Velocity-gradient statistics along particle trajectories in turbulent flows: The refined similarity hypothesis in the Lagrangian frame

We present an investigation of the statistics of velocity gradient related quantities, in particular energy dissipation rate and enstrophy, along the trajectories of fluid tracers and of heavy/light particles advected by a homogeneous and isotropic turbulent flow. The refined similarity hypothesis (RSH) proposed by Kolmogorov and Oboukhov in 1962 is rephrased in the Lagrangian context and then tested along the particle trajectories. The study is performed on state-of-the-art numerical data resulting from numerical simulations up to ${\text{Re}}_{\lambda }\sim 400$ with ${2048}^3$ collocation points. When particles have small inertia, we show that the Lagrangian formulation of the RSH is well verified for time lags larger than the typical response time ${\tau }_p$ of the particle. In contrast, in the large inertia limit when the particle response time approaches the integral time scale of the flow, particles behave nearly ballistic, and the Eulerian formulation of RSH holds in the inertial range.

Figure 1

(Color online) Joint probability density function $\mathcal{P}\left(Q^{\ast },R^{\ast }\right)$ of $Q^{\ast }\equiv Q/{⟨Q^2⟩}^{1/2}$ and $R^{\ast }\equiv R/{⟨Q^2⟩}^{3/4}$ for particles of different types. Contour lines are drawn at values ${10}^{-z}$ with $z=0$, 1, 2, 3, 4, 5, and 6 (from the center to the outside of the figure). The thick line traces the curve: ${\left(R/2\right)}^2+{\left(Q/3\right)}^3=0$ (Viellefosse line), discriminating between complex (above) and real (below) eigenvalues of $\mathcal{A}$. Data come from ${\text{Re}}_{\lambda }=180$ calculations.

Figure 2

(Color online) Temporal autocorrelation function, i.e., $C_X\left(\tau \right)\equiv ⟨X^{\prime }\left(t\right)-X^{\prime }\left(t+\tau \right)⟩/⟨X^{\prime 2}⟩$ with $X^{\prime }\left(t\right)=X\left(t\right)-⟨X⟩$, of the enstrophy $X=\Omega$ (top) and the energy-dissipation-rate $X=\epsilon$ (bottom) for fluid tracers and for inertial particles with $\text{St}=0.5$, $\beta =0,3$, at ${\text{Re}}_{\lambda }=180$.

Figure 3

(Color online) Test of LRSH along the trajectories of tracers and heavy particles at ${\text{Re}}_{\lambda }=400$. (a) For $p=4$ we show Eq. (5) for $\text{St}=0$ (circles) and $\text{St}=2$ (squares). (b) We show the validity of Eq. (6) for the same values of $p$ and St. Straight lines correspond to the theoretical scaling prediction. Data come from ${\text{Re}}_{\lambda }=400$ calculations.

Figure 4

(Color online) Test of LRSH along the trajectories of tracers $\left({\text{Re}}_{\lambda }=400\right)$. It is plotted $⟨{\left({\delta }_{\tau }v\right)}^4⟩/\left({⟨{\left({\delta }_{\tau }v\right)}^2⟩}^2⟨X_{\tau }^2⟩{⟨X_{\tau }⟩}^{-2}\right)$: (circles) represent the case $X=\epsilon$, (squares) the case $X=\Omega$, and (triangles) the case $X=\text{const}$. In the inset the joint pdf: $P\left({ϵ}_{\tau }^{\prime },{\left|{\delta }_{\tau }v\right|}^{\prime }\right)$ and $P\left({\Omega }_{\tau }^{\prime },{\left|{\delta }_{\tau }v\right|}^{\prime }\right)$ at $\tau =13{\tau }_{\eta }$; (note that the prime symbol denotes variables normalized respect to their mean values, i.e., $x^{\prime }\equiv x/⟨x⟩$).

Figure 5

(Color online) Same as in Fig. 4, here also along the trajectories of heavy particles $\left({\text{Re}}_{\lambda }=400\right)$. It is plotted $⟨{\left({\delta }_{\tau }v\right)}^4⟩{⟨{\epsilon }_{\tau }⟩}^2/\left(⟨{\epsilon }_{\tau }^2⟩{⟨{\left({\delta }_{\tau }v\right)}^2⟩}^2\right)$. Particles with $\text{St}=0.6$, 1, 2, and 10 are compared with the result for tracers (solid black line). The LRSH is satisfied both in the inertial and in the dissipative ranges. The prefactors, $A_I$ and $A_D$, however differs in the two regions. Notice that for the largest Stokes, $\text{St}=10$, the smallest time lags where LRSH is verified, as expected, roughly $\tau \sim 10{\tau }_{\eta }$. In the inset, the behavior of ratio of prefactors $A_D/A_I$ is plotted vs St. For small St values a behavior as ${\text{St}}^{-0.38}$ is found (solid line in the inset).

Figure 6

(Color online) Same as in Fig. 4 along the trajectories of heavy/light inertial particles with $\text{St}=1.0$ $\left({\text{Re}}_{\lambda }=180\right)$. It is plotted $⟨{\left({\delta }_{\tau }v\right)}^4⟩{⟨{\epsilon }_{\tau }⟩}^2/\left(⟨{\epsilon }_{\tau }^2⟩{⟨{\left({\delta }_{\tau }v\right)}^2⟩}^2\right)$. Particles with $\beta =0$, 0.5, 0.75, 1.0, 1.5, and 3.0 are compared with the result for tracers (solid black line). The LRSH is satisfied both in the inertial and in the dissipative ranges. As for the case of heavy particles the prefactors differs in the two regions. In the inset, the behavior of ratio of prefactors $A_D/A_I$ is plotted vs $\beta$.

Figure 7

(Color online) Particles with very large inertia do not verify the LRSH (6) but follow the Eulerian version (7). We show this for the order $p=6$ on particle trajectories with $\text{St}=70$, at ${\text{Re}}_{\lambda }=400$. As it can be seen, in the inertial range, the Eulerian RSH compensate better than LRSH.