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Geometric quasi-similarity: Case of nozzles with quadrant-shaped inlet
Accepted Manuscript Title: GEOMETRIC QUASI-SIMILARITY: CASE OF NOZZLES WITH QUADRANT-SHAPED INLET Author: V´aclav Tesaˇr PII: DOI: Reference:
S0924-4247(16)30339-9 http://dx.doi.org/doi:10.1016/j.sna.2016.07.008 SNA 9752
To appear in:
Sensors and Actuators A
Received date: Revised date: Accepted date:
20-8-2015 11-7-2016 11-7-2016
Please cite this article as: V´aclav Tesaˇr, GEOMETRIC QUASI-SIMILARITY: CASE OF NOZZLES WITH QUADRANT-SHAPED INLET, Sensors and Actuators: A Physical http://dx.doi.org/10.1016/j.sna.2016.07.008 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
GEOMETRIC QUASI-SIMILARITY: CASE OF NOZZLES WITH QUADRANT-SHAPED INLET Václav Tesař Institute of Thermomechanics v.v.i., Czech Academy of Sciences, Prague, Czech Republic Street address: Dolejškova 1402/5 182 00 Praha 8
HIGHLIGHTS Characterisation of nozzles by invariants of their behaviour is discussed New concept introduced of geometric quasi-similarity, applied to a nozzle family New invariants based on the idea of boundary layer displacement thickness Another set of invariants are based on idea of additive correction term Universal characteristic found, valid for any nozzle shape
Abstract: Geometric similarity (i.e. difference only in size scale) is generally believed to an unavoidable condition without which it is impossible to apply the Buckingham theorem to two flowfields. Recently introduced idea of secondary invariants can bypass this limitation and accept geometric quasi-similarity — cases of different values of ratio geometric parameters. In this paper this new approach is demonstrated on in an example case of a single-parameter family of nozzles mutually not geometrically fully similar. Keywords: Nozzles, local geometric similarity, quasi-similarity, Reynolds number, Euler number, quadrant nozzles
Nomenclature A
coefficient of power-law fit
(-)
a
values of invariants
(-)
b
values of invariants
(-)
Bo
Boussinesq number
(-)
cT
invariant coefficient
(-)
d
nozzle exit diameter
(m)
Eu
Euler number
(-)
e
specific energy
(J/kg )
e
drop in fluid specific energy
(J/kg )
l
length of constant-diameter exit channel
(m)
mass flow rate
(kg/s)
Ha
Hagenbach correction term
(-)
Ha0
Hagenbach term for fully developed flow
(-)
P
pressure
(Pa)
P
pressure drop
(Pa)
Q
quadratic dissipance
(m2/kg2)
Qt
loss-less dissipance
(m2/kg2)
Re
Reynolds number
(-)
r
radius of quadrant-shaped inlet dimensionless parameter for convenient presentation of the universal law
(m)
volume flow rate
(m3/s)
Te
(-)
v
fluid specific volume
(m3/kg)
Greek alphabet letters *
displacement thickness of boundary layer
(m)
fluid viscosity
(m2/s)
1. Introduction The ultimate target of research in physical sciences is discovery of invariants of the investigated problem. Once discovered, invariants are the key to formulation of governing laws. In fluid mechanics, the invariants are usually dimensionless complexes set up from problem variables. The primary invariant for a particular problem, Fig. 1, is the numerical value of basic complex set up from problem variables. It is generally known that similarity – the powerful tool for approaches to solution of problems in fluid mechanics – admits only size scale. Only this makes the invariants transferable to different size. If there is, however, only a limited number of geometric parameters that are different in investigated cases, it is possible to extrapolate the basic ideas of similarity and evaluate the corresponding secondary invariants. These then make possible full description of both geometrically quasi-similar cases. This resembles the basic idea of the local similarity applied successfully in [1] to the problem of analytic solution of turbulent jet. Instead of a single universally valid similarity solution for the whole jet, the cases of quasi-similarity leads to continuous series of locally valid results, in the jet gradually varying with increasing axial distance. In this paper, this idea is extended to geometric quasi-similarity. It is demonstrated on fluid flows inside nozzles. This is a case of flows in which the question of invariants was until recently complicated by the fact that the used characterisation parameter – discharge coefficient (or the related Euler number) - is not realy invariant. Only recently the true primary invariant for nozzles, shape parameter cT, was derived in [2]. Discussed method of identifying secondary invariants is based on measurements of family of laboratory nozzle models. Parameters cT for individual nozzles formed together the dependence, analogous to Fig. 2. This was used to derive the secondary invariants. While primary invariant characterises by a single value a particular family member, the
family as a whole is characterised by two secondary invariants. The obvious advantages gained by the secondary invariants is the possibility of solving such tasks like optimum nozzle geometry for a particular application. The methodology of identifying secondary invariants thus consisted of three steps: 1) Setting up a mathematical model based on the hypotheses formulated in [2] and identifying the nozzle shape parameter cT as the primary invariant. 2) Experimental work: measurement of hydraulic losses for incompressible, low Mach number flows over a very wide range of Reynolds numbers and verification of the primary invariance. 3) Then the relative lengths l/d of nozzle exit channels formed the parameter of the family. Indentification of the secondary invariant for the geometric quasi-similar family of nozzles by analysis of the dependence cT = f( l/d ) .
2. Nozzles Fluid flow supplied into a nozzle leaves there the closed conduits upstream and issues as a jet. Practical importance of nozzles is due to extensive use of jets in many engineering tasks like cooling, drying, or heating by jet impingement [3]. Jets are also used in agitating suspensions, in burners, jet pumps, and more recently also in nomoving-part fluidic jet-type devices like amplifiers [4, 5] and oscillators [6, 7]. Especially the oscillators found recently popularity intensification of various processes, like those in chemical engineering. Nozzles used in this paper to demonstrate the search for secondary invariants were studied in [2]. They are of axisymmetric shape as presented in Fig 4, operated at very low Mach numbers, i.e. in flows not influenced by compressibility. Important component of these nozzles as presented in Fig. 3 are their exit channels. Members of the discussed nozzle family differ in their relative lengths l/d of this channel. The jet after leaving the nozzle exit is slowed down by momentum exchange with the surrounding fluid. To counter this slowing and to get the jet reaching the required axial distance, the fluid is almost always accelerateed inside the nozzle by an area contraction (- part A in Fig. 3). This contraction is one of the three basic nozzle components. The exit channel C is another component, positioned downstream from the contraction.. It is characterised by its constant internal cross section. Its task is to stabilise the jet flow and to secure its aiming into the desirable direction. Another part of the nozzle (B in Fig. 3) is the inlet into the exit channel. In some simple, inexpensively made nozzles [9] this part may be missing, but this inevitably leads to higher hydraulic losses. It should be noted that even with the usual smooth shape of the channel inlet B , the curvature radius of fluid flow trajectories there suddenly increases. It is most pronounced for the near-wall trajectories as an increase from a finite upstream value ru to the infinite r = inside the exit channel. This changeof the radiusis necessarily associated with large local side force acting on the fluid. Discontinuity from ru to r = theoretically means an infinitely large acting force. In fact the force magnitude decreases due to presence of viscous boundary layer at the wall. Nevertheless, even after this decrease it influences the resultant character of the velocity profiles in the nozzle exits.. Suppressing this discontinuity effect is one of the reasons why presence of the exit channel is beneficial for making the jet flow pathlines straight and parallel. The character
of exit velocity profile is usually requested to be near to rectangular. Sometimes it is one of the requested performance criteria in the nozzle design. Another, usually more important nozzle behaviour parameter, of main interest in this paper, is the low overall pressure loss across the nozzle,. This is also very much dependent on the relative exit channel length l/d. Because of the wide range of their uses, nozzles exist in innumerably many shape variants. The idea of finding a universal law valid for all nozzles may thus seem to be rather hopeless. A feature that makes this task easier is most of the variants differing in the large contraction region A (Fig. 3). because of the large cross sections of the flowfield there, velocities there are usually small. With the low local velocities, there is also small local contribution to the overall pressure loss. The pressure drop is thus dominated by the conditions inside the exit channel C and perhaps in its inlet B. This, together with simple geometric character (constant radius) of the exit channel, makes the idea of universal description possible. Important factor is the smoothness of the nozzle internal shape. It is quite common on extreme end of the spectrum of shapes — the carefully contoured nozzles. They are usually of large size, typically designed for applications that require a particular shape of the exit velocity profile [7, 8]. The other extreme of nozzle shapes is encountered usually in small size nozzles for applications with generally decreased requirements on performance. Economy of manufacturing these nozzles is primary factor. Another aspect of the small size is also difficult access into the inner surfaces for standard machining tools. As a result, the inexpensive nozzles are often characterised by presence of internal sharp edges [9]. This causes flow separation from the wall and formation of stationary vortex rings usually immediately downstream from such edges. Nozzles are very rarely used alone. They are mostly an element in a fluidic (hydraulic or pneumatic) system. Designing the system containing a nozzle calls for nozzle characterisation of properties by an invariant. This, however, is a problem. The standard characterisation parameters known from textbooks – the discharge coefficient (or the related Euler) – is not invariant. It varies with the magnitude of the flow rate. A solution to the characterisation problem avoiding this value variation was introduced by the present author in [2]. Its derivation is based on two hypotheses. The first one is the idea of the unique dependence of nozzle pressure drop on the displacement effect of the boundary layer that forms inside the nozzle. Conditions (in particular the streamwise pressure gradient) existing in nozzles are such that this boundary layer is almost always laminar and very thin. The second hypothesis in [2] then describes the variations of the nozzle loss with flow rate by a Reynolds-number governed growth law. The above mentioned two cases of the nozzle flow - with and without the vortex ring at the channel inlet - result in a dichotomy of approaches to the search for the nozzle invariants. Apart from the approach assuming smooth nozzle contour, as introduced and discussed in [2], it was necessary to introduce in [9] another approach, suitable for the nozzle geometries characterised by the vortex ring presence. In the present paper, the discussion is demonstrated on family of simply shaped axisymmetric subsonic nozzles with geometry presented in Fig. 4. These nozzles are characterised by constant-radius shape (a quadrant of a circle) of the exit channel inlet and further downstream by channels of various lengths. Although simple, these nozzles are of practical importance because of relative ease of their manufacturing. The exit channel is made by simple drilling (or similar method). To obtain the shapes from Fig. 4
then suffices to round the channel entrance. The small velocity over most of the rounded inlet means that this nozzle part has only very small influence on the overall pressure drop. The relative magnitude r/d (Fig. 4) of the rounding radius is identical in the whole discussed nozzle family. Thus the geometric parameter that identifies a particular nozzle in the family is its relative channel length l/d. As shown in the upper part of Fig. 4, for generality of the obtained results it was decided to include into the present investigations also the extreme case, the Koennecke’s quadrant orifice [2, 10], i.e. the shape completely lacking the exit channel. It is necessary to emphasise that in spite of seemingly membership in the family, the absence of exit channel makes this extreme case significantly different. The critical conditions at the downstream end of the quadrant inlet, caused by the above mentioned large jump in trajectory curvature radius, is in the Koennecke [10] case not smoothed in the constantdiameter channel. As a result, conditions there make the hydrodynamics different from other, non-zero length family members.
3. Variables of the problem As is the case with many other components of fluidic systems, also the character of flow in the discussed nozzles is determined by the magnitude of Reynolds number Re. Of its several exisiting definitions, in the nozzles discussed here is Re evaluated from the nozzle exit diameter d [m] as the characteristic distance and from the average velocity in the exit, so that ... (1) - this means in experiments an evaluation from the flowmeter-measured mass flow rate [m2/s] . [kg/s] of air with specific volume v [m3/kg] and kinematic viscosity Nozzles in most engineering applications usually operate in a range of flows corresponding to Reynolds numbers within a decimal order magnitudes, typically from Re = 1 000 to slightly above Re = 10 000. Higher values, up to the order 106 , may be encountered in high-velocity flows in very large size facilities (e.g., in fire-fighting nozzles) – while lower values near to 1 000 and less [11] are nowadays increasingly often encountered in microfluidics [5]. The latter is valid especially in the viscositydominated flows of biological samples, currently often required [12]. For characterisation of nozzles with long exit channels at low Reynolds number it is useful to apply another characterising non-dimensional parameter. It is based on loss law of long tubes, published already in 1897 by Boussinesq [13], using his observations of laminar pipe flow. In the fully developed regime the shape of velocity profile is second-order parabola, not varying in streamwise direction. Condition of the full development at the beginning of the pipe was found by Boussinesq to be determined not by Reynolds number but by Boussinesq number ... (2) - where l [m] is the streamwise length of the pipe. Engineers designing hydraulic or pneumatic circuits with nozzles need an information about them most importantly in the form of dependence of energy dissipation on the flow rate. The dependence is called characteristic of the nozzle.
... (3) [kg/s] as the through parameter and specific energy drop - with mass flow rate [J/kg] (consisting of sum of pressure and kinetic energy components) as the across parameter. Because of convenience in experimental measurements (quantities directly measured by instruments: manometer and flowmeter) the characteristics of nozzles eq. (3) are in practice often replaced by the closely related functional dependence ... (4) [Pa] as a function of volume flow rate [m3/s]. The disadvantage - of pressure drop of these two latter variables is their not obeying a conservation law - in contrast to the perhaps less convenient but conservative variables in eq. (3). measured across the Dominant component of the specific energy difference nozzle – with, as usual, the downstream measurement in the zero velocity location, i. e. outside the issuing jet - is the total dissipation of the jet kinetic energy. This, of course, is conversion of kinetic energy of organised jet flow into the kinetic energy of thermal chaotic motion. The characteristic eq. (3) may be quite simply evaluated in the one-dimensional approximation neglecting the fluid friction, as ... (5) with the multiplicative factor, quadratic dissipance Qt [m2/kg2] representing the total loss of jet kinetic energy. The quadratic dependence in eq. (5) is due to the fact that the kinetic energy is proportional to the second power of velocity – which, in turn, is directly proportional to the mass flow rate . Thus this simplfied proportionality constant Qt valid for rectangular velocity profile in the nozzle exit is ... (6) This quantity, easily computed, may be a useful first approximation, although rather rough, for characterisation of a particular nozzle. The magnitude of the energetic drop may be non-dimensionalised into Euler number ... (7) -
and used in the non-dimensional version of the characteristic
... (8) Eu is here the non-dimensional measure of the across variable while Re has the role of non-dimensionalised through variable, the fluid flow rate. Real nozzles have slightly more complex exit velocity profiles so that also the generated jet has when considering the two- or three-dimensional character of the flow not simply rectangular velocity profile shape at the nozzle exit. It is also necessary, for higher approximation, to include into the energetic loss in eq. (3) also the losses (mainly
of friction character) taking place inside the internal cavities of the nozzle. The equation reflecting reality and replacing the idealised eq.(5) is therefore ... (9) with the quadratic dissipance Q [m2/kg2] ... (10) This quadratic dissipance Q is a quite useful characterisation quantity for description of nozzle properties (including the character of the particular fluid) and its behaviour in a fluidic system. In fact, because of its advantage of simplicity, there are in literature many uses of Q in this role of characterisation parameter. There are, however two factors somewhat limiting its usefulness: a) Because of the presence of specific volume v in Qt as defined in eq. (7), the value Qt is not a property of only the nozzle alone. It depends also on the properties of the fluid passing through it. In engineering practice this is actually not a serious problem. The different specific volume would cause complications in a fluidic system with several nozzles, each operating with a different fluid. This is highly exceptional. In most applications whole fluidic circuit – and all the nozzles in it - operate with the same fluid. Also any changes of properties of the same fluid between different locations in the circuit due to compressibility are not causing significant problem in the subsonic nozzles that are here the object of discussion. (b) A much more severe problem, especially nowadays — with interests shifting towards low Re typical in microfluidics — is the fact that Eu and Q vary with the fluid flow rate passing through the nozzle. In the example presented in Fig. 5 are fitted smooth lines to experimental Eu data obtained for the dimensionless characteristic (eq. 8) of the family of nozzles from Fig. 4. The variations of Eu as a function of Re are there seen to be especially apparent at small values of the Boussinesq number eq. (2). Another example of non-constant Eu is the assemblage of experimental data for quadrant nozzles is presented in Fig. 6. The main task in solving the nozzle characterisation problem at a higher precision than with the dissipance Q is to identify a set of nozzle behaviour invariants so that a particular value of the invariant is associated with a particular nozzle.
4. Two hypotheses for smooth nozzle shapes The ideal of representing nozzles as components of fluidic circuits required a new approach to the traditional problem of suitable characterisation quantity. This quantity should associate with particular nozzle a unique numerical value irrespective of the flow rate passing through. The non-constancy of Q (or Eu) made them in principle unsuitable for the role. This was not really serious in standard size flows in engineering applications, typically at Reynolds number above Re ~ 10 000. The usual rather short exit channel lengths l/d the example Fig. 5 and the usual tolerance in engineering applications made until recently the small variations of Eu negligible from practical point of view. However, the same Fig. 5 clearly demonstrates inapplicability of Eu (and hence Q) in conditions of low Re that are nowadays becoming of increasing importance, especially in the field of fluidics and microfluidics.
A solution of the problem of characterisation invariant was found in [14, 2, 9]. The starting point were two hypotheses introduced as follows: (1) Hydraulic losses in nozzles are in a unique manner dependent on the displacement thickness * of boundary layer in the nozzle exit. In particular, the Euler number Eu of the nozzle may be expressed, according to the hypothesis in [2, 9] as ... (11) (2) The expression for the displacement thickness * of the boundary layer, to be inserted into eq. (11), assuming laminar character of the boundary layer, is
... (12) - introducing an invariant coefficient cT. This hypothesis maintains that eq. (12) is applicable universally, irrespective of the nozzle geometry. The geometry only influences the numerical value of the constant cT — which thus has the role of the sought characterisation invariant.
5. Extreme case of similar channel-less quadrant orifices Although the search for the characterisation invariants is in this paper applied to the family of nozzles with exit channel as shown in the bottom part of Fig. 4, author accumulated extensive experimental data also for the channel-less quadrant orifices [10, 2] as shown in the upper part of Fig. 4. This has led to inclusion also of this case into the present discussion. It must be emphasised that the absence of exit channel — and hence absence of the sudden change in the trajectory curvature radius for near-wall flow — inevitably leads to a behaviour that cannot be simply compared with the non-zero length channel nozzles. The data for quadrant orifices were obtained with family of 5 nozzles as shown in the upper part of Fig. 6. Their nominal exit diameters were from the smallest d = 5 mm (exact values, presented in the bottom part of Fig. 6, were obtained by Zeiss workshop microscope measurements) to the largest 9 mm. The disks shown in top parts of Fig. 6 as well as Fig. 7 were of thickness h equal to the entrance radius r — which was also equal to the exit diameter d. Experimental determination of the dissipance were very simple. The fluid was air, sucked into the nozzle from the atmosphere as suggested in the bottom part of Fig. 7. The flow was driven by an exhaustor with electronic speed control. Measured values were volume flow rate (converted into the mass flow rate using measured temperature and barometric pressure) and pressure drops between the atmosphere and the space downstream from the nozzle (with correction, very small, for air kinetic energy in this space). Because the pressure levels in the nozzle were not for the insertion much different from the atmospheric, it was possible to evaluate into eq. (7) – and into the characteristic eq. (8) - simply as =v ... (13)
Bottom part of Fig. 6 shows the typical variation of the quadratic dissipance Q as defined in eq. (10). The data scatter in the illustration is due to sensitivity of the evaluation procedure (note the very small range of Eu values). In a family of nozzles differing only in their size, as is the case in Fig. 6, all family members should exhibit the same magnitude of the shape parameter cT defined in eq. (12). Indeed, the diagram in Fig. 7, presenting results obtained in different measurements than in Fig. 6, shows no systematic deviation from the statistically mean constant value
cT00 = 1. 258 ... (14) of the characterisation coefficient cT, computed from Reynolds and Euler number values as ... (15)
6. Coefficients cT for nozzles with different exit channels In the same way as described above in the experiments with the quadrant orifices, the values of the coefficient cT were accumulated with nozzles having non-zero lengths l, as shown in the bottom part of Fig. 4. The exit diameter of all models was the same, nominally 7 mm. Different lengths of the exit channel were obtained by addition to the basic short nozzle (at left in Fig. 8) of extension component(s). Workshop drawings of the extensions are shown on the right side of Fig. 8. Below them is photograph of an example. The basic nozzle at left had its exit channel l = 3d long. The increase was by alternative use of the four extensions A, B, C and D — each of different length and used in various combinations. Some lengths obtained with simultaneous use of several extensions were obviously beyond any rational lengths of ordinary nozzles. Nevertheless this exercise was considered an opportunity for obtaining additional information outside the usual nozzle lengths range. Maximum length of the exit channel, with all four extensions added simultaneously, could be as large as 160 mm, i.e. the exit channel length was l = 22.9 d as is shown in Fig. 12. Euler numbers of nozzle loss obtained in these measurements at various flow rates in quadrant-entrance nozzles are compared in the example in Fig. 9 with earlier author’s measurements [8] of the simplest nozzles with drilled, sharp edge exit channel. The sharp 90 deg edge without the rounding of the entrance was, of course, expected to exhibit larger hydraulic loss. The rounding should make the fluid flow easier and thus the Eu values were expected significantly lower. Figure 9 shows that this is true only at large Reynolds numbers – especially if the behaviour is compared with the higher c regime of the two flow regimes (with transition between them) found in [9]. At Reynolds numbers lower than Re ~ 2000 — which is certainly not very low value considering the situation in present-day microfluidics [4, 16] — in the lower c regime of the sharp-edge geometry, the losses are seen in Fig. 9 to be surprisingly the same. This may be an important message for nozzle designers in microfluidics: dominance of friction inside the exit channel at low Re makes superfluous any attempt at making easier the fluid flow entry into the exit channel.
Examples of the coefficient cT values obtained in the experiments are presented in Figs. 10, 11, and 12. As expected, the values of the coefficient cT for each nozzle are indeed constant over a quite wide range of Reynolds numbers. This coefficient is thus indeed an invariant for each of the members of the nozzle family. The next diagram, presented in 13, shows the cT values plotted as a function of the relative exit channel lengths l /d. The functional dependence is converted into a linear proportionality simply by plotting the data points in logarithmic co-ordinates. The resultant power law ... (16) has thus yield, by linear regression, two secondary-type invariants of this nozzle family
cT0 = 1.748
... (17)
and
n = 0.40
...(18)
The power law eq. (16) allows an extrapolation to zero exit channel length l / d = 0 i.e. to the expected quadrant orifice case. They might be expected governed simply by eq. (17). While not substantially different from the value in eq.(14), this magnitude obtained by extrapolation to zero in Fig. 13 is evidently higher. This is an important fact: the absence of the exit channel does not make energy dissipation in a nozzle lower. The channel, if it is short, actually decreases the hydraulic loss. It may be useful to note that that characteristics of all nozzles characterised by the invariant cT may be converted into a single universal characteristic. This is demonstrated in Fig. 14 for two extreme cases – the channel-less orifice of Sect. 5 and a nozzle from the present Sect. 6 with very long channel l/d = 13.9. For expressing the universal nozzle law it is useful to introduce the auxiliary variable ... (19) - with which the universal dependence (Fig. 14) is ... (20) Despite the two experimentally investigated cases in Fig. 14 are the extreme ends of the range, they are both covered by the same line. Of course, this line is also valid for all nozzles between these extremes.
7. Limits of applicability It should be kept in mind that the second one of the two invariance hypotheses presented in Sect. 4, eq. (12) was formulated on the basis of Blasius solution of laminar boundary layer. While for all practical nozzle designs the resultant law eq. (20) is valid (at not very small Reynolds numbers, as discussed below), there is an upper limit to its validity. With extremely long channels at high Reynolds numbers, as seen in Fig. 12, the values cT cease to be constant and show an increase with increasing flow rate. Explanation for this deviation is an influence of transition into turbulence – due to the lack of stabilising favourable pressure gradient in the constant-section channel, which
ceases to keep the boundary layer laminar above and to the right from the limit line drawn in Fig. 12. Higher cT means higher friction – and this is in agreement with the idea of transition into turbulence with its more intensive transversal momentum transport towards the exit channel wall. Since the phenomenon was found in nozzle models having excessive channel lengths (so long, in fact, that there is some difficulty in describing them as nozzles), practical meaning of this increase of cT is negligible. More influential is the opposite effect, the loss of constancy by decreasing cT values with increasing fluid flow rate, seen in Figs. 10 and 11 below Reynolds numbers roughly Re ~ 2000. That the validity of the hypotheses formulated in Sect. 4 cannot be extended arbitrarily to very low Re flows is evident from the nozzle law equation (20) and its graphical presentation in Fig. 14. The equation eq. (20) predicts the impossible infinite Eu value at Te = Te = 4 . The reason for the deviations seen in Figs. 10 and 11 is apparent from Fig. 15 in which are plotted – for two different cases of exit channel length – the relative displacement thicknesses *. At small Reynolds numbers at the left-hand side of the diagram the assumption of the very thin boundary layer is no more valid. It should be also noted that the effective thinness of * is only a small percentage of the total physical transversal distance from the wall to the core flow. From Fig. 15 it is apparent that at the distances from the wall corresponding to Re ~ 2000 the outer parts of the boundary layer on the mutually diametrally opposite locations in the exit channel must inevitably interact. The concept of the boundary layer loses there its meaning.
8. Alternative approach: Hagenbach’s additive term The characterisation by cT is based on the idea of boundary layer on exit channel walls. Inside, between the layers on the opposite wall sides, is assumed a friction-less, constant-velocity core. At very low and decreasing Re or with very long and increasing relative channel length l /d — i.e. at low Bo, eq. (2) — this core disappears. The radial momentum transport caused by the wall friction influences not only the layer but the whole cross section of the channel. The velocity distribution with decreasing Bo gradually approaches more and more the well-known Poiseuille’s parabolic velocity profile [17]. The loss asymptotically approaches the Hagen-Poiseuille [17, 18] fully developed laminar pipe flow law ... (21) This is a useful starting point for the alternative approach to studies of the flow in the nozzle exit channels at very low Re, below the validity of the hypotheses eqs.(11) and (12). Complicating fact is the character of the regime in which the velocity profile is not yet developed [19, 20, 21, 24]. Historically, this presence of flow development caused trouble in capillary viscometry – an approach to fluid viscosity measurement and evaluation from pressure drop across a length of a capillary. In these flow measurements the development occupied the initial part of the capillary. Hagenbach [22, 21] found a good correspondence with experimental data if eq. (19) was supplemented by an additive correction term Ha: ... (22)
Inadequacy of Hagenbach’s approach (which resulted in his correction term being no more used and practically forgotten) was caused by his idea of the additive term being a constant – with the value ... (23) If, instead, Ha is considered a variable dependent on the Boussinesq number Bo, the eq. (22) becomes actually very useful. Author’s experimental data plotted in Fig. 16 show a very good correlation with power law ... (24) Conversion into the linear dependence by using logarithmic co-ordinates in Fig. 16 has again yielded two secondary invariants
A = 2.1577
...(25)
and
nH = -1 / 6
...(26)
To demonstrate that this useful result applicable to nozzles at low Bo values is not perhaps a consequence of author’s experimental technique, the next Fig. 17 compares several author’s data points with information in literature [23, 27]. The last diagram, Fig. 18, concentrates on the transition between the two approaches, that of eqs. (21) and ( 22). It also tests the possibilities of obtaining a widerrange expression for the nozzle characteristic by inclusion of additional terms to eq. (22).
9. Conclusions Recently introduced ideas about characterisation of nozzles in [8, 9] are here extended from the point of view of local geometric similarity – or dissimilarity [25]. Applied to an example single-parameter family of simply-shaped axisymmetric subsonic nozzles, it is demonstrated that it is possible to characterise the whole family by only two values of secondary invariants. Described are actually two different approaches, each useful in its range of operating conditions. So far standard description of nozzle behaviour is presenting a table of many numerical values – the cases presented in [30] are typical. An alternative, in principle the same, is diagram of graphical representation of these values. These numbers usually define the dependence of traditional discharge coefficient on magnitude of Reynolds number. If the object of interest is a whole family of nozzles, e.g. differing mutually in a magnitude of their particular geometric parameter, it is necessary to use for the description also a whole family of such tables. This is inconvenient and interpolation within the tables is laborious. The introduced concept of invariants of a nozzle and secondary invariants of a nozzle family makes possible representing fully the all the tables (or diagrams) by only one or two numbers. Apart from the use in circuit design, this simplified characterisation may find applications such as, e.g., the matching and optimisation calculations, similar to those in [26]. The nozzles used as an example of invariants of fluidic elements have constantcross-section exit channel and a quadrant-shaped rounded inlet into it. One of the discussed approaches to finding the invariants, useful at higher Reynolds numbers, is based on the idea of displacement thickness of the laminar boundary layer formed on
channel exit walls. The other approach, for low and very low Re, where the idea of boundary layer is not applicable, was developed starting from the law of fully developed pipe flow modified by an additive correction term. Both approaches are applicable to any nozzle shape, provided there are not internal sharp edges that would cause flow separating from the wall and generating there a standing vortex ring. As an addition, the new approach enabled also new point of view to the quadrant orifices, as used for small flow rate measurements. Their behaviour is compared with the extrapolation to zero l/d of the discussed nozzles having an exit channel. Acknowledgements: The research discussed in this paper was supported by the grant Nr. GA1323046S obtained form GACR – the grant agency of the Czech Republic. Author also received institutional support RVO: 61388998
References [1] Tesař V., Kordík J.: “Quasi-Similarity Model of Synthetic Jets” Sensors and Actuators A-Physical, Vol.149, p. 255, 2009 [2] Tesař V.: „Characterisation of Subsonic Axisymmetric Nozzles“, Chemical Engineering Research and Design, Vol. 86, p. 1253, 2008 [3] Chapter “Impinging jets”, in monoghraph “Vortex rings and Jets”, eds. New D. T. H., Yu S. C. M., Springer Singapore 2015 [4] Raghu S., Raman G.: “Miniature fluidic devices for flow control”, Proceedings of the ASME Fluids Engineering Division Summer Meeting, FEDSM 99-7526, 1999 [5] Tesař V.: "Pressure-driven microfluidics" , Artech House Publishers, Norwood, MA USA, ISBN-10: 1596931345, July 2007 [6] Gregory J. W., Tomac M. N.: “A review of fluidic oscillator development and application for flow control”, p. 2474, AIAA Cong. San Diego, 2013 [7] Thwaites B.: “On the design of contractions for wind tunnels”, A.R.C. technical report, R & M 2278, H.M. Stationery Office, 1949 [8] Smith R.H., Wang C. T.: “Contracting cones giving uniform throat speeds”, Journal of Aeronautical Sciences, Vol. 11, p. 356, 1944 [9] Tesař V.: “Characterisation of inexpensive, simply-shaped nozzles” Chemical Engineering Research and Design, Vol. 88, p. 1433, 2010 [10] Koennecke W.: “Neue Düsenformen für kleinere und mittlere Reynoldszahlen”, Forschung im Ingenieurwesen Vol. 9, p. 109, 1938
[11] Tesař V., et al.: "Subdynamic asymptotic behavior of microfluidic valves" Journal of Microelectromechanical Systems, Vol. 14, p. 335, 2005 [12] “Microfluidics for biological applications”. eds. Tian W.-C., Finehour E., Springer 2009 [13] Boussinesq J. V.: „Théorie de l'écoulement tourbillonnant et tumultueux des liquides dans les lits rectilignes a grande section", Gauthier-Villars, 1897 [14] Tesař V.: "Law for friction losses in nozzles of round exit cross section" (in Czech), Application PO 17-18, Czechoslovak Office for Inventions and Discoveries, Prague, Czech Rep., April 1985 [15] Downie-Smith J. R., Steel S.: "Rounded approach orifices – their use at low Reynolds numbers", Mechanical Engineering, Vol. 57, p. 760, 1935 [16] Tesař V., Bandalusena H.: „Bistable diverter valve in microfluidics”, Experiments in Fluids, Vol. 50, p. 1225, 2011 [17] Poiseuille J. L. M. : „Recherches experimentales sur le movement des liquides dans les tubes de tres petits diametres“, Compte Rendu Academie des Sciences, p. 961, 1840 [18] Hagen G.: "Über die Bewegung des Wassers in engen zylindrischen Röhren", Annalen der Physik und Chemie, Vol. 46, p. 423, 1839 [19] Campbell W.D., Slattery J. C.: "Flow in the entrance of a tube", Trans. ASME – Journal of Basic Engineering, Vol. 41, p. 41, 1963 [20] Shapiro A.H., Siegel R., Kline S. J. : " Friction factor in the laminar entry region of a round tube", Proc.of 2nd Nat. Congr. of Appl. Mechanics, ASME, p. 733, 1954
[21] Rieman W.: " The value of the Hagenbach factor in the determination of viscosity by the efflux method", Journal of the American Chemical Soiciety, p. 46, 1928 [22] Hagenbach E., : „Über die Bestimmung der Zähigkeit einer Flüssigkeit duch den Ausfluss aus Röhren“, Annalen der Physik und Chemie, Vol. 109, p. 25, 1860 [23] Carley jr. C. T., Smetana F. O.: "Experiments on transition regime flow through a short tube with a bellmouth entry", AIAA Journal, Vol. 4, 1966 [24] Shapiro A.H., Smith R. D.: " Friction coefficients in the inlet length of smooth round tubes", NACA Report Nr. TN 1785, 1948 [25] Tesař V.: “Invariants of simple nozzles”, Proc. of 22nd International Conference Engineering Mechanics 2016, Svratka, Czech Republic, May 2016 [26] Tesař V.: “Fluidic Control of Reactor Flow – Pressure Drop Matching” Chemical Engineering Research and Design, Vol. 87, p. 817, 2009 [27] Miller D.S.: “Internal flow systems”, BHRA Fluid Engineering: Cranfield, Bedford U.K., 1984 [28] Ramamoorthy M.V., Seetharamaiah K.: “Losses in quadrant edge orifice meters”, La Houille Blanche, p. 129, 1965 [29] Nagamatsu S., et al.: “Measurement with quadrant nozzles of air flow at low Reynolds numbers”, Chemical Engineering, Vol. 25, p. 2, 1961 [30] “Measurement of fluid flow by means of pressure differential devices inserted in circular cross-section conduits running full – Part 3 Nozzles and Venturi Nozzles”, Norm ISO 5167-3, International Organization for Standardization, various years
CV
Prof. Ing. Václav Tesař CSc. — received his degree in mechanical engineering and later CSc (an equivalent of PhD) from CTU Czech Technical University in Prague, Czech Republic. From 1984 he was Head of the Department of Fluid Mechanics and Thermodynamics at the Faculty of Mechanical Engineering at CTU. He was Visiting Professor at Keio University in Japan and later Visiting Professor at Northern Illinois University, DeKalb, USA. Between 1999 and 2005 he was Professor at the Department of Chemical and Process Engineering at the University of Sheffield in United Kingdom. Since 2006 is Senior Research Scientist at the Institute of Thermomechanics, Academy of Sciences of the Czech Republic. His research interests are focused on shear flows – in particular jets and wall jets – and also their applications in fluidics. An author of 4 textbooks, more than 400 papers, and monograph “ Pressure-Driven Microfluidics” , published in the U.S.A. In 2010 he was awarded in Britain the Moulton Medal. He is named as the inventor on more than 200 Patents, mainly on various fluidic devices.
Figure captions Fig. 1 Schematic representation of identifying a primary invariant from experimental or computed data. The invariant is the value of the basic variable X at which the data attain their highest probability. Fig. 2 Meaning of the secondary invariants: they are the constants of dependence on the primary invariant of a family of studied mutually not exactly similar cases. The invariants are preferably defined by the parameters of least-squares linear fit to the suitable transformed dependence. . Fig. 3 Components of a typical nozzle. Most nozzle designs differ in the shape of their contraction part A — which, due to the low flow velocity prevailing there, has very little effect on the overall nozzle pressure loss behaviour. Fig. 4
The particular case of nozzles discussed in this paper as example of a family with
determined secondary invariants. It is a single-parameter family with the same rounding radius r = d of entrance into exit channels of different lengths l. The quadrant nozzle (top), known from its use as low-Re flowmetering orifice [9, 17], may be thought of as the extreme zero-length case. Fig. 5
Behaviour of the nozzles presented in Fig. 4 at low Re region, evaluated by
interpolation fitted to author’s experimental data. The inconveniently large Eu variations should be particularly noted. Fig. 6
Pressure losses were evaluated experimentally also for family of r/d = 1
quadrant nozzles (orifices with absent exit channel). Individual family member nozzles are practically fully mutually similar, so that the dependence is characterised by a single value of invariant. Fig. 7 The coefficient cT evaluated experimentally for the quadrant nozzles without exit channels. With low-velocity conditions at larger radial distances from the nozzle axis, all five nozzles of the family may be considered mutually fully similar and hence possessing a common value of the characterising invariant cT. Fig. 8 The basic tested nozzle and the four extension components (workshop drawing and photograph, at right) of different lengths that were attached to the basic nozzle (left) in various length combinations.
Fig. 9 Reynolds number dependence of measured Eu numbers of the rounded-entrance nozzles (Fig. 8) compared with sharp-edge inlet geometry discussed in ref. [10]. Perhaps surprisingly, these results show that dominance of friction loss at low Reynolds numbers Re < 2000 makes useless any effort at making the channel entry easier by rounding the edge. Fig. 10 The characterisation coefficient cT evaluated as the invariant of the basic nozzle shown in Fig. 8, with its typical rather short exit channel length l. Fig. 11 The same coefficient cT as in Fig. 10 evaluated for another nozzle - one with exceptionally long exit channel. The deviation from invariance at the small Reynolds numbers (roughly below Re ~2000) is caused by the mutual interaction of too thick boundary layer parts on the opposite sides of the nozzle exit. Fig. 12 Values of the coefficient cT evaluated from experiments with nozzles shown in Fig. 8, including those with extremely long exit channels. Those with lengths beyond reasonable limits are found here to lose the invariance of the coefficient cT, very probably as a consequence of transition into turbulence. Fig. 13 The secondary invariants 1.748 and 0.40 evaluated from experimental data for the single-parameter family of geometrically dissimilar nozzles (Fig. 4) – differing in their various exit channel lengths. Fig. 14
The model behaviour with invariant cT makes possible a general, energy drop
law Eu = f (Te) universally valid for all nozzles – of course, the two hypotheses eqs. (9) and (10) cease to be valid at small Te = Re/cT2 as is obvious from the law Eu = f (Te) leading to infinitely large resistance to flow. Fig. 15 The loss of constancy of the coefficient cT at very low Reynolds numbers, apparent in Figs. 10 and 11, is due to the extremely rapid growth of boundary layer thickness * as Re decreases. The layer becomes so thick that its parts on opposite sides of the channel mutually interact, so that and the simple boundary-layer model eqs. (9), (10) ceases to be valid and the infinite Eu from Fig. 14 does not occur. Fig. 16 Another quasi-similarity law for the discussed nozzles was identified in the region of very small Reynolds numbers and large exit channel lengths. The law fitted here to the data is useful for the conditions where the concept of thin boundary layer is no more applicable.
Fig. 17 Comparison of the low Boussinesq number law and author’s data presented in Fig. 16 with available experimental data from ref. [23]. Fig. 18
The power law of Fig. 16 is a mere approximation, but attempts at its extension
providing a more precise formulation of the dependence, as shown here, are not worth the increased complexity of the expression because they are also not valid everywhere.. Fig. 19
Specific energy loss characteristic of nozzles in universal logarithmic co-
ordinates. Note the deviations from the quadratic ideal limit case and the two auxiliary variables evaluated for given nozzle size, shape characterisation parameter, and fluid properties. Fig. 20
Practical working with the newly introduced invariant quantities with an
engineer’s typical task: computing the pressure drop.
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S0924-4247(16)30339-9 http://dx.doi.org/doi:10.1016/j.sna.2016.07.008 SNA 9752
To appear in:
Sensors and Actuators A
Received date: Revised date: Accepted date:
20-8-2015 11-7-2016 11-7-2016
Please cite this article as: V´aclav Tesaˇr, GEOMETRIC QUASI-SIMILARITY: CASE OF NOZZLES WITH QUADRANT-SHAPED INLET, Sensors and Actuators: A Physical http://dx.doi.org/10.1016/j.sna.2016.07.008 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
GEOMETRIC QUASI-SIMILARITY: CASE OF NOZZLES WITH QUADRANT-SHAPED INLET Václav Tesař Institute of Thermomechanics v.v.i., Czech Academy of Sciences, Prague, Czech Republic Street address: Dolejškova 1402/5 182 00 Praha 8
HIGHLIGHTS Characterisation of nozzles by invariants of their behaviour is discussed New concept introduced of geometric quasi-similarity, applied to a nozzle family New invariants based on the idea of boundary layer displacement thickness Another set of invariants are based on idea of additive correction term Universal characteristic found, valid for any nozzle shape
Abstract: Geometric similarity (i.e. difference only in size scale) is generally believed to an unavoidable condition without which it is impossible to apply the Buckingham theorem to two flowfields. Recently introduced idea of secondary invariants can bypass this limitation and accept geometric quasi-similarity — cases of different values of ratio geometric parameters. In this paper this new approach is demonstrated on in an example case of a single-parameter family of nozzles mutually not geometrically fully similar. Keywords: Nozzles, local geometric similarity, quasi-similarity, Reynolds number, Euler number, quadrant nozzles
Nomenclature A
coefficient of power-law fit
(-)
a
values of invariants
(-)
b
values of invariants
(-)
Bo
Boussinesq number
(-)
cT
invariant coefficient
(-)
d
nozzle exit diameter
(m)
Eu
Euler number
(-)
e
specific energy
(J/kg )
e
drop in fluid specific energy
(J/kg )
l
length of constant-diameter exit channel
(m)
mass flow rate
(kg/s)
Ha
Hagenbach correction term
(-)
Ha0
Hagenbach term for fully developed flow
(-)
P
pressure
(Pa)
P
pressure drop
(Pa)
Q
quadratic dissipance
(m2/kg2)
Qt
loss-less dissipance
(m2/kg2)
Re
Reynolds number
(-)
r
radius of quadrant-shaped inlet dimensionless parameter for convenient presentation of the universal law
(m)
volume flow rate
(m3/s)
Te
(-)
v
fluid specific volume
(m3/kg)
Greek alphabet letters *
displacement thickness of boundary layer
(m)
fluid viscosity
(m2/s)
1. Introduction The ultimate target of research in physical sciences is discovery of invariants of the investigated problem. Once discovered, invariants are the key to formulation of governing laws. In fluid mechanics, the invariants are usually dimensionless complexes set up from problem variables. The primary invariant for a particular problem, Fig. 1, is the numerical value of basic complex set up from problem variables. It is generally known that similarity – the powerful tool for approaches to solution of problems in fluid mechanics – admits only size scale. Only this makes the invariants transferable to different size. If there is, however, only a limited number of geometric parameters that are different in investigated cases, it is possible to extrapolate the basic ideas of similarity and evaluate the corresponding secondary invariants. These then make possible full description of both geometrically quasi-similar cases. This resembles the basic idea of the local similarity applied successfully in [1] to the problem of analytic solution of turbulent jet. Instead of a single universally valid similarity solution for the whole jet, the cases of quasi-similarity leads to continuous series of locally valid results, in the jet gradually varying with increasing axial distance. In this paper, this idea is extended to geometric quasi-similarity. It is demonstrated on fluid flows inside nozzles. This is a case of flows in which the question of invariants was until recently complicated by the fact that the used characterisation parameter – discharge coefficient (or the related Euler number) - is not realy invariant. Only recently the true primary invariant for nozzles, shape parameter cT, was derived in [2]. Discussed method of identifying secondary invariants is based on measurements of family of laboratory nozzle models. Parameters cT for individual nozzles formed together the dependence, analogous to Fig. 2. This was used to derive the secondary invariants. While primary invariant characterises by a single value a particular family member, the
family as a whole is characterised by two secondary invariants. The obvious advantages gained by the secondary invariants is the possibility of solving such tasks like optimum nozzle geometry for a particular application. The methodology of identifying secondary invariants thus consisted of three steps: 1) Setting up a mathematical model based on the hypotheses formulated in [2] and identifying the nozzle shape parameter cT as the primary invariant. 2) Experimental work: measurement of hydraulic losses for incompressible, low Mach number flows over a very wide range of Reynolds numbers and verification of the primary invariance. 3) Then the relative lengths l/d of nozzle exit channels formed the parameter of the family. Indentification of the secondary invariant for the geometric quasi-similar family of nozzles by analysis of the dependence cT = f( l/d ) .
2. Nozzles Fluid flow supplied into a nozzle leaves there the closed conduits upstream and issues as a jet. Practical importance of nozzles is due to extensive use of jets in many engineering tasks like cooling, drying, or heating by jet impingement [3]. Jets are also used in agitating suspensions, in burners, jet pumps, and more recently also in nomoving-part fluidic jet-type devices like amplifiers [4, 5] and oscillators [6, 7]. Especially the oscillators found recently popularity intensification of various processes, like those in chemical engineering. Nozzles used in this paper to demonstrate the search for secondary invariants were studied in [2]. They are of axisymmetric shape as presented in Fig 4, operated at very low Mach numbers, i.e. in flows not influenced by compressibility. Important component of these nozzles as presented in Fig. 3 are their exit channels. Members of the discussed nozzle family differ in their relative lengths l/d of this channel. The jet after leaving the nozzle exit is slowed down by momentum exchange with the surrounding fluid. To counter this slowing and to get the jet reaching the required axial distance, the fluid is almost always accelerateed inside the nozzle by an area contraction (- part A in Fig. 3). This contraction is one of the three basic nozzle components. The exit channel C is another component, positioned downstream from the contraction.. It is characterised by its constant internal cross section. Its task is to stabilise the jet flow and to secure its aiming into the desirable direction. Another part of the nozzle (B in Fig. 3) is the inlet into the exit channel. In some simple, inexpensively made nozzles [9] this part may be missing, but this inevitably leads to higher hydraulic losses. It should be noted that even with the usual smooth shape of the channel inlet B , the curvature radius of fluid flow trajectories there suddenly increases. It is most pronounced for the near-wall trajectories as an increase from a finite upstream value ru to the infinite r = inside the exit channel. This changeof the radiusis necessarily associated with large local side force acting on the fluid. Discontinuity from ru to r = theoretically means an infinitely large acting force. In fact the force magnitude decreases due to presence of viscous boundary layer at the wall. Nevertheless, even after this decrease it influences the resultant character of the velocity profiles in the nozzle exits.. Suppressing this discontinuity effect is one of the reasons why presence of the exit channel is beneficial for making the jet flow pathlines straight and parallel. The character
of exit velocity profile is usually requested to be near to rectangular. Sometimes it is one of the requested performance criteria in the nozzle design. Another, usually more important nozzle behaviour parameter, of main interest in this paper, is the low overall pressure loss across the nozzle,. This is also very much dependent on the relative exit channel length l/d. Because of the wide range of their uses, nozzles exist in innumerably many shape variants. The idea of finding a universal law valid for all nozzles may thus seem to be rather hopeless. A feature that makes this task easier is most of the variants differing in the large contraction region A (Fig. 3). because of the large cross sections of the flowfield there, velocities there are usually small. With the low local velocities, there is also small local contribution to the overall pressure loss. The pressure drop is thus dominated by the conditions inside the exit channel C and perhaps in its inlet B. This, together with simple geometric character (constant radius) of the exit channel, makes the idea of universal description possible. Important factor is the smoothness of the nozzle internal shape. It is quite common on extreme end of the spectrum of shapes — the carefully contoured nozzles. They are usually of large size, typically designed for applications that require a particular shape of the exit velocity profile [7, 8]. The other extreme of nozzle shapes is encountered usually in small size nozzles for applications with generally decreased requirements on performance. Economy of manufacturing these nozzles is primary factor. Another aspect of the small size is also difficult access into the inner surfaces for standard machining tools. As a result, the inexpensive nozzles are often characterised by presence of internal sharp edges [9]. This causes flow separation from the wall and formation of stationary vortex rings usually immediately downstream from such edges. Nozzles are very rarely used alone. They are mostly an element in a fluidic (hydraulic or pneumatic) system. Designing the system containing a nozzle calls for nozzle characterisation of properties by an invariant. This, however, is a problem. The standard characterisation parameters known from textbooks – the discharge coefficient (or the related Euler) – is not invariant. It varies with the magnitude of the flow rate. A solution to the characterisation problem avoiding this value variation was introduced by the present author in [2]. Its derivation is based on two hypotheses. The first one is the idea of the unique dependence of nozzle pressure drop on the displacement effect of the boundary layer that forms inside the nozzle. Conditions (in particular the streamwise pressure gradient) existing in nozzles are such that this boundary layer is almost always laminar and very thin. The second hypothesis in [2] then describes the variations of the nozzle loss with flow rate by a Reynolds-number governed growth law. The above mentioned two cases of the nozzle flow - with and without the vortex ring at the channel inlet - result in a dichotomy of approaches to the search for the nozzle invariants. Apart from the approach assuming smooth nozzle contour, as introduced and discussed in [2], it was necessary to introduce in [9] another approach, suitable for the nozzle geometries characterised by the vortex ring presence. In the present paper, the discussion is demonstrated on family of simply shaped axisymmetric subsonic nozzles with geometry presented in Fig. 4. These nozzles are characterised by constant-radius shape (a quadrant of a circle) of the exit channel inlet and further downstream by channels of various lengths. Although simple, these nozzles are of practical importance because of relative ease of their manufacturing. The exit channel is made by simple drilling (or similar method). To obtain the shapes from Fig. 4
then suffices to round the channel entrance. The small velocity over most of the rounded inlet means that this nozzle part has only very small influence on the overall pressure drop. The relative magnitude r/d (Fig. 4) of the rounding radius is identical in the whole discussed nozzle family. Thus the geometric parameter that identifies a particular nozzle in the family is its relative channel length l/d. As shown in the upper part of Fig. 4, for generality of the obtained results it was decided to include into the present investigations also the extreme case, the Koennecke’s quadrant orifice [2, 10], i.e. the shape completely lacking the exit channel. It is necessary to emphasise that in spite of seemingly membership in the family, the absence of exit channel makes this extreme case significantly different. The critical conditions at the downstream end of the quadrant inlet, caused by the above mentioned large jump in trajectory curvature radius, is in the Koennecke [10] case not smoothed in the constantdiameter channel. As a result, conditions there make the hydrodynamics different from other, non-zero length family members.
3. Variables of the problem As is the case with many other components of fluidic systems, also the character of flow in the discussed nozzles is determined by the magnitude of Reynolds number Re. Of its several exisiting definitions, in the nozzles discussed here is Re evaluated from the nozzle exit diameter d [m] as the characteristic distance and from the average velocity in the exit, so that ... (1) - this means in experiments an evaluation from the flowmeter-measured mass flow rate [m2/s] . [kg/s] of air with specific volume v [m3/kg] and kinematic viscosity Nozzles in most engineering applications usually operate in a range of flows corresponding to Reynolds numbers within a decimal order magnitudes, typically from Re = 1 000 to slightly above Re = 10 000. Higher values, up to the order 106 , may be encountered in high-velocity flows in very large size facilities (e.g., in fire-fighting nozzles) – while lower values near to 1 000 and less [11] are nowadays increasingly often encountered in microfluidics [5]. The latter is valid especially in the viscositydominated flows of biological samples, currently often required [12]. For characterisation of nozzles with long exit channels at low Reynolds number it is useful to apply another characterising non-dimensional parameter. It is based on loss law of long tubes, published already in 1897 by Boussinesq [13], using his observations of laminar pipe flow. In the fully developed regime the shape of velocity profile is second-order parabola, not varying in streamwise direction. Condition of the full development at the beginning of the pipe was found by Boussinesq to be determined not by Reynolds number but by Boussinesq number ... (2) - where l [m] is the streamwise length of the pipe. Engineers designing hydraulic or pneumatic circuits with nozzles need an information about them most importantly in the form of dependence of energy dissipation on the flow rate. The dependence is called characteristic of the nozzle.
... (3) [kg/s] as the through parameter and specific energy drop - with mass flow rate [J/kg] (consisting of sum of pressure and kinetic energy components) as the across parameter. Because of convenience in experimental measurements (quantities directly measured by instruments: manometer and flowmeter) the characteristics of nozzles eq. (3) are in practice often replaced by the closely related functional dependence ... (4) [Pa] as a function of volume flow rate [m3/s]. The disadvantage - of pressure drop of these two latter variables is their not obeying a conservation law - in contrast to the perhaps less convenient but conservative variables in eq. (3). measured across the Dominant component of the specific energy difference nozzle – with, as usual, the downstream measurement in the zero velocity location, i. e. outside the issuing jet - is the total dissipation of the jet kinetic energy. This, of course, is conversion of kinetic energy of organised jet flow into the kinetic energy of thermal chaotic motion. The characteristic eq. (3) may be quite simply evaluated in the one-dimensional approximation neglecting the fluid friction, as ... (5) with the multiplicative factor, quadratic dissipance Qt [m2/kg2] representing the total loss of jet kinetic energy. The quadratic dependence in eq. (5) is due to the fact that the kinetic energy is proportional to the second power of velocity – which, in turn, is directly proportional to the mass flow rate . Thus this simplfied proportionality constant Qt valid for rectangular velocity profile in the nozzle exit is ... (6) This quantity, easily computed, may be a useful first approximation, although rather rough, for characterisation of a particular nozzle. The magnitude of the energetic drop may be non-dimensionalised into Euler number ... (7) -
and used in the non-dimensional version of the characteristic
... (8) Eu is here the non-dimensional measure of the across variable while Re has the role of non-dimensionalised through variable, the fluid flow rate. Real nozzles have slightly more complex exit velocity profiles so that also the generated jet has when considering the two- or three-dimensional character of the flow not simply rectangular velocity profile shape at the nozzle exit. It is also necessary, for higher approximation, to include into the energetic loss in eq. (3) also the losses (mainly
of friction character) taking place inside the internal cavities of the nozzle. The equation reflecting reality and replacing the idealised eq.(5) is therefore ... (9) with the quadratic dissipance Q [m2/kg2] ... (10) This quadratic dissipance Q is a quite useful characterisation quantity for description of nozzle properties (including the character of the particular fluid) and its behaviour in a fluidic system. In fact, because of its advantage of simplicity, there are in literature many uses of Q in this role of characterisation parameter. There are, however two factors somewhat limiting its usefulness: a) Because of the presence of specific volume v in Qt as defined in eq. (7), the value Qt is not a property of only the nozzle alone. It depends also on the properties of the fluid passing through it. In engineering practice this is actually not a serious problem. The different specific volume would cause complications in a fluidic system with several nozzles, each operating with a different fluid. This is highly exceptional. In most applications whole fluidic circuit – and all the nozzles in it - operate with the same fluid. Also any changes of properties of the same fluid between different locations in the circuit due to compressibility are not causing significant problem in the subsonic nozzles that are here the object of discussion. (b) A much more severe problem, especially nowadays — with interests shifting towards low Re typical in microfluidics — is the fact that Eu and Q vary with the fluid flow rate passing through the nozzle. In the example presented in Fig. 5 are fitted smooth lines to experimental Eu data obtained for the dimensionless characteristic (eq. 8) of the family of nozzles from Fig. 4. The variations of Eu as a function of Re are there seen to be especially apparent at small values of the Boussinesq number eq. (2). Another example of non-constant Eu is the assemblage of experimental data for quadrant nozzles is presented in Fig. 6. The main task in solving the nozzle characterisation problem at a higher precision than with the dissipance Q is to identify a set of nozzle behaviour invariants so that a particular value of the invariant is associated with a particular nozzle.
4. Two hypotheses for smooth nozzle shapes The ideal of representing nozzles as components of fluidic circuits required a new approach to the traditional problem of suitable characterisation quantity. This quantity should associate with particular nozzle a unique numerical value irrespective of the flow rate passing through. The non-constancy of Q (or Eu) made them in principle unsuitable for the role. This was not really serious in standard size flows in engineering applications, typically at Reynolds number above Re ~ 10 000. The usual rather short exit channel lengths l/d the example Fig. 5 and the usual tolerance in engineering applications made until recently the small variations of Eu negligible from practical point of view. However, the same Fig. 5 clearly demonstrates inapplicability of Eu (and hence Q) in conditions of low Re that are nowadays becoming of increasing importance, especially in the field of fluidics and microfluidics.
A solution of the problem of characterisation invariant was found in [14, 2, 9]. The starting point were two hypotheses introduced as follows: (1) Hydraulic losses in nozzles are in a unique manner dependent on the displacement thickness * of boundary layer in the nozzle exit. In particular, the Euler number Eu of the nozzle may be expressed, according to the hypothesis in [2, 9] as ... (11) (2) The expression for the displacement thickness * of the boundary layer, to be inserted into eq. (11), assuming laminar character of the boundary layer, is
... (12) - introducing an invariant coefficient cT. This hypothesis maintains that eq. (12) is applicable universally, irrespective of the nozzle geometry. The geometry only influences the numerical value of the constant cT — which thus has the role of the sought characterisation invariant.
5. Extreme case of similar channel-less quadrant orifices Although the search for the characterisation invariants is in this paper applied to the family of nozzles with exit channel as shown in the bottom part of Fig. 4, author accumulated extensive experimental data also for the channel-less quadrant orifices [10, 2] as shown in the upper part of Fig. 4. This has led to inclusion also of this case into the present discussion. It must be emphasised that the absence of exit channel — and hence absence of the sudden change in the trajectory curvature radius for near-wall flow — inevitably leads to a behaviour that cannot be simply compared with the non-zero length channel nozzles. The data for quadrant orifices were obtained with family of 5 nozzles as shown in the upper part of Fig. 6. Their nominal exit diameters were from the smallest d = 5 mm (exact values, presented in the bottom part of Fig. 6, were obtained by Zeiss workshop microscope measurements) to the largest 9 mm. The disks shown in top parts of Fig. 6 as well as Fig. 7 were of thickness h equal to the entrance radius r — which was also equal to the exit diameter d. Experimental determination of the dissipance were very simple. The fluid was air, sucked into the nozzle from the atmosphere as suggested in the bottom part of Fig. 7. The flow was driven by an exhaustor with electronic speed control. Measured values were volume flow rate (converted into the mass flow rate using measured temperature and barometric pressure) and pressure drops between the atmosphere and the space downstream from the nozzle (with correction, very small, for air kinetic energy in this space). Because the pressure levels in the nozzle were not for the insertion much different from the atmospheric, it was possible to evaluate into eq. (7) – and into the characteristic eq. (8) - simply as =v ... (13)
Bottom part of Fig. 6 shows the typical variation of the quadratic dissipance Q as defined in eq. (10). The data scatter in the illustration is due to sensitivity of the evaluation procedure (note the very small range of Eu values). In a family of nozzles differing only in their size, as is the case in Fig. 6, all family members should exhibit the same magnitude of the shape parameter cT defined in eq. (12). Indeed, the diagram in Fig. 7, presenting results obtained in different measurements than in Fig. 6, shows no systematic deviation from the statistically mean constant value
cT00 = 1. 258 ... (14) of the characterisation coefficient cT, computed from Reynolds and Euler number values as ... (15)
6. Coefficients cT for nozzles with different exit channels In the same way as described above in the experiments with the quadrant orifices, the values of the coefficient cT were accumulated with nozzles having non-zero lengths l, as shown in the bottom part of Fig. 4. The exit diameter of all models was the same, nominally 7 mm. Different lengths of the exit channel were obtained by addition to the basic short nozzle (at left in Fig. 8) of extension component(s). Workshop drawings of the extensions are shown on the right side of Fig. 8. Below them is photograph of an example. The basic nozzle at left had its exit channel l = 3d long. The increase was by alternative use of the four extensions A, B, C and D — each of different length and used in various combinations. Some lengths obtained with simultaneous use of several extensions were obviously beyond any rational lengths of ordinary nozzles. Nevertheless this exercise was considered an opportunity for obtaining additional information outside the usual nozzle lengths range. Maximum length of the exit channel, with all four extensions added simultaneously, could be as large as 160 mm, i.e. the exit channel length was l = 22.9 d as is shown in Fig. 12. Euler numbers of nozzle loss obtained in these measurements at various flow rates in quadrant-entrance nozzles are compared in the example in Fig. 9 with earlier author’s measurements [8] of the simplest nozzles with drilled, sharp edge exit channel. The sharp 90 deg edge without the rounding of the entrance was, of course, expected to exhibit larger hydraulic loss. The rounding should make the fluid flow easier and thus the Eu values were expected significantly lower. Figure 9 shows that this is true only at large Reynolds numbers – especially if the behaviour is compared with the higher c regime of the two flow regimes (with transition between them) found in [9]. At Reynolds numbers lower than Re ~ 2000 — which is certainly not very low value considering the situation in present-day microfluidics [4, 16] — in the lower c regime of the sharp-edge geometry, the losses are seen in Fig. 9 to be surprisingly the same. This may be an important message for nozzle designers in microfluidics: dominance of friction inside the exit channel at low Re makes superfluous any attempt at making easier the fluid flow entry into the exit channel.
Examples of the coefficient cT values obtained in the experiments are presented in Figs. 10, 11, and 12. As expected, the values of the coefficient cT for each nozzle are indeed constant over a quite wide range of Reynolds numbers. This coefficient is thus indeed an invariant for each of the members of the nozzle family. The next diagram, presented in 13, shows the cT values plotted as a function of the relative exit channel lengths l /d. The functional dependence is converted into a linear proportionality simply by plotting the data points in logarithmic co-ordinates. The resultant power law ... (16) has thus yield, by linear regression, two secondary-type invariants of this nozzle family
cT0 = 1.748
... (17)
and
n = 0.40
...(18)
The power law eq. (16) allows an extrapolation to zero exit channel length l / d = 0 i.e. to the expected quadrant orifice case. They might be expected governed simply by eq. (17). While not substantially different from the value in eq.(14), this magnitude obtained by extrapolation to zero in Fig. 13 is evidently higher. This is an important fact: the absence of the exit channel does not make energy dissipation in a nozzle lower. The channel, if it is short, actually decreases the hydraulic loss. It may be useful to note that that characteristics of all nozzles characterised by the invariant cT may be converted into a single universal characteristic. This is demonstrated in Fig. 14 for two extreme cases – the channel-less orifice of Sect. 5 and a nozzle from the present Sect. 6 with very long channel l/d = 13.9. For expressing the universal nozzle law it is useful to introduce the auxiliary variable ... (19) - with which the universal dependence (Fig. 14) is ... (20) Despite the two experimentally investigated cases in Fig. 14 are the extreme ends of the range, they are both covered by the same line. Of course, this line is also valid for all nozzles between these extremes.
7. Limits of applicability It should be kept in mind that the second one of the two invariance hypotheses presented in Sect. 4, eq. (12) was formulated on the basis of Blasius solution of laminar boundary layer. While for all practical nozzle designs the resultant law eq. (20) is valid (at not very small Reynolds numbers, as discussed below), there is an upper limit to its validity. With extremely long channels at high Reynolds numbers, as seen in Fig. 12, the values cT cease to be constant and show an increase with increasing flow rate. Explanation for this deviation is an influence of transition into turbulence – due to the lack of stabilising favourable pressure gradient in the constant-section channel, which
ceases to keep the boundary layer laminar above and to the right from the limit line drawn in Fig. 12. Higher cT means higher friction – and this is in agreement with the idea of transition into turbulence with its more intensive transversal momentum transport towards the exit channel wall. Since the phenomenon was found in nozzle models having excessive channel lengths (so long, in fact, that there is some difficulty in describing them as nozzles), practical meaning of this increase of cT is negligible. More influential is the opposite effect, the loss of constancy by decreasing cT values with increasing fluid flow rate, seen in Figs. 10 and 11 below Reynolds numbers roughly Re ~ 2000. That the validity of the hypotheses formulated in Sect. 4 cannot be extended arbitrarily to very low Re flows is evident from the nozzle law equation (20) and its graphical presentation in Fig. 14. The equation eq. (20) predicts the impossible infinite Eu value at Te = Te = 4 . The reason for the deviations seen in Figs. 10 and 11 is apparent from Fig. 15 in which are plotted – for two different cases of exit channel length – the relative displacement thicknesses *. At small Reynolds numbers at the left-hand side of the diagram the assumption of the very thin boundary layer is no more valid. It should be also noted that the effective thinness of * is only a small percentage of the total physical transversal distance from the wall to the core flow. From Fig. 15 it is apparent that at the distances from the wall corresponding to Re ~ 2000 the outer parts of the boundary layer on the mutually diametrally opposite locations in the exit channel must inevitably interact. The concept of the boundary layer loses there its meaning.
8. Alternative approach: Hagenbach’s additive term The characterisation by cT is based on the idea of boundary layer on exit channel walls. Inside, between the layers on the opposite wall sides, is assumed a friction-less, constant-velocity core. At very low and decreasing Re or with very long and increasing relative channel length l /d — i.e. at low Bo, eq. (2) — this core disappears. The radial momentum transport caused by the wall friction influences not only the layer but the whole cross section of the channel. The velocity distribution with decreasing Bo gradually approaches more and more the well-known Poiseuille’s parabolic velocity profile [17]. The loss asymptotically approaches the Hagen-Poiseuille [17, 18] fully developed laminar pipe flow law ... (21) This is a useful starting point for the alternative approach to studies of the flow in the nozzle exit channels at very low Re, below the validity of the hypotheses eqs.(11) and (12). Complicating fact is the character of the regime in which the velocity profile is not yet developed [19, 20, 21, 24]. Historically, this presence of flow development caused trouble in capillary viscometry – an approach to fluid viscosity measurement and evaluation from pressure drop across a length of a capillary. In these flow measurements the development occupied the initial part of the capillary. Hagenbach [22, 21] found a good correspondence with experimental data if eq. (19) was supplemented by an additive correction term Ha: ... (22)
Inadequacy of Hagenbach’s approach (which resulted in his correction term being no more used and practically forgotten) was caused by his idea of the additive term being a constant – with the value ... (23) If, instead, Ha is considered a variable dependent on the Boussinesq number Bo, the eq. (22) becomes actually very useful. Author’s experimental data plotted in Fig. 16 show a very good correlation with power law ... (24) Conversion into the linear dependence by using logarithmic co-ordinates in Fig. 16 has again yielded two secondary invariants
A = 2.1577
...(25)
and
nH = -1 / 6
...(26)
To demonstrate that this useful result applicable to nozzles at low Bo values is not perhaps a consequence of author’s experimental technique, the next Fig. 17 compares several author’s data points with information in literature [23, 27]. The last diagram, Fig. 18, concentrates on the transition between the two approaches, that of eqs. (21) and ( 22). It also tests the possibilities of obtaining a widerrange expression for the nozzle characteristic by inclusion of additional terms to eq. (22).
9. Conclusions Recently introduced ideas about characterisation of nozzles in [8, 9] are here extended from the point of view of local geometric similarity – or dissimilarity [25]. Applied to an example single-parameter family of simply-shaped axisymmetric subsonic nozzles, it is demonstrated that it is possible to characterise the whole family by only two values of secondary invariants. Described are actually two different approaches, each useful in its range of operating conditions. So far standard description of nozzle behaviour is presenting a table of many numerical values – the cases presented in [30] are typical. An alternative, in principle the same, is diagram of graphical representation of these values. These numbers usually define the dependence of traditional discharge coefficient on magnitude of Reynolds number. If the object of interest is a whole family of nozzles, e.g. differing mutually in a magnitude of their particular geometric parameter, it is necessary to use for the description also a whole family of such tables. This is inconvenient and interpolation within the tables is laborious. The introduced concept of invariants of a nozzle and secondary invariants of a nozzle family makes possible representing fully the all the tables (or diagrams) by only one or two numbers. Apart from the use in circuit design, this simplified characterisation may find applications such as, e.g., the matching and optimisation calculations, similar to those in [26]. The nozzles used as an example of invariants of fluidic elements have constantcross-section exit channel and a quadrant-shaped rounded inlet into it. One of the discussed approaches to finding the invariants, useful at higher Reynolds numbers, is based on the idea of displacement thickness of the laminar boundary layer formed on
channel exit walls. The other approach, for low and very low Re, where the idea of boundary layer is not applicable, was developed starting from the law of fully developed pipe flow modified by an additive correction term. Both approaches are applicable to any nozzle shape, provided there are not internal sharp edges that would cause flow separating from the wall and generating there a standing vortex ring. As an addition, the new approach enabled also new point of view to the quadrant orifices, as used for small flow rate measurements. Their behaviour is compared with the extrapolation to zero l/d of the discussed nozzles having an exit channel. Acknowledgements: The research discussed in this paper was supported by the grant Nr. GA1323046S obtained form GACR – the grant agency of the Czech Republic. Author also received institutional support RVO: 61388998
References [1] Tesař V., Kordík J.: “Quasi-Similarity Model of Synthetic Jets” Sensors and Actuators A-Physical, Vol.149, p. 255, 2009 [2] Tesař V.: „Characterisation of Subsonic Axisymmetric Nozzles“, Chemical Engineering Research and Design, Vol. 86, p. 1253, 2008 [3] Chapter “Impinging jets”, in monoghraph “Vortex rings and Jets”, eds. New D. T. H., Yu S. C. M., Springer Singapore 2015 [4] Raghu S., Raman G.: “Miniature fluidic devices for flow control”, Proceedings of the ASME Fluids Engineering Division Summer Meeting, FEDSM 99-7526, 1999 [5] Tesař V.: "Pressure-driven microfluidics" , Artech House Publishers, Norwood, MA USA, ISBN-10: 1596931345, July 2007 [6] Gregory J. W., Tomac M. N.: “A review of fluidic oscillator development and application for flow control”, p. 2474, AIAA Cong. San Diego, 2013 [7] Thwaites B.: “On the design of contractions for wind tunnels”, A.R.C. technical report, R & M 2278, H.M. Stationery Office, 1949 [8] Smith R.H., Wang C. T.: “Contracting cones giving uniform throat speeds”, Journal of Aeronautical Sciences, Vol. 11, p. 356, 1944 [9] Tesař V.: “Characterisation of inexpensive, simply-shaped nozzles” Chemical Engineering Research and Design, Vol. 88, p. 1433, 2010 [10] Koennecke W.: “Neue Düsenformen für kleinere und mittlere Reynoldszahlen”, Forschung im Ingenieurwesen Vol. 9, p. 109, 1938
[11] Tesař V., et al.: "Subdynamic asymptotic behavior of microfluidic valves" Journal of Microelectromechanical Systems, Vol. 14, p. 335, 2005 [12] “Microfluidics for biological applications”. eds. Tian W.-C., Finehour E., Springer 2009 [13] Boussinesq J. V.: „Théorie de l'écoulement tourbillonnant et tumultueux des liquides dans les lits rectilignes a grande section", Gauthier-Villars, 1897 [14] Tesař V.: "Law for friction losses in nozzles of round exit cross section" (in Czech), Application PO 17-18, Czechoslovak Office for Inventions and Discoveries, Prague, Czech Rep., April 1985 [15] Downie-Smith J. R., Steel S.: "Rounded approach orifices – their use at low Reynolds numbers", Mechanical Engineering, Vol. 57, p. 760, 1935 [16] Tesař V., Bandalusena H.: „Bistable diverter valve in microfluidics”, Experiments in Fluids, Vol. 50, p. 1225, 2011 [17] Poiseuille J. L. M. : „Recherches experimentales sur le movement des liquides dans les tubes de tres petits diametres“, Compte Rendu Academie des Sciences, p. 961, 1840 [18] Hagen G.: "Über die Bewegung des Wassers in engen zylindrischen Röhren", Annalen der Physik und Chemie, Vol. 46, p. 423, 1839 [19] Campbell W.D., Slattery J. C.: "Flow in the entrance of a tube", Trans. ASME – Journal of Basic Engineering, Vol. 41, p. 41, 1963 [20] Shapiro A.H., Siegel R., Kline S. J. : " Friction factor in the laminar entry region of a round tube", Proc.of 2nd Nat. Congr. of Appl. Mechanics, ASME, p. 733, 1954
[21] Rieman W.: " The value of the Hagenbach factor in the determination of viscosity by the efflux method", Journal of the American Chemical Soiciety, p. 46, 1928 [22] Hagenbach E., : „Über die Bestimmung der Zähigkeit einer Flüssigkeit duch den Ausfluss aus Röhren“, Annalen der Physik und Chemie, Vol. 109, p. 25, 1860 [23] Carley jr. C. T., Smetana F. O.: "Experiments on transition regime flow through a short tube with a bellmouth entry", AIAA Journal, Vol. 4, 1966 [24] Shapiro A.H., Smith R. D.: " Friction coefficients in the inlet length of smooth round tubes", NACA Report Nr. TN 1785, 1948 [25] Tesař V.: “Invariants of simple nozzles”, Proc. of 22nd International Conference Engineering Mechanics 2016, Svratka, Czech Republic, May 2016 [26] Tesař V.: “Fluidic Control of Reactor Flow – Pressure Drop Matching” Chemical Engineering Research and Design, Vol. 87, p. 817, 2009 [27] Miller D.S.: “Internal flow systems”, BHRA Fluid Engineering: Cranfield, Bedford U.K., 1984 [28] Ramamoorthy M.V., Seetharamaiah K.: “Losses in quadrant edge orifice meters”, La Houille Blanche, p. 129, 1965 [29] Nagamatsu S., et al.: “Measurement with quadrant nozzles of air flow at low Reynolds numbers”, Chemical Engineering, Vol. 25, p. 2, 1961 [30] “Measurement of fluid flow by means of pressure differential devices inserted in circular cross-section conduits running full – Part 3 Nozzles and Venturi Nozzles”, Norm ISO 5167-3, International Organization for Standardization, various years
CV
Prof. Ing. Václav Tesař CSc. — received his degree in mechanical engineering and later CSc (an equivalent of PhD) from CTU Czech Technical University in Prague, Czech Republic. From 1984 he was Head of the Department of Fluid Mechanics and Thermodynamics at the Faculty of Mechanical Engineering at CTU. He was Visiting Professor at Keio University in Japan and later Visiting Professor at Northern Illinois University, DeKalb, USA. Between 1999 and 2005 he was Professor at the Department of Chemical and Process Engineering at the University of Sheffield in United Kingdom. Since 2006 is Senior Research Scientist at the Institute of Thermomechanics, Academy of Sciences of the Czech Republic. His research interests are focused on shear flows – in particular jets and wall jets – and also their applications in fluidics. An author of 4 textbooks, more than 400 papers, and monograph “ Pressure-Driven Microfluidics” , published in the U.S.A. In 2010 he was awarded in Britain the Moulton Medal. He is named as the inventor on more than 200 Patents, mainly on various fluidic devices.
Figure captions Fig. 1 Schematic representation of identifying a primary invariant from experimental or computed data. The invariant is the value of the basic variable X at which the data attain their highest probability. Fig. 2 Meaning of the secondary invariants: they are the constants of dependence on the primary invariant of a family of studied mutually not exactly similar cases. The invariants are preferably defined by the parameters of least-squares linear fit to the suitable transformed dependence. . Fig. 3 Components of a typical nozzle. Most nozzle designs differ in the shape of their contraction part A — which, due to the low flow velocity prevailing there, has very little effect on the overall nozzle pressure loss behaviour. Fig. 4
The particular case of nozzles discussed in this paper as example of a family with
determined secondary invariants. It is a single-parameter family with the same rounding radius r = d of entrance into exit channels of different lengths l. The quadrant nozzle (top), known from its use as low-Re flowmetering orifice [9, 17], may be thought of as the extreme zero-length case. Fig. 5
Behaviour of the nozzles presented in Fig. 4 at low Re region, evaluated by
interpolation fitted to author’s experimental data. The inconveniently large Eu variations should be particularly noted. Fig. 6
Pressure losses were evaluated experimentally also for family of r/d = 1
quadrant nozzles (orifices with absent exit channel). Individual family member nozzles are practically fully mutually similar, so that the dependence is characterised by a single value of invariant. Fig. 7 The coefficient cT evaluated experimentally for the quadrant nozzles without exit channels. With low-velocity conditions at larger radial distances from the nozzle axis, all five nozzles of the family may be considered mutually fully similar and hence possessing a common value of the characterising invariant cT. Fig. 8 The basic tested nozzle and the four extension components (workshop drawing and photograph, at right) of different lengths that were attached to the basic nozzle (left) in various length combinations.
Fig. 9 Reynolds number dependence of measured Eu numbers of the rounded-entrance nozzles (Fig. 8) compared with sharp-edge inlet geometry discussed in ref. [10]. Perhaps surprisingly, these results show that dominance of friction loss at low Reynolds numbers Re < 2000 makes useless any effort at making the channel entry easier by rounding the edge. Fig. 10 The characterisation coefficient cT evaluated as the invariant of the basic nozzle shown in Fig. 8, with its typical rather short exit channel length l. Fig. 11 The same coefficient cT as in Fig. 10 evaluated for another nozzle - one with exceptionally long exit channel. The deviation from invariance at the small Reynolds numbers (roughly below Re ~2000) is caused by the mutual interaction of too thick boundary layer parts on the opposite sides of the nozzle exit. Fig. 12 Values of the coefficient cT evaluated from experiments with nozzles shown in Fig. 8, including those with extremely long exit channels. Those with lengths beyond reasonable limits are found here to lose the invariance of the coefficient cT, very probably as a consequence of transition into turbulence. Fig. 13 The secondary invariants 1.748 and 0.40 evaluated from experimental data for the single-parameter family of geometrically dissimilar nozzles (Fig. 4) – differing in their various exit channel lengths. Fig. 14
The model behaviour with invariant cT makes possible a general, energy drop
law Eu = f (Te) universally valid for all nozzles – of course, the two hypotheses eqs. (9) and (10) cease to be valid at small Te = Re/cT2 as is obvious from the law Eu = f (Te) leading to infinitely large resistance to flow. Fig. 15 The loss of constancy of the coefficient cT at very low Reynolds numbers, apparent in Figs. 10 and 11, is due to the extremely rapid growth of boundary layer thickness * as Re decreases. The layer becomes so thick that its parts on opposite sides of the channel mutually interact, so that and the simple boundary-layer model eqs. (9), (10) ceases to be valid and the infinite Eu from Fig. 14 does not occur. Fig. 16 Another quasi-similarity law for the discussed nozzles was identified in the region of very small Reynolds numbers and large exit channel lengths. The law fitted here to the data is useful for the conditions where the concept of thin boundary layer is no more applicable.
Fig. 17 Comparison of the low Boussinesq number law and author’s data presented in Fig. 16 with available experimental data from ref. [23]. Fig. 18
The power law of Fig. 16 is a mere approximation, but attempts at its extension
providing a more precise formulation of the dependence, as shown here, are not worth the increased complexity of the expression because they are also not valid everywhere.. Fig. 19
Specific energy loss characteristic of nozzles in universal logarithmic co-
ordinates. Note the deviations from the quadratic ideal limit case and the two auxiliary variables evaluated for given nozzle size, shape characterisation parameter, and fluid properties. Fig. 20
Practical working with the newly introduced invariant quantities with an
engineer’s typical task: computing the pressure drop.
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