Local estimates of Hölder exponents in turbulent vector fields

It is still not known whether solutions to the Navier-Stokes equation can develop singularities from regular initial conditions. In particular, a classical and unsolved problem is to prove that the velocity field is Hölder continuous with some exponent $h<1$ (i.e., not necessarily differentiable) at small scales. Different methods have already been proposed to explore the regularity properties of the velocity field and the estimate of its Hölder exponent $h$. A first method is to detect potential singularities via extrema of an “inertial” dissipation $D*={lim}_{\ell \to 0}{D}_{\ell }^I$ that is independent of viscosity [Duchon and Robert, Nonlinearity 13, 249 (2000)]. Another possibility is to use the concept of multifractal analysis that provides fractal dimensions of the subspace of exponents $h$. However, the multifractal analysis is a global statistical method that only provides global information about local Hölder exponents, via their probability of occurrence. In order to explore the local regularity properties of a velocity field, we have developed a local statistical analysis that estimates locally the Hölder continuity. We have compared outcomes of our analysis with results using the inertial energy dissipation $D_{\ell }^I$. We observe that the dissipation term indeed gets bigger for velocity fields that are less regular according to our estimates. The exact spatial distribution of the local Hölder exponents however shows nontrivial behavior and does not exactly match the distribution of the inertial dissipation.

Figure 1

Exponent $\tau$ as a function of the parameter $q$ for the global (red circles), longitudinal (green squares), and perpendicular (blue triangles) components as well as for the native method (black diamonds) applied to 100 fields of fBm in two dimensions with three components of velocity at $H=1/3$. The three components are generated independently before enforcing the divergence-free condition. The theoretical curve from Eq. (26) is materialized by the solid magenta line. Inset: Corresponding parametric plot of $D\left(q\right)$ with respect to $h\left(q\right)$. The dashed line materializes the theoretical value of $H=1/3$.

Figure 2

Box-counting dimension of the isocontours of the global (solid red), longitudinal (dashed green), and perpendicular (dotted blue) components of the wavelet-based velocity increments for $\ell =2.8\times {10}^{-3}$ for the fBm.

Figure 3

Values of the threshold ${\mathcal{S}}_p\left(\ell \right)$ for the global increments as a function of scale for several $p$ for the fBm. The values of $p$ increase from bottom to top.

Figure 4

Box-counting dimension of the isocontours of the different components of the wavelet-based velocity increments for $\ell =8.2\times {10}^{-4}$ for the fBm. The color code for the different components is the same as that for the previous figure.

Figure 5

Pseudo “multifractal spectrum” reconstructed from the box-counting dimensions of isovalues of velocity increments at scale $\ell =8.2\times {10}^{-4}$ and the scaling laws over the $S_p\left(\ell \right)$. The red circles, green squares, and blue triangles correspond to the spectra reconstructed using the global, longitudinal, and perpendicular increments, respectively. The solid vertical line materializes the theoretical $h=1/3$.

Figure 6

Energy spectra for the two simulations. The solid blue (dashed red) curve stands for the simulation in the inertial (dissipative) range. The black line materializes the $k^{-5/3}$ slope for the inertial range. The horizontal lines of the same color as the curves materialize the corresponding fitting range used when computing power laws. The vertical lines materialize the $k_{\lambda }$ corresponding to each simulation.

Figure 7

Multifractal spectra for the velocity fields from simulation in the inertial region. The solid red, dashed green, and dotted blue curves corresponds to the global, longitudinal, and perpendicular increments, respectively. The dash-dotted black curve corresponds to the native method. The vertical line materializes the expected exponent of $h=1/3$ according to K41. The error bars correspond to a shift of the fitting range by $5\%$ for the power laws. Inset: Exponents $\tau$ as a function of $q$ for the global (red circles), longitudinal (green squares), and perpendicular (blue triangle) increments, as well as the native method (black diamonds).

Figure 8

Multifractal spectra for the velocity fields from simulation in the dissipative regime. The solid red, dashed green, and dotted blue curves corresponds to the global, longitudinal, and perpendicular increments, respectively. The dash-dotted black curve corresponds to the native method. The error bars correspond to a shift of the fitting range by $10\%$ for the power laws. Inset: Exponents $\tau$ as a function of $q$ for the global (red circles), longitudinal (green squares), and perpendicular (blue triangle) increments, as well as the native method (black diamonds).

Figure 9

Box-counting dimension of the isosurfaces of the different components of the wavelet-based velocity increments for $\ell \approx 0.24$ for the simulation in the inertial range. The color code for the different components is the same as that for the previous figures.

Figure 10

Values of the threshold ${\mathcal{S}}_p\left(\ell \right)$ for the global increments as a function of scale for several $p$ for the simulation in the inertial range.

Figure 11

Mapping function from the global velocity increments at scale $\ell \approx 0.24$ to the coefficients $\tilde{h}$.

Figure 12

Pseudospectra reconstituted using the coefficient $c_p$ computed at scale $\ell \approx 0.24$ and the box-counting dimensions of the velocity increments computed at scales $\ell \approx 0.12$ (dashed green curve), $\ell \approx 0.24$ (solid red curve), and $\ell \approx 0.46$ (dotted blue curve). The scales are also materialized by vertical lines of the same color in Fig. 10.

Figure 13

(a) Arrows stand for in-plane velocity; color plot for the singularity exponent $\tilde{h}\left(\mathit{x}\right)$. The gray areas correspond to increments outside of the domain of the mapping function. (b) Arrows stand for in-plane velocity; color plot for the Duchon-Robert term $D_{\ell }^I\left(\mathit{x}\right)$. (c) 3D representation of the same event. The green surface is an isosurface of $\tilde{h}\left(\mathit{x}\right)$ at $\tilde{h}=0.14$, and the red surface is an isosurface of $D_{\ell }^I\left(\mathit{x}\right)$ at $D_{\ell }^I=0.5$. All computations are done for a value of $\ell \approx 0.24$.

Figure 14

Joint PDF of $\tilde{h}$ and the Duchon-Robert energy transfer term $D_{\ell }^I$ over 50 fields regularly spaced over approximatively 30 turnover times. For the sake of better displaying the correlation of those terms, the PDF is rescaled such that every vertical line is a conditional PDF of $D_{\ell }^I$ at a given $\tilde{h}$.