LI Dan-xun, LIN Qiu-sheng, ZHONG Qiang, WANG Xing-kui

State Key Laboratory of Hydroscience and Engineering, Tsinghua University, Beijing 100084, China, E-mail: lidx@tsinghua.edu.cn

(Received February 14, 2012, Revised May 28, 2012)

BIAS ERRORS INDUCED BY CONCENTRATION GRADIENT IN SEDIMENT-LADEN FLOW MEASUREMENT WITH PTV*

LI Dan-xun, LIN Qiu-sheng, ZHONG Qiang, WANG Xing-kui

State Key Laboratory of Hydroscience and Engineering, Tsinghua University, Beijing 100084, China, E-mail: lidx@tsinghua.edu.cn

(Received February 14, 2012, Revised May 28, 2012)

Sediment-laden flow measurement with Particle Tracking Velocimetry (PTV) introduces a series of finite-sized sampling bins along the vertical of the flow. Instantaneous velocities are collected at each bin and a significantly large sample is established to evaluate mean and root mean square (rms) velocities of the flow. Due to the presence of concentration gradient, the established sample for the solid phase involves more data from the lower part of the sampling bin than from the upper part. The concentration effect causes bias errors in the measured mean and rms velocities when velocity varies across the bin. These bias errors are analytically quantified in this study based on simplified linear velocity and concentration distributions. Typical bulk flow characteristics from sediment-laden flow measurements are used to demonstrate rough estimation of the error magnitude. Results indicate that the mean velocity is underestimated while the rms velocity is overestimated in the ensemble-averaged measurement. The extent of deviation is commensurate with the bin size and the rate of concentration gradient. Procedures are proposed to assist determining an appropriate sampling bin size in certain error limits.

bias error, velocity, concentration, gradient

Introduction

Image-based techniques are widely used for flow velocity measurement[1-3]and Particle Tracking Velocimetry (PTV) finds increasingly more applications in two-phase flow experiments[4-6]. The PTV technique is very simple in principle, i.e., it evaluates flow velocity by tracking the movements of tracing particles seeded in the flow. Typical PTV procedures consist of illuminating the targeted flow, recording a pair of tracer images at two successive instants, processing the image pair, and quantifying velocity vectors. In sediment-laden flow experiments, PTV enables simultaneous yet separate measurement of the two phases, making it suitable for investigating the coupling mechanism between sediment particles and the carrier water. It is worth noting that the PTV technique determines randomly located instantaneous Lagrangianvelocities across the flow field. PTV can be used for sediment-laden flow measurement only at low concentrations.

In typical turbulent flow measurement, an experiment with PTV generally involves two steps[6], i.e., instantaneous velocity vectors (raw data) are firstly obtained through image acquisition and PTV, and mean and root mean square (rms) velocities are then calculated through ensemble average (statistical analysis). Due to the fact that raw velocity data are randomly distributed across the flow, it is extremely difficult to establish a statistically significant sample at a fixed point in the flow field. In practice, PTV users choose to conduct sampling over a finite-sized bin instead of at a fixed point, i.e., treating velocity data“falling” into a sampling bin as true members of a single population. In open channel flow experiments, a series of rectangular sampling bins stacked over the flow are often used to perform sampling, and the ensemble-averaged results are attributed to the center of each bin[6,7].

When the flow is not uniform within the sampling bin, the use of finite-size bin for data collection and analysis violates a fundamental statistical rule(sampled data must be representative of the same population) and results in bias errors. For example, velocity gradients lead to bias errors in flow measurement with the Pitot probes, hot-wire probe, ADV, and PIV/ PTV[8-11]. Analysis indicates that velocity gradient causes an underestimation of mean velocity and an overestimation of rms velocity in the measurement of steady, uniform open channel flows[11].

The presence of concentration gradient constitutes another form of non-uniformity across a sampling bin. Due to the presence of concentration gradient, sediment particles are not uniformly distributed along the vertical. As a result, the sample established through collecting velocity vectors over a bin is biased towards having a larger proportion of data from the lower part of the bin than from the upper part.

The effect of concentration gradient on measurement accuracy first drew attention when PTV was applied for investigating the particulate phase?s velocity in sediment-laden flow experiments[6]. This study presents analytical quantification of the concentrationinduced bias errors. The analysis is based on linear velocity and concentration profiles for simplicity. Typical bulk flow characteristics from sediment-laden flow measurements are used to estimate the magnitude of the errors. Procedures are also proposed to assist determination of appropriate sampling bin size in specified error limits.

Fig. 1 Sketch of PTV sampling within a bin

1. Measured mean and rms velocities

The PTV method provides randomly located velocity data across the entire flow field. Figure 1 illustrates a sampling bin centered at ymwith a height of Δh (the streamwise size equals the length of the image). Due to concentration gradient, there are more sampled data in the lower part of the bin than in the upper part.

Mean and rms velocities are calculated based on the instantaneous velocities sampled in the bin, i.e.,

For analysis, the entire sampling bin is virtually divided into a number of M sub-bins with equal height of Δh/M, and the ith sub-bin has Nidata points “falling” into it,i.e.,. Then Eqs.(1) and (2) chang e into

where Uijis the jth instantaneous velocity within the ith sub-bin, expressed as

where yi=ym-Δh/2+(i-0.5)Δh/M , and ymis the distance from bin center to channel bed. Define a continuous function f(y) mathematically equivalent to Ni/N as M→∞

By definition f(y)is proportional to concentration distribution whilemeeting the following requirement

Based on Eqs.(8-12), Eqs.(6) and (7) change into their continuous forms as follows:

Equations (13) and (14) are the analytical relations hips for ensemble-averaged mean and rms velocities, respectively. The mean velocity is a weighted average of the actual velocity across the sampling bin. The rms velocity consists of two parts, with the first part being indicative of actual turbulence level whereas the second part merely an artifact of the averaging procedure. This added artifact is likely to mislead a careless researcher to report turbulence in a laminar flow. Note that Eqs.(13) and (14) reduce to analytical results of clear water flows in which f(y)=1/ Δh[11].

Givenconcentration and velocity profiles, Eqs.(13) and (14) can be used to estimate bias errors or to determine a proper size for a sampling bin. Generally, some mathematical complexities are involved due to the nonlinear characteristics of velocity and concentration distributions. For simplicity, linear velocity and concentration profiles across a sampling bin are used for analysis in the present study.

2. Bias errors in the mean and the rms velocities

2.1 Bias error in the mean velocity

Define UEas the deviationof the ensembleaveraged mean ve locity,in Eq.(13), from actual velocity at the bin?s center,U(ym)

Assume that velocity and concentration are both linearly distributed within the sampling bin of (ym-Δh/2≤y≤ym+Δh/2), i.e.,

where C1,U1are the concentration and velocity at the lower boundary of the bin, C2, U2are the concentration and velocity at the upper boundary. Note that Eqs.(16) and (17) define linear relationships only for the local sampling bin, not for the overall flow. The local linear assumption is reasonable at a small sampling bin size.

Based on its definition and the specified linear concentration profile, the weighting function f(y) for the sampling bin centered at ymis determined as follows

Substituting Eqs.(16)-(18) into Eq.(13) yields the ense mble-averaged mean velocity

where ΔUis the mean velocity difference between the upper and lower boundaries of the sampling bin, and λ is a dimensionless measure of concentration gradient, i.e.,

Substitutingy=yminto Eq.(17) yields the actual velocity at the bin center

Substituting Eqs.(19) and (22) into Eq.(15) and divid ing the shear velocity, u*, gives the dimensionless bias error in mean velocityas follows

The bias error in mean velocities is proportional to b oth λ and ΔU, the measures of concentration gradient and velocity gradient, respectively. Commonly encountered two-phase open channel flows are characterized by UE/u*＜0 as ΔU＞0 and λ＞0, resulting in an underestimation of the actualmean velocity. Equation (23) also shows that the bias error reduces to zero whenλ=0 (no concentration gradient), in consistence with analytical results of clear water flows with linear velocity distributions[11].

2.2 Bias error in the rms velocity

Similar procedure in the preceding section can be applied to quantify bias errors in the rms velocity. Assume that the actual rms velocity within the sampling bin of (ym-Δh/2≤y≤ym+Δh /2) is also linearly distributed, i.e.,

where u1＇, u2＇are the rms velocities at the lower and upper boundaries of the sampling bin, respectively.

Substituting Eqs.(16)-(19) and (24) into Eq.(14) yields the ensemble-averaged rms velocity

The first term in Eq.(25) reflects true turbulence levels in the sampling bin while the last two terms are artifacts of the ensemble-averaging process. For commonly-encountered open channel flows, both the turbulence intensity and the concentration decrease with the distance above channel bed, i.e.,＇＞and 0＜λ＜1. Therefore, these two terms are both positive, leadingto an overestimated turbulence level in the measurement. The extent of overestimation is closely related to velocity and concentration gradients as well as the bin size.

Substituting y=yminto Eq.(24) yields actual rms velocity at the bin center

Subtracting Eq.(26) from Eq.(25) gives the following bias error,, defined as the deviation offrom u＇2(ym)

From Eq.(28) one can see that a new term is adde d to the measured rms velocity of the particulate phase compared with clear water flows, this term is positive and its presence leads to an increase in the measured rms velocity. Taking root-mean-square of Eq.(28) and dividing the shear velocity yields the dimensionless bias error in rms velocity as follows

The basis for derivation of Eqs.(23) and (29) is with out major approximation as the linear behavior can be considered approximately true at a small sampling bin. Estimation of the error can be made by using Eqs.(23) and (29), respectively, when the bulk flow characteristics are known.

2.3 Estimating bias errors based on bulk flow characteristics

For illustration, bulk flow characteristics reported by Muste et al.[6]are used to make a rough estimation of the concentration-induced bias errors. Following is a brief description of the measurement, NB2. The velocities of natural sand (specific gravity is 2.65) in two-phase open channel flows were measured through PTV, and the mean and rms velocities were then calculated by using a number of sampling bins stacked along the vertical of the flow. In order to maximize the number of particles and to minimize velocity gradient effect, the streamwise length of the sampling bins was set equal to the image length while the vertical height of the bins was non-uniform, i.e., sampling bins were finer (0.00005 m) near the bed and coarser (0.0012 m) in the upper flow (see Fig.2).

Fig.2 Non-uniform bin size used in the experiment by Muste et al.[6]

Fig.3 Bias errors in mean and rms velocities

Given the measured bulk flow characteristics including mean velocity profile, turbulent intensity profile, and concentration distribution profiles, estimations of the bias errors are made by using Eqs.(23) and (29). Figure3(a) plotsthebiaserrors under original bin size used by Muste etal.[6]whileFig.3(b)shows how the bias errors increase when the bin size doubles.

Figure 3 exhibits three prominent features of the biaserrors: (1) the error in the rms velocity far exceeds the error in the mean velocity, i.e., the former is about 10-40 times larger than the latter, (2) the errors increase with bin size, and (3) as distance from channel bed increases, the errors drop.

Figure 3 reveals that the error inthe mean velocityis pretty small even at a relatively large sampling bin size, e.g.,at y/h＞0.1. This is good news for PTVusers to make a trade-off between maximizing the number of samples in each bin and minimizing the gradient effect.

The error in the rms velocity, however, is relativel y large, e.g., uE/u*=5%-20% at the original bin size and increases to uE/u*=10%-60% as the bin size doubles. Errors ofsuch magnitude should not be neglected because they are of the same order in magnitude as that of the rms velocities.

Fig.4 Relative magnitude of the bias errors

To further illustrate the significance of the errors, the d ata used in plotting Fig.3(b) are scaled with local mean velocity and turbulent intensity, respectively. The results are shown in Fig.4. The relative error in the mean velocity is smaller than 1% across the water depth, and such errors can be safely neglected. The relative error in the rms velocity, ranging from 10% (water surface) to 20% (channel bed), should be fully addressed.

3. Determination of appropriate sampling bin sizes

Of practical interest to PTV users is to determine an appropriate sampling bin size to confine the bias error within an allowable limit. To this end, one needs to know the actual profiles of velocity and concentration distributions of the flow. This is quite deceptive as the velocity and concentration profiles are often the goals of measurement. Nevertheless, one can get a rough estimate of the bin size by using assumed velocity and concentration profiles based on bulk flow characteristics. In the measurements of sediment-laden open channel flow, abundant knowledgeof velocity and concentration distributions has been gathered. For illustration, typical profiles of velocity, concentration, and turbulent intensity distributions are used to quantify a proper sampling bin size under allowable uncertainty intervals.

3.1 Analytical formulations

For open channel flows, the vertical concentrationdistribution can be described by the Rouse equation[13]

where C(y) and C(a)are the concentrations aty and a,respectively, his the water depth, z is the Rouse number of the suspended particles.

Solving Eq.(30) for C1and C2andsubstituting them into Eq.(21) yield the following

For clear open channel flows, ΔU in the stream-wise direction can be estimated by using the log-law[13]

where u*is the shear velocity, and κ is Karman coefficient (κ=0.4). It is assumed in this study that ΔU for theparticulate phase can also be roughly estimated by Eq.(32). Substituting Eqs.(31) and (32) into Eq. (23) and solving for Δh/h yield

The sampling bin size can be estimated by using Eq.(33) at an allowable error of UE/u*in mean velocity.

In sediment-laden flows, turbulent intensities for the solid a nd liquid phases are different[6,7]. As a first estimation, such difference is neglected here and the turbulence intensity for the solid phase is estimated by referencing experimental results of clear water flows[14]

whereα,βar e the coefficients, and α=5.11, β=-0.57for the stream-wise direction.

Substituting Eqs.(31), (32) and (34) into Eq.(29) and solvingfor Δh/h yield

Sampling bin size can be estimated by using Eq.(35) at an allowable error uE/u*in rms velocity.

3.2 Numerical illustrations

Figure 5 plots the sampling bin sizes calculated with Eqs.(33) and (35) at various allowable errors for the case of z=1. The curves show a general increase from channel bed to water surface along the vertical, but the functions are not monotonically increasing. The maximum occurs at about y/h=0.5-0.7 instead of at water surface, which contradicts common practice of enlarging the sampling bins all theway from channel bed to water surface.

Fig.5 Variation of the sampling bin size along the vertical at fixed errors

Figure 5 also indicates that the rms velocity requires a smaller bin than the mean velocity at the same error magnitude. This is equivalent to the findings in Fig.3 that the error in rms velocity is larger than that in mean velocity.

4. Concluding remarks

PTV measurement of sediment-laden flow evaluates mean and rms velocities by sampling and analyzing data in finite-sized sampling bins. Bias errors are introduced in the procedure when gradient in velocity and concentration present across the bin. Simplified analytical formulation is presented in this study to quantify such bias errors. Error magnitudes are estimated based on bulk flow characteristics measured in sediment-laden flow experiment. Procedures are also proposed for determination of an appropriate sampling bin size at allowable errors. Major findings are summarized as follows:

(1) The mean velocity estimated for the sampling bin deviates from actual velocity by a product of proportional velocity and concentration gradients. The deviation leads to an underestimation of the actual mean velocities in PTV measurement.

(2) The rms velocity estimated for the sampling bin overestimates the actual turbulence level, and the extent of departure depends on concentration gradient, velocity gradient and the sampling bin size.

(3) The error in rms velocity far exceeds the error in the mean velocity. To ensure the same level of measurement accuracy, the rms velocity requires a much finerbin.

The described approaches provide an effective tool for PTV users to estimate concentration-induced errors or to confine measurement uncertainty in measurement of sediment-laden flows.

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10.1016/S1001-6058(11)60290-4

* Project supported by the National Natural Science Foundation of China (Grant No. 50779023).

Biography: LI Dan-xun (1970-), Male, Ph. D.,

Associate Professor