GB2142439A

GB2142439A - Vorticity measuring apparatus

Info

Publication number: GB2142439A
Application number: GB08412629A
Authority: GB
Inventors: Michael Missenden Freestone
Original assignee: City University of London
Current assignee: City University of London
Priority date: 1983-05-24
Filing date: 1984-05-17
Publication date: 1985-01-16
Also published as: GB2142439B; GB8412629D0

Abstract

Vorticity measuring means including an array of two-tube yawmeter probes 2,3; 4,5; 6,7; 8,1 arranged whereby the differences in pressure registered by the two tubes of each probe are combined to produce an output responsive primarily to the vorticity component along the longitudinal axis of the array. The vorticity measuring means may include three arrays of two-tube yawmeter probes arranged with the probes of respective arrays inclined with respect to the x,y and z axes each array producing an output responsive primarily to the vorticity component along the axis of the particular array or to produce the vorticity vector from the vorticity components produced. The array may be moved across an air flow. <IMAGE>

Description

SPECIFICATION Improvements in and relating to vorticity measuring means This invention relates to vorticity measuring means.

The vorticity vector 4, which is related to the velocity vector V by -= V x V (1) is a fundamental variable in fluid flows and it is an object of the present invention to provide an instrument which permits either the vector 5 or one or more of its components to be determined by direct measurement.

Throughout the specification the word vorticity will be generally used to mean the longitudinal, or streamwise component of vorticity.

Vorticity is twice angular velocity. This is a helpful relationship because it is easy to grasp the physical significance of angular velocity, which is simply rate of spin. The units for vorticity and angular velocity are the same, namely rad/s. It is natural to think of a wing vane instrument which is free to spin about its longitudinal axis, responding to vorticity and spinning round ideally at a rate equal to half the vorticity. In order to provide an instrument which could similarly respond to spin in the flow a pressure probe known as a Conrad yawmeter was considered.In conditions where flow direction varies mainly in one plane, it is known to provide such a yawmeter consisting of two chamfered tubes with provision for their rotation in the plane in which they lie, to determine wind speed and direction of flow in the one plane; but this instrument fails completely to give any indication concerning vorticity.

The feature of the Conrad probe is that the difference in the pressures registered by the two tubes is almost proportional to the angle (P) the local oncoming stream makes with the horizontal plane of symmetry of the Conrad probe.

This approximate proportionality has been investigated for a variety of two-tube geometries, and seems to be well established for angles of between ± 10". In particularthe pressure difference is nearly independent of the angle of pitch a, at least in the range + 12". In Pressure probe methods for determining wind speed and flow direction by D. W. Bryer and R. C. Pankhurst (National Physical Laboratory H. M. Stationery Office 1971) information is provided which would help suitable geometries of two-tube probes to be selected so as to ensure as large ranges as possible for the near constancy of proportionality with P and the independence of a.

For the purpose of the present invention, this characteristic behaviour of the two-tube probe is utilised to show how a number of two-tube probes may be arranged so that their pressure differences are, as an ensemble, sensitive to the vorticity component of the flow about the probe axis.

Thus, according to one aspect of the present invention there is provided vorticity measuring means including an array of two-tube yawmeter probes arranged whereby the differences in pressure registered by the two tubes of each probe are combined to produce an output responsive primarily to the vorticity component along the longitudianl axis of the array.

According to another aspect of the invention there is provided vorticity measuring means including three arrays of two-tube yawmeter probes arranged with the probes of respective arrays aligned with the x, y and z axes whereby the differences in pressure registered by the two tubes of each probe of an array are combined to produce an output responsive primarily to the vorticity component along the axis of the particular array or to produce the vorticity vector (5) from the vorticity components produced.

The invention will now be described by way of example only with particular reference to the accompanying drawings wherein: Figure la is a schematic view of a typical two-tube probe; Figure lb is a plan view of the probe of Figure 1a; Figure 2 is a schematic view of an array of two-tube probes of the vorticity measuring means of the present invention; Figure 3 is a diagram of the notation and axis system for analysis of the array of Figure 2; Figure 4 shows pitot and static tubes in combination with two tube yawmeters to obviate the need to use overall mean pressure from the yawmeters; Figure 5a is a plan view of means for testing an aerofoil section in a wind tunnel using the vorticity measuring means of the present invention;; Figure 5b is a side view of the testing means of Figure 5a with the aerofoil section mounted vertically in a low speed wind tunnel; Figure 5c is a view upstream of the tunnel; and Figure 5d is a view downstream thereof; Figure 6 illustrates typical geometries for two tube probes to determine the lateral components of vorticity and Figure 7 is a diagrammatic illustration to explain how pressures are averaged by a plenum manifold.

Referring to the drawings, a typical two-tube probe is shown in Figure 1 a and 1 b and for the purposes of the present invention, the essential feature of such a two-tube probe is that the difference in the pressures Pt, Pw, registered by the two tubes is almost proportional to the angle (g) the local oncoming stream makes with the horizontal plane of symmetry S of the probe. In Figures 1 a, 1 b, the flow direction is indicated by the arrow x,the angle of chamfer of the tubes of the probe as 6 and the internal and external diameters of the tubes, by d, D, respectively.

This approximate proportionality appears to be well established for angles of 4, between 1 10 . In particular, the pressure difference (PwV-Pf ) is nearly independent of the angle of pitch, at least in the range 12 The above characteristics of tNo-tube probes viz the near constancy of proportionailty with # and the independence of a is represented by the following equation, in which K is a constant, and typically about 2.0.

where pa is the pressure registered by the more windward tube 2 and p by the more leeward tube 1.

U is the component of the local velocity in the direction of the axis of the probe and the local density is p.

Figure 2 shows an arrangement using four'two-tube' probes 2,3;4,5;6,7;1,8; arranged so that their pressure di-Sçrsnces pw - p, are as an ensemble or array, sensitive to the 'spin component' of the flow about the probe axis.

With such an arrangement the pressure differences are combined so as to yield an output which responds primarily, to the spin or vorticity component. By connecting the eight tubes to two manifolds (not shown), it can be eh3tnfn that the difference between these two manifold pressures is proportional to the desired pressure -ifference. As shown in Figure 2, the instrument is provided with a pitot tube 9 on its axis - so that, by suitab@e calibration, the local axiai velocity may be found using this pitot pressure and the average pressure of the eight surrounding tubes. This effectively yields the value of the term pU2 of equation (2) at the axis of the eight tube probe.

The performance of the meter is analyzed with the aid of the notation and axis system shown in Figure 3.

' .e x-axis is taken to be along the axis Gf the probe, with y and z making up a righthanded Cartesian coordinate system arranged with respect 0 the probe as shown. The plane x = 0 is taken to be the pane of the probe tube ends. The velocity components along the (x,y,z) axes, in the absence of the probe, are denoted by (U - u, v, w). By introducing U in this way, we may take u = O at the origin. It is assumed that the probe is at least roughly aligned with the stream direction so that v/U and wiU will be small quantities, say of the order of e. in practice they may be of the order of 0.1, or even more.

Introducing the probe into the flow will affect the velocity field. However, it is assumed here that the influence of an; pair of tubes of the probe is negligible in the plane x = O at any other pair. For this to be so viii ;-equire Lhe probe and its supporting mechanism to be sufficiently slender.

It will be assumed v and ware not zero at the origin; rather the parameters v0 and w0 are introduced for the values at the origin of these important cross velocity components.

The analysis which follows will take into account spatial variations in the (u,v,w) velocity components over the x = 0 plane in the locality of the probe. For simplicity these variations are only expressed here in terms up to and including the First spatial derivatives since the results of the simpier analysis are adequate for most purposes.

For example the v component in the x = O plane will be written as

It will be seen that using equation (2) the two tubes of the probe with tubes numbered 2 and 3 in Figure 2 will yield a pressure difference given by

i.e.

Similarly considering the other three pairs of tubes: 4,5;6,7;8,1;

Using equations 5 to 8 the difference of "the sum of the odd numbered pressures" and "the sum of the even numbered pressures" is formed.

This yields, to first order,

Or denoting the left hand side by ASp,

where x is the component of vorticity along the x-axis.

If all the terms in equations 5 to 8 are taken into account, an additional term appears, and the exact form of equation (10) is

Thus, if the probe is regarded as a vorticity meter, it is seen that the principal error is the second term on the right hand side of equation (11). For most flows envisaged for experimental investigation the principal error should be relatively small since, if all the velocity derivative terms are regarded as likely to be of the same order, then the error term in relation to the vorticity is smaller by a factor of order E, which although it was stated E could be around 0.1 in practice, it is only likely to achieve this magnitude in strong vortex flows, or if the probe axis is markedly inclined to the local flow direction.

Where greater accuracy is required, or when a measure of the relative error is desired, it is an easy matter to tilt the probe axis slightly and re-measure the pressures. This will have the effect of changing and WO U and U by quite accurately known amounts. If either vO and w0 can be made very small, or a series of measurements taken at different inclinations, then the error will be negligible or capable of estimation. An alternative method of estimating the error, by making measurements at a series of probe rotational positions may be considered. It is emphasized that for most applications these procedures should not be needed.

It is to be noted that in flows where there is a very weak variation of axial velocity in the normal plane, i.e.

du du ay' 3z are very small, then changing the direction of the probe axis will permit measurements to be made of the vorticity component in different directions, so that in principle the actual vorticity vector, and not just one of its components, can be found.

As revealed by equations 10 and 11 the data which may be gathered from the nine-tube probe described, gives rise to a coefficient, referred to as Cpp (probe pressure coefficient), which is predominantly related to the component of vorticity along the probe axis, i.e.

Now K, as has been mentioned, is of the order of 2.0 and is effectively independent of probe size so that for a given flow the value of Cpp found will be proportional to 6, the probe "radius". Clearly this is to be expected since the vorticity is a spatial differential property and its measurement inevitably relies on the spatial extent of any static detector. The following question arises: Is there a useful range of probe sizes, as described by the radius 6, for which Cx may be found in sufficient detail (in terms of variations of Cx resulting from changes in the x,y,z coordinates), and for which the output Cpp is adequately measurable? In practical terms the answer to this question is affirmative, and with ti of the order of 1 mm to 10 mm a wide range of flows may be studied.Before dealing briefly with some further practical questions concerning application of such a vorticity probe, some simple means, referred to briefly earlier, are considered, whereby the pressures required to give A p and 1/2pU2 (in equation 10) may be presented for virtually direct measurement on either liquid or electrical manometers.

Consider the odd numbered tubes, see Figure 2, to be connected by equal lengths of uniform tubing to a manifold plenum, denoted as ODD, and the even numbered tubes similarly to be connected to a manifold plenum, denoted as EVEN.

The pressures in the two plena are considered. Provided the pressure losses in the flow in the tubing and manifold are governed by Poiseuille flow pressure losses in the tubing, and providing the four odd tube probe pressures are not vastly different, nor the four even tube probe pressures vastly different, then it is easy to show that in a steady situation

and

Referring to Figure 7, consider a number of tubes 1 ,2....n of equal length e and bore (internal radius a) kept at different pressures pr, P2...Pn at one of each of their ends (the left hand end of Figure 7) and with their other ends connected into a plenum chamber Ct as shown.

In the example described with reference to Figure 7, p1 etc. would refer to either the four odd numbered tubes, or the four even numbered tubes.

Let the plenum pressure be denoted by pc. Let the mass flow rates down the tubes be denoted by m1, m2 ....,mn with the sign convention that flux from left to right is positive, and flux from right to left is negative.

The fluid viscosity is A and the fluid density is p. Then provided the density, p, of the fluid is sensibly constant in the system, which is certainly so for the applications, and the flow in the tubes is Poiseuille flow, and end effects are neglected, we have

P2 - Pe = s rh2 - p = 5 rn Al When steady conditions are established

Hence in that case, by adding the equations Al we find:

Consequently, in steady conditions the pressure in the plenum Ct (or manifold) is simply the average of the pressures at the open ends of the tubes.

The flow out of or into the probe tube tips in the plane x = 0 will be negligible in its effect on probe pressures. Consequently, referring to equation 10, AP = p000 PEVEN = Ăp. (14) So

is the equation equivalent to equation (10).

The "1/2pU2" term may be found, as suggested earlier, by calibration of the difference between the central and perimeter tubes. This could similarly be accomplished by another manifold but this manifold would affect PODO and PEVEN slightly (in a "calibratable" way) unless very long thin tubes were used.

An alternative method of obtaining pU2 would be to use a combined pitot static tube on the probe axis.

This would mean that there would be no need to use the pressure from the yaw probe tube pairs for any purpose other than yielding AP. Two tubes, one from each owt the pitot and static taps would give, via a calibration, the value of pU2 to the desired accuracy. Figure 4 shows the geometric features for such an arrangement and also a similar one consisting of separate static and pitot tubes.

Pressure differences such as AP and that between the central pitot and the eight tube average may readily be found using standard laboratory methods, such as by using liquid manometers or electrical micromanometers.

For most purposes the flow environment of a liquid or gas may be characterized, so far as it concerns a local probe such as the one described, in terms of one or more of the following parameters (a) Reynolds number, P (b) Mach number U a, where a is the speed of sound at the pitot position in the undisturbed flow.

(c) Turbulence structure at the probe, including space and time scales of the velocity disturbances,and their intensities.

In considering the effects of these parameters use can be made of the known results concerning their influence on two-tube yawmeters and on a pitot tube's pressure, for example, as given in the Bryer and anr reference referred to hereinbefore. Thus it would seem that the calibration constant K would need to be found for the particular Mach number range and possibly for the (rough) Reynolds number range, and also that turbulence structure should have only a slight effect, provided turbulence levels are less than 10 per cent in terms of fluctuating velocity component magnitude relative to mean flow speed.

Moreover it would seem that a particular problem would arise from Mach numbers near unity, because of the enhanced levels of interference between tube pairs resulting from such situations. However with due care and sufficiently detailed investigation one would expect the probe to be capable of calibration even at Mach numbers near unity.

Many detailed effects could be taken and considered, but possibly the three that are most important are those due to (a) imperfect manufacture and/or wear.

(b) Finite response time.

(c) Interference between probe elements.

Imperfect geometric detail of the nine-tube probe might, one would guess, lead to poor results. Thus one would expect that unless the angle 4, see Figure 1 b, was maintained precisely as designed in all the eight perimeter tubes, one would find (i) the calibration constant K would depend on the particular instrument and possibly also (ii) that a non-zero value of Cpp would be measured even in an irrotational (6 = 0) part of a flow.

Nevertheless it is not too difficult to keep + in any instrument consistently within 0.5 deg. of the value desired, and this should certainly be adequate for most purposes.

The response time of the instrument is mainly determined by the tube internal diameters and lengths.

Naturally the pressure registering instruments used in conjunction with the probe are considered to be sufficiently rapid acting so as not to govern the response time.

The response time could fairly readily be sufficiently short, provided relatively short tubulations and a relatively large probe are employed, for pressure fluctuations with frequency components of up to 100 Hz to be accurately conveyed to the pressure registering instruments, and applications to aerodynamic testing of helicopter rotors could be viable.

Experience in testing dynamic models in windtunnelswould be relevant to determining the likely frequency response of any particular probe. In many cases only very low frequency components (up to 0.1 Hz) would be of interest, and these present no difficulty.

In instruments made with 8 as small as 1 mm (say) it is probable that the yaw probe elements would be relatively bulky in relation to the spaces between them. In such cases interference between the flows around the individual yaw elements might be appreciable. However there is reason to believe that even with appreciable interference of this type that calibration procedures would permit the determination of Cx to be made. One can envisage using flows having accurately known vorticityforthis purpose.

Some results of a preliminary nature have been obtained with a probe of the type described which employed four pairs of "two-tube" yawmeters. In this example probe 8 was 5 mm and + = 45 deg. The internal and external tube diameters d, and D, see Figure 1 b, were 0.7 mm and 1.2 mm. In this model the pitot tube on the axis was absent and so the values of 1/2pU2 (see equations 10 and 11) would have had to be found separately (i.e. in different runs under the same test conditions).

As shown in Figure 5a, the vorticity meter was traversed in the wake of a model wing 14 (oo=0 ) on which was mounted a relatively large vortex generator 15, and also in the region downstream of a row of much smaller vortex generators (not shown). The main wind speed of most of these tests was approximately 20 ms- (U:o say).

The track of the tip of the probe in horizontal traverse is indicated by the dotted line T-T. A indicates the vortex resulting directly from the large vortex generator 15 and B is the vortex resulting in the wake of the main wing 14 produced by the surface cross flow induced by vortex A shedding into the wake. The air speed is indicated by arrow A/S in Figure 5b and of the order of 19 m/s. The wing 14 is mounted vertically in a low speed wind tunnel and the angle of incidence I is 16 degrees.

Values of (Cpp) = (defined as)

of up to 0.25 were found from the probe in tracking through the vortex produced by the large vortex generator. Values for the smaller generators were limited to ICppl less than 0.1.

It was found that consistent levels of Cppx as low as 0.01 could be registered and related to particular features (recesses) on the wing surface. In virtually irrotational regions of the flow the output from the probe was consistently much less than 10-2 times its output in moderately strong vortical flow. This gives some indication of the dynamic range of the probe, which is probably of the order of 103.

Vorticity is certainly a fundamentai parameter in at least very many fluid flow situations and a simple instrument capable of giving reliable measurements of it, or, since it is a vector, of one of its components, should gain wide acceptance.

A particular model size and type would be optimum for one range of low speed windtunnel tests, and perhaps a smaller one for work in boundary layers and near wakes. At least a partial optimization would seem to be of value.

Clearly a variety of geometries, some having three Conrad yawmeters; some five, six or more could be considered in this optimization exercise.

A pressure probe permitting determination of the axial component of vorticity has been described together with the exposition of its principles of construction, analysis and use.

Error estimation has been dealt with in principle.

In orderto determine the other two vorticity components, namely

and

7. Vorticity measuring means as claimed in claim 6 wherein the said first tubes are connected by tubing of equal lengths and bore to a first manifold plenum, and said second tubes are connected to a further manifold plenum, the tubes each being of different pressure at the ends remote from the plenum, the pressure at the outputs of the two manifolds being

and

where ODD and EVEN refer to the first and second manifolds respectively, the pressure difference A p being expressed as AP = PooD - PEVEN = Ăp.

giving

where Cx is the vorticity component along the axis axis of the array.

8. Vorticity measuring means as claimed in claim 2 wherein the velocity components along the x,y,z axes are denoted by (U+u) v,w, the velocity component in the x=o plane parallel to they axis being expressed by

the probes of each array including tubes p1 to pn and the difference of the sum of the pressures of the odd numbered tubes and the sum of the pressures of the even numbered tubes yielding

where 5 oc is the vorticity component along the x axis of the array and the components of vorticity along they and z axes being given by

and

9. Vorticity measuring means substantially as hereinbefore described with reference to the accompanying drawings.

**WARNING** end of DESC field may overlap start of CLMS **.

Claims

**WARNING** start of CLMS field may overlap end of DESC **. 7. Vorticity measuring means as claimed in claim 6 wherein the said first tubes are connected by tubing of equal lengths and bore to a first manifold plenum, and said second tubes are connected to a further manifold plenum, the tubes each being of different pressure at the ends remote from the plenum, the pressure at the outputs of the two manifolds being and where ODD and EVEN refer to the first and second manifolds respectively, the pressure difference A p being expressed as AP = PooD - PEVEN = Ăp. giving where Cx is the vorticity component along the axis axis of the array. 8. Vorticity measuring means as claimed in claim 2 wherein the velocity components along the x,y,z axes are denoted by (U+u) v,w, the velocity component in the x=o plane parallel to they axis being expressed by the probes of each array including tubes p1 to pn and the difference of the sum of the pressures of the odd numbered tubes and the sum of the pressures of the even numbered tubes yielding where 5 oc is the vorticity component along the x axis of the array and the components of vorticity along they and z axes being given by and 9. Vorticity measuring means substantially as hereinbefore described with reference to the accompanying drawings. it is possible to devise arrangements of combinations of two-tube yawmeters (which would provide the response to and pitot-static tubes or possibly pitot tubes would be adequate, which would provide the response to au au Oz and respectively. Typical geometries are shown in Figure 6. A pair of probes consisting of tubes 16,17; 18,19; are shown with a pair of pitot tubes 20,21 by way of example. CLAIMS

1. Vorticity measuring means including an array of two-tube yawmeter probes arranged whereby the differences in pressure registered by the two tubes of each probe are combined to produce an output responsive primarily to the vorticity component along the longitudinal axis of the array.

2. Vorticity measuring means including three arrays of two-tube yawmeter probes arranged with the probes of respective arrays inclined with respect to the x,y and z axes whereby the differences in pressure registered by the two tubes of each probe of an array are combined to produce an output responsive primarily to the vorticity component along the axis of the particular array or to produce the vorticity vector (C) from the vorticity components produced.

3. Vorticity measuring means as clairned in claim 1 or 2 wherein the differences in pressure registered by the two tubes of each probe is related to the angle (4,) the local oncoming stream of fluid makes with the horizontal plane of symmetry of the probe as represented by Pw - P@ = K 1/2pU2/a .tan # a where Pw and Pe are the respective pressures registered by the more windward and the more leeward tubes; Ua is the component of the local velocity in the direction of the axis of the probe, and the local density of the fluid is p.

4. Vorticity measuring means as claimed in any preceding claim wherein the probes are arranged such that all pressure differences are, as an ensemble or array sensitive to the "spin component" of the fluid flow about the probe axis, the tubes of the probes being arranged such that the pressure differences are combined to provide an output from the array which responds primarily to the spin or vorticity component.

5. Vorticity measuring means as claimed in claims 3 and wherein a first tube of each probe is connected to one manifold and a second tube of each probe is connected to a second manifold, the difference between the two manifold pressures being proportional to the required pressure difference, a first tube being included in the centre of the array such as to be surrounded by the tubes of the probes whereby by utilising the pitot tube pressure and the average pressure of the tubes surrounding the pitot tube, the value of the term pU2 is obtained.

6. Vorticity measuring means as claimed in claim 2 and 5 wherein the velocity components along the x,y and z axes are denoted by U+u; v and w respectively, the difference of the sum of the pressures of said first tubes and the sum of the pressures of the second tubes yielding to first order:

which by denoting the left hand side by A E: p produces:

where (= is the component of vorticity along the axis of the array.