## Abstract

Rayleigh’s inflection point theorem and Fjortoft’s theorem provide necessary conditions for inviscid temporal instability of a plane parallel flow. Although these theorems have been assumed to hold in the spatial framework also, a rigorous theoretical basis for such an application is not available in the literature. In this paper, we provide such a basis by carrying out a formal analysis of the Rayleigh equation. We prove that, under certain conditions satisfied by a wide class of flows, the Rayleigh and Fjortoft theorems are applicable to the spatial stability problem also. This work thus fills the lacuna in the spatial stability theory with regard to these classical theorems.

## 1. Introduction

Two of the most celebrated results in the classical inviscid stability theory are the Rayleigh inflection point theorem and the Fjortoft theorem, which provide necessary conditions for the existence of inviscid instability for plane parallel flows. It should however be noted that these theorems were proved by Rayleigh [1] and Fjortoft [2] for temporally growing disturbances, i.e. for waves having real wavenumber and complex frequency. A review of the relevant literature shows that equivalent necessary conditions for spatially growing disturbances (i.e. for waves with real frequency and complex wavenumber) have not been formally proposed and proved (e.g. [3]). Despite the absence of a theoretical basis, the inflection point theorem has been tacitly assumed to hold for spatial instability also (e.g. [4,5] among others)—an expectation which was justified only *a posteriori*. In this connection, Michalke [6] has remarked about the lacuna in the spatial stability literature due to non-availability of the standard stability theorems, such as Rayleigh’s theorem. This motivated us to carry out this work.

An alternative approach which uses spatio-temporal formulation for stability analysis [7] has, to some extent, touched upon this issue. In this approach, both wavenumber and frequency are taken complex and this allows a classification into absolutely and convectively unstable flows. The spatio-temporal formulation shows that the integration contours (for the inverse Fourier transform) in the complex wavenumber and frequency planes can be continuously deformed, without changing the result of the integration, provided certain conditions such as analyticity, causality and convergence are satisfied in the region under consideration (see [8]). For convectively unstable flows what this means is that the contour along the real wavenumber axis can be deformed until the contour in the frequency plane coincides with the real frequency axis. Thus, the above framework provides a way of transforming the temporal stability formulation into a spatial one (and vice versa), and as a result one may expect that the classical stability theorems like the Rayleigh and Fjortoft carry over to the spatial stability problem. This line of reasoning seems to indirectly support the tacit assumption mentioned above that these theorems hold for spatial instability also.

However, this argument cannot be taken as a theoretical proof of these theorems for the following reason. The above equivalence between the temporal and spatial formulations is obtained by integrating over all the wavenumbers and frequencies in the spectral space at a given cross-flow location in the physical space. On the other hand, the classical stability theorems (as will be seen below) are derived by integrating the Rayleigh equation in the cross-flow direction in the physical space for a given wavenumber or frequency. Furthermore, the characteristics of spatial modes such as the number and shape of branches, their propagation behaviour and the eigenfunctions associated with them can be (and *are* [8]) very different from those of temporal modes even for convectively unstable flows (see figure 1 for a schematic illustration). As a result, the correspondence between the two formulations, although quite powerful in enabling one to go back and forth from one to the other, does not imply that the theorems or conditions which apply to an individual temporal mode carry over to a spatial mode. In fact, it is well known that the Howard semicircle theorem, which has been proved for inviscid temporal instability, need not hold for spatial instability [8]. For example, the spatial stability analysis of the hyperbolic tangent velocity profile (involving reverse flow) shows that there can exist modes on the upstream travelling branch whose phase velocities fall outside the range dictated by the base velocity profile (see fig. 7 in [7]). It is therefore clear that, from a theoretical point of view, a direct and rigorous proof to the Rayleigh and Fjortoft theorems for spatial instability is still warranted.

In this paper, we provide such a proof. Starting from the Rayleigh equation for two-dimensional disturbances, we first present the consequences of a formal analysis (in the spatial framework) in the most general form. We then outline the conditions under which the spatial analogues of the Rayleigh and Fjortoft theorems can be recovered. Note that even though the analysis presented here is classical in nature, it is still relevant as it provides a rigorous theoretical basis for applying the theorems to the spatial stability problems.

## 2. Analogue of rayleigh necessary condition for spatial instability

The Rayleigh equation governing the linear inviscid stability of a plane parallel incompressible base flow with respect to two-dimensional disturbances is given by
*U*(*y*) is the base velocity profile, *ω* is the circular frequency, *α* is the streamwise wavenumber, *ϕ* is the amplitude of the disturbance streamfunction given by *y* (*x* being the streamwise coordinate). Let *y*_{1} and *y*_{2} be the lower and upper boundaries of the flow domain, respectively. The boundary conditions for equation (2.1) are

For temporal instability (*α* real and *ω* complex; *ω*=*ω*_{r}+i*ω*_{i}), Rayleigh [1] derived the following condition starting from equation (2.1):
*ω*_{i} being the temporal growth/decay rate. It then follows from equation (2.3) that for (*ω*_{i}/*α*)>0 (i.e. for a temporally growing mode), *U*′′ must vanish somewhere in the flow implying the necessity of an inflection point.

In the present analysis, we follow similar steps starting from the Rayleigh equation (2.1) but consider *ω* real and *α* complex (*α*=*α*_{r}+i*α*_{i}) to suit spatial stability considerations. We rewrite equation (2.1) as
*c*=*ω*/*α* is the complex wave velocity with *c*_{ph}=(*ω*/*α*_{r}) is in general different from *c*_{r} for *α*_{i}≠0 and *c*_{r}→*ω*/*α*_{r} as *α*_{i}→0 for a finite *α*_{r}, i.e. as a neutral wave is approached. Multiplying both sides of equation (2.4) by *ϕ** (complex conjugate of *ϕ*) and integrating from *y*=*y*_{1} to *y*=*y*_{2}, we get

At this stage, with *α* and *c* complex, we split the complex quantities into their real and imaginary parts. This gives
*c*_{i} in the above equation, we obtain

For non-neutral spatial disturbances, *α*_{i}≠0. The sign of *α*_{i} for a growing spatial wave depends upon its group velocity d*ω*/d*α*. For a downstream travelling growing wave *α*_{i}<0, and for an upstream travelling wave *α*_{i}>0. Note that, for each growing wave, a decaying wave is also a solution to the Rayleigh equation as both the wavenumber and its complex conjugate satisfy the equation. Thus, the following analysis would equally apply to decaying waves (although such waves may not be physically relevant; see §4 for a brief discussion). With this observation, we proceed further. A slight rearrangement of equation (2.6), after cancellation of *α*_{i} from both sides, yields

Looking at equation (2.7), it is clear that the sign of *U*′′ on the r.h.s. is dictated by the sign of *ω*/*α*_{r}(=*c*_{ph}) on the l.h.s. Thus, the situation for spatial instability is more complicated than that for temporal instability, for which the requirement on the sign of *U*′′ is independent of any particular unstable wave (see equation (2.3)). This is understandable because in the Rayleigh equation *ω* (complex in the temporal problem) is raised to a power of unity, whereas *α* (complex for the spatial problem) is not. Thus, the most general form of the necessary condition in the spatial framework can be stated as follows.

### Theorem 2.1

*The necessary condition for inviscid spatial instability of a plane parallel base flow with respect to a given unstable wave is the existence of a region in the flow where the curvature of the velocity profile has the same sign as that of the phase velocity of the wave.*

In this form, the theorem does not seem to necessitate the presence of an inflection point in the flow. It is therefore of interest to find out if there exist conditions under which an inflection point would be necessary. Referring back to equation (2.7), for a given flow situation, there are three possibilities with regard to the type of waves that may appear.

(1) The flow supports non-neutral waves with

*c*_{ph}both positive and negative. For*c*_{ph}>0, l.h.s.>0; this implies r.h.s.>0, which demands that*U*′′>0 somewhere in the flow. Similarly, for*c*_{ph}<0, the requirement is that*U*′′<0 somewhere in the flow. Thus, when both types of waves are present, for spatial instability, there must be a point in the flow where*U*′′=0, i.e. a point of inflection.(2) The flow supports non-neutral waves with

*c*_{ph}>0 alone. For this case, l.h.s.>0 for all waves and equation (2.7) requires that*U*′′>0 somewhere in the flow.(3) The flow supports non-neutral waves with

*c*_{ph}<0 alone. For this case, l.h.s.<0 for all waves and equation (2.7) requires that*U*′′<0 somewhere in the flow.

The second as well as the third possibility suggest that there can be flows that are spatially unstable but without a point of inflection. Now the contraposition of Rayleigh’s theorem states that without an inflection point a flow *cannot* be temporally unstable. Thus, the last two possibilities above point to (possibly bizarre) flows that are spatially unstable but temporally stable. This sounds improbable though it may not be entirely impossible. (This incidentally shows the merit of deriving the necessary conditions in the spatial framework directly rather than indirectly inferring them from the temporal framework—it points to possibilities for spatial instability that may not exist for temporal instability.) Anyhow, if we restrict ourselves to flows which are both temporally and spatially unstable, then the first possibility above gives us the spatial analogue of Rayleigh’s inflection point theorem, i.e.

*For flows which are inviscidly spatially (and temporally) unstable, the necessary condition is that there exists a region in the flow where the curvature of the velocity profile vanishes.*

Now almost all the spatially unstable flows (both wall-bounded and wall-free) analysed in the literature so far, as far as the author knows, have been found to be temporally unstable also (a typical example would be the convectively unstable flows). Therefore, the restricted form of the necessary condition above gives rigorous theoretical support to applying the inflection point criterion for spatial instability for these flows.

It is important to note that the necessary condition presented above is valid for *α*_{i}≠0, i.e. for non-neutral disturbances. For a neutral disturbance, Foote & Lin [10] showed (in the temporal framework) that, if the jump in the Reynolds stress across the critical layer is zero, the base profile must contain a point of inflection whose location coincides with that of the critical layer. In other words, for such flows, the wave speed of the neutral disturbance matches with the base-flow velocity at the inflection point, which makes the critical point regular (as against singular). Following the procedure of Foote and Lin, it is easy to show that the same result essentially carries over to a neutral mode in the spatial framework. This is not surprising as, for a neutral disturbance, the spatial or temporal framework is irrelevant—the growth rate (temporal or spatial) being identically zero.

## 3. Analogue of fjortoft necessary condition for spatial instability

We next move to the analogue of the Fjortoft theorem. Note that the following analysis applies only to disturbances belonging to possibility 1 above (i.e. those which necessitate an inflection point in the flow). The real part of equation (2.5) reads

Again following Fjortoft’s line of analysis, we multiply equation (2.7) by (*c*_{r}−*U*_{s}) and add it to equation (3.1). Here, *U*_{s} is the velocity at the inflection point and *c*_{r}−*U*_{s}≠0 for a spatially growing wave. This gives
*is* negative definite and the second term is if *K*>0. For a general spatially growing wave, it is not possible to determine the sign of *K* *a priori* unless we know something about the dispersion relation. However, it is instructive to look at what happens for the near-neutral disturbances.

Now it is well known that the presence of an inflection point is a sufficient condition for the existence of an inviscid neutral wave [11]. To proceed further, we assume that the dispersion relation is analytic around the neutral point. This sounds reasonable although the pathological cases involving an isolated neutral point cannot be ruled out, for which the following analysis will not hold. (Incidentally, the same assumption was also made by Tollmien (see [11], p. 134) towards showing that an inflection point is a sufficient condition for temporal instability for a certain class of velocity profiles.) If such an analyticity is granted, we can expand *c*_{r} using Taylor’s series around the neutral point, i.e. for *α*_{i}→0 (with *α*_{r} finite, as dictated by the dispersion relation),
*c*_{rn} is the wave speed of the neural disturbance. Taking the inverse of the above expression and using binomial expansion, we get
*K*. This gives
*c*_{rn}=*U*_{s}. Substituting this in the above expression and arranging the terms in increasing order of *α*_{i}, we get
*α*_{i}→0, *K*→1 and therefore in this limit the l.h.s. of equation (3.3) is negative definite. This implies *U*′′(*U*−*U*_{s})<0 somewhere in the flow. Thus, the spatial analogue of the Fjortoft theorem can be stated as follows.

### Theorem 3.1

*The necessary condition for inviscid spatial instability of a plane parallel base flow with respect to near-neutral disturbances (provided the neutral mode is not isolated) is the existence of a region in the flow where U′′(U−U*_{s}*)<0.*

Note that even though the above condition is shown to be valid only for near-neutral disturbances that does not make it less general. This is because in a spatially unstable flow, near-neutral disturbances can be expected to be always present (except in the rare cases in which the neutral mode is isolated). Again the above theorem is the reason why almost all of the flows analysed so far in the spatial framework have been found to obey the Fjortoft condition.

## 4. Discussion

The analysis presented above in the spatial framework has an immediate relevance to contemporary research, as many of the flows of interest to physicists and engineers are spatially evolving. The relevance of the inviscid framework, however, may not be as clear and warrants a discussion. From a mathematical point of view, the connection between the perfect world of inviscid flows and the real world of viscous flows can be made if the Rayleigh equation is viewed as an asymptotic limit of the Orr–Sommerfeld equation as the Reynolds number tends to infinity [12]. As a result, only those solutions of the Rayleigh equation are physically valid which are also the asymptotic solutions of the Orr–Sommerfeld equation. For example, decaying inviscid waves in the temporal framework (which are complex conjugates of the growing waves) are not physically valid for the same reason [12]. Furthermore, in this limit, inviscid solutions do not hold in critical and wall layers, wherein the effect of viscosity (or nonlinearity) needs to be included.

From a more practical point of view, the Reynolds number is always finite and the role of viscosity cannot be entirely neglected. For free shear layers, the primary effect of viscosity is to damp out the disturbances. However, it is a weak effect at high Reynolds numbers and the calculated inviscid amplification rates in such flows have been found to be quite insensitive to the changes in the Reynolds number [13]. For wall-bounded flows which have an inflectional velocity profile viscosity also plays a destabilizing role in the near-wall region, and as a result both inviscid and viscous instabilities are operative. If the inflection point is close to the wall, the viscous mechanism is dominant. On the other hand, if it is sufficiently away from the wall (e.g. a separated flow), the inviscid mechanism dominates, and the results of the Rayleigh and Orr–Sommerfeld calculations are almost identical (see [14] for more details). Thus for flows in which the inflectional instability mechanism is strong, the inviscid stability analysis is of physical relevance even when the Reynolds number is finite (provided it is sufficiently high).

## 5. Summary

We have formally proved here the analogues of Rayleigh’s inflection point theorem and Fjortoft’s theorem in the spatial stability framework. Although certain restrictive conditions apply, they do not limit the generality of the theorems in terms of their applicability to a wide class of flows. The present analysis thus provides a rigorous theoretical basis for applying the Rayleigh and Fjortoft theorems for spatial instability—a basis which hitherto has only been implicitly assumed.

## Acknowledgements

The author thanks Prof. O. N. Ramesh from the Indian Institute of Science (IISc), Bangalore, for suggesting the problem, going through an earlier draft of the manuscript and for useful discussions. Thanks are due to an anonymous referee for making useful suggestions which have helped improve the paper. This work was initiated while the author was a PhD student at the Department of Aerospace Engineering, IISc, whose support is gratefully acknowledged.

- Received August 7, 2014.
- Accepted October 28, 2014.

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