^{1}

^{2}

Erosion around a submerged cylinder is a well-studied problem, and is of particular interest in bridge pier scour applications. Particles erode when lift and drag forces overcome a critical threshold. These forces are typically studied from above the water-riverbed interface and are related to geometry and surficial processes. The present study maps hyporheic pressure fluctuations as they are related to surface water velocity fluctuations. Relatively, high-pressure events in the subsurface promote a destabilizing force from within the riverbed and increase the potential for the mobilization of sediment. Differential pressure transducers were fitted within a vertical cylinder in a movable bed flume. The pressure ports were flush with the cylinder surface and below the water-sand interface. The three-orthogonal components of velocity were recorded synchronously with differential pressure measured over a 15 mm depth. As expected, results show decay in pressure fluctuations as a function of depth.

Many sediment transport problems are empirically approached using sediment transport equations with a large amount of uncertainty. One such problem is local scour around bridge piers. Local accelerations in river velocity increase the ability for a river to erode sediment. Bridge support structures at river crossings create local acceleration. Removing enough sediment from the river bottom near bridge piers or abutments can cause the bridge to become unstable, increasing the risk of failure. Over the last few decades, statistical and physical modeling dominated scour research with the goal of relating hydrodynamics, geometry and sediment data to scour depth. In practice, scour, as well as other sediment transport processes are estimated with empirical equations with functional forms dependent on surficial processes and geometry. The goals of this work are to measure patterns of pressure fluctuations below the sand-water interface in three positions around a cylinder (0, 45 and 90 degrees relative to the stream wise flow direction) and use it to describe how pressure fluctuations change with depth and modify an existing formula describing these fluctuations as a function of depth.

Understanding the intricacies of sediment transport and incipient motion requires knowledge of forces applied to individual particles. Multiple studies exist measuring pressures in the interstices of gravel and relate it to sediment transport [

The focus of the investigation is two-fold: first, use the synchronous measurement of velocity and the hyporheic pressure field around a vertical cylinder to adapt existing pressure fluctuation decay equations. The modified equation is for applications in unimodal sediment where a circular hydraulic structure is present. Second, describe and compare the distribution of hyporheic pressure fluctuation at three radial locations around the cylinder.

Under favorable conditions, subsurface pressure fields reduce the apparent weight on individual particles, destabilizing and mobilizing the bed. This occurs when localized zones of high-pressure fluctuation develop in a relatively deep stratum while a localized zone of low-pressure fluctuation appears in a more elevated stratum [

The present experimental design positions the pressure sensors inside the cylinder and below the water-sand interface, eliminating the disturbance of flow around the cylinder. Since flow around a cylinder is already well studied, theoretical results and previous experiments are used to help validate this work.

The experimental portion of the project comprised nine experimental runs to capture sub-surface pressure fluctuations around a vertical cylinder. Experiments gathered data at three radial locations around the cylinder. The pressure ports on the cylinder were aligned at 0-degrees, 45-degrees and 90-degrees relative to the longitudinal centerline of the flume. Three trials were conducted for each alignment. The cylinder was placed approximately 7 meters from the head of the flume. Synchronous near-bed velocities and subsurface pore-water pressure measurements were recorded at three different elevations on the cylinder at each sampling location, radially spaced at 0, 45, and 90-degrees from the stream wise orientation as shown in

The low-pressure port for pressure sensor three was fully exposed to the flow (i.e. it was at the sand/water interface but fully above it) at the start of the experiment,

This research used differential, dry-gas pressure sensors from Sensor Technics^{®}. The pressure sensors output a zero-to-five volt analog signal corresponding to a differential pressure head of zero-to-five inches of water. The pressure sensors have an associated accuracy of 0.6% full scale or 0.8mm of water. The dry gas pressure sensors were mounted in a box above the flume and connected to the cylinder with flexible Tygon^{®} tubing with an inside diameter of 3 mm (6 mm outside diameter). The pressure sensors were connected to the cylinder in two columns: a high-pressure side and a low-pressure side, sampled at 1000 Hertz and smoothed with a moving average with period of 100 samples.

A Nortek Vectrino II^{®} down-looking profiler was used to measure three dimensional velocity components around the cylinder at a sampling frequency of 100 Hz. Velocities were collected as close to the cylinder as possible. Accommodating the geometry of the probe head required approximately 30 mm of horizontal space between the cylinder and the center of the sample volume. Velocities were collected over 30 mm range in the vertical at 1 mm resolution. A Gaussian filter was applied to both the velocity and pressure time series to remove outliers. When a velocity or pressure difference was determined an outlier, it was removed from the time series and a cubic polynomial was used to interpolate these missing data, this affected less than three percent of all measurements. Prior to the cylinder being placed in the flume, the friction velocity, U_{τ}, was determined using the method described by [

Three trials were conducted for each pressure port orientation (0, 45 and 90 degrees). Identical flow conditions were used for all nine experiments. Velocity and pressure measurements were recorded for five minute trials. Spectral densities of each signal were examined with the Welch method [

This research was conducted in a 91 cm wide, 13-meter long flume. The flume has clear acrylic sides and an engineered plywood bottom. A 60 Hz variable speed pump and a 20,000 liter reservoir supply water to the flume. The flume has a flow straightener, elevated sediment test section and a v-notch weir to control the flow (_{c} is the critical velocity for particle entrainment, U_{τ} is the friction velocity and d_{50} is the median grain size diameter. A uniform, angular, sand with mean sediment size (d_{50}) of 0.85 mm and uniformity coefficient of 1.4 was used in this experiment. The grain size distribution is provided in

Pier diameter (m) | Flume width (m) | Depth (m) | U (m/s) | U/U_{c } | U_{τ } (m/s) | D_{50} (mm) | Reynolds Number | Particle Reynolds Number |
---|---|---|---|---|---|---|---|---|

0.11 | 0.91 | 0.08 | 0.20 | 0.66 | 0.0198 | 0.85 | 22,000 | 15 |

Velocities were carefully measured and velocity distribution parameters determined with the Krogstad fit were compared to expected values, [

The Strouhal number is a dimensionless parameter important in oscillating flows [

where St is the Strouhal number, ω is the frequency of oscillation associated with vortex shedding, l is the characteristic length (diameter of cylinder) and v is the average velocity. At high Reynolds numbers, the Strouhal number is 0.21 for flow around a cylinder [

The characteristics of subsurface pressure fluctuations change with radial position and depth.

Several investigators, including [_{th} out the presence of a structure, is approximately three times the boundary shear stress for the smooth or rough flat bed case. In the present investigation, pressure sensor three is 3mm beneath the surface of the sand/water interface and is useful for comparisons between near-bed pressures and shear stress similar to those by [

pressure fluctuations is approximately seven times larger than the boundary shear stress.

The present investigation seeks to adapt an equation to describe the decay in the magnitude of pressure fluctuations as a function of depth below the water/sand interface for a given flow condition around a cylinder. The starting point for this work is based on current literature describing the decay, absent any hydraulic structure, as exponential [

Reference [

where z is the depth of cover and c is a wave number associated with the structure of the flow. Equation (2) is a generalized solution to the Navier-Stokes Equation for flow within a granular layer [

Equation (2) has no physical meaning and [

In the present work, Equation (3) was adapted to describe pressure fluctuation decay as a function of depth of cover in the presence of a vertical cylinder and unimodal sand. The Detert and Parker equation (Equation (3)) was used as the foundation for this development. However, there are some notable differences between the conditions of the present investigation and those of the Detert and Parker investigation that resulted in Equation 3. The Detert and Parker investigations used bimodal bed material, while the present experiments took place in a sand bed with uniform grain size distribution. The present study included a cylinder in the flow field while the Detert and Parker work used an unobstructed flow field. Since the present work used uniform grain size, the equivalent grain size appearing in Equation (3) was changed to the median grain size

where

d_{50} is the median grain size; The relationship expressed by Equation (4) is shown alongside the measured values in

When instantaneous pressure fluctuations above a particle are reduced relative to the average differential pressure at the same point, the particle requires a reduced shear to initiate motion. These measurements define the pressure distribution at the surface as well as through the first several millimeters of sand. The subsurface pressure distributions change with radial position and depth of cover; however, at all radial locations, the intensity of the pressure fluctuations decreases with depth as seen in

Pressure sensors located at a 0-degree alignment had an approximate normal distribution at all depths as shown in

¤ | b1¤ | b2¤ | MSE¤ | R-squared¤ |
---|---|---|---|---|

Zero degree ¤ | 9.55¤ | 0.10¤ | 0.70¤ | 0.89¤ |

45 degree ¤ | 8.41¤ | 0.08¤ | 0.34¤ | 0.93¤ |

90 degree ¤ | 14.8¤ | 0.11¤ | 1.1¤ | 0.93¤ |

with flow around a cylinder and described by [

This work helps understand how fluctuating pressure fields, in the presence of cylinders, interact with and decay through the hyporheic zone. Previous investigations incorporated the forces on particles due to hyporheic process only for the case of materials with relatively high hydraulic conductivity and coarse size fraction (gravels), and only in flat beds with no flow obstruction. The present investigations clearly document the ability to measure and quantify hyporheic pressure fluctuations for much finer material and in the presence of a hydraulic structure. These fluctuations play an important role in the force balance associated with incipient motion of individual grains and these measurements help identify the processes responsible for generating additional lift.

Subsurface pressure fluctuations were measured and used to modify existing functions describing pressure fluctuations in the hyporheic zone; however, models were modified to better reflect the processes specific to this research such as the inclusion of a circular cylinder. The modeled data clearly follow the exponential decay model proposed by [

Timothy Calappi,Carol Miller, (2016) Decay of Pressure Fluctuation in the Hyporheic Zone around a Cylinder. World Journal of Mechanics,06,159-168. doi: 10.4236/wjm.2016.64013

The following symbols are used in this paper:

P_{rms} root mean square of pressure fluctuation^{ }

St Strouhal Number

U stream wise velocity

U_{c} critical velocity for particle entrainment

U_{τ} friction velocity

V average velocity

b_{i} regression parameter

c wave number

d_{50} median grain size

k roughness height

l characteristic length, diameter of cylinder

U_{τ} friction velocity

y distance above the bed

z depth of cover

δ boundary layer thickness

ε shift in velocity origin associated with rough bed

σ_{p}_{ }Standard deviation of pressure fluctuations

τ_{0}_{ }boundary shear stress_{ }

ω frequency of vortex shedding off the cylinder