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Abstract
NavierStokes Equations; Viscosity; Laminar Flow; Transition; Turbulence Closure; Fluctuation; TimeAverage; Ensemble; Instability In the theory of hydrodynamic stability, the procedure to decompose an incompressible flow field into its basic motion and disturbances is imprecise and problematic because the disturbances, infinitesimal or finite, are illdefined quantities in analysis. The linearised equations contravene the first principles of classical mechanics while the disturbancedriven nonlinear formulation can hardly be considered as exact science. The notion that unstable and amplified disturbances precipitating the early stages of laminarturbulent transition is vague, speculative or fundamentally flawed. Similarly, turbulence is unjustifiably assumed to involve a statistical mean, and zeroonaverage fluctuations. This simplistic postulation is unpromising, and inevitably renders turbulent flows to a recondite dynamics of selfcontradiction. By consequence, the closure problem of turbulence modelling the Reynolds stresses deals with issues of no physical relevance.
NavierStokes Dynamics]On flowfield decomposition in fluid dynamics F. Lam]F. Lam
Concepts that have proven useful in classifying viewpoints readily establish imposing authority upon us, so that we forget their secular roots, and accept them as timehonoured rules. … The path of scientific progress is often blocked for a long time by such misapprehensions.
1 Background
Since the theory of hydrodynamic instability was first developed in the nineteenth century, its relevance to real flow phenomena has been under scrutiny. There has not been a single (generic) flow where the instability analyses are able to provide satisfactory explanations to the initial process of the laminarturbulent transition. Thus continual efforts have been made to reinvent the theory. We notice that one or more stability terminologies appear after every jumpstart attempt of the revival. (In Lin’s book (1955), there are or legacy instabilities; the book by Drazin (2002) summarises more than exotic kinds.) Alarmingly, the plausibilities implying unstable motions suggest that the full NavierStokes dynamics is less significant than the perturbationoriented specifications. As an unstable flow may not exist or cannot be observed in Nature or in laboratory, an instability proposition is unfalsifiable in a sense. In this note, we give a critical analysis on the very gist of the instability theory. We also extend our discussions to turbulent flow.
The equations of motions for incompressible flows in domain with boundary read
(1.1) 
where the velocity and space variables are denoted by and respectively. All the symbols have their usual meanings in fluid dynamics. The dynamic and kinematic viscosities are denoted by and respectively. We use the shorthand for transport for vector quantities and . Solutions of the equations are sought with known initial solenoidal velocity , and the noslip boundary condition . The vorticity, , is governed by the vorticity equation
(1.2) 
Its initial data must be consistent with the velocity .
The pressure Poisson equation,
(1.3) 
where the subscripts denote differentiation, shows that the pressure is an auxiliary variable as it is driven by the shears.
The study of flow instability branches out from the takenforgranted postulation that every fluid motion satisfying (1) is decomposable as a basic flow and disturbances:
(1.4) 
Examples of basic flows include plane Couette flow, plane Poiseuille flow, Blasius’ boundary layer and a parabolic profile in a straight pipe of circular cross section. Among these generic flows, only the Blasius solution is nonlinear and its timedependence is implicitly expressed in a similarity rule. An instability dynamics may be derived from (1) by direct substitution:
(1.5) 
because the basic flow is considered as a solution of (1). The noslip condition applies to . Linear stability theory hypothesises that equations (1) may be linearised for small or infinitesimal disturbances so that the nonlinear momentum term may be neglected:
(1.6) 
The linearised equations can be solved by the method of normal modes which reduces the stability analysis to an eigenvalue problem. Each of the modes is proportional to an exponential growth factor
(1.7) 
where is complex, and the tilde variables denote the eigenfunctions. A basic flow is said to be unstable if the (discrete) eigenvalue has positive real parts as the disturbances will amplify exponentially in time as .
If the eigenvalue spectrum is complete, an initial velocity disturbance can be expanded as the linear superposition of the eigenfunctions
(1.8) 
where denotes the real part, and the expansion determines the coefficients . It follows that any arbitrary disturbance can be represented by
(1.9) 
If the spectrum is incomplete or consists of continuous eigenvalues, additional justifications may be required to ensure the convergence of the normal mode expansions.
2 Experiment on transition
Reynolds (1883) observed the phenomenological nature of the transition from the laminar flow to turbulent state in a circular pipe. From the quantitative point of view, the experiments by Schubauer & Skramstad (1947) epitomize the dynamic principles that underpin the stability or instability of fluid motion. Figure 1 is a summary of the transition in a flatplate boundary layer.
We register several remarks here:

First, a disturbance in (1) is not a Taylorseries approximation of the basic flow at a fixed location (say), and in this context, the superposition specifies something rather ad hoc. The disturbances have been considered to be an inherent part of the NavierStokes dynamics as they come into existence neither from the initial data nor via the boundary condition. This inherent nature is described as the naturally excited oscillations.

Figure 1 suggests that, at least in unidirectional basic flows, the allowable disturbances may generate different laminarflow patterns prior to the transition. For stronger disturbances, say between and , the intrinsic transition has been modified. However, the modification is not a restart process but proceeds from the groundlevel transition.

Laboratory measurements are rarely free of imperfections. Thus the recorded fluctuations given in figure of Schubauer & Skramstad showed the excitations of the freestream residual disturbances. These oscillations were not the naturally excited oscillations implied by the linear theory. Logically, in Wells’ experiments where the acoustic disturbances have been minimised, we expect that the oscillations at any identical are much weaker. On the other hand, the supplementary tests demonstrated how the externally forced excitations of known specifications might be used to perturb the boundary layer (see, for instance, their figure ).

Within the framework of an initialboundary value problem, the flowfield in the whole pipe (domain ) in Reynolds’ experiment is absolutely timedependent, as shown in figures 3 to 5 on p942 of Reynolds (1883), though the local flows close to the pipe entrance (subsets of ) appear to be steady.

It must be frequent events during flow evolution that the velocity or the vorticity over a short period of time at a fixed space location is increasing or even surging. The flow may be described locally by a function for and (say). Strictly, the value of does not coincide with any of the eigenvalues of the stability theory. In view of the Weierstrass theorem, the shortintime flow can also be approximated by a polynomial of . Thus the temporal growth has no instability implications at all.
Significance of SchubauerSkramstad experiments
The experimental work of Schubauer & Skramstad (1947) is instrumental to our understanding of the laminarturbulent transition in boundary layers. Their measurements have vindicated the fact that the transition is an intrinsic process of flow evolution and its initiation is independent of any ad hoc infinitesimal disturbances. Strong external excitations or finite disturbances may trigger premature breakdown of the laminar state, thus modifying the intrinsic transition.
3 Analysis
Let us return to (1). It is critical to notice that the reduction procedure merely requires the disturbances to be “infinitesimally small” or “sufficiently small”. In practice, experimentalists deal with a few percent of Blasius’ boundary layer, and the disturbances from a loud speaker or a vibrating ribbon effectively cover a range of different magnitudes. Then we immediately encounter a conceptual difficulty. If a is small, so are and . If equation (1) admits a solution for for a basic flow, it may well admit solutions for disturbances, and , as both may be small. In general, if is infinitesimal, so is where the size parameter is a real, and (say). Similar arguments hold for the pressure perturbation. In the case of a boundary layer where the noslip condition is imposed, the values of in the vicinity of the wall and at the edge of the layer, in all likelihood, are distinct.
Likewise, should we consider a “finite” disturbance or , its meaning is even more inexact as both and are finite for arbitrary finite . The possibility, , is plausible. This extreme scenario may not have been intended in the first place.
For finite and in the order of unity, the linear decomposition can be generalised to
(3.10) 
for infinitesimal or finite disturbances. Clearly, the continuity holds for the mean flow , and for if is a constant. The nonlinear term in momentum (1) is revised as
(3.11) 
Linearisation
To linearise (3), the second and third terms on the last displayed line are retained while the last one is neglected on the ground that the product of the perturbation is a lowerorder term. Subtracting the contributions for the mean motion, the equation for the velocity disturbances becomes
(3.12) 
In view of (1) and the decomposition (3), we see that the disturbance pressure has the velocity derivatives which are proportional to both and . The pressure may be linearised to the same order.
It is easy to see that, by repeating the normalmode analysis, the size parameters become simple multiplication factors in the linearised equations and in fact both drop out the determination of the eigenvalues. Thus we may make use of the eigenvalue spectrum and the normalised eigenfunctions of the case . Instead of (1), we now specify
(3.13) 
For every , there is no limitation on the choice of and , as long as they are in the order of unity. For initial data (1), s are known. Then the righthand side of the eigenfunction expansion (1) now becomes
(3.14) 
This series always converges as arbitrary values of s can be suitably chosen. As there are infinite distinct s on any finite set of the real line according to Cantor’s diagonal analysis, the expansion (3) is nonunique and it approximates no general disturbances of infinitesimal or finite size. For an incomplete spectrum, it is sufficient to form the summation (3) only over the finite discrete eigenvalues.
Second, the reason to neglect the nonlinearity or is now a matter of ambiguity as, on infinitesimal scales, the magnitudes of , and the nonlinearity are in the same order. Nevertheless, the undisputed fact is that, as wellknown, the process of the linearisation compromises the conservation principle of linear and angular momenta at all times for any fixed and . Popular instability criteria resulting from the linearisation are conjectural and incomplete unless we disregard the laws of physics. (For inviscid flows, the conservation principles refer to the Euler equations of motion.)
Perturbed nonlinearity
Proposals have been put forward to retain the disturbance nonlinearity in the equations of motion. At least, it is expected that the improved theory should enable us to understand the linear counterpart. It follows that, without any approximation, the nonlinear instability theory is derived as
(3.15) 
subject to the initial data and the noslip condition. The Poisson equation for the pressure is explicitly given by
(3.16) 
Clearly the solution for fixed . Yet there is no connection between and . Thus can be specified independent of .
For given initial data, there are no particular a priori reasons to restrict the magnitudes of the perturbations, as discussed earlier. Consider one specific example . Then the viscous and pressure effects are opposite to those of the “solution” of . We struggle to make “correct sense” out of these scenarios.
In principle, the size parameters may be generalised to and as long as the disturbances remain sufficiently small or finite. Specifically, if we assign different s to the velocity components, the continuity reads
which, in general, does not hold at every given time. At any rate, the perturbationspecified momentum equation (3) or (1) is not uniquely formulated. The indeterminacy is a direct consequence of the vague quantitative notion of the disturbances. Regrettably, no specific physical meanings can be given to the hypothesised nonlinear instability theory. To reiterate our key observation, the evolution of the NavierStokes dynamics is spatiotemporal definite and ascribes no instability stages.
Discussion
Darrigol (2005) remarked that:
In nineteenthcentury parlance, kinetic instability broadly meant a departure from an expected regularity of motion. In hydrodynamics alone, this notion included unsteadiness of motion, nonuniqueness of the solutions of the fundamental equations under given boundary conditions, sensibility of these solutions to infinitesimal local perturbation, sensibility to infinitesimal harmonic perturbations, sensibility to finite perturbations, and sensibility to infinitelysmall viscosity.
In the lowturbulence test environment, typically , (Schubauer & Skramstad 1947), the leading edge of the flatplate was tailormade so that the selfsimilarity of the Blasius solution was experimentally exemplified in the downstream by an ingenious manner. As discussed earlier, the scheme theorising the natural TollmienSchlichting waves does not conserve linear momentum, and this is why the experiments could not verify the stability neutral curve (their figure 13). But the experimental data of the forced excitations were used to gauge the calculations of (1) and (1) for one or two locations so that a consistent calibration applied to the theory and the measurements. It happened that the scaling was very effective in compensating the theoretical momentum deficit thanks to the similarity property of Blasius’ profile. Because the actual sizes of the excitations were wellcontrolled in the tests, plausibly, this rectification sustained over a wide range of the wavenumbers and frequencies of the artificial wavetrains.
In addition, most boundary layers in practical applications do not possess any similarity. Prandtl’s boundary layer approximation works well in the vicinity of purposedesigned leading edges. Thus the local similarity profiles are an exception. Conceptually, the (mis)interpretation that the linear stability theory describes the evolution of forced disturbances cannot be substantiated. It is an established fact that the linear theory is hopeless in explaining the observed flow phenomena in pipe and plane Poiseuille flows under forced or unforced excitations. In plane Couette flow, a push to include the effects of stratification does not help clarify the fundamental issue of unspecified disturbance. In the numerical simulations, unsteadiness or temporal growth is frequently seen as evidence of instability. The unfortunate reality is that, despite the relentless concerted efforts to revise the stability theory over the past century, the contemporary arguments and answers to the instability riddles remain ‘tentative, controversial, or plainly wrong’.
4 Turbulence
The analyses of the previous section have implications in the random nature of turbulent flows because turbulence is traditionally treated as a superposition of a mean component and a fluctuating part:^{*}^{*}*Readers may consult the book by Darrigol on the idea of decomposing turbulence from a historical perspective.
(4.17) 
where the overbar stands for the time average and the prime for the fluctuations. The only restriction is that every component of the fluctuations is a zero timeaveraged quantity. All the notations in the present section refer to turbulence. If the component is averaged as
(4.18) 
for to prevail, then
(4.19) 
for arbitrary having a finite timeaverage. The limit in (4.18) simply indicates that we must take a sufficiently long period of time for the fluctuations to average out. Thus the decompositions,
(4.20) 
are also valid reductions for finite and arbitrary , and . These zoom factors may even be functions of the means. Conceptually, (4.17) and (4.20) are equivalent.
By virtue of (4.20), the mass conservation for demands
The incompressibility of the mean motion does not guarantee the fluctuation continuity. If we substitute the decomposition (4.20) into the momentum equation in (1) and take the mean, we obtain counterpart systems of the Reynoldsaveraged equations (equations (15) and (16) in Art. of Reynolds 1895) for the mean and the fluctuations. Unfortunately, the dynamics is obscure as the revised equations are now functions of the zoom factors. In particular, none of the stress terms like
is definitive except at spacetime locations where . Moreover, the integrals of energy (his equations (41) and (42) in Art.) are indeterminate because both depends on the fluctuation terms. The ambiguous nature of these stresslike terms causes difficulties in understanding the physics. If we consider a control volume in turbulence, the instantaneous momentum flux across any surface due to two fluctuating components represents a quantity of uncertainty.
However, the coefficient of correlation between and is independent of the size of the fluctuations . Analogous equations for the coefficient dynamics may be derived but they still contain the arbitrary zoom factors.
Reynolds (1895) assumed that, at every spacetime location in all turbulent flows, the case, , holds. This assumption has never been fully justified (see, for instance, Goldstein 1938). In order for the mean () to be unique, the zeroonaverage fluctuations must be in place. As implied in (4.19), the linearity loosely defines their sizes. By experiment, we may be able find a probable mean but we do not know the extent of data’s deviations from the mean. Throughout a turbulent flow, timeaveraging is a bold instrument to “measure” the inexplicable randomness whose origin is beyond the macroscopic continuum.
The regularity theory of the NavierStokes equations shows that the fluid dynamics operates at the spatiotemporal scales or finer. Significantly, the NavierStokes equations enshrine the complete description of laminar flow, the transition process, and turbulent state as long as the fine structures have been accounted for. In practice, the tour de force of the subdivision samples the spatiotemporal dynamics by large numbers of scales so as to stipulate the mean quantities as well as to smooth out the fluctuations. To reconcile the theory and experiments, laboratory measurements ought to be conducted with the comparable levels of precision. Detailed flow field surveys within a turbulent flow are notoriously difficult. For instance, the size of a hotwire probe is practically in excess of the finescales at small viscosity of common turbulence. Systematic errors due to the intrusion of a measuring probe can be destructive. Use of a laser anemometer certainly minimises the intrusive detriment only if the measuring volume is in the order of the finescales of test flow. Nevertheless, it is known that the integral properties such as the surface shear forces and the lift of a body can be reliably measured.
Summary
Because of the arbitrariness of the Reynolds stresses, the proposed turbulence equations are de facto nonunique and unproven. The same conclusion holds if we work with ensemble averages. We draw our attention to the fact that the reductionist view of splitting turbulence into a mean flow and fluctuations (4.17) or (4.20) is not inspired by a rule or a law of physical or statistical nature. In hindsight, it is not known whether a random field is always linearly decomposable. Two examples are given in the appendix A to highlight the issue. In applications, the and models, and the socalled oneequation approximations simulating the stress terms must be of minimal validity, if not empty.
The key ideas from the present analysis also apply to other fluidrelated physical systems such as magnetohydrodynamics and to branches of astrophysics involving turbulence dynamics.
Appendix Appendix A Numerical examples
Let us look at the hypothesis (4.17) from analysis. Suppose that we have made some “measurements” in two different turbulent flows at fixed locations, see figures 2 and 3 where the measured variable is a velocity and appropriate boundary conditions are satisfied. How universal is the reduction (4.17)?
The measurements in figure 2 show the hallmarks of turbulence  irregular bounded variations over time about an apparently wellfounded mean  a statistically stationary flow. These features seem to be independent of the sample rates. The graphs are defined by a simple random function:
(A 1) 
where the fluctuation is
Variables and are bounded by , and they are output of consecutive calls to the intrinsic function random_number in FORTRAN 90/95. The nonzero quantity controls the size of the fluctuations ( in figure 2). Different values of give the identical mean but distinct fluctuating amplitudes. Then “turbulence” (A 1) is not linearly decomposable in time (a composite function in time).
Similarly, the unsteady field in figure 3 is generated by an exponential function:
(A 2) 
for the random seeds and . In this case, the randomness constituents a multiplicative function on the mean motion.
References
 [1] Darrigol, O. 2005 Worlds of flow  A History of Hydrodynamics from the Bernoullis to Prandtl. Oxford University Press.
 [2] Goldstein, S. (Editor) 1938 Modern Developments in Fluid Dynamics, volume I. Oxford: Clarendon Press.
 [3] Reynolds, O. 1883 An experimental investigation of the circumstances which determine whether the motion of water shall be direct or sinuous, and of the law of resistance in parallel channels. Phil. Trans. Roy. Soc. 174, 935982.
 [4] Reynolds, O. 1895 On the dynamical theory of incompressible viscous fluids and the determination of the criterion. Phil. Trans. Roy. Soc. 186, 123164.
 [5] Schubauer, G.B. & Skramstad, H.K. 1947 Laminar boundarylayer oscillations and transition on a flat plate. J. Res. Nat. Bur. Stand. 38, 251292.
 [6] Wells, C.S. 1967 Effect of freestream turbulence on boundarylayer transition. AIAA J. 5, 172174.
25 April 2018
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