Convection in Scaled Turbine Internal Cooling Passages With Additive Manufacturing Roughness

Additive manufacturing (AM) technologies enable the creation of parts with geometries more complex than those possible using more traditional subtractive manufacturing technologies. Because of the novel geometries allowed, AM is being explored for the potential to improve the design and manufacturing of gas turbine blades and components.

Maximizing the initial cost-effectiveness of a gas turbine system requires maximizing the turbine inlet temperature of the combustion products exiting the combustor, with current technology allowing for temperatures in excess of 1500 °C [1]. However, these temperatures are close to the melting temperature for the metals which make up the blades. Thus, the performance of the turbine blades under extreme heat becomes a major limiting factor in engine design and effectiveness.

To maintain blade integrity during high-temperature operation, cooler air from the compressor is directed through small interior cooling passages manufactured within each individual blade. These passages provide increased heat transfer away from the blade and thus serve to maintain the life and maximize the efficiency of the engine. AM has the potential to improve turbine blade cooling by enabling the manufacture of complex cooling passages in situ. Thus, AM would reduce the need for the complex cores and core removal process required for cooling passage creation when turbine blades are constructed using metal casting.

The AM process most commonly used to manufacture turbine components is known as direct metal laser sintering (DMLS). DMLS involves the use of a laser to systematically melt layers of fine metal powder, thereby creating the complex geometries of the part. However, the process is known to produce a unique form of roughness on the surface of the AM part [2]. Generally, this roughness could be corrected with post-processing of the surface, but due to the in situ manufacturing and placement of the passages, little can be done to improve the interior surface quality. Additionally, the roughness produced during DMLS varies with the build orientation of the part because of gravitational effects on the heat-affected zone or melt area of the metal powder [2].

Previous work in this space has shown the potential for significant design and performance improvements. Kirsch and Thole [3] additively manufactured wavy microchannels of varying wavelength at the engine component scale in an exploration of potential design improvements to cooling channel design. The work showed performance enhancement related to heat transfer for longer wavelength channels and shed light on the vast design space available with DMLS. Stimpson et al. [4,5] studied AM tolerances and performance, again at the engine scale, using ten test coupons made of CoCr and Inconel 718 at various build orientations. The study presents roughness comparisons between build orientations and proposes new Nusselt number correlations based on experimental heat transfer results.

Hanson et al. [6] furthered the study by scaling roughness geometries by 102x and performing friction and velocity measurements within a custom wind tunnel designed to simulate cooling channel flow. Tests were also completed using an analog roughness made up of randomly placed ellipsoids which were generated to match the roughness statistics of the real surface. The scale of the surface roughness being at wind tunnel size allowed for more detailed flow measurements and served to justify the frictional correlations of Stimpson et al. [5]. McClain et al. [7] presented additional wind tunnel scale measurements of friction and velocity using surface roughnesses that had been created with AM with different build orientations. The work provides detailed frictional and flow measurements for a variety of AM roughness ceiling and floor combinations using hot-film anemometry.

While previous works have provided both friction and heat transfer measurements using AM coupons at the engine component scale [3–5] and efforts have been made to produce detailed friction and velocity measurements at the wind tunnel scale by Hanson et al. [6] and McClain et al. [7], no work has yet been done to provide detailed heat transfer measurements of AM surface roughness at the wind tunnel scale. This study aims to produce those heat transfer measurements, which are essential to the understanding of the thermal performance of spatially varying AM roughness, as well as to characterize the frictional losses and heat transfer enhancement provided by the AM roughness.

To better measure the frictional and heat transfer effects of the surface roughness within a turbine blade cooling air passage, a test channel was constructed with both channel dimensions and roughness elements scaled up to improve measurement quality. One smooth surface and three rough surfaces were created from aluminum plates. The roughness geometries were created using high-fidelity computed tomography (CT) scans or structured light scans as described by McClain et al. [7]. The rough aluminum test plates were used to perform two distinct experiments: (1) friction factor measurements were taken of each surface at Reynolds number intervals of 10,000 using hotwire anemometry and (2) convection coefficients and resulting Nusselt numbers were measured at the same Reynolds number intervals using infrared surface images and steady-state heat transfer approaches.

The measurements were made with the goal of calculating the “Reynolds analogy performance parameter” and “global thermal performance parameter” used by Ligrani [8]. The performance parameters served as useful metrics for quantifying the increase in convective heat transfer relative to the increase in friction factor. The performance parameters may additionally be used to provide a more direct comparison between the thermal performance and frictional cost associated with AM roughness and those associated with the current forms of convection enhancement currently used on the interior of gas-turbine blades such as rib turbulators, surface dimples, or pin fins.

In the following sections, the facilities and methods used to achieve the experimental objectives are described. The data reduction and the uncertainty assessment approaches are also presented.

The roughness internal flow tunnel (RIFT) is an internal flow rig housed at Baylor University, which was constructed to simulate the flow within an interior cooling passage of a gas turbine blade. The body of the tunnel is made almost entirely of printed ABS plastic and has a highly modular design, with a cross section measuring 228.6 mm by 35.56 mm (9 in. by 1.42 in.), which gives the tunnel a hydraulic diameter of 62.3 mm (2.45 in.). The RIFT has a settling region of 506-mm (L_s/D_h = 11.2) and a test section measuring 914 mm (36 in.) in length. Previous efforts to study friction effects and turbulent flow characteristics within the tunnel involved the use of additively manufactured ABS plastic surfaces as the floor and ceiling of the test section [6,7]. Back and side views of the RIFT test section are presented in Fig. 1, and more details regarding the RIFT may be found in Refs. [6,7].

For this investigation, a heated test stand was constructed by machining a 12.7-mm (0.5-in.) deep by 228.6-mm (9-in.) wide inset channel from a 304.8-mm (12-in.) by 457.2-mm (18-in.) acrylic plate with a thickness of 34.93 mm (1.375 in.). A second inset was cut from the acrylic to form space for heating elements. Spanwise channels were then cut using a 1/8-in. ball mill to form 1.59-mm deep channels where thermocouples were installed to determine the conduction losses through the acrylic base. E-type thermocouples were installed in the channels with butt-welded beads placed at the center span of the acrylic base. Top and isometric views of the acrylic base section are shown in Fig. 2. As depicted in Fig. 1, the heated test is placed such that the heated region is the last 457.2 mm (18 in.) of the length of the RIFT test section.

Inside the heater inset, a layer of neoprene was attached to the acrylic, and two 8″ by 8″ Omega Polyimide insulated flexible resistance heaters were attached to the top of the neoprene layer to heat the test surfaces. The heaters were powered by two BK-Precision 9104 variable DC power supplies, which were manually set before each test. On the top of the heaters, a sheet of 0.31-mm steel shim was added to maintain thermal contact and evenly distribute energy to the test plate. Finally, the test plates were constructed from aluminum 6061 plate and were machined such that the mean thickness was 12.7 mm (0.5 in.). A cut-away of the heated test section is presented in Fig. 3. As noted in Fig. 3, an additional set of thermocouples was attached to the bottom of the test plate directly below the internal thermocouple channels to determine the conduction losses through the acrylic base.

The surfaces used in the investigation have been described in the prior investigations of Hansen et al. [6] and McClain et al. [7]. The rough surfaces were created by scaling either X-ray tomography scans or structured light scans of the internal passages of direct-metal laser sintering (DMLS) coupons constructed in the START Lab at Pennsylvania State University [2]. The coupons were printed with the passages making a 45-deg angle to the build plane. Two of the surfaces came from an Inconel 718 coupon, and the other came from a Hastelloy coupon. Scaling values of 50× and 102× were chosen to provide a geometric progression (×2.5) of roughness characteristics.

The surfaces investigated in this study are referred to as the Inco718_Upskin, the Inco718_Downskin, and the Real_x102 surface. Figure 4 presents a top view of the solid-model topography of each rough surface.

Because of the size of the heat transfer area and the size of the roughness elements investigated, comparing the heat transfer surface coupon roughness to the solid model characterizations of Table 1 presents challenges. Most profilometry systems are intended to measure much smaller roughness than employed in this study, especially for the Real_x102 surface which has a peak-to-valley maximum of 1.84 cm (0.72 in.). Additionally, laser scanner systems that can conceivably characterize that size roughness are not typically certified for roughness measurements following the ISO 4287:1997 standard.

To provide an example comparison of the machined surface to the computational model, a Keyence VR-3100 wide-area structured-light profilometry system with a height measurement range of ±5 mm was used to measure a 24 mm by 18 mm (0.95 in. by 0.71 in.) area of the Inco718_Downskin surface. Figure 5 presents a comparison of the solid model geometry used to create the measured Inco718_Downskin aluminum surface. Figure 5 demonstrates the CNC machining process results in a surface that generally represents the solid model, but tool marks are clearly visible on the surface.

Table 2 presents a comparison of the roughness statistics for the model surface section and the measured aluminum surface. Generally, the peak-to-valley and root-mean-square roughness are within 6% of the model, but the skewness of the aluminum surface is 10.4% higher than the skewness for the surface computational model.

A FLIR SC4000 infrared camera supported 279.4-mm (11-in.) above the test section was used to measure the surface temperature of the rough aluminum test plates. Figure 1 depicts the position of the IR camera above the test section. The IR camera was allowed to view the surface downward through an IR viewing window set in the ceiling of the tunnel. The FLIR SC4000 has an indium-antimonide (InSb) detector and resolution of 320 by 256 pixels. A 25-mm lens is used producing an imaged area of 10.72 cm by 8.58 cm at the distance of 27.94 cm from the mean test surface. The camera was placed with the infrared chip parallel to the test surface, and a 76-mm (3-in.) “C-Range” circular infrared window was installed in the RIFT ceiling panel. A 0.76-mm (0.030-in.) lip was CNC machined out of an aluminum ceiling panel to hold the IR window in place during testing.

During testing, the infrared camera captured 100 frames of temperature measurements at 30 frames per second. Since the IR window was circular and the IR images captured a larger rectangular area, the IR images included temperature features that were not from the heat transfer surface. A matlab code was created which generated a mask for use with the IR surface images. The surface mask served to isolate only the portion of the images which gave relevant temperature data by disregarding certain portions of each image. The masked-out regions included the ceiling of the tunnel in which the IR viewing window was set, as well as a reflection of the IR sensor off of the viewing window, which was visible in the center of every surface image.

The percentage of pixels masked in the infrared images changes slightly from case to case because of slight changes in the camera location following the changing of the surface panels. However, for a typical case, 4.77% (3905 pixels) are masked for the center reflection, and 44.5% (36,455 pixels) are masked for the ceiling of the tunnel. Thus, approximately 50.73% of the infrared image pixels were used for the convection calculations. Complete details of the masking and infrared image reduction process may be found in Stafford [10].

Following temperature calibration, the power supplies for all other instrumentation were turned on, and the remaining sensors were allowed to warm up for approximately 30 min. During this time, a labview program was started to monitor and log the tunnel’s Reynolds number, flowrate, pressure, as well as all relevant tunnel temperatures and room conditions.

Using a variable autotransformer, the desired Reynolds number of the flow was then set. Test Reynolds numbers ranged from 10,000 to 70,000 in increments of 10,000. Because of fan performance limitations, tests for the Real_x102 and Inco718_Downskin surfaces ended at 60,000.

After the flow conditions were set, a steady-state heat approach was used to determine the surface heat transfer coefficients. Two DC power supplies were turned on which powered the resistance heaters beneath the aluminum test plate. For each test, the heater power was set such that the temperature difference between surface and air was about 15 °C. After the heaters were turned on, the plate was allowed to reach a steady-state temperature. This point was determined to be when plate temperatures had a change of less than 0.1 °C over the preceding 30 min. Once the test plate settled at steady-state temperatures, a final surface temperature image set was taken using the IR camera. Following the test IR acquisition, either the next Reynolds number case was selected or the shutdown procedure was followed.

One reason for performing flow loss and velocity profile measurements using the aluminum heat transfer plates was to verify the frictional similarity of the surfaces created for this study to the rough surfaces used by McClain et al. [7]. The same surface geometry files were used to create both sets of surfaces. However, as noted in Fig. 5 and Table 2, the subtractive machining approach used for the aluminum heat transfer plates created a visibly different surface texture than was created using the computational models from the study of McClain et al. [7].

The uncertainties associated with the calculations of the convection coefficient and the Nusselt number were determined using the large sample size approach of Coleman and Steele [11], which is an extension of the approach of Klein and McClintock [12]. The uncertainty associated with the friction factor measurements was found using a combination of the method of Coleman and Steele and the Moffat [13] method of uncertainty. Because of the compounded fluid property dependencies on the fluid temperature, the perturbation approach of Moffat [13] was more convenient for the friction factor evaluation. All uncertainties were determined using a 95% level of confidence. Table 3 reports the systematic and random uncertainties associated with each of the directly measured quantities.

Inspecting Table 3, the systematic uncertainties of the temperature measurements are significant. However, the in situ calibration scheme described by Eqs. (7) and (8) correlates the systematic uncertainties of all of the temperature measurements. The in situ calibration process significantly reduces the resulting uncertainties in the calculated convection coefficients and Nusselt numbers [11].

The results are presented in four sections. In the first section, friction factors and velocity profiles are compared to the measurements made for the ABS plastic surfaces used by McClain et al. [7]. In the second section, the heat transfer measurements are presented. In the third section, the measurements are compared to the performance of internal heat transfer enhancement typically employed in gas turbine cooling passages. Finally, the current data are compared to the measurements of Stimpson et al. [5], which were acquired in the START Lab using DMLS coupons with internal channels.

As noted earlier, the heat transfer plates were constructed by machining aluminum plates. While the same geometry file was used to create the surfaces used in this study, differences between the heat transfer plates and the ABS plastic plates used by McClain et al. [7] were observed.

The choice of aluminum for the heat transfer investigation was made to minimize the effect of temperature change along the length of the roughness elements. As noted by McClain et al. [14,15], roughness elements behave as fins on a surface. Because of internal thermal resistance, temperature change along the length of the roughness element affects the apparent fin efficiency of the surfaces. Aluminum was chosen to have a low surface resistance as would be expected for metal surfaces created using DMLS printers. The low thermal conductivity ABS surfaces used for the study of McClain et al. [7] would have exhibited significant temperature change along the length of the roughness elements and would not be expected to replicate the enhancement of additive manufacturing roughness.

Figure 6 presents the resulting friction factors from the current study and from the ABS surfaces of McClain et al. [7] versus the Reynolds number based on the hydraulic diameter. In Fig. 6, the overlapping uncertainty bars for the aluminum heat transfer or “HT” surfaces, and the ABS surfaces demonstrate agreement for the two roughest surfaces in the investigations. However, significant differences were observed for the Inco718_Upskin surface and the Smooth HT surface, which were compared to the acrylic floor for the hydraulically smooth reference.

The mean velocity profiles measured over the heat transfer surfaces are presented in Fig. 7 for Re_Dh values of 30,000 and 60,000. The profiles shown in Fig. 7 reflect similar trends as the velocity profiles presented by McClain et al. [7] for the ABS surfaces.

Figures 6 and 7, when considered with the velocity profiles presented in McClain et al. [7], demonstrate that for Inco718_Downskin and Real_x102 surfaces, the heat transfer plates provide the same flow effects as the ABS surfaces. For the Inco718_Upskin and Smooth surfaces, the machining and painting of the aluminum heat transfer plate produce different flow interactions than the ABS or acrylic surface. The difference between the smooth heat transfer plate and the acrylic smooth reference is surprising, but the flat-black paint has a texture that can be observed and felt by touch.

Figure 8 also includes repeatability measurements for the Real_x102 surface. Three measurements of the average Nusselt number were made at Re_Dh = 40,000, and five measurements of the average Nusselt number were made at Re_Dh = 30,000 using different heater power levels. The scatter in the repeated measurements is consistent with the calculated uncertainties associated with the Nusselt number measurements.

Figure 9 presents the percentage enhancement. Figure 9 also demonstrates that while there were some hydraulic differences between the smooth heat transfer plate and the acrylic plate used in McClain et al. [7], the convection measurements for the smooth plate agree with Dittus–Boelter equation by 15%. At Re_Dh = 20,000 and above, the agreement is within 11% and generally improves with increasing Reynolds number.

Figure 9 demonstrates that the enhancement of the rough surfaces is relatively consistent with the Reynolds numbers investigated. For the Inco718_Upskin surface, the enhancement is 17–20%. For the Inco718_Downskin surface, the enhancement is consistently 70–81%, and for the Real_x102 surface, the enhancement is 130–140%.

Tables 4–7 include the calculated Biot number based on the mean thickness of the heated roughness plates. The Biot number is reported to demonstrate that mean wall-normal temperature gradients or temperature changes along the roughness elements should not be significantly affecting the measured convection coefficients.

However, the use of single-sided roughness convection measurements complicates the comparison of the roughness measurements to the traditional enhancement methods. Single-sided measurements were used for the current study because of the requirement for IR window access on the opposite wall from the roughness. Because roughness exists on only one wall, the flow migrates toward the smooth wall for the rough surfaces as shown in Fig. 7.

Figure 11 presents the global thermal performance ratios for the surfaces of this study. The inset of Fig. 11 presents a chart from Ligrani [8] reporting the global performance ratios for different types of internal passage convection enhancement methods. The box in the inset indicates the range of global thermal performance values reported in Fig. 11. Comparing the data of the current study to the data compiled by Ligrani [8], many passage enhancement methods provide greater global thermal performance than the additively manufactured roughness investigated in this study. For the AM roughness investigated, the maximum global thermal performance is on the order of 1.4, while the data compiled by Ligrani [8] reach as high as 2.25 for swirl chambers, dimples, and ribs. The comparison of Fig. 11 demonstrates that AM surfaces will enhance the heat transfer in internal passages; however, additional enhancements mechanisms may still be required in the additive manufacturing geometries to reach global thermal performance values achieved by traditional convection enhancement methods.

The adjusted convection enhancements were also compared to the measurements of Stimpson et al. [4] for engine-scale additively manufactured passages created and tested in the START Lab at Pennsylvania State University. Data covering the range of enhancement ratios (Nu/Nu₀) from Stimpson et al. [4] are included in Fig. 10 for the Reynolds analogy comparison. Figure 10 demonstrates that the engine-scale data follow a similar trend as the measurements presented in this study for scaled passages and are scattered about the trend observed for the scaled passages.

The correlation of Eq. (26) follows a Norris [18] style approach, but Eq. (26) results in a different exponent for the friction ratio (b + 1 = 0.401 versus n = 0.68Pr^0.215 = 0.633 for the correlation provided by Norris [18]).

Figure 12 presents a comparison of the predictions for the scaled passage enhancement ratios of this study and for the measurements of Stimpson et al. [4] as a function of (f/f₀)^0.401. Figure 12 demonstrates that the modified enhancement ratios measured for scaled passages with AM roughness capture the trend of the Stimpson et al. [4] measurements. However, the agreement with the scaled passage data is ±10%, while the agreement with the data from Stimpson et al. [4] is ±50%.

The scaled passage convection measurements in the RIFT were designed to isolate the nature of additively manufactured roughness. Figure 12 indicates that the overall trend of friction enhancement and convection enhancement caused by AM roughness is captured by the RIFT measurements. However, physics not captured in the scaled passage measurements are clearly affecting the engine-scale measurements. Important phenomena not included in the RIFT measurements that are present in the engine-scale tests are (1) compressibility effects and (2) long-wavelength effects caused by the additive manufacturing process such as concentricity, circularity, and total runout [2,19].

The data reduction equations used by Stimpson et al. are based on the incompressible modified Bernoulli equation and constant specific heat energy equation. Stimpson [20] presents an investigation of the friction factor measurements considering compressible flow behavior in the channels and found that compressibility effects should only be important at the highest flowrates investigated. However, the heated coupon cases at the high Reynolds numbers are experiencing a more complicated Fanno–Rayleigh situation that may not be captured appropriately using the data reduction equations of Stimpson et al. [4] or Stimpson [20].

Inspecting Fig. 12, two features may point to a correction for real, engine-scaled passages. In Fig. 12, the scaled AM measurements in the RIFT and the correlation run through the middle of the engine-scale data. However, the RIFT enhancement ratios are essentially constant, while engine-scale measurements decrease with increasing friction factor ratio. Revisiting Stimpson et al. [4], the engine-scale measurements are in the lower half of Reynolds numbers investigated in the RIFT. The measurements of Stimpson et al. [4] also appear to cross the correlation as the Reynolds number increases. The fact that the engine-scale data from the START Lab cross the correlation as the Reynolds number increases suggest that a velocity-based or Reynolds-based feature is affecting the measurements in a way that is not present in the RIFT data. The fact that the RIFT data are also in the “fully-rough” regime where the friction factor is no longer changing with Reynolds number, while the engine-scale tests of Stimpson et al. appear to still be in the “transitionally rough” regime. In any case, explaining the differences between the Stimpson et al. [4] data and the scaled RIFT data is an active area of research.

Convection measurements were made in a scaled passage using rough surfaces with characteristics of additively manufactured roughness. Steady-state convection measurements were made using machined aluminum surfaces to minimize the effects of temperature change along the length of the roughness elements. The primary findings of the investigation are as follows:

1. Frictional differences between the aluminum heat transfer roughness plates and the ABS plates created for the investigation were negligible for the two surfaces with larger roughness. For the surface with the smallest roughness characteristics and for the smooth heat transfer surface, fluid friction differences were more significant.

2. The percentage enhancements for the scaled surfaces relative to the Dittus–Boelter correlation were essentially constant for cases investigated.

3. The scaled AM passage measurements were modified for comparison to the data from Ligrani [8]. Following modification to represent channels with roughness around the perimeter, the AM roughness behaves similar to traditional internal convective enhancement devices based on the Reynolds analogy parameter. However, when compared using the global thermal performance ratio, the AM surfaces did not perform as well as swirl chambers, dimples, and ribs.

4. The Reynolds analogy ratio collapses the scaled AM roughness convection measurements when reported versus the friction enhancement ratio suggesting a Norris [18] style correlation. A correlation was provided and produces agreement levels within ±10% of the measurements in the RIFT.

5. The scaled AM passage enhancement ratios and resulting correlation are in line with the engine-scale test data from Stimpson et al. [4]. However, the scatter about the correlation is ±50% for the engine-scale measurements of Stimpson et al. [4].

This material is based upon work supported by the Department of Energy under Award Number(s) DE-FE0001730. Mr. Ashley Orr’s efforts in machining the aluminum heat transfer plates are greatly appreciated.

This report was prepared as an account of work sponsored by an agency of the United States Government. Neither the United States Government nor any agency thereof, nor any of their employees, makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise does not necessarily constitute or imply its endorsement, recommendation, or favoring by the United States Government or any agency thereof. The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States Government or any agency thereof.

There are no conflicts of interest.

The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request. The authors attest that all data for this study are included in the paper. Data provided by a third party are listed in Acknowledgment.

h =

convection coefficient

u =

the velocity component in the flow direction

y =

location in the wall-normal location in the fluid domain

H =

the channel height 36.07 mm (1.42 in.)

Q =

volumetric flowrate

Y =

wall-normal direction and ordinate of a specific surface location

$Q˙$ =

heat rate

$Y¯$ =

mean surface location

Y′ =

surface elevation relative to the mean elevation

A_H =

the heated platform area (top-view projected area)

A_ts =

test section cross-sectional area = 8200 mm²

B_X =

systematic uncertainty of the X measurement

C₀ =

nozzle or Venturi discharge coefficient

D_h =

the RIFT hydraulic diameter = 62.3 mm (2.45 in.)

D_m =

the modified hydraulic diameter adjusting for the velocity profile shift caused by roughness on one wall of the RIFT

f_DW =

Darcy-Weisbach friction factor

k_Al =

the thermal conductivity of aluminum 6061

k_s,FS =

the predicted equivalent sand-grain roughness height using Flack and Schultz (2010) correlation

k_s,ST =

the predicted equivalent sand-grain roughness height using Stimpson and Thole correlation

t_a =

thickness of the acrylic base

$t¯p$ =

average thickness of the heated test plate 12.7 mm (0.5 in.)

w_p =

width of the heated test plate 228.6 mm (9 in.)

L_P =

length of the heated test plate 457.2 mm (18 in.)

L_s =

length of the flow settling region from the nozzle exit to the first pressure tap

L_ts =

length of the RIFT pressure-drop metered section = 685.8 mm (27.0 in.)

N_p =

number of points in surface point cloud

R_a =

the arithmetic mean roughness

R_PV =

the peak-to-valley roughness

R_q =

the root-mean-square roughness

R_X =

random uncertainty of the X measurement

T_cal =

instrument raw temperature during calibration

T_in =

temperature of the air at the RIFT inlet

T_IR =

unmasked infrared temperature measurement

T_lower =

temperature of thermocouples attached to the bottom of the acrylic base

T_m =

instrument raw temperature during measurement

T_ref =

calibration reference temperature

T_run =

calibrated temperature during measurement

T_um,cal =

temperature of unmasked infrared measurement region during calibration

T_upper =

temperature of thermocouples in recessed channels machined in acrylic base and just below heaters

U_mean =

average flow speed over the tunnel cross section

ΔP_v =

gage pressure at the nozzle or Venturi exit

ΔP_ts =

pressure drop from first to last pressure taps

Bi =

$Biotnumber=ht¯p/kAl$

Re_Dh =

Reynolds number based on the hydraulic diameter, Re_Dh = ρU_meanD_h/μ

Skw =

roughness height relative skewness distribution

μ_air =

the molecular viscosity of air

ρ_air =

the density of air

$τ¯$ =

the perimeter averaged shear

τ_R =

the shear on the rough perimeter of the channel

m =

measured quantity

$(⋅)¯$ =

time-averaged quantity