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Exp Fluids (2015) 56:39 DOI 10.1007/s00348-014-1850-1

RESEARCH ARTICLE

Corrections on LIFPA velocity measurements in microchannel with moderate velocity fluctuations

Wei Zhao · Fang Yang · Jamil Khan · Ken Reifsnider · Guiren Wang

Received: 8 May 2014 / Revised: 24 October 2014 / Accepted: 26 October 2014 © Springer-Verlag Berlin Heidelberg 2015

in microfluidics at low Reynolds number (Wang et al. 2014). However, there is currently no velocimeter that can measure velocity fluctuation of high frequency in unsteady flows with sufficiently high spatial resolution (Burghelea et al. 2004), such as in electrokinetic flows (Posner and Santiago 2006; Chen and Cho 2008; Pacheco et al. 2008; Posner et al. 2012). This situation becomes even worse in turbulent flows in microfluidics, such as micro electroki-netic (EK) turbulence (Wang et al. 2014). For turbulence in microfluidics, the flow pattern is usually irregular with the existence of large spatial and temporal velocity gradi-ents. Therefore, to measure the velocity variation, both high spatial resolutions and temporal resolutions are required simultaneously.

For this reason, Kuang et al. (2009, 2011) have devel-oped a velocity measurement technique—laser-induced fluorescence photobleaching anemometer (LIFPA). Ricka (1987) proposed a bleaching-based method for qualitative measurement of flow velocity. Sugarman and Prud’homme (1987) and Wang (2005) had applied LIFPA method to measure the bulk flow velocity. Compared with other well-known velocity measurement methods in microflu-idics, such as micro particle imaging velocimetry (μPIV) and hot-wire-based velocity diagnostic techniques, this technique has several advantages: (1) noninvasive; (2) high simultaneous spatiotemporal resolution; (3) capa-ble for far-field nanoscopic measurement (Kuang and Wang 2010); (4) can have sufficient tracer molecules in nanoscopic resolution for long-time sampling; and (5) neutral tracer can faithfully follow fluid in contrast to the microparticles that more or less have charge and cannot accurately monitor flow velocity of fluids. Recently, this technique has been successfully used to measure velocity power spectrum in a micro EK turbulent flow (Wang et al. 2014). LIFPA is similar to hot-wire anemometer (HWA),

Abstract Laser-induced fluorescence photobleaching anemometer (LIFPA) has been developed in order to meas-ure velocity fluctuations of the unsteady micro electroki-netic turbulent flows in microfluidics. The statistical errors of LIFPA measurement, because of 3-D flows and Taylor’s hypothesis (compared with local Taylor’s hypothesis Pin-ton and Labbé in J Phys II 4:1461–1468, 1994), are theo-retically estimated and compared to hot-wire anemometer (HWA) measurement that is used for conventional turbu-lence measurement. The correction factor in the direction parallel to the laser beam is estimated, and the influence of directional correction factors of LIFPA is also investigated. It is found that in our investigation, the error due to Taylor’s hypothesis is negligible. The influence of 3-D flows on the first derivative variance of velocity fluctuations in LIFPA is smaller than that in HWA measurement.

1 Introduction

Unsteady chaotic flows are widely used in microfluidics when mixing enhancement of fluids is required. Recently, we have even found that it is possible to generate turbulence

Electronic supplementary material The online version of this article (doi:10.1007/s00348-014-1850-1) contains supplementary material, which is available to authorized users.

W. Zhao · F. Yang · J. Khan · K. Reifsnider · G. Wang (*) Department of Mechanical Engineering, University of South Carolina, Columbia, SC, USAe-mail: guirenwang@sc.edu

G. Wang Biomedical Engineering Program, University of South Carolina, Columbia, SC, USA

Exp Fluids (2015) 56:39

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but noninvasive. Before application, a calibration relation is required between flow velocity and fluorescence inten-sity. The fluorescence intensity will be indicated by the voltage of data acquisition system of LIFPA and propor-tional to the instant module of velocity vector. This rela-tion can be explored to measure flow velocity. However, similar to a single-wire HWA, the present LIFPA tech-nique cannot distinguish individual velocity components in different directions. Also, owing to relatively small aspect ratio, intuitively, the 3-D flow effect is pronounced which could cause significant bias errors on: (1) mean velocity; (2) velocity fluctuations and the higher order statistics; and (3) first derivative variance (FDV) of velocity fluctuation, which creates discrepancies on calculating characteristic spatial scales of turbulence. These errors should be eval-uated when LIFPA is used to measure unsteady flows in microfluidics.

In the present investigations, we focus on the correction of measured statistical velocity quantities through LIFPA in a microflow with moderate velocity fluctuations. The theories on corrections of mean flow velocity and standard deviation of velocity fluctuations are introduced in Sect. 2. Then, in Sect. 3, the FDV of velocity fluctuation is first evaluated by local Taylor’s hypothesis (LTH) suggested by Pinton and Labbé (1994) and compared to the traditional Taylor’s hypothesis (TH). Meanwhile, the statistical errors due to 3-D flow effect at different turbulent intensities are also estimated. The directional correction factor in the laser beam direction is theoretically evaluated in Sect. 4. The experimental details are introduced in Sect. 5 which is fol-lowed by the results in Sect. 6.

2 Correction of velocity measurement

2.1 Basic relations

LIFPA could be considered as an optical version of HWA with much higher spatial resolution. There is no effect of applied electric field because electrically neutral dye is used as a molecular scale tracer. The measured instant velocity can be determined as (Ewing 2004):

where the subscript “m” is the measured value of relevant physical quantity. ui is the actual instantaneous velocity. u1 is in the direction (x1) of mean flow velocity; u2 is in the transverse direction (x2), i.e., perpendicular to both the laser beam and x1. And u3 is parallel to the laser beam (x3). ai is the correction factor in each direction for LIFPA measurement. The repeated subscripts indicate summation convention from 1 to 3, if not specified. In this manuscript,

(1)um =(

a2i u2

i

)1/2

we investigate the flow where the mean velocity is per-pendicular to the laser beam. Hence, the velocity com-ponents can be expressed as ui = Uδ1i + u′

i, where U is mean flow velocity. For clarity and simplification without arbitrary, we assume the mean flow is in the streamwise of microchannel.

2.2 Relations of kinetic energy

Take the square of Eq. (1), and because um = Um + u′m

(where Um and u′m are the measured mean velocity and

velocity fluctuation, respectively), we have the equation for kinetic energy:

where ⟨ ⟩ indicates ensemble averaging.

2.3 Corrections of U and ⟨

u′21

⟩

Based on Eqs. (1) and (2), the actual mean velocity U can be estimated by the following processes. Rewrite Eq. (1) in dimensionless form and apply ui = Uδ1i + u′

i, we have:

where u′i = u′

i/U and

When |M| < 1 is satisfied, by applying binomial expan-sion, we have:

where u′ = u′/U =(

u′iu

′i/3

)1/2. In isotropic flow, the mean value of measured velocity Um and the actual mean velocity U has the relation as below:

Or alternatively in dimensional expression,

Plug Eq. (2) into (7), and we can easily find,

(2)U2m +

⟨

u′2m

⟩

= U2 +⟨

a2i u′2

i

⟩

(3)

um

a1U

=

(

1 + 2u′1 + u

′21 +

a22

a21

u′22 +

a23

a21

u′23

)1/2

= (1 + M)1/2

(4)M = 2u′1 + u′2

1 +a2

2

a21

u′22 +

a23

a21

u′23

(5)um

a1U= 1 + u′

1 +

a22

a21

u′22

2+

a23

a21

u′23

2+ O

(

u′3)

(6)Um

U= a1 +

⟨

u′21

⟩

2a1

(

a22 + a2

3

)

(7)a1U2 − UmU +1

2a1

(

a22 + a2

3

)⟨

u′21

⟩

= 0

(8)U =C2Um ±

√

C3U2m − C1

(

a22 + a2

3

)⟨

u′2m

⟩

C1

Exp Fluids (2015) 56:39

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where C1 = 2a14 + 2a1

2a22 + 2a1

2a32 − a2

2 − a32,

C2 = a1(a12 + a2

2 + a32) and C3 = a1

3 + (a1 − 1)(a22 + a3

2). In the solution, only the positive sign is selected to meet the restriction |M| < 1.

For the anisotropic case, the corrections of mean velocity and velocity variance are also available. This can be simply

achieved by replacing ai (i = 2, 3) with ai

√

⟨

u′2i

⟩/⟨

u′21

⟩

to

count the anisotropy into directional correction factors.After calculating U,

⟨

u′21

⟩

can be obtained as

The relevant experimental results will be introduced in Sect. 6.

3 Correction on FDV of velocity fluctuations

3.1 The influence of LTH

In the flow with moderate fluctuations, the FDV of u′1,

i.e., �(

∂u′1/∂x1

)2�, is an important parameter for calculat-

ing the dissipation rate and characteristic spatial or tempo-ral scales. However, similar to HWA, it is impossible for LIPFA to directly measure u′

1, and an error will be produced if using u′

m instead of u′1 to calculate FDV. The origin of the

error may be from the improper use of TH due to the rela-tively large velocity fluctuations and the 3-D velocity fluc-tuations due to the indistinguishable velocity components in LIFPA. In this section, the influence of the former on �(

∂u′1/∂x1

)2� is investigated.

Conventionally, the calculation of �(

du′m/dx1

)2� by TH

can be expressed as (Monin and Yaglom 1975):

This is a reasonable approximation when the local turbu-lent intensity is much smaller than 1 (Tennekes and Lumley 1972). However, if this condition is not satisfied, TH cannot be applied arbitrarily, as the spatial velocity distributions may be not accurately calculated by velocity time series and its mean value. Hence, the influence of relatively large-scale velocity fluctuations, which determines the spatial position of velocity vector, should be evaluated by LTH.

LTH is first suggested by Pinton and Labbé (1994) while transferring the energy spectrum of velocity from fre-quency to wavenumber domain. In this scheme, the spatial position is determined by the time interval and advective average velocity, which is calculated by:

(9)⟨

u′21

⟩

=U2

m + u′2m − U2

aiai

(10)

⟨

(

du′m

dx1

)2⟩

=

⟨

(

du′m

Umdt

)2⟩

(11)Um(t) =

1

T

∫ t+T/2

t−T/2

um(τ )dτ

where T is a characteristic time scale. In the publication (Pinton and Labbé 1994), the authors defined T as the period of an external force. However, in many practical cases, the forcing frequency is higher than the cutoff fre-quency of velocity signal to be measured in the flow, such as in high-frequency AC EK flow (Wang et al. 2014). We cannot simply use the period of the electric field as T. Instead, an arbitrary time scale related to large-scale energy containing structure has been used. If Kolmogorov spec-tra exist, T is the reciprocal of the frequency, where −5/3 slope starts. If not, such as in cases of chaotic flows, the reciprocal of frequency, where steeper descending of power spectrum (related to the viscous diffusion region) starts, is applied. Then, the FDV of velocity fluctuation via LTH can be calculated as:

where um(t) = Um(t) + u′m(t). The results of FDV calcu-

lated by TH and LTH will be introduced in Sect. 6. Here, the reference mean velocity of TH is Um, instead of U, for comparing with Um(t) used in LTH.

3.2 3-D flow effect

In LIFPA measurements, the coefficient a3 is not negligi-ble because the aspect ratio (the ratio between the depth of focus and the diameter of focus of a laser beam) is relatively small (4–5), which is mainly determined by the microscope objective lens. Therefore, the contribution of u′

3 cannot be ignored without proper evaluation of its influ-ence. Here, we expand the work of Ewing and George (2000) from HWA to LIFPA to calculate the 3-D veloc-ity fluctuations and their derivatives with the influence of directional correction factor ai. This leads to:

where l, n, i, j are summed from 1 to 3. u′i = u′

i,LW + u′i,HW

and ui,LW = Uδ1i + u′i,LW. Here, subscript “LW” means

the low-wavenumber components of velocity fluctua-tions which are related to the energy containing structures, i.e., large-scale vortex structures. This part dominates the kinetic energy of velocity fluctuations. “HW” indicates the high-wavenumber components of velocity fluctua-tions which is in inertial and dissipation subrange (Lum-ley 1965). It dominates the FDV of velocity fluctuations. Applying binomial expansion to the third order, we can

(12)

⟨

(

du′m

dx1

)2⟩

=

⟨

(

du′m

Um(t)dt

)2⟩

(13)

⟨

(

∂u′m

∂x1

)2⟩

=1

U2

⟨

a2

l a2naiajul,LWun,LWui,LWuj,LW

a2

1u2

1,LW+ a2

2u2

2,LW+ a2

3u2

3,LW

⟩

⟨

∂u′l,HW

∂xi

∂u′n,HW

∂xj

⟩

Exp Fluids (2015) 56:39

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39 Page 4 of 10

rewrite the denominator of the first part on the right side of Eq. (13) as:

Truncated to the second order of u′i,LW, Eq. (13)

becomes:

(14)

1

a2

1u

2

1,LW+ a

2

2u

2

2,LW+ a

2

3u

2

3,LW

=1

a2

1U

2

[

1 − 2

(

u′1,LW

U

)

+ 3

(

u′1,LW

U

)2

− 4

(

u′1,LW

U

)3

−a

2

2

a2

1

(

u′2,LW

U

)2

−a

2

3

a2

1

(

u′3,LW

U

)2

+4a

2

2

a2

1

(

u′1,LW

U

)(

u′2,LW

U

)2

+4a

2

3

a2

1

(

u′1,LW

U

)(

u′3,LW

U

)2

+ . . .

]

Similarly, as the FDV is normally dominated by high-wavenumber fluctuations (Lumley 1965; Wyngaard and Clifford 1977), we have:

Assume the flow is homogeneous and isotropic, the fourth-order tensor of velocity gradient can be expressed by (Monin and Yaglom 1975; Pope 2000):

Then, Eq. (15) can be simplified as,

(17)⟨

u′i,LWu′

j,LW

⟩

≈⟨

u′iu

′j

⟩

(18)

⟨

∂u′l,HW

∂xi

∂u′n,HW

∂xj

⟩

≈

⟨

∂u′l

∂xi

∂u′n

∂xj

⟩

(19)

⟨

∂u′l

∂xi

∂u′n

∂xj

⟩

= 2

⟨

(

∂u′1

∂x1

)2⟩

[

δijδln −1

4

(

δilδjn + δinδjl

)

]

(15)

⟨

(

∂u′m

∂x1

)2⟩

= a4

1

(

1 +

⟨

u′21,LW

U2

⟩

−a

2

2

a2

1

⟨

u′22,LW

U2

⟩

−a

2

3

a2

1

⟨

u′23,LW

U2

⟩)⟨

(

∂u′1,HW

∂x1

)2⟩

+

⟨

u′22,LW

U2

⟩[

a4

2

⟨

(

∂u′2,HW

∂x1

)2⟩

+ a2

1a

2

2

⟨

(

∂u′1,HW

∂x2

)2⟩

+ 2a1a3

2

(⟨

∂u′1,HW

∂x2

∂u′2,HW

∂x1

⟩

+

⟨

∂u′1,HW

∂x1

∂u′2,HW

∂x2

⟩)

]

+

⟨

u′23,LW

U2

⟩[

a4

3

⟨

(

∂u′3,HW

∂x1

)2⟩

+ a2

1a

2

3

⟨

(

∂u′1,HW

∂x3

)2⟩

+ 2a1a3

3

(⟨

∂u′1,HW

∂x3

∂u′3,HW

∂x1

⟩

+

⟨

∂u′1,HW

∂x1

∂u′3,HW

∂x3

⟩)

]

+ 2

⟨

u′1,LW

u′2,LW

U2

⟩[

a2

1a

2

2

⟨

∂u′1,HW

∂x1

∂u′2,HW

∂x1

⟩

+ a3

1a2

⟨

∂u′1,HW

∂x1

∂u′1,HW

∂x2

⟩]

+ 2

⟨

u′1,LW

u′3,LW

U2

⟩[

a2

1a

2

3

⟨

∂u′1,HW

∂x1

∂u′3,HW

∂x1

⟩

+ a3

1a3

⟨

∂u′1,HW

∂x1

∂u′1,HW

∂x3

⟩]

+ 2

⟨

u′2,LW

u′3,LW

U2

⟩[

a1a2

2a3

(⟨

∂u′1,HW

∂x3

∂u′2,HW

∂x1

⟩

+

⟨

∂u′1,HW

∂x1

∂u′2,HW

∂x3

⟩)

+ a1a2a2

3

(⟨

∂u′1,HW

∂x2

∂u′3,HW

∂x1

⟩

+

⟨

∂u′1,HW

∂x1

∂u′3,HW

∂x2

⟩)

+ a2

1a2a3

⟨

∂u′1,HW

∂x2

∂u′1,HW

∂x3

⟩

+ a2

2a

2

3

⟨

∂u′2,HW

∂x1

∂u′3,HW

∂x1

⟩]

+ H.O.T.

Here, “H.O.T.” means high-order terms. Also,

Because the correlation between LW and HW parts can be ignored (Lumley 1965) and also �u′

i,HWu′j,HW� is much

smaller than �u′i,LWu′

j,LW�, the Eq. (16) can be rewritten as:

(16)

⟨

u′iu′j

⟩

=⟨

(

u′i,LW

+ u′i,HW

)

(

u′j,LW

+ u′j,HW

)⟩

=⟨

u′i,LW

u′j,LW

⟩

+⟨

u′i,LW

u′j,HW

⟩

+⟨

u′i,HW

u′j,LW

⟩

+⟨

u′i,HW

u′j,HW

⟩

It can be seen that the measured FDV of the veloc-ity fluctuation will be affected by both the 3-D velocity

(20)

⟨

(

∂u′m

∂x1

)2⟩

= a4

1

⟨

(

∂u′1

∂x1

)2⟩

+

⟨

u′21

⟩

U2

[

a4

1+ a

4

2+ a

4

3+ a

2

2(a1 − a2)

2

+a2

3(a1 − a3)

2

]

⟨

(

∂u′1

∂x1

)2⟩

Exp Fluids (2015) 56:39

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Page 5 of 10 39

components and the different direction factors of LIFPA. This will inevitably introduce an error when ai (i = 1, 2, 3) is close to 1, even though |M| < 1 is satisfied. However, in comparison to the previous works (Wyngaard and Clif-ford 1977; Ewing and George 2000), Eq. (20) shows a rel-atively smaller error, although ai (i = 1, 2, 3) is assumed to be 1. If a3 ≪ 1, the error could be even smaller. This means in isotropic flow, even if all the component effects are considered, the error is not that large as mentioned by Wyngaard and Clifford (1977), Ewing and George (2000). Here, we only expand Eq. (15) to the second order of u′ and include all the 3-D components, irrespective of fre-quencies. Compared with the approach in Ewing and George (2000), which approximates Eq. (15) to the third order of LW components but neglect the HW components of velocity fluctuations in x3 direction, the present pro-cessing may cause the error being underestimated, when u′ is not small. But, instead, the HW velocity compo-nents in x3 direction are included. If a3 is not negligible, the influence of HW velocity components in x3 direction may be more important than the higher order terms of LW velocity components, which are ignored in our investiga-tion. Hence, the underestimation of Eq. (20) due to the neglecting of higher order terms of LW velocity compo-nents is insignificant.

In fact, Ewing and George (2000) have also taken into account of the TH influence of moderate turbulent intensi-ties. But compared with the LTH, the restriction of bino-mial expansion limits the application of their work in higher turbulent intensity regime. Hence, the method from Ewing and George (2000) and the Eq. (20) of the present manuscript are appropriate in the case of moderate or small velocity fluctuations. By comparison, LTH can be applied for flows with large velocity fluctuations, even though there can be reverse flows, as introduced by Pinton and Labbé (1994).

4 Estimation of the coefficients ai

The coefficients ai are normally determined by the pho-tobleaching character of LIFPA system, which depends on various parameters, such as dimension and power den-sity of laser focus, fluorescent dye and flow velocity. As the calibration curve between velocity and fluorescence intensity in LIFPA normally has good linearity, the varia-tions of fluorescent intensity due to velocity fluctuations in x1 and x2 directions would be the same. Also, according to the noninvasive character of LIFPA and symmetry of laser beam, the velocity fluctuations in x1 and x2 directions have same photobleaching process. Hence, both the stream-wise factor, a1, and transverse factor, a2, are assumed to be unity. Due to the low aspect ratio of laser focus, in LIFPA

measurement, the correction factor of parallel-beam direc-tion, i.e., a3, may have apparent difference from the HWA. In a HWA measurement, the smaller the aspect ratio of the hot wire, the larger the value of a3. The flow parallel to the hot wire will have larger contribution to velocity measure-ment, a case that is undesirable. Also due to the existence of prongs, the flow in x3 direction can generate additional flow disturbance in other two directions, and the measured velocity signals in the transverse direction will be influ-enced. However, in LIFPA measurement, even though the aspect ratio is only 4–5, the influence of flow in parallel-beam direction is still negligible. One of the main reasons is that the dye molecules flowing along axial direction of the laser beam have already been partially photobleached before they flow into the detection volume. The value of a3 can be estimated by the comparing photobleaching process of the dye in perpendicular and parallel directions of the laser beam.

The fluorescence intensity decreases with time exponen-tially as below:

where I0 is the initial intensity at t = 0. In the case where the flow is uniform and perpendicular with laser beam (i.e., x1 or x2 direction) with velocity magnitude u⊥, as shown in Fig. 1a, c, the spatial distribution of I in the laser focus is:

where l⊥ is the distance from where the dye exposed to the laser beam, along the direction of the flow velocity. How-ever, if the flow direction is parallel to the laser beam (i.e., x3 direction with magnitude of u∥), before the dyes flow into the focus region, they have been more or less pre-photobleached (as schemed in Fig. 1b, d). Assume the pre-photobleach length is dpp, the spatial distribution of fluores-cence intensity along the laser focus is:

As an approximation, the focus region is supposed to be a rectangular cube having an uniform laser intensity, with transverse width df and depth of focus dpf. Then, in the per-pendicular case, the total fluorescence intensity can be cal-culated by:

While for the parallel case, the total fluorescence intensity is:

(21)I = I0e−t/τ

(22)I⊥(l⊥) = I0e−l⊥/u⊥τ

(23)I�(l�) = I0e−(l�+dpp)/u�τ

(24a)

I⊥ = dpfdf

∫ df

0

I⊥(l⊥)dl⊥ = u⊥τdpfdfI0

(

1 − e−

dfu⊥τ

)

(24b)I� = d2f

∫ dpf

0

I�(

l�)

dl� = u�τd2f I0e

−dppu�τ

(

1 − e−

dpfu�τ

)

Exp Fluids (2015) 56:39

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39 Page 6 of 10

For the case where both u⊥τ ≫ dpf ∼ dpp > df and u∥τ ≫ dpf ∼ dpp > df, expand the exponential term of Eqs. (24a, 24b) to the second order, we have:

And, the ratio (R) of I∥ and I⊥ can be simplified as:

For the case while both u⊥τ ≪ df < dpf ∼ dpp and u∥τ ≪ df < dpf ∼ dpp:

To evaluate the influence of different photobleaching in parallel and perpendicular directions of laser beam, we assume u⊥ = u∥ = u and then Eq. (26) becomes:

(25a)I⊥ ≈ dpfd2f I0

(

1 −df

2u⊥τ

)

(25b)I� ≈ dpfd2f I0

(

1 −dpp

u�τ−

dpf

2u�τ

)

(26a)R =I�

I⊥=

1 − dpp/u�τ − dpf/2u�τ

1 − df/2u⊥τ

(26b)R =I�

I⊥=

u�dfe−

dppu�τ

(

1 − e−

dpfu�τ

)

u⊥dpf

(

1 − e−

dfu⊥τ

)

(27a)

R ≈ 1 −dpf + 2dpp − df

2uτ − df

(for uτ ≫ dpf ∼ dpp > df)

(27b)R ≈df

dpf

e−dppuτ (for uτ ≪ df < dpf ∼ dpp)

R is equivalent to the correction factor in x3 direction, i.e., a3. This can be simply proved as below. Because the velocity calibration curve is normally calibrated in the x1 direction, i.e., u1 = u1(I⊥), a velocity in x3 direc-tion with actual magnitude of u3 can exhibit an apparent value as:

In LIFPA, as in most ranges of calibration curve, espe-cially at low velocity magnitude, the calibration curve has excellent linearity, hence:

Clearly, a3 = R. Hence, a3 is a function of local velocity. This is different from the case in a HWA, where a3 is nor-mally treated as a constant (Lomas 1986).

5 Experimental setup

The experiments are carried out in a quasi T-shaped microdiffuser, which have 5° divergent angle (Fig. 2b), where a pressure-driven flow is forced electrokineti-cally, as shown in Fig. 2a. The channel initial width is 130 μm, and the height and length are 240 μm and 5 mm, respectively. Two streams with the same concen-tration (20 µM) of a neutrally charged fluorescent dye (Coumarin 102) but different conductivities are sepa-rated by a trailing edge at the entrance of the microchan-nel and delivered into the channel via a syringe pump (Harvard Apparatus PHD 2000). The conductivities of

(28)u1

[

I�(u3)]

= u1[RI⊥(u3)]

(29)u1[RI⊥(u3)] = Ru1[I⊥(u3)] = Ru3 = a3u3

Fig. 1 Schemes of pho-tobleaching process. a Laser beam is orthogonal to the bulk flow. b Laser beam is parallel to the bulk flow. c Variation of fluorescence intensity with l⊥ when laser beam is orthogo-nal to the flow. d Variation of fluorescence intensity with l∥ when laser beam is parallel to the flow. The influence of pre-photobleaching can be clearly illustrated

Exp Fluids (2015) 56:39

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solutions in the two streams are 1 and 5,000 μS/cm, respectively. An AC electric field is applied vertically to the streamwise direction through two electric conductive sidewalls of the channel. In this setup, a micro EK turbu-lent flow has been achieved at bulk flow Reynolds num-ber (Re = Ubd/υ, where Ub is the bulk flow velocity; d is the hydraulic diameter of channel at the entrance; and υ is the kinematic viscosity) in the order of just unity (Wang et al. 2014). The flow rate of each stream is 2 μL/min.

The details of the LIFPA setup have been introduced in previous publications (Wang et al. 2014) and are schemed in Fig. 2a. In this research, a home-developed confocal microscope that consists of continuous wave 405 nm UV laser (50 mW), Olympus PlanApo 100× NA 1.4 oil immer-sion microscope objective lens and optical system was used. The fluorescent signal is captured by a photomulti-plier (PMT, Hamamatsu R-928). After proper amplifica-tion and electric filtering, the voltage signal is recorded on a computer through our A/D convertor from National Instrument.

From previous investigations (Kuang et al. 2009), the dimension of laser beam at focus was df = 203 nm and dpf = 812 nm. The time constant τ is estimated to be <10 μs (Kuang et al. 2011). Hence, in the flow where in x3 direc-tion, the velocity fluctuation is limited, and on the order of 1 mm/s, uτ ≪ df < dpf is satisfied. If we further assume dpf = dpp, for a typical velocity fluctuation of 1 mm/s, a3 (Eq. 26b) is approximately to be 0. In fact, even dpp is down to uτ, a3 is still no more than 0.1. This clearly indicates that even though the aspect ratio of LIFPA is much smaller than HWA, a3 of LIFPA is not larger than that of HWA, due to the short photobleaching time of LIFPA. Furthermore, as LIFPA is a noninvasive velocimeter, there is no additional flow disturbance generated. The velocity fluctuation compo-nents in the direction orthogonal to laser beam can be reli-ably evaluated.

Below this section, we use x1, x2 and x3 to represent streamwise, spanwise and vertical positions, respectively. The mean flow velocity is assumed to be in streamwise direction.

6 Experimental results

6.1 Errors of U and u′1_rms

The mean velocity and the root-mean-square (rms) values of velocity fluctuations at different voltages are investigated, respectively. In the ideal case, where the laser beam is axial symmetry without directional distortion and the veloc-ity calibration curve maintains linearity at the sufficiently large range, a2 can be assumed to be equal to a1. However, in practical applications, this may not be true since: (1) The laser beam may not be circular, but elliptical or eccentric circle, as a result of laser beam distortion or poor laser qual-ity; (2) The calibration curve has a small range of linear-ity, when the components of LIFPA system (such as fluo-rescent dye, laser wavelength and optics.) are not carefully selected and adjusted. In this case, the contribution of u1 and u2 in LIFPA measurement will be different, since normally streamwise velocity is higher than transverse velocity. Both of the reasons will cause the difference between a1 and a2. Hence, in this section, we neglect the assumption a1 = a2 and evaluate the influence of varying a2.

Assuming the flow is isotropic to simplify the analy-sis, the influences of different a2 and a3 on measurement error (Eq. 30) are investigated and plotted in Fig. 3. Here, we use the relative error between measured values (such as

Um and u′m_rms =

√

�u′2m�) and actual values (such as U and

u′1,_rms

=

√

�u′21�) to evaluate their difference, as shown

below:

It can be seen at 10 Vp-p, the measured turbulent inten-sity u′

m_rms/Um is only 1.7 %. In this case, the relative errors of both mean velocity and rms values are very small. The influence of 3-D flow on velocity fluctuations is even smaller than that on U. Compared with the corresponding measured values, the actual magnitude of U is normally smaller (i.e., U < Um), with relatively larger u′

1,_rms (i.e., u′

1,_rms > u′m,_rms).

(30)Error (%) =

∣

∣

∣

∣

actual value − measured value

actual value

∣

∣

∣

∣

× 100

Fig. 2 Schematic of the experimental setup. a LIFPA measurement system and b microchannel

Exp Fluids (2015) 56:39

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39 Page 8 of 10

When the voltage is increased to 20 Vp-p, larger veloc-ity fluctuations can be generated. The measured turbulent intensity is about 11.1 %. In this case, the errors in the measurements of mean velocity and velocity fluctuations are <2 %, as shown in Fig. 3c, d. These errors are limited and will not cause much difference from the actual values.

6.2 FDV of velocity fluctuations by TH and LTH

Both the FDV of velocity fluctuations by TH and LTH are investigated at different streamwise posi-tions and voltages. As plotted in Fig. 4, the differ-ence between TH and LTH is very small and irrespec-tive of varying streamwise positions (20 Vp-p) or the

applied voltages. The relative errors, which are defined as (∣

∣

∣

∣

[

�(du′m

/dx1)2� − �(

du′m

/ dx1

)2�]

/

�(du′m

/dx1)2�

∣

∣

∣

∣

× 100 %

)

,

are <5 %. In the case of low turbulent intensity, using TH will not cause large error, which is consistent with that mentioned by Tennekes and Lumley (1972). Compared with the error caused by 3-D flows, the LTH will not help much in improving the corrections, even if at x1 = 10 μm where turbulent intensity is high. Normally, correction of TH by using LTH is not necessary, except at ultra high tur-bulent intensities, which are not achievable in the present work.

As the calculation of TH is normally based on the meas-ured mean velocity, Um, not U (note Um > U), the FDV

Fig. 3 Relative error of U and u′1,rms

influenced by a2 and a3, meas-ured at x1 = 100 μm from the trailing edge, and x2 = x3 = 0, i.e., the centerline of channel. a a2 influences when a3 = 0.01, 10 Vp-p.

b a3 influences when a2 = 1, at V = 10 Vp-p. c a2 influences when a3 = 0.01, 20 Vp-p. d a3 influences when a2 = 1, at V = 20 Vp-p

Fig. 4 Comparison of TH and LTH for calculating �(du

′m/dx1)

2�. a Compared under different voltages at x1 = 100 μm and x2 = x3 = 0. b Relative errors corresponding to (a). c Compared at different streamwise positions at 20 Vp-p, x2 = x3 = 0. d Relative errors corresponding to (c)

Exp Fluids (2015) 56:39

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calculated by TH should be smaller than the actual value. The actual relative error between TH and LTH is expected to be larger than the present estimation. Compared with the 3-D flow effect that will be introduced, LTH is not restricted by |M| < 1. Hence, LTH can be applied even in high turbulent intensity flow if there is a specific large energy injection scale.

6.3 Effect of cross flow on the FDV of velocity fluctuations

Assuming the flow to be homogeneous and isotropic, the relative error of FDV of velocity fluctuations can be esti-mated from Eq. (20) as:

The relative errors influenced by both a2 and a3 are investi-gated for three different cases, and results are plotted in Fig. 5. It can be seen that error is monotonically increasing with a2 and a3. Below 10 Vp-p at x1 = 100 μm, the error attributed to the 3-D velocity fluctuations is very small due to the small velocity fluctuations and can be ignored. However, as the voltage increased to 20 Vp-p, �u

′21 �/U2 is enhanced due to the

larger EK force. The actual turbulent intensity after correction increases to 11.3 %, which results in a larger error around 3 %. At x1 = 10 μm, i.e., adjacent to the entrance of channel, where the flow is highly disturbed, the error becomes signifi-cant. While forced at 20 Vp-p, the actual turbulent intensity is about 26.4 % which is comparable to that in turbulent mixing layer (George and Hussein 1991). The error of FDV can be up to 26 % depending on a2 and a3. In such cases, the errors can-not be arbitrarily ignored and should be carefully corrected, especially when uτ ≫ dpf ∼ dpp > df is satisfied (or in other words, a3 approaches 1). In our experiments, because of the noninvasive nature of LIFPA and fast photobleaching of the dye, a3 is always and apparently smaller than 0.2. In the range

(31)

Error (%) =

∣

∣

∣

∣

∣

1 − a41 −

[

a41 + a

42 + a

43 + a

22(a1 − a2)

2

+ a23(a1 − a3)

2] �u′2

1�

U2

∣

∣

∣

∣

∣

× 100

of a3 from 0 to 0.2, although the error of FDV increases with a3, the changing is very small and difficult to be distinguished as shown in Fig. 5. However, if the dye photobleaches slowly, a3 may be much larger than what we estimated here. In that case, the error will increase with a3.

Compared with the relevant works of Lumley (1965) and Ewing and George (2000), the measurement error estimated from Eq. (31) of our investigation is smaller. Because Lum-ley (1965) and Ewing and George (2000) did not investigated the influence of directional correction factor while calculat-ing FDV (which is equivalent to assume all the three direc-tional correction factor to be 1); for comparison, we assume ai = 1 (i = 1, 2 and 3). It is found at x1 = 10 μm and V = 20 Vp-p, the errors estimated from Lumley (1965) and Ewing and George (2000) are 34.8 %, and 27.9 %, respectively, due to the turbulent intensity of 26.4 %. Both of the errors are larger than our calculation, which is at most 23.9 %. As in this LIFPA system, a3 is actually much smaller than the cor-responding correction factor of a HWA and far from 1. The error of FDV of velocity fluctuations caused by 3-D flow can be reduced to 13.7 % when a3 approaches 0. Therefore, the FDV measurement using LIFPA is intrinsically more accu-rate than that using HWA. (Note that we cannot directly compare LIFPA and HWA measurement technique in our experiments, because HWA currently cannot be used for flow field measurement in microfluidics. Here, we assume the flows are the same, in order to evaluate the influence of the 3-D effect on the velocity measurement.)

In Eq. (20) and assuming a1 is unity, it can be seen �(

∂u′m/∂x1

)2� is always larger than �

(

∂u′1/∂x1

)2�. This

is completely different from the variation of �u′21 � that is

always larger than �u′2m�. It indicates the different influ-

ence of the 3-D effect on calculating large- and small-scale velocity components, and the actual Taylor Scale should be underestimated.

6.4 Discussion on moderate velocity fluctuations

The analysis in the present work focuses on the moderate velocity fluctuation. The concept of the moderate fluctuation

Fig. 5 Relative error of �(

∂u′1/∂x1

)2� influenced by a2 and a3. a a2 influences when a3 = 0.01. b a3 influences when a2 = 1

Exp Fluids (2015) 56:39

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39 Page 10 of 10

relies on the validity of binomial expansion. For different probability density function of velocity fluctuations, the intensity of velocity fluctuations that can be called “moder-ate” is different. Ewing and George (2000) suggested that the turbulent intensity should be below 35 %. However, this value could be an overestimation. As introduced in Eq. (4), |M| < 1. Neglecting the higher order terms of u′

i (i = 1, 2 and

3), we can approximately have the relation:

Then, it is found:

For a flow in which the velocity fluctuations have a Gaussian distribution, if we require 90 % of velocity fluctu-ations that satisfy |M| < 1 (the cumulative distribution func-tion of the Gaussian distribution around the mean value is 0.9), i.e.,

∣

∣u′1

∣

∣/u′1,_rms ≤ 1.645, then:

This means, for a strict and reliable correction, the criti-cal value of measured velocity fluctuation intensity for moderate fluctuation should be below 30.4 %.

7 Conclusion

In this manuscript, we theoretically analyze the error of velocity measurement by LIFPA, due to both TH and 3-D flow. Then, the correction factor in x3 direction is theoreti-cally evaluated. It is found:

1. The small aspect ratio of laser focus does not gener-ate a large a3, indicating that the measurement is not significantly affected by the velocity fluctuations in x3 direction. This result is completely different from that in existing HWA, in which the aspect ratio has to be large enough to avoid parallel flow effect. The small a3 could be due to two reasons: (1) noninvasive character of LIFPA; (2) the fast photobleaching of dye and small photobleaching time.

2. The actual mean flow velocity is smaller than the measured value, while the actual velocity fluctuations in streamwise direction are larger than the measured values. But in the investigated range of turbulent inten-sities (below 11.1 %), the errors are negligible.

3. LTH will not have much improvement on the correc-tion of FDV of velocity fluctuations compared with using TH in our investigation where turbulent intensity is smaller than 26.4 %.

(32)2∣

∣

u′1

∣

∣ < 1 or∣

∣

u′1

∣

∣ < U/2

(33)

∣

∣u′1

∣

∣

u′1,_rms

<U

2u′1,_rms

(34)u′

m,_rms

Um

<u′

1,_rms

U<

1

3.29= 30.4 %

4. The error of FDV of velocity fluctuations measured by LIFPA due to the 3-D flow is not negligible, especially when the turbulent intensity is not small, such as at x1 and 20 Vp-p in our investigation. However, compared with HWA, LIFPA suffers a smaller influence of 3-D flows on the measurement of FDV of velocity fluctuations.

Acknowledgment The authors are grateful to discussion with Ronald Adrian. The work was supported by NSF under Grant No. CAREER CBET-0954977, MRI CBET-1040227 and NSF EPSCoR award, EPS-1317771.

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