A statistical theory for homogeneous helical turbulence is developed under the condition of strong symmetry. The latter describes reflectional symmetry in planes through and normal to the helical unit vector eξ, which can be achieved by demanding that the mean velocity is zero. The two-point velocity correlation, the pressure–velocity correlation, and the two-point triple correlation are expressed by scalar functions in the helical unit vector system. By introducing the continuity equation for the correlations, the number of unknown functions can be reduced to such an extent that ultimately, a single scalar transport equation remains. Furthermore, a two-point Poisson equation is derived to express the pressure–velocity correlation in terms of the triple correlation. From the two-point version of the transport equation, the single-point limit is derived. Using the single-point equation, it can be shown that all velocity components are generally non-zero. Therefore, it is concluded that the phenomenon of vortex stretching is present in the helical coordinate system. Finally, the theory of axisymmetric turbulence is derived as a limiting case of helical turbulence to show consistency with former work.

We study the dynamics and interactions of elliptic active particles in a two dimensional solvent. The particles are self-propelled through prescribing a fluid stress at one half of the fluid-particle boundary. The fluid is treated explicitly solving the Stokes equation through a discontinuous Galerkin scheme, which allows to simulate strictly incompressible fluids. We present numerical results for a single particle and give an outlook on how to treat suspensions of interacting active particles.

We derive a closed system of equations modeling an active suspension using the Eulerian spatial averaging theory under the assumption of a low-Reynolds flow Re≪1. The suspension consists of a Newtonian fluid and multiple identical active, non-spherical Janus particles. The volume-averaged mass, linear momentum, angular momentum, and orientation balance equations are derived for the fluid and solid phases separately. The focus of the present work is to derive closure relations for the resulting equations, based on fluid–particle and particle–particle interactions. Also included is a numerical study of a channel flow, driven by the active forces of the particles and a pressure gradient or/and a moving wall. The numerical results indicate the importance of the Saffman effect for an active suspension.

A new approach to studying active suspensions is presented. They exhibit a specific behavior pattern, sometimes referred to as active turbulence. Starting from first principles, we establish a description for an active suspension, consisting of a Newtonian fluid and active Janus particles. The fluid phase is described by Navier–Stokes equations and the particles by Newton–Euler equations. A level set approach is used to separate the two phases, well-known from the representation of sharp interfaces in various numerical schemes. By introducing the multi-point probability density function (PDF)-approach known from hydrodynamic turbulence, we obtain a hierarchical ordered infinite set of linear statistical equations. However, the equations for the K-point PDF depend on the K + 1 and K + 2-point PDF, exposing the closure problem of active turbulence. As all statistical moments can be formed from the PDF, the latter set of equations already includes every statistical model for an active suspensions. To illustrate this, we derive the Eulerian spatial averaging theory from the hierarchy of multi-point PDF-equations.