wind tunnel velocity profiles generated by differentially-spaced flat plates

Journal of Wind Engineeringand Industrial Aerodynamics 80 (1999) 253—262

Wind tunnel velocity profilesgenerated by differentially-spaced flat plates

J.C. Phillips!,",*, N.H. Thomas", R.J. Perkins!,1, P.C.H. Miller#! Department of Applied Mathematics and Theoretical Physics, University of Cambridge,

Silver Street, Cambridge CB3 9EW, UK" FAST Team, School of Chemical Engineering, University of Birmingham,

P.O. Box 363, Birmingham B15 2TT, UK# Silsoe Research Institute, Wrest Park, Silsoe, Beds MK45 4HS, UK

Received 11 April 1997; received in revised form 30 July 1998; accepted 1 October 1998

Abstract

Production of linear shear with low turbulence level in a wind tunnel provides a convenientenvironment for testing the results of computational fluid dynamics simulations and equipmentcalibration. Boundary layer flow over a flat plate at zero incidence provides controlleddeceleration of the approach flow according to plate length, with interacting boundary layersbetween adjoining flat plates merging to provide fully developed duct flow. In this way, an arrayof differentially-spaced flat plates can be used to modify a uniform wind tunnel velocity field toa specified velocity profile. A one-step iterative scheme is offered to determine plate spacings forsimulation of weakly sheared flows, constrained by zero vertical pressure gradient in thedownstream flow (representative of boundary layer conditions). The scheme is tested forrealisation of uniform shear flow (maximum velocity variation $10% of centreline velocity) bywind tunnel simulation, and produces reasonable results, at least comparable with previousstudies. ( 1999 Published by Elsevier Science Ltd. All rights reserved.

Keywords: Wind tunnel; Velocity profile; Flat plates; Iterative scheme; Weak shear

1. Introduction

Wind tunnel experiments remain the primary tool for realistic investigation of theinteraction of atmospheric boundary layer flows with the built environment [1]. Inthis context, modification of uniform upstream flow to provide a boundary layer flowwhich adequately reproduces the full-scale characteristics is usually achieved by

*Corresponding author. Present address: Centre for Environmental and Geophysical Flows, School ofMathematics, University of Bristol, University Walk, Bristol BS8 1TW, UK. E-mail: [email protected] address: Laboratoire de Mecanique des Fluides et d’Acoustique, Ecole Centrale de Lyon, 36avenue Guy de Collongue, BP 163, 69131 Ecully, France.

0167-6105/99/$ — see front matter ( 1999 Published by Elsevier Science Ltd. All rights reserved.PII: S 0 1 6 7 - 6 1 0 5 ( 9 8 ) 0 0 2 0 7 - 4

introducing an array of barriers to artificially thicken the boundary layer withina reasonable streamwise distance, supplemented by mixing devices to tune theresulting mean and turbulence profiles and even turbulence spectra [2—5]. Moreexotically, arrays of jets or fans have been used to achieve the same effect [6,7],although necessarily at greater expense, both as equipment and operational costs.However, there are a number of applications which require the production of linearshear flow or weakly sheared flow with low turbulence levels, for which a versatile yetsimple experimental method is required. These include linear shear flows for experi-mental testing of computational fluid dynamics predictions and flow measurementequipment calibration, and full-scale investigation of droplet dispersion from agricul-tural sprays [8,9], where mean shear controls initial droplet removal [10]. Wedescribe here a flexible, low cost technique for generating mean velocity profiles usingan array of parallel flat plates to introduce the appropriate momentum deficit, andshow its application to the benchmark case of weak uniform shear flow.

The method described here is a variation on that presented by Lloyd [11], whoprovided a design procedure based on uniform fully developed channel flow betweeneach pair of plates in the array. His scheme proceeds in a stepwise manner from thewind tunnel floor, specifying the spacing of each pair of plates to produce channel flowwith velocity matching the required downstream profile as sole design condition, thusyielding the number of plates required. As a variation on this approach, we offer analternative design procedure in terms of frictional losses associated with boundarylayer development along the surfaces of each plate in the array subject to an overallconstraint of zero vertical pressure gradient in the emerging downstream flow.Perhaps the most significant consideration here is that the plates are thin comparedwith the boundary layer thickness in order to minimise discrete wake deficits in thedownstream profile. As a consequence flat plates cannot produce high turbulencelevels in the downstream flow by vortex shedding, limiting their application toproblems with low representative turbulence levels described above. However, thepresent method is demonstrated to be an adequate approximation for weaker shearflows, requiring a smaller number of plates without risking separation of the upstreamapproach flow. In Section 2 we outline the design approach and calculations, inSection 3 we report the experimental method and instrumentation, with Section 4describing the experimental results and finishing with broader discussion and mainconclusions in Sections 5 and 6.

2. Design calculations

The principle is illustrated in the schematic of Fig. 1, our purpose being to deliverthe desired downstream velocity profile using plate spacings within an array of thinplates all of equal length as the design parameter. Neglecting hydrostatic variations, asis usual, and introducing a Fanning friction factor to describe the head loss over eachsurface, the mechanical energy balance over plate n (see Fig. 1) can be written as follows:

p1o#

u212"

p2no

#

u22n2#

C&l((u

1#u

2n)/2)2

2sn

, (1)

254 J.C. Phillips et al./J. Wind Eng. Ind. Aerodyn. 80 (1999) 253–262

Fig. 1. Schematic diagram of the array of differentially-spaced flat plates in the wind tunnel.

where p1is the upstream uniform static pressure, p

2nthe downstream static pressure at

plate n, u1

the upstream uniform mean velocity, u2n

the prescribed downstreamvelocity at plate n, o the fluid density, C

&the friction coefficient for flow over plate n,

l the plate length and snthe spacing between plates n and n#1.

The frictional term in Eq. (1) is based on mean velocity over the plate length((u

1#u

2n)/2), assumed representative for velocity modification from u

1to u

2n. The

friction factor C&is assigned according to the Reynolds number at each plate based on

this mean flow velocity, following established empirical correlations for laminar andturbulent flows [12]. Re-arranging explicitly in terms of plate spacings

sn"C

p1!p

2no

#

u21!u2

2n2 D

~1 C&l(u

1#u

2n)2

8. (2)

For present purposes we regard u1

as prescribed uniform approach flow, u2n

asprescribed (uniformly-sheared) exit flow, p

1as prescribed uniform entry pressure,

p2n

as unknown exit pressure, with sn, the plate spacing, as solution variable. Clearly

we require additional closure constraint to specify p2n

, here being the requirement ofzero vertical pressure gradient in the downstream flow. For weak shear flows con-sidered here, we used the mean downstream pressure defined according topN2"+p

2n/n as an estimate of constant downstream pressure.

An iterative argument was adopted, described in outline as follows. Starting withn plates spaced equally across the control volume (s

nassigned), the downstream exit

pressure p2n

was calculated for each plate using Eq. (2) and the mean determined. Thiswas substituted back into Eq. (2) in place of the local value p

2nto specify a new

spacing for each plate in the array based on the prescribed local exit velocity u2n

andmean downstream pressure. Downstream exit pressures were recalculated fromEq. (2), and a new mean and set of spacings determined. From Eq. (2) it can be seenthat plate spacings are simply proportional to the inverse values of pressure anddynamic head differences, and as the latter term is prescribed, the iteration processsimply corresponds to modification of plate spacing by the factor (p

1!p

2n)/(p

1!pN

2),

J.C. Phillips et al./J. Wind Eng. Ind. Aerodyn. 80 (1999) 253–262 255

calculated at each time step. The iteration was repeated until the plate spacingscalculated for consecutive steps varied by less than 0.5%. In weak shear simulations,with shear parameter (maximum velocity variation as a proportion of centrelinevelocity) less than $15% the maximum downstream pressure variation (p

2n/ p

2) at

each iterative step was about 10%, determined at the duct walls.The convergence of the scheme is linear, as indicated by the proportionality of plate

spacing to pressure head shown by Eq. (2). Convergence of the iteration scheme ischaracterised by approximately constant static pressure conditions both upstreamand downstream of the plate array, with changes in dynamic pressure to match theprescribed shear balanced by changes in drag loss due to variation in plate spacing. Inassuming this behaviour, we note the limitation of this technique to simulation ofweakly sheared flows where the upstream static pressure remains uniform across theplate array, and dependence on the number of plates specified. In principle, it ispossible to increase the range of flow shear simulated by using streamlined plates tominimize blockage of the approach flow, maintaining uniform upstream static pres-sure, and by using plate length as a variable to allow greater drag loss.

The calculation scheme was compared with that due to Lloyd [11] for productionof a weak nominally uniform shear, characterised by shear parameter (mean velocitydifference at the tunnel wall expressed as a fraction of the centreline velocity)of $10%. Lloyd’s scheme calculates plate number explicitly, and so was solvedfor a range of shear parameters to provide an estimate of the number of platesrequired for weak shear simulations (Fig. 2). Both schemes were solved numericallyfor 19 plates 305 mm long and 2 mm wide with upstream approach flow velocityof 10 m s~1, yielding plate spacings shown in Fig. 3. The spacings calculatedfor the present iterative scheme show a uniform increase across the wind tunnel,whereas spacings calculated using Lloyd’s method increase at an increasing rate nearthe wind tunnel top. This region is most poorly reproduced in Lloyd’s originaluniform shear simulation [11], with plate spacings showing similar trends at thetunnel top.

3. Experimental method

The experiments were conducted in an 18A square section open-circuit NPL-typewind tunnel (windspeeds up to 15 m s~1), working section length 3 m [13], in whichthe plate array was located just downstream of the entry section. Plate arrays wereconstructed from dural plates 305 mm long and 2 mm thick (Sharpe and Fisher Ltd.)using spacings calculated for the two schemes above (see Fig. 3). A mixing grid (celldimension 11 mm]11 mm, bar diameter 1 mm) was added immediately downstreamof the plate array to allow velocity profile relaxation. Quality of the downstream flowswas gauged by measuring the mean velocity profiles using a Dantec 55P11 singlehot-wire probe in conjunction with 55M01 hot-wire anemometer mounted on a 3-Dstepper motor-driven traverse 1.3 tunnel heights downstream of the plate array. Theprobe was traversed across the tunnel height at a spanwise position corresponding tothe tunnel centreline. At each measuring station, 300 voltage readings were taken,


Fig. 2. The number of plates required to simulate a uniform flow with specified shear parameter (maximumvelocity variation as a proportion of centreline velocity), calculated using Lloyd’s scheme [11].

Fig. 3. Comparison of plate spacings calculated for simulation of weak shear flow (maximum velocityvariation $10% of centreline velocity) using the present iterative scheme and Lloyd’s scheme [11].

using the ASYST data acquisition package sampling at 500 Hz. These were convertedto velocity data within the ASYST package using probe calibrations against pitot-static measurements taken using a Betz manometer.


4. Experimental results

Mean streamwise velocity profiles are shown for the present iterative scheme(Fig. 4) and Lloyd’s design calculation (Fig. 5). The effects of plate wakes are manifestas velocity defects in the downstream flow. Fitting trends in these data by eye suggests

Fig. 4. Mean velocity profile produced by flat plates specified using the present iterative calculation schemeto simulate weak shear flow (maximum velocity variation $10% of centreline velocity).

Fig. 5. Mean velocity profile produced by flat plates specified using Lloyd’s calculation scheme [11] tosimulate weak shear flow (maximum velocity variation $10% of centreline velocity).


production of shear flows with parameter about $9% for both schemes. Addition ofthe mixing grid enabled the downstream velocity profile to satisfactorily relax,although reducing the final shear parameter simulated. This is shown in Fig. 6 for thepresent scheme, with reduction in final shear parameter to $8.6%, determined usinglinear regression. Streamwise turbulence intensity (Ju@2/uN ) was calculated from thevelocity data, shown in Figs. 7 and 8 for the present iterative scheme and Lloyd’scalculation, respectively. For both schemes, the turbulence intensity level produced wasaround 3% over the tunnel depth, broadly in accordance with results obtained by Lloyd[11] (recall introduction). Peak values of 10% at the walls simply indicate the presenceof wall boundary layers in the upstream flow. A smoke tracer (CFT generator, ShellOndina oil) was used to visualise flow through the plate array, shown in Fig. 9 for thepresent scheme. For both calculation schemes with and without mixing grid attach-ment, smoke streams followed approximately horizontal trajectories, indicating ap-proximately zero vertical pressure gradient in the downstream velocity field.

5. Discussion

Use of flat plates for mean velocity profile simulation is only suited to applicationswhich require production of weak shear and low turbulence levels, and production ofuniform shear still provides a useful reference case in wind tunnel experiments. Withinthis framework, a simple calculation scheme is attractive. The scheme offered hereconsists of a single-step iteration which requires the minimum of computation. Giventhe limited 2-D view of flow dynamics adopted here, there will be a range of convergentsolutions for the calculation scheme. For representative weak shear simulation, the

Fig. 6. Mean velocity profile produced by flat plates specified using the present iterative calculation schemeto simulate weak shear flow (maximum velocity variation $10% of centreline velocity) with mixing gridadded.


Fig. 7. Streamwise turbulence intensity profile produced by flat plates specified using the present iterativecalculation scheme to simulate weak shear flow (maximum velocity variation $10% of centreline velocity).

Fig. 8. Streamwise turbulence intensity profile produced by flat plates specified using Lloyd’s calculationscheme [11] to simulate weak shear flow (maximum velocity variation $10% of centreline velocity).

mean downstream static pressure appears to provide a reasonable estimate of theconvergent solution, typically within 10% of the local downstream pressure at anypoint. Modification of plate spacing is subject to maintaining uniform upstreampressure. This condition will not hold for stronger shear, where small plate spacing willlead to blockage of the upstream approach flow. However, use of flat plates as outlined


Fig. 9. Smoke visualisation of flow produced by flat plates specified using the present iterative calculationscheme. Side view of the flat plate array in the wind tunnel with flow direction left to right, showingnegligible deflection of the smoke tracer indicating zero vertical pressure gradient in the downstream flow.

here may prove a useful platform from which to enhance the range of simulatedconditions to higher levels of turbulence and shear. Lloyd [11] added turbulencegenerating barriers to flat plates in an ad hoc manner. We suggest that, with barrierssized to provide desired turbulence characteristics, empirical determination of the Fan-ning friction coefficient for a plate-plus-barrier arrangement will allow direction incorpo-ration into the iterative scheme. Addition of grids to tune turbulence characteristics mayprove more attractive than barriers given detailed treatment in the literature, withincorporation of their additional flow resistance into the friction term as before. However,we again caution the requirement of constant upstream pressure for convergence. Simula-tion of stronger shear may be approached by allowing plate length to vary independentlyof plate spacing, although this is beyond the scope of our preliminary investigation.

Perhaps the biggest weakness of the present iterative scheme is not calculatingthe plate number required for a given shear, although the graphical correlationpresented as Fig. 2 provides a reasonable estimate. Although inferior to Lloyd’sscheme in this respect, the present iterative scheme does explicitly simulate zerovertical pressure gradient (suitable for boundary layer applications) and appears to bemore reliable in uniform weak shear simulation (Fig. 3). The increasing plate spacingnear the tunnel top appears to be characteristic for Lloyd’s design scheme [11]although it is not made clear why this is the case. The effect of plate wakes as velocitydefects in the downstream flow is undesirable, although at only 1.3 tunnel heightsdownstream the measuring station is closer to the array than is standard for wind


tunnel flow experiments [1]. As demonstrated here, any uniform grid will allow thevelocity defects to relax, with grid dimensions effecting down-stream distance overwhich this occurs. Streamlined plates would also minimize this effect.

6. Conclusions

A calculation scheme has been developed to specify the spacings within an array offlat plates to simulate weak shear flows with zero vertical pressure gradient, producingresults comparable to the earlier design calculation of Lloyd [11]. An estimate of platenumber to create a given shear flow is required, provided here graphically in Fig. 2. Thebalance of dynamic head to frictional loss in flow between any pair of plates in the arrayallows this scheme to be extended to flat plates with turbulence generators attached viaan empirically determined friction coefficient. Restriction to weak shear is a conse-quence of requiring uniform upstream static pressure; this may be maintained for morestrongly sheared profiles by introduction of plate length as an independent variable.

Acknowledgements

This work was supported by Agriculture and Foods Research Council (now theBiotechnology and Biological Sciences Research Council) grant LRG 103. We aregrateful to Mark Davidson for useful discussion and technical assistance.

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wind tunnel velocity profiles generated by differentially-spaced flat plates

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