William W. Fowlis

Numerical solutions for spin-up from rest in a cylinder

Numerical solutions for the impulsively started spin-up from rest of a homogeneous fluid in a cylinder for small Ekman numbers are presented. The basic analytical theory for this spin-up flow is due to Wedemeyer (1964). Wedemeyers solution shows that the interior flow is divided into two regions by a moving front which propagates radially inward across the cylinder. The fluid ahead of the front remains non-rotating, while the fluid behind the front is being spun up. Experimental observations have shown that Wedemeyers model captures the essential dynamics of the azimuthal flow, but that it is not a quantitative model. Wedemeyer made several assumptions in formulating an Ekman compatibility condition, and inconsistencies exist between these assumptions and his solution. Later workers attempted to improve the analytical theory, but their work still included the same basic assumptions made by Wedemeyer. No previous work has provided a comprehensive and accurate set of three-dimensional flow-field data for this spin-up problem. We chose to acquire such data using a numerical model based on the Navier–Stokes equations. This model was first checked against accurate laser-Doppler measurements of the azimuthal flow for spin-up from rest. New flow-field data over a range of Ekman numbers 9·18 × 10 −6 [les ] E [les ] 9·18 × 10 −4 are presented. Diagnostic studies, which reveal the various contributions to spin-up of the separate inviscid and viscous terms as functions of radius and time, are also presented. The plots of the viscous-diffusion term reveal the moving front, which is identified as a layer of enhanced local viscous activity. Immediately after the impulsive start, viscous diffusion is seen to be the major contributor to spin-up, then the nonlinear radial advection term takes over, and, finally, when spin-up is well progressed, the linear Coriolis force dominates. In the vicinity of the front, the inward radial flow is a maximum, and the vertical velocity is very small. Strong radial gradients of the vertical velocity are observed across the front and behind the front at the edge of the Ekman layer, and the azimuthal flow behind the front shows strong departures from solid-body rotation. These results enable us to fill in details of the flow not accurately given by Wedemeyers model and its extensions.

Experimental and theoretical analysis of the rate of solvent equilibration in the hanging drop method of protein crystal growth

Abstract In the hanging drop method of protein crystal growth, a water droplet containing protein, buffer and a precipitating agent, such as ammonium sulfate, is suspended from a glass coverslip above a well containing an aqueous solution of the precipitating agent at a concentration double that in the drop. We present a comprehensive theoretical study of the rate of water evaporation in the hanging drop method. We find that in earths gravity the rate controlling step in the evaporation is the rate of diffusion of water vapor across the air space separating the drop from the well. Using ammonium sulfate as the precipitating agent, we have made careful measurements at both 4°C and 25°C of the evaporation times for some 25 μL droplets at various concentrations. These results are in good agreement with our theory. As determined by the theory, the parameters affecting the rate of evaporation include the temperature, the vapor pressure of water, the ionization constant of the salt, the volume of the droplet, the contact angle between the droplet and the coverslip, the number of moles of salt in the droplet, the number of moles of water and salt in the well, the molar volumes of water and salt, the distance from the droplet to the well, and the coefficient of diffusion of water vapor through air. These parameters do not act independently; rather, they combine to form three dimensionless groups upon which the rate of evaporation depends. We evaluate numerically 18 different drop and well arrangements commonly encountered in the laboratory. In all cases considered at 25°C, the number of moles of water in the droplet achieves 95% of its final value in 3—30 h, after which further evaporation is quite slow. Our experiments confirm this. We consider qualitatively the effect of weightlessness (spaceflight) on the rate of evaporation and find it to be most likely controlled by the rate of the interdiffusion of salt and water in the droplet.

Numerical solutions for the spin-up of a stratified fluid

The model of Warn-Varnas et al. (1978) is used to numerically examine the spin-up flow of a thermally stratified fluid in a cylinder with an insulating side wall, and comparison of the results with the laser-Doppler measurements of Lee (1975) shows excellent agreement. It is shown that flow gradients are created in the interior of the fluid during the meridional circulation spin-up phase, and that the azimuthal flow decayed faster than has been predicted by Wallin (1969). It is established that viscous diffusion in the interior, arising from the interior-flow gradients, is the cause of the discrepancy with Wallins theory.

Confinement of thermocapillary floating zone flow by uniform rotation

Abstract The microgravity environment of an orbiting vehicle permits crystal growth experiments with greatly reduced buoyant convection in the liquid melt. Crystals grown in ground-based laboratories do not achieve their potential properties because of dopant variations caused by flow in the melt. The floating zone crystal growing system is widely used to produce crystals of silicon and other materials. However, in this system the temperature gradient on the free sidewall surface of the melt is the source of a thermocapillary flow which does not disappear in a low gravity environment. Smith and Greenspan examined theoretically the idea of using a uniform rotation of the floating zone system to confine the thermocapillary flow to the melt sidewall leaving the interior of the melt passive. These workers considered a half zone, a cylinder of fluid with a constant axial temperature gradient imposed on the cylindrical sidewall. They examined the linearized, axisymmetric flow in the absence of crystal growth. They found that rotation does confine the linear thermocapillary flow. In this paper the simplified model of Smith and Greenspan is extended to a full zone with a more realistic temperature distribution imposed on the sidewall and both linear and nonlinear thermocapillary flows are studied theoretically. Analytical and numerical methods are used for the linear flows and numerical methods for the nonlinear flows. We found that the linear flows in the full zone have more complicated and thicker boundary layer structures than in the half zone, and that these flows are also confined by the rotation. For the model considered and for realistic values for silicon, however, the thermocapillary flow is not linear. We examined the nonlinear flows by first computing a weakly nonlinear flow and then computing the fully nonlinear flow. The weakly nonlinear flow is steady, has less boundary layer character, and penetrates more deeply into the interior than the linear flow but still shows some rotational confinement. The fully nonlinear flow is strong and unsteady (a weak oscillation is present) and it penetrates the interior. Some non-rotating flow results are also presented. Since silicon has a large value of thermal conductivity, we would expect the temperature fields to be determined by conduction alone. This is true for the linear and weakly nonlinear results, but for the stronger nonlinear flow the results show that temperature advection is also important. Thus, this work reveals that for the nonlinear flow, a radiative sidewall boundary condition would be an improvement over the specified temperature boundary condition used in this paper and previously by others. Such a boundary condition would weaken the sidewall axial temperature gradient and hence the thermocapillary flow, allowing the confining effect of rotation to play a stronger role. Hence, uniform rotation may still be a means of confining the flow, and the results obtained define the procedure to be used to examine this hypothesis.

Particle orbits in a rotating liquid

Monodisperse latex microspheres ranging in size from submicrometer to several micrometers in diameter can be prepared in the laboratory. The uniformity of diameter is important for instrument calibration and other applications. However it has proved very difficult to manufacture commercial quantities of mondisperse latex microspheres with diameters larger than about 3 micrometers owing to buoyancy and sedimentation effects. In an attempt to eliminate these effects NASA sponsored a Space Shuttle experiment called the Monodisperse Latex Reactor (MLR) to produce these monodisperse microspheres in larger sizes in microgravity. Results have been highly successful. Using technology gained from this space experiment, a ground-based rotating latex reactor has been fabricated in an attempt to minimize sedimentation without using microgravity. The entire reactor cylinder is rotated about a horizontal axis to keep the particles in suspension. In this paper we determine the motion of small spherical particles under gravity, in a viscous fluid rotating uniformly about a horizontal axis. The particle orbits are approximately circles, with centres displaced horizontally from the axis of rotation. Owing to net centrifugal buoyancy, the radius of the circles increases (for heavy particles) or decreases (for light particles) with time, so that the particles gradually spiral inward or outward. For a large rotation rate, the particles spiral outwards or inwards too fast, while for a small rotation rate, the displacement of the orbit centre from the rotation axis is excessive in relation to the reactor radius. We determine the rotation rate that maximizes the fraction of the reactor cross-section area that contains particles that will not spiral out to the wall in the experimental time (for heavy particles), or that have spiralled in without hitting the wall (for light particles). Typically, the rate is close to 1 r.p.m., and design rotation rate ranges should span this value.

Flow regimes in a shallow rotating cylindrical annulus with temperature gradients imposed on the horizontal boundaries

Experimental flow regime diagrams are determined for a new rotating cylindrical annulus configuration which permits a measure of control over the internal vertical temperature gradient. The new annulus has radial temperature gradients imposed on plane horizontal thermally conducting endwalls (with the cylindrical sidewalls as insulators) and is considered to be more relevant to atmospheric dynamics studies than the classical cylindrical annulus. Observations have revealed that, in addition to the axisymmetric flow and nonaxisymmetric baroclinic wave flow which occur in the classical annulus, two additional nonaxisymmetric flow types occur in the new annulus: boundary-layer thermal convection and deep thermal convection. Flow regime diagrams for three different values of the imposed vertical temperature difference are presented, and explanations for the flow transitions are offered. The new annulus provides scientific backup for the proposed Atmospheric General Circulation Experiment for Spacelab. The apparatus diagram is included.

Laboratory experiments in a baroclinic annulus with heating and cooling on the horizontal boundaries

Abstract Experiments have been performed in a cylindrical annulus with horizontal temperature gradients imposed upon the horizontal boundaries and in which the vertical depth was smaller than the width of the annulus. Qualitative observations were made by the use of small, suspended, reflective flakes in the liquid (water). Four basic regimes of flow were observed: (1) axisymmetric flow, (2) deep cellular convection, (3) boundary layer convective rolls, and (4) baroclinic waves. In some cases there was a mix of baroclinic and convective instabilities present. As a “mean” interior Richardson number was decreased from a value greater than unity to one less than zero, axisymmetric baroclinic instability of the Solberg type was never observed. Rather, the transition was from non-axisymmetric baroclinic waves, to a mix of baroclinic and convective instability, to irregular cellular convection.

Three-dimensional baroclinic instability of a Hadley cell for small Richardson number

A three-dimensional linear stability analysis of a baroclinic flow for Richardson number Ri of order unity is presented. The model considered is a thin, horizontal, rotating fluid layer which is subjected to horizontal and vertical temperature gradients. The basic state is a Hadley cell which is a solution of the Navier–Stokes and energy equations and contains both Ekman and thermal boundary layers adjacent to the rigid boundaries; it is given in closed form. The stability analysis is also based on the Navier–Stokes and energy equations; and perturbations possessing zonal, meridional and vertical structures were considered. Numerical methods were developed for the solution of the stability problem, which results in an ordinary differential eigenvalue problem. The objectives of this work were to extend the previous theoretical work on three-dimensional baroclinic instability for small Ri to a more realistic model involving the Prandtl number σ and the Ekman number E , and to finite growth rates and a wider range of the zonal wavenumber. The study covers ranges of 0.135 [les ] Ri [les ] 1.1, 0.2 [les ] σ [les ] 5.0, and 2 × 10 −4 [les ] E [les ] 2 σ 10 −3 . For the cases computed for E = 10 −3 and σ ≠ 1, we found that conventional baroclinic instability dominates for Ri > 0.825 and symmetric baroclinic instability dominates for Ri E [ges ] 5 × 10 −4 and σ = 1 in the range 0.3 [les ] Ri [les ] 0.8, conventional baroclinic instability always dominates. Further, we found in general that the symmetric modes of maximum growth are not purely symmetric but have weak zonal structure. This means that the wavefronts are inclined at a small angle to the zonal direction. The results also show that as E decreases the zonal structure of the symmetric modes of maximum growth rate also decreases. We found that when zonal structure is permitted the critical Richardson number for marginal stability is increased, but by only a small amount above the value for pure symmetric instability. Because these modes do not substantially alter the results for pure symmetric baroclinic instability and because their zonal structure is weak, it is unlikely that they represent a new type of instability.

Baroclinic Instability of a Rotating Hadley Cell

Abstract The stability of a thin fluid layer between two rotating plates which are subjected to a horizontal temperature gradient is studied. First, the solution for the stationary basic state is obtained in a closed form. This solution identifies Ekman and thermal layers adjacent to the plates and interior temperature and velocity fields which are almost linear functions of height. Then the stability of that basic state with respect to infinitesimal zonal waves is analyzed via the solution of the complete viscous linear equations for the perturbations. The character of the growth rates is found to be similar to the growth rates of the classical baroclinic waves. The neutral stability curves for these waves possessed a knee in the Rossby-Taylor number plane to the left of which all perturbations are stable. The region of instability is found to depend on the Prandtl number, the vertical stratification parameter, and both the meridional and zonal wavenumbers. It is found in general that the flow is unstabl...

Baroclinic instability with variable static stability—a design study for a spherical atmospheric model experiment

Abstract Exact solutions are obtained for a quasi-geostrophic baroclinic stability problem in which the rotational Froude number (inverse Burger number) is a linear function of the height. The primary motivation for this work was to investigate the effect of a radially-variable, dielectric body force, analogous to gravity, on baroclinic instability for the design of a spherical, synoptic-scale, atmospheric model experiment for a Spacelab flight. Such an experiment cannot be realized in a laboratory on the Earths surface because the body force cannot be made strong enough to dominate terrestrial gravity. Flow in a rotating, rectilinear channel with a vertically variable body force and with no horizontal shear of the basic state is considered. The horizontal and vertical temperature gradients of the basic and reference states are taken as constants. Consequences of the body force variation and the other assumptions of the model are that the static stability (Brunt-Vaisala frequency squared) and the vertica...