1. Scuola di Dottorato
11-12 Giugno 2009, Politecnico di Milano, Italia
LARGE EDDY SIMULATION Part 1: Introduction to turbulence and formulations
Filippo Maria Denaro
Dipartimento di Ingegneria
Aerospaziale e Meccanica,
Seconda Università di Napoli
Web:

2. OUTLINE Suggested books:
Part I – Introduction: Phenomenology and mathematical description of governing equations;
- Solution methodologies: Direct Numerical Simulation (DNS), Reynolds Averaged Navier-Stokes (RANS), Unsteady RANS (URANS), Large Eddy Simulation (LES) formulations in Computational Fluid Dynamics (CFD). Why so many acronyms ?
Part II – LES theory: Decomposition of the flow field in resolved (Large Eddies) and unresolved (Sub-Grid Scales - SGS) scales of the motion. Types of filtering in physical and wavenumbers space.
- Implicit and explicit filtering on the Navier-Stokes equations for incompressible flows (Mach = 0), the closure problem statement.
- SGS modelling type: eddy-viscosity, scale-similarity, mixed models. The dynamic procedure and approximate deconvolution modelling.
- The importance of the accuracy order of time-space discretization. Computational codes “own-made” or commercial?
- Project strategies: choice of the computational grid, discretization and SGS models; execution time (statistical steady state or not); Post-processing (qualitative, quantitative) and statistical analysis.
Suggested books:
P. Sagaut, Large Eddy Simulation for Incompressible Flows. An introduction. III Edition, Springer, 2005.
L.Berselli, T.Iliescu, W. Layton, Mathematics of Large Eddy Simulation of Turbulent Flows, Springer, 2005.
M. Lesieur, O. Métais, P. Comte, Large-Eddy Simulations of Turbulence, Cambridge University Press, 2005.
B. Guerts, Elements of Direct and Large-Eddy Simulation, Edwards, 2004.
F.F. Grinstein, L.G. Margolin, W.J. Rider, (Ed.s), Implicit Large Eddy Simulation, Cambridge, 2007.
J. Mathieu, J, Scott, An Introduction to Turbulent Flows, Cambridge University Press, 2000.
S.B. Pope, Turbulent flows, Cambridge, 2000.
J.H. Ferziger, M. Peric, Computational Methods for Fluid Dynamics, Springer, 2002.
Notes: These presentations are written with Microsoft Office PowerPoint 2003 with MathType 5.0 Equation Editor. Executing the presentation view allows use the proper hyperlinks, indicated by , to deepening sections containing details on the issues. From such sections, one returns to the main outline by clicking on the symbol

3. Phenomenology and mathematical description
Numerical simulation of turbulence is an extremely complex task. This is due to the presence of a wide range of space-temporal characteristic scales of the motion, generated by the non-linearity and not purely deterministic  Non-deterministic dynamics = chaos? 1
Turbulent motion (by definition 3D and unsteady) is characterized by a kinetic energy distribution over a range of scales, coming from the integrals ones (L, T), wherein energy is provided to the system. Then energy correlates along an inertial cascade (~k-5/3 in wavenumber space), without any effect of molecular viscosity (Theory K41), until reaching the small Kolmogorov scales (η, τ) where, viscosity becomes relevant due to high gradients and kinetic energy is irreversibly converted (dissipated) in internal energy.
Characteristic scales: the Kolmogorov scale, is the smallest one over kinetic energy is still dissipated. It depends on the viscosity:
Often one considers the Taylor micro-scale λ, the biggest of the dissipative scales, that is the scale for which the viscous stress starts to become comparable to the convective flux of the m.q.
N.B.: the magnitude of a certain scale expressed in the physical space can be transformed in the wavenumber space by taking the reciprocal, e.g., kη=O( η-1).
N.B.: Turbulence does not imply any change in the molecular properties of a fluid! In the validity of the continuum, that is O(m.f.p) << O(η), the equations system governing the motion of any fluid does not differ from laminar regime to turbulent regime
 The governing equations remain the classical Navier-Stokes (NS). 2

4. DNS, RANS, URANS, LES in CFD methods - 1
Direct Numerical Simulation (DNS) is the “simplest” CFD method for simulating turbulent flows. It is practically the virtual counterpart of a real experiment in laboratory. But … it is computationally onerous as well as not fully feasible for engineering applications (computing hours have some cost …)
DNS consists in determining the numerical solution of the NS equations for 3-D, unsteady flows, without further models or simplifications.
DNS requires a computational grid having a space-temporal resolution (h, t steps) sufficiently fine to be capable of describing the turbulence dynamic over any characteristic scales of the motion, from the largest to the smallest.
Therefore, it is plausible to assume the DNS parameters h  η, t  τ .
Consequently, DNS requires, even in the simplest case of isotropic homogeneous turbulence, a number of grid-nodes equal to L/η=O(ReL3/4) for each directions. As the time-step is proportional to the space one, the computational effort is  O(ReL3).
More critical situation for non-homogeneous turbulence:
- DNS of complete airplane would require 1015 computational nodes! Ref: Moin & Mahesh, Ann. Rev. Fluid.Mech., 30, ,1998,
- R. Moser, January 31, 2008 in: "What Can Peta-scale Direct Numerical Simulation Contribute to the Solution of the Turbulence Problem?“ available at:

5. DNS, RANS, URANS, LES in CFD methods - 2
Thus, even if DNS is the simplest method it requires a careful discretization of the equations (FD, FV, SM) as well as a well-refined grid. One wonders, how great must be the accuracy order in performing DNS?
The large scales characterizing the motion, of O(L) (or kL=O(L-1) in the wavenumber space), are set by the boundary conditions of the problem and are strongly anisotropic and energetic;
 Dependence on ReL; inertial energy cascade in the middle range frequency;
 But every consistent discretizations have necessarily a good accuracy at low wavenumber!
Energy spectra at different flow condition and Reynolds number (Ref: Saddoughi & Veeravalli, 1994)
The small scales characterizing the motion, of O(η) (or kη =O(η-1) in the wavenumber space), are much less energetic and are independent from the largest;
 Rapid energy equilibrium time and behavior almost isotropic;
 But, not every consistent discretizations guarantee a good accuracy at high wavenumber; anyway errors have effect on low energy wavenumber components! Therefore … 3

6. DNS, RANS (brief), URANS, LES in CFD methods - 3
For engineering problems, it is of interest to estimate mainly some mean parameters needed to project (Example: the evaluation of the mean force acting on a surface or a mean efflux coefficient through a valve).
The requisite of evaluating the temporal details of the flow field, that is the unsteady property of the average field, is eliminated. Actually, without the unsteady character it is no longer possible performing statistical analysis of high order…
RANS: in case of a statistically steady field, a variable f is assumed to be decomposed in a mean value and a fluctuation around the mean:
wherein the Reynolds average is defined as
being < f > an ensemble average.
One substitutes in NS eq.s, then (see other lectures…) f t P
N.B.: By means of such decomposition RANS, no isolate events are predictable.

7. DNS, RANS, URANS (brief), LES in CFD methods - 4
When the flow does not admit a statistical steady state, the time-mean is not suitable and one has to use the ensemble average < f > on the variable.
Thus, some of the low frequency components of the motion are retained in the average field.
 Some low-order statistical analysis can be performed.
URANS: For a statistically unsteady flow, any variable is decomposed in a mean value and two fluctuations around the mean value:
wherein the ensemble average is defined as
being N the number of realizations.
One substitutes in NS eq.s, then (see other lectures…) f t P
The mathematical model of URANS allows for some unsteady character; it is suitable when such character is caused by an external action, e.g.: compression-expansion cycles in internal combustion engines;
Still present some limitations on the prediction of specific events having life over temporal scales smaller than those where the average takes effect;

8. DNS, RANS, URANS, LES (intro) in CFD methods - 5
Large Eddy Simulation (LES) is a methodology based on the idea of performing a local scales separation (in general in space and time) of the characteristic scales of the motion (see Part II).
One introduces the filtering operation on the flow variables, in order for large scales of the motion to be resolved and only the effect of the small ones to be modeled. But…first it is necessary defining what is a scales separation in LES!
 One can perform high order statistical analysis and identify specific events (very similarly to DNS …). f
x,t P
LES: each variable is decomposed in a filtered variable and a fluctuation around it:
wherein the filtered variable is defined as
being G the so-called filter function.
One substitutes in NS …(object of part II)
N.B.: by means of LES, one can study theoretical problems as well as some engineering application at moderate Reynolds number.

9. f t DNS t RANS URANS t ti tf LES
DNS, RANS, URANS, LES in CFD methods - 6 A sketch of the usual levels of approximation f t
DNS t
RANS 4
URANS t ti tf 5
t,x
LES 6

10. The unresolved tensor terms in the averaged equations (RANS, URANS) versus the filtered ones (LES) - 1
Some observations are necessary to highlight the differences existing between the Navier-Stokes equation upon application either of the Reynolds average or of the LES filtering. Specifically, an unresolved tensor originates from the same non-linear quadratic term of the NS governing equations, but …
The steady or unsteady (RANS/URANS) equations, governing the averaged quantity are defined in two or three dimensions, according to the chosen problem. For the RANS equations, one has
where one can decompose the non linear convective flux according to
being for the definition of average Reynolds operator
Hence, the unresolved tensor takes into account for the effect of the turbulence fluctuations, over all the scales of motion, on the averaged ones.

11. The unresolved tensor terms in the averaged equations (RANS,URANS) versus the filtered ones (LES) - 2
On the other hand, the LES equations govern locally filtered quantities (on a spatial length ), considered unsteady and three-dimensional, the LES equations are (for homogeneous filters)
Apparently (but only apparently…), it seems to deal with similar governing equations both in RANS and in LES. Perhaps, the meaning of the resolved variable as well as of the unresolved tensor is very different! One can decompose the non linear convective flux according to (Ref: Sagaut, 2005)
since in general, for a (smooth) LES filter operator one has
L = Leonard tensor (resolved scales, it can be totally computed!);
C = Cross tensor (contains resolved and unresolved scales);
R = Reynolds tensor (contains only unresolved scales);
The SGS tensors C and R take into account for the effect of the turbulent fluctuations, associated only to the small scales of the motion, on the filtered quantities . Very different from the RANS-based Reynolds tensor!
N.B.: Only in case of an idempotent LES filter, such as the Fourier cut-off, one has G*v’=0.

12. Conclusions
DNS is mainly devoted to a theoretical approach to the study of turbulence;
DNS is useful in supporting the development of SGS modelling;
At present, applications performed on computational grids with unknown;
Perspective of future applications in DNS? Dependent on HPC progresses (Hardware + Software);
RANS/URANS still widely used in industry but LES starts to be considered feasible.
Fine presentazione Parte 1

13. * Let us try to figure it out...
1- Casualness and determinism. Turbulent flows are characterized as being strongly unsteady and three-dimensional. In principle, if one would perform several realizations (that is several either experiments or numerical simulations) each one having exactly the same initial and boundary conditions, if the governing system were of deterministic type, then one would expect identical solutions (perhaps, there are not rigorous proofs of existence and uniqueness of the solution for the Navier-Stokes equations in the most general case). As a matter of fact, one gets that, for long time duration, each realization will depart from the others because the solution is strongly dependent on small perturbations on the conditions.
- Apparently, same initial and boundary conditions would produce very different the solutions.
- A representation of a velocity component as function of space or time would induce us thinking of a purely chaotic (random) behavior of the turbulence. This is not exactly true! As a matter of fact, turbulence has not a purely chaotic behavior, at least regarding the meaning recently addressed for chaos (linked to Lyapunov exponent for dynamic systems).
- Turbulent flows are characterized as being formed by well-identifiable sequences of events, each one associated to the presence of the so-called coherent structures, that are particular vortical structures (N.B.: the presence of vorticity DOES NOT imply necessarily presence of vortices) possessing the largest part of the energy content, organized on length scales quite large, in repeatable way and of quite deterministic character.
- Perhaps, the tools provided by the statistical analysis allow us getting a clearer framework …
Large scales structures Small scales structures
Turbulent mixing layer: Flow is organized in vortical structures of different characteristic scales, among others one can recognize the so-called coherent structures. (from: Brown & Roshko, J. Fluid Mech.64, 1974)
However, the coexistent “random” part of turbulence (associated to the non-linear character of the convective terms in Navier-Stokes eqs.) causes that such events, despite of being repeatable, differs each others for the localization, dimensions and frequency of generation. In conclusion, it is such a hybrid aspect between determinism and casualness to make turbulence a scientific problem of great interest.
* Let us try to figure it out... 1

14. and when they are seen in frequency space ?
These are two signals in physical space… which one is the turbulent one?
Hard?…
and when they are seen in frequency space ?
Random signal Turbulent signal (white rumour) (correlated) *

15. 2- The governing equations: Navier-Stokes system
Turbulence does non imply a change in molecular properties of the fluid. Within the hypothesis of validity of continuum, the system of equations governing the motion of a fluid in turbulent regime is the same of that one valid for laminar regime, that is the system of Navier-Stokes equations written in integral or differential form, for example, by considering the state vector of the extensive variables
where F(w) is the tensor of the total flux (convective e diffusive).
For a fluid having homogeneous density 0 and homotermal, (Mach = 0) the system written in differential conservative form is:
(1)
where v is the velocity field, p is the “pressure” (perhaps it has no thermodynamic meaning), I is the identity matrix. Suitable initial and boundary conditions allow for the mathematical problem to be well posed.
The equation system (1) was written in vector form from which it is possible to deduce the representations valid for any reference system. In particular, it is possible rewriting the momentum equation by taking into account the constraint of the continuity equation,
Thus the momentum equation can be re-written in non-conservative form (equivalent to the equilibrium Newton law for a fluid particle) by writing the Lagrangian derivative according to
N.B.: Actually, even if mathematically equivalent in the continuous differential form, the non-conservative form is not advisable to be used for discretizations (e.g.: R.J.LeVeque, Finite Volume Method for Hyperbolic Problems, Cambridge, 2004).

16. 2- The governing equations: the Burgers model equation and the turbulence genesis -I
The genesis of turbulence is inherent to the character of the momentum equation: the quadratic non-linearity of the convective terms. As a simple model equation, that can help us clarifying the role of the non-linearity, consider
called Burgers equation from the name of the scientist J.M. Burgers which proposed in 1948 a model study of turbulence. Such equation, despite of the fact that the pressure gradient was neglected as well as a single spatial direction is considered, shows the same character of a quadratic non-linear term as in (1).
Consider in un domain of length L=1 the case of an initial condition prescribed by
that is a regular initial condition (the only non-vanishing wavenumber component corresponds to the wavenumber q=1), as happens for example in case of laminar flows.
Furthermore, let us suppose a fluid with very small viscosity, namely the convection is prevalent over diffusion on the integral scale L. In such a case, the energy dynamics is practically equivalent to that of the pure non-viscous counterpart, at least until high velocity gradient are generated.
But, what about the causes of generation of high velocity gradients starting from a smooth initial solution?
0.25

17. N.B.: But in 1D there is no vorticity dynamic …
2- The governing equations: the Burgers model equation and the turbulence genesis - II
A time-sequence of the numerical solution is plotted in the physical space (left) and in frequency space (right).
0.25
During the early times, the solution shown in physical space seems to be still quite similar to the initial one (t=0). On the other hand, one can see in the frequency space a substantial increasing of the spectral energy content at wavenumbers greater than the initial q=1.
Such behavior can be understood by considering that the initial condition is expressible as u=U1exp(i2x), thus the convective terms u2 generates a contribution to a doubled frequency! In the next times the quadratic non-linear term repeats its action, creating contribution at frequencies that are via via greater, as one can see at t=0.05.
As the time passes (t=0.1), the velocity gradient becomes stronger and also the small molecular viscosity can produce a diffusive term of magnitude comparable to the convective one.
Such a simple example allows us to understand that even a smooth flow condition (typical of laminar regime) can generate turbulence by enlarging the energy spectra over higher frequency
N.B.: But in 1D there is no vorticity dynamic …

18. 2- The governing equations: vorticity and coherent vortical structure
Turbulent flows are characterized by strong vorticity intensity, being the vorticity vector field
which is mainly associated to the high fluctuations of velocity.
Actually, the presence of a non-vanishing vorticity in a flow field does not imply that within that zone one has a vortex core. Vorticity and vortex are not the same thing! One can easily think of a laminar flow, e.g. a 2D Poiseuille motion, which has vorticity distributed in all the field, maximum at walls, but has no presence of vortical structures. On the other hand, the presence of a separation region of the flow (a “vortex”) does not imply that it is turbulent (for example, at low Reynolds number, a flow around a plane cylinder shows both vorticity and separation but it is laminar).
The instantaneous (static) detection of a vortex is based on some specific criterion and was object of several proposals (e.g., Jeong J, Hussain F. On the identification of a vortex. J. Fluid Mechanics 1995; 285), each one having some intrinsic approximation. In general, the vortex identification exploits the analysis of the vorticity magnitude along with some other measures (such as pressure gradient, eigenvalues of the strain and rotation matrices, etc.)
Even a more complex task is to define a coherent vortex. For example, Lesieur proposes defining it as a closed space region within:
1) It exists a high vorticity magnitude such that one can suppose (but it is not ensured) the existence of recirculation zones of the fluid;
2) A coherent vortex must approximately retain its shape for such a long time to be comparable to the characteristic rotation time |ζ |-1;
3) A coherent vortex should not be deterministically predictable.
The coherence of a vortex is, hence, a property associated to statistical peculiarities of the vortex dynamic and requires the preliminary “static” detection of possible candidate coherence regions.
Thus, coherent vortices are particular structures of turbulent motion of a fluid that are highly energized and have anisotropic behavior (in other words they depend on the type of the problem). DNS and LES simulations are expected to resolve such structures, RANS/URANS to model them.

19. 2- The governing equation: the Helmholtz equation and vorticity dynamic
While applying the curl operator to the momentum equation, by taking into account that
one writes the Helmholtz equation
(2)
being the so-called stretching term. Therefore, the variation of vorticity along a path-line dx/dt = v is due to two causes: 1) contribution of the stretching; 2) contribution of diffusion.
The main appearance associated to the vorticity dynamics in a turbulent flow has to be focused on the stretching, one the most relevant physical mechanisms through which intensity of turbulence is modulated. Stretching can be in effect only if the flow field is three-dimensional, because in 2D the vorticity vector is, by definition, orthogonal to the velocity gradient and the stretching term vanishes.
Consider a vortical tube V defined, analogously to a flux tube, as the tube having as surface the envelop of the vorticity vector. When the tube is stirred, the vorticity intensity increases because the vorticity field obeys the divergence-free constraint,
being A a section along the tube.
If one consider a vanishing viscosity then (2) rewrites as
Therefore, the only way the vorticity vector can change its direction is by means of the stretching action. The stretching term is responsible (even for vanishing viscosity) of the evolution of a vortical tube, being subject to change in the characteristic dimension A.
This way, vortical tubes can be stirred going from the integral scale L until to reach the characteristic small length over which the diffusive term in (2) becomes relevant (e.g., the Taylor micro-scale) and causes a rapid energy dissipation of the vortical structures (this scale is characterized by producing a unitary order Reynolds number).
N.B.: Length scales l can be easily converted in the Fourier space in terms of frequency components, i.e., k=O(l-1). Large energetic scales are hence associated to low-frequency components of the motion, the smallest scales (dissipative) to the high-frequency ones. A1 A2

20. 2 - Dynamics of vorticity in 3D.
2) Stirring and change of direction of vortical structures (stretching): the lowering of the dimension of the area increases the vorticity intensity. The effect of viscosity is still disregardful over such dimensions.
N.B.: Large structures, energetic and anisotropic.
Direction of flow
(inflow)
1) Generation of vortical tubes: caused by flow separation induced by the geometry (integral scale L=H).
N.B.: Large structures are energetic and anisotropic.
3) Fragmentation and dissipation of vortical structures: the continue action of the stretching diminishes via via the characteristic dimensions of the vortical tubes until reaching the small length (Taylor micro-scale) over which the vorticity diffusion starts. Now, energy is dissipated until the Kolmogorov scale (Re =O(1)).
N.B.: Small structures have low energy and almost isotropic behavior. .
The height of the step is half of the channel height H.
Example of the dynamics of vortical structures behind a backward facing-step of height H/2: generation, stretching, fragmentation and dissipation. The vortical structures are detected by means of the criterion proposed by Jeong.
(from: De Stefano, Denaro, Riccardi, Int. J. Num. Meth. Fluids, 37, 7, )

21. 2 - Dynamics of vorticity in 2D. (no action of the stretching)
(TEMPORAL MIXING LAYER, Iannelli, Denaro De Stefano, IJNMF, 2001)
Genesis: formation of a continuous energy spectrum due to quadratic non-linearity
Pairing: fusion from 4 to 2 vortical structures (backscatter)
Pairing: fusion from 2 to 1 vortical structure (backscatter) 2

22. 3- Criteria required for analyzing and choosing the accuracy of the discretization - I
Getting a DNS solution does not merely signify to solve the NS equation without supplying a turbulence model, over an arbitrary computational grid!
The sufficient requirement to realize a DNS is to ensure that the highest frequency contained in the turbulent energy spectra (that is the component of the motion corresponding to the Kolmogorov scale) is enclosed in the range of frequencies resolved by the discretization of the physical domain. In other words, assuming a discretization step h, uniform in each direction, in a “real” DNS the Nyquist frequency (see: Theorem of Nyquist) kN=/h must be chosen such that kN ≥O(η-1).
N.B.: Some authors have analyzed if such constraint is or not too restrictive. They deduced that a DNS is realizable as long as one resolves at least the initial part of the dissipative energy range (Taylor micro-scale). This is justified because high frequencies are subject to numerical errors but are much less energetic than largest…
Considering the problem on the length scale h of the computational cell, the relative importance between convection and diffusion of the momentum quantity must be such that the cell Reynolds number (using a suitable velocity, for example |S|l) is of unitary magnitude order. This way, also the characteristic time of diffusion and convection are comparable (consequently, the time-step t is chosen).
The accuracy analysis of a scheme is performed in terms of the analysis of the local truncation error (LTE), both in physical and in frequencies space. In particular, by performing the analysis of the LTE of a Finite Difference-based derivative in the Fourier space, allows us understanding what to expect from a discretization of a certain accuracy order.
The exact analytical derivative (D=d/dx) is obtained by taking a single generic q-th Fourier component
Conversely, a Finite Difference derivative shows a modified wavenumber keff
The modified wavenumber analysis (Ref: Ferziger & Peric, Springer, 2002) drives us to determine the error distribution in terms of modulus (dissipative effect) and phase (dispersive effect) as a function of the resolved frequency. Thus, one can highlight the way in which the error is expected to distribute along the frequency range of a real turbulent spectrum.

23. N.B.: A different conclusion can be drawn for LES analysis… Real part
3 - Criteria required for analyzing and choosing the accuracy of the discretization - II
Let us show some examples. One starts expressing keff for the central second order FD-based derivative,
Then let us express for the asymmetric first order (backward) derivative,
Thus, the central formula shows a real modified wavenumber while for the asymmetric one it is complex!! The same thing would happen for higher order formulas, e.g., fourth order central or second asymmetric ones → asymmetric derivatives add artificial dissipation.
Modified wavenumber keff for FD formulas, centered (left) and asymmetric (right) versus exact wavenumber. N.B.: Spectral methods have, by definition, exact representation up to the Nyquist frequency.
In conclusion, the numerical error is as higher as the Nyquist frequency is approached (some improvement for higher accuracy order of the formulas). Thus, the greatest errors affect the components associated to the dissipative part of the turbulence spectrum, at low energy content. Hence, possibly, DNS is feasible with schemes of relatively low accuracy order but with a sufficiently refined computational grid.
N.B.: A different conclusion can be drawn for LES analysis…
Real part
Imaginary part . 3

24. Total Modelled + = Resolved
E(k) k
Total
E(k) k
Modelled
E(k) + = k
Resolved
Symbolic representation of the decomposition of the turbulent energy spectra in a numerical solution associated to the RANS approach (from: Sagaut, Springer, III Ed. 2005) 4

25. + = Total Resolved Modelled
E(k)
E(k)
E(k) + = k k k
Total
Resolved
Modelled
Symbolic representation of the decomposition of the turbulent energy spectra in a numerical solution associated to the URANS approach (from: Sagaut, Springer, III Ed. 2005) 5

26. + = Total Resolved Modelled
E(k)
E(k)
E(k) + = k k k
Total
Resolved
Modelled
Symbolic representation of the decomposition of the turbulent energy spectra in a numerical solution associated to the LES approach (from: Sagaut, Springer, III Ed. 2005) 6

27. The Nyquist theorem - I
Such theorem states someway a “thermodynamic” constraint in representing a continuous function on a discrete set of nodes. It fixes the ultimate limitation to which any numerical method must subjugate…
In order to understand first qualitatively such theorem, let us make an example of a simple representation of the function f(x) = sin(q2x) in x [0, 1], being the wavenumber q an integer value. Consider, for example in Matlab, the plot of the function (using the instruction fplot) for q=10, 100, 500.
What happened when the wavenumber q= 500 is used? Why is the function unrecognizable and its magnitude between maximum and minimum values -1 and +1 appears wrong (there is a vanishing magnitude, however, note that also q=100 causes a slight diminished magnitude)?
The problem is that, without any other instruction, Matlab becomes not able to provide the real behavior of the function when the oscillations are too rapidly repeating in the interval to be correctly represented.
But it is sufficient to plot the function in
a smaller range, for example [0, 0.1], and
the representation turns out to be (quite) correct!!
We have simply improved the representation
of the correct oscillations by diminishing the
sample interval. Matlab recovers sufficient
resolution. Therefore …

28. The Nyquist theorem - II
In the previous example, we have shown how Matlab reached a limit of representation of the correct behavior of a of the single harmonic function when the frequency is high. Now, we explain why.
Consider a real periodic function, expressed in terms of the discrete Fourier series
being kq=2q/L (be careful in distinguishing the wavenumber q and the frequency being, actually, kq the frequency having as dimension the reciprocal of the length), L the measure of the period and N (even) the number of subdivisions of L in steps x.
Furthermore, consider the interval [-, [, so that the period L corresponds to the maximum wavelength L=λmax=2, resolvable in the interval. Conversely, the smallest resolvable wavelength is λmin=2x (namely, three grid-nodes, or two steps x, are the minimum number of nodes wherein it is possible to represent the sine function of wavelenght λmin). It follows that the highest wavenumber that is resolvable on the grid, corresponds to the ratio between the highest to the smallest wavelength, that is qmax=λmax/λmin=/x. The associated maximum frequency kmax is called Nyquist frequency.
Hence, if one tries to represent, over a computational grid,
some function having a number of harmonics greater than
the maximum given by the Nyquist frequency, the discrete
sampling does not allow to describe such harmonics. In
other words, the grid acts as a spectral cut-off filtering on
the harmonics greater than qmax:
Such principle, is quite general, being applicable also to
experimental measurements whereas the frequency response
of the measure tool implicitly defines the Nyquist frequency.
N.B.: in the previous example (qmax=λmax/λmin=1/2x), Matlab
requires to have at least N=L/x=1/(1/2q) sampling nodes, that
is N=20 at q=10 but N=1000 at q=500! This is only to simple
represent a functions in a correct way…