## Abstract

The Eulerian 3-terminal gyrator is a hypothetical element whose novel features relate to systems dominated by dynamic forces. In mechanical linkages, turbomachines, flow networks and many other systems forces are proportional to the square of velocities. This square-law is fundamental in the Eulerian 3-terminal gyrator, unlike the linearity of the conventional gyrator. A gyrator embodies non-reciprocity and, without power loss, exchanges pressure-like and flow-like variables. The name derives from the gyroscope, for which torque and precession rate are linearly related if the gyro speed is constant, but this speed would have to vary to produce a square law relationship. Such so-called ‘internal modulation’ has been viewed askance in the definition of an ideal element. However, despite potential mathematical complexity, a neat definition follows from a coordinate transformation and mapping relating to 3-terminal elements. The resulting characteristics are well-behaved functions. Formulae are derived for the small-signal *Z*- and *Y*-parameters and these enable the non-reciprocity to be demonstrated. The large-signal characteristics are compared with those of a fluidic reverse flow diverter applied to process fluid handling. In general, the Eulerian gyrator is proposed as a natural prime element in analytical modelling because it builds in the modulation that must be applied to a linear gyrator in representing systems beyond the confines of electrical networks.

## 1. Introduction

Before Tellegen's paper (1948), electrical circuits were analysed or represented in the context of the synthesis of a limited set of ideal components. These included 2-terminal resistors, inductors and capacitors, and a single irreducible ‘four pole’ or 4-terminal element; the ideal transformer. A 4-terminal element, generically like that in figure 1, was conceived as having two distinct ‘ports’ 1 and 2, hence the alternative name ‘2-port’. ‘Port’ means a terminal-pair at which inputs or outputs can be made. A seemingly infinite number of 2-ports could be made by combinations of the ideal components. However, Tellegen showed that all the resulting circuits were subject to the condition of reciprocity and this limited the phenomena that the ideal circuits could represent. Using Tellegen's original example of a 2-port characterized by(1.1)reciprocity implies that *Z*_{12}=*Z*_{21}. To escape from this constraint, the gyrator, as symbolized as in figure 2*a*, was defined by(1.2)

This gave it both ‘ideal’ non-reciprocity (the transfer resistances being opposite, not equal) and zero power dissipation (the *IV*-products summing to zero).

Although usually represented as a 2-port element, Shekel (1952, 1953) showed that a 3-terminal representation was appropriate for characterizing amplifying devices such as transistors that physically had three connections. Transistors may be represented as having two fixed ports but this imposes a particular set of variables and characterizing parameters (gains, transfer ratios, etc.), which are dependent on the arbitrary choice of datum terminal. A 3-terminal gyrator is obtained by amalgamating two terminals, as in figure 2*b*. This has a symmetry that is independent of the choice of datum, so Shekel proposed the symbol in figure 2*c* to emphasize this. Shekel (1953, 1954) showed that a gyrator was essential in any model of a real 3-terminal amplifier (such as a transistor) and that it represented an orientation-independent intrinsic property. Other components such as positive and negative resistances characterized the asymmetric attributes.

Although its early destiny was electric circuits and small-signal linearization, the gyrator's conception was more fundamental, as its name suggests. A suitably mounted gyro-wheel spinning at constant speed conforms closely to the defining equations of the gyrator in the analogous relationships between bearing torques and precession speed:

These relationships are simple but special outcomes of Newtonian dynamics. The simplicity is because the gyro is constrained by its mounting (like the front wheel of a bicycle). The precession and torque vectors are orthogonal so no power is dissipated, and the signs of the constants impose ideal non-reciprocity. For both large and small signals, this mechanical gyrator exchanges force-like and velocity-like variables. The ‘exchange-rate’, defined by a constant *modulus*, is a characteristic of the ideal linear gyrator. The broader significance of the gyrator as a ‘prime element’ in dynamic systems modelling was emphasized by Paynter (1961) in his development of bond graphs as a general method encompassing electrical, mechanical and thermo-fluid systems. In this endeavour, the question of ‘modulation’ became an issue. If the gyro speed were to vary, then the modulus would change and the corresponding gyrator would be ‘modulated’. ‘Internal modulation’ is a phenomenon where the modulus is a function of the input or output variables. Mathematical models of even simple mechanical systems require such modulation, but the general consequences of this are regarded with suspicion. As discussed by Breedveld (1985), the suspicion arises from the thought that ‘if *any* form of modulation is allowed, the gyrator loses its distinction’. One might call it a ‘box of functions’. Despite these warnings, the general thrust of this paper is that the particular form of non-linearity, which necessarily involves ‘internal modulation’ embodied in the Eulerian 3-terminal gyrator (E3TG), is fundamentally natural and does not lead to wayward behaviour. Indeed, it could be regarded as an essential ‘prime element’ in the context of fluid flow networks and perhaps, more generally, for mechanisms where dynamic non-viscous forces are dominant. Perhaps, the full gamut of effects can be observed in a bicyclist for which the relevant ‘forces’ include the dynamic forces throughout the moving parts and structure, the air resistance owing to turbulent eddies and any, perhaps vestigial, aerodynamic lift or drag forces ascribable to an ideal flow field. All of these are proportional to the square of the road speed.

## 2. Representation of fluid‐dynamic devices

The method of representing the characteristics of a device is important because the definition of the E3TG comes from an unusual method of characterization that uses a polar coordinate system. Conventionally, in characterizing an aerofoil, for example, two dependent variables, lift and drag, are represented as functions of two independent variables, *angle* (of attack) and *speed* (translation velocity relative to still air). To represent the results on a two-dimensional graph, speed is held constant and the lift and drag are plotted as functions of the angle. A set of characteristics is needed to accommodate different air speeds. Basically, one of the independent variables is regarded differently from the others in the characterization and, indeed, during experiments. A good reason for this is that, for subsonic flow, the aerofoil's non-dimensional characteristics are largely independent of air speed. This means that if the lift and drag are divided by the dynamic pressure of the air, then the resulting characteristics coalesce into two non-dimensional functions of the angle of attack.

The lift is the product of a constant coefficient×air-speed×circulation. The lift is orthogonal to the airspeed vector, so in this relationship, speed is ‘converted’ to force. The circulation is in fact proportional to airspeed, so from the product of these comes the square-law dependence of lift on airspeed. Paynter (1972) promoted the term ‘Eulerian’ to describe the consequent form of similitude that is encountered in fluid systems dominated by Euler's equations, which essentially apply to mechanical-energy-conserving operating regimes of efficient turbomachines and aerofoils, for example. For turbomachines, shaft-speed and flow are conventionally treated as independent variables, whereas torque and pressure are dependent. Suitably non-dimensionalized, these become largely independent of shaft speed. Within a reasonable tolerance, a pair of non-dimensional characteristics describes performance. However, the surprising fact (even to fluid dynamicists) is that even in completely chaotic turbulent flows in which mechanical energy is grossly dissipated, the ‘Eulerian’ square-law similitude still applies. This effect of non-dimensionalizing in terms of a judiciously chosen dynamic pressure is so fundamental that it is taken for granted in fluid mechanics to such an extent that the generic term *Eulerian similitude* is rarely used. In general, the non-dimensionalizing variable is held constant during experiments and in plotting results. In effect, the assumption is that Eulerian similarity will prevail, if not perfectly then adequately, for practical use. Consider the phrase ‘the drag coefficient’ for a car, not a ‘speed-drag-graph’ or indeed, the drag-coefficient in general. It is remarkable (though convenient) that drag-producing effects also conform reasonably to Eulerian similitude. For this much wider applicability of similitude to real fluids (not inviscid ideal fluids), the limits of applicability must be known and characterized. These limits are primarily in terms of the Reynolds number (*Re*), so this is important, despite ‘Eulerian’ suggesting otherwise. In real flows, other limits may be needed, perhaps characterized by Mach or Cavitation numbers, but these would be used in the same way as for any device.

The conventional representation of characteristics based on fixing one of the independent variables has a disadvantage when both positive and negative values need to be covered. In a pump-turbine, both flow and shaft speed can cover positive and negative ranges. These include the pumping regime, turbine regime and regimes where all mechanical energy is dissipated, for example, when flow is forced the ‘wrong way’ through a pump. Again this is a regime very far from being dominated by Euler's equations. Although a bunch of conventional graphs can represent this, the result is a fragmented picture of the continuum of operating states. To overcome this, von Karman and Knapp (1937) suggested a representation in which loci of constant torque and pressure are *mapped* as contours on a flow-shaft-speed plane. Usually known as the Karman–Knapp circle diagram, it provides the basis for a more general method to show all operating regimes as parts of a continuum. The circle diagram was relevant to the control of turbomachines, which were shown by Paynter (1972) to include strong gyrator-like characteristics. For the turbomachine, the gyrator was appropriately represented by a 2-port element that highlights the clear distinction between the two physical ports: the shaft and the pair of pipe connections admitting and discharging flow.

No-moving-part fluidic devices rely on the interaction and mixing of flow streams so the natural prime element for representing these is a ‘3-terminal element’ with three pipe connections. Practical devices are jet pumps, flow diverters, vortex amplifiers and various junctions. These are characterized in Cartesian coordinates by two dependent variables as functions of two independent variables. A jet pump can be characterized by output pressure and jet pressure as functions of entrained flow, while the jet flow is held constant. Results at different jet flows give different sets of these characteristics, but if Eulerian similitude prevails, then they condense into a single pair of dimensionless characteristics. The jet pump can be applied in many ways so other characterizations are necessary.

Unlike the turbomachine, input and output ‘ports’ are subjected to a degree of choice. Contrary to intuition, the merging of terminals in the 2-port representation to yield three terminals *increases* the generality by allowing the two ports to be allocated in three ways, not just in one way. Hence, the potential variety is increased. As shown by Tippetts (1984), there are up to 216 ways of allocating *constant*, *independent*, and *dependent* variables for a 3-terminal element. Some of these give very different appearances to what are the same fundamental characteristics. In principle, a coordinate transformation can convert one to another, but even with easy digital manipulation, conventional characterization does not smoothly cover all operating states. For example, how can one graph describe the succession of states where flow in each terminal is zero?

## 3. Indefinite circle diagram

The ‘indefinite circle diagram’ was proposed by Tippetts (1974*a*,*b*) to do these things: (i) to represent all flow states as a continuum, (ii) to give an isometric representation of homogeneous variables (*q* or *e*), (iii) to eliminate the representational bias owing to different choices of a datum terminal, (iv) to show separately, *power use* and *signal amplification*, and (v) to take a step towards a canonical method of characterization.

The third objective and, indeed, the name were inspired by Shekel's (1952) indefinite (or ‘zero-sum’) admittance matrix as related to transistors.

In order to obtain relationships that are relatively independent of the choice of datum terminal, it is helpful to use the unusual allocation of variables and notation shown in figure 3. The flows are inflows and the pressure-differences (henceforth ‘pressures’) are defined as positive cyclically so they sum to zero. The flow and pressure subscripts are also peculiar because in a sense, they are opposite to each other. However, this leads to a satisfactory geometric and algebraic uniformity.

The state of a device is fully described by two flows and two pressures. Unlike usual graphical characterizations, the operating state of the device can be described by using the flows to define a point on a flow plane and the pressures to define a point on a pressure plane. Normally, a point on a plane is specified using orthogonal coordinates but this builds in a bias because two out of the three flows or pressures must be the axes. However, because of the zero-sum property of the variables, an isometric grid having three axes set at 120° intervals can represent all three variables equally. A point in such a grid is specified by the intersection point of normals from the point to the axes (which have positive and negative branches). Figure 4 represents the characteristics by a mapping from the flow plane to the pressure plane. Hence, the arbitrary choice of a datum terminal and particular set of flows or pressures for characterization does not introduce a bias in the appearance or subsequent use of the characteristics.

Characterization involves three functions: (i) an independently specified locus in the flow plane, (ii) the mapping of the locus onto the pressure plane, and (iii) a function showing the angular relationship between the mapped points.

Polar coordinates shown in figure 4 are convenient for representing gyrators. Thus, *θ* and *r* define points in the flow plane, and *φ* and *s* define points in the pressure plane. Two of six similar relationships between the variables are: *q*_{x}=*r* cos *θ* and *q*_{y}=*r* cos (2π/3−*θ*). The independently defined locus in the flow plane is simply a circle, so our concern is how this maps onto the pressure plane for a 3-terminal gyrator (3TG). Consider first a *linear* 3TG. This has the following characteristics: power must be conserved so the power input must equal the power output. This can be expressed by choosing any datum, such as *z* and the adjacent terminal-pairs. Thus, the power entering at terminal-pair *x*–*z* is −*e*_{y}*q*_{x} and the power delivered from terminal-pair *y*–*z* is −*e*_{x}*q*_{y}. Equating these gives(3.1)

These ratios define angles corresponding to the relationships *ϕ*=*θ* or *θ*+π. For a linear gyrator with a resistance * s*, the magnitudes of the flows and pressures are related by

*r*=

**s***s*. Consequently, the linear 3TG is represented by circles in the flow plane mapping into proportionally sized circles in the pressure plane, and with operating points at the same polar angles or diametrically opposite. (Note that a conductance or ‘modulus’

*G*, the inverse of

*, has become more usual in defining the linear gyrator.)*

**s**## 4. Definition and characteristics of the E3TG

For the Eulerian gyrator, power must certainly be conserved so *ϕ*=*θ* or *θ*+π. The magnitudes, however, are related by *s*=*kr* where *k* is a square-law constant equivalent in a fluid-dynamic embodiment to a quadratic or orifice-like resistance coefficient having the dimensions of N s^{2} m^{−8} in SI units. (*k* is the square-law equivalent of Tellegen's * s*). Note that there is only one physical kind of 3-terminal gyrator despite the apparent choice of

*ϕ*−

*θ*options that result simply from the allocation of terminals

*x*,

*y*, and

*z*. For example, if

*x*and

*y*are interchanged, then

*ϕ*=

*θ*+π rather than

*ϕ*=

*θ*.

Choosing *k* to be constant means that a circle in the flow plane still maps into a circle in the pressure plane, but the circles' diameters are related by square-law proportionality. For the linear gyrator, all loci map identically but for the Eulerian gyrator, only circles map into circles. Other loci are distorted in the mapping, as can be seen by considering how straight lines map from one plane to the other. Two examples can be seen in how the line representing *q*_{z}=unity maps into the pressure plane, and how the line *e*_{z}=unity maps into the flow plane. Letting *k* be unity, the underlying equations can be found from figure 5 which shows the relationships between a point 1 on the line and the mapped point 2. In figure 5*a* point 1 is located by the polar radius *r* and its *z*-value of unity. Point 2 in the pressure plane is located at a radius *s*, an ordinate of *z* and abscissa *x*, which is the distance normal to the *z*-axis. This implied *x*-axis is purely for convenience in describing the form of the mapped functions.

The analysis is facilitated by two relationships seen from figure 5*a*, i.e. *s*^{2}=*z*^{2}+**x**^{2}, and *z*/*s*=1/*r*. By definition, *s*=*r*^{2} so both *s* and *r* can be eliminated ultimately leading to the mapping relationship:

*q*_{z}=1 in the flow plane maps into(4.1)in the pressure plane, where *x* is distance normal to the *z*-axis. The function shown in figure 6 is asymptotic to a parabola centred at the origin with its axis normal to the *z*-axis.

The pressure-to-flow mapping is derived in a similar way by consideration of the coordinates of points 1 and 2 in figure 5*b*. The necessary relationships are: *r*^{2}=**x**^{2}+*z*^{2}, *z*/*r*=1/*s* and *s*=*r*^{2}. Eliminating *s* and *r* leads to the required formula:(4.2)

Shown in figure 6, this is asymptotic to the line *q*_{z}=0.

In the context of linear systems, the distortion implies modulation, but both mapped functions are unimodal and not particularly threatening. It may be noted that a set of flow curves, such as equation (4.2) covering a range of fixed pressure values would correspond to Knapp's constant pressure and torque contours in the pump circle diagram.

## 5. Questioning the definition

It must be acknowledged, that one aspect of the definition of the E3TG is arbitrary, in that, the square-law modulus *k* is chosen to be *constant* for all flow states. However, it could be some *function* of the flow angle *θ*, thereby describing a shape other than a circle in the pressure plane. The main justification for *k* being constant is that it seems that anything else would produce enormous mathematical complexity but with no apparent advantage. Supporting this is the fact shown below, that a measure of the E3TG's non-reciprocity has a particularly neat form if *k* is constant.

## 6. Characteristics in Cartesian coordinates

To compare the E3TG with other devices in detail, it is useful to be able to describe its characteristics in conventional Cartesian coordinates. For these, a datum is defined and then pressures relative to the datum and the flows at the non-datum-terminals are used as variables. In the following, terminal *z* is the datum. It will be obvious how subscripts could be re-assigned to correspond to any other datum. The two formats considered are analogous to impedance and admittance characterizations.

In deriving the following relationships, it is helpful to note that the quadratic terms within the square root define an ellipse having its major axis along the lines *q*_{y}=*q*_{x} or *e*_{y}=*e*_{x} in conventional *x*−*y* coordinates. In the isometric coordinates the ellipse is ‘squashed’ into the desired circle. The terms outside the square-root scale the circle so that, with *k* equal to unity, a unit-radius-circle in the flow plane is mapped to an identical circle in the pressure plane.

Pressures as functions of flows (‘impedance’)(6.1)

(6.2)

Flows as functions of pressures (‘admittance’):(6.3)

(6.4)

To understand the action of the E3TG, its characteristics are plotted as if resulting from a hypothetical experiment whose objective is to get output characteristics describing the power delivered from one port when power is supplied to the other port. For this, terminal *z* is the datum, terminal *x* is the supply and terminal *y* the output. We note that, so far, the notation has been satisfactory in producing convenient sets of relationships. However, by demanding more conventional characteristics, the variables have unfamiliar signs and subscripts. Thus the *supply* variables are *q*_{x} (an inflow) and −*e*_{y}, implying a positive ‘supply’ pressure. The *output* variables are −*q*_{y} (an outflow) and the pressure *e*_{x}. If the products of these two sets of variables are positive then power is supplied to the device and becomes available as useful power at the output. For the graphs shown in figures 6 and 7, *k* is unity. For the impedance characterization (figure 7) the ‘supply’ flow *q*_{x} is unity, and for the admittance characterization (figure 8) the supply pressure, −*e*_{y} is unity. The graphs are obtained by putting these values into the appropriate equations (6.1)–(6.4).

The constant supply flow (impedance) characteristics in figure 7 contrast sharply with those of many real devices, but this is as expected for gyrators. The linear device is just as unusual when observed by way of its large-signal behaviour. In fact, plotted on the same graph as the characteristics in figure 7, the linear gyrator would have straight-line characteristics given by *e*_{x}=−*q*_{y} and *e*_{y}=−1, which can be regarded as straightened-out versions of those of the Eulerian gyrator. Straight lines would also characterize a linear gyrator if described by the format of figure 8, so in this respect, the devices are qualitatively similar. Considering again figure 7, for both linear and Eulerian gyrators, the output characteristic (*e*_{x}) does not decrease thereby maintaining 100% efficiency at all points, while the supply pressure −*e*_{y} varies strongly and reflects the increasing power demand of the output as the outflow increases. Many devices such as jet pumps or electrical sources have falling output characteristics with very weak coupling between output and input, but this is usually because for much of the operating range they are highly inefficient. However, some devices do come closer to the gyrator's unusual behaviour. Certain centrifugal pumps demand increasing torque to maintain constant speed as the outflow is increased, and some regions of the output characteristic may have a positive gradient mimicking that of the −*e*_{y} curve in figure 7. In general, it may be said that the E3TG's characteristics are not outlandish. Rather, they can be regarded as mild distortions of those of the linear gyrator which always did have strange characteristics.

## 7. Small‐signal parameters

Again, for comparison with conventional characterizations it is useful to derive the most common small-signal parameters. As a first example, consider the impedance parameters (or more strictly *resistance* parameters because no time-dependency is included in the ideal E3TG). In generic notation, these are *Z*_{11}, *Z*_{12}, *Z*_{21} and *Z*_{22} quantifying the pressure changes in terms of flow changes at notional *ports* 1 and 2 as defined for the 2-port in equation (1.1). The driving-point impedance (or ‘input’ impedance) *Z*_{11} gives the change in *e*_{y} caused by a change in *q*_{x}, while *q*_{y} is constant. Partially differentiating equation (6.2) gives(7.1)but because of the sign ascribed to *e*_{y}, the differential corresponds to −*Z*_{11}.

The transfer impedance *Z*_{12} quantifies the change in input pressure *e*_{y}, because of a change in output flow *q*_{y}, with the inflow *q*_{x} held constant. Again, partially differentiating equation (6.2) gives(7.2)which is actually −*Z*_{12} because of the sign ascribed to *e*_{y}.

The other impedance parameters are obtained similarly for the output port by partially differentiating equation (6.1). In this case, no sign changes are needed so the differentials are identical to the *Z*-parameters. Thus, *Z*_{21} is given by(7.3)and *Z*_{22} is given by(7.4)

In principle, any set of small-signal parameters, including hybrid parameters and gain-factors can be derived from any set, such as the impedances, by matrix manipulation. However, it is helpful to have explicit formulae for the *Y*-parameters because these are often used as the basis for numerous aspects of circuit theory.

The admittance (*Y*) parameters (strictly, *conductances*) quantify the flow changes as functions of pressure changes. Again, for direct comparison with conventional 2-port theory, terminal *z* of the E3TG is used as the datum, terminal *x* is regarded as ‘input’ (terminal 1) and *y* as ‘output’ (terminal 2). The corresponding admittances are obtained by partially differentiating equations (6.3) and (6.4) Thus, −*Y*_{11} is given by(7.5)

The transfer admittance −*Y*_{12} is given by(7.6)

The other elements of the matrix are identical to the partial differentials derived from equation (6.4), hence, *Y*_{21} is:(7.7)and *Y*_{22} is(7.8)

## 8. Non-reciprocity

The transfer impedances *Z*_{12} and *Z*_{21} must be different for the device to exhibit non-reciprocity. The linear gyrator can be said to have a ‘perfect form’ of non-reciprocity because the relevant parameters are not just different, they have equal magnitudes but opposite signs. The difference between them is constant (i.e. 2* s* for Tellegen's impedance formulation or 2

*G*in terms of the gyrator modulus). For the E3TG, being non-linear, the transfer impedances vary with the operating point as do all the other small‐signal parameters. Hence, quantifying and, more importantly, verifying non-reciprocity is less obvious. It is clear that the transfer functions are all different from one another so

*some*degree of non-reciprocity is assured, but there might be a point at which the relevant functions become equal thereby defeating (if only singularly) the purpose of the E3TG. A suitably clinching relationship is the difference between the transfer impedances, i.e.

*Z*

_{12}−

*Z*

_{21}. Taking care with the signs, this means that

*adding*equations (7.2) and (7.3), yields(8.1)

It is gratifying to note that the variable component of this expression (the root) is generally always positive. Its minimum value is zero but only when all the flows are zero. The square-root in fact represents the flow vector *r* used in defining the E3TG, and so it is constant for all the operating states defined by circles in the flow plane. This has a satisfactory simplicity, which is perhaps the counterpart of the ‘perfect’ non-reciprocity of the linear gyrator.

Reinforcing this feature is the fact that the difference between the transfer admittances, obtained by adding equations (7.6) and (7.7), is also constant on the circular characterizing loci. The loci in this case are those with a constant pressure vector in the pressure plane:(8.2)

## 9. As a general fluid‐dynamic device

So far, apart from *k*, the variables could have been non-specific representing even amps and volts perhaps, although the *q* always implied flow of an incompressible medium. If the variables are confined to a fluid system, then the characterization can be generalized in the sense that the flow plane can be defined as one of *Re* and the pressure plane can be generalized to one of pressure coefficients or, in other words, Euler Number (*Eu*). If one imagines that a hypothetical physical fluidic E3TG has a characteristic dimension of *d* metres, working with a fluid having properties *ν* (kinematic viscosity) *ρ* (density), then the polar radii would be related by

However, for the ideal E3TGY, *s*=*kr*^{2}, so

Along with the power conservation equations this would be the definition of the general *fluid‐dynamic* E3TG expected to have a limited range of ideal performance. A truly ideal element would be unaffected by viscosity but by defining *Re*, the ideal E3TG can be brought into the real world of non-ideal devices that typically have operating ranges (where *Re* has a minimal effect) usually between limits definable by *Re* or other parameters.

## 10. The E3TG and process fluid handling

What would be the use of an E3TG? The question can be answered in the form of a parable. By misfortune, omitted from Gulliver's travels was his visit to Euleria. Now, for the first time, figure 9 shows how the Eulerians produce their elixir ‘eul’. The process has two phases. In phase 1, water is pumped through an E3TG to a reactor vessel at a pressure of *H*. Subnuclear reactions quickly produce *eul*. After a batch of *eul* has been produced, the pump is switched off and the *eul* flows back under gravity through the E3TG and then through a long delivery pipe with significant square-law resistance *k*_{p} to the stock tank.

The E3TG fulfils a vital function. In phase 1, it transmits all of the flow to the reactor against the pressure head of *H* without any entrainment or leakage in the pipe-connection to the stock tank. In this phase the flow state (in terms of the circle diagram) corresponds to *θ*=−30° because *q*_{y} is an outflow equal to the inflow *q*_{x}, and *q*_{z} is zero. The pressures correspond to *φ*=−30° because −*e*_{y}, the pump pressure, is equal to the output pressure *H*, and *e*_{z}, the pressure-drop between *x* and *y*, is zero. In phase 2, despite zero pressure from the pump, the gyrator diverts all of the flow through the resistance of the delivery pipe without any leakage or entrainment through the pump. The flow and pressure states correspond to *θ*=*φ*=90° (*e*_{x} being zero and *e*_{y}=−*e*_{z}=*H*). Therefore acts like a perfect reverse flow diverting valve but it has no moving parts, operating entirely by fluid-dynamic interactions.

The perfection of flow transmission and diversion depends on matching (i.e. sizing) the E3TG to the other components in the system. The pressure, *H*, and the pipe resistance, *k*_{p} determine the flow, *q*=√(*H*/*k*_{p}), for the system. The pump must deliver *q* at pressure *H* in phase 1 and the gyrator must generate the pressure *H* at the relevant pair of terminals when transmitting the flow *q* in both phases. To match this demand, the value of *k* for the E3TG is given by(10.1)

In fact, the E3TG would be a valuable component in the real world, especially if the reactor were producing a toxic but valuable compound. Alternatively, it could be regarded as a safety device alleviating the consequences of pump failure, in which case the contents of the reactor would be diverted, perhaps to a safe zone, rather than back through the failed pump. The hypothetical example uses the E3TG as a ‘heavy current’ device operating in two widely spaced flow states separated by 120° in the *θ*−*r* flow plane. Stability or deviation from the ideal operating states could be analysed by using the small-signal parameters or by inspection of the characteristics.

## 11. Real fluid‐dynamic and fluidic devices

Can an E3TG physically be made? It seems that the best that can be expected is that certain devices can mimic parts of the E3TG's operating range. For example, a jet pump with an area ratio near unity can direct the flow as in phase 1 of the hypothetical *eul* plant, however, the output pressure *H* would only be some 70% of the pump supply pressure. In phase 2, with reverse flow, a normal jet pump would not divert the flow so the pump would be subject to a high back pressure and high back flow. However, a greatly modified jet-pump called an ‘X-type reverse flow diverter’ (RFD), shown in figure 10, can divert reverse flow even against a back pressure analogous to that of the resistive pipe in figure 9. Forward and reverse flow states correspond to phases 1 and 2 in the *eul* plant. Forward-state flow supplied at port *a* passes through the nozzle and is recovered in the conical diffuser from which it discharges through a second tangentially aligned diffuser. In the reverse state, flow enters the tangential diffuser at *b*, forms a swirling flow in the conical diffuser and discharges through the radial diffuser recovering some pressure, thereby producing a relatively low pressure at port *a*. Qualitatively, this RFD can operate usefully in the phase 1 and 2 flow states; however, its performance as quantified by the pressure states falls well short of the ideal. If a real X-type-RFD, as described by Tippetts (1974*a*,*b*) were used instead of the E3TG in the *eul* plant, inserted so that ports *a*, *b* and *c* correspond to *x*, *y* and *z* of the gyrator, the following pressures would be obtained:

In phase 1, the pump pressure would have to be 2.4 H. In phase 2, the pressure drop in the pipeline would be 0.464 H, the rest of the available reverse driving pressure being lost in the RFD. Simultaneously, however, the pressure at port *a*, connected to the inactive pump, would be a *suction* relative to atmosphere of 0.256 H. This implies that a small flow would be drawn through the pump, bad perhaps for the *eul* plant, but if *backflow* in the pump is to be avoided then this ensures a safety margin. If a reduced reverse flow (0.88*q*) were acceptable, then by having a higher pipeline resistance, port *a* could be matched precisely to atmospheric pressure resulting in zero pump throughflow, thereby satisfying the Eulerians.

The characteristics of E3TG and the RFD at constant supply flow are compared in figure 11. These have been scaled so that the supply pressures (−*e*_{y} and *e*_{a}) for both devices are unity at an outflow of unity. This corresponds to operation in phase 1 and shows the obvious deficit in the RFD's output pressure (*e*_{b}) by comparison with *e*_{x} of the E3TG. At smaller outflows the output pressures become closer but the supply pressures increasingly diverge with the greatest divergence being in the region of negative outflow. Not shown in figure 11, is the situation corresponding to phase 2, in which the supply flows are zero and reverse flow dominates. Like the forward flow state, performance is quantified by pressure recovery. Here, it is only about half of the ideal, however, qualitatively the overall characteristics are gyrator-like in the vicinity of the desired operating states but at other states, considerable difference exists.

A device matching the E3TG for all flow states seems to be a difficult objective but the need for a physical device having perfect symmetry and able to operate over all flow states is probably very rare. The main purpose of the E3TG is for setting benchmarks or limiting-values for the performance of real devices. Topics of this nature were discussed by Tippetts (1974*a*,*b*, 1977, 1989) using the circle diagram. Real fluidic (no-moving-part) devices are confined between two of the gyrator lines in figure 4 (otherwise they generate power) but the closer and more aligned their characteristics come to these ‘bounds’ then, generally, the more we can expect greater gyrator-like performance and possibly more useful applications. An optimized combining flow junction (‘FJ’) was one device investigated. It was suggested by Tippetts (1977) that the ideal FJ was a gyrator in which the modulus switched sign at the state of equal inflow. A wide range of real and hypothetical large-signal flow controllers was discussed by Tippetts (1989), but all this was constrained given that the detailed characteristics of the ideal E3TG had not been fully worked out.

In broader areas of fluid mechanics, the turbomachine, already shown to have gyrator-like characteristics, can be related to the Eulerian paradigm. For the aerofoil, the lift (force) is produced by gyrational coupling from the speed but the relationship is Eulerian because the lift derives from the product of speed and circulation, which itself is proportional to the speed, thereby producing the ubiquitous square-law relationship. In mechanics there are relevant examples, including the flywheel in a vehicle rounding a bend, which experiences gyroscopic torques, but these derive from the product of precession speed and flywheel speed, both of which are geared to road speed and so leading again to a square-law. These examples are special consequences of Newton's laws. Any mass moving on a curved path must be constrained by a centripetal force orthogonal to its path. Being orthogonal, the force does no work. Its magnitude is proportional to the rate of change of momentum and orthogonal to the path and this is proportional to the square of the speed along the path. These are the lossless (‘non-energic’) and square-law characteristics idealized in the E3TG. The 3-terminal embodiment and the characterization using *Eu* and *Re,* roots it in fluid‐dynamics and flow networks. Changes to the definition of force-like and flow-like variables would give it wider application as a Eulerian 2-port gyrator, perhaps with closer relevance to mechanical systems.

## 12. Conclusion

The E3TG is recommended as an addition to the existing range of ideal or *gedanken* devices. It fulfils the requirement for a network element that embodies non-reciprocity and behaves like ideal and real fluid-dynamic devices in that pressures, flows, scaling-in-size and fluid density all vary as if controlled by Euler's equations. While the effects of scaling and fluid density do not make any theoretical demands, the ‘Eulerian’ square-law pressure-flow relationship and non-reciprocity needed special scrutiny. The proposed element has a simple definition but only in terms of an unusual coordinate system. The all-important square-law nonlinearity has not caused severe consequences despite what could be termed ‘internal modulation’, the necessary variation of the gyrator modulus abjured by some specialists. Its intrinsic square-law means that the element may have uses outside fluid dynamics for systems in which dynamic forces are dominant. It is acknowledged that an arbitrary choice means that the defining locus is a circle rather than some other shape. It seems that anything else would be a mathematical mess, so neatness prevailed. Things that look neat geometrically often turn out to be useful.

## Footnotes

- Received January 9, 2004.
- Accepted September 14, 2004.

- © 2005 The Royal Society