## Abstract

In this work, laminar transonic weakly three-dimensional flows at high Reynolds numbers in slender channels, as found in microsupersonic nozzles and turbomachines of micro-electro-mechanical systems, are considered. The channel height is taken so small that the viscous wall layers forming at the channel walls start to interact strongly rather than weakly with the inviscid core flow and, therefore, the classical boundary layer approach fails. The resulting viscous–inviscid interaction problem is formulated using matched asymptotic expansions and found to be governed by a triple-deck structure. As a consequence, the properties of the predominantly inviscid core region and the viscous wall layers have to be calculated simultaneously in the interaction region. Weakly three-dimensional effects caused by surface roughness, upstream propagating flow perturbations, boundary layer separation as well as bifurcating solutions are discussed. Representative results for subsonic as well as supersonic conditions are presented, and the importance of these flow phenomena in technical applications as, for example, a means to reduce shock losses through the use of deformed geometry is addressed.

## 1. Introduction

Classical boundary layer theory has been used extensively in the past to calculate high Reynolds number flows through nozzles and turbomachines. In this approach, the inviscid core region is considered first followed by the analysis of viscous wall layers inside which, to a first approximation, the pressure distribution is imposed by the outer inviscid flow.

If, however, following present tendencies to miniaturization made feasible by the rapidly expanding capability of micromachining technology the size of such devices is reduced this so-called hierarchical boundary layer concept fails as inviscid and viscous flow regions now interact strongly rather than weakly and thus cannot be investigated in successive steps but have to be treated simultaneously. Examples are provided by micronozzles (thrusters) for space propulsion of satellites and micro-electro-mechanical system (MEMS) gas turbine engines with improved compact power sources for portable electronics.

Length scales in the millimetre range imply that a device with room temperature inflow will operate at Reynolds numbers of the order of 10^{4}, whereas nozzles with higher gas temperatures will operate at Reynolds numbers of a few thousand and thus in the laminar flow regime. Consequently, the stage at which strong interaction between predominantly inviscid and viscous flow regions comes into play can be identified by inspection analysis based on the triple-deck concept, which also provides the relevant geometrical scalings as well as the magnitude of the associated perturbations of the various field quantities. An interesting example is provided by flows through slender channels that were first investigated by Kluwick & Bodonyi [1] assuming supersonic conditions of perfect gases, whereas transonic effects including more general gas behaviour were also studied later [2–4].

In this study, a first step towards the understanding of three-dimensional flow features is taken which allows among others to capture the influence of localized surface roughness representing an important aspect of small-scale flows. To simplify, the analysis considerations will be restricted to weakly three-dimensional flows of perfect gases which, however, clearly indicate how it can be extended to more complex fluids.

## 2. Problem formulation

We consider the transonic flow through a channel with height with a region of strong interaction, caused by rapid changes of the area of cross section, forming a distance from the inlet (figure 1). The flow is described by non-dimensional quantities as defined as
2.1
Here, denotes the position vector with components in the streamwise, the wall normal and the lateral directions. The corresponding components of the velocity vector are . Furthermore, denotes the time, the vector describing the geometry of the channel walls, the pressure, the density, the temperature, the specific enthalpy, the specific heat at constant pressure, the dynamic viscosity, the thermal conductivity and the speed of sound. The subscript 0 characterizes flow quantities immediately upstream of the local interaction region. The Navier–Stokes equations then are given by
2.2
where ** τ** is the viscous stress tensor of a Newtonian fluid and

**is the heat flux satisfying Fourier's law. The non-dimensional groups entering the governing equations (2.2) are the Reynolds number , the Prandtl number and the Eckert number . An important additional non-dimensional group entering the problem via the upstream boundary conditions is the Mach number . In the following, it is assumed that**

*q**M*

_{0}deviates only slightly from its critical value

*M*

_{0}=1 and that the Prandtl and Eckert numbers are of . Because the channel is assumed to be symmetric with respect to its axis, it is sufficient to consider the lower half of the configuration. For large values of the Reynolds number considered here the local interaction process taking place is found to be given by the interplay of three regions (or decks) exhibiting significantly different flow behaviour and, as in the case of strictly two-dimensional flow [2,3], the relevant perturbation parameter is governed by 2.3 In contrast to the aforementioned studies, however, the shape of the isolated surface roughness is allowed to vary in the spanwise (

*z*)-direction. Balancing inertia and pressure gradient terms in the

*x*and

*z*momentum equations then shows that a self-consistent theory capturing three-dimensional effects can be constructed if significant changes in the surface geometry occur over distances , as summarized in figure 2. Inside the upper deck comprising the whole inviscid core the flow is almost two dimensional and described by nonlinear dynamics. Nonlinear effects are of importance also within the lower deck buried deep inside the oncoming boundary layer, so that induced velocity perturbations are of the same order of magnitude as the velocities in the unperturbed flow. Upper and lower deck regions are separated by the main deck, across which information is exchanged in an almost passive manner.

The fact that in the structure just outlined the upper deck region spans the whole channel width and thus is common to the two ‘triple decks’ present in the upper and lower half planes is crucial and distinguishes the present investigation from other triple-deck studies of external transonic flows where the relevant perturbation parameter *ε* typically is of rather than of .

The scalings of the streamwise, the lateral distance and the time are the same in all three decks
2.4
whereas the scalings of the wall normal distance in the lower, main and upper decks are
2.5
By substituting the expansions of the various flow quantities
2.6
into the Navier–Stokes equations, one finds that the leading-order main deck problem can be solved in closed form, which in turn allows us to formulate the matching conditions that have to be satisfied by the field quantities between adjacent decks analytically. Consequently, the essence of the local interaction process can be condensed into a fundamental lower deck problem by introducing suitably transformed quantities and making use of Prandtl's transposition theorem, Kornfeld [5],
2.7
The continuity and momentum equations in the streamwise and lateral directions are supplemented with the no-slip condition at the wall where *S*(*X*,*Z*) defines the geometry of the surface roughness as well as matching conditions in the limit and at the base of the main deck involving the (negative) perturbation displacement thickness . The problem is closed by the interaction relationship, which contains three non-dimensional constants *Γ*, *K* and λ. The first one is a thermodynamic property measuring the curvature of isentropes in the pressure-specific volume diagram. For perfect gases, which will be considered here exclusively, *Γ* assumes the value (*γ*+1)/2 with *γ* being the ratio of the specific heats at constant pressure and volume. However, the fundamental lower deck problem remains valid also for more general fluids provided that the absolute value of . The second constant , also taken to be of , is the transonic similarity parameter that controls the dependence of the perturbation mass flux *G* on *P*, whereas , again of , characterizes the strength of the interaction between the inner and outer flow regions.

Before going into further details, it should be noted that the continuity equation is of two-dimensional form and that the momentum equations in (2.7) do not include variations of *U* and *W* in the *Z*-direction, which expresses the weakly three-dimensional nature of the interaction process. Also note that the key role in the interaction law is played by the algebraic function *G*, the perturbation mass flux known from the theory of inviscid gas dynamic flows through slender channels/nozzles in definite contrast to external flows where integral relationships of Hilberth or Ackeret type come into play.

The formulation of the interaction problem (2.7) includes steady as well as unsteady flow phenomena. In the following, however, we concentrate almost entirely on steady flows but indicate how unsteadiness affects the bifurcation analysis in §6.

## 3. Solutions of the linearized problem

The fundamental lower deck problem has to be solved numerically in general. In order to obtain first insights into the resulting flow structure it is useful, however, to consider the case of steady flow and discuss flow disturbances caused by a single localized hump of sufficiently small height, so that a linear treatment is appropriate. Introducing the Fourier transform *Q*^{**} of any flow quantity *Q* with wavenumbers *k* and *l*
3.1
the solution in Fourier space can be expressed in closed form
3.2
It includes the function with *r*=(*ik*)^{1/3}*Y* , which appeared first in two papers by Smith [6,7] and thus provides a welcome link between the present study and earlier work by J. T. Stuart. No attempt was made to carry out the back transformation analytically. Rather it proved more efficient to use a fast Fourier numerical approach, which also has the advantage that it is not restricted to particular hump geometries or other simplifications.

A quite common smooth surface deformation also investigated in related studies of external flows, for example by Smith *et al.* [8], is the cosine-squared hump geometry
3.3
which will be used as the basis for the following discussion.

As a representative example, we consider subsonic flows with *K*=1, λ=1.25 and a hump height *h*_{hump}=0.1. The graph in figure 3*a* then displays lines of constant lateral wall shear stress *σ*=∂*W*/∂*Y* |_{Y =0} with a spacing of 0.005 in the *X*,*Z* plane. As the hump geometry is symmetric with respect to *Z*=0, *σ* vanishes at the centreline. At the front part of the hump, *σ* increases with increasing values of *Z*, leading in turn to a local maximum, whereas a region of negative *σ* forms at the rear part of the hump. As in the work of Smith *et al.* [8] four regions of different flow behaviour have to be distinguished. At the front part of the hump, region I, the fluid moves away from the centreline at all distances *Y* from the wall. Further downstream, region II, the flow direction close to the wall reverses, thus leading to the formation of a vortical structure. Even further downstream in region III at the rear part of the hump the entire lower deck flow takes on a sink-like form. Finally, in region IV, downstream of the hump the fluid close to the wall is seen to move away from the line of symmetry and to move inwards at larger values of the wall normal distance *Y*, thereby again generating a vortical motion.

Figure 4 displays the distributions of the pressure perturbations *P* and the streamwise component *τ*=∂*U*/∂*Y* |_{Y =0} of the wall shear stress on the line of symmetry *Z*=0. As the fluid approaches, the hump *P* increases, but this phase of deceleration is followed by regions of rapid acceleration and again deceleration on the front and rear parts of the hump. The latter results in a pressure overshoot which slowly decays further downstream in the limit . As expected, *τ* increases/decreases in the regions of favourable/adverse pressure gradient, thus indicating the possibility of boundary layer separation at the lee side of the hump if its height is sufficiently large. In both figure 4*a* and figure 4*b*, the extent of the hump in the streamwise direction is indicated by yellow strips that clearly show that its presence is felt by the fluid upstream of the surface deformation. This is in sharp contrast to the case of strictly two-dimensional flow in which, as proved by Kluwick & Meyer [3], upstream influence is possible for supersonic inlet conditions only and thus represents a genuine three-dimensional effect.

## 4. Boundary layer separation

According to linear theory, the local minimum of the wall shear stress distribution shown in figure 3 becomes more pronounced with increasing hump height, thus pointing to the occurrence of boundary layer separation if *h*_{hump} is sufficiently large. In order to follow this development up to and beyond incipient separation, it is, however, necessary to account for nonlinear effects which have been neglected so far. To this end, a pseudo-spectral method, cf. Canuto *et al.* [9], Duck & Burggraf [10], Gottlieb & Orszag [11], Biringen & Kao [12], is used that requires much less memory resources than other numerical techniques (e.g. finite difference schemes), which is of crucial importance for the study of three-dimensional flows. In addition, a major advantage of spectral methods is to capture regions of reversed flow (i.e. boundary layer separation) without any additional approximations (e.g. the FLARE approximation for finite difference methods). The physical problem can be solved very effectively and allows the usage of standard numerical algorithms (e.g. the fast Fourier transform algorithm) by mapping the unbounded physical domain onto a bounded domain *x*_{n}×*z*_{n}∈[0,*π*]×[0,*π*], cf. Boyd [13] and Cain *et al.* [14].

As an example, figure 5 displays results for subsonic flow with *K*=1 and λ=1.25 as before but with a much larger value of *h*_{hump}=2.25. The wall shear stress distribution at the centreline *Z*=0 of the hump is qualitatively similar to that in figure 4 but *τ* now assumes negative values between *X*=0.57 and *X*=0.98, i.e. almost up to the rear end of the hump. The associated region of reverse flow is clearly visible in figure 6, which depicts streamlines in the plane of symmetry. Even though the cross flow velocity component is much smaller than the streamwise velocity component, it strongly influences the flow behaviour in regions of small shear stress *τ*. In particular, it is seen that the separation streamline does not connect with the reattachment point, resulting in turn in the formation of an open rather than a closed separation bubble.

In order to study the geometrical properties of streamlines close to the singular points of detachment/reattachment taken for the sake of simplicity to be located at *X*=0 we, following Oswatitsch's [15] study based on the full Navier–Stokes equations, Taylor expand the pressure disturbance *P* and the velocity components *U*,*V*,*W* about *X*=*Y* =*Z*=0. Taking into account the symmetry relations *τ*(*X*,0,*Z*)=*τ*(*X*,0,−*Z*), *σ*(*X*,0,*Z*)=−*σ*(*X*,0,−*Z*) and *P*(*X*,*Z*)=*P*(*X*,−*Z*) expansions satisfying the lower deck continuity and momentum equations take the form of
4.1
Streamlines in the vicinity of *X*=*Y* =*Z*=0 then satisfy
4.2
From the first relationship, one infers that *τ*=0 is a necessary condition for separation/reattachment and that the separation/reattachment angle *ϑ*_{r/s} for weakly three-dimensional flows is unaffected by *σ*_{Z}, in contrast to general three-dimensional flows [15], and thus agrees with the result holding for two-dimensional configurations
4.3
By integration of the slope conditions (4.2), one obtains the projections of streamlines onto the *X*,*Z*-plane
4.4
with arbitrary positive constants *C*_{1}, *C*_{2} and by elimination of *Y* the expression
4.5
for the wall streamlines. Evaluation for *τ*_{X}<0, *σ*_{Z}<0 and *τ*_{X}>0, *σ*_{Z}<0 yields the characteristic streamline patterns associated with a separation node and a reattachment saddle, which is seen to be qualitatively similar to the behaviour of wall streamlines calculated numerically (figure 7). In addition, the positions of the separation streamline calculated analytically (equation (4.3)), and numerically are compared in figure 7, and close agreement is observed.

## 5. Mixed flow

It is a distinguishing feature of transonic flows that they may exhibit properties of mixed subsonic–supersonic character, for example the formation of a supersonic pocket in an otherwise subsonic environment or the presence of a local subsonic region under overall supersonic conditions. An example of the first type is considered in figure 8, which displays pressure distributions at the centreline of a cosine-squared hump with *h*_{hump}=1 for subsonic inlet conditions and different values of *K*>0. For large positive values of *K*, the flow is purely subsonic and inspection of the interaction relationship, being part of the fundamental lower deck problem (2.7), indicates that the pressure disturbances generated by the hump are small and linear theory applies. Evaluation of the analytical result for *P*^{**} in equation (3.2) in the limit then yields that *P*(*X*,*Z*) is proportional to the deformation *S*(*X*,*Z*) of the channel walls
5.1
which is found to be in excellent agreement with the numerical computations. As *K* decreases, i.e. as the Mach number in the inviscid core region approaches the critical value 1, the flow reacts increasingly sensitively to the displacement exerted by the hump. The pressure disturbances increase in magnitude and eventually sonic conditions, which as will be shown below correspond to *P*=−1, are reached. Further decrease of *K* then leads to the formation of a supersonic pocket near the crest of the hump.

According to classical gas dynamics local supersonic regions are in general terminated by shock discontinuities, and it is therefore of interest to investigate how shocks are incorporated in the present theory. To this end, we consider the interaction relationship specialized to the case of two-dimensional steady flow
5.2
As pointed out before, herein *G*(*P*;*K*,*Γ*) represents the perturbation mass flux. The graph *G* versus *P* is plotted in figure 9*a* for supersonic inlet conditions *K*<0. Using results known from classical gas dynamics, it is easily shown that the difference between the local Mach number *M* and its critical value 1 is proportional to d*G*/d*P* [2,3]. Thus, one infers immediately that under subsonic (*K*>0) or supersonic (*K*<0) inlet conditions the sonic state is reached for *P*=−1 or *P*=1, respectively. In figure 9 where we consider supersonic upstream conditions *K*<0, therefore, the sonic line *P*=1 separates regions of subsonic and supersonic flow.

Across gas dynamic shocks the pressure changes discontinuously, whereas the mass flux remains constant. Corresponding pressure pairs therefore are selected by the intersection points with straight lines parallel to the *G*-axis. If the upstream state is taken to be undisturbed *P* jumps from 0 to *P*_{1}, and the resulting shock discontinuity is sketched in figure 9*b*. In interacting flows, however, such a discontinuity is impossible as subsequent fluid states must follow the *G* versus *P* graph as indicated by the arrow. In this way, the shock discontinuity is resolved into a continuous shock profile [2]. Such pseudo-shocks, which mathematically represent eigensolutions of the interaction problem, can be generated experimentally by placing a hump of critical height inside the channel, so that the flow is choked. If the height of the hump is reduced, then the pseudo-shock moves closer to the hump, and only part of the complete profile is realized. No steady solutions exist for humps exceeding the critical height and presumably periodic oscillations known as buffeting will set in, but this is still an open question [4].

Next, let us investigate how pseudo-shocks react to weakly three-dimensional perturbations. Specifically, we consider supersonic flow past a wavy hump
5.3
as shown in figure 10. Here, the wavelength in the *Z*-direction as well as the hump height are kept constant, whereas the parameter *α* which controls the waviness in the *X*-direction varies. The height of the hump is taken to be smaller than the critical value *h*_{hump}=1.58. Therefore, as pointed out before, only part of the complete pseudo-shock profile indicated by the broken line in figure 11 is realized. In the case of strictly two-dimensional flow *α*=0, however, the associated constriction of the area of cross section is large enough to trigger the transition from supersonic to subsonic flow far upstream of the hump before the fluid is accelerated again and eventually returns to the unperturbed state *P*=0 in the limit . As the parameter *α* increases, weakly three-dimensional effects come into play, and two interesting features can be observed. First, the induced pressure disturbances are reduced rather rapidly and for *α*=1 the values of *P* are so small that a subsonic region in the inviscid core of the channel is not formed any more. Second, it is seen that the shape of the pressure distribution upstream of the hump is found to be almost unaffected by increasing values of *α* apart from a downstream shift. In other words, the flow there remains almost two-dimensional and increasing waviness in weakly three-dimensional channel flow has the same effect as decreasing hump heights in strictly two-dimensional configurations and thus can be used very effectively to reduce shock losses.

## 6. Non-uniqueness

As pointed out before, there exists an upper bound of hump height under two-dimensional supersonic flow conditions for which the channel is choked, and a fully developed pseudo-shock is formed far upstream. Similarly, such a maximum hump height leading to choked flow exists also for two-dimensional subsonic flow. In the inviscid limit, it is calculated easily from the requirement that the value of the local Mach number assumes its critical value 1 at the crest of the hump. The resulting result for *h*_{hump,max} has to be modified if viscous–inviscid interaction is important, but the basic prediction that steady-state solutions do not exist if the hump height exceeds a critical value remains intact. Next, we ask if this is true also for weakly three-dimensional subsonic flows and numerical experiments indicate that this is the case. As an example, figure 12 shows the variation of *P* in the plane of symmetry and *X* fixed with increasing hump height. Specifically, *X*=0.85 has been chosen where the bifurcation behaviour—which occurs at other locations as well—is most pronounced. *P* initially drops before the computations come to an abrupt end and steady-state solutions cannot be found anymore using the numerical scheme adopted so far. In addition, it is observed that the pressure distribution exhibits a vertical tangent at the point of breakdown and that its shape is almost parabolic in the vicinity of this point. This points to the existence of a second solution branch, which was verified by modifying the numerical algorithm appropriately (figure 13). In addition, figure 14 displays the pressure and wall shear distributions for two subcritical values of *h*_{hump} where solid and dashed curves correspond to the upper and lower branch, respectively. For *h*_{hump}=2.4, the pressure distributions corresponding to the two branches differ only slightly, whereas the differences in the distributions of the wall shear stress are more pronounced, and this trend intensifies as *h*_{hump} is decreased further. For *h*_{hump}=2.3, the shape of the wall shear stress distribution associated with the upper branch is qualitatively similar to that for a hump with larger height *h*_{hump}=2.4. The distributions connected with the lower branch, however, differ significantly. In particular, the formation of a second wall shear stress minimum inside the separated flow region *τ*<0 is clearly visible for *h*_{hump}=2.3, which is reflected also by the first signs of a pressure plateau.

In order to support this numerical finding, it is desirable to perform a local analysis of the flow behaviour near the bifurcation point. As in related cases of marginally separated flows (see [16,17]), the appropriate perturbation parameter is given by the square root of the difference between the critical and the actual value of the relevant control parameter
6.1
Flow quantities *U,V,P,A* represented as components of a vector ** r** then are expanded in the form
6.2
The leading-order term

*r*_{c}describes the critical steady flow state. Perturbations of this state depend on the spacial coordinates and the time suitably scaled, which allows the steady as well as the unsteady flow behaviour to be captured. By inspection of the first-order problem 6.3 one infers that

*r*

_{1}can be written as the product of a function

*c*(

*t*) and the right eigenvector

**of the associated singular operator matrix 6.4**

*m**c*(

*t*) remains arbitrary at this level of approximation but is determined by the solvability condition emerging at second order 6.5 Here, Δ

*h*measures the difference between the critical and the actual value of

*h*

_{hump}, whereas

*α*,

*β*,

*γ*are positive constants.

In the case of subcritical flow where Δ*h*>0, the evolution equation (6.5) for *c*(*t*) exhibits two real stationary points
6.6
which correspond to upper and lower branch solutions, respectively. Moreover, inspection of the phase diagram versus *c* indicates that upper branch solutions are stable, whereas lower branch solutions are unstable. Whether such unstable flow situations can be realized experimentally and thus are of practical interest is not known at the present.

A comparison between numerical and analytical results is provided in figure 15, where *P*(1.3,0) is plotted as a function of *h*_{hump} close to the critical value 2.588… It is seen that the solid curve obtained numerically agrees quite well with the prediction from the first-order analysis just outlined. In addition, it is seen that extending the analytical considerations to higher order leads to a significant improvement.

## 7. Summary and conclusion

In this study, it has been shown that viscous–inviscid interactions of weakly three-dimensional transonic flows in slender channels in the high Reynolds number range, as occurs, for example, in micronozzles and in turbomachines of MEMSs, can be described properly by means of matched asymptotic expansions and the triple-deck concept. To gain insights into the general behaviour of such flows, perturbations which are so small that the fundamental lower deck problem can be linearized are considered first. The resulting closed-form solution provides a complete picture of the flow response to localized surface roughness over the whole Mach number range covered by the theory, extending from purely subsonic to purely supersonic conditions characterized by large positive and negative values of the transonic similarity parameter *K*.

Following the discussion of results obtained by linearization of the governing equations nonlinear effects such as boundary layer separation, the occurrence of mixed flows with local supersonic/subsonic regions and the existence of non-unique solutions which require a numerical treatment are investigated. As a representative situation leading to boundary layer separation subsonic flows past a cosine-squared hump are considered. The height of the hump is taken so large that the reverse flow region forming at the lee side is substantial and covers almost one-quarter of its length in the streamwise direction. The recirculation region is found to be of open type, resulting in a singular behaviour of streamlines at the points of detachment and reattachment. Local solutions covering the neighbourhood of these points are obtained by modifying the analysis presented in the seminal study of Oswatitsch [15] and compared with numerical results both qualitatively and quantitatively.

Subsonic flows past cosine-squared humps also form the starting point for the discussion of mixed flows. To this end, the transonic similarity parameter *K*>0 initially is taken to be large, so that subsonic conditions hold throughout, and then reduced continuously. Decreasing values of *K* cause the local Mach number near the crest of the hump to increase and eventually sonic conditions are reached. Further decrease of *K* leads to the formation of local supersonic pockets embedded in the otherwise subsonic flow. The investigation of flows with *K*<0 past surface roughness is seen to lead to complementary effects, the formation of local subsonic regions in otherwise supersonic flow. These are associated with the presence of (pseudo-)shocks upstream of the deformed wall geometry. Most interestingly, it is found that the magnitude of the resulting pressure rise can be reduced very effectively by introducing weakly three-dimensional perturbations which, therefore, provide a simple means to reduce shock losses.

As is well known, the reduction of the channel area of cross section under strictly two-dimensional subsonic conditions eventually results in choking, and steady flow is not possible beyond a critical limit. Surprisingly, a similar flow behaviour is observed also in subsonic flows past localized surface roughness, having the possibility to encircle the obstacle rather than to pass over it. But, obviously, this additional freedom is of limited capacity, and, again, steady flow is not possible if the hump height exceeds a critical value, whereas two solution branches exist for subcritical hump heights. This numerical finding is supported by a local bifurcation analysis.

As pointed out by one of the referees, the use of the small perturbation parameter for flows where the Reynolds number is casts doubt on the usefulness of the analysis, which should be commented on by the authors. We respond by citing Stewartson [18], one of the founders of triple-deck theory, who in his seminal treatise Multistructured boundary layers on flat plates and related bodies wrote: ‘Comparison between theory and experiment should therefore be made in a spirit of humility and we should always be prepared for the possibility that, however exact the theory, it is relevant to situations which are of no interest to the engineer’ (p. 188). But, then he hastens to add a number of reasons why ‘such a comparison is still useful’. Oswatitsch, the teacher of one of us (A.K.), has condensed an analogous line of reasoning into its most concise form: ‘Eine gute Theorie ist besser als ihr formaler Gültigkeitsbereich’ (a good theory is better than its formal range of validity).

## Funding statement

This work was supported by the Austrian Science Fund in the framework of the WK differential equations.

## Footnotes

One contribution of 15 to a Theme Issue ‘Stability, separation and close body interactions’.

- © 2014 The Author(s) Published by the Royal Society. All rights reserved.