Turbulent flow and mixing are widespread in nature and in technical applications. Understanding and modeling turbulent flows presents considerable experimental, computational, and theoretical challenges due to the wide range of time and length scales involved. Turbulence and mixing are studied in canonical and simple flow geometries that retain important features present in nature and in technical devices, such as instabilities, mean shear, and external intermittency.

In our group, these flows are investigated via direct numerical simulation (DNS), solving over the whole set of time and length scales that characterize the physical processes. The results obtained with DNS can be considered exact and are comparable to those obtained from accurate measurements. In addition to providing insight into the physics of turbulence and mixing, the resulting databases are invaluable in the development of models for Large Eddy Simulation (LES) and Reynolds Average Navier-Stokes (RANS) modeling of turbulence and turbulent combustion problems.

**Goals**

- Investigate turbulence and turbulent mixing in high Reynolds number flows in relevant configurations with mean shear.

- Investigate universal scaling laws for velocity and passive scalar in flows with shear and external intermittency.

- Formulate models for LES and RANS modeling of turbulence, mixing, and combustion.

**Methods**

Our work is based on the DNS of the Navier-Stokes equations; an equation for a passive scalar (Schmidt number approximately equal to one) is also solved. The parallel flow solver “NGA” developed at Stanford University is used to solve the transport equations. The solver implements a finite difference method on a spatially and temporally staggered grid with the semi-implicit fractional-step method. Velocity and scalar spatial derivatives are discretized with a second order finite differences centered scheme. The time step size is calculated so as to have a Courant-Friedrichs-Lewy (CFL) number of one. A pressure-correction step involving the solution of a Poisson equation ensures mass conservation discretely.

The code decomposes the computational domain over a number of processors and implements a distributed memory parallelization strategy using the message-passing interface. The solution of the Poisson equation on massively parallel machines is performed by the library HYPRE using the preconditioned conjugate gradient iterative solver coupled with one iteration of an algebraic multigrid preconditioner. The solver shows an excellent scalability on massively parallel supercomputers, up to the whole Blue Gene/P system ``Shaheen’’ (65,536 processing cores) available at the KAUST Supercomputing Laboratory. The domain was discretized with 3072 × 940 × 1024 ≈ 3 billion grid points, yielding a resolution comparable to the Kolmogorov scale.

**Results**

The statistical features of turbulence in the self-similar region were analyzed in terms of longitudinal velocity structure functions, with scaling exponents estimated by applying the extended self-similarity concept. In the small-scale range (60 < r/η < 250), the scaling exponents display the universal anomalous scaling observed in homogeneous isotropic turbulence. At larger scales (r/η > 400), the mean shear and large coherent structures result in a significant deviation from predictions based on homogeneous isotropic turbulence theory. In this second scaling range, the numerical values of the exponents agree quantitatively with those reported for a variety of other flows characterized by strong shear, such as boundary layers and channel and wake flows [Attili and Bisetti, Phys. Fluids 24, 035109 (2012)].

Passive scalar spectra show inertial ranges characterized by scaling exponents −4/3 and −3/2 in the streamwise and spanwise directions, in agreement with a recent theoretical analysis of scaling of passive scalars in shear flows [Celani et al., J. Fluid Mech. 523, 99 (2005)]. Scaling exponents of high-order structure functions in the streamwise direction show saturation of intermittency with an asymptotic exponent ζ∞ = 0.4 at large orders. Saturation of intermittency is confirmed by the self-similarity of the tails of the probability density functions of the scalar increments at different scales r with the scaling factor r−ζ∞ and by the analysis of the cumulative probability of large fluctuations. Conversely, intermittency saturation is not observed for the spanwise increments and the relative scaling exponents agree with recent results for homogeneous isotropic turbulence with mean scalar gradient [Attili and Bisetti, Phys. Rev. E 88, 033013 (2013)].

The thin interface separating the inner turbulent region from the outer irrotational fluid was analyzed in the mixing layer. A vorticity threshold was defined to detect the interface separating the turbulent from the non-turbulent regions of the flow, and to calculate statistics conditioned on the distance from this interface. The conditional statistics for velocity were in remarkable agreement with the results for other types of free shear flow available in the literature, such as turbulent jets and wakes. In addition, an analysis of the passive scalar field in the vicinity of the interface was performed. We found that the scalar has a jump at the interface, even stronger than that observed for velocity. The strong jump for the scalar has been observed previously in the case of high Schmidt number. In the present study, such a strong jump was observed for a scalar with Schmidt number close to one. Conditional statistics of kinetic energy and scalar dissipation were analyzed. While the kinetic energy dissipation has its maximum far from the interface, the scalar disspation is characterized by a strong peak very close to the interface. Finally, we show that the geometric features of the contorted interfaces correlate with relatively large-scale structures that can be visualized using low-pressure isosurfaces [Attili, Cristancho, and Bisetti J. Turb. 15, 555 (2014)].

**References**

- A. Attili, J. C. Cristancho, and F. Bisetti. Statistics of the turbulent/non- turbulent interface in a spatially developing mixing layer. J. Turb., 15, 555 (2014). doi: 10.1080/14685248.2014.919394

- A. Attili and F. Bisetti. Fluctuations of a passive scalar in a turbulent mixing layer. Phys. Rev. E, 88, 033013 (2013). doi: 10.1103/PhysRevE.88.033013

- A. Attili and F. Bisetti. Statistics and scaling of turbulence in a spatially developing mixing layer at Reλ = 250. Phys. Fluids, 24, 035109 (2012). doi: 10.1063/1.3696302

- A. Attili and F. Bisetti. Structure function scaling in a Reλ = 250 turbulent mixing layer. In Journal of Physics: Conference Series, 318, 042001 (2011). doi: 10.1088/1742-6596/318/4/042001