Fractally Fourier decimated homogeneous turbulent shear flow in noninteger dimensions

Time evolution of the fully resolved incompressible homogeneous turbulent shear flow in noninteger Fourier dimensions is numerically investigated. The Fourier dimension of the flow field is extended from the integer value 3 to the noninteger values by projecting the Navier-Stokes equation on the fractal set of the active Fourier modes with dimensions $2.7\le d\le 3.0$. The results of this study revealed that the dynamics of both large and small scale structures are nontrivially influenced by changing the Fourier dimension $d$. While both turbulent production and dissipation are significantly hampered as $d$ decreases, the evolution of their ratio is almost independent of the Fourier dimension. The mechanism of the energy distribution among different spatial directions is also impeded by decreasing $d$. Due to this deficient energy distribution, turbulent field shows a higher level of the large-scale anisotropy in lower Fourier dimensions. In addition, the persistence of the vortex stretching mechanism and the forward spectral energy transfer, which are three-dimensional turbulence characteristics, are examined at changing $d$, from the standard case $d=3.0$ to the strongly decimated flow field for $d=2.7$. As the Fourier dimension decreases, these forward energy transfer mechanisms are strongly suppressed, which in turn reduces both the small-scale intermittency and the deviation from Gaussianity. Besides the energy exchange intensity, the variations of $d$ considerably modify the relative weights of local to nonlocal triadic interactions. It is found that the contribution of the nonlocal triads to the total turbulent kinetic energy exchange increases as the Fourier dimension increases.

Figure 1

Schematic representation of homogeneous turbulent shear flow (HTSF) configuration, with $U_1$ as the mean velocity, $S$ the imposed uniform mean shear, and $(L_1,L_2,L_3)$ representing the domain size.

Figure 2

Schematic representation of the remeshing procedure in the $(k_1,k_2)$ plane. The straight arrows inside the lattice represent the shifts of the Fourier modes after each remeshing according to relabelling (13). The magnitudes of these arrows are proportional to the magnitude of the shift for each column. The circles in black (white) are active (decimated) Fourier modes. The grey circles outside the lattice indicate the exited active (decimated) Fourier modes, which are re-entered as the active (decimated) modes from the opposite boundary, as indicated with big arrows around the lattice.

Figure 3

Temporal evolution of (a) the Reynolds number based on the Taylor microscale ${\text{Re}}_{\lambda }$; (b) the streamwise integral length scale $\mathcal{L}$, normalized by the computational domain size; (c) the Kolmogorov length scale $\eta$ normalized by the grid spacing in the cross-stream direction $\mathrm{\Delta }{x}_2$.

Figure 4

The snapshots of the energy isosurfaces $q^2\approx 1.7$ of the flow fields for (a) $d=3.0$ and (b) $d=2.7$. The snapshots of the enstrophy isosurfaces ${\omega }^2\approx 160$ of the flow fields for (c) $d=3.0$ and (d) $d=2.7$. All flow fields at $St=10$.

Figure 5

The three-dimensional energy spectrum at $St=10$, developed from a $k^4$-exponential initial energy spectrum, indicated by the grey curve.

Figure 6

Time evolutions of the normalized (a) turbulent kinetic energy $q^{*2}$, and (b) enstrophy ${\omega }^{*2}$. In the insets: the asymptotic values of the nondimensional growth rates of the (a) energy ${\sigma }_{\text{eng}}\equiv {\dot{q}}^2/\left(S{q}^2\right)$, and (b) enstrophy ${\sigma }_{\text{ens}}\equiv {\dot{\omega }}^2/\left(S{\omega }^2\right)$.

Figure 7

Temporal evolution of (a) the normalized turbulent kinetic energy production $-\mathcal{P}$ and (b) the ratio of the turbulent kinetic energy production to dissipation $-\mathcal{P}/\epsilon$.

Figure 8

Time evolution of three normalized diagonal components of the Reynolds stress tensor: (a) normal $R_{22}/q^2$, (b) spanwise $R_{33}/q^2$, and (c) streamwise component $R_{11}/q^2$.

Figure 9

Time evolutions of the pressure-strain diagonal components (a) ${\mathrm{\Pi }}_{33}$, (b) ${\mathrm{\Pi }}_{22}$, and (c) ${\mathrm{\Pi }}_{11}$.

Figure 10

Plot of the Lumley triangle in the plane of invariants ($\eta , \xi$) of the Reynolds stress anisotropy tensor. The vertices correspond to two-component (2C) and one-component (1C) turbulence, as indicated. The inset in the upper side presents the final stage of the anisotropy levels for all cases.

Figure 11

Time development of different enstrophy dynamic terms. The time evolutions of (a) the vortex stretching ${\mathrm{VS}}_{\mathrm{Turb}}$, (b) the vortex stretching ${\mathrm{VS}}_{\mathrm{Mean}}$, and (c) the enstrophy dissipation ${\epsilon }_{\mathrm{ens}}$.

Figure 12

Time development of the velocity derivative skewness $S_{u_x}$.

Figure 13

(a) The spectral energy transfer $T\left(k\right)$, and (b) the spectral energy flux $\mathrm{\Pi }\left(k\right)$ for all test cases at $St=10$. The inset of (b) presents the maximum of the energy flux $\mathrm{\Pi }\left(k\right)$ vs the flow field dimension $d$.

Figure 14

Splitting energy flux $\mathrm{\Pi }\left(k\right)$ into two components ${\mathrm{\Pi }}^{+}\left(k\right)$ and ${\mathrm{\Pi }}^{-}\left(k\right)$ for all test cases at $St=10$. ${\mathrm{\Pi }}^{+}\left(k\right)$ appears with positive values while ${\mathrm{\Pi }}^{-}\left(k\right)$ takes negative value, both contributing to the forward energy flux.

Figure 15

The normalized triad energy transfer function $\overline{S}\left(l\right|n,m)$ for $k=50$. (a) $d=2.7$, and (b) $d=3.0$, at $St=10$. The contour levels are $\pm 1/2^m,m=0,\cdots ,20$ that are the same for all pictures.

Figure 16

The energy transfer locality function for all test cases at $St=10$ and $k=50$. The inset presents the variation of the locality of triads involved in $50\%$ energy transfer.

Figure 17

Comparison of time evolutions of (a) the normalized turbulent kinetic energy $q^2/q_0^2$, and (b) the normalized energy dissipation rate $\epsilon /{\epsilon }_0$, obtained with remeshing process (Rogallo) and shear periodic method (Brucker et al. [33]), for the test case $d=2.7$.