Stochastic models for the droplet motion and evaporation in under-resolved turbulent flows at a large Reynolds number

In this work we introduce a Lagrangian stochastic model for particle motion and evaporation to be used in large-eddy simulations (LES) of turbulent liquid sprays. Effects of small-scale intermittency, usually under-resolved in LES, are explicitly included via modelling of the energy dissipation rate seen by a droplet along its trajectory. Namely, the dissipation rate is linked to the norm of the droplet sub-filtered acceleration which is included in the droplet motion equation. This norm, along with the direction of the droplet sub-filtered acceleration, is simulated as a stochastic process. With increasing Reynolds number, the distribution of the sub-filtered acceleration develops longer tails, with a slower decay in auto-correlation functions of the norm and direction of this acceleration. The stochastic models are specified for particles larger and smaller the Kolmogorov length scale. The assumption of the droplet evaporation model is similar, i.e. the evaporation rate is strongly enhanced when a droplet is subjected to very localized zones of intense velocity gradients. Thereby, the overall evaporation process is assumed to be a succession of two steady-state sub-processes with equal intensities, i.e. evaporation and vapour mixing. Then the stochastic properties of the overall evaporation rate are also controlled by fluctuations of the energy dissipation rate along the droplet path, and with increasing Reynolds number, the intensity of fluctuations of this rate is also increasing. The assessment of the presented stochastic models in LES of high-speed non-evaporating and evaporating sprays show the accurate prediction of experimental data on relatively coarser grids along with a remarkably weaker sensitivity to the grid spacing. The joint statistics and Voronoi tessellations exhibit strong intermittency of evaporation rate. The intensity of turbulence along the droplet pathway substantially promotes the vaporization rate.

Growth of vortical disturbances entrained in the entrance region of a circular pipe

The development and growth of unsteady three-dimensional vortical disturbances entrained in the entry region of a circular pipe is investigated by asymptotic and numerical methods for Reynolds numbers between $1000$ and $10\,000$ , based on the pipe radius and the bulk velocity. Near the pipe mouth, composite asymptotic solutions describe the dynamics of the oncoming disturbances, revealing how these disturbances are altered by the viscous layer attached to the pipe wall. The perturbation velocity profiles near the pipe mouth are employed as rigorous initial conditions for the boundary-region equations, which describe the flow in the limit of low frequency and large Reynolds number. The disturbance flow is initially primarily present within the base-flow boundary layer in the form of streamwise-elongated vortical structures, i.e. the streamwise velocity component displays an intense algebraic growth, while the cross-flow velocity components decay. Farther downstream the disturbance flow occupies the whole pipe, although the base flow is mostly inviscid in the core. The transient growth and subsequent viscous decay are confined in the entrance region, i.e. where the base flow has not reached the fully developed Poiseuille profile. Increasing the Reynolds number and decreasing the frequency causes more intense perturbations, whereas small azimuthal wavelengths and radial characteristic length scales intensify the viscous dissipation of the disturbance. The azimuthal wavelength that causes the maximum growth is found. The velocity profiles are compared successfully with available experimental data and the theoretical results are helpful to interpret the only direct numerical dataset of a disturbed pipe-entry flow.

Linear instability of lid- and pressure-driven flows in channels textured with longitudinal superhydrophobic grooves

It is known that an increased flow rate can be achieved in channel flows when smooth walls are replaced by superhydrophobic surfaces. This reduces friction and increases the flux for a given driving force. Applications include thermal management in microelectronics, where a competition between convective and conductive resistance must be accounted for in order to evaluate any advantages of these surfaces. Of particular interest is the hydrodynamic stability of the underlying basic flows, something that has been largely overlooked in the literature, but is of key relevance to applications that typically base design on steady states or apparent-slip models that approximate them. We consider the global stability problem in the case where the longitudinal grooves are periodic in the spanwise direction. The flow is driven along the grooves by either the motion of a smooth upper lid or a constant pressure gradient. In the case of smooth walls, the former problem (plane Couette flow) is linearly stable at all Reynolds numbers whereas the latter (plane Poiseuille flow) becomes unstable above a relatively large Reynolds number. When grooves are present our work shows that additional instabilities arise in both cases, with critical Reynolds numbers small enough to be achievable in applications. Generally, for lid-driven flows one unstable mode is found that becomes neutral as the Reynolds number increases, indicating that the flows are inviscidly stable. For pressure-driven flows, two modes can coexist and exchange stability depending on the channel height and slip fraction. The first mode remains unstable as the Reynolds number increases and corresponds to an unstable mode of the two-dimensional Rayleigh equation, while the second mode becomes neutrally stable at infinite Reynolds numbers. Comparisons of critical Reynolds numbers with the experimental observations for pressure-driven flows of Daniello et al. (Phys. Fluids, vol. 21, issue 8, 2009, p. 085103) are encouraging.

The pitfalls of investigating rotational flows with the Euler equations

Small viscous effects in high-Reynolds-number rotational flows always accumulate over time to have a leading-order effect. Therefore, the high-Reynolds-number limit for the Navier–Stokes equations is singular. It is important to investigate whether a solution of the Euler equations can approximate a real flow at large Reynolds number. These facts are often overlooked and, as a result, the Euler equations are used to simulate laminar rotational flows at large Reynolds number. Based on the Fredholm alternative, an asymptotic perturbation theory is described to establish secularity conditions determined by viscosity for an inviscid solution to approximate a real viscous fluid. Four important classical inviscid solutions are investigated using the theory with the following conclusions. The Stuart cats’ eyes and Mallier–Maslowe vortices are inconsistent with any real fluid at high Reynolds number; whereas Hill's spherical vortex is confirmed to be consistent with a steady state in the spherical core region and the Lamb–Chaplygin dipole is found to be consistent with a quasi-steady state in the circular core region. These solutions have been widely used for analysing the stability of vortex flows and wakes, and their interactions with shock waves or bubbles. Serendipitously, we have revealed an original exact solution of the Navier–Stokes equations which is time dependent, has non-zero nonlinear convective terms and is restricted to a finite domain with the decay rate depending on dipole radius.

Numerical simulations of the agitation generated by coarse-grained bubbles moving at large Reynolds number

We present a numerical method for simulating the flow induced by bubbles rising at large Reynolds number. This method is useful to simulate configurations of large dimensions involving a great number of bubbles. The action that each bubble exerts on the liquid is modelled as a volume source of momentum distributed over a few mesh-grid elements. The flow in the vicinity of the bubbles is thus not finely resolved. The bubbles are treated as Lagrangian particles that move under the influence of the hydrodynamic force exerted by the liquid. The determination of this force on a given bubble requires knowledge of the liquid flow that is undisturbed by this bubble. A model is developed to accurately estimate this disturbance for large-Reynolds-number objects and get rid of any spurious self-induced effect. Thanks to that, a homogeneous swarm of rising bubbles is simulated. Comparisons with experiments show a good agreement with the flow scales larger than the bubbles, which turn out to be controlled by the interactions between bubble wakes and rather independent of unresolved smaller scales. This method can be used to study the coupling between bubble-induced agitation and large-scale motions, such as those produced in industrial bubble columns.

Analysis and Control of Combustion Instabilities in Rocket Engines

Sound is the transmission of energy that is produced when two particles or objects undergo collision. It is one of the forms in which energy modulates and travels in a medium. Vibration is a phenomenon in which an object of mass executes a periodic oscillatory displacement when certain energy is transferred to it. Combustion is a chemical reaction in which a lot of molecules collide. Combustion instabilities occur in a reacting flow. It is a physical phenomenon. Combustion instabilities have mostly been studied in a particular flow but they also occur in real life. Real engines often feature specific unstable modes such as azimuthal instabilities in gas turbines or transverse modes in rocket chambers. Eddy simulation has been the major tool to study this type of instabilities so far but recently it has been proved to be insufficient to completely understand the complex nature of these instabilities [3]. These instabilities involve large Reynolds number, high pressure, densities in real engines. Combustion instabilities can occur at any part of the rocket propulsion system like nozzle, combustion chamber, injectors, feed systems and lines. Theory plays an important role in understanding and analysing these instabilities and the amount of damage they can cause to a particular object. In this paper we will look at different types or modes of combustion instabilities and active and passive ways to control them in real situations.

Numerical study on the contribution of surface and volume components of flow-induced noise in baffle silencers

Baffle silencers are a well-known solution for noise mitigation in industrial applications. One of the issues concerning these devices is the flow-inducted noise produced when a non-laminar flow of the medium in the duct occurs. These situations occur, for example, in dedusting installations or exhaust systems with the high-speed flow (large Reynolds number of the turbulence and small Mach number). This kind of installation has a complex shape that causes a turbulent flow in the medium. Installing a baffle silencer in these conditions causes additional noise. This noise cannot be predicted by using a standard approach with equations for laminar flow conditions. This paper presents the first step of the research in this field. The first step is to find a relation between CFD simulations' results and self-noise of the baffle silencer. In this work, we use the formulation proposed by Proudman in 1952 to calculate the sound power generated by the flow. The formulation is based on the turbulent kinetic energy k and dissipation rate ε of the flow, which is calculated by CFD simulations. The resulting sound power level needs to be calibrated. The calibration method is developed and presented. The aim of this research is to design an experimental setup.

Receptivity of inviscid modes in supersonic boundary layers to wall perturbations

The present paper investigates the receptivity of inviscid first and second modes in a supersonic boundary layer to time-periodic wall disturbances in the form of local blowing/suction, streamwise velocity perturbation and temperature perturbation, all introduced via a small forcing slot on the flat plate. The receptivity is studied using direct numerical simulations (DNS), finite- and high-Reynolds-number approaches, which complement each other. The finite-Reynolds-number formulation predicts the receptivity as accurately as DNS, but does not give much insight to the detailed excitation process, nor can it explain the significantly weaker receptivity efficiency of the streamwise velocity and temperature perturbations relative to the blowing/suction. In order to shed light on these issues, an asymptotic analysis was performed in the limit of large Reynolds number. It shows that the receptivity to all three forms of wall perturbations is reduced to the same mathematical form: the Rayleigh equation subject to an equivalent suction/blowing velocity, which can be expressed explicitly in terms of the physical wall perturbations. Estimates of the magnitude of the excited eigenmode can be made a priori for each case. Furthermore, the receptivity efficiencies for the streamwise velocity and temperature perturbations are quantitatively related to that for the blowing/suction by simple ratios, which are of $$O(R^{-1/2})$$ O ( R - 1 / 2 ) and have simple expressions, where R is the Reynolds number based on the boundary-layer thickness at the centre of the forcing slot. The simple leading-order asymptotic theory predicts the instability and receptivity characteristics accurately for sufficiently large Reynolds numbers (about $$10^4$$ 10 4 ), but appreciable error exists for moderate Reynolds numbers. An improved asymptotic theory is developed by using the appropriate impedance condition that accounts for the $$O(R^{-1/2})$$ O ( R - 1 / 2 ) transverse velocity induced by the viscous motion in the Stokes layer adjacent to the wall. The improved theory predicts both the instability and receptivity at moderate Reynolds numbers ($$R=O(10^3)$$ R = O ( 10 3 ) ) with satisfactory accuracy. In particular, it captures well the finite-Reynolds-number effects, including the Reynolds-number dependence of the receptivity and the strong excitation occurring near the so-called synchronisation point.