Lagrangian and Eulerian accelerations in turbulent stratified shear flows

The Lagrangian and Eulerian acceleration properties of fluid particles in homogeneous turbulence with uniform shear and uniform stable stratification are studied using direct numerical simulations. The Richardson number is varied from $\mathrm{Ri}=0$, corresponding to unstratified shear flow, to $\mathrm{Ri}=1$, corresponding to strongly stratified shear flow. The probability density functions (pdfs) of both Lagrangian and Eulerian accelerations have a stretched-exponential shape and they show a strong and similar influence on the Richardson number. The extreme values of the Eulerian acceleration are stronger than those observed for the Lagrangian acceleration. Geometrical statistics explain that the magnitude of the Eulerian acceleration is larger than its Lagrangian counterpart due to the mutual cancellation of the Eulerian and convective acceleration, as both vectors statistically show an antiparallel preference. A wavelet-based scale-dependent decomposition of the Lagrangian and Eulerian accelerations is performed. The tails of the acceleration pdfs grow heavier for smaller scales of turbulent motion. Hence the flatness increases with decreasing scale, indicating stronger intermittency at smaller scales. The joint pdfs of the Lagrangian and Eulerian accelerations indicate a trend to stronger correlations with increasing Richardson number and at larger scales of the turbulent motion. A consideration of the terms in the Navier-Stokes equation shows that the Lagrangian acceleration is mainly determined by the pressure-gradient term, while the Eulerian acceleration is dominated by the nonlinear convection term. A similar analysis is performed for the Lagrangian and Eulerian time rates of change of both fluctuating density and vorticity. The Eulerian time rates of change are observed to have extreme values substantially larger than those of their Lagrangian counterparts due to the advection terms in the advection-diffusion equation for fluctuating density and in the vorticity equation, respectively. The Lagrangian time rate of change of fluctuating vorticity is mainly determined by the vortex stretching and tilting term in the vorticity equation. Since the advection-diffusion equation for fluctuating density lacks a quadratic term, the Lagrangian time rate of change pdfs of fluctuating density show a more Gaussian shape, in particular, for large Richardson numbers. Hence, the Lagrangian acceleration and time rates of change of fluctuating density and vorticity reflect the dominant physics of the underlying governing equations, while the Eulerian acceleration and time rates of change are mainly determined by advection.

Figure 1

Evolution of the turbulent kinetic energy $K$ in nondimensional time $St$ (left) and dependence of the normalized production rate $P/\left(SK\right)$, buoyancy flux $B/\left(SK\right)$, and dissipation rate $\epsilon /\left(SK\right)$ on the Richardson number at $St=10$ (right).

Figure 2

Evolution of the ratio of turbulent potential to kinetic energy $K_{\rho }/K$ in nondimensional time $St$ (left) and dependence of this ratio on the Richardson number at $St=10$ (right).

Figure 3

Pdfs (top) and pdfs normalized with the corresponding standard deviations (bottom) of Lagrangian acceleration ${\mathit{a}}_L$ (left) and Eulerian acceleration ${\mathit{a}}_E$ (right) at nondimensional time $St=10$. Note that all pdfs are estimated using histograms with 100 bins, and they are plotted in log-lin representation. Note that pdfs for the vector quantities are shown.

Figure 4

Pdfs of the shear (top left), buoyancy (top right), pressure-gradient (bottom left), and advection (bottom right) terms in the Navier-Stokes equations at nondimensional time $St=10$.

Figure 5

Joint pdfs of Lagrangian acceleration ${\mathit{a}}_L$ and Eulerian acceleration ${\mathit{a}}_E$ for Richardson numbers $\mathrm{Ri}=0.1$ (left) and $\mathrm{Ri}=1$ (right) at nondimensional time $St=10$ using a linear color scale.

Figure 6

Pdfs of the cosine of the angles between different acceleration contributions: $cos(a_L,a_E)$ (top left), $cos(a_L,a_C)$ (top right), $cos(a_E,a_C)$ (bottom left), and $cos(a_L,a_P)$ (bottom right) for different Ri values at nondimensional time $St=10$.

Figure 7

Scale-dependent normalized pdfs of Lagrangian acceleration ${\mathit{a}}_L$ (left) and Eulerian acceleration ${\mathit{a}}_E$ (right) for Richardson numbers $\mathrm{Ri}=0.1$ (top) and $\mathrm{Ri}=1$ (bottom) at nondimensional time $St=10$. Note that the flat sections in the pdfs around 0 are artifacts of an even number of bins chosen in the computation of the pdfs. Additionally, pdfs of quantities with large variances only show fewer bins in the figure.

Figure 8

Scale-dependent joint pdfs of Lagrangian acceleration ${\mathit{a}}_L$ and Eulerian acceleration ${\mathit{a}}_E$ for Richardson numbers $\mathrm{Ri}=0.1$ (left) and $\mathrm{Ri}=1$ (right) and at large scale with scale index $j=3$ (top) and at small scale with $j=7$ (bottom) at nondimensional time $St=10$ using a linear color scale.

Figure 9

Pdfs (top) and normalized pdfs (bottom) of Lagrangian time rate of change of fluctuating density $s_L$ (left) and Eulerian time rate of change $s_E$ (right) at nondimensional time $St=10$.

Figure 10

Pdfs of the buoyancy (left) and advection (right) terms in the advection-diffusion equation for fluctuating density at nondimensional time $St=10$.

Figure 11

Pdfs (top) and normalized pdfs (bottom) of Lagrangian time rate of change of vorticity $c_L$ (left) and Eulerian time rate of change $c_E$ (right) at nondimensional time $St=10$. Note that pdfs for the vector quantities are shown.

Figure 12

Pdfs of the shear (top left), buoyancy (top right), vortex tilting and stretching (bottom left), and advection (bottom right) terms in the vorticity equation at nondimensional time $St=10$.

Figure 13

Comparison of normalized vector pdfs with their corresponding normalized component pdfs for Lagrangian acceleration (left) and Eulerian acceleration (right) for $\mathrm{Ri}=0.1$ (top) and $\mathrm{Ri}=1$ (bottom) at nondimensional time $St=10$.