Commercial water tunnels typically generate a momentum thickness based Reynolds number (Re* _{θ}*) ∼1000, which is slightly above the laminar to turbulent transition. The current work compiles the literature on the design of high-Reynolds number facilities and uses it to design a high-Reynolds number recirculating water tunnel that spans the range between commercial water tunnels and the largest in the world. The final design has a 1.1 m long test-section with a 152 mm square cross section that can reach speed of 10 m/s, which corresponds to $Re\theta =15,000$. Flow conditioning via a tandem configuration of honeycombs and settling-chambers combined with an 8.5:1 area contraction resulted in an average test-section inlet turbulence level <0.3% and negligible mean shear in the test-section core. The developing boundary layer on the test-section walls conform to a canonical zero-pressure-gradient (ZPG) flat-plate turbulent boundary layer (TBL) with the outer variable scaled profile matching a 1/7th power-law fit, inner variable scaled velocity profiles matching the log-law and a shape factor of 1.3.

## Introduction

Typical commercial recirculating water tunnels achieve a momentum thickness based Reynolds number (Re* _{θ}*) on the order of 10

^{3}, which is slightly above that required for laminar to turbulent transition. This is not ideal for studying Reynolds number dependent turbulent flow phenomena, such as velocity profile modifications from drag reducing polymer solutions [1,2] or helicopter wake structures [3]. Consequently, much of this work is performed in extremely large government owned water tunnels such as the U.S. Navy Large Cavitation Channel (LCC) [4,5] or the Garfield Thomas Water Tunnel (GTWT) [6,7]. The LCC, the world's largest water tunnel, can achieve $Re\theta \u223c105$, but the operation cost is extremely high. Consequently, facilities that can achieve $Re\theta \u223c104$ are ideal for studying Reynolds number dependent turbulent phenomena, which was the primary specification for the current facility.

There are several established facilities that fit this operation range [8–16], but they were built as part of major research programs (i.e., sufficient funds for prototype testing components) and/or refurbished facilities (i.e., key design considerations already fixed). Furthermore, the limited literature on these designs is difficult to obtain since it is typically in technical reports, conference papers, and/or theses. Thus, the current work compiles information from a wide range of facilities (see Ref. [17] for complete list), which is a guide to the overall design of a relatively low-cost water tunnel capable of achieving Reynolds numbers comparable to any nongovernment owned facility and verifies the design with characterization of the as-built tunnel performance. The completed facility (schematically shown in Fig. 1) bridges the gap between commercial water tunnels that are barely turbulent and the world's largest facilities [4–7,18–22]. The test-section specifications were (i) Re* _{θ}* ≥ 10

^{4}, (ii) maximize optical access, and (iii) minimize inlet flow nonuniformity. The remainder of this paper covers the design of individual components (Sec. 2), characterization of the completed facility (Sec. 3), and conclusions (Sec. 4). The interested reader is directed to the previous work [17,23,24] for more detailed discussion on structural design, fabrication, and installation.

## Water Tunnel Design and Construction

### Test Section.

Test-section design is driven by the application, operation range, and instrumental suite (optical and mechanical access). The current facility focuses on modifications to canonical turbulent boundary layers (TBLs). The maximum length (∼1 m) was set to achieve the required rigidity, flatness, and surface smoothness without excessive costs. Momentum integral analysis with a 1/7th velocity profile on a zero-pressure-gradient (ZPG) flat-plate [25] estimates the outlet boundary layer thickness ($\delta /x=0.16Rex\u22121/7$) to be ≥16 mm. Here, Re* _{x}*$(=Uex/\nu )$ is the downstream-based Reynolds number,

*ν*is the kinematic viscosity (∼10

^{−6}m

^{2}/s),

*x*is the downstream distance from the inlet, and

*U*is the local freestream (external) speed. The TBL overlap region is unaltered from the log-law $u+=lny+/\kappa +B$ when the dimensionless acceleration parameter $K\u2032=\nu /Ue2dUe/dx<1.62\xd710\u22126$ [26]. Here, $y+(=y/l\nu )$ is the inner variable scaled wall-normal distance, $u+(=u/u\tau )$ is the inner variable scaled velocity, $u\tau (=\tau w/\rho )$ is the friction velocity, $l\nu (=\nu /u\tau )$ is the viscous wall unit,

_{e}*τ*is wall shear stress,

_{w}*ρ*is the fluid density,

*κ*(≈0.41) is the Kármán constant, and

*B*(≈5.0) is the intercept constant. Accounting for displacement thickness $\delta */x=0.02Rex\u22121/7$, the same analysis can estimate $dUe/dx$. The $K\u2032$ criterion is achieved with a $15\u2009mm\xd715\u2009mm$ cross section, but the final size was increased $152\u2009mm\xd7152\u2009mm$ because recent high-Reynolds number findings [27,28] suggest that mild pressure-gradients can alter velocity profiles.

The as-built test-section has a flange-to-flange total length of 1.1 m with a viewable length of 0.914 m (Fig. 2) on all sides except the top, which has a $114\u2009mm\xd776\u2009mm$ panel to accommodate an injection plate [17]. Edge fairings were not used and the frame designed to facilitate direct line-of-sight of the tunnel wall. A trip of uniformly distributed 122 *μ*m silicon carbide grit was located around the test-section inlet perimeter. When pressurized $pmax=276\u2009kPa$, the acrylic walls have a maximum displacement of 115 *μ*m [17]. The resulting local curvature has a negligible impact on the velocity profile but is nontrivial compared to *l _{ν}*. Thus, the wall location must be identified for each operating condition. The stainless steel frame has an average roughness height (

*R*) of 0.8

_{a}*μ*m. Converting

*R*to Colebrook type roughness (

_{a}*k*) [29,30] and noting $l\nu \u22652.0\u2009\mu m$, the maximum $k+=kc/l\nu $ is 0.6 to 1.7. This is acceptable since hydraulically smooth is $k+<4$ and the majority of the TBL develops on the smoother acrylic windows.

_{c}### Flow Conditioning.

The ideal test-section inlet velocity would be steady and uniform with negligible turbulence. This is never realized because 90 deg elbows introduce swirl, flow-straighteners remove swirl while introducing turbulence, and boundary layers cause local flow acceleration. Consequently, no ideal flow conditioning system exists and the final components in this work are not considered optimized. In the current work, flow nonuniformity was mitigated with a tandem honeycomb/settling-chamber configuration, an 8.5:1 contraction, and gradual expansion in diffusers (Fig. 1). Design/selection of the flow conditioning components assumed that flow enters some distance upstream of the test section with a high turbulence and swirl. Swirl, which is difficult to remove, is generally done with pressure-drop via a combination of screens, baffles, and/or honeycomb. Honeycomb was selected because screens in water tunnels typically need to be tight meshed, which can foul resulting in variable tunnel performance.

A tandem honeycomb configuration was selected because turning vanes were not used and the upstream turbulence level was unknown. The honeycomb sizing used experimental turbulence reduction factors and pressure-drop coefficients for a wide range of cell length and diameters [31]. An iterative process [17,23] resulted in the selection of a 610 mm (length) by 19 mm (cell diameter) polycarbonate hexagonal honeycomb (PCFR750W24, Plascore, Zeeland, MI) followed by 152 mm (length) by 6.35 mm (diameter) stainless steel honeycomb (76 *μ*m foil, Benecor, Wichita, KS). It is recommended that 30–40 cell diameter long settling-chambers be positioned downstream of the honeycomb to utilize viscous decay to suppress their turbulent wakes [32,33]. Thus, 594 mm (31 cell diameters) and 254 mm (40 cell diameters) long settling-chambers were placed downstream of the first and second honeycomb, respectively.

*X*is the streamwise coordinate measured from the test-section inlet toward the contraction inlet,

_{c}*Y*is the transverse direction from the centerline,

_{c}*L*is the contraction length, and

_{c}*Y*/

_{o}*Y*is half the test-section/contraction inlet height, respectively. The constants were determined from the contraction ratio and setting the first and second derivatives of the inlet/outlet profile to zero. Selection of

_{L}*L*is ideally done via a parametric analysis [37], but the current design leveraged the fact that seven low-turbulence water tunnels (LCC, UNH HiCaT, mini-LCC, Michigan 9 in, GTWT, PSU-ARL 12 in, and St. Anthony Falls Water Tunnel) use the same profile shape. The average $Lc/2YL=1.44$ (standard deviation = 0.33), which corresponds to $Lc=737\u2009mm$ for the current facility

_{c}Diffusers are required to complete the loop from the minimum area at the test section through the pump and back to the contraction inlet. Increasing the area minimizes pressure losses, but the expansion half-angle should not exceed 4 deg [16,32]. Separation is likely at higher angles, which will produce unsteady flow that will propagate to the test section. The fiberglass test section diffuser follows Nedyalkov [16] to expand with the square-to-round shape change. This diffuser expanded to a 254 mm diameter over a length of 762 mm, which gives an effective half-angle of 3 deg. The other diffuser sections were located on the lower leg and fabricated from 304 stainless steel. The diffuser upstream of the pump transitioned from $254\u2009mm$ to $356\u2009mm$ over 1.8 m length, which gives a half-angle of 1.6 deg. The diffuser was oversized (pump inlet diameter = 305 mm) to allow a short contraction section at the pump inlet to promote uniform flow into the pump. The diffuser downstream of the pump transitions from the pump outlet (254 mm) to the flow straightener diameter (495 mm) over a length of 2.5 m, which gives a 2.8 deg half angle.

### Pump Sizing.

The pump supplies the pressure differential (Δ*p*) to overcome the system losses at the desired flowrate, which requires an iterative design process. The volumetric flowrate (0.232 m^{3}/s) is readily available given the cross section and maximum speed (10 m/s). However, the total system losses are often severely underestimated due to nonuniform flow entering components [38] and assembly imperfections. From experience, the calculated total pressure losses (91 kPa) [23] are typically 3 to 4 times smaller than the as-built. Thus, the pump specifications were $\Delta p=300\u2009kPa$ at 0.279 m^{3}/s (i.e., increased Δ*p* and flowrate). While axial-flow pumps are preferred due to their efficiency and lack of a radial velocity, significant cost savings was possible with a centrifugal pump. Thus, a horizontal split case centrifugal pump (S10B12A-4, Patterson, Toccoa, GA) with a 112 kW (150 hp) motor (MP44G3909, Baldor, Fort Smith, AR) was selected because it (i) achieved the operating condition, (ii) had a low minimum flowrate, (iii) low cost, and (iv) compact design.

## Tunnel Characterization

### Experimental Methods.

The tunnel performance was primarily assessed with PIV. An image plane was illuminated with a 532 nm Nd:YAG laser (Gemini-200, New Wave, Fremont, CA) beam formed into a sheet. Light scatter from 18 *μ*m hollow-glass-sphere tracer particles (iM30K, 3M, Maplewood, MN) was recorded with a 2560 × 2160 pixel sCMOS camera (Imager, LaVision, Göttingen, Germany). The typical field-of-view (see Fig. 2) was nominally 49 × 41 mm and images were recorded in double-frame, double-pulse mode. The velocity vector-fields were computed using standard cross-correlation methods (DaVis8, LaVision) with a final interrogation window of 16 × 16 pixels with 50% overlap. At least 100 independent vector-field realizations were used for mean velocity profiles. The *x*-, *y*-, and *z*-axes are aligned with the streamwise, vertical, and horizontal (completing the right-handed system) directions, respectively. The origin was placed on the centerline at the test-section inlet, but note that this is not the boundary layer origin. Also, note that TBL measurements are presented in wall normal coordinates (*y _{n}*) as shown in Fig. 2. The displacement uncertainty was ∼0.1 pixels, which corresponds to ∼1% uncertainty in velocity.

Since the inlet turbulence intensity was <1%, a single-component hot-film-anemometer (HFA; MiniCTA-54T42, Dantec, Skovlunde, Denmark) with a cylindrical hot-film probe (55R15, Dantec) was used. The probe was positioned at $x=60mm$ and sampled at 500 Hz. Mean HFA signals were calibrated in situ with simultaneous PIV measurements and recorded with tunnel operation conditions (temperature, pressure, and pump frequency) via a data acquisition (DAQ) card (USB-6218-BNC, NI, Austin, TX) and commercial daq software (LabView15.0.1, NI). Water temperature was measured via a temperature controller (CNI3253, Omega, Norwalk, CT) with a T-Type thermocouple (TC-T-1/4NPT-U-72, Omega) located 0.92 m upstream of the contraction inlet. Tunnel static pressure was monitored with a pressure transducer (PX2300-50DI, Omega) located 76 mm upstream of the contraction inlet at the test-section centerline elevation. The pump motor frequency (*f _{p}*) was manually controlled with a variable-frequency-drive (EQ7-4150C, Teco, Round Rock, TX). Calibration between

*U*at the center of the test-section and

_{e}*f*$Ue=0.3363fp\u22120.0106$ validates the pump selection and overall facility design with the maximum $Ue=10.1m/s$.

_{p}### Inlet Turbulence.

The accuracy of the inlet TI was assessed by comparing the spectra to the classic high-Reynolds number isotropic turbulence scaling (i.e., K41 theory). The temporal single-sided power spectra $Suu(\omega )$ were transformed to spatial spectra via Taylor's frozen turbulence hypothesis ($kU\u2248\omega $; $Suu(k)\u2248U\u2009Suu(\omega )$), where *k* is the wavenumber $k=2\pi /\lambda $, *λ* is the wavelength, *U* is the mean convection velocity, $\omega =2\pi f$ is the angular frequency, and *f* is the temporal frequency. K41 theory suggests $Suu(k)/(\nu 5\epsilon )1/4=g(k\lambda k)$, where *g* is an unknown function, *ε* is the dissipation rate of turbulent kinetic energy per unit mass (proportional to production $\epsilon \u221dUe3/Hc$ with *H _{c}* the test-section height), and $\lambda k=(\nu 3\epsilon )1/4$ is the Kolmogorov length scale. Four orders of magnitude separate the production $Hc=152\u2009mm$ and dissipation $14.5\u2264\lambda k\u226427.5\u2009\mu m$ length scales, which suggests that there should be a measurable inertial range $(Suu(k)\u221d\epsilon 2/3k\u22125/3)$. Spectra for the current facility are shown in Fig. 3 with much of the data following the

*k*

^{−5/3}slope.

The spectra are truncated to omit frequencies above the Strouhal shedding frequency (*f _{s}*) for the HFA support $d=6.3mm$. The Strouhal number $(St\u2261fsd/Ue)$ is ∼0.21 [39] for the Reynolds number range, and spectral peaks are observed at frequencies slightly above

*f*. This limited measurements to $fp<4.0HzUe=1.3\u2009m/s$. In addition, the spectra at $3<fp<4Hz$ were contaminated from a pump natural frequency at ∼3.4 Hz. The root-mean-square velocity (

_{s}*u*

_{rms}) and TI for each condition are shown in Table 1. Note that the reported

*u*

_{rms}is without filtering out the Strouhal or pump natural frequency content. Thus, the mean inlet TI of ∼0.30% (±0.04%) is an upper limit. A bandpass Butterworth filter, with cutoff frequencies of 8 Hz and

*f*, reduces the TI to ∼0.13%. Note that the freestream TI from PIV was ∼1% for all test conditions, which is the PIV uncertainty. This indicates that over the entire operation range, the inlet $TI<1%$, which is a common requirement for a low-turbulence facility.

_{s}### Mean Velocity Distribution.

Scaled mean velocity profiles at $x\u223c0.5m$ and $z=0$ are given in Fig. 4 to quantify the mean shear outside of the developing boundary layer. These profiles are a composite of three independent measurements (top, middle, and bottom). The maximum standard deviation of the three measurements of *U _{e}* for any test speed was 1.3% of the mean. This is comparable to the PIV uncertainty, thus there is negligible mean shear outside of the boundary layer. The top boundary layer profiles at $Ue\u22653.2m/s$ are scaled with outer variables in Fig. 5. Lower speeds do not collapse due to pressure-gradient effects, as predicted by the $K\u2032$ criteria. The power-law fit in Fig. 5, $u/Ue=1.01yn/\delta 1/7.03$, supports the design assumption that the velocity profile can be approximated as a 1/7th power-law profile.

The momentum thickness (*θ*), shape factor, and other boundary layer parameters are provided in Table 2. Beyond $x=0.3\u2009m$, *θ* showed excellent collapse (Fig. 6(a)) using traditional Reynolds number scaling, $\theta /x=0.00878Rex\u22120.1086$. Note that while these scaled results closely follow the canonical curve [25], the boundary layers were thicker (virtual origins were computed to confirm). The curve fit combined with flat-plate momentum integral analysis, $Cf=2d\theta /dx$, allows an estimate of *τ _{w}*, where $Cf\u2261\tau w/0.5\rho Ue2$ is the coefficient of friction. The resulting inner variable scaled profiles are in Fig. 6(b). For reference, dashed lines corresponding to the viscous sublayer $(u+=y+)$ and the log-law area included. The measurements do not extend to the viscous sublayer, which is typical of high-Reynolds number TBL measurements. However, there is a significant overlap region that follows the log-law with the higher Reynolds number profiles extending to larger

*y*values. The scatter within the overlap region is due to the limited accuracy of the

^{+}*θ*-gradient (<2% change in

*u*collapses the data and the uncertainty of

_{τ}*u*is ∼5%).

_{τ}## Conclusions

The current work offers a guide to the overall design of a high-Reynolds number recirculating water tunnel that bridges the gap between commercial water tunnels that are barely turbulent ($Re\theta \u223c103$) and the world's largest facilities ($Re\theta \u223c105$). The design specifications for the current facility were to achieve $Re\theta \u2265104$ (see Fig. 7), maximize test-section optical access, and minimize flow nonuniformity at the test-section inlet. The final design resulted from an iterative process matching pump selection and total system losses. PIV measurements showed the maximum test section speed of 10 m/s, which confirmed the facility design. The inlet flow quality was conditioned with a tandem honeycomb configuration (with settling chambers) sized assuming that the turbulent integral length scale was proportional to the pipe diameter and empirical relationships from the literature [31]. Following the flow straighteners, an 8.5:1 area contraction with a fifth-order polynomial contraction shape reduced TI and increased the flow speed. The overall flow conditioning design was assessed with measurements of the inlet turbulence and mean velocity. Turbulence spectra followed the *k*^{−5/3} slope for isotropic turbulence and the average TI was <0.3%. Variation in the mean velocity outside of the boundary layer was ∼1.3%. The test section was sized using momentum integral analysis for a flat-plate ZPG TBL. The final cross-sectional area $152\u2009mm\xd7152\u2009mm$ was selected to minimize pressure-gradient effects. The mean boundary layer profiles scaled with outer and inner variables conform to the traditional 1/7th profile and the log-law, respectively. Overall, the test-section wall TBL conform to the classical ZPG TBL (e.g., outer variable scaling, inner variable scaling, and shape factor) over the majority of operating conditions, which confirms the overall facility design.

## Acknowledgment

The authors would like to thank Dan Diehl at Diehl Aero-Nautical and Eddie Bryan at Weamco for going above and beyond to ensure that the final product matched the design objectives. In addition, we thank CEAT technical staff (Gary Thacker, Ron Markum, and Gerry Battles) and EFPL students (Dalton Dunlap, Shahrouz Mohagheghian, Jacquelyne Baade, Marcus Lander, Bret Valenzuela, and Shannon Maher) who contributed to the installation, shakedown, and characterization.

## Funding Data

The National Science Foundation (Grant No. 1604978) (Dr. Ronald Joslin, Program Manager).