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A 12-sensor hot-wire probe to measure the velocity and vorticity vectors in turbulent flow

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1996 Meas. Sci. Technol. 7 1451

(http://iopscience.iop.org/0957-0233/7/10/016)

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Meas. Sci. Technol. 7 (1996) 1451–1461. Printed in the UK

A 12-sensor hot-wire probe tomeasure the velocity and vorticityvectors in turbulent flow

Petar Vukoslav cevi c† and James M Wallace ‡† Department of Mechanical Engineering, The University of Montenegro,81000 Podgorica, Yugoslavia‡ Department of Mechanical Engineering, The University of Maryland,College Park, MD 20742, USA

Received 22 March 1996, in final form 3 July 1996, accepted for publication3 July 1996

Abstract. A 12-sensor hot-wire probe, capable of simultaneously measuring thevelocity and vorticity vectors, is described. The probe is a further step in vorticityand velocity hot-wire measurement technique development, extending thenine-sensor hot-wire probe. Results from tests of the probe performance arepresented to show that the velocity field is measured with sufficient accuracy inorder to determine the velocity gradients and, therefore, the vorticity vector field.The probe also has been tested in a turbulent boundary layer in order to compareits performance with that of the nine-sensor probe. The advantage of this newprobe, compared with the nine-sensor probe, is clearly demonstrated for flow fieldswith a high ratio of cross stream to streamwise velocity components, as well as forthe measurement of higher statistical moments of the turbulence.

1. Introduction

The fundamental role that vorticity plays in turbulent flowshas been emphasized by many authors during the past30 years. Different methods and instruments have beendeveloped to measure this defining property of turbulence.Wallace and Foss (1995) have reviewed all the techniques(methods and instruments) available to date. In order tomeasure components of the vorticity vector it is usuallynecessary to measure velocity components at two or morepoints over small distances of the same order as the smallestlength scales that can be expected in turbulent flows. Tominimize the vorticity measurement errors it is necessary tomeasure the velocity components much more precisely thanis usually required for other turbulent flow characteristics.This paper presents the design, construction, calibration andoperation of a 12-sensor hot-wire probe which is capableof simultaneously measuring both the velocity and vorticityvectors in turbulent flows. The performance of the probein a turbulent boundary layer is briefly described.

The 12-sensor probe is a logical next step followingthe nine-sensor hot-wire probe developed by Vukoslavcevicet al (1991), which was capable of simultaneousmeasurements of all three components of velocity andvorticity. Indeed, 12-sensor probes have been built andtested by Tsinoberet al (1992) and Honkan (1993).The probe described herein is configured and constructeddifferently from those probes and the data reduction method

is completely different. The present probe was previouslyused, with permission of Vukoslavcevic, by Nguyen (1993)for measurements in the near wake of a circular cylinder.The data for that study were reduced with an errorminimization algorithm described by Marasliet al (1993).

2. The nine-sensor probe

The nine-sensor probe used by Vukoslavcevic et al (1991)consists of three identical arrays each having three hot-wire sensors oriented at 45◦ to the mean flow, as seen inthe sketches in figure 1. The total dimension of the probesensing area is 1.7 mm vertically and 2.2 mm horizontallywith an average distance,h, between supporting prongsof 0.5 mm. The diameter of the nine tungsten sensors is2.5 µm with a length of about 0.7 mm, giving a length-to-diameter ratio of about 280. The probe was operated in theconstant-temperature mode.

Bearing in mind that the cooling intensity of any sensoris a function of all three velocity components, it is necessaryto use at least three sensors in order to measure the velocityvector at a given point. However, with only three sensors itis necessary to neglect the velocity gradients over the spaceoccupied by sensors. To measure the gradient of a velocitycomponent in a given direction, the velocity component hasto be determined simultaneously at one additional closepoint in that direction. This was the main idea that theconstruction of the nine-sensor probe was based on. Three

0957-0233/96/101451+11$19.50 c© 1996 IOP Publishing Ltd 1451

P Vukoslavcevic and J M Wallace

Figure 1. Schematic diagrams of the nine-sensor probe, showing the end view and a side view of one array.

arrays of three sensors are separated by small distancesin the y and z directions in order to measure the velocitygradients∂ui/∂y and∂ui/∂z (i = x, y, z). The assumptionof a uniform velocity field over the probe sensing area,which is unavoidable with only three sensors, is replacedby the assumption of a linear velocity gradient over theprobe.

The cooling law of anykth sensor can be expressed ingeneral form as

Fk(u, v, w, Ek) = 0 (1)

whereu, v and w are the velocity components at a givenlocation in the mean flow, wall normal and spanwisedirections, respectively, andEk are the voltage responses ofeach sensor at the same location. The shape of this functionis not universal. In the nine-sensor probe algorithm,Vukoslavcevic et al (1991) used a special form of thisfunction based on effective angles. For three sensors(k =1, 2, 3), two of the velocity components in equation (1)can be eliminated and a single expression for the third, forexamplev, remains in the following form:

F(v, E1, E2, E3) = 0. (2)

The shape of this function, for different values ofv,is shown in figure 2 (for each of the curves shown infigure 2 the voltagesE1, E2 and E3 are constant, soF(v, E1, E2, E3) = F(v)). It is qualitatively similarfor different types of cooling expressions (1) and differssomewhat for different sensor orientations. For negativevalues of v, only one root (the intersection with theabscissa) occurs so there is only one physical solution.However, forv/u > 0 the curve turns downwards, givingtwo possible real roots. One of these solutions is alwayslarger than a value corresponding to the maximum of thecurveF(v) = 0, which is denoted as(v/u)cr . This criticalvalue determines the range of uniqueness of the probe.This means that the probe can be used in a flow in whichthe largest value ofv/u is smaller than the critical value.Otherwise it is impossible to recognize which of the twopossible solutions physically occurs in the flow. The criticalvalue can be determined analytically. In the boundary layerinvestigation of Vukoslavcevic et al (1991) it varied in

Figure 2. Function F (V ) curves for various V /U ratios.

the range 0.36–0.27 (for lower values of theu velocitycomponent). The fact thatv/u can be close to the criticalvalue may also lead to an additional problem. Any errorin measuring theEk voltage values or the dependence ofthe calibration constants on velocity variations that is nottaken into account moves theF(v) curve up or down. Thischanges its intersection with the abscissa, resulting in anerror in determiningv. It is obvious that the error will behigher close to(v/u)cr , where the curve is flat. The curvecan even fall under the abscissa, giving no solution. Thenumber of no-solution occurrences increases as the wallis approached due to the decreasingu velocity componentwhich increases the(v/u) ratio. If the number of non-converged data is too large, then essential information aboutthe flow field will be lost and the statistics will be biased.

3. The fourth sensor

An interesting characteristic of an array of three sensorsis that the critical value ofv/u appears only in the rangev > 0, for the geometrical arrangement shown in figure 1(with sensor 1 below sensors 2 and 3 in the negativey

direction). It is easy to show that the critical range willmove tov < 0, if sensor 1 is above sensors 2 and 3 (in the

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A 12-sensor hot-wire probe

Figure 3. The function F (V ) for different positions ofsensor 1: Fl (V ), full line and Fu(V ), broken line.

positivey direction). Both cases are shown in figure 3, inwhich Fl(v) denotes sensor 1 below sensors 2 and 3 andFu(v) denotes sensor 1 above sensors 2 and 3. The criticalvalues in the second case are negative, and the maximumerror and the no-solution occurrences will be forv < 0.This fact provided the impetus to add an additional sensorto each array. Having at the same time sensors both in thepositive and in the negativey directions, both theFl(v)

and theFu(v) curves are available. Depending on the signof v, the appropriate curve can be chosen and the criticalrange avoided.

4. Design and construction of the 12-sensorprobe

The probe consists of three arrays of four hot-wire sensorsoriented at 45◦ to the mean flow, as seen in the sketches infigure 4. Besides the fourth sensor of each array, there isan additional main difference from the nine-sensor probe.The common prong for each array of the nine sensor probehas been replaced by four separated thin central prongs aswas done by Tsinoberet al (1992) and Honkan (1993).Therefore this probe consists of 12 sensors and 24 prongsplaced in a sensing area of approximately 2.5 mm diameter.In fact the dimensions of the probe are practically the sameas for the nine-sensor probe although the distance betweenthe prongs is a little smaller, being 0.44 mm compared to0.5 mm for the nine-sensor probe. However, this is largeenough to give a reasonableI/d ratio. For the tungstensensors of 2.5 µm diameter this ratio is about 250. Theprongs are made of stainless steel of 0.25 mm diameter;they are copper- and silver-plated all the way up to theirtips to reduce their resistance. The prong tips are not platedto make welding of the wire to the prongs easier and morerobust. To reduce the blockage of the sensing area the tipsof the prongs are tapered to about 50µm. The prongs aredrawn through ceramic insulating tubes, which are placedin a stainless steel tube and plastic probe holder and aresealed.

The choice of probe dimensions has been guided bythe same two resolution criteria as for the nine-sensor

probe. The first criterion is that each sensor lengthshould be not much larger than the size of the smallestturbulent length scale encountered in the flow, namely theKolmogorov microscaleη. From the direct measurementsof turbulent dissipation rate with the nine-sensor probe byBalint et al (1991), which compare well to direct numericalsimulation values of Spalart (1988), the Kolmogorov scaleis 0.192 mm aty+ = 11.2 and Rθ = 2685, for anominally zero-pressure-gradient boundary layer. Thus thesensor length of the 12-sensor probe will be 3.2 timesthe measured Kolmogorov microscale at this location andRθ . For positions further from the wall,η becomes largeras the kinetic energy dissipation rateε becomes smaller,giving better sensor resolution. Obviously the sensors donot resolve the smallest turbulence scales, as is the casefor virtually all turbulence measurements. The secondresolution criterion is that the probe should be able toresolve adequately the velocity gradient field. The measurefor this criteria is the average spacing between sensorcentres over which gradients are estimated. This spacingfor the 12-sensor probe is about 6.3η at y+ = 11.2 andRθ = 2685, just as it was for the nine-sensor probe. Basedon the analysis of Wyngaard (1969) using Pao’s (1965)spectrum and the experimental investigation of Klewickiand Falco (1990), it is expected that measurements ofRMS values ofωz will be attenuated by about 10–15% atRθ = 2100, just as was the case for the nine-sensor probe(Balint et al 1991).

From these considerations it is expected that the 12-sensor probe will be able to resolve all but the smallestturbulent scales in the thick boundary layer used in thepresent study. Considering that the turbulent dissipationrate frequency spectrum peaks at length scaleU/(f d) ≈10, where U is the local mean velocity andd is theapproximate diameter of the probe’s sensing area, this isquite good resolution.

5. The equations governing the operation of theprobe

One of the possibilities to express the effective velocitycooling a sensor is based on Jorgensen’s (1971) definition

U2e = U2

n + k2t U

2t + k2

bU2b (3)

where Ue is the effective cooling velocity,Un is thevelocity component normal to the sensor in the plane ofits supporting prongs,Ut is the component tangential to thesensor in the same plane andUb is the component normal tothe sensor in the plane normal to the plane of the prongs (thebinormal component). The coefficientskb andkt take intoaccount, among other unknown effects, the aerodynamicblockage resulting from the sizes and arrangements of thesensors and prongs. The effective cooling velocity canbe related to the measured voltage by King’s law or bya polynomial fit.

For the coordinate system shown in figure 4, in whichx–y and x–z planes will be denoted as the vertical and

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P Vukoslavcevic and J M Wallace

Figure 4. A schematic diagram of the 12-sensor probe, showing the end view and a side view of one array.

horizontal planes, respectively, the normal, tangential andbinormal components can be expressed as

Un = n1u + n2v + n3w

Ut = t1u + t2v + t3w

Ub = b1u + b2v + b3w (4)

and the constantsnp, tp and bp depend on the sensor’sgeometrical orientation. For example, for a sensor in thevertical plane the normal velocity component is

Un = u cosα − v sinα (5)

so thatn1 = cosα, n2 = − sinα and n3 = 0. Combining(3) and (4), the effective cooling velocity,Ue, is given by

U2e = s0u

2 + s1v2 + s2w

2 + s3uv + s4uw + s5vw (6)

where sp are the combined constants replacing thenp,tp, bp, kt and kb products. Dividing bys0 and takingAq = sq/s0 andU2

es = U2e /s0 we have

U2es = u2 + A1v

2 + A2w2 + A3uv + A4uw + A5vw (7)

or for any sensorj , of arrayi

U2esij

= u2 + A1ijv2 + A2ij

w2 + A3ijuv

+A4ijuw + A5ij

vw. (8)

For a sensor in the vertical (x–y) plane of fig-ure 4

U2esv

= u2 + A1vv2 + A2v

w2 + A3vuv (9)

and for a sensor in the horizontal (x–z) plane

U2esh

= u2 + A1hv2 + A2h

w2 + A4huw (10)

because the coefficientsA4v, A5v

, A3hand A5h

are identi-cally zero. Equations (9) and (10) can be re-arranged sothat

U2esv

=(

u + A3v

2v

)2

+ A2vw2 +

(A1v − A2

3v

4

)v2 (11)

U2esh

=(

u + A4h

2w

)2

+ A1hv2 +

(A2h − A2

4h

4

)w2. (12)

Comparing equation (11) to the analogous expressionbased on effective anglesU2

ev= (u cosαe + v sinαe)

2 +kbw

2, analysed by Vukoslavcevic et al (1991), it followsthat the last terms in equations (11) and (12) shouldbe zero for sensors that are exactly in the horizontalor vertical planes, respectively. For sensors that arenearly in the horizontal or vertical plane these termsas well as the constantsA4v

, A5v, A3h

and A5hshould

be close to zero. In that case the expression forthe effective cooling velocity, for any sensor close tothe vertical plane, can be written in the general formas

U2esij

= (u + kij1v)2 + kij2w2 + εij (13)

whereεij = kij3v

2 + kij4uw + kij5vw. (14)

For a sensor close to, but not exactly in the horizontalplane, similar equations are obtained by exchangingv andw in equations (13) and (14). The coefficientskijk aresimple arithmetic combinations of theAkij

coefficients from(8).

Equation (13) for the effective cooling velocity ofa sensor in the vertical plane is similar to the nine-sensor probe expression, but it incorporates the terms inequation (14) that were not previously accounted for. Itis valid for any sensor oriented nearly in the verticalplane (or in the horizontal plane ifv is exchanged forw).This is a great advantage compared to the approach usedfor the nine-sensor probe by Vukoslavcevic et al (1991),in which any misalignment of the sensor from its planeresults in a significant deviation from the assumed coolingexpression.

The effective cooling velocityUesijof the 12-sensor

probe can be defined in terms of the velocity componentsu0, v0 and w0 at the probe centrec0 (see figure 4), theirgradients in the plane normal to the probe axis passingthrough the sensor centres and the distances of each sensor’scentre from the probe centre. Expanding the velocitycomponentsu0, v0 andw0 to first order in a Taylor seriesaboutc0 gives the following set of 12 nonlinear algebraicequations in general form, with nine velocity and cross

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A 12-sensor hot-wire probe

stream velocity gradient component unknowns:

U2esij

= [u0 + gupij + kij1(v0 + gvpij )]2

+kij2(w0 + gwpij )2 + εij (15)

where

εij = kij3(v0 + gvpij )2 + kij4(u0 + gupij )(w0 + gwpij )

+kij5(v0 + gvpij )(w0 + gwpij ) (16)

and the velocity increments from the probe to the sensorscentres are

gupij = cij

∂u

∂z+ dij

∂u

∂y

with u replaced byv and w for gvpij and gwpij . Theconstantscij and dij are thez or y coordinates of theij th sensor centre with respect to the probe centre. Asfor equations (13) and (14),v is replaced byw for j = 2and 4 in (15) and (16).

The effective cooling velocities can also be definedin terms of the velocity componentsu0i , v0i and w0i atthe centre of theith array and the corresponding velocityincrements from that array centre to the sensor centres:

U2esij

= [u0i + guaij + kij1(v0i + gvaij )]2

+kij2(w0i + gwaij )2 + εij (17)

where

εij = kij3(v0i + gvaij )2 + kij4(u0i + guaij )(w0i + gwaij )

+kij5(v0i + gvaij )(w0i + gwaij ) (18)

and

gu(v, w)aij = eij

∂u(v, w)

∂z+ fij

∂u(v, w)

∂y. (19)

The constantseij andfij are thez andy coordinates of thesensors centres with respect to the array centres. It is clearthat eij < cij , fij < dij and, therefore,

gu(v, w)aij � gu(v, w)pij . (20)

Using this formulation, a method of successive iteration,which clearly illustrates the character of the equations andlimitations of the probe, can be applied to solve equation(17). In the first iteration the velocity increments from thearray centres to the sensor centres,gu(v, w)aij , as wellas εij can be taken equal to zero, giving three sets offour equations with three velocity components at each arraycentre unknown:

U2esi1

= (u0i + ki11v0i )2 + ki12w

20i

U2esi2

= (u0i + ki21w0i )2 + ki22v

20i

U2esi3

= (u0i + ki31v0i )2 + ki32w

20i

U2esi4

= (u0i + ki41w0i )2 + ki42v

20i . (21)

These equations can be solved for the three velocitycomponents at the centre of each array, in the first iteration.The first iterative approximation of the velocity gradientscan then be found as the velocity differences betweenarrays divided by the corresponding distances between thearray centres, and the small quantitiesgu(v, w)aij andεij (expressions (19) and (18)) determined. The velocity

components in the second iteration then can be obtainedby solving the set of equations (17), and the velocitygradients in the second iteration can be calculated bydividing the velocity component differences between arraysby corresponding distances. The calculation should proceeduntil the differences between two successive gradient orvelocity iterations become negligible. The method hasbeen found to converge very quickly, requiring only a fewiterations.

It is obvious that four equations are available to findthree unknowns at each array. The guideline to chose thebest three out of four is given in section 3. Combiningthe second and fourth equation with the first or thirdrespectively, and bearing in mind thatU2

esikis related to

Eik through King’s law or a polynomial fit, two functionsof the v0i velocity component can be obtained:

F1i (v0i , Eij ) = 0 or Fu(v0i , Eij ) = 0F3i (v0i , Eij ) = 0 or Fl(v0i , Eij ) = 0. (22)

Here the subscripts 1 oru are used to indicate when thisnumbered sensor is used, and the subscripts 3 orl indicatewhen this numbered sensor is used together with the twosensors in the horizontal plane to find thev0i velocitycomponent. From the analysis given in section 3 above, itfollows that, for large positivev0i , the functionF1i will givebetter results, whereas, for large negativev0i , the functionF3i is better. The sign ofv0i can be determined in advanceby examining the value ofF1i or F3i for v0i = 0. If v0i < 0thenF1i (0, Eij ) < 0 andF3i (0, Eij ) > 0 and vice versa. Asimilar combination of the first and third equations with thesecond and fourth, respectively, can be made for the caseof a largew0i value. If v0i andw0i are small, well insidethe range of uniqueness, then any combination of three outof four equations will give the same results. A numericalalgorithm and corresponding software have been developedin order to make an optimal choice of 3× 3 out of 3× 4equations for each array and iteration.

6. The probe calibration method

The constants that need to be defined areAkij, k = 1, 5,

i = 1, 3 andj = 1, 4, from equation (8), as well asthe constants that will appear in the expressions relatingthe effective cooling velocitiesUesij

to the correspondingvoltage outputEij . Different relations have been examinedfor the latter (Petrovic 1996). One convenient choice is touse a fourth-order polynomial fit

U2sij =

r=4∑r=0

BijrErij . (23)

On combining this with equation (8), it follows that

Bij0 + Bij1Eij + Bij2E2ij + Bij3E

3ij + Bij4E

4ij

= u2ij + Aij1v

2ij + Aij2w

2ij + Aij3uij vij + Aij4uijwij

+Aij5vijwij . (24)

By placing the probe in a nominally irrotational calibrationjet flow and pitching and yawing it, an optimal numbern

of different flow realizations,uij = u0n, vij = v0n, wij =w0n, can be induced. These known flow realizations and

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P Vukoslavcevic and J M Wallace

Figure 5. Induced and measured velocity and vorticity components during pitch variation (a), (b) and (c); and simultaneouspitch and yaw variation (d ), (e) and (f ): (♦), induced values; (+), 12-sensor probe; and (�), nine-sensor probe.

measured voltages yield a corresponding number of linearequations (24) withAkij

and Bkijcoefficients unknown.

The number of equations should, of course, be equalto or greater than the number of unknown calibrationconstants for a sensor. In this case there are ten calibrationconstants for each sensor, requiring at least ten differentflow realizations for the 12-sensor probe.

The calibration coefficients should be determined forflow conditions that are as close as possible to the turbulentflow conditions. The probe was pitched and yawed at

anglesϕ and ϑ in an irrotational air jet core stream atvarying speeds. The speed was varied around the meanvelocity that was expected at a given distance from thewall in the turbulent boundary layer. The pitch and yawrange was±20◦ in steps of 5◦, at each constant meanspeed, inducing variousw and v values in the rangevor w = ±0.34u. In this way 45 differentu, v, w

combinations were induced, giving 45 linear equations.Using a least squares fit these equations were solved forthe ten unknown coefficients for each sensor, giving a

1456

A 12-sensor hot-wire probe

set of 120 calibration coefficients. With the knowledgethat calibration coefficients are affected by the effectivecooling velocity when it varies over a wide range (Bruunand Tropea 1980), different sets of calibration coefficientswere obtained for a set of mean velocity values that wereexpected at different locations in the turbulent boundarylayer. One optimal choice of calibration coefficients sets,especially for lower speed, has been the subject of a detailedinvestigation (Petrovic 1996).

7. Testing the accuracy of the probe

As was done for the nine-sensor probe (Vukoslavcevic et al1991), the probe was tested in order to evaluate the velocitycomponent errors as well as the spurious velocity gradientsand spurious vorticity components measured by the probein irrotational flow, in which the latter should, in principlebe zero. In fact, this test is a good measure of the accuracyof the measuring system. The measured values, shown infigure 5, are those obtained using measured voltages andcalibration coefficients found as described in the previoussection. The probe was tested for flow conditions coveringcombinations ofu, v and w that are expected in theturbulent boundary layer. Figures 5(a)–(c) show the resultsobtained for a yaw angle of zero(w = 0) and differentpitch angles in the range±20◦ with U∞ = 1.43 m s−1. Infigures 5(d)–(f ), the results for yaw angles equal to pitchangles in the range±20◦ are presented for the sameU∞.Similar results are obtained for zero pitch and variable yawangle as well as for pitch angles equal to negative yawangles.

It is evident that the velocity values measured by the12-sensor probe match the induced values within about0.7%. To compare the relative accuracy of the 12- and nine-sensor probes the signals from sensor 3 of each array wereneglected, transforming the 12-sensor probe into a nine-sensor one. The corresponding data are shown togetherwith 12-sensor probe data in the same figures. For pitchand yaw angles smaller than 15◦, the results agree well,but above 15◦ an error of up to 5% appears in figures 5(a)and (b) for the nine-sensor probe. Moreover, there aresome pitch and yaw angles for which the nine-sensor probealgorithm does not converge at all, such asϕ = ϑ = −15◦

and ϕ = ϑ = −20◦ in figures 5(d)–(f ). This is notsurprising because this is the most critical range, in whichthe induced cross stream components of velocity are of theorder of 0.5u, corresponding to a cone angle of attack ofabout 28◦. This is well outside the range of uniquenessof the nine-sensor probe (about 17◦ at this speed). Similarresults are obtained neglecting the signals from sensor 3 ofeach array.

The superior performance of the 12-sensor probecompared to the nine-sensor probe (with the signal fromsensor 3 neglected) is also clearly seen from the slopes ofF1(v) and F3(v) shown in figure 6. TheF1(v) curve, forthe case of an induced value ofv = −0.240 m s−1, has twointersections; one of them atv = −0.232 m s−1, giving thephysical solution, whereas the other is atv = −0.5 m s−1.The solution is obviously not unique. In contrast toF1(v),the F3(v) curve has only one intersection, giving a unique

Figure 6. The function F (V ) for different V values: F1(V ),broken lines and F3(V ), full lines.

and accurate solution. On increasing the induced valueof v up to −0.463 m s−1, the F1(v) curve becomes moretangential to the abscissa, giving no intersections, whereasF3(v) gives a unique and accurate solution.

The spurious streamwise components of vorticity�x

measured by the 12-sensor probe in the irrotational flowduring calibration (figures 5(c) and (f )) were under 10 s−1,except for one case(ϕ = ϑ = 15◦), in which it was18 s−1. This corresponds to an accuracy of 0.7% invelocity measurements. That there are simultaneous errorsof +0.7% and −0.7% at two arrays, with a spacing of1.14 mm, results in a gradient error of 18 s−1. Themaximum nine-sensor probe error was of the same order butoccurred much more often, with some cases not convergingat all, such asϕ = ϑ = −20◦, ϕ = ϑ = −15◦,ϕ = ϑ = 20◦. Thus it is obvious that a small velocityerror corresponds to a significant spurious vorticity error.

Figure 7 shows the probability density distribution ofthe spurious vorticity components measured in the core ofthe calibration jet with the 12-sensor probe. The widthof this distribution is another measure of the noise ofthe entire measuring system. The stream-wise vorticitycomponent has rare occurrences of spurious values of about25 s−1, with a probability of 98.74% of being in the rangeof ±15 s−1. Similar results are obtained simulating thenine-sensor probe by neglecting the signal from sensor 1or sensor 3 of each array. The maximum spurious valuewas about 30 s−1, with a probability of 95.65% of beingwithin the range of±15 s−1. The spurious RMS value of�x is 6.3 s−1 for the 12-sensor probe and 7.5 s−1 for thenine-sensor one. These values indicate a high signal-to-noise ratio when compared to peak values of�x of about±1000 s−1 and RMS�x values of about 200 s−1 near thewall in the boundary layer. The maximum spurious valueof 90 s−1 that Balintet al (1991) obtained is reduced by afactor of three in this experiment due to improvements inprobe design and calibration procedures.

8. The probe’s performance in a boundary layer

In order to compare the new 12-sensor probe performanceto the performance of the nine-sensor probe, data were

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P Vukoslavcevic and J M Wallace

Figure 7. The probability density function of thestreamwise vorticity component; (+), the 12-sensor probeand (�), the nine-sensor probe.

taken at several positions near the wall in a low-speedopen return wind tunnel which was designed to create athick turbulent boundary layer. The free stream speed inthe tunnel can be varied over a 1–7 m s−1 range with afree-stream turbulence level of about 0.5%.

The 12 sensors of the probe were heated by constanttemperature anemometer circuits built by AA lab systems.The frequency response of the system for the velocityrange studied was about 4000 Hz, well enough over thehighest frequency at which there is any significant energyin this flow. The data were amplified and digitized witha 12-bit Data Translation simultaneous sample and holdAD converter. As was described above, the 12-sensorprobe was treated as a nine-sensor probe here also bynot using data from sensor 1 or 3 of each array. Inthis way a simultaneous analysis of the same data in 12-and nine-sensor probe configurations was possible and astraightforward comparison of the two probes under thesame flow and experimental conditions could be achieved.Additional to this, the results are compared to the previousmeasurements with the nine-sensor probe of Balintet al(1991). The free stream velocity was 3.47 m s−1 which,at room temperature of 20◦C, gives a friction velocity ofuτ = 0.135 m s−1 at the measurement station as determinedby the Clauser method.

One of the characteristics that demonstrates the probeperformance is the number of non-converged time stepsin the data. As discussed in section 2, when the flowconditions approach the critical value two problems appear:the accuracy is decreased and no convergence for highv/u or w/u values can occur. The number of these non-converged time steps is a good indication of the flow beingclose to the critical condition and, therefore, one indicationof measurement accuracy. These non-converged data arelost, influencing the measured statistical properties of theturbulence. Additionally, as the critical point is approachedthe accuracy decreases and the measurement error for theconverged data is largest. Figure 8 shows the percentage ofnon-converged data from the total data sample at differentlocations from the wall, for the 12- and nine-sensor probes.The percentage of non-converged data for the 12-sensor

Figure 8. Percentage non-converged data (NCD) atdifferent locations from the wall: (+), the 12-sensor probeand (�), the nine-sensor probe.

Figure 9. The instantaneous U -velocity component:(——), 12-sensor probe and (�), nine-sensor probe.

probe goes from 5% aty+ = 14.2 to 2.4% aty+ = 18.7and approaches zero further from the wall. In the case ofthe nine-sensor probe the percentage of non-converged datais about three times larger.

Figure 9 presents aU -velocity component short timeseries of data from the near-wall region measured with 12-and nine-sensor probes. Although there are significant gapsin the data because of non-convergence with the nine-sensorprobe, the nine-sensor converged values compare well tothe 12-sensor values. The same conclusion applies to thev

andw velocity components except for rare occasions. Thisdemonstrates a good characteristic of the nine-sensor probe;if the data converge, they generally converge to the correctsolutions.

To examine the influence of non-converged data onthe measurement accuracy, several statistical propertiesare compared for the two probes. Figures 10 and 11show the mean velocity and mean velocity gradient∂U/∂y

distributions obtained both from the 12- and from the nine-sensor probes for the boundary layer with the distance fromthe wall normalized by the friction velocity,uτ , and thekinematic viscosity,v. The mean values of Balintet al(1991) are also shown. The agreement of the mean velocity

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Figure 10. The mean velocity at the geometric centre ofthe probe: (——), Coles (1962); (– – –), Spalding (1961);(+), 12-sensor probe; (�), nine-sensor probe; and (×),Balint et al (1991).

Figure 11. The mean velocity gradient: (– – –), Spalding(1961); (+), 12-sensor probe; and (×), Balint et al (1991).

distribution between the 12- and nine-sensor probes (withthe data from sensor 1 or 3 neglected) measurements isgood, and both compared rather well to the measurementsof Balint et al (1991). All the data match the logarithmiclaw given by Coles (1962) and that by Spalding (1961) forthe region between the linear sublayer and the logarithmiclayer. The mean velocity gradients∂U/∂y, obtained byaveraging instantaneous∂U/∂y values, are compared infigure 11 to the derivatives of the Spalding (1961) meanvelocity expressions as well as to the measurements ofBalint et al (1991). Good agreement shows the ability ofthe 12-sensor probe to resolve this instantaneous velocitygradient accurately. The fact that 12- and nine-sensormeasurements do not differ significantly shows that non-converged data from the nine-sensor probe did not influencesignificantly the mean velocity and mean velocity gradientvalues. The same conclusion applies to the RMS values ofthe three fluctuating velocity and vorticity components.

The agreement of the skewness and flatness factordistributions of the velocity components from both the 12-and the nine-sensor probes, with the signal of sensor 3(M9U) or of sensor 1 neglected (M9L), as well as from

Figure 12. Skewness factor distributions: (+), 12-sensorprobe; (∗), M9U; (M), M9L; and (×), Balint et al (1991).

the previous measurements of Balintet al (1991), is quitegood for theu andw components, as seen in figures 12(a)and (c) and figures 13(a) and (c). M9U and M9L valuesare practically the same. This is not the case for thev

component skewness and flatness values, for which theresults differ significantly.

The skewness factor distribution for thev component infigure 12(b) differs for the 12- and nine-sensor probes. Theresults obtained with the nine-sensor probe with the signalfrom sensor 3 neglected (M9U) or sensor 1 neglected (M9L)are also different from each other. This is because of thecritical region being shifted from the high negative valuesto the high positive values of thev component. In the first

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P Vukoslavcevic and J M Wallace

Figure 13. Flatness factor distributions: (+), 12-sensorprobe; (∗) M9U; (M), M9L; and (×), Balint et al (1991).

case, the non-converged data contain high negativev valueswhich make the skewness more positive and vice versa forthe second case. In the case of skewness nearly zero, bothconfigurations of the nine-sensor probe will give resultswith the same accuracy. However, in the case of negativeskewness ofv, it is better to have the critical region wherev is positive and vice versa. In the case presented theskewness is negative and this is why the 12-sensor probeand the nine-sensor probe with sensor 3 operating (M9L)compare well. However, the data do not compare well tothe measurements of Balintet al (1991). It is well knownthat the results ofS(v) are characterized by large differences

Figure 14. Skewness factor distributions of ωx : (+),12-sensor probe; (∗), M9U; (M), M9L; and (×), Balint et al(1991).

for various investigations. Relying on the highly resolvedLDV measurements by Karlsson and Johansson (1988) whoobtained a value of about−0.2 in the region ofy+ = 3–20, crossing over to positive values aty+ ≈ 40, the presentdata look more accurate than those obtained by Balintet al(1991).

The flatness factor for thev component is presented infigure 13(b). The nine-sensor data, for the case in whichthe signal from the lower sensor was neglected (M9U), fallbelow the 12-sensor probe data near the wall. This is, atfirst sight, surprising because the previous measurementswith a nine-sensor probe were much higher. However, ifthe signals from sensor 1 are neglected, then the results(M9L) are similar to these obtained by Balintet al (1991).This is logical because the nine-sensor probe used in thatexperiment was equivalent to the case (M9L) in whichthe signals from sensor 1, of the 12-sensor probe, wereneglected.

The fact that the flatness and skewness factors of theother velocity components are practically the same for the12- and nine-sensor probes shows that the high positive ornegative values ofv are not correlated to high positive ornegative values ofu andw.

The skewness factor distributions ofωx also comparereasonably well for the 12-sensor probe and the nine-sensorprobe simulated by neglecting either wire 1 or 3. It variesfor the 12-sensor probe from 0.12 near the wall to zero aty+ ≈ 40 in figure 14. The results are significantly differentfrom those obtained by Balintet al (1991), in which theskewness is of the order of−0.5 in the near-wall region.Bearing in mind that the skewness of theωx vorticitycomponent should be zero because of the symmetry of flowaround thex axis, the present results appear to be muchmore accurate than those of Balintet al (1991).

In figure 15 the flatness factor of theωx vorticitycomponent is presented for the 12- and both combinationsof nine-sensor probes. These data are close to each other,showing that high values ofv are not strongly correlatedto the high value of the∂v/∂z gradient. The resultsare above the Gaussian value of 3 for random functions,showing the intermittent character of vorticity above the

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Figure 15. Flatness factor distributions ωx : (+), 12-sensorprobe; (∗), M9U; (M), M9L; and (×), Balint et al (1991).

buffer layer, but the results are slightly lower than theprevious measurements by Balintet al (1991).

9. Conclusion

This paper describes a small 12-sensor hot-wire probecapable of simultaneous measurement of the velocity andvorticity vectors in turbulent flows. The probe is animprovement over nine-sensor probes. The advantage ofa fourth sensor in each array is clearly demonstrated forthe case in which the ratio of cross stream to streamwisevelocity components is greater than 0.3. The dimensionsof the probe are about the same as the previous nine-sensorprobe of Vukoslavcevic et al (1991), and it is thus ableto resolve the small scales in turbulent flows to the sameextent.

Tests of the probe in an irrotational flow withu, v

andw components similar to those that occur in turbulentflows showed that this probe is capable of measuring theu,v andw components in the range ofu > 0.6 m s−1 with amaximum absolute error of about 0.01u. This error resultsin maximum vorticity component errors of about 10% of theaverage RMS vorticity fluctuations in the near-wall regionof a turbulent boundary layer. This was the average errorfor the nine-sensor probe, but is the maximum error for the12-sensor probe.

Some of the results obtained with the nine-sensor probe,that were not consistent with the constraints of symmetry,are now consistent for measurements with the 12-sensorprobe. For measuring mean and RMS values, both the12- and nine-sensor probes have acceptable accuracy.However, for some higher moments the 12-sensor probeis much more accurate, especially in the flows with a high

ratio of v/u or w/u. It appears that the upper value ofthis ratio for the nine-sensor probe is 0.3; for the 12-sensorprobe it is about 0.45. Therefore, the 12-sensor probe isbetter able to measure statistical properties of the boundarylayer velocity and vorticity fields, especially in the near-wall region in which the nine-sensor probe does not performas well.

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