coherent (Kolmogorov) & incoherent (Heisenberg) turbulent flow decomposition
In (BrK2) we provide an alternative (quantitative model to Kolmogorov's purely qualitative statistical turbulence model. It takes into account the quantitative fluid behavior as its described by the Euler or the Navier-Stokes equations. In order to highlight and focus on the new conceptual elements (and to avoid technical difficulties) we restrict ourself to the one-dimensional Constantin-Lax-Majda (CLM) vorticity equation with a viscosity term.
In the following we provide further information related to (BrK2).
The first building element is the CLM-vorticity equation with viscosity term considered in a H(-1/2) physical (Hilbert) space framework (in sync with the H(1/2) solution concept of our 3D-NSE solution). The Hilbert scale framework is enabled by the well known Generalized Fourier Transform (GFT) concept. The second building element is about the Continuous Wavelet Transform (CWT) (as proposed by (FaM)) with its natural Hilbert space framework H(-1/2) (due to its admissibility condition). As a consequence of those two building elements the physical and the wavelet framework is identical. The third building element combines the CWT with the Hilbert transform concept to enable "space-scale decomposition" of the H(-1/2) Hilbert space leading to "localized" Heisenberg uncertainty inequality restricted to the (complementary to the H(0)-test space) closed subspace of H(-1/2). This means that the "Heisenberg uncertainty" which is about the interaction of "fluid elements" and the "fluid momentum", is only valid in the closed H(-1/2)-H(0) subspace of H(-1/2).
Working titles of the following early thoughts
" A turbulent flow field as superposition of fractional Hilbert wavelets modelling coherent structures (vortices, shocklets) and incoherent noise" (FaM1)
"Space-scale decomposition of turbulent fields into localized wavelet oscillations of finite energy" (FaM1)
"A Dawson (wavelet) function based turbulence model"
"A simple one-dimensional turbulent flow (weak H(-1/2) variation equation) model based on a revisited CLM model for the vorticity equation with viscosity term" (MaA) 5.2
“A single answer to the two “Heisenberg (relativity-turbulence) questions?”
“when H(0) conservation of law (w/o contributions from the non-linear term) meets H(-1/2) conservation law (with contribution of the non-linear term reflecting how divergence affects the velocity)”
“when today's H(0) low- and high-pass filtering (turbulence) concept meets H(-1/2) space-scale decomposition wavelet (turbulence) concept, (FaM)”
“when H(-1/2) (physical & wavelet) space meets non-stationary random functions with finite variance and related H(1/2) spectrum (FrU) 4.5”
((FaM) 5.1): “The definition of the appropriate “object” that composes a turbulent field is still missing. It would enable the study how turbulent dynamics transports these space-scale “atoms”, distorts them, and exchanges their energy during the flow evolution. If the appropriate “object” has been defined that composes a turbulent field it would enable the study how turbulent dynamics transports these space-scale “atoms”, distorts them, and exchanges their energy during the flow evolution”
The story line
1. The trilinear form of the non-linear NSE is antisymmetric. Therefore the energy inequality of the NSE with respect to the physical H(0) space does not take into account any contribution from the non-linear term. At the same time the regularity of the non-linear term cannot be smoother than the linear term. In (BrK2) an alternative physical space H(-1/2) is proposed to overcome those drawbacks providing the adequate variation equation framework to guarantee a unique 3D-NSE solution in H(1/2).
2. (FaM1): the turbulent regime develops when the non-linear term of the NSE strongly dominates the linear term. Superposition principle holds no more for non-linear phenomena. Therefore turbulent flows cannot be decomposed as a sum of independent subsystems that can be separately studied. A wavelet representation allows analyzing the dynamics in both space and scale, retaining those degrees of freedom which are essential to compute the flow evolution.
3. Methods based on wavelet (Galerkin) expansions in L(2) framework face the issue that in Galerkin methods the degrees of freedom are the expansion coefficients of a set of basis functions and these expansion coefficients are not in physical space (means in wavelet space). First map wavelet space to physical space, compute non-linear term in physical space and then back to wavelet space, is not very practical (MeM).
4. The admissibility condition is basically the norm of the H(-1/2) Hilbert space. Therefore ct(x) is a candidate for a wavelet as element of H(-1/2)-L(2). The Hilbert transform is an isomorphism on any Hilbert scale H(b), b real. Therefore the Hilbert transformed ct(x) distributional H(-1/2)-“function” is a wavelet, as well ((WeJ).
5. (FaM1): the turbulent H(-1/2)-signal can be split into two contributions: coherent bursts, corresponding to that part of the signal which can be compressed in a L(2)-wavelet basis, plus incoherent noise, corresponding to that part of the signal which cannot be compressed in a L(2)-wavelet basis, but in the H(-1/2)-wavelet basis. For the n=1 periodic case the later one corresponds to the alternative zero-state energy model of the harmonic quantum oscillator.
A revisited (H(-1/2-) CLM vorticity turbulence model with viscous term would enable non-stationary random functions with finite variance and related spectrum ((FrU) (4.54)) with respect to the corresponding H(1/2) energy norm/spectrum space.
If the solution of the Euler equation is smooth then the solution to the slightly viscous NSE with same initial data is also smooth. Adding diffusion to the CLM model makes the solution less regular (MuA). As a consequence of this the CLM model lost most of the interest in the context of NSE analysis. In (MuA) a nonlocal diffusion term is proposed removing this drawback.
The (MuA)-(DeS) proposal results in unbalanced energy norm terms. This drawback can be removed applying the Symm operator A also to the non-linear term (resulting in with same H(1/2) energy norm). This is nothing else then changing from the weak variation original equation (MuA) (10) in a H(0) framework to the weaker H(-1/2) representation (BrK). The spectral coefficient analysis and the proof of the required properties of the nonlinear operator F(w):=w*H[w] follow analog to (SaT) with reduced Hilbert scale number by -1/2.
The today's generalized revisited CLM equations (e.g. (WuM)) affect only the "energy" relevant change to the linear (dissipative) term while the non-linear terms are not adapted correspondingly to conserve the energy balance. In case the nonlinear term governs the linear term (which is the case for the turbulent flow phenomenon) this "modification" to the physical model is at least worth to be challenged.
The building concept for a turbulent flows model is about an alternatively proposed physical H(-1/2)Hilbert space which enables
- a Dawson function based hydrodynamics (including gases dynamics, (HoE)) statistics and
- a space-scale decomposition of turbulent fields into wavelet oscillations of finite energy (FaM1).
The Dawson function replaces the Gaussian function. The L(2) subspace of H(-1/2) governs the coherent flow structures while the closed subspace H(-1/2)-L(2) governs the incoherent ("noise") structures (eigendifferentials, wave packages).
The alternatively proposed Hilbert space might also enable a common model for statistics of gases and highly turbulent fluid flows (HoE). The concept of (ChF) might provide the appropriate (Nitsche-based domain decomposition) method for the solution of the underlying hypersingular integral equations.
The building of a physical (weak) H(-1/2) NSE representation can go along with an integral representation of the NSE. By this approach the successfully (RH solution & zero point energy modelling approach) applied (hypergeometric) Kummer functions enter the stage from the other (fluid ensemble statistics and Hilbert transformed probability (Gaussian) function) side. For the related Fourier analysis we refer to (PeR). For relaxed "vanishing at infinity" conditions in case of unbounded turbulent flow applying generalized harmonic analysis we refer to ((MoA), (LuJ).
“A single answer to the two “Heisenberg (relativity-turbulence) questions?”
The "transform" tool set is about the Generalized Fourier Transform (GFT), the Windowed FT (WFT), the Continuous Wavelet Transform (CWT) and the Hilbert Transform (HT). GFT, WFT and CWT do not allow localization in the phase space. All of them can be interpreted as phase-space representations. The WFT and the CWT (resp. its inverse transforms) were known in mathematics since quite a while as a "derivative" of Calderon's reproducing formula. From a group theory perspective WFT and CWT are the same, as both are built in the same way:
- WFT is built on Weyl-Heisenberg group
- CWT is built on affine-linear groups (---> affine connexions).
Both are facing the same consequences from the Heisenberg uncertainty inequality. The Heisenberg principle states that the product of the variances of localization and momentum (of a quantum or a fluid) is always greater than a positive constant. Therefore localization & momentum cannot be measured with arbitrarily exactness at the same point in time. The function which has minimal uncertainty in the neighborhood of the expectation values of localization and momentum is the Gaussian function. This impacts the "quality" of the WFT.
The advantage of the CWT against the WFT is, that the product of the variances does not "a priori" depends from the wavelet function itself, but "only" from the "zoom" parameter "a" governing the so-called "zoom effect" property of the CWT.
The answer to Heisenberg's question above is related to the "quantum gravity" problem. This is about the interaction of quanta (in our case the interaction of fluids) and the corresponding momentum of the continuum (in our case the fluid).
There is no way out from the mathematical constraint of the Heisenberg uncertainty inequality. However our solution concept take advantage of properties of the Hilbert transform (e.g. the constant Fourier coefficient of a Hilbert transform vanishes; the Hilbert tranform of a wavelet is a wavelet) which go along with the admissibility condition of a wavelet. The Hilbert transform of the ("WFT-Heisenberg-optimal") Gaussian function (which is not a wavelet) conserve the properties of the Gaussian (in a L(2)-sense), while it becomes a wavelet. The "natural" Hilbert space of (Hilbert transformed) wavelets is H(-1/2) (due to the admissibility condition). By this there is a Hilbert space framework given with identical physical and wavelet space enabling Galerkin-wavelet representation and supporting the turbulence model solution concept of (FaM).
We emphasis that the regularity of the Dirac function requires a fractional Hilbert space with scale factor -n/2+e (e := "epsilon" >0), i.e. the regularity of the Dirac function depends from the space-dimension and the regularity of the Dirac function in case of n=1 is "nearly" a H(-1/2) "function", while for n>1 an H(-1/2) "alternative Dirac" function is more regular than its origin. The "H(-1/2) physical space" concept can also be applied to the QED, e.g. for an alternative model of a quantizied Dirac particle in a given Maxwell field overcoming today's underlying vaccum state modelling challenge (absorption and emission operators).
The extensions to 3D NSE and Reynolds number
In the following we recall corresponding central supporting statements and concepts which allow to extent the concept above to space dimension n > 1:
- Hilbert transform --> Riesz transforms
- Hermite polynomials --> Hilbert transformed Hermite polynomials
- wavelet transform --> time-frequency analysis of functions with symmetry properties
(FaM1): “The notion of “local spectrum” is antinomic and paradoxical when we consider the spectrum as decomposition in terms of wave numbers for as they cannot be defined locally. Therefore a “local Fourier spectrum” is nonsensical because, either it is non-Fourier, or it is nonlocal. There is no paradox if instead we think in terms of scales rather than wave numbers. Using wavelet transform then there can be a space-scale energy be defined ((FaM) (51)-(58)) with a correspondingly defined scale decomposition in the vicinity of location x and a correspondingly defined local wavelet energy spectrum. By integration this defines a local energy density and a global wavelet energy spectrum. The global wavelet spectrum can be expressed in terms of Fourier energy spectrum. It shows that the global wavelet energy spectrum corresponds to the Fourier spectrum smoothed by the wavelet spectrum at each scale.
The concept of (FaM) enables the definition of a space-scale Reynolds number, where the average velocity is being replaced by the characteristics root mean square (rms) velocity Re(l,x)) at scale l and location x. At large scale (i.e. l ~ L) Re(L) coincides with the usual large-scale Reynolds number, where Re(L) is defined as the integral of Re(l,x) over all x of the R(n) space (FaM) (60)-(61).”
Note 1: for a given geometrical shape of the boundaries the Reynolds number is the only control parameter of the flow. If friction forces are small compared to inertia forces this corresponds to large Reynolds numbers (ScH). In (FaM) an alternative scale "l" and location "x" dependent definition of a "Reynolds" control data is proposed. It is an enabler for a turbulence model building on fluid "H(-1/2)-objects" and "H(1/2)-fluid flow" properties overcoming variance & power-law spectrum divergence challenges of current (basically K41, nonsensical “local Fourier spectrum”) turbulence theory. In (FaM) it is proposed "to study the energy spectrum of turbulent flows (which are statistically stationary (in time) and homogeneous (in space) using a wavelet (energy spectrum) representation, alternatively to the energy spectrum given by the modulus of the Fourier transform of the velocity auto-correlation (FaM1). The wavelet representation allows analyzing the dynamics in both space and scale. The wavelet analysis and synthesis can be performed locally, in contrast to the Fourier transform where the local nature of the trigonometric functions does not allow performing a local analysis. Since wavelet transforms conserve the energy and preserve locality in physical space, one can extend the concept of energy spectrum and define a local energy spectrum (FaM1) (4). Although the wavelet transform analyzes the flow using localized functions rather than complex exponentials, one can show that the global wavelet spectrum converges towards the Fourier energy spectrum provided the analyzing wavelet has enough vanishing moments. The global wavelet spectrum, defined by integrating local energy spectrum over all positions gives the correct exponent for a power-law Fourier energy spectrum of order (-b) if the analyzing wavelet has at least M > (b-1)/2 vanishing moments."
Note 2: The Gaussian (normal distribution) function is no wavelet. The Mexican hat wavelet is built out of the Gaussian function as its 2nd derivative, i.e. the Mexican hat is a wavelet of order 2. Also the first derivative of the Gaussian function is a wavelet. The Gaussian function itself does not fulfill the admissibility and the zero-average conditions (vanishing constant Fourier term) The later (also missing) property is the baseline for the alternatively proposed Zeta function theory to prove the Riemann Hypothesis (BrK), using the fact that the Hilbert transform has always a vanishing constant Fourier term. Both functions, the Gaussian function and its Hilbert transform (the Dawson function) are L(2)-norm equivalent due to the related property of the Hilbert transformation operator. The commutator of the Hilbert transform operator applied to odd function vanishes, as well, i.e. especially for the Dawson function it holds, (xH-Hx)(F)(x)=0.
From (WeJ) we quote:
"... the Hilbert transform is the unique, bounded linear operator in L(2) that commutes with translation, dilation and reflection. Therefore the HT is the unique operator that preserves the L(2) solution space of the one-dimensional, linear translation-dilation equations. The higher dimensional scaling relations involve translations, dilations and rotations of the argument vector. The appropriate bounded linear operators on L(2)(R(n)) that maps solutions into solutions would be the Riesz operators. We suggest that the Hilbert wavelets may have several useful numerical applications. These include exterior boundary value problems and the inversion of the Radon transform. The inverse Radon transform requires evaluation of derivatives and Hilbert transforms. A Galerkin approximation could be a natural application of the Hilbert wavelets."
The Galerkin method based on wavelet expansion requires (ongoing) mappings between wavelet and physical space (during computing process) which is not practical if both spaces are different which is the case for most of current (weak) variation PDE representations in a L(2)-Hilbert space framework (MeM). The proposed H(-1/2) (weak) physical space concept enables identical wavelet and physical Hilbert spaces, while at the same time enabling the full power of Galerkin method computing non-linear terms in this (newly) physical space. This points back to the "solution" section of this page with the weak H(1/2)-NSE solution of the corresponding weak NSE representation in the H(-1/2) Hilbert space and the (H(0)=L(2)-based) physical principles of quantum theory (HeW).
Note 3 (GrK): "Time-frequency analysis is a modern branch of harmonic analysis that uses the structure of translations and modulations (or time-frequency shifts) for the analysis of functions and operators. It is a form of local Fourier analysis that treats time and frequncy simultaneously and symmetrically. ... It originates in the early development of quantum mechanics by H. Weyl, E. Wigner, and J. von Neumann around 1930 ....
The ideal time-frequency representation of f would provide the occurring frequency spectrum at each instant x. The main obstruction to this ideal is the uncertainty principle:
the uncertainty principle makes the concept of an instantaneous frequency impossible
... To find the frequency spectrum of a signal f at a time x, one localizes f to a neighborhood of x and takes its Fourier transform. This leads to the short-time Fourier transform (STFT). THe localization procedure is parametrized by a window function g. ... Pseudodifferential operators can be represented as superpositions of translations and modulation operators. ....
Gaussians, its time-frequency shift properties, its Fourier transform and Plancherel's theorem play a very special role in time-frequency analysis. ... conceptually one can think of f and its Fourier transform as two different, equivalent representations of the same object f. Each of these representations contains the same information, but each one makes visible and accessible rather different features of f. ... In time-frequency analysis we search for representations that combine the features of both f and its Fourier transform into a single function, a so-called time-frequency representation. ... It is clear that neither f nor its FT alone can accomplish the tasks of of a time-frequency representation. ...
The musical score is a useful analogy for illustrating several .. concepts in time-frequncy analysis.
... in raw, qualitative form, the uncertainty principle states that
a function f and its Fourier transform cannot be supported on arbitrarily small sets. ....
... the uncertainly principle is often loosley formulated as follows:
A realizable signal occupies a region of area at least one in the time-frequency plane.
... The analogy between "time-frequency" and "location-momentum" leads to many similarities between signal analysis and quantum mechanics. ...
... in order to obtain information about local properties of f, in particular about some "local frequency spectrum", we restrict f to an interval and take the Fourier transform of this restriction. Since a sharp cut-off introduces artificial discontinuities and can create unwanted problems, we choose a smooth cut-off function as a "window".
Note 4: The Riesz operators are the natural generalization of the Hilbert transform. They are singular integral operators which are not continuous at the origin, because of the singularity at the origin. They are related to the Leray-Hopf (Helmholtz-Weyl) operator. They are idempotent, commute with translations and homothesis and they are "rotation" invariant (StE). Therefore for the Riesz transformed Gaussian the same properties are valid as for the Gaussian (time-frequency shift, density in L(2), Plancherel theorem, uncertainty principle) while ensuring rotation invariance and vanishing constant Fourier terms. This enables the application of harmonic analysis techniques of radial functions (commutative hypergroups) with its relationship to the continuous wavelet transform (RaH). The related Hankel transform (Fourier-Bessel transform which is essentially the Fourier transform of a radial function) provides the tool set to define corresponding Hilbert scales. The radial wavelet transform coincides (up to normalization) with the wavelet transform of the Bessel-Kingman hypergroup of index (n-2)/2 with corresponding Haar measure ((RaH) 1.4). For commutative hypergroups it is possible to develop harmonic analysis analogous to the one on locally compact Abelian groups. One prominent example is the group of homogenous isotropic random field generated by the parallel shifts of rotation and reflection. A stationary phase distribution is called in ergodic theory an invariant measure. In (HoE) the average of a phase function with respect to a given phase distribution function is defined by a corresponding Stieltjes integral.
In (ShA) the group of isometric transformations on R(n) (generated by the parallel shifts, the rotations and the reflections) with the homogeneous isotropic random field (and its special case of a stationary process) is considered. For homogeneous isotropic fields the covariance function is defined by Bessel functions with index (n-2)/2 combining with a non-negative random measure which provides the link back to the above.
Note 5 (MoA) p.18: In (HoE) the use of the techniques of "characteristic functionals" of a turbulent velocity field in an incompressible field is proposed. These characteristic functionals define uniquely e.g. a probability distribution P(dw) in the phase space of a turbulent flow, and hence finding them would give a complete solution of the problem of turbulence. The conceptual elements are (stationary) phase distribution and an average of a phase function F(u) with respect to a given phase distribution defined by a Lebesgue-Stieltjes integral over F(u)dP(du) where P(du) denotes the "differential" of the probability distribution P. In case the phase distribution is stationary, then the average of any phase function stays constant in time.
The proposed Hopf equation is linear, although the dynamics of fluid is nonlinear, the fundamental problem of statistical fluid mechanics. The Stieltjes integral approach is in line with the Plemelj concept. Hopf's equation is formally similar to the Schwinger equations of quantum field theory, which are equations in functional derivatives for the Greens function of interacting quantum fields. In this context we again refer to the proposed alternative zero point energy model and the related Hilbert space framework (isomorph to H(-1/2)) with inner product in the form ((du,v)).
Note 5: The "Energy Gradient Theory" of H.-S. Dou is based on an alternative laminar flow NSE representation in terms of the gradient of the total mechanical energy. It applies the concept of the magnitude and the direction of the gradient of the total mechanical energy for unit volumetric fluids. It enables time averaged NSE for turbulent flows. It is related to the concept of a "local Reynolds number K", which represents the direction of the gradient of the total mechanical energy for pressure driven flows controlling the stability of a flow under certain disturbance.
The magnitude and the direction of the gradient of the total mechanical energy are singular (boundary layer) integral operators with to-be-appropriately-defined domains, which provide the linkage to the Calderon-Zygmund theory and the approximation theory in Hilbert scales.
This means that Dou's energy gradient theory has its weak representation in appropriate Sobolev space H(a) (energy) Hilbert space framework enabling (complementary: principle of Thomson) variation resp. energy operator minimization (Trefftz, Prager and Synge, Friedrich) methods. The singular integral operator representing of this laminar flow NSE indicates a less energy Hilbert scale factor than 1 (e.g. in sync with the regularity requirements of the Plemelj normal derivative.
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"When I meet God, I am going to ask him two questions: Why relativity? And why turbulence? I really believe he will have an answer for the first."
“Fourier transform would be the appropriate tool to analyze the intrinsic structure of a turbulent flow if and only if the turbulent flow field is a superposition of waves. Only in this case are wave numbers well defined and the Fourier energy spectrum is meaningful for describing and modeling turbulence. If, on the contrary, turbulence were a superposition of point vortices then the Fourier spectrum in this case would be meaningless. The problem we still face in turbulence theory is that we have not yet identified the typical “object” that composes a turbulent field.