M. V. Zagarola

Mean-flow scaling of turbulent pipe flow

Measurements of the mean velocity profile and pressure drop were performed in a fully developed, smooth pipe flow for Reynolds numbers from 31×10 3 to 35×10 6 . Analysis of the mean velocity profiles indicates two overlap regions: a power law for 60 y + <500 or y + R + , the outer limit depending on whether the Karman number R + is greater or less than 9×10 3 ; and a log law for 600 y + R + . The log law is only evident if the Reynolds number is greater than approximately 400×10 3 ( R + >9×10 3 ). Von Karmans constant was shown to be 0.436 which is consistent with the friction factor data and the mean velocity profiles for 600 y + R + , and the additive constant was shown to be 6.15 when the log law is expressed in inner scaling variables. A new theory is developed to explain the scaling in both overlap regions. This theory requires a velocity scale for the outer region such that the ratio of the outer velocity scale to the inner velocity scale (the friction velocity) is a function of Reynolds number at low Reynolds numbers, and approaches a constant value at high Reynolds numbers. A reasonable candidate for the outer velocity scale is the velocity deficit in the pipe, U CL − Ū , which is a true outer velocity scale, in contrast to the friction velocity which is a velocity scale associated with the near-wall region which is ‘impressed’ on the outer region. The proposed velocity scale was used to normalize the velocity profiles in the outer region and was found to give significantly better agreement between different Reynolds numbers than the friction velocity. The friction factor data at high Reynolds numbers were found to be significantly larger (>5%) than those predicted by Prandtls relation. A new friction factor relation is proposed which is within ±1.2% of the data for Reynolds numbers between 10×10 3 and 35×10 6 , and includes a term to account for the near-wall velocity profile.

Log laws or power laws: The scaling in the overlap region

The scaling in the overlap region of turbulent wall-bounded flows has long been the source of controversy, and until recently this controversy could not be addressed because measurements did not span a sufficient range of Reynolds number. Mean velocity surveys performed in a new pipe flow experiment span a very large range of Reynolds numbers, 31×103 to 35×106 (based on average velocity and diameter). Here, these experimental data are used to evaluate theories on the scaling in the overlap region. At sufficiently high Reynolds numbers, the mean velocity profile in the overlap region is found to be better represented by a log law than a power law. These results suggest a theory of complete similarity instead of incomplete similarity, contradicting the theories recently developed by Barenblatt et al.

A new friction factor relationship for fully developed pipe flow

The friction factor relationship for high-Reynolds-number fully developed turbulent pipe flow is investigated using two sets of data from the Princeton Superpipe in the range 31×10^3 ≤ ReD ≤ 35×10^6. The constants of Prandtl’s ‘universal’ friction factor relationship are shown to be accurate over only a limited Reynolds-number range and unsuitable for extrapolation to high Reynolds numbers. New constants, based on a logarithmic overlap in the mean velocity, are found to represent the high-Reynolds-number data to within 0.5%, and yield a value for the von Karman constant that is consistent with the mean velocity profiles themselves. The use of a generalized logarithmic law in the mean velocity is also examined. A general friction factor relationship is proposed that predicts all the data to within 1.4% and agrees with the Blasius relationship for low Reynolds numbers to within 2.0%.

Friction factors for smooth pipe flow

Friction factor data from two recent pipe flow experiments are combined to provide a comprehensive picture of the friction factor variation for Reynolds numbers from 10 to 36,000,000.

Experiments in high Reynolds number turbulent pipe flow

An experimental facility was constructed to investigate fully-developed turbulent pipe flow over an unprecedented range of Reynolds numbers (approximately 32 x 10 to 35 x 10 based on average velocity and diameter). The maximum Reynolds number investigated exceeds the highest previously measured by an order of magnitude. To attain high Reynolds numbers at reasonable cost, compressed air (up to 220 aim} was used as the working fluid. The results show that the friction factor depends on Reynolds number logarithmically, as proposed by Prandtl, but a set of new constants are found. The friction factor data show von K&rmans constant to be 0.44. If the velocity profile is normalized using inner scaling variables, a log-law with this slope and an additive constant of 6.3 is in excellent agreement with the data. If the velocity profile is normalized using outer scaling variables, a log-law with this slope and an additive constant of 1.5 is in good agreement with the data. It is also shown that the average velocity occurs at a location approximately 1/4 radius from the wall, independent of Reynolds number. vious studies (for example, [1, 2, 3, 4, 5]), it is difficult to find data at very high Reynolds numbers where many industrial systems operate, or across a large range of Reynolds numbers over which subtle Reynolds number effects can become apparent and basic scaling dependencies can be unambiguously established. Previous investigations have shortcomings due to the limited range of Reynolds numbers covered by the data, the uncertainty in the quality of the surface finish, the uncertain accuracy of the measurements, or an insufficient development length. The present experiment was designed to provide accurate measurements over a very large range of Reynolds numbers in a single apparatus. The lowest Reynolds number investigated was approximately 32 x 10, and the highest exceeded 35 x 10, corresponding to an order of magnitude increase over the highest Reynolds number for mean flow measurements [4], and two orders of magnitude increase over the highest Reynolds number for turbulence measurements [3]. This paper presents the mean flow measurements for this range of Reynolds numbers and addresses the scaling of the mean velocity profile and friction factor.

Applications of dense gases to model testing for aeronautical and hydrodynamic applications

Here we review the advantages and disadvantages of using highly compressed air and other working fluids to test submarines and aircraft at full-scale Reynolds numbers. The conclusions are based on the design and implementation of two facilities at Princeton: the DARPA/ONR Superpipe apparatus built at Princeton to enable accurate pipe flow measurements over three orders of magnitude in Reynolds numbers, and the ONR High Reynolds Number Testing Facility (HRTF), designed to test submarine models at 1/5th full-scale Reynolds numbers.

Calibration of the Preston probe for high Reynolds number flows

Experiments were performed in order to calibrate Preston probes for high Reynolds number applications. The results extend previous calibrations to Reynolds numbers that are approximately 30 times larger. The new calibration is valid in the range 6.4<x*<11.3, that is, 4.3<y*<8.7 and 280<D+<45?000. Within this range, the wall shear stress may be found to within better than ?0.8%, as long as inner layer scaling applies up to a wall distance equal to the probe diameter.

Design of a High Reynolds Number Testing Facility using compressed air

Preliminary design studies are presented for a High Reynolds Number Testing Facility (HRTF) using compressed air as the working fluid. The facility is intended to achieve Reynolds numbers comparable to those experienced by Seawolf class submarines under cruise conditions. 1 Full-scale Reynolds number Here we present a design study for a High Reynolds Number Testing Facility (HRTF) using compressed air as the working fluid. This wind tunnel concept is designed to test vehicles such as submarines at fullscale Reynolds numbers, and its design is largely based on the successful DARPA/ONR Superpipe apparatus built at Princeton to enable accurate pipe flow measurements across a range of Reynolds numbers spanning three orders-of-magnitude. In that facility also, compressed air was chosen as the working fluid to reduce costs. A closed-loop system was built with the test pipe located inside high-pressure piping (see figure 1). The test pipe had a nominal diameter of 129 mm, with a length-to-diameter ratio of 200. For further details of the Superpipe facility, see [1] and [2]. For a submarine of length £, and speed V: the Reynolds numbers based on length is given by To estimate the Reynolds number requirement for the new HRTF, two conditions will be considered, * Associate Fellow, AIAA t Associate Member AIAA ^Copyright

The mean velocity profile in turbulent pipe flow

An experimental investigation was conducted to determine the mean-flow scaling in a fully-developed, smooth pipe flow. Measurements of the mean velocity profiles and friction factors were performed over a large range of Reynolds numbers (31 × 103 and 35 × 106 based on average velocity and diameter). Analysis of the mean velocity profiles indicate two types of overlap regions; one which scales as a power-law and one which scales as a log-law. The log-law is only evident if the Reynolds number is greater than approximately 300 × 103 which only a handful of other experiments have achieved. A new scaling is proposed that describes this behavior in both overlap regions. The scaling requires a velocity scale for the outer region such that the ratio of the outer velocity scale to the inner velocity scale (the friction velocity) is a function of Reynolds number at low Reynolds numbers, and approaches a constant value at high Reynolds numbers. The outer velocity scale is believed to be the difference between the centerline velocity and the average velocity. This velocity scale is used to normalize the velocity profiles in the outer region and is found to give better agreement between different Reynolds numbers than the friction velocity. At sufficiently high Reynolds numbers, the scaling in the overlap region is found to be logarithmic and the outer velocity scale was found to be proportional to the friction velocity.

Heat transfer enhancement in a transitional channel flow

Abstract Flow modification by an array of spanwise eddy-promoters oriented normal to the flow direction is sometimes used to enhance the convective heat transfer rate from a heated wall. Here, we study the effect of such eddy-promoters on the heat transfer and dissipation in a fully-developed channel flow with one heated wall, in the laminar and transitional flow regimes. The experiment shows that flows with eddy-promoters result in a significant decrease in dissipation for a given heat transfer rate at all supercritical Reynolds numbers investigated. When the results are compared to other eddy-promoter geometries, they support a scale-matched destabilization method. The primary enhancement mechanism in the subcritical flows is travelling waves, but for supercritical flows vortex shedding is an equally important mechanism.