Abstract
This chapter is devoted to isotropic incompressible turbulence. The main features of related theories are discussed, along with the recent results: energy spectrum and two-point correlations and related models, closures for non-linear terms in both physical and Fourier space, theories for grid turbulence decay including fractal grid case, non-equilibrium effects, advanced spectral theories and models for kinetic energy cascade and dynamics in physical space.
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Notes
- 1.
It is recalled that the Taylor microscale is associated with scales at which the spectrum of kinetic-energy dissipation, or equivalently the enstrophy spectrum, exhibits its maximum.
- 2.
Let us recall that the flatness factor F(a) and the skewness factor S(a) of the random field \(\varvec{a}\) are defined as
$$\begin{aligned} F(a) \equiv \frac{\langle a ^4 \rangle }{\langle a ^2 \rangle ^2}, \quad S(a) \equiv \frac{\langle a ^3 \rangle }{\langle a ^2 \rangle ^{3/2}}. \end{aligned}$$(4.8)If \(\varvec{a}\) is a Gaussian field, then
$$\begin{aligned} F(a) = 3, \quad S(a) =0. \end{aligned}$$(4.9)Still assuming that \(\varvec{a}\) is a random Gaussian field, and defining \(\varvec{\omega }_a = \text {curl} \varvec{a}\) and \({{\mathbf {\mathsf{{S}}}}}_a = \frac{1}{2} ( \nabla \varvec{a}+ \nabla ^T \varvec{a})\), one has
$$\begin{aligned} F( \omega _a )= 5/3 , \quad F ( S_a ^2 ) = 7/5. \end{aligned}$$(4.10)Another important point is that almost all non-linear functions of \(\varvec{a}\) will exhibit a non-Gaussian behavior.
- 3.
It can be shown Falkovich and Lebedev (1997) that a Gaussian random forcing having a correlation scale \(l_F\) and a time scale \(\tau _F\) yields velocity pdf tails of the form
$$\begin{aligned} \ln P ( u ) \propto - u ^4 \, \, \text {for} \, \, u \gg \max ( u_{\text {rms}} , l_F / \tau _F ), \end{aligned}$$where P(u) is the pdf of the velocity fluctuation u. For a short-correlated forcing such that \(\tau _F \ll l_F /u_{\text {rms}}\), one obtains
$$\begin{aligned} \ln P ( u ) \propto - u ^3 \, \, \text {for} \, \, l_F / \tau _F \gg u \gg u_{\text {rms}}. \end{aligned}$$Therefore, it is seen that the interplay between the external forcing and the turbulence non-linearity leads to an automatic breakdown of Gaussianity for very intense events.
- 4.
The term quasi-isotropic refers here to a state in which at least second-order statistical moments are isotropic. But some anisotropic effects due to turbulence memory may remain on higher-order moments.
- 5.
Let us emphasize here the physical meaning of the sign of T(k). The net effect of nonlinearity on modes k such that \(T(k)>0\) is a kinetic energy gain (which must be balanced by viscous effects in the statistically steady case \(\partial E(k,t) / \partial t\)), while modes such that \(T(k) < 0\) lose more kinetic energy than they gain through nonlinear interactions (these scales must be fed by a forcing term to obtain a statistically steady state). At last, scales such that \(T(k) =0\) are in equilibrium, in the sense that they don’t lose or gain kinetic energy on the mean.
- 6.
The term cascade was coined by Onsager in the late 1940s.
- 7.
This scaling is consistent with the content of the papers published by Kolmogorov in 1941. But it is worth noting that Kolmogorov never worked in the Fourier space. The expression of the turbulent kinetic spectrum was given by his PhD student A. Obhukov, and almost independently rendered popular by Heisenberg.
- 8.
It is important to keep in mind that Kolmogorov inertial range theory is a priori derived assuming that turbulence scales are at equilibrium, i.e. that they are statistically steady. Consequences of non-equilibrium, e.g. in freely decaying turbulence, are discussed in Sect. 4.5.6.3.
- 9.
Other symmetries, such as the mirror symmetry, exist but are not one-parameter symmetries.
- 10.
This analysis is performed considering the following one-parameter (Lie-group) transformation:
Once \(\varvec{X}\) is known, all elements of the symmetry group \(T_a\) can be calculated.
- 11.
An adequate choice for \(v^{\prime 2}_0\) yields \(v^{\prime 2}(t) = \mathcal{K}(t)\).
- 12.
- 13.
The value \(n=-1.38\) for \(\sigma =4\) is associated to a time varying Loitsyanski integral: \(\mathfrak {I}\sim t^{0.16}\). This results conflicts the most recent DNS results Ishida et al. (2006). This can be understood looking at the expansion of the nonlinear transfer term mediated by strongly non-local triadic interactions in the limit of very small wave numbers retrieved from two-point closures (e.g. EDQNM):
$$\begin{aligned} T(k \rightarrow 0)= \partial E/\partial t \sim A k^4 - 2 \nu _{turb} k^2 E, \end{aligned}$$where \(\nu _{turb}\) is an eddy viscosity. An error on the constant A may yield an error on the energy balance at very small wave numbers, inducing a spurious time-evolution of \(\mathfrak {I}\).
- 14.
One recognizes here \(\sqrt{\nu t}\) which is the similitude variable that appears in the dimensional analysis of the diffusive problems.
- 15.
This is consistent with the fact that it is the only solution which is fully consistent with the symmetry analysis at finite Reynolds number.
- 16.
Looking at previous results, such a theory should fill the gap that exists for \(0.1-1 \le Re_\lambda \le 100-300\).
- 17.
The Gâteaux derivative of a function \(\Psi \) at E in the direction F, with both E and F in the space spanned by the energy spectrum, is defined as:
$$ \langle \frac{\partial \Psi }{\partial E} (E), F \rangle = \left. \frac{\partial \Psi }{\partial E}\right| _{E}(F)=\lim \limits _{\epsilon \rightarrow 0} \frac{d}{d \epsilon }\Psi (E+\epsilon F). $$ - 18.
Weizsäcker and Heisenberg did not know the works of Oboukhov at this time.
- 19.
Thermalized is used here by analogy with the random molecular motion in which the macroscopic quantities such as temperature and pressure originate.
- 20.
While the observation of these structure is recent, it is worth noting that the idea that turbulent dissipation can be tied to a random distribution of vortex tubes and vortex sheets goes back to Townsend in 1951.
- 21.
This can be directly seen looking at the analytical expressions of the eigenvalues obtained for the Burgers vortex:
$$\begin{aligned} \hat{\lambda } _{\pm } = \frac{\alpha }{2} \left( -1 \pm Re_\Gamma \left( \frac{4 \nu }{\alpha r^2} \left( 1 - e^{ - \alpha r^2 / 4 \nu } \right) - e^{ - \alpha r^2 / 4 \nu } \right) \right) , \end{aligned}$$(4.364)$$\begin{aligned} \hat{\lambda } _z = \alpha , \end{aligned}$$(4.365)where \(Re_\Gamma = \Gamma / 4 \pi \nu \) is the circulation-based Reynolds number. If \(Re_\Gamma \) is high enough, the crossover between \(\hat{\lambda } _+\) and \(\hat{\lambda } _z\) occurs, i.e. there exists a region with \( \hat{\lambda } _+ \ge \hat{\lambda } _z\).
- 22.
These authors also define a strong vortex tube as a region with large vorticity and small strain rate.
- 23.
Another possible physical process for this scaling law, the interactions between vortex tubes, is shown to be much weaker than self-stretching.
- 24.
Uncoherent structures are defined here as structures with a characteristic vorticity weaker than the background vorticity \(\omega '\).
- 25.
High is to be understood as a relative notion, the reference being the wave numbers at which turbulent kinetic energy is injected/created by external forcing or hydrodynamic instabilities.
- 26.
In the one-dimensional case, the divergence-free constraint simplifies into a null space derivative constraint, leading to uniform solutions in space.
- 27.
Exact relations are \(u^{(1)} = - \hat{\omega }_\parallel /k\) and \(u^{(2)} = - \hat{u}_\parallel \) in the horizontal wave plane (\(k_\parallel =0\)).
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Sagaut, P., Cambon, C. (2018). Incompressible Homogeneous Isotropic Turbulence. In: Homogeneous Turbulence Dynamics. Springer, Cham. https://doi.org/10.1007/978-3-319-73162-9_4
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