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Low & high prandtl number approaches for heat transfer in laminar flow of incompressible fluids with constant material properties
The objective is to study quantitatively thermocapillary flows in two-dimensional and axisymmetric geometries in order to learn the impact of such flows on float-zone refining configurations. The work begins with the analytical study of steady flows in slots and zones to obtain predictions of surface curvature, flow and heat transport. It then examines the instability of such flows. Three new instability mechanisms have been identified and the results show that thermocapillary flows of high Prandtl number fluid become convectively unstable while those of low Prandtl number fluid become hydrodynamically unstable. The work includes numerical simulations of steady, high Marangoni number flows in unit aspect ratio boxes and asymptotic solutions in large boxes. Furthermore, the effect of surface contamination on the steady thermocapillary flows was examined. This fundamental study focused on fluid dynamical systems used to understand events in the flow of melts in float-zone geometries both on Earth and in microgravity environments.
In the microgravity environment, the effect of gravity on fluid motion is much reduced. Hence, secondary effects such as vibrations, jitters, surface tension, capillary effects, and electromagnetic forces become the dominant mechanism of fluid convection. Numerous studies have been conducted to investigate fluid behavior in microgravity with the ultimate goal of developing processes with minimal influence from convection. Industrial applications such as crystal growth from solidification of melt and protein growth for pharmatheutical application are just a few examples of the vast potential benefit that can be reaped from material processing in space. However, a space laboratory is not immune from all undesirable disturbances and it is imperative that such disturbances be well understood, quantifiable, and controlled. Non-uniform and transient accelerations such as vibrations, jitters, and impulsive accelerations exist as a result of crew activities, space vehicle maneuvering, and the operations of on-board equipment. Measurements conducted on-board a U.S. Spacelab showed the existence of vibrations in the frequency range of 1 to 100 Hz with a dominant mode of 17 Hz and harmonics of 54 Hz. The observed vibration is not limited to any coordinate plane but exists in all directions. Similar situation exists on-board the Russian MIR space station. Due to the large structure of its design, the future International Space Station will have its own characteristic vibration spectrum. It is well known that vibration can exert substantial influence on heat and mass transfer processes, thus hindering any attempts to achieve a diffusion-limited process. Experiments on vibration convection for a liquid-filled enclosure under one-g environment showed the existence of different flow regimes as vibration frequency and intensity changes. Results showed the existence of a resonant frequency, near which the enhancement is the strongest, and the existence of a high frequency asymptote. Numerical simulations of vibration convection have been conducted by Yurkov, Fu and Shieh, and by Wang. These analyses considered a two-dimensional air-filled cell under weightlessness condition and showed results similar to those of the experiments. It is not yet known whether resonance convection can be triggered by jitter alone or whether it requires the interaction of jitter with other convective forces in low gravity. An order of magnitude analysis, however, can be used to show the dependence of the resonance frequency on the fluid Prandtl number. Even though the onset of resonance convection may depend on other factors, results indicates that fluids with low Prandtl numbers are more susceptible to resonance than those with high Prandtl numbers. The current study is aimed at gaining additional insights to this problem using germanium as working fluid. Germanium was chosen for this analysis because of its common usage in solidification process and its relatively low Prandtl number (Pr = 0.02).
A comprehensive analytical, numerical and experimental study of convective flow in multilayer immiscible liquids has been performed. Studies include transparent high Prandtl number of liquid as well as opaque low Prandtl metallic melts. A new radioscopic flow visualization has been developed for the latter studies.
It is well established that residual flows exist in contained liquid metal processes. In 1-g processing, buoyancy forces often drive these flows and their magnitudes can be substantial. It is also known that residual flows can exist during microgravity processing, and although greatly reduced in magnitude, they can influence the properties of the processed materials. Unfortunately, there are very few techniques to visualize flows in opaque, high temperature liquid metals, and those available are not easily adapted to flight investigation. In this study, a novel technique is developed that uses liquid tin as the model fluid and solid-state electrochemical cells constructed from Yttria-Stabilized Zirconia (YSZ) to establish and measure dissolved oxygen boundary conditions. The melt serves as a common electrode for each of the electrochemical cells in this design, while independent reference electrodes are maintained at the outside surfaces of the electrolyte. By constructing isolated electrochemical cells at various locations along the container walls, oxygen is introduced or extracted by imposing a known electrical potential or passing a given current between the melt and the reference electrode. This programmed titration then establishes a known oxygen concentration boundary condition at the selected electrolyte-melt interface. Using the other cells, the concentration of oxygen at the electrolyte-melt interface is also monitored by measuring the open-circuit potentials developed between the melt and reference electrodes. Thus the electrochemical cells serve to both establish boundary conditions for the passive tracer and sense its path. Rayleigh-Benard convection was used to validate the electrochemical approach to flow visualization. Thus, a numerical characterization of the second critical Rayleigh numbers in liquid tin was conducted for a variety of Cartesian aspect ratios. The extremely low Prandtl number of tin represents the lowest value studied numerically. Additionally, flow field oscillations are visualized and the effect of tilt on convecting systems is quantified. Experimental studies of the effect of convection in liquid tin are presented. Three geometries are studied: (1) double electrochemical cell with vertical concentration gradients; (2) double cell with horizontal concentration gradients; and (3) multiple cells with vertical temperature gradients. The first critical Rayleigh number transition is detected with geometry (1) and it is concluded that current measurements are not as affected by convection as EMF measurements. The system is compared with numerical simulations in geometry (2), and oscillating convection is detected with geometry (3).
Experiments on boundary layer transition with flat, concave and convex walls and various levels of free-stream disturbance and with zero and strong streamwise acceleration have been conducted. Measurements of both fluid mechanics and heat transfer processes were taken. Examples are profiles of mean velocity and temperature; Reynolds normal and shear stresses; turbulent streamwise and cross-stream heat fluxed; turbulent Prandtl number; and streamwise variations of wall skin friction and heat transfer coefficient values. Free-stream turbulence levels were varied over the range from about 0.3 percent to about 8 percent. The effects of curvature on the onset of transition under low disturbance conditions are clear; concave curvature leads to an earlier and more rapid transition and the opposite is true for convex curvature This was previously known but little documentation of the transport processes in the flow was available
In connection with Pohlhausen's solution for the temperature field on the flat plate, a series of formulas were indicated by means of which the velocity and temperature field for variable physical characteristics can be computed by an integral equation and an iteration method based on it. With it, the following cases were solved: On the assumption that the viscosity simply varies with the temperature while the other fluid properties remain constant, the velocity and temperature field on the heated and cooled plate, respectively, was computed at the Prandtl numbers 12.5 and 100 (viscous fluids). A closer study of these two cases resulted in general relations: The calculations for a gas of Pr number 0.7 (air) were conducted on the assumption that all fluid properties vary with the temperature, and the velocities are low enough for the heat of friction to be discounted. The result was a thickening of the boundary layers, but no appreciable modification in shearing stress or heat-transfer coefficient.
Instabilities in a fluid with a constant density gradient that is subject to arbitrarily oriented oscillatory accelerations are considered. With the Boussinesq approximation and for the case of an unbounded fluid, transformation to Lagrangian coordinates allows the reduction of the problem to an ordinary differential equation for each three-dimensional wavenumber. The problem has three parameters: the nondimensional amplitude R of the base-state oscillation, the nondimensional level of background steady acceleration, which for some cases can be represented in terms of a local (in time) Richardson number Ri, and the Prandtl number Pr. Some general bounds on stability are derived. For Pr = 1 closed-form solutions are found for impulse (delta function) accelerations and a general asymptotic solution is constructed for large R and general imposed accelerations. The asymptotic solution takes advantage of the fact that at large R wave growth is concentrated at 'zero points'. These are times when the effective vertical wavenumber passes through zero. Kelvin-Helmholtz instabilities are found to dominate at low R, while Rayleigh-Taylor instabilities dominate at high R. At high R, the uniform shear of the Kelvin-Helmholtz case tends to distort and weaken instability waves. With unsteady flows, Ri = 1/4 is no longer an instability limit. Significant instabilities have been found for sinusoidal forcing for Ri up to 0.6.
Two papers are included in this progress report. In the first, the compressible Navier-Stokes equations have been used to compute leading edge receptivity of boundary layers over parabolic cylinders. Natural receptivity at the leading edge was simulated and Tollmien-Schlichting waves were observed to develop in response to an acoustic disturbance, applied through the farfield boundary conditions. To facilitate comparison with previous work, all computations were carried out at a free stream Mach number of 0.3. The spatial and temporal behavior of the flowfields are calculated through the use of finite volume algorithms and Runge-Kutta integration. The results are dominated by strong decay of the Tollmien-Schlichting wave due to the presence of the mean flow favorable pressure gradient. The effects of numerical dissipation, forcing frequency, and nose radius are studied. The Strouhal number is shown to have the greatest effect on the unsteady results. In the second paper, a transition model for low-speed flows, previously developed by Young et al., which incorporates first-mode (Tollmien-Schlichting) disturbance information from linear stability theory has been extended to high-speed flow by incorporating the effects of second mode disturbances. The transition model is incorporated into a Reynolds-averaged Navier-Stokes solver with a one-equation turbulence model. Results using a variable turbulent Prandtl number approach demonstrate that the current model accurately reproduces available experimental data for first and second-mode dominated transitional flows. The performance of the present model shows significant improvement over previous transition modeling attempts.