Bjoern Hof

Streamwise-localized solutions at the onset of turbulence in pipe flow

Although the equations governing fluid flow are well known, there are no analytical expressions that describe the complexity of turbulent motion. A recent proposition is that in analogy to low dimensional chaotic systems, turbulence is organized around unstable solutions of the governing equations which provide the building blocks of the disordered dynamics. We report the discovery of periodic solutions which just like intermittent turbulence are spatially localized and show that turbulent transients arise from one such solution branch.

Scale Invariance at the Onset of Turbulence in Couette Flow

Laminar-turbulent intermittency is intrinsic to the transitional regime of a wide range of fluid flows including pipe, channel, boundary layer, and Couette flow. In the latter turbulent spots can grow and form continuous stripes, yet in the stripe-normal direction they remain interspersed by laminar fluid. We carry out direct numerical simulations in a long narrow domain and observe that individual turbulent stripes are transient. In agreement with recent observations in pipe flow, we find that turbulence becomes sustained at a distinct critical point once the spatial proliferation outweighs the inherent decaying process. By resolving the asymptotic size distributions close to criticality we can for the first time demonstrate scale invariance at the onset of turbulence.

Experimental investigation of laminar turbulent intermittency in pipe flow

In shear flows, turbulence first occurs in the form of localized structures (puffs/spots) surrounded by laminar fluid. We here investigate such spatially intermittent flows in a pipe experiment showing that turbulent puffs have a well-defined interaction distance, which sets their minimum spacing as well as the maximum observable turbulent fraction. Two methodologies are employed. Starting from a laminar flow, puffs are first created by locally injecting a jet of fluid through the pipe wall. When the perturbation is applied periodically at low frequencies, as expected, a regular sequence of puffs is observed where the puff spacing is given by the ratio of the mean flow speed to the perturbation frequency. At large frequencies however puffs are found to interact and annihilate each other. Varying the perturbation frequency, an interaction distance is determined which sets the highest possible turbulence fraction. This enables us to establish an upper bound for the friction factor in the transitional regime, which provides a well-defined link between the Blasius and the Hagen-Poiseuille friction laws. In the second set of experiments, the Reynolds number is reduced suddenly from fully turbulent to the intermittent regime. The resulting flow reorganizes itself to a sequence of constant size puffs which, unlike in Couette and Taylor–Couette flow are randomly spaced. The minimum distance between the turbulent patches is identical to the puff interaction length. The puff interaction length is found to be in agreement with the wavelength of regular stripe and spiral patterns in plane Couette and Taylor–Couette flow.

An experimental study of the decay of turbulent puffs in pipe flow.

As reported in a number of recent studies, turbulence in pipe flow is transient for Re<2000 and the flow eventually always returns to the laminar state. Generally, the lifetime of turbulence has been observed to increase rapidly with Reynolds number but there is currently no accord on the exact scaling behaviour. In particular, it is not clear whether a critical point exists where turbulence becomes sustained or if it remains transient. We here aim to clarify if these conflicting results may have been caused by the different experimental and numerical protocols used to trigger turbulence in these studies.

A hybrid MPI-OpenMP parallel implementation for pseudospectral simulations with application to Taylor–Couette flow

Abstract A hybrid-parallel direct-numerical-simulation method with application to turbulent Taylor–Couette flow is presented. The Navier–Stokes equations are discretized in cylindrical coordinates with the spectral Fourier–Galerkin method in the axial and azimuthal directions, and high-order finite differences in the radial direction. Time is advanced by a second-order, semi-implicit projection scheme, which requires the solution of five Helmholtz/Poisson equations, avoids staggered grids and renders very small slip velocities. Nonlinear terms are evaluated with the pseudospectral method. The code is parallelized using a hybrid MPI-OpenMP strategy, which, compared with a flat MPI parallelization, is simpler to implement, allows to reduce inter-node communications and MPI overhead that become relevant at high processor-core counts, and helps to contain the memory footprint. A strong scaling study shows that the hybrid code maintains scalability up to more than 20,000 processor cores and thus allows to perform simulations at higher resolutions than previously feasible. In particular, it opens up the possibility to simulate turbulent Taylor–Couette flows at Reynolds numbers up to O ( 10 5 ) . This enables to probe hydrodynamic turbulence in Keplerian flows in experimentally relevant regimes.

A Lagrangian approach to the interface velocity of turbulent puffs in pipe flow

Turbulent puffs in pipe flow are characterized by a sharp laminar-turbulent interface at the trailing edge and a more diffused leading interface. It is known that these laminar-turbulent interfaces propagate at a speed that is approximately equal to the flow rate. Our results from direct numerical simulation show that, locally, the interface velocity relative to the fluid (i) counteracts the advection due to the laminar velocity profile so that the puff can preserve its characteristic overall shape, (ii) is very small in magnitude, but involves a large interface area so that the global propagation velocity relative to the mean flow can be large and (iii) is determined by both inertial and viscous effects. The analysis provides some new insights into the mechanisms that sustain or expand localized turbulence and might be relevant for the design of new control strategies.

Hydrodynamic turbulence in quasi-Keplerian rotating flows

We report a direct-numerical-simulation study of the Taylor–Couette flow in the quasi-Keplerian regime at shear Reynolds numbers up to O(105). Quasi-Keplerian rotating flow has been investigated for decades as a simplified model system to study the origin of turbulence in accretion disks that is not fully understood. The flow in this study is axially periodic and thus the experimental end-wall effects on the stability of the flow are avoided. Using optimal linear perturbations as initial conditions, our simulations find no sustained turbulence: the strong initial perturbations distort the velocity profile and trigger turbulence that eventually decays.

Interaction of turbulent spots in pipe flow

The process of transition from laminar to turbulent regime in shear driven flows is still an unresolved issue. Localized turbulent regions or spots occur in pipe flow for Reynolds numbers around 2000. Typically in this regime an intermittent change between laminar and turbulent flow is observed (Wygnanski). Indeed, even if a large section of the laminar flow is uniformly perturbed localized turbulent spots emerge rather than an extended region of turbulence. A good understanding of this localization process is crucial for the comprehension of the transition to turbulence. We investigate the interaction of such turbulent spots in pipe flow for Reynolds numbers from 1900 to 2500. Turbulence is created locally by injecting a jet of water through a small hole in the pipe wall. For small perturbation frequencies the spacing of the turbulent spots downstream is inversely proportional to the frequency. It is observed that for distances less than approximately 20 pipe diameters turbulent spots start to interact and annihilate each other. The interaction distance is measured as a function of Reynolds number. We are also studying the effect of amplitude of the perturbations on the mutual interaction of the puffs. This investigation is closely related to spatially turbulent laminar periodic patterns which were earlier observed in other shear driven flows like Taylor-Couette or plane Couette (Prigent et al), (D. Barkley and L. Tuckerman).

Forced localized turbulence in pipe flows

Low Reynolds number turbulence is manifested in shear flows in the form of disordered patches of fluid motion embedded in laminar flow. Here, we investigate the mean properties of these patches in pipe flow and present a new method to influence their physical mechanisms.

From localized to expanding turbulence

In the transitional regime turbulence in pipe, channel and Couette flow appears in localized patches, sometimes called spots or puffs. These have a fixed length and a propagation speed which depends on the Reynolds number. It has been demonstrated in recent studies of pipe flow that such turbulent puffs have a finite lifetime and that their decay is a memoryless process [1, 2]. Generally this behaviour is consistent with the assumption that the turbulent state forms a chaotic saddle in phase space.