Degeneracy of velocity strain-rate tensor statistics in random isotropic incompressible flows

The article considers the strain-rate tensor distribution in various isotropically distributed incompressible flows. By means of a fortunate choice of variables, a strong degeneracy in the probability distribution of strain-rate tensor characteristics is found in numerical simulations of isotropic turbulence. This allows us to reduce the probability density function (PDF) of the strain-rate tensor to a function of one variable. Also it appears that for those particular parameters that reflect the ratio of different eigenvalues of the strain tensor $(s$ and $\beta$ parameters), the shapes of their probability distributions are universal and do not depend on the specific shape of distribution. Furthermore, it is also shown analytically that for all time-reversible statistical isotropic flows the probability distribution of $s$ is uniform, which generalizes previous numerical calculations.

Figure 1

The histogram of invariants ${\xi }_{+}$ and ${\xi }_{-}$ in the JHTDB distribution case (i.e., obtained from JHU database). The red lines show the symmetry. Note that the horizontal scale differs from the vertical scale, and the levels of the function are skewed to the horizontal axis.

Figure 2

Dependence of $f_s^{-1/2}$ on $s$ in the JHTDB distribution case and linear fitting (17) of $f_s^{-1/2}$ on $s$ in the JHTDB distribution case with its analytical expression.

Figure 3

PDFs (22) and (23) of parameters $s$ and $\beta$ in the case of time-reversible distribution.

Figure 4

The histogram of invariants ${\xi }_{+}$ and ${\xi }_{-}$ in the zero region in the JHTDB distribution.

Figure 5

The domain of the maximum strain-rate tensor eigenvalue ${\lambda }_1$ and the minimum strain-rate tensor eigenvalue ${\lambda }_3$ (grayscale).

Figure 6

$D_{Q_S{R}_S}$: the domain of $Q_S$ and $R_S$ invariants (grayscale).

Figure 7

Joint PDF of the maximum and the minimum strain-rate tensor eigenvalues in the Gaussian distribution case. The axes are made dimensionless through dividing by $\langle {\omega }^2\rangle$ in the corresponding degrees: This makes the PDF contour shape universal. The difference between every two neighbor contour levels is one decade. The exponents of the decade level are the following: 0 for the red point (the maximum of PDF), $-1$ for the pink thickest contour, $-2$ for the orange dashed one, and $-3$ for the green dot-dashed one.

Figure 8

Joint PDF of invariants $Q_S$ and $R_S$ in the Gaussian distribution case. The axes labels are made dimensionless by means of dividing by $\langle {\omega }^2\rangle$ in corresponding powers: This makes the PDF contour shape universal. The difference between every two neighbor contour levels is one decade. The exponents of the decade level are the following: 0 for the red point (the maximum of PDF), $-1$ for the pink thickest contour, $-2$ for the orange dashed one, $-3$ for the green dot-dashed one, and $-4$ for the blue thick one.

Figure 9

Joint PDF (B5) of invariants ${\xi }_{+}$ and ${\xi }_{-}$ in the case of Gaussian distribution with $\langle {\omega }^2\rangle =125$. The axes labels are made dimensionless by means of dividing the function arguments by ${\langle {\omega }^2\rangle }^{3/2}$. The function levels are shown in the legend. The white region corresponds to the function singularity. The levels shape shows $f_{{\xi }_{+}{\xi }_{-}}$ dependence on the linear combination of its arguments (i.e., $\frac{x_{+}+x_{-}}{2})$ only, which means that for the Gaussian distribution the parameter of symmetry (10) is equal to $a=0$.