that for the desired turbulence intensity (refer to Eq. (6)). In the present study, analysis was performed with a constant r1 (or, that for T u = 7% for all Tu ≥ 7%) to demonstrate the potential for errors when a probe calibrated at a Tu ~ 0% is used to determine the total temperature in a highly turbulent flow situation.
ST u =
Tr + (1 − rT u )Td − Tt,∞ Td,∞
(6)
Fig. 4 shows the profile of the modified energy separation factor overlaid on the profile of the standard energy separation factor. From this figure, it is evident that the energy separation factor (or, the total temperature) can be seriously underestimated if the correction to recovery factor due to turbulence is not applied. 5. Conclusions Effect of freestream turbulence on the recovery factor of a buttjoint thermocouple probe is investigated for turbulence intensities of 0.25% and 7%. The probe recovery factor is found to be lower at a higher freestream turbulence intensity, which in turn can affect the total temperature measurement. The effect of such erroneous measurements is examined for energy separation results in the wake of a cylinder. These results show that the estimated
Fig. 5. Normalized velocity and turbulence intensity profile in the wake of cylinder (Re = 920 0 0 , x/D = 1 ).
K.S. Kulkarni, U. Madanan and R.J. Goldstein / International Journal of Heat and Mass Transfer 152 (2020) 119498
total temperature is substantially underestimated if the effect of freestream turbulence intensity on probe recovery factor is not considered. Thus, it is evident that the total temperature calculated from the measured recovery temperature of the probe needs to be corrected properly when used under highly turbulent flow conditions. Declaration of Competing Interest None. References [1] R.J. Moffat, Gas Temperature Measurement; Temperature: its Measurement and Control in Science and Industry, Reinhold Publishing Corporation, NY, 1962. [2] H.A. Stine, R. Scherrer, Experimental investigation of turbulent-boundary-layer temperature-recovery factor on bodies of revolution at Mach numbers from 2.0 to 3.8, NACA Technical note 2664, 1952. [3] J.R. Stalder, M.W. Rubesin, T. Tendeland, Determination of laminar-, transitional-, and turbulent-boundary-layer temperature-recovery factors on flat plate in supersonic flow, NACA Technical note 2077, 1950. [4] W.R. Wimbrow, Experimental investigation of temperature recovery factors on bodies of revolution at supersonic speeds, NACA Technical note 1975, 1949. [5] H.C. Hottel, A. Kalitinsky, Temperature measurements in high-velocity air streams, J. Appl. Mech. 12(1) (1945) A25–A31. [6] C.F. Dewey Jr, A correlation of convective heat transfer and recovery temperature data for cylinders in compressible flow, Int. J. Heat Mass Transf. 8(2) (1965) 245–252.
5
[7] A. Hirschberg, J.C. Van Muiswinkel, A correlation of recovery temperature data for cylinders in a compressible flow at high Reynolds numbers, Int. J. Heat Mass Transf. 20(6) (1977) 669–674. [8] A. Zukauskas, J. Ziugzda, Heat transfer of a cylinder in crossflow, Hemisphere Publishing Corp., Washington, DC, 1985, p. 219. [9] M. Stinson, R.J. Goldstein, Effect of freestream turbulence on recovery factor of a cylindrical temperature probe, In ASTFE Digital Library, Begell House Inc., 2015. [10] R.J. Goldstein, K.S. Kulkarni, Energy separation in the wake of a cylinder, J. Heat Transf. 130 (6) (2008) 061703-1-9. [11] H. Tennekes, J.L. Lumley, A First Course in Turbulence, MIT Press, 1972. [12] E.R.G. Eckert, W. Weise, Messugn der temperaturverteilung auf der oberflache schnell angestromter unbeheizter korper, Der deutscgen Luftahrtforschung 2 Jahrbuch (1940) 25–31. [13] E.R.G. Eckert, O. Drewiz, Die berechnung des temperaturfeldes in der laminaren grenzschicht schnell angestromter, unbeheizter korper, Luftfartforschung 19 (1941) 189–196. [14] W.S. Seol, R.J. Goldstein, Energy separation in a jet flow, J. Fluids Eng. 119(1) (1997) 74–82. [15] B. Han, R.J. Goldstein, H.G. Choi, Energy separation in shear layers, Int. J. Heat Mass Transf. 45 (2002) 47–55. [16] M. Kurosaka, J.B. Gertz, J.E. Graham, J.R. Goodman, P. Sundaram, W.C. Ringer, H. Kuroda, W.L. Hankey, Energy separation in a vortex street, J. Fluid Mech. 178 (1987) 1–29. [17] E.R.G. Eckert, Energy separation in fluid streams, Int. Commun. Heat Mass Transf. 13 (1986) 127–143. [18] K.S. Kulkarni, R.J. Goldstein, Energy separation in the wake of cylinder: effect of Reynolds number and acoustic resonance, Int. J. Heat Mass Transf. 52 (2009) 3994–40 0 0.
Contents lists available at ScienceDirect
International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/hmt
Effect of freestream turbulence on recovery factor of a thermocouple probe and its consequences K.S. Kulkarni, U. Madanan∗, R.J. Goldstein Department of Mechanical Engineering, University of Minnesota, 111 Church St, MN-55455, USA
a r t i c l e
i n f o
Article history: Received 29 September 2019 Revised 28 January 2020 Accepted 11 February 2020
Keywords: Thermocouple Temperature measurement Recovery factor Freestream turbulence Energy separation
a b s t r a c t The temperature recovery factor of a thermocouple probe is important when being used to measure fluid temperature of a high-speed flow. Typically, the recovery factor is assumed to be dependent on the probe geometry and Prandtl number of the fluid, and invariant with freestream turbulence. In the present study, recovery factor of a butt-joint thermocouple probe is determined for two freestream turbulence conditions (0.25% and 7%). Results show that the recovery factor decreases with an increase in the freestream turbulence, which can lead to significant errors if not taken into consideration. The consequences of not taking into account the effect of freestream turbulence on the probe recovery factor is demonstrated with an application to energy separation in the wake of a circular cylinder.
1. Introduction When a stationary thermocouple is placed inside a fluid flowing at high speed, it measures an equilibrium temperature which is neither the local static temperature nor the total temperature of the flow, but a value called the recovery temperature [1]. Due to the irreversible behavior of the fluid, caused by its viscosity and thermal conductivity, this recovery temperature is usually lower than the total temperature (or, the ideal value that would exist if the fluid is stagnated adiabatically). Thus, only a fraction of the kinetic energy is recovered by the thermocouple probe, called the recovery factor of the probe. Significant research has been carried out to determine the recovery factor for a flat plate (or, bodies of revolution) immersed in high-speed flows due to its far-reaching real-world applications such as understanding the excessive heating during re-entry of space vehicles [2–4]. The recovery factor is postulated to be a function of probe geometry and fluid Prandtl number (Pr). Many other researchers have also investigated into the effect of various other factors (such as probe design, probe orientation, Reynolds number (Re), and Mach number (Ma)) that can affect the probe recovery factor [5–7]. They have often also proposed correlating equations in terms of either Mach number and Reynolds number [6] or Reynolds number and Prandtl number [7]. However, studies taking into account the effect of freestream turbulence on the probe recovery factor are scarce. Zukauskas and ∗
Corresponding author. E-mail address: [email protected] (U. Madanan).
https://doi.org/10.1016/j.ijheatmasstransfer.2020.119498 0017-9310/© 2020 Elsevier Ltd. All rights reserved.
© 2020 Elsevier Ltd. All rights reserved.
Ziugzda [8] have studied the effect of freestream turbulence and blockage factor on the flow structure around a cylinder in crossflow. Since some thermocouple probes can be approximated as a cylinder in cross-flow, their findings can have a direct consequence on the cylinder (probe) recovery factor. Stinson and Goldstein [9] have recently reported that the probe recovery factor decreases slightly as the freestream turbulence is increased. They have performed experiments with thermocouple probes of varied wire gauges and covering a wide range of freestream turbulence intensities (1 − 6%) for a fixed Mach number of 0.2. They have proposed a set of correlations to estimate the probe recovery factor for the investigated range freestream turbulence intensities. The present study aims to validate this decreasing trend of recovery factor with freestream turbulence observed in the literature and illustrate the magnitude of potential errors that can surface in an application as a consequence. These objectives are achieved by: (i) performing experiments with a butt-joint thermocouple probe at two different freestream turbulence levels (0.25% and a higher turbulence of 7%) and varied Mach numbers (0.2 ≤ Ma ≤ 0.3) to assess the effect of freestream turbulence intensity on the probe recovery factor and (ii) analyzing the implications of improperly correcting for this effect when determining the total temperature from the probe-measured temperature in a highly turbulent flow situation, i.e., energy separation in the wake of a circular cylinder. 2. Experimental setup and procedures The present study uses a butt-joint K-type thermocouple probe made of 30 gauge ( ≈ 0.25 mm) shielded wires (see Fig. 1). Two
2
K.S. Kulkarni, U. Madanan and R.J. Goldstein / International Journal of Heat and Mass Transfer 152 (2020) 119498
Nomenclature Cp D L Ma Pr Re r r1 rp S S1 STu Td
Specific heat at constant pressure for the fluid [J/Kg · K] Diameter of circular cylinder [m] Wire mesh spacing [m] Mach number Prandtl number Reynolds number Recovery factor of temperature probe for negligible turbulence (0.25%) Recovery factor of temperature probe for 7% turbulence Recovery factor of temperature probe at an arbitrary freestream turbulence Energy separation factor (Eq. (4)) Modified energy separation factor (Eq. (5)) Modified energy separation factor at a higher arbitrary freestream turbulence, rTu (Eq. (6)) Local Dynamic temperature in the wake of cylinder [K] 2 U∞ 2C p ) 2 U00 2C p ) U12 2C p )
Td,∞
Dynamic temperature (=
Td,00
Dynamic temperature (=
Td,1
Dynamic temperature (=
Tr
Recovery temperature measured by the probe in the wake of cylinder [K] Recovery temperature measured by the probe in the absence of cylinder or wire mesh [K] Recovery temperature measured by the probe downstream of wire mesh [K] Static temperature [K] Total (stagnation) temperature [K] Total (stagnation) temperature at wind tunnel inlet [K] Turbulence intensity Instantaneous stream-wise velocity [m/s] Stream-wise velocity in the absence of cylinder or wire mesh [m/s] Spatially and temporally averaged stream-wise velocity upstream of cylinder [m/s] Spatially and temporally averaged stream-wise velocity upstream of wire mesh [m/s] random fluctuations in stream-wise (x) velocity [m/s] random fluctuations in cross stream-wise (y) velocity [m/s] Distance in stream-wise direction [m] Distance in cross stream-wise direction [m]
Tr,00 Tr,1 Ts,00 Tt Tt,∞ Tu U U00 U∞ U1 u v x y
[K] [K] [K]
circular steel tubes support the thermocouple junction and also carry the wires into a long steel tube. This design is based on the suggestions of Moffat [1] and Hottel and Kalitinsky [5] and the details of the probe construction can be found in Goldstein and Kulkarni [10]. A subsonic suction type wind tunnel is used to generate high-speed air flows necessary for the present study. A maximum speed of 95 m/s can be achieved with a low freestream turbulence of ~ 0.25% inside a test section of 203 mm × 120 mm cross section. The total temperature of the flow (Tt,∞ ) is measured with the help of five K-type thermocouples placed at the entrance of the wind tunnel (609 mm × 609 mm cross section), where the dynamic temperature is negligible. The thermocouple probe is then introduced into the test section to measure the recovery temperature (Tr,00 ). The freestream velocity (U00 ) is measured using a
Fig. 1. Schematic diagram of recovery temperature probe (all dimensions in mm) [10].
pitot tube placed close to the thermocouple probe during calibration. The recovery factor of this temperature probe can then be calculated using Eq. (1). Tr,00 − Ts,00 Tr,00 − Ts,00 Tr,00 − Tt,∞ r= = =1+ (1) Tt,∞ − Ts,00 Td,00 Td,00 The calibration is repeated with a uniform freestream turbulence of 7%. A wire mesh of 1 mm diameter steel wire, with square mesh spacing (L) of 6 mm, is used to generate a uniform turbulent flow inside the test section. The wire mesh is fixed on a steel frame and covers the entire cross section of the test section resulting in a blockage ratio of 15%. The distance from the wire mesh where the freestream turbulence is uniform and constant in the cross stream and span wise directions is calculated using theoretical arguments [11] and trial experimental observations. All measurements are carried out at this distance (x), which is determined to be 5.5L for a uniform freestream turbulence of 7%. A TSI 1218-T 1.5 standard boundary layer hot-wire probe is used to measure the average velocity and turbulence intensity at various locations along the cross streamwise direction. Once the hot-wire probe is placed at a desired stream-wise location (x/L), an automated traverse (refer to [10] for details) is used to capture the various cross stream-wise (y/L) measurements. A large number of samples (typically ~ 20 0 0 0) are recorded for the hot-wire voltage data, with the help of a Keithley 194A high-speed voltmeter, to compute the instantaneous velocity. This instantaneous velocity is used to estimate the corresponding time-averaged velocity and random fluctuations. Turbulence intensity is then calculated from the random fluctuations and the timeaveraged velocity. Fig. 2 shows the average velocity and turbulence intensity profiles at xL = 5.5 for an upstream uniform velocity (U1 )
K.S. Kulkarni, U. Madanan and R.J. Goldstein / International Journal of Heat and Mass Transfer 152 (2020) 119498
3
Fig. 2. Normalized velocity and turbulence intensity profile in the wake of cylinder (x/L = 5.5 , U00 = 60 m/s ).
1. The recovery factor of the probe is a constant for constant and uniform turbulence conditions. 2. Under turbulent conditions, there is no significant dependence observed for the recovery factor on the freestream velocity for the investigated range. 3. The recovery factor estimated under turbulent conditions is found to be lower than that measured under negligible turbulence conditions. The present study, thus, highlights the importance of accounting for the freestream turbulence while determining the total temperature of high-speed gas flows with a recovery temperature probe. The significance of the error introduced when measuring the total temperature can be clearly seen in highly turbulent flow applications such as energy separation in the wake of a circular cylinder, which is discussed in the following section. 4. Application to energy separation in the wake of a cylinder Fig. 3. Calibration of recovery temperature probe (circles 0.25% Tu, squares 7% Tu).
of 60 m/s. It is evident from this figure that the turbulence intensity at this downstream location is uniform. The recovery temperature probe is placed at this location and the recovery factor is calculated using the dynamic temperature of the upstream uniform flow, as given by Eq. (2).
r1 = 1 +
Tr,1 − Tt,∞ Td,1
(2)
3. Discussion of results The estimated probe recovery factor values are observed to follow a decreasing trend with the freestream turbulence. Although the freestream turbulence intensities fall slightly outside the range used for their study, these estimated probe recovery factor values are also found to be within 10% of the predictions of Stinson and Goldstein [9] (see Eq. (3)).
rp =
√ √ P r + (1 − P r )(−1.25 − 6.41T u )
(3)
The results of the two calibrations are plotted in Fig. 3. A few noteworthy observations from the comparison of recovery factors for these two cases are:
Energy separation is a spontaneous redistribution of total energy (enthalpy) in a fluid without external work or heat transfer, which results in some portions of fluid having higher total energy (enthalpy) and other portions having lower total energy (enthalpy) than the surrounding fluid. Eckert and Weise [12] are the first to observe this phenomenon in the case of a circular cylinder. Further studies on the energy separation phenomenon have been conducted by many other researchers for different flows such as laminar and turbulent boundary layers [13], impinging jets [14], shear layers [15], and flow behind bluff bodies [10,16]. Most of the literature agree that, in general, the cause of energy separation is convection of unsteady vortices. Kurosaka et al. [16] and Eckert [17] further attribute the reason behind the energy separation in the wake of a circular cylinder to Kármán vortex street that forms behind the cylinder at high Reynolds numbers. For circular cylinders, time-averaged total temperature measurements can be utilized to verify the extent of this energy separation. Time-averaged total temperature measurements show the existence of a central wake region, where the total temperature is found to be lower than that of the freestream, and two symmetrical regions near the outer wake, where the total temperature is higher than the freestream total temperature [10,18]. In general, this total temperature is determined using a recovery temperature probe and expressed in terms of an energy separation factor (S), as given in Eq. (4). Fig. 4 shows the profile of this energy sep-
4
K.S. Kulkarni, U. Madanan and R.J. Goldstein / International Journal of Heat and Mass Transfer 152 (2020) 119498
To resolve this discrepancy, a modified energy separation factor (S1 ) is introduced based on the present experimental work. An equation (Eq. (5)) is proposed to calculate this modified energy separation factor (S1 ) using the recovery factor under significantly turbulent conditions (or, r1 ; see Eq. (2)). For lower turbulence intensities, i.e., Tu < 7%, the general expression (Eq. (4)) may be utilized. Thus, a comparison of Eqs. (4) and (5) will help us appreciate the discrepancy in the estimated energy separation factor that would prevail under turbulent situations when using a probe calibrated at a near-zero turbulence intensity.
S1 =
Fig. 4. Effect of recovery factor on energy separation factor profile in the wake of cylinder (Re = 920 0 0 , x/D = 1 ).
aration factor at Dx = 1 in the cross stream wise direction, nondimensionalized with the cylinder diameter (i.e., Dy ). Note that the present study utilizes an adiabatic hollow cylinder, made of phenolic, with 19 mm outside diameter and 10.2 mm inside diameter. This yields a Reynolds number (Re) of ≈ 920 0 0 at which, for the flow past a circular cylinder, formation of a turbulent wake is evident. Additional information about the experimental setup used to study the energy separation in the wake of a cylinder can be found in Goldstein and Kulkarni [10].
S=1+
Tt − Tt,∞ Tr + (1 − r )Td − Tt,∞ = Td,∞ Td,∞
(4)
The recovery factor of the probe used to calculate S is typically calibrated at a low turbulence intensity. But, the energy separation problem in the wake of a circular cylinder is a highly turbulent flow situation. As is apparent from Fig. 5, showing the measured mean velocity and turbulence intensity at a location Dx = 1, the turbulence intensity is much higher (∼ 40 − 50%) than the value at which the recovery factor probe is typically calibrated. This can introduce significant variation in the energy separation factor.
Tr + (1 − r1 )Td − Tt,∞ Td,∞
(T u ≥ 7% )
(5)
In this analysis, the recovery factor (r1 ) is assumed to be constant for turbulence intensities equal to or higher than 7%. Although this equation (Eq. (5)) is a general expression, for turbulence intensities much higher than 7%, using the same r1 value as that for the 7% Tu case may yield lower predictions for the energy separation factor. Thus, it is recommended that, for accurate predictions for the energy separation factor, the value of r1 (for the 7% Tu case in the present analysis) needs to be replaced with that for the desired turbulence intensity (refer to Eq. (6)). In the present study, analysis was performed with a constant r1 (or, that for T u = 7% for all Tu ≥ 7%) to demonstrate the potential for errors when a probe calibrated at a Tu ~ 0% is used to determine the total temperature in a highly turbulent flow situation.
ST u =
Tr + (1 − rT u )Td − Tt,∞ Td,∞
(6)
Fig. 4 shows the profile of the modified energy separation factor overlaid on the profile of the standard energy separation factor. From this figure, it is evident that the energy separation factor (or, the total temperature) can be seriously underestimated if the correction to recovery factor due to turbulence is not applied. 5. Conclusions Effect of freestream turbulence on the recovery factor of a buttjoint thermocouple probe is investigated for turbulence intensities of 0.25% and 7%. The probe recovery factor is found to be lower at a higher freestream turbulence intensity, which in turn can affect the total temperature measurement. The effect of such erroneous measurements is examined for energy separation results in the wake of a cylinder. These results show that the estimated
Fig. 5. Normalized velocity and turbulence intensity profile in the wake of cylinder (Re = 920 0 0 , x/D = 1 ).
K.S. Kulkarni, U. Madanan and R.J. Goldstein / International Journal of Heat and Mass Transfer 152 (2020) 119498
total temperature is substantially underestimated if the effect of freestream turbulence intensity on probe recovery factor is not considered. Thus, it is evident that the total temperature calculated from the measured recovery temperature of the probe needs to be corrected properly when used under highly turbulent flow conditions. Declaration of Competing Interest None. References [1] R.J. Moffat, Gas Temperature Measurement; Temperature: its Measurement and Control in Science and Industry, Reinhold Publishing Corporation, NY, 1962. [2] H.A. Stine, R. Scherrer, Experimental investigation of turbulent-boundary-layer temperature-recovery factor on bodies of revolution at Mach numbers from 2.0 to 3.8, NACA Technical note 2664, 1952. [3] J.R. Stalder, M.W. Rubesin, T. Tendeland, Determination of laminar-, transitional-, and turbulent-boundary-layer temperature-recovery factors on flat plate in supersonic flow, NACA Technical note 2077, 1950. [4] W.R. Wimbrow, Experimental investigation of temperature recovery factors on bodies of revolution at supersonic speeds, NACA Technical note 1975, 1949. [5] H.C. Hottel, A. Kalitinsky, Temperature measurements in high-velocity air streams, J. Appl. Mech. 12(1) (1945) A25–A31. [6] C.F. Dewey Jr, A correlation of convective heat transfer and recovery temperature data for cylinders in compressible flow, Int. J. Heat Mass Transf. 8(2) (1965) 245–252.
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