H. J. Rath

Convective instability mechanisms in thermocapillary liquid bridges

The primary instability of axisymmetric steady thermocapillary flow in a cylindrical liquid bridge with non‐deformable free surface is calculated by a mixed Chebychev‐finite difference method. For unit aspect ratio the most dangerous mode has an azimuthal wavenumber m=2. The physical instability mechanisms are studied by analyzing the linear energy balance of the neutral mode. If the Prandtl number is small (Pr≪1), the bifurcation is stationary. The associated neutral mode is amplified in the shear layer close to the free surface. For large Prandtl number (Pr=4), the basic state becomes linearly unstable to a pair of hydrothermal waves propagating nearly azimuthally. Both mechanisms are compared with those previously proposed in the literature.

Experimental study on high-pressure droplet evaporation using microgravity conditions

Evaporation of an individual fuel droplet at high pressures and high temperatures has been studied experimentally under microgravity conditions. A suspended n -heptane droplet was used in the experiments at pressures in the range of 0.1–5.0 MPa and temperatures varying from 400 to 800 K. Temporal variations of the droplet diameter were measured with a computer-aided image analysis system. Microgravity conditions, which were produced by using 5-m and 110-m drop towers and parabolic flights, were employed to prevent natural convection that complicates the phenomena. It was observed that dense fuel vapor surrounded a droplet and the droplet surface became obscure at high pressures and high temperatures. The slope of the temporal variations of the squared droplet diameter initially increases but later becomes approximately constant at ambient pressures below the critical pressure of the fuel. At a pressure of 5.0 MPa and temperatures below the critical temperature, the slope becomes less in the latter half of the evaporation lifetime. The ratio of the initial heat-up time to the evaporation lifetime was used as a measure of unsteadiness of droplet evaporation. The ratio is almost independent of ambient temperature at an ambient pressure of 0.1 MPa, but, as ambient pressure is increased, its tendency to rise with ambient temperature becomes noticeable. Corrected evaporation lifetime t c decreases monotonically as ambient temperature is increased. The slope of its curve becomes steeper as ambient pressure increases. Dependence of t c on ambient pressure changes according to ambient temperature. Above 550 K, t c decreases as ambient pressure is increased. Below 450 K, t c tends to increase as ambient pressure is increased. It is suggested that there exists a certain ambient temperature at which ambient pressure has little effect on t c .

Three-dimensional centrifugal-flow instabilities in the lid-driven-cavity problem

The classical rectangular lid-driven-cavity problem is considered in which the motion of an incompressible fluid is induced by a single lid moving tangentially to itself with constant velocity. In a system infinitely extended in the spanwise direction the flow is two-dimensional for small Reynolds numbers. By a linear stability analysis it is shown that this basic flow becomes unstable at higher Reynolds numbers to four different three-dimensional modes depending on the aspect ratio of the cavity’s cross section. For shallow cavities the most dangerous modes are a pair of three-dimensional short waves propagating spanwise in the direction perpendicular to the basic flow. The mode is localized on the strong basic-state eddy that is created at the downstream end of the moving lid when the Reynolds number is increased. In the limit of a vanishing layer depth the critical Reynolds number approaches a finite asymptotic value. When the depth of the cavity is comparable to its width, two different centrifugal-in...

Flow in two-sided lid-driven cavities : non-uniqueness, instabilities, and cellular structures

The steady flow in rectangular cavities is investigated both numerically and experimentally. The flow is driven by moving two facing walls tangentially in opposite directions. It is found that the basic two-dimensional flow is not always unique. For low Reynolds numbers it consists of two separate co-rotating vortices adjacent to the moving walls. If the difference in the sidewall Reynolds numbers is large this flow becomes unstable to a stationary three-dimensional mode with a long wavelength. When the aspect ratio is larger than two and both Reynolds numbers are large, but comparable in magnitude, a second two-dimensional flow exists. It takes the form of a single vortex occupying the whole cavity. This flow is the preferred state in the present experiment. It becomes unstable to a three-dimensional mode that subdivides the basic streched vortex flow into rectangular convective cells. The instability is supercritical when both sidewall Reynolds numbers are the same. When they differ the instability is subcritical. From an energy analysis and from the salient features of the three-dimensional flow it is concluded that the mechanism of destabilization is identical to the destabilization mechanism operative in the elliptical instability of highly strained vortices.

Hydrodynamic instabilities in cylindrical thermocapillary liquid bridges

The hydrodynamic stability of steady axisymmetric thermocapillary flow in a cylindrical liquid bridge is investigated by linear stability theory. The basic state and the three-dimensional disturbance equations are solved by various spectral methods for aspect ratios close to unity. The critical modes have azimuthal wavenumber one and the most dangerous disturbance is either a pure hydrodynamic steady mode or an oscillatory hydrothermal wave, depending on the Prandtl number. The influence of heat transfer through the free surface, additional buoyancy forces, and variations of the aspect ratio on the stability boundaries and the neutral mode are discussed.

Capillary driven flow in circular cylindrical tubes

Capillary-driven flow of a perfectly wetting liquid into circular cylindrical tubes is studied. Based on an analysis of previous approaches, a comprehensive theoretical model is presented which is not limited to certain special cases. This model considers the meniscus reorientation, the dynamic contact angle as well as inertia, convective, and viscous losses inside the tube and the reservoir. The capillary-driven flow is divided into three successive phases where first inertia then convective losses and finally viscous forces counteract the driving capillary force. This leads to an initial meniscus height increase proportional to the square of time followed by a linear dependence and finally the Lucas–Washburn behavior where the meniscus height is proportional to the square root of time. The three phases are separated by two characteristic transition times which are determined by the Ohnesorge number and the inertia of the liquid. Experiments were carried out under microgravity condition in a carefully chosen range of Ohnesorge numbers and initial liquid heights to cover the complete process from the initial meniscus development to the final Lucas–Washburn behavior. Good agreement of experimental and theoretical data is found throughout the complete range of experiment parameters. The existence of all three flow regimes predicted by the theory is verified by the experiments.

Three-dimensional numerical simulation of thermocapillary flows in cylindrical liquid bridges

The dynamics of thermocapillary flows in differentially heated cylindrical liquid bridges is investigated numerically using a mixed finite volume/pseudo-spectral method to solve the Navier–Stokes equations in the Boussinesq approximation. For large Prandtl numbers ( Pr = 4 and 7) and sufficiently high Reynolds numbers, the axisymmetric basic flow is unstable to three-dimensional hydrothermal waves. Finite-amplitude azimuthally standing waves are found to decay to travelling waves. Close to the critical Reynolds number, the former may persist for long times. Representative results are explained by computing the coefficients in the Ginzburg–Landau equations for the nonlinear evolution of these waves for a specific set of parameters. We investigate the nonlinear phenomena characteristic of standing and pure travelling waves, including azimuthal mean flow and time-dependent convective heat transport. For Pr [Lt ] 1 the first transition from the two-dimensional basic flow to the three-dimensional stationary flow is inertial in nature. Particular attention is paid to the secondary transition leading to oscillatory three-dimensional flow, and this mechanism is likewise independent of Pr . The spatial and temporal structure of the perturbation flow is analysed in detail and an instability mechanism is proposed based on energy balance calculations and the vorticity distribution.

Three-dimensional instability of axisymmetric buoyant convection in cylinders heated from below

The stability of steady axisymmetric convection in cylinders heated from below and insulated laterally is investigated numerically using a mixed finite-difference/Chebyshev collocation method to solve the base flow and the linear stability equations. Linear stability boundaries are given for radius to height ratios γ from 0.9 to 1.56 and for Prandtl numbers Pr = 0.02 and Pr = 1. Depending on γ and Pr , the azimuthal wavenumber of the critical mode may be m = 1, 2, 3, or 4. The dependence of the critical Rayleigh number on the aspect ratio and the instability mechanisms are explained by analysing the energy transfer to the critical modes for selected cases. In addition to these results the onset of buoyant convection in liquid bridges with stress-free conditions on the cylindrical surface is considered. For insulating thermal boundary conditions, the onset of convection is never axisymmetric and the critical azimuthal wavenumber increases monotonically with γ. The critical Rayleigh number is less then 1708 for most aspect ratios.

DETAILED NUMERICAL SIMULATIONS OF THE MULTISTAGE SELF- IGNITION PROCESS OF n-HEPTANE ISOLATED DROPLETS AND THEIR VERIFICATION BY COMPARISON WITH MICROGRAVITY EXPERIMENTS

Detailed understanding of the basic physical and chemical processes of self-ignition phenomena for technical fuel sprays is required for many technical combustion applications. Because single droplets as the basic elements of fuel sprays allow the study of fundamental ignition behavior, this work focuses on the multistage self-ignition behavior of Φ 0.7 mm single n -heptane droplets in air. The investigated ambient conditions are temperatures from 580 to 1000 K and pressures from 0.3 to 1 MPa. A substantial physical and chemical model is developed for the detailed numerical simulation. The chemical reaction mechanism consists of a 62-step kinetic model with special consideration of the low-temperature reaction branch. The program was validated by comparison with results from microgravity experiments at Drop Tower Bremen, which allow clarification of the ignition process free from natural convection. Cool flame and hot flame appearances were obtained from non-intrusive interferometric measurement in a well-tested experimental setup. The calculated ignition delays resulting from temperature and temperature gradient criteria, respectively, were compared with these experimental results. Furthermore, the cool flame temperature was measured with a K-type thermocouple of Φ 25 μ m at the place of its appearance and was also compared with the numerical simulations. A quantitative good agreement for first and total ignition delays as well as for the cool flame temperature could be achieved. With this detailed numerical model, the multistage ignition behavior was analyzed, and the ignition criteria employed in interferometric measurement, which are temperature and temperature gradient, respectively, were confirmed.

Elliptic instability in two-sided lid-driven cavity flow

Abstract The recently discovered concentration of vorticity in slender vortex tubes in turbulent flow fields has motivated the investigation of a class of vortices with elliptical streamlines. As a prototype of this flow, long vortices confined in a rectangular cavity and driven by tangentially moting walls are studied. These vortices are characterized by a large rate of plane strain at the core. The quasi-two-dimensional flow is found to be unstable at small Reynolds numbers, if the eccentricity of the streamlines, i.e. the strain rate, is sufficiently large. The three-dimensional supercritical flow is found to be steady with a wavelength of the order of the vortex core diameter. The flow pattern appears in the form of rectangular cells that are very robust. Good agreement between experiment and numerical calculations is obtained. It is argued that the instability found is due to the elliptic instability mechanism.