turbulent flow viscosity

8/18/2019 Turbulent Flow Viscosity

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15th Int Symp on Applications of Laser Techniques to Fluid MechanicsLisbon, Portugal, 05-08 July, 2010

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Wall shear stress distribution in a turbulent channel flow

Omid Amili1, Julio Soria

2

1: Laboratory for Turbulence Research in Aerospace and Combustion, Department of Mechanical and AerospaceEngineering, Monash University, VIC 3800, AUSTRALIA

[email protected]

2: Laboratory for Turbulence Research in Aerospace and Combustion, Department of Mechanical and AerospaceEngineering, Monash University, VIC 3800, AUSTRALIA

[email protected]

Abstract A film-based wall shear stress sensor has been used to measure dynamic wall shear stress

distribution in a turbulent channel flow. The sensor which used to directly measure the wall shear stress

consists of mounting a thin flexible film on the solid surface. The sensor is made of a homogeneous,

isotropic, and incompressible material. The geometry and mechanical properties of the film are measured,

and particles with the nominal size of 11 µm in diameter are embedded on the film’s surface to act as

markers. An optical technique is used to measure the in-plane film deformation caused by the flow. The filmhas typically deflection of less than 2% of the material thickness under maximum loading. The sensor

sensitivity can be adjusted by changing the thickness of the layer or the shear modulus of the film’s material.

For this study, a film with the shear modulus of 80 Pa and 2 mm in thickness was used. The sensor is

statically calibrated by applying a constant shear stress to the top surface of the film showing an excellent

linear stress-strain relationship. The dynamic response of the sensor has been measured using a dynamic

calibration apparatus and procedure developed specifically for this purpose. The sensor transfer function

shows a low-pass behavior. Measurements have been performed in a fully developed turbulent channel flow

at the Reynolds number of 130,000 based on the bulk velocity and channel full height. The results, which are

compared with the available results in the literature, show the capacity of the technique to dynamically

measure wall shear stress over an extended area of the surface. Distribution measurement of the fluctuating

wall shear stress reveals the imprint of small-scale hairpins or counter-rotating streamwise vortex pairs

existing in the near-wall region.

1. Introduction

In turbulence research, to understand the dynamics of the near wall momentum transfer, high spatial

resolution measurements of dynamic wall shear stress distribution are needed. The measurements

should be performed in a non-intrusive way that does not affect the flow field itself. In addition, the

measurement field of view should be extensive enough to detect the largest turbulent structures. It is

also important that measurements can detect a wide range of scales of the shear stress fluctuations.

Furthermore, it is desired that the shear stress measuring techniques possess an adequate dynamic

bandwidth to detect all frequency contents usually in the range of a few kHz depending on the fluid

and flow properties. As a result, determination of wall shear stress using a sensor which fulfills

these characteristics is a critical and challenging task.

Due to the high demand on the ability to measure the magnitude, direction, and distribution of wall

shear stress, a large number of studies have been performed. The shear stress measurement

techniques are usually divided into direct and indirect methods (Naughton and Sheplak, 2002;

Fernholz et al, 1996). The large group of indirect shear stress measuring techniques includes the

obstacle-in-flow methods, velocity profile measurement, and application of heat or mass transfer

analogies. On the other hand, the small group of direct skin friction measuring techniques currently

available is mostly based on a floating element which responds to the force applied by the fluid on

the element. Direct measuring methods are preferable as they directly measure the applied force bythe fluid with no assumption of the flow field. However, they are accompanied with different errors


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and limitations which are well descried in Winter (1977) and Haritonidis (1989). Micro-electro-

mechanical systems (MEMS) technology has removed many of the limitations existing in

conventional mechanical fabrications (Schmidt et al, 1988; Lofdahl and Gad-el Hak, 1999; Sheplak

et al, 2004). However, the nature of the sensors is not suitable for unclean environments and they

also need further development to become standard and reliable measuring tools (Naughton and

Sheplak, 2002).Among different developed techniques, the thin-oil-film and liquid-crystal-coating techniques

which are quasi-direct means of shear stress measurement are able to determine stress distribution.

Thin-oil-film techniques which have been widely used for the last two decades are based on the

behavior of the oil film under shear loading. The oil thinning rate responds to changes of shear

stress with a bandwidth in the 10 kHz range (Naughton and Brown, 1999). The major uncertainty

arising from the difficulty of precisely measuring the oil viscosity was reported within 4% by

Fernholz et al (1996) and was evaluated less than 2.4% in the range of investigated shear stress by

Zanoun et al (2003). In liquid-crystal technique, the temporal resolution of measuring the two-

dimensional wall shear stress distribution is in the range of a few kHz (Fujisawa et al, 2003).

Although it was introduced more than three decades ago, different problems including measurement

uncertainty of 5% in the shear stress magnitude have made the technique unsuitable for manyturbulent flow applications (Naughton and Sheplak, 2002).

A novel type of sensor which has the potential to measure the mean and fluctuating wall shear stress

distribution using a flexible material has been proposed in recent years. Micro-pillar shear stress

sensor (MPS3), able to measure dynamic wall shear stress in turbulent flows, is based on flexible

micro-pillars immersing in the viscous sublayer (Grosse and Schroder, 2008a,b). Depending on the

pillar dimensions and mechanical properties, length scales better than 50 µm and time scales in the

range of a few kHz can be resolved. The sensor length to diameter ratio is between 15 and 25 and

the Young’s modulus is in the order of a few MPa. The optical detection and array of micro pillars

allow the high spatial resolution determination of the two-dimensional shear stress (Grosse, 2008).

The surface shear sensitive film (S3F) is another sensor which takes advantage of using a linear

elastic material. This technique was first introduced by Tarasov and Orlov in the early 1990s as a

direct method for measuring wall shear stress (cited in Tarasov et al (1997)). A thin elastomer layer

is mounted on the solid surface subjected to a flow field and its resulting viscous shear stress is

determined using Hooke’s law from the film deformation. The experimental setup similar to the

pressure sensitive paint (PSP) technique includes a surface or volume-distributed transducer, a light

source, and an image and data acquisition system (Fonov et al, 2006a,b, 2007). However, unlike

PSP which can only be used in air, this technique can be used in most fluids since it does not

depend on oxygen quenching. McQuilling et al (2008) and Crafton et al (2008) claimed sensor

fabrication with frequency response up to 1 kHz with the shear modulus ranging from 30 Pa to a

few hundred kPa. In these studies, the film response estimation is based on finite element modeling

of the film under unit normal and tangential applied loads. McQuilling et al (2008) verified thetechnique using an oil film shear stress measurement in the range of 0.25 to 1.5 Pa and observed a

difference of 0.25-0.33 Pa. Crafton et al (2008) estimated the accuracy in the tangential

displacement detection of 0.05 px (equivalent to 0.3 µm) corresponding to an error of 25 Pa in the

shear stress measurements in the range of 50 to 500 Pa.

The present study is aimed at measuring dynamic wall shear stress in a turbulent channel flow using

an in-house developed film-based shear stress sensor at the Laboratory for Turbulence Research in

Aerospace and Combustion (LTRAC) at Monash University. The feasibility of the technique

concept to detect mean wall shear stress has been demonstrated by developing and applying the

sensor to the same flow facility at Reynolds numbers in the range of 90,000-130,000 based on the

bulk velocity and channel full height (Amili et al, 2009; Amili and Soria, 2009; Amili and Soria,

2010). A brief description of the sensor, film fabrication, static and dynamic calibration, and flowfacility is given here. More detailed information can be found in the cited works. The sensor


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capability of dynamically detecting wall shear stress has been assessed by measuring instantaneous

wall shear stress distribution.

2. Sensor description

The film-based shear stress sensor technique is based on mounting a thin layer film made of anelastic material on the solid surface of the model or the test section of the flow facility. The elastic

layer is created by forming the material at room temperature into a flat rectangular cavity with a

smooth surface made of glass as fabricated in the study by Amili et al (2009) and Amili and Soria

(2010). The cavity which is machined into a flat plate was designed to fit into the floor of the wind

tunnel test section. Depending on the experimental condition, the thickness of the film can be

adjusted between 0.5 and 3 mm by varying the cavity depth. The geometry and mechanical

properties of the flexible material were accurately measured. Immediately after the film formation,

11 µm diameter Potters spherical particles acting as markers are embedded on the film’s top surface

flush with that surface to minimize the roughness. An optical technique is used to measure the film

deformation caused by the flow field.

The film is formed from a material which satisfies the linear elastic solid characteristics within the

desired range of operation: the deformations are very small; the relationship between applied loads

and deformations is linear; upon removal of applied loading, deformations are completely removed;

the loading rate has no effect on deformations. Owing to the fact that the film layer is made of a

liquid-based elastic material, it is deformed when loading under incompressibility condition and

returns to its initial shape after the load is removed. In addition, since the film used for the sensor is

homogeneous and isotropic, the shear modulus is independent of the location and direction.

The film deformation is measured using the images of the film while loaded with respect to its

initial unloaded condition. Then, the shear stress distribution over the film is determined by

implementing the shear stress-strain relationship which is based on linear continuum mechanics

(Lai et al, 1993). More information about the governing equations can be found in Amili et al(2009). The film deformation is a function of the applied load, film’s shear modulus, and thickness.

The sensor geometry and mechanical properties are carefully selected so that the deformation does

not exceed more than 2% of the film’s thickness under maximum predicted flow loading. In

addition, the sensor sensitivity can be adjusted by changing these parameters based on the estimated

shear stress range to be measured. An important advantage of this film-based technique is that

depending on the employed material, it is applicable to air, water, or any environment where the

film is not chemically or physically modified by the working fluid.

In this study we are only concerned with measuring the tangential shear stress and not the surface

pressure, hence only the in-plane strain deformation of the film is required. The 2D in-plane film

deformation is measured by means of a 2C-2D cross correlation PIV algorithm (Soria, 1998).

3. Sensor calibration

3.1. Static Calibration

The static calibration of the sensor involves applying a constant shear load on the film’s surface and

measuring the film deformation under different loading conditions. Detailed information about the

static calibration stage and the procedure can be found in Amili et al (2010). All the associated

errors in the calibration step including specifying the loading angle, mass of the load, load contact

area, and the cross-correlation resolvable displacement have been considered and the overall

uncertainty of measuring shear modulus based on the 95% level of confidence is estimated to be

better than 2% of the measured modulus for G


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Figure 1a shows excellent linear behavior with a negligible hysteresis effect for the films developed

at our laboratory. The non-zero intercept, which pre-loading, hysteresis or any other bias error in the

measurement system may contribute to, falls within the overall measurement uncertainty. The shear

modulus is determined from the linear least squares fit to the measured shear stress-strain

relationship. The Measurements of the shear modulus in three arbitrary directions yields similar

values within the measurement uncertainty confirming the isotropic assumption. In a similar way, performance of the static calibration at different arbitrary locations of the film proves that the

homogeneous assumption is true. To date, sensors with the shear modulus between 50 Pa and 3000

Pa have been developed which possess similar behavior to the sensor shown in Figure 1a.

Owing to the fact that the sensor is planned to be applied to turbulent shear flows, it is important to

conduct the calibration in the range of expected wall shear stress values. As a result, calibration is

carried out in the range of expected mean wall shear stress considering the fluctuations. From the

probability density function of the wall shear stress or near wall velocity fluctuations in turbulent

flows reported in the literature (such as Colella and Keith (2003); Monty et al (2007)), fluctuations

reaching up to four times the mean value can be expected. However, a static calibration curve as

shown in Figure 1a with a larger range of the applied loading shows in fact that the material

behaves linearly over a larger range of deformations. Although the calibration is staticallyconducted, a constant shear modulus can be inferred at low loading rates.

3.2. Dynamic calibration

The effect of the loading rate on the sensor behavior was investigated via transfer functions

obtained using a dynamic calibration apparatus and procedure developed for this specific purpose.

The setup consists of an electro-magnetic vibration exciter (B&K type 4809) capable of producing

vibrations in the range of 10 Hz to 20 kHz driven by a B&K power amplifier. A HP 33120A

function generator is used as a high fidelity signal input source and loading is applied via a known

mass placed on the top surface of the film. A force transducer, and an accelerometer both from

B&K with the sensitivity of 110 mV/N and 10 mV/ms-2 respectively are used to measure the

applied force and corresponding acceleration of the mass. These signals are amplified and then

recorded by means of a computer using an A/D converter. In case of forcing with white noise, a

low-pass filter (at 2 kHz) is used to create the desired frequency bandwidth. A sample length of

600,000 for the signals is recorded at 16 kHz rate to ensure a high enough temporal resolution.

Finding the transfer function to indicate the relation between the input force and the output film

displacement consists of decomposing the film response into its frequency contents and determining

the oscillation amplitudes using Fourier analysis. As a result, the amplitude of the film oscillation,

normalized by a reference amplitude, is determined in terms of frequency. It is worth noting that the

effective mass applied to the film has been accurately measured via a no-film experiment first.

The dynamic calibration was conducted for different sensors developed in our laboratory made of

different materials and different thicknesses. Figure 1b shows the transfer function for the sensorused for this experiment. The transfer function shows low-pass filter behavior with the cut-off

frequency of 120 Hz. The bandwidth calculation is based on the frequency where the sensor gain

falls to -3 dB. In the dynamic calibration, the point should be highlighted that all measurements

have been performed in air and for sensor application in different flow facilities, there is a need for

calibration in the same working fluid. It is expected that sensors show more damped behavior in

water because of the damping nature of the fluid.

For an accurate measurement of the instantaneous wall shear stress, it is important that the sensor

does not show a gain greater than 1 (i.e. a resonant frequency) in the expected range of frequencies

that may exist in the wall shear stress signal. Therefore, selection of a suitable sensor with a proper

shear modulus and thickness is made based on the working fluid and flow properties. The effect of

geometry and shear modulus of the film on the frequency response of the sensor has beenconsidered. To date, developed sensors in our lab have low-pass filter behavior with the cut-off


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frequency up to 240 Hz. The measurements by Grosse (2008) in a turbulent duct and pipe flow

show that the highest frequencies of the wall shear stress fluctuations in the streamwise direction

are less than 500 Hz for comparable Reynolds numbers to this study. This indicates that the sensors

developed until now in our laboratory are not able to dynamically detect all significant frequencies

contained in the instantaneous wall shear stress signal. While a sensor with low-pass filter behavior

has been developed and implemented, further development of a sensor with an increased bandwidthis in progress. It is worth mentioning that to avoid any possible errors caused by changes of the

film’s mechanical properties or thickness, calibration tests were performed before and after the wall

shear stress measurement.

Figure 1. The shear stress-strain curve in a static calibration for the film-based shear stress sensor with the thickness of2 mm. The measurement accuracy is estimated to be better than 0.035 Pa for the applied shear stress and 1.7×10

−4 Rad

for the shear strain, (a). The dynamic transfer function for the sensor with the cut-off frequency of 120 Hz, (b).

4. Experimental setup

4.1. Flow facility

Wall shear stress measurements were performed in the open-circuit wind tunnel facility located inthe Laboratory for Turbulence Research in Aerospace and Combustion (LTRAC) at Monash

University. The flow in the facility is generated by a three-phase 5.5 KW electrical motor coupled

with an in-line centrifugal blower. The flow rate is accurately controlled by adjusting the motor

rotational speed by means of a motor controller. The working section is 4.6 m long, 1 m wide, with

the aspect ratio of 9.75:1. The channel has a large enough aspect ratio which ensures that secondary

flows in the channel corners do not affect the mean streamwise velocity profile and the turbulent

fluctuations. The sensor is flush mounted on the lower wall of the test section at a position

approximately 41 channel height downstream of the test section entrance. The film was formed into

a cavity with dimensions of 100×70 mm and depth of 1 mm using a perspex plate with dimensions

of 320×320 mm. The plate was mounted flush with the tunnel wall to prevent any disturbance to the

local flow field. The experiment was conducted at the Reynolds number of 130,000 based on the bulk flow velocity and the channel full height. At this Reynolds number, based on the PIV

measurements along the streamwise-wall normal plane, the bulk velocity is 18.55 m/s. The

turbulence intensity at the centerline of the channel is approximately 1.7%.

4.2. Optical setup

Deformation of the film was imaged at 1 KHz using an 8-bit Motion Pro X3 high speed camera at

full CCD size of 1280×1024 px in combination with a 200 mm Nikkon Micro-Nikkor lens. The F-

stop of 4 was selected to reduce the depth of field and an array of LEDs was used for particle

illumination. It is worth noting that to remove any possible rigid-body motion of the sensor caused

by vibrations of the wind tunnel, a reference pattern attached to the bottom surface of the sensor

plate was simultaneously recorded. In order to image the reference pattern with the same recordingsystem, a combination of mirrors and prisms were used to split the field of view. As a result, a

(a) (b)


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narrow strip of the CCD with the approximate size of 256×1024 px was dedicated for this purpose

and the rest was used for the shear stress measurement. Those movements were subtracted from the

displacement of the tracer particles on the film’s surface to measure the film’s deflection under flow

loading. Firstly, particle positions at no-flow condition were recorded as a reference, and then 3270

images at flow-on state were taken. The sensor used for the experiment has the shear modulus of 80

Pa and thickness of 2 mm. The schematic of the sensor and imaging system is illustrated in Figure2. The measurements have been performed at T=20°.

Figure 2. The schematic of the experimental setup for the shear stress measurement.

5. Wall shear stress measurements

The multi-grid cross-correlation digital particle image velocimetry (MCCDPIV) algorithm

developed by Soria (1996, 1998) was used to measure the instantaneous wall shear stressdistribution. Multi-passing with the final interrogation window size of 32×32 px with the step size

of 16 px at a high sub-pixel accuracy using 2D Gaussian peak-fitting function enables

measurements with a high dynamic range. This leads to a vector spacing of 185 µm corresponding

to 10.2 wall units using the imaging resolution of 11.66 µm/px. The field of view is 11.4×11.4 mm

which is equal to 630×630 wall units.

5.1. Mean wall shear stress

The mean film deformation, wall shear stress, friction velocity, and skin friction coefficient have

been measured at six Reynolds numbers, ReH=90,000-130,000 (Amili et al, 2010). The mean wall

shear stress calculation is based on the time-average and also the spatial-average of the stress over

the film located in the entire field of view. The mean stress measurements are in good agreementwith indirect measurements by means of a logarithmic fit to the mean streamwise velocity profile

obtained from PIV experiments and application of the Clauser method (Amili et al, 2010). The

measured skin friction compares favorably with the measurements by Zanoun et al (2009),

Christensen (2001), Monty (2005) and also the logarithmic skin friction relation by Zanoun et al

(2007).

In this study, the mean film deflection is approximately 1.5 pixel corresponding to ~1% of the filmthickness. The mean wall shear stress and friction velocity are estimated to be 0.827 Pa and 823

mm/s respectively. The accuracy of the shear stress measurements is determined by the sub-pixel

accuracy of the cross-correlation algorithm and the accuracy of the shear modulus determination

which has been assessed in section 3. The uncertainty of measuring wall shear stress based on the

95% level of confidence is estimated to be better than 2.5% of the measured stress for τ


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simultaneous measurements of the rigid-body motion of the sensor. The relative movement of the

sensor with respect to the camera can cause a DC error. As a result, it is independent of the applied

loading during the experiments and consequently has a greater effect on smaller displacements.

Figure 3. Example results of the instantaneous fluctuating wall shear stress distribution using a film-based shear stresssensor. The contours indicate the strength of the streamwise fluctuating wall shear stress, τx’/τx,rms in (a) and (c), and

spanwise fluctuating wall shear stress, τz’/τz,rms in (b) and (d). The field of view is normalized using the friction velocity.

5.2. Two-dimensional wall shear stress distribution

Two example fields of the instantaneous wall shear stress fluctuation distribution are shown in

Figure 3. The vector field represents the instantaneous wall shear stress fluctuations. The contoursindicate the normalized streamwise fluctuating wall shear stress, τx’/τx,rms (left) and spanwise

fluctuating wall shear stress, τz’/τz,rms (right). The sensor detects the existence of low- and high-

shear regions aligned in the streamwise direction. By measuring the local wall shear stress

distribution, the imprint of small-scale hairpins or counter-rotating streamwise vortex pairs existing

in near-wall region can be captured. These coherent structures reported in the inner layer (Adrian

(2007); Herpin et at (2008); Jimenez et al (2004)) contribute to the evolution of the fluctuating

shear stress on the surface. The instantaneous wall shear stress fluctuation distribution is helpful to

reveal the width of the coherent motions and their spacing.

5.3. Two-point correlations

In addition to the qualitative presentation of the fluctuating wall shear stress distribution shown in

section 5.2, two-point correlations of the fluctuations shows some of the structures’ characteristics.

(a) (b)

(c) (d)


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The coherent motions dominate the two-point correlations of the streamwise and spanwise wall

shear stress fluctuations. Figure 4 shows the two-point correlation of the streamwise wall shear

stress fluctuations, R τx’τx’ at the wall. Elongated positive correlation region in the streamwise

direction is accompanied by negative correlation behavior at spanwise sides.

Figure 4. Contour map of the two-point correlation of the streamwise wall shear stress fluctuations, R τx’ τx’(∆x+

,∆z+

).

The two-point correlations of the streamwise wall shear stress fluctuations at ∆z+=0 and at ∆x

+=0

are shown in Figure 5a and 5b respectively. While a smooth decay in the correlation is observed in

the streamwise direction, a strong change in correlation’s sign in the spanwise direction is indicated.

This supports the findings in pipes, channels, and boundary layers (Hutchins and Marusic, (2007);

Monty et al (2007); Jeon et al (1999)). In Figure 5b, the spanwise extent of the positive correlation

provides the average width of the coherent structures in the near wall region. This characteristic

width of eddies is a function of the Reynolds number and distance from the wall. At the wall, this

scale is in the order of 100z+. It is reported that it increases as wall distance increases

(Chernyshenko and Baig (2005); Monty et al (2007)). Furthermore, at a fixed distance from the

wall, the width scale increases with increasing in Reynolds number (Monty et al, 2007). Based onthe using a correlation threshold of 0.05, the width scale of 244 ℓ

+ is estimated (Reτ=2730).

Figure 5. The two-point correlations of the streamwise wall shear stress fluctuations in the streamwise direction,R τx’τx’(∆z

+=0), (a) and in the spanwise direction, R τx’τx’(∆x+=0), (b). The two-point correlations of the spanwise wall

shear stress fluctuations in the streamwise direction, R τz’τz’(∆z+=0), (c) and in the spanwise direction, R τz’τz’(∆x

+=0), (d).

(c) (d)

(a) (b)


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In addition, the two-point correlations of the spanwise wall shear stress fluctuations in the

streamwise and spanwise directions are shown in Figure 4c and 4d respectively. In Figure 5c, a

stronger decay in the correlation coefficient of R τz’τz’(∆x+) than R τx’τx’(∆x

+) is observed. The

spanwise two-point correlations of the streamwise and spanwise wall shear stress fluctuations with

outer scaling are shown in Figure 6a and 6b for comparison with other wall shear stress estimationsfound in the literature. In spite of the fact that the qualitative behavior of the correlations is similar,

good collapse can be seen only for ∆z/H>0.05.

Figure 6. Spanwise two-point correlations of the streamwise wall shear stress fluctuations R τx’τx’(∆x=0), (a) and thespanwise wall shear stress fluctuations, R τz’τz’(∆x=0), (b). The length scale is normalized using the channel full height,H.

5.4. Probability density functions

The PDFs of the normalized streamwise and spanwise wall shear stress fluctuations are presented in

Figure 7a and 7b respectively. The probability density functions are scaled in a way that the integral

of the PDF by τx’/τx,rms (or τz’/τz,rms) over the range of the fluctuations be 1. The skewness and the

flatness of the streamwise shear stress fluctuations are calculated as S ƒ =0.21 and F ƒ =3.42respectively. While the flatness is in good agreement with the findings of Colella and Keith (2003),

and Sheng et al (2008), the fluctuations are less skewed. In the cited works, a longer positive tail for

the streamwise PDFs, indicating more frequent positive fluctuations, is reported. However,

measurements by Herpin et at (2008) shows that the streamwise velocity fluctuations near the wall

(y+


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6. Conclusions

Instantaneous wall shear stress distribution in a fully developed turbulent channel flow using a film-

based sensor has been measured. The wall shear stress sensor with the working principle based on

the deformation of a thin elastic film has been developed and investigated. The advantage of this

non-intrusive technique is the ability to measure dynamic shear stress distribution based on theevaluated dynamic response measured and demonstrated here. The technique is applicable to air or

water and its sensitivity can be tuned for different flow conditions by using an available wide range

of thickness and shear modulus. The technique allows high-spatial resolution measurements of wall

shear stress with an uncertainty better than 2.5% of the measured stress for


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turbulent flow viscosity

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