proceedings - topical problems of fluid mechanics 2015 · note how the cavity shapes of inside the...

MECHANICALLY MODULATED NOZZLES

V. Tesař

Institute of Thermomechanics of the Academy of Sciences of the Czech Republic v.v.i., 182 00 Praha, Dolejškova 5, Czech Republic

Abstract

There are applications in which jet-generating nozzles are requested to have variable properties, responding to a mechanical input action. The paper studies nozzles with a needle - a component with tapered end inserted into the nozzle exit channel. Axial movement of the needle varies the cross-section area available for the flow. Author made laboratory and computational investigations of the simplest shape: a sharp-point cone inserted into a sharp-edged, constant-diameter exit channel. Presented examples from accumulated data discuss pressure measurements: overall pressure loss, pressure distribution on the cone surface, and on surfaces of the exit channel.

Keywords: nozzles, needle nozzles, flow separation, internal separation bubbles

1. Introduction Nozzles generate jets by allowing the fluid to leave confined cavities and issue into a more or less open space. There are applications in which it is requested to adjust the nozzle properties – or perhaps to vary or modulate them continuously. The idea of doing this variation by placing into the nozzle a movable component that more or less obstructs (depending on its motion) the nozzle exit channel, Fig. 1, has been known since the early history of technology. As more demands became placed on the hydrodynamic quality of the jet, it was obvious that simple inserted plugging component as shown in the example A in Fig. 2 causes instabilities in the jet and uncertainty of the direction into which the jet is aimed. It was already in Middle Ages when the benefits obtained by addition of the downstream cone were recognised (B in Fig. 2). With small apex angle cone precision of flow rate adjustment was very much increased. Thus the configuration shown in Fig. 1 soon became standard – especially for adjustment of small flow rates such as in dosage of liquid fuels. The correspondingly small-diameter movable component, tapering to a sharp point, received the name “needle”. Such “needle nozzles”, the subject of this paper, have been widespread devices in all laboratories handling fluids. This does not mean that nozzles with cone-shaped downstream part of the movable body are limited to the small scales – Fig. 3 presents two examples of existence of quite large sized adjustable nozzles (those shown in Fig. 3 were intended for flow adjustment in rocket engines). Early "needles" were made with small apex angles of its conical end and this, together with the often very small diameter, has led to the name under which they are known. Nowadays, however, there is a general trend to choosing larger apex angles. The π / 2 case in Fig. 1 is thus rather a rule rather than exception. There are several reasons for this course of developments. (a) One of them is the attempt at minimising the friction component of hydraulic loss in the nozzle flow . This is done by only a small region in which the fluid accelerated to high velocity to flow parallel with surfaces. The friction effect is not welcome because it depends upon viscosity and this, in turn, upon temperature. With friction-type loss, any accidental variation in temperature or fluid composition causes deviation from the adjusted jet flow rate. This is particularly unwelcome with highly viscous and non-newtonian liquids as they are currently increasingly often handled in biomedical fluidics. In fact, already ordinary machine oils exhibit significant dependence of properties on temperature. With

Figure 1: Most needle-type adjustable nozzles are small. This is particularly necessary for fast periodic flow modulation, where the small size means small inertial mass. Together with the sharp point at its end this small size was the reason why this component obtained its name ”needle”.

TOPICAL PROBLEMS OF FLUID MECHANICS 199_______________________________________________________________________

Figure 2: Adjusting or continuously modulating an outflow from the needle nozzle. A - Historically earliest solution, partial plugging of the exit by simply shaped plug. B – Later addition to the plug of a cone which stabilises the flowfield - and also makes possible more precise flowrate adjustment.

Figure 3: Examples of large-scale designs of flow-rate adjustment valves, benefiting from the effect of flow stabilising downstream cone. Note how the cavity shapes of inside the valve bodies keep in the open regime low internal loss cD.

the large apex angle of the cone only a short flowpath is in the region of smallest cross section influenced by the friction loss. (b) Of course, the very thin sharp tips are also avoided because they are rather vulnerable to an accidental damage, e.g. when handled in manufacturing and assembly. (c) Development of precise positioning mechanisms makes it now possible to get fine flow control by small movements of the large apex angle needle. (c) In applications calling for time-varying modulation of the nozzle flow – either a fast sudden change or periodic modulation - the short travelled distances obtained with the large apex angle allow for smaller acceleration forces and therefore higher upper frequency range limit. The above mentioned trend to minimise the friction component of hydraulic loss - to avoid variation of the adjusted flow rate due to variations of viscosity – is also the reason for the choice of sharp edge at the entrance into the exit channel. Rounded or bevelled entrance would, of course, lead to lower hydraulic loss. However, it would make longer the high-velocity flow past walls and hence the friction component of hydraulic loss. With the sharp edge and flow separating from it the adjusted flow rate remains constant, a more important factor in the flow-adjustment devices. In some application the requirement may be not only to vary the available flowpath but also to have the capability of closing it completely. It may be tempting to use the needle valve also for

Figure 4 (above): Simple shape of the nozzles discussed in this paper and the four geometry definition parameters adjusted or varied in the investigations.

Figure 5 (right): Importance of the relative length λ of the constant cross-section exit channel. Most applications cannot accept the separated-flow of case A because the generated jet exhibits unsteadiness. Preference is given for the case B, with re-attachment to the exit channel wall downstream from separation bubble.

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this purpose – and with rounded or bevelled exit channel entrances (or with an inserted deformable ring of soft material as is the case in the bottom nozzle in Fig. 3) this may indeed be done. However, with the 90 deg sharp-edge entrance, the subject of discussion in this paper, such closing contact should be avoided. Due to the inevitably high contact stresses in the tiny areas the contact would easily lead to disfiguring either the needle surface or the entrance edge (or very probably both). The basic sharp-edged geometry and its four parameters are presented in Fig. 4, where there is also the definition of the dimensionless ratios ζ and λ. Despite the dominance of the edge and cone core of the needle-modulated nozzle, other components may play non-negligible role in determining the character of generated jet flow [4, 5]. Figure 5 demonstrates the importance of the relative length λ of the exit channel. The short case A is generally undesirable because the jet it generates is not sufficiently well directed by the flow past the cone (moreover, as will be discussed below, there may be near the needle tip a region of hydrodynamic instability). It is therefore recommended to design the nozzle not only with safe re-attachment of the flow to the exit channel wall, but providing a sufficient length downstream from this location for proper directing of the generated jet flow. Experience has shown that this demands adding to the re-attachment distance an appended length equal to at least two channel diameters d.

2. Features of the flowfield 2.1 Numerical flowfield computations The primary investigation method discussed in this paper was laboratory pressure measurement – both overall pressure drops on the nozzle and the detailed pressure distributions on the internal surfaces for which was provided a quite dense distribution of pressure tap holes. In addition, numerical solutions of flowfield equations were also made. They were mainly intended to obtain an insight into the internal details of the flowfield – considering the simplicity of the geometry, the flow was found quite complex. Typical examples of computed velocity in the central part of the modulated nozzle is presented in Fig. 6. In the other example, Fig. 7, typical pressure distribution is presented. The velocity flowfield has a character that could be expected – the only complicating feature is the recirculation bubble downstream from the entrance into the exit channel, which is also quite predictable because the fluid cannot suddenly change its flow direction at the sharp edge of the entrance. On the other hand, the pressure field in Fig. 7 with several local extremes would be difficult to expect. Especially the isobars that in the location E are parallel to the flowlines. An important advantage for the computation [8, 9] is the axial symmetry of the flowfield. This made possible performing the computations as two-dimensional and hence fast enough for obtaining extensive data about large number of geometric (cone position) and Reynolds number variants. The computations were performed using commercial software Fluent 6.3.26 (positive experience with which was much better than with the newer version, like Fluent 15, which were also tried). The FLUENT computation is based on the finite volumes approach. The triangular discretisation mesh was set up using Gambit 2.4.6 software package. The mesh was unstructured and this allowed its refinements in the course of the computations, applied in those regions where local gradient of velocity magnitude surpassed a limit value that was gradually decreased as the computation progressed. Typically, the computation was begun with ~35 000 triangular elements, with their number gradually increased so that final convergence the number of elements was ~ 120 000. The inflow from atmosphere was modelled by constant velocity boundary condition sufficiently far away from the central critical region. For this purpose, a sector-shaped auxiliary volume of 300 mm radius and (π − β)/2 apex angle from the axis was added to the solution domain at its upstream end. The inlet flow into this extended domain was defined on the 300 mm radius surface of the auxiliary volume. Similarly, a sector-shaped auxiliary volume was also added to the exit end of the domain. On its surface was defined pressure outlet boundary condition. Turbulence was modelled by standard two-equation (k-epsilon) model with the low turbulence Reynolds number regions resolved using the RGN approach. The model was used with the standard values of the model constants, as provided by software supplier. Of course, if there were adjustments of model constants made, they could lead to a better conformity with experimental data. This was, however, not considered necessary because of the role of the computations being mainly to explain qualitatively the character of the flowfield and its details. Generally speaking, the qualitative agreement of computational results with experimental data was very good – with the only exception (to be discussed below) where in the experiment there were quite substantial mutual deviations in the vicinity of the cone tip. These were found only in some data series. The experimental data values in these cases were chaotic there. The strange behaviour was expected as a consequence of flow separation from the cone surface and the chaotic character there was due to instability of the separation.


Figure 6: An example of computed velocity flowfield in the critical part of the modulated nozzle. Typical length of the time-mean separation bubble is not much larger than the diameter d, although for secure attachment the channel is made significantly longer.

Figure 7: Computed pressure field of the same flow as in Fig. 6. Important feature is the local pressure maximum at the tip of the needle, caused by the centripetal collision of flow - a vestigial case of the collision in the case A from Fig.1.

2.2 Local extremes a,b, and c in the flowfield The simple geometry of adjustable nozzles may lead to wrong impression that similarly simple are also the internal flowfields, needing no special attention. Perhaps because of this underestimation, details of needle-nozzle hydrodynamics have remained largely unknown in available literature – despite the old history of this device and its engineering importance. At any rate, references on the subject are far from extensive and mostly remain at rather superficial level. In fact, the flowfield is complicated by rather complex phenomena, especially associated with flow separation, inescapable since the flow separation, after all, is considered desirable for the minimisation of undesirable friction effects. The example of velocity distribution presented in Fig. 6 was computed for the nozzle with the β = π / 2 needle tip taper. The needle is there lifted to the height ζ = 0.335 above the position of cone contact (extrapolated since there should be no real contacts) with the exit channel entrance edge. Dominant feature of the flowfield is the rather thin annular region of recirculating fluid downstream from the channel entrance edge. The pathlines in this region (there are, of course, no streamlines in three-dimensional flowfields) are in Fig. 6 shown as smooth and continuous. It is important to keep in mind that this is only a consequence of the averaging on which is based the computational mathematical model, neglecting the fact that in reality the flowfield is highly unsteady. Apart of the averaged unsteadiness of turbulent eddies, the separation bubble is the region in which are intermittently formed vortices — some annular, others occupying only a part of the circumference of the annulus — that separate quite irregularly and are shed, being carried away with the flow. The pressure field presented in Fig. 7 is more complex than the velocity field, especially at the surface of the conical end part of the needle. Presented in this picture are isobars computed for the same conditions as the velocity field of Fig. 6. Especially difficult to predict beforehand would be the isobars near the centre of the entrance flow, in the region indicated as E. As was already mentioned above, the isobars there are in the first approximation more or less parallel with the pathlines - rather than perpendicular to them as might be expected. What may be also not immediately apparent is the existence of the local pressure minimum a at the cone surface (as indicated in Fig. 7). Another local extreme is the further downstream formed local pressure maximum b near the tip of the needle. It is a consequence of the pressure increase caused by conversion from fluid kinetic energy in the region where the velocity of colliding flows must rapidly decrease. The local pressure extremes vary in an interesting manner with the changes in the axial position of the needle. An example of these changes is presented in the combined Fig. 9, which consists of three diagrams (each at different relative lift ζ) of pressure distribution computed along rather special, sharply bent axis Xax defined in Fig. 8. The pressure difference ∆P = P – Pexit plotted on the vertical axes in Fig. 9 is the difference between the pressure P at a particular position and the pressure at the exit Pexit from the nozzle. To make ∆P, widely varying with the cone lift ζ, mutually comparable in the three cases, the plotted values are non-dimensionalised by being related to the pressure drop ∆Ptotal across the whole nozzle. In all three cases there is clearly visible the local pressure maximum b at the tip of the needle. The conditions there are, however, further complicated by rapid velocity increase (and hence

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Figure 8 (above): Co-ordinate Xax - somewhat unusual because of the sudden direction discontinuity at the cone tip - provides a useful approach to presenting pressure distributions in the critical region of the nozzle. It is used in the three cases presented in Fig. 9.

Figure 9 (right): Variation of the pressure distributions along the axis Xax (plotted according to Fig. 8) caused by the changes of relative cone lift ζ. The minimum a near the entrance into the exit channel - and the local pressure maximum b at the needle tip - become more prominent as the nozzle is closed (i.e. the relative lift ζ decreases).

pressure decrease) downstream from the tip. The local pressure extreme b is therefore an isolated peak. Its relative importance decreases with increasing cone lift ζ. Immediately downstream from the cone tip the velocity increase is so rapid that the positive peak is there followed by the local minimum. Also the importance of this minimum increases with nozzle closure, i.e. with decreasing cone lift ζ. Further downstream there is yet another pressure recovery to a value slightly above zero ∆P. After that, pressure decreases again by simple friction loss in the exit channel. It is obvious that the pressure field complexity is much higher than what might be expected from the inconspicuously looking distribution of velocity in Fig. 6. The distribution of pressure, such like those shown in Fig. 9, and its variation are of large practical engineering importance. Integrating the axial components of local forces determines the force balance on the needle.

3. Experiments: pressure distributions on the cone at a constant lift

3.1 Pressure coefficient cP and area ratio ϕϕϕϕ Since the measured and as well as evaluated pressure differences depend roughly on the square of the flow rate, their values obtained at different needle lifts ζ differ so much in magnitude that they would be difficult to plot in a single diagram – and their mutual comparisons base on graphical presentation would not be particularly convincing. In Fig. 9 the problem was solved by non-dimensionalisation with respect to the overall pressure drop ∆Ptotal. This, however, is not a particularly good choice. A better approach is introduction of the pressure coefficient cP

(1) where v [m3/kg] is the specific volume, for gas as the fluid evaluated as

(2) r [J/kg K] is gas constant, T [K] temperature and Pb [Pa] is barometric pressure, and wr [m/s] is reference velocity evaluated from the mass flow rate οM [kg/s] measured by flowmeter and from the cross-sectional area Fopen = π d2 / 4 of unobstructed exit channel of diameter d (Fig. 4).

οM = Fopen wr / v (3)


Figure 10: Important parameter for evaluation of the conditions in the nozzle is the relative area of flowpath cross section remaining free as the cone is closing the entrance. In data presentation in the present paper the values are related to the approximate value ϕ, which differs from the exact ϕM (evaluated by means of Pappus’s theorem) in a significant measure only at very large relative lifts ζ. The needle lift z (or its relative value ζ) is a parameter of high practical importance because it determines the extent of the flow acceleration inside the nozzle. If there were not the variations in internal nozzle geometry and the inevitable but generally small effect of varying Reynolds number (due to the friction component of the hydraulic loss), the measured pressure values multiplied by square of the flow rate would be constant – a desi-

rable situation since the general target of all investigations in fluid mechanics may be characterised as search for invariants. In the non-dimensionalised presentations, such near-constant values would be obtained by multiplying the pressure coefficient cP by non-dimensionalised area Fmin , the smallest one that remained available around the cone for the fluid flow through the nozzle. The obvious non-dimensionalisation is made by relating this minimal (and hence dominant) area to the unobstructed area Fopen (eq.(3)) ϕM = Fmin / Fopen (4) Presented in Fig. 10 is the dependence of this parameter ϕM on the relative lift ζ ζ ζ ζ , computed for the case of needle with β = π / 2 taper angle. It was evaluated from the Pappus’s theorem. Unfortunately, the resultant exact expression is inconveniently complex, even for a single included angle β value presented in Fig. 10. A general expression that would be valid for all possible β values is even more complex. Considering the fact that the value is in this paper needed merely for visual comparisons of curves in a diagram, it was decided to replace it in practical use by the simple approximation ϕ that

neglects the gradual increase of the angle ϑ (Fig. 10).

( )βζ−βζ=ϕ sin22

sin2 (5)

Presented in this paper are only results obtained for two apex angle values, β = π / 2 and β = π / 3. The corresponding expressions of eq. (5) for these angles are

( )ζ−ζ=ϕ 22 (6)

and

( )2/32 ζ−ζ=ϕ (7)

3.2 Pressure distributions on the cone surface Experimental investigations were made with laboratory models the key parts of which were two bodies movable precisely relative to one another. It was on one hand the cone and on the other hand the body with sharp-entranced cylindrical hole representing the nozzle exit channel. Lather laborious aspect of manufacturing the laboratory models of the modulated nozzles was providing the cone as well as exit channel surfaces with small (0.3 mm) pressure tap holes. Because of the large number of the taps — and because the ferrules for the tubing that was used to connect the taps inside the cone to the manometer could not be made small - the size of the model had to be rather large. The inner diameter of the exit

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Figure 11: Photograph of the central part of the experimental setup of a cone – here with the β = π/2 - axially movable into the entrance of constant-diameter d = 30.11 mm channel. Oversize cone and plate diameters were chosen to correspond to the ideal entrance conditions defined for the computations.

channel was d = 30.11 mm. Of course, the large scale made possible higher precision of the experiments. Moreover, to maintain the overall simplicity of the basic geometry and also to avoid any differences from the simple boundary conditions in the computational approach, the two bodies (cone and exit channel entrance, Fig. 11) were made very large compared to the diameter d. The outer diameter of the plane faceplate was 600 mm, i.e. ~ 20 d. Because of the very laborious fabrication, only two interchangeable cone models were used. The one with apex angle β = π / 2 , shown in Fig. 11, had its largest diameter 340 mm. The other one, of β = π / 3 shown in Figs. 12 and 13 was somewhat smaller, its largest diameter being 125 mm. Material of the central part of the faceplate as well as of both cones was aluminium alloy. The pressure tap holes of 0.3 mm diameter were drilled in small brass cylindrical components made separately and then inserted into the bodies before the final machining of the whole body surface. Using the axial symmetry of the investigated geometry, the taps were arranged in spirals (cf. e.g. Fig. 12) to avoid any possible influencing of a downstream tap by an upstream one. Measurements were made with air. It entered the model from the outside atmosphere and was flowing past the conical surface of the needle into the exit channel. From there the air passed into a large settling chamber and through an orifice flowmeter (replaced by a rotameter for very small flow rates) measuring the air flow rate. The air flow was driven by a variable-speed exhauster at the end of the setup. Pressure tap readings were made with capacitive pressure sensor Barocell model 590D-100T-3Q8-H5X-4D used together with 1450 Electronic Manometer. The stated accuracy of the Barocell set was 0.05% of the reading plus 0.001% of the full scale. Measurement runs were made with both cone

Figure 12: Photograph of the entrance into the exit channel of the model with the more acute, β = π/3 cone. The pressure tap holes of 0.3 mm diameter were drilled in the brass inserts (dark in the photograph) fixed into the aluminium alloy bodies. The holes are distributed along spirals to avoid influencing a downstream tap by an upstream one (and also in order to get smaller axial distances between the taps than could be possible if inserts were in a straight rows).


Figure 13: Photograph of the internal layout of the β = π/3 cone, with Tygon tubes used for pressure reading transfer from the tap holes into the micromanometer.

variant β in the first test series at a number of adjusted lifts, from ζ = 0.05 to ζ = 0.5, where at each ζ value the flow rate could be varied to result in a practically constant Reynolds number. In the second pressure distribution test series the lift ζ was varied continuously and the pressure reading was taken from only one selected pressure tap. In both series the Reynolds numbers could be adjusted from the lowest value Re ≈ 10 .103 to the highest 50 .103. Larger values of the relative lift, ζ > 0.5, were not investigated because in these conditions (cf. Fig. 10) the needle ceases to limit effectively the available cross-sectional area for flow inside the nozzle. 3.3 Measured pressure distributions on 90 deg cone As a typical example of the pressure distribution measurement results, Fig. 14 presents the non-dimensionalized pressure along the surface of the β = π / 2 (= 90 deg) cone obtained at a relatively large air flow rate with different relative cone lifts. The aim was to keep constant Reynolds number throughout all measurement runs, but there were inevitable small deviations because of impossibility to keep constant temperature and barometric pressure. The pressure was measured in 17 pressure tap holes and there were six runs with different relative cone lifts ζ. On the horizontal coordinate in Fig. 14, relative distance from the extrapolated contact point is plotted ξ1C = X1C / d (8) - defined in the left-hand upper corner insert into the picture. It should be noted that coordinate origin X1C = 0 is here different than in Fig. 8. On the vertical coordinate of the diagram are the pressure coefficient values cP (eq. (1)) multiplied by square of the approximate area ratio ϕ2. Using the expression eq.(6), the plotted data are 2cpζ2(2-ζ)2 - non-dimensionalised pressure differences ∆P measured between the interrogated pressure tap on the cone surface and the inlet pressure Pin (cf. Fig. 7). Note that because

Figure 14: Typical pressure distribution data obtained experimentally with the β = π/2 cone at various lift values ζ and more or less constant Reynolds number. In some cases (marked "S") the pressure distribution failed to recover from the near-tip local minimum a to the local maximum b (cf. Fig.9).

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Figure 15: Another set of experimental results obtained by measurement of pressure at the surface of the needle tip cone, similar to previous Fig. 15 but at lower Reynolds number. The failure of pressure to recover towards the cone tip is indicated again by "S".

of the absolute value in the definition eq. (1), the plotted data values in Fig. 14 are all positive. In each curve fitted to the data points there is a distinct peak – and these peaks are connected by a straight line. This corresponds to the local pressure minima a introduced in Fig. 9. The cone model had a sharp point on the nozzle axis. In this location it was not possible to make a pressure tap hole to measure and evaluate the local relative pressure cP ϕ2 in the local pressure maximum b (Fig. 9). The impossibility of placing at the tip a pressure tap is unfortunate, because the measured data in Fig. 14 show in the region of the tip strange deviations from the computational results presented in Fig. 9. These deviating data points are marked "S". Their deviation is of the character of failing to get the expected pressure rise (which, because of the absolute value in the definition eq. (1) is in Fig. 14 the failure to go down). The Reynolds number that was attempted to maintain constant was defined as

ν= /dwRe r (9)

with viscosity ν [m2/s] and the reference velocity wr [m/s], the latter evaluated from the measured flow rate by means of the definition eq. (3). The mean Reynolds number value of the experiment in Fig. 14 was Re = 38.5⋅103 and the standard deviation ± 0.36⋅103 from the mean value. Another example of pressure distribution measurement, very similar but obtained by measurements at much lower Reynolds numbers (mean Reynolds number value Re = 27⋅103) is presented in the next Fig. 15. Deviations form the dependence in the flows near to the cone tip, marked "S", are seen there again – though they are not present in all measurement runs. The most probable explanation of this effect is separation of flow from the cone. Separations in general are processes associated with some chaotic character – and this may explain the irregularity in absence or presence of the deviation "S". In general the separation was in both measurements absent in the configurations with large cone lifts ζ. As the lift was decreased, here as well as in previous Fig. 14, the flow separation did occur - but then it fails to take place in the configurations of extremely small values of the lift ζ. This may be explained by the high velocity at the same Reynolds number and very small slit remaining under the sharp entrance into the exit channel. To test whether there is some regularity in this observation, the measurements at Re = 27⋅103 in Fig. 15 were supplemented by another run at an even smaller relative lif t value ζ = 0.058, practically one half of the previous smallest relative lift ζ = 0.101. Apparently, the centripetal wall jet issuing through the very narrow slit had very high kinetic energy, the inertia of which overcomes the trend to the separating effect. It is also remarkable that in the region marked “m” in Fig. 15 the adverse pressure gradient along the surface is initially small and therefore easy to overcome. It increases rapidly as the flow gets near to the cone tip, which with the larger slit could not take place. 3.4 Universal pressure distribution law for 90 deg cone Proper final result of investigations in Fluid Mechanics should be a formulation of a law of the dependence between the investigated variables – valid as much as possible universally. In the present case, a law of pressure distribution would be of high importance for designers of the modulated nozzles – the more so because such a law formula may be integrated over the cone surface, leading to a universal formula for the axial hydrodynamic force acting on the needle.


Figure 16: Universal-law curve obtained by similarity transformation of the curves from Fig. 15. Strictly speaking, this particular curve is valid only for the indicated Reynolds number (right-hand top corner). However, the influence of Re is weak and in many cases its effect on the curve is smaller than the experimental data scatter.

Very often, such generally valid laws may be found by noting a visually apparent mutual similarity between graphical presentations. This visible similarity may be the basis for formulation of the universal law also in the present case. For example, there is an obvious similarity of the shapes of the plotted curves for different ζ in the above discussed Fig. 15. This means that introductions of a horizontal scaling factor a*, eq. (10), and analogous vertical factor b*, eq.(11), can convert the family of curves from Fig. 15 into a practically single universal curve valid for any lift ζ . 061.0395.1827.0*a 2 +ζ+ζ−= (10)

898.154.2*b +ζ−= (11)

The curves reduced by the application of these factors are presented in Fig. 16. The data taken from experiments exhibit in this similarity transformed presentation only a small scatter - apart, of course, from the positions near the local pressure minimum a (as it was introduced in association with Fig. 9), where there are deviations that were later explained to be a consequence of the irregular flow with separation near the cone tip. The curve in Fig. 16, in order to be applicable both with and without separation, was evaluated from all cases, including those with the separation causing a deformation of the pressure distributions. Universal curves similar to that shown in Fig. 16 were found also for different Reynolds numbers and different angle β.

4. Pressure variation on the cone tip in response to change of lift The large number of pressure taps and manual connecting and disconnecting one-by-one of them by Tygon tubes to the input of micromanometer made the measurement very time consuming. This made a complete investigation of the effect of lift ζ too laborious. To obtain an information about this dependence within a reasonable time, it was necessary to limit the number of interrogated pressure taps. In particular, the measurements could progress faster if the number of taps was limited to a single one - to avoid the step of pressure-line disconnecting. Of particular interest for this experiment were the taps at or near to the cone tip, where the experiments discussed above have revealed the existence of the initially unexpected flow separation. As in the previous Section 3, also here the variable evaluated from the measured pressure differences was the dimensionless complex cP ϕ2. It was plotted there as a function of the relative distance ξ1C that was there the independent variable. Now, in investigation discussed in Section 4, the independent variable as much as possible continuously adjusted was the relative lift ζ. The first example of such results is presented in Fig. 18. It presents the data evaluated from the measurements with β = π/2 cone at a constant Reynolds number Re ~ 27.4 103. As shown in the right-hand top insert picture (Fig. 18), the tap hole from which the measured pressure was delivered into the micromanometer was the one nearest to the cone tip, in 2.5 mm it was introduced in association with Fig. 9). Downstream from this local maximum a the wall jet flow past the cone slows down and, if there were not the flow separation, the pressure would increase at the expense of decreasing kinetic energy of the fluid. This pressure increase - or here in non-dimensionalised expression, the decrease of cP ϕ2 values plotted in Fig. 19 – is presented as the vertical distance between the maxima a upstream and the measured

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Figure 17 (above): Photograph of the β = π/3 cone tip.

The sharp point, made of PMMA (polymethylmetacrylate) is a component that could be exchanged for another alternative one, with pressure tap hole in place of the tip. The version with the hole (however small, of only 0.3 mm diameter) was used

only for the measurements of the cone-tip conditions – otherwise it was not considered sharp enough.

Figure 18 (right): Results of pressure measurements in only one tap hole at very small steps of the relative lift ζ . Selected for this measurement run was the pressure

tap hole nearest to the cone tip of the β = π/2 cone. Also plotted (dashed line without data points) is the straight line of cP ϕ2 values in the pressure minima

locations a (Fig. 9).

Figure 19: The separation phenomena found by pressure measurement in the tip of the sharper, β = π/3 cone. Similar to Fig. 18, the details here became apparent by plotting in the logarithmic co-ordinates.

data points further downstream. Also the measured data points may be fitted by a straight line – but only in Fig. 19 inside the range of large lifts, from ζ =ζ =ζ =ζ = 0.2 to ζ = ζ = ζ = ζ = 0.5. On approaching from above the critical lift value ζζζζcrit = = = = 0.1507 the wall-jet flow separates. In the separated flow, the measured data show considerable chaos. Results of the second example of such measurement runs is presented in Fig. 19. Again, it was a measurement in the single-tap configuration – this time performed with the β = π/3 cone from Fig. 17 (shown there with the inserted sharp tip component – while the experiments now discussed were made with the sharp tip replaced by another component, having in the tip a small pressure tap hole). The behaviour presented in Fig. 19 is actually similar to the one in Fig. 18 – although somewhat less chaotic - but the visual appearance of the diagram is here quite different because of the used plotting in logarithmic coordinates. At the right-hand side there is the large-lift dependence of the pressure at the tip on the cone lift (corresponding to the pressure growth region between ζ = 0.1507 to ζ = 0.5 in Fig. 18). As the relative lift ζ is decreased below this range, the separation with the cone of smaller apex angle is less dramatic – initially, the pressure remains constant. The constancy is at still smaller lift ζ replaced by


Figure 20: Photograph of the entrance part of the exit channel from outside. The ferrules for connecting Tygon tubes lead to pressure taps on internal wall and on the flat face (the latter taps are well visible in Fig. 12). Also shown in this photograph is the manual traverser for investigation of velocity profiles in the gap between the exit channel and the cone.

decrease that could be in Fig. 19 fitted by the power law (with slope 1/3 in the logarithmic co-ordinates). There is then another region of constancy and at extremely small lift ζ ζ ζ ζ yet another sloping region that is in Fig. 19 fitted again by the 1/3 power law.

5. Pressure distributions on the exit channel walls A system of pressure taps analogous to those on the surface of movable cones was made also on the surfaces (the flat face surface and the cylindrical surface of diameter d = 30.11 mm) of the inlet part of the exit channel. As was the case on the cone, also these tap holes were made in precise brass inserts arranged into spirals. The pressure taps on the flat face are shown in Fig. 12 seen from above. The view of the inlet into the exit channel from below is seen in the photograph Fig. 20. Clearly recognisable there with pressure measurement on the cone surface, also here the working fluid was air sucked in from the surrounding atmosphere (at upstream pressure Pin = Pb). Measured pressure on the exit channel surfaces are the ferrules intended for pulling over them 1 mm i.d. Tygon tubes – in the photograph they are not yet in place. The entrance part of the model was rather inaccessible so that some ferrules lead to the tap holes through rather complex drilled cavities. Axial distance between the taps was 3 mm, i.e. 0.0996 d. Also the same distance 3 mm upstream from the sharp entrance edge was the nearest tap hole on the flat face – and 3 mm downstream from the edge was the first tap on the cylindrical surface inside the exit channel. An example of measurement results is presented in Fig. 21. On the horizontal co-ordinate there is plotted the relative axial distance ξξξξ1 - the absolute distance X1 measured from the edge divided by the diameter d. Plotted on the vertical axis is the pressure coefficient defined in (1). As in the above discussed experiments

Figure 21: Example of measured pressure distributions on the exit channel surface. Immediately apparent are the bulges of low pressure in the locations downstream from the sharp entrance edge, where the cross-section available for the flow is reduced by stationary separation bubble. For comparison, data (lower curve) obtained without the cone are also plotted.

210 Prague, February 11-13, 2015_______________________________________________________________________

was below the atmospheric level (pressure decreased from the atmospheric level because of conversion in the kinetic energy), but the values plotted in Fig. 21 are positive because of the absolute value in the definition. The main set of data in Fig. 21 was obtained with the larger β = π /2 apex angle cone positioned so that the cone tip was level with the entrance edge – in other words, the relative lift was ζζζζ = 0.5. For comparison purposes there is also another plotted set, one obtained with the cone removed. Both curves are qualitatively similar. There is a slow growth of the pressure coefficient as the atmospheric air approached the entrance edge (note that the pressure difference ceases to be measurable at a distance 15.06 mm upstream from the edge). Just at the edge, where manufacturing constraints made not possible positioning there a pressure tap, the measured pressure difference raised rapidly and stayed high for a streamwise distance ~ 9 mm (i.e. for the 3 spaces between the pressure taps). Thereafter it begun to decrease, being stabilised finally on the level of developed flow in the exit channel (very slowly gradually rising again because of the friction inside the channel). The axial extent of the high values indicates the length of the separation bubble immediately downstream from the entrance edge. Demonstrated in Fig. 21 is the fact that the cone placed upstream from the inlet into the channel increases the size of the separation bubble.

6. Specific energy drop on the nozzle 6.1. Small model The large models of exit channel diameter d = 30.11 mm, as they are shown in Figs. 11, 12, and 13, were excellent devices for high resolution measurements of pressure distribution on the surfaces. They could be also used for measuring at least some overall nozzle pressure drop values, but there was a distinct limit at the lower end of Reynolds numbers range, where it was not possible to measure precisely the very small pressure differences (at small air flow velocities). It became obvious that to obtain data at lower Re levels it was necessary to use a smaller-scale model in which the same Reynolds number were obtained at higher velocity. Such a model was made and its two main components are shown in the photograph Fig. 22. The basic geometry – i.e. the sharp-pointed cone and the cylindrical exit channel with sharp-edged entrance – were the same as before. The model was made with the larger β = π / 2 apex angle. Internal diameter of the exit channel was d = 7 mm. The model was again designed for operation with atmospheric air sucked by a vacuum pump into the dominant small space between the cone and the exit channel. The exit channel component is shown at the right-hand side of the photograph, in a view oriented in the air flow direction, with clearly visible d = 7 mm diameter in the centre. The exit channel component is fixed inside a cage of 60 mm internal diameter. The cage is provided with a large number of large holes so that its internal pressure is for all practical purposes identical to the atmospheric Pin = Pb. Nevertheless, there was a pressure tap measuring the conditions inside the cage – to the associate ferrule of which was connected one input terminal of the micromanometer. At the left hand side in Fig. 22 is the "needle" component. It was made using a commercially available micrometer distance-measurement instrument (with reading accuracy to 0.01 mm). The b = π/2 apex angle cone was fixed to the axially movable spindle of the micromanometer – while fixed to the body of the micromanometer was the cylinder (Fig. 22) smoothly fitted to the end of the cage. The air flow rate was measured by a rotameter. The length of the 7 mm dia. exit channel in the basic configuration was 28 mm, so that the relative length was λλλλ = 4.0.

Figure 22: Photograph of the two components of the small d = 7 mm model designed for evaluating the overall pressure drop across the nozzle in terms of Euler number values.

6.2. Small model with extensions – large λλλλ There was no particular reason for just this relative length. Already the earliest experiments have shown that the value of the overall pressure drop depends in a rather non-trivial manner on the length of the exit channel – mainly because of the existence of the separation bubble and the dependence of its


configuration on the length of insertion of the needle. The effect was investigated experimentally and has shown an unexpected influence even of very long exit channels. To obtain a configuration suitable for the investigation of the exit channel length influence, the small model was provided with a set of four extension components. These could be attached to the base M (Figs. 23, 24) so that in their alternative combinations exit channels of various lengths – some of them very long indeed (Fig. 25) became available. The extensions, called A, B, C, and D, were made according to the workshop drawing presented in Fig. 24.

Figure 23: Photograph of the d = 7 mm model with extension B. The extensions made possible measurements with various values λ of the exit channel lengths. The working fluid, air, was sucked from atmosphere into the cage (right) and from there enters the exit channel inlet more or less obstructed by the cone (at left, removed from the cage).

Figure 24: Workshop drawings of the extension used in the small, d = 7 mm model for testing of overall Euler number. In their various length combinations these extensions made possible measurements at large relative lengths λ.

Figure 25: An example of assembled d = 7 mm nozzle model with three extensions (A, B and C from Fig. 24) as well as the base M. In the configuration shown here the exit channel length l (cf. Fig. 4) was 145 mm. Note that despite the large holes in the cage walls, there is a pressure tap (and associate ferrule) to which was connected one input terminal of the micromanometer measuring the pressure inside the cage..

6.3. Characterisations of nozzles Measurements of pressure drop ∆P [Pa] across the nozzle models [10], together with the simultaneous measurements of the mass flow rate οM [kg/s], provided the raw data. These have to be later processed – with the aim to identify dependences or quantities characterising individual nozzle configuration. Ideally, the result of the data processing would be an invariant quantity from which a nozzle designer can compute the properties of any nozzle he wishes to design or produce. If this ideal of an invariant quantity is not attainable, then the relaxed aim is to obtain, as a result of the processing, a single universal law of the dependence between the variable (i.e. measured) quantities ζ, λ,ζ, λ,ζ, λ,ζ, λ, and Re.

212 Prague, February 11-13, 2015_______________________________________________________________________

Investigations of nozzles [6, 7] show that their behaviour is more complex than is generally believed. A description taking into account all complexities leads to expressions rather difficult to handle – and in some application of secondary importance uselessly laborious. It is therefore advisable to posses a hierarchy of mathematical models of increasing complexity and select from them just the model suitable for the particular task. The models follow the historical development of nozzle behaviour theories.

(I) The zero order approach is characterisation of the nozzle neglecting the dissipation of energy that takes place inside the nozzle – and considering only the dissipation of the kinetic energy of the generated jet. The real fluid is for this purpose replaced by an idealised loss-less approximation. This description of nozzle behaviour with this fluid assumes that the pressure loss (or, more exactly, specific energy dissipation) is due to the complete dissipation of all kinetic energy of the jet flow inside the space into which the jet issues (by generating vortices that gradually cease to rotate – but this mechanism is outside the interest of the model).

(II) The higher, first-order characterisation takes into account not only the dissipation of the jet kinetic energy, but also the dissipation of some of the fluid energy inside the nozzle. At this level of approximation, the internal dissipation is characterised by a constant quantity. This is the usual present-day approach to the problem. The characterisation quantity, a parameter considered capable of characterising the nozzle, may be introduced in several ways – for example it may be the discharge coefficient or the drag or loss coefficient cD.

(I I I) Of course, real nozzles do not have their coefficients of internal loss constant. Instead, This is because of the presence (and sometimes even dominance) of friction type loss, with the coefficient varying with varying Reynolds number. A useful approach was introduced in [6] and [7]. It expresses the

internal loss effect by converting it into equivalent growth of the displacement thickness *δ in the nozzle exit, which decreases the effective exit cross section. The size of the displacement thickness is a function of Reynolds number. In [6, 7] it is assumed that the thickness growth is governed by the laws of laminar boundary layers. This is a plausible assumption even at quite high Reynolds numbers because the fluid inside the nozzle is accelerated and this favourable pressure gradient suppresses the transition into turbulence. 6.3.(I) Loss-less dissipance Nozzle is characterised by dissipance. This is the proportionality constant between the square of mass flow rate οM [kg/s] and the drop in specific energy ∆e [J/kg]. The dissipated energy is the pressure energy. When the fluid compressibility may be neglected, the specific energy drop ∆e is simply proportional to the pressure drop ∆P ∆e = v ∆P (13) - where v is the specific volume of air. Evaluating the dissipance of a given nozzle is not straightforward; it may necessitate computing the internal flowfield, clearly not possible at the initial design stages when the nozzle details are yet to be decided upon. In the loss-less approach it is easy to evaluate the approximation based on the one-dimensional expression for the flow rate, eq. (3). The velocity of the fluid leaving the nozzle is

open

r FMv

wο= (14)

Specific kinetic energy (per unit of mass) of the flow leaving the nozzle is

22

2

2r M

d

v421

2

wο

π= (15)

- and if this energy is completely dissipated in the space into which the fluid issues,

22

2M

d

v421

e ο

π=∆ (16)

so that compared with the definition of the ideal loss-less nozzle dissipance [1, 2, 3] as the proportionality constant between οM2 and ∆e :

2th MQe ο=∆ (17)

there is

2

2thd

v421

Q

π= (18)


6.3.(II) Dissipance of a nozzle with internal loss The inclusion of internal losses into the account leads to an increase of the dissipance value, which is then defined by an analogy to eq.(17)

∆e = Q οM2 (19) The mutual relation between Qth of eq. (17) and Q of eq.(19) is defined by the Euler number Q = Eu Qth (20) This means a nozzle with the losses taking place inside may be characterised at a particular constant geometry (which here means a fixed, non-movable needle) at the first-order level by its dissipance Q [m2/kg2]:

2

2d

v42

EuQ

π= (21)

The internal losses inside the nozzle are evaluated, as usual, as the proportion cD of the kinetic energy of the flow

2

Wce

2r

D=∆ (22)

The fact that the overall pressure loss (or, more exactly, the specific energy loss) is a sum of two loss components – on one hand the dissipance Qth characterising the total dissipation of kinetic energy of the jet, and on the other hand the specific energy decrease ∆e due to the internal dissipation, is reflected in the expression for dissipance, which may be written as a sum

2

2D

2

2 d

v42

c

d

v421

Q

π+

π= (23)

or, from another point of view – eq. (20) DU c1E += (24)

2

2D

2

2 d

v42

c1

d

v42

EuQ

π+

=

π= (25)

6.3.(III) Nozzle with Re-dependent dissipance Dissipance Q evaluated as described in the previous Section is a very useful characterisation parameter. Examples of its use are demonstrated e.g. in references [6, 7, 16,17]. In computations of states and their changes in fluidic circuits this parameter Q characterises nozzle properties, on the assumption that it is a constant. This constancy may be a plausible assumption for many cases, because the variations of Q with fluid flow rate is quite small and very often indeed negligible. In more or less exceptional situations it may be necessary, however, to apply a yet higher, second-order characterisation. This takes into the account the fact that the internal loss inside the nozzle body is mainly of the frictional character – and, as such, it decreases with increasing Reynolds number. There are two alternative approaches to characterising axisymmetric nozzles at this level. In both cases the parameter of similarity is not Reynolds number Re (as is the case in most other problems of Fluid Mechanics), but Boussinesq number

d/l

ReBo = (26)

6.3. (IIIa) Hagenbach number This approach is based on the empirical finding that there is a universal dependence between parameters Ha – the Hagenbach number – and Bo. The starting point is the well-known Hagen-Poiseuille law of the loss of laminar flow in a pipe according to which (as may be easily derived) Eu = 64 / Bo (27) which is valid for the fully developed flow (i.e. the flow with constant parabolic-shape velocity profiles everywhere along the round channel length l. Real flows cannot be exactly described by eq. (27) because there is to be an additional term in the expression describing the influence of the flow development at the entrance. The longer is the channel, the better approximation the eq. (17) is, but - of course - even if the length is extremely long, the additional loss due to the initial flow development cannot be just forgotten and has to be included into the resultant expression. The first researcher who expressed the behaviour of long channels using this idea was E. Hagenbach in 1860 – so that the additional term may be called Hagenbach number Ha. It is defined as

Ha2Bo/64Eu += (28)

214 Prague, February 11-13, 2015_______________________________________________________________________

Hagenbach's approach stemmed from his interest in capillary viscometers with the typically very long channels - and thus in very low Bo values. At extremely small Bo, Hagenbach was led to consider wrongly Ha to be a constant, Hao - in fact also his value of this asymptotic value was not correct. Author of the present paper found by analysis of very large ensemble of data for simply shaped orifices (no entrance cone) that the asymptote is Hao = 1.125 for very low Bo. At Bo higher than 50 the Ha values decrease with increasing Bo. This decrease was also found in Fig. 26 for data with constant cone position z = 0.5 d (= the "grazing tip" condition). This behaviour is more complex than found for channels without the cone at their entrance, as shown for the line "b" obtained from measurements of a channel with the same l and d without the cone.

Figure 26: Losses in mechanically modulated needle nozzle presented in terms of Hagenbach number, here plotted for a constant cone lift z as a function of Boussinesq number, eq.(26). Note the asymptotic value at left and the family of curves with constant cTL and different values cδ.

6.3. (IIIb) The boundary-layer approach The Hagenbach number approach has the disadvantage of being based purely on empirical finding, without any direct physical approximation. The problem of characterisation at this second-order level with physical justification was solved in [6] and [7] by an approach based on the consideration of boundary layer. The friction inside the nozzle results according to this idea in decreased available area in the exit, because it generated a boundary layer that partially blocks the nozzle cross section. The parameter characterising this effect is the displacement thickness *δ of the boundary layer. Its presence decreases the nominal (open exit channel) area

4d

F2

open

π= (29)

to the smaller ( )

4*2d 2δ−π

.

This decrease of area at the same flow rate inevitably increases the kinetic energy of the generated jet – and increases also the dissipated kinetic energy downstream from the nozzle. The loss-less expression eq. (18) may be used to describe the loss effect due to smaller area

( )

2

2

2

2 *2d

v421

d

v42

EuQ

δ−π=

π= (30)

By comparing the two expressions in eq.(30), the Euler number of the loss caused by the decreased effective nozzle exit area is

( ) 4d/*21Eu −δ−= (31)

The next step is to evaluate the expression governing the variation of the displacement thickness *δ in response to the varied flow rate. References [6, 7] have demonstrated that a good approach to this expression is to assume applicability of the Blasius growth law for laminar layer

Re/1~*δ (32)


The laminar character of the boundary layer is a plausible assumption because the streamwise length of the layer inside the nozzle is inevitably quite short. Moreover, the fluid flow is inside the nozzle accelerated and this favourable pressure gradient suppresses any trend to transition into turbulence (note that even inside the constant-diameter exit channel there is an acceleration because of the growing thickness of the boundary layer). Of course, the Blasius solution is, strictly speaking, valid for laminar boundary layer on a flat plate. In the case now discussed, there is the complicating factor of the transversal curvature of the layer. It is characterised by the nozzle exit diameter d. Let us therefore define a special expression for the layer growth

Re/cd/* T=δ (33)

with introduced coefficient Tc as the constant characterising the growth in a nozzle of a given shape.

This expression eq. (33) has been quite successful in the case of well rounded internal shape of the nozzle, not causing flow separation. In the present problem of characterising the energy drop across the whole nozzle there is the complicating presence of the separation bubble in the inlet into the exit channel of the nozzle. The problem thus caused was solved in [7] by addition of a term that simulates the bubble presence by addition of non-zero entrance thickness iδ . Thus the growth of the laminar boundary layer inside the exit

channel of the nozzle (not varying the cone lift) is described by

Bo/cd/d/* TLi +δ=δ (34)

( ) 4

TLi Bo/c2d/21Eu−

+δ−= (35)

It is useful to introduce coefficient cδ = δi / d which is a constant of a nozzle of invariable geometry. Extensive data accumulated in [7] show that cTL = 0.6853. 6.4. Dissipance of a needle nozzle In the individual parts of previous Sect. 6.3. the expressions defining nozzle characterisation quantities, eq. (16), (25), and (35) were all valid under the assumption of no internal obstruction – i.e. in the case of the needle-modulated nozzle for its fully open state

2opend

v4F

π= .

If the needle is axially translated, the most important resultant change is the variation of the minimum cross section area Fmin , non-dimensionalised according to eq. (4). Since the interest is likely to concentrate on small cone lifts, it may be acceptable to adopt the simplification for area ratio ϕ≈ϕM (36)

- because the difference between these two expressions for area ratio (Fig. 10) becomes significant only at large ζζζζ where the blockage of the exit channel by the cone is small. The dissipation of fluid energy is conversion between kinetic energy of organised motion into heat, i.e. the kinetic energy of chaotic motion of fluid particles. To express the organises kinetic energy necessitates evaluation of the velocity w in the dominant (i.e. smallest) cross section inside the nozzle. From mass conservation

minopenr F/Fw/w = (37)

so that ϕ= /ww r (38)

(I) The loss-less approximation In this simplest characterisation, the dissipated energy is only the kinetic energy of the flow inside the dominant cross section [15, 16]

2

2

min

2

MF

v21

2w

e ο

==∆ (39)

the dissipance [17, 18] is 2

2

2

minth

d

v421

Fv

21

Q

ϕπ=

= (40)

(II) The "constant coefficient" approximation In the invariant geometry nozzle, discussed in Sect. 6.3.(II), the characterisation by dissipance was a simple extension of thQ and the same approach may be applied here:

216 Prague, February 11-13, 2015_______________________________________________________________________

2

2D

2

2U

d

v42

c1

d

v42

EQ

πϕ+

=

ϕπ= (41)

- the non-constancy of DU c1E += in the invariant-geometry cases is usually small, caused by the

Reynolds-number dependent growth of the boundary layer. In the needle nozzle, however, the translation of the needle varies the internal geometry and this – mainly by varying the separation bubble in the channel entrance – causes in general a more pronounced variance of Eu. The extent of these changes is evaluated by processing of the experimental overall loss data to obtain

2red EuEu ϕ= (42)

In the geometry retaining shape similarity the reduced Euler number would be a constant for any cone lift

ζζζζ. With the translation of the needle the evaluated Eu multiplied by 2ϕ may result into the desirable

result – the characterisation curve. Plausibility of this idea is demonstrated in Fig. 27, there the data processed this way indeed show the universal characterisation curve for overall pressure loss across the nozzle

17/12Bo787.1Eu −−ϕ= (43)

Presentation in logarithmic co-ordinates led in Fig. 27 to a straight line, fitted by the least squares regression. The slope -1/17 = -0.0588 is very small, obviously due to the separation on the channel entrance edge in the dominant cross section. This demonstrates that the fluid flow in contact with the wall, generating the friction, is minimised. The data for very small relative cone lifts (note that with the diameter d = 7 mm the relative lift ζ = 0.01 means the absolute width of the slit only 0.05 mm) deviate in Fig 27 from the general law eq. (43). This is obviously a consequence of interaction with the behaviour of the boundary layer which is of comparable thickness. The character of this deviation is apparent from the next Fig. 28, where the reduced Euler number is plotted (for a single nozzle) as a function of relative lift. The region of universality is here seen from to be the flat part of the curve from ζζζζ ~ 0.03 to ζζζζ ~ 0.2 (the main, heavy line). The deviating data in Fig. 27 are obviously away from this region, on the downwards sloping part of the curve at the left-hand side in Fig. 28.

Figure 27: Properties of the needle nozzles. Typical feature of friction-type losses is Euler number variation with varied Reynolds number. For nozzles, this is reflected in the variation of the reduced Euler number

2red EuEu ϕ= with varying

Boussinesq number Bo. The dependence presented here is universal, valid for all relative lengths l/d as long as the relative lift (Fig. 28) is above ζζζζ ~ 0.03 and below ζζζζ ~ 0.2.

Of course, the evaluation of the reduced Euler number according to eq.(42) is based upon one-dimensional approximation and this may be also a reason for some deviations. The dashed fitted line in the top part of the diagram Fig. 28 shows an alternative in which the reduction by multiplying by ϕϕϕϕ2 is applied not to the whole Euler number, but instead only to the drag coefficient cD (simply evaluated by subtracting the 1.0) In Fig. 28 this has led to an almost constant dependence, with very small deviations. This might be an interesting approach – not forgetting, of course, the approximation character of the both approaches. (III) Boundary layer approach Also the total nozzle loss across the nozzle may be analysed using the expression eq.(35) – applying it to evaluation of the reduced Eu ϕ2. The axially shifted cone of the needle is likely to lead to variation of the coefficient cδ (i.e. relative magnitude of the nonzero starting boundary layer thickness simulating the influence of the separation bubble). In contrast to constant-geometry nozzles, for which cδ


Figure 28: Example of characterising the nozzle properties by means of reduced Euler number. Measurement results obtained at quite high and nearly constant Reynolds number are here plotted for two different exit channel lengths l as a function of the relative cone lift ζ.

is practically a constant characterising the nozzle, in the here discussed needle nozzles this coefficient varies with the lift. In fact, as presented in the last Fig. 29, the Reynolds number independence is here only very approximate (Fig. 29 shows a small increase with increasing Bo). As was emphasised above in 6.3.(IIIb), the second-order approach to nozzle characterisation is here – based on in [6, 7, 13] - founded on the assumption of laminar character of the boundary layer. While it has led to interesting and useful results, it is, of course, necessary not to forget that at very high Reynolds numbers the flow may undergo a transition into turbulence. This is the explanation author can provide for the sudden change of the character shown in Fig. 29. The transition causes a sudden raise from the nearly constant value of the coefficient cδ (cf. the lines of cδ constant in Fig. 26).

7. Conclusions Paper discusses adjustable nozzles of the needle type, modulated by mechanical input – axial motion of a needle component, the tapered end of which partly blocks the entry into the exit channel of the nozzle. Laboratory measurements and computational investigations of the simplest shape, a sharp-point cone inserted from upstream into a constant-diameter exit channel, found the pressure filed inside the nozzle more complex than might be expected – with several local pressure extremes. Interest concentrated on pressure distributions on the surface of the cone [11, 14], for which was found universal dependence, based on the similarity transformation. Another important finding, obtained by measurement of the pressure at the cone tip at its different axial positions, was the existence of flow separation at some regimes. Measurements of overall pressure drop across the nozzle identified three approximations at different complexity levels, from the loss-less simplification to an approach based on the study of boundary layer at the wall of the nozzle exit.

Figure 29: Measured overall pressure drop on the needle nozzle with constant cone position z/d = ζ = 0.5 are here shown converted to the equivalent additional inlet boundary layer displacement thickness δi for two different exit channel lengths l. The nozzle properties are expressed based on the boundary-layer model of nozzle properties [7]: the dependence on the Boussinesq number Bo is characterised by the effective initial thickness thickness δi causing the same area blockage in the nozzle exit.

218 Prague, February 11-13, 2015_______________________________________________________________________

Acknowledgments Authors’ research was supported by grant 13-2304S obtained from GAČR. They were also recipients of institutional support RVO: 61388998.

Nomenclature Bo [-] Boussinesq number

cD [-] Drag (or loss) coefficient

cT [-] Coefficient of boundary layer displacement thickness in the nozzle exit

cTL [-] Coefficient of boundary-layer streamwise growth

cδ [-] Coefficient indicating the equivalent inlet boundary layer thickness, simulating the effect of entrance separation bubble

d [m] Nozzle exit diameter

∆e [J/kg] Specific energy drop across the nozzle

Eu [-] Euler number

Eured [-] Reduced Euler number

Eu*red [-] Alternative reduced Euler number Eu*red = cD ϕ ϕ ϕ ϕ2 +1

F [m2] Cross sectional area

Fmin [m2] Dominant (minimum) cross

section area in the flowpath

Ha [-] Hagenbach number

Hao [-] Low Bo Hagenbach factor at Bo< 50

1 [m] Exit channel length

oM [kg/s] Mass flow rate oM = F w / v

∆P [Pa] Pressure drop across the nozzle

Q [m2/kg2] Dissipance, Q = Eu Qth

Qth [m2/kg2] Dissipance of total jet

dissipation

Re [-] Reynolds number

v [m3/kg] Specific volume of fluid

w [m/s] Mean (bulk) velocity

z [m] Needle (cone) lift

β [rad] Cone vertex angle

*δ [m] Boundary layer displacement thickness

iδ [m] Equivalent initial boundary

layer thickness

[-] Relative cone lift = z / d

[m2/s] Viscosity (kinematic)

[-] Area ratio

References [1] Tesař V.: A system of quantities for description of states occurring in fluidic circuits, Proc. of the

VIth „Jablonna Conference on Fluidics, page 16, Moscow, 1976 [2] Tesař V.: Some basic solutions of fluidic circuits involving the quadratic dissipance, Proc. of the

VIth „Jablonna Conference on Fluidics, page 45, Moscow, 1976 [3] Chapters: Tesař V.: Introductory notes on microfluidics, pp. 185-220, Tesař V.: Characterization

of two-terminal devices, pp. 223-253, Tesař V.: Fluidic Circuits, pp. 255-304, in book Microfluidics: History, Theory, and Applications, ed. W. B. Zimmerman, CISM Courses and Lectures No. 466, ISBN-10-3-211-32994-3, Springer-Verlag, Wien - New York, 2006

[4] Tesař V.: Hydrodynamics of an idealised poppet valve, Proc. of 2nd Internat. Symposium on Fluid Control, Measurement, Mechanics and Flow Visualisation FLUCOME '88. Sheffield, Great Britain, September 1988

[5] Kent J.C., Brown G.M.: Nozzle exit flow characteristics for square-edged and rounded inlet geometries, Combustion Science and Technology, Vol. 30, p. 121, 1983

[6] Tesař V.: „Characterisation of subsonic axisymmetric nozzles“, Chemical Engineering Research and Design, Vol. 86, p. 1253, 2008

[7] Tesař V.: Characterisation of inexpensive, simply-shaped nozzles, Chemical Engineering Research and Design Vol. 88, p. 1433, 2010

[8] Zhang S. B., Zhu J. M.: Numerical simulation of adjustable nozzles, Materials Science and Engineering Vol. 52, 072014, 2013


[9] Zhua H., Pan Q., Zhang W., Feng G., Li X.: CFD simulations of flow erosion and flow-induced deformation of needle valve: Effects of operation,structure and fluid parameters, Nuclear Engineering and Design Vol. 273, p. 396, 2014

[10] Tesař V.: Losses in a mechanically modulated nozzle, Proc. of XIth International Conference on Fluidics, Varna, Bulgaria, October 1988

[11] Tesař V.: Surface pressure distribution in a poppet valve with central plug body having rounded tip, Acta Polytechnica – Journal of Advanced Engineering, II, 3, 1991

[12] Tesař V.: Pressure-driven microfluidics, Artech House Publishers, Norwood, MA, USA, 2007

[13] Tesař V.: Frictional losses in nozzles, PNEU-HYDRO ’87, Proc. of 6th Colloquium on Pneumatic and Hydraulics, October 1987, Györ, Hungary

[14] Tesař V.: Pressure on the conical surface of needle valves, Sensors and Actuators A: Physical, Vol. 220, p. 1, 2014

[15] Tesař V.: Effective hydraulic resistance of actuator nozzle generating a periodic jet, Sensors and Actuators A - Physical, Vol. 179, p. 211, 2012

[16] Tesař V., Kordík J.: Two forward-flow regimes in actuator nozzles with large-amplitude pulsation, Sensors & Actuators A: Physical, Vol. 191, p. 34, 2013

[17] Tesař V., Kordík J.: Effective hydraulic resistance of a nozzle in an electrodynamic actuator generating hybrid-synthetic jet – Part I: Data acquisition, Sensors & Actuators: A. Physical, Vol.199 , p. 379, 2013

[18] Tesař V., Kordík J.: Effective hydraulic resistance of a nozzle in an electrodynamic actuator generating hybrid-synthetic jet – Part II: Analysis and Correlations, Sensors & Actuators: A. Physical Vol. 199, p. 391, 2013

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