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A novel open-source anisotropic k − ε − v 2 − f model is presented for turbulent viscoelastic duct flow with dilute polymeric solutions described by the finitely extensible nonlinear elastic-Peterlin constitutive model. The turbulence model for channel and square duct flow of Newtonian fluids is adapted to incorporate the polymeric terms within the governing equations. All the required non-linear terms are validated with simple closure models and are assessed a priori against independent direct numerical simulation data in fully developed channel flow. The NLTij term, which accounts for the interaction between fluctuating components of the conformation tensor and the velocity gradient tensor, is modeled with the mean flow direction, ti, and wall-normal, ni, present in the Newtonian model, based on the streamwise alignment of mean polymer stretch. The implicit polymer effects on pressure–strain are assessed with a simple ad hoc closure accounting for the reduced near-wall production of turbulent kinetic energy. The same closure is also adapted for the spanwise Reynolds stress predictions of polymer-enhanced secondary flow. The model performs well in channel flow and captures low, intermediate, and high drag reduction features for a wide range of rheological parameters. The capabilities are extended for square ducts (or any regular polygon) due to the symmetric modeling of the closure models, which can predict the mean streamwise and secondary flow features associated with second normal Reynolds stress differences. Accessible codes and models are crucial for the advancement and improvement of turbulent viscoelastic models, and an OpenFOAM C++ code package is developed and freely available on GitHub (https://github.com/MikeMcDermott-Code/v2f). [ABSTRACT FROM AUTHOR]

We establish the mixing property for a family of Lebesgue measure preserving toral maps composed of two piecewise linear shears, the first of which is non-monotonic. The maps serve as a basic model for the ‘stretching and folding’ action in laminar fluid mixing, in particular flows where boundary conditions give rise to non-monotonic flow profiles. The family can be viewed as the parameter space between two well-known systems, Arnold’s Cat Map and a map due to Cerbelli and Giona, both of which possess finite Markov partitions and straightforward to prove mixing properties. However, no such finite Markov partitions appear to exist for the present family, so establishing mixing properties requires a different approach. In particular, we follow a scheme of Katok and Strelcyn, proving strong mixing properties with respect to the Lebesgue measure on two open parameter spaces. Finally, we comment on the challenges in extending these mixing windows and the potential for using the same approach to prove mixing properties in similar systems. [ABSTRACT FROM AUTHOR]

The Rayleigh Ohnesorge jetting extensional rheometer (ROJER) enables measurement of very short relaxation times of low viscosity complex fluids such as those encountered in ink-jet printing and spraying applications. This paper focuses on the design and operation of the ROJER. The performance of two nozzle designs is compared using Newtonian fluids alongside a study using computational fluid dynamics. Subsequently, a disposable nozzle is developed that overcomes issues of blockage and cleaning. The operability of this design is subject to a focused study where low viscosity polymer solutions are characterized. The test fluid materials are ethyl hydroxy-ethyl cellulose and polyethylene oxide mixed with water/glycerol solutions. Results obtained by the disposable nozzle are encouraging, paving the way for a more cost-efficient and robust ROJER setup. [ABSTRACT FROM AUTHOR]

A simple experimental setup to study the impact and coalescence of deposited droplets is described. Droplet impact and coalescence have been investigated by high-speed particle image velocimetry. Velocity fields near the liquid-substrate interface have been observed for the impact and coalescence of 2.4 mm diameter droplets of glycerol/water striking a flat transparent substrate in air. The experimental arrangement images the internal flow in the droplets from below the substrate with a high-speed camera and continuous laser illumination. Experimental results are in the form of digital images that are processed by particle image velocimetry and image processing algorithms to obtain velocity fields, droplet geometries, and contact line positions. Experimental results are compared with numerical simulations by the lattice Boltzmann method. [ABSTRACT FROM AUTHOR]

Eddy formation and presence in a plane laminar shear flow configuration consisting of two infinitely long plates orientated parallel to each other is investigated theoretically. The upper plate, which is planar, drives the flow; the lower one has a sinusoidal profile and is fixed. The governing equations are solved via a full finite element formulation for the general case and semianalytically at the Stokes flow limit. The effects of varying geometry (involving changes in the mean plate separation or the amplitude and wavelength of the lower plate) and inertia are explored separately. For Stokes flow and varying geometry, excellent agreement between the two methods of solution is found. Of particular interest with regard to the flow structure is the importance of the clearance that exists between the upper plate and the tops of the corrugations forming the lower one. When the clearance is large, an eddy is only present at sufficiently large amplitudes or small wavelengths. However, as the plate clearance is reduced, a critical value is found, which triggers the formation of an eddy in an otherwise fully attached flow for any finite amplitude and arbitrarily large wavelength. This is a precursor to the primary eddy to be expected in the lid-driven cavity flow, which is formed in the limit of zero clearance between the plates. The influence of the flow driving mechanism is assessed by comparison with corresponding solutions for the case of gravity-driven fluid films flowing over an undulating substrate. When inertia is present, the flow generally becomes asymmetrical. However, it is found that for large mean plate separations the flow local to the lower plate becomes effectively decoupled from the inertia dominated overlying flow if the wavelength of the lower plate is sufficiently small. In such cases the local flow retains its symmetry. A local Reynolds number based on the wavelength is shown to be useful in characterizing these large-gap flows. As the mean plate separation is reduced, the form of the asymmetry caused by inertia changes and becomes strongly dependent on the plate separation. For lower plate wavelengths which do not exhibit a kinematically induced secondary eddy, an inertially induced secondary eddy can be created if the mean plate separation is sufficiently small and the global Reynolds number is sufficiently large. [ABSTRACT FROM AUTHOR]

The formation and presence of eddies within thick gravity-driven free-surface film flow over a corrugated substrate are considered, with the governing equations solved semianalytically using a complex variable method for Stokes flow and numerically via a full finite element formulation for the more general problem when inertia is significant. The effect of varying geometry (involving changes in the film thickness or the amplitude and wavelength of the substrate) and inertia is explored separately. For Stokes-like flow and varying geometry, excellent agreement is found between prediction and existing flow visualizations and measured eddy center locations associated with the switch from attached to locally detached flow. It is argued that an appropriate measure of the influence of inertia at the substrate is in terms of a local Reynolds number based on the characteristic corrugation length scale. Since, for small local Reynolds numbers, the local flow structure there becomes effectively decoupled from the inertia-dominated overlying film and immune from instabilities at the free-surface; the influence of inertia manifests itself as a skewing of the dividing streamline (separatrix). It is shown that the formation and presence of eddies can be manipulated in one of two ways. While an decrease/increase in the corrugation steepness leads to the disappearance/appearance of kinematically induced eddies, an increase/decrease in the inertia present in the system leads to the appearance/disappearance of inertially induced eddies. A critical corrugation steepness for a given film thickness is defined, demarking the transition from a kinematically to an inertially induced local eddy flow structure and vice versa. [ABSTRACT FROM AUTHOR]

The effects of localized two-dimensional disturbances on the structure of shear flows featuring a stagnation line are investigated. A simple superposition of a planar Couette flow and Moffatt’s [J. Fluid Mech. 18, 1–18 (1964)] streamfunction for the decay of a disturbance between infinite stationary parallel plates shows that in general the stagnation line is replaced by a chain of alternating elliptic and hyperbolic stagnation points with a separation equal to 2.78 times the half-gap between the plates. The flow structure associated with each saddle point consists of a homoclinic separatrix and two other separatrices which locally diverge but become parallel far from the disturbance. This basic structure repeats to give a sequence of nested separatrices permitting the streamfunction to approach that of simple shear flow far from the disturbance. Using the finite-element method, the specific disturbance caused by a stationary cylinder placed on the stagnation line is considered, and results confirm the existence of the stagnation point chain, with computed separations and velocity damping ratios in very good agreement with those obtained from the Couette-Moffatt superposition. Numerical solutions also illustrate that while Reynolds number greatly affects the stagnation point separation and velocity damping ratio, these two quantities are the same for any pair of adjacent stagnation points in a given chain. Insight gained from the analysis of planar shear flows is applied to the flow in a half-filled horizontal annulus between rotating coaxial cylinders, and is used to explain why only certain flow patterns from the range of mathematically possible structures arise in previous numerical solutions. By way of contrast, the concentric annulus solution is then perturbed to allow for a small eccentricity. The nonuniformity of the intercylinder gap is shown to destroy the chain of stagnation points, but also to unfold additional flow structures not realizable when the gap is uniform. [ABSTRACT FROM AUTHOR]

A range of two- and three-dimensional problems is explored featuring the gravity-driven flow of a continuous thin liquid film over a non-porous inclined flat surface containing well-defined topography. These are analysed principally within the framework of the lubrication approximation, where accurate numerical solution of the governing nonlinear equations is achieved using an efficient multigrid solver. Results for flow over one-dimensional steep-sided topographies are shown to be in very good agreement with previously reported data. The accuracy of the lubrication approximation in the context of such topographies is assessed and quantified by comparison with finite element solutions of the full Navier-Stokes equations, and results support the consensus that lubrication theory provides an accurate description of these flows even when its inherent assumptions are not strictly satisfied. The Navier-Stokes solutions also illustrate the effect of inertia on the capillary ridge/trough and the two-dimensional flow structures caused by steep topography. Solutions obtained for flow over localized topography are shown to be in excellent agreement with the recent experimental results of Decré & Baret (2003) for the motion of thin water films over finite trenches. The spread of the "bow wave", as measured by the positions of spanwise local extrema in free-surface height, is shown to be well-represented both upstream and downstream of the topography by an inverse hyperbolic cosine function. An explanation, in terms of local flow rate, is given for the presence of the "downstream surge" following square trenches, and its evolution as trench aspect ratio is increased is discussed. Unlike the upstream capillary ridge, this feature cannot be completely suppressed by increasing the normal component of gravity. The linearity of free-surface response to topographies is explored by superposition of the free surfaces corresponding to two "equal-but-opposite" topographies. Results confirm the findings of Decré & Baret (2003) that, under the conditions considered, the responses behave in a near-linear fashion. [ABSTRACT FROM AUTHOR]