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All sourcesNASA NTRSDOE OSTI
NASA NTRS2010

Ocean Turbulence, III: New GISS Vertical Mixing Scheme

We have found a new way to express the solutions of the RSM (Reynolds Stress Model) equations that allows us to present the turbulent diffusivities for heat, salt and momentum in a way that is considerably simpler and thus easier to implement than in previous work. The RSM provides the dimensionless mixing efficiencies Gamma-alpha (alpha stands for heat, salt and momentum). However, to compute the diffusivities, one needs additional information, specifically, the dissipation Epsilon. Since a dynamic equation for the latter that includes the physical processes relevant to the ocean is still not available, one must resort to different sources of information outside the RSM to obtain a complete Mixing Scheme usable in OGCMs. As for the RSM results, we show that the Gamma-alpha s are functions of both Ri and Rq (Richardson number and density ratio representing double diffusion, DD); the Gamma-alpha are different for heat, salt and momentum; in the case of heat, the traditional value Gamma-h = 0.2 is valid only in the presence of strong shear (when DD is inoperative) while when shear subsides, NATRE data show that Gamma-h can be three times as large, a result that we reproduce. The salt Gamma-s is given in terms of Gamma-h. The momentum Gamma-m has thus far been guessed with different prescriptions while the RSM provides a well defined expression for Gamma-m(Ri,R-rho). Having tested Gamma-h, we then test the momentum Gamma-m by showing that the turbulent Prandtl number Gamma-m/Gamma-h vs. Ri reproduces the available data quite well. As for the dissipation epsilon, we use different representations, one for the mixed layer (ML), one for the thermocline and one for the ocean;s bottom. For the ML, we adopt a procedure analogous to the one successfully used in PB (planetary boundary layer) studies; for the thermocline, we employ an expression for the variable epsilon/N(exp 2) from studies of the internal gravity waves spectra which includes a latitude dependence; for the ocean bottom, we adopt the enhanced bottom diffusivity expression used by previous authors but with a state of the art internal tidal energy formulation and replace the fixed Gamma-alpha = 0.2 with the RSM result that brings into the problem the Ri, R-rho dependence of the Gamma-alpha; the unresolved bottom drag, which has thus far been either ignored or modeled with heuristic relations, is modeled using a formalism we previously developed and tested in PBL studies. We carried out several tests without an OGCM. Prandtl and flux Richardson numbers vs. Ri. The RSM model reproduces both types of data satisfactorily. DD and Mixing efficiency Gamma-h(Ri,Rq). The RSM model reproduces well the NATRE data. Bimodal epsilon-distribution. NATRE data show that epsilon (Ri < 1) approximately equals 10epsilon(Ri > 1), which our model reproduces. Heat to salt flux ratio. In the Ri much greater than 1 regime, the RSM predictions reproduce the data satisfactorily. NATRE mass diffusivity. The z-profile of the mass diffusivity reproduces well the measurements at NATRE. The local form of the mixing scheme is algebraic with one cubic equation to solve.

Canuto, V. M.↗
NASA NTRS1984

Forced convection heat transfer to air/water vapor mixtures

Heat transfer coefficients were measured using both dry and humid air in the same forced convection cooling scheme and were compared using appropriate nondimensional parameters (Nusselt, Prandtl and Reynolds numbers). A forced convection scheme with a complex flow field, two dimensional arrays of circular jets with crossflow, was utilized with humidity ratios (mass ratio of water vapor to air) up to 0.23. The dynamic viscosity, thermal conductivity and specific heat of air, steam and air/steam mixtures are examined. Methods for determining gaseous mixture properties from the properties of their pure components are reviewed as well as methods for determining these properties with good confidence. The need for more experimentally determined property data for humid air is discussed. It is concluded that dimensionless forms of forced convection heat transfer data and empirical correlations based on measurements with dry air may be applied to conditions involving humid air with the same confidence as for the dry air case itself, provided that the thermophysical properties of the humid air mixtures are known with the same confidence as their dry air counterparts.

Richards, D. R.↗
NASA NTRS1985

A catalogue of potentially bright close binary gravitational wave sources

This is a current print-out of results of a survey, undertaken in the spring of 1985, to identify those known binary stars which might produce significant gravitational wave amplitudes at earth, either dimensionless strain amplitudes exceeding a threshold h = 10(exp -21), or energy fluxes exceeding F = 10(exp -12) erg cm(exp -2) s(exp -1). All real or putative binaries brighter than a certain limiting magnitude (calculated as a function of primary spectral type, orbital period, orbital eccentricity, and bandpass) are included. All double degenerate binaries and Wolf-Rayet binaries with known or suspected orbital periods have also been included. The catalog consists of two parts: a listing of objects in ascending order of Right Ascension (Equinox B1950), followed by an index, listing of objects by identification number according to all major stellar catalogs. The object listing is a print-out of the spreadsheets on which the catalog is currently maintained. It should be noted that the use of this spreadsheet program imposes some limitations on the display of entries. Text entries which exceed the cell size may appear in truncated form, or may run into adjacent columns. Greek characters are not available; they are represented here by the first two or three letters of their Roman names, the first letter appearing as a capital or lower-case letter according to whether the capital or lower-case Greek character is represented. Neither superscripts nor subscripts are available; they appear here in normal position and type-face. The index provides the Right Ascension and Declination of objects sorted by catalogue number.

Webbink, Ronald F.↗
NASA NTRS1974

Steady simple shear flow past a circular cylinder at moderate Reynolds numbers - A numerical solution

The two-dimensional steady flow of an incompressible viscous fluid past a circular cylinder, placed symmetrically in a simple shear field, has been studied for both the stationary and the freely rotating case by solving numerically the Navier-Stokes equations for values of the Reynolds number R in the range from 0.047 to 70. At R = 0.047, the results obtained are in substantial agreement with the analytic small-R perturbation solution given by Robertson and Acrivos (1970). Inertia effects were found, however, to play a significant role even at R = 1, and hence the calculated flow pattern for R greater than or equal to 1 differs significantly from that of the creeping-flow solution. Specifically, for the freely rotating case, the region of closed streamlines decreases rapidly in extent with increasing R, two symmetrically placed wakes are formed on either side of the cylinder, and the dimensionless rotational speed of the freely suspended cylinder decreases as the reciprocal of the square root of R.

Kossack, C. A.↗
NASA NTRS1990

Heat transfer in rotating serpentine passages with smooth walls

Experiments were conducted to determine the effects of buoyancy and Coriolis forces on heat transfer in turbine blade internal coolant passages. The experiments were conducted with a large scale, multi-pass, smooth-wall heat transfer model with both radially inward and outward flow. An analysis of the governing flow equations showed that four parameters influence the heat transfer in rotating passages (coolant-to-wall temperature ratio, Rossby number, Reynolds number and radius-to-passage hydraulic diameter ratio). These four parameters were varied over ranges which are typical of advanced gas turbine engine operating conditions. It was found that both Coriolis and buoyancy effects must be considered in turbine blade cooling designs and that the effect of rotation on the heat transfer coefficients was markedly different depending on the flow direction. Local heat transfer coefficients were found to decrease by as much as 60 percent and increase by 250 percent from no rotation levels. Comparisons with a pioneering stationary vertical tube buoyancy experiment showed reasonably good agreement. Correlation of the data is achieved employing dimensionless parameters derived from the governing flow equations.

Wagner, J. H.↗
NASA NTRS1992

Low Reynolds number multiple-time-scale turbulence model and calculations of steady and pulsating shear layers

A low Reynolds number multiple-time-scale turbulence model (LMS) and its application to fully developed turbulent channel flows and pulsating pipe flows are presented. The LMS can describe the inequilibrium turbulence phenomena down to the viscous sublayer. The calculated fluid flow and turbulence fields for the channel flows are in better agreement with the direct numerical simulation (DNS) results than those obtained using a Reynolds stress turbulence model, and the calculated near-wall dissipation rates are in qualitatively correct agreement with the DNS results. The LMS also successfully predicts the rapidly varying phase-lead of the wall shearing stress that occurs in a narrow range of the dimensionless frequency omega (+) = (omega x nu)/(upsilon x tau(exp2)) for the pulsating pipe flows while various other turbulence models fail to predict this phenomenon, and the LMS yields significantly improved numerical results for a wide range of the dimensionless frequency compared with those obtained using a rapid distortion theory (RDT).

Kim, Sang-Wook↗
NASA NTRS1980

A hill-sliding strategy for initialization of Gaussian clusters in the multidimensional space

A hill sliding technique was devised to extract Gaussian clusters from the multivariate probability density estimate of sample data for the first step of iterative unsupervised classification. Each cluster was assumed to posses a unimodal normal distribution. A clustering function proposed distinguished elements of a cluster under formation from the rest in the feature space. Initial clusters were extracted one by one according to the hill sliding tactics. A dimensionless cluster compactness parameter was proposed as a universal measure of cluster goodness and used satisfactorily in test runs with LANDSAT multispectral scanner data. The normalized divergence, defined by the cluster divergence divided by the entropy of the entire sample data, was utilized as a general separability measure between clusters. An overall clustering objective function was set forth in terms of cluster covariance matrices, from which the cluster compactness measure could be deduced. Minimal improvement of initial data partitioning was evaluated by this objective function in eliminating scattered sparse data points. The hill sliding clustering technique developed herein has the potential applicability to decomposition any multivariate mixture distribution into a number of unimodal distributions when an appropriate distribution function to the data set is employed.

Park, J. K.↗
NASA NTRS1994

Coherent-phase or random-phase acceleration of electron beams in solar flares

Time structures of electron beam signatures at radio wavelengths are investigated to probe correlated versus random behavior in solar flares. In particular we address the issue whether acceleration and injection of electron beams is coherently modulated by a single source, or whether the injection is driven by a stochastic (possibly spatially fragmented) process. We analyze a total of approximately = 6000 type III bursts observed by Ikarus (Zurich) in the frequency range of 100-500 MHz, during 359 solar flares with simultaneous greater than or = 25 keV hard X-ray emission, in the years 1890-1983. In 155 flares we find a total of 260 continuous type III groups, with an average number of 13 +/- 9 bursts per group, a mean duration of D = 12 +/- 14 s, a mean period of P = 2.0 +/- 1.2 s, with the highest burst rate at a frequency of nu = 310 +/- 120 MHz. Pulse periods have been measured between 0.5 and 10 s, and can be described by an exponential distribution, i.e., N(P) varies as e (exp -P/1.0s). The period shows a frequency dependence of P(nu)=46(exp-0.6)(sub MHz)s for different flares, but is invariant during a particular flare. We measure the mean period P and its standard deviation sigma (sub p) in each type III group, and quantify the degree of periodicity (or phase-coherence) by the dimensionless parameter sigma (sub p)P. The representative sample of 260 type III burst groups shows a mean periodicity of sigma (sub p/P) = 0.37 +/- 0.12, while Monte Carlo simulations of an equivalent set of truly random time series show a distinctly different value of sigma (sub p)P = 0.93 +/- 0.26. This result indicates that the injection of electron beams is coherently modulated by a particle acceleration source which is either compact or has a global organization on a timescale of seconds, in contrast to an incoherent acceleration source, which is stochastic either in time or space. We discuss the constraints on the size of the acceleration region resulting from electron beam propagation delays and from Alfvenic synchronization during a pulse period. We discuss two periodic preocesses in flares, which potentially control quasi-periodic particle acceleration: (1) MHD oscillations, and (2) current sheets with oscillatory dynamics.

Aschwanden, Markus J.↗
NASA NTRS1996

Mixing of Multiple Jets with a Confined Subsonic Crossflow in a Cylindrical Duct

This paper summarizes NASA-supported experimental and computational results on the mixing of a row of jets with a confined subsonic crossflow in a cylindrical duct. The studies from which these results were derived investigated flow and geometric variations typical of the complex 3-D flowfield in the combustion chambers in gas turbine engines. The principal observations were that the momentum-flux ratio and the number of orifices were significant variables. Jet penetration was critical, and jet penetration decreased as either the number of orifices increased or the momentum-flux ratio decreased. It also appeared that jet penetration remained similar with variations in orifice size, shape, spacing, and momentum-flux ratio when the number of orifices was proportional to the square-root of the momentum-flux ratio. In the cylindrical geometry, planar variances are very sensitive to events in the near wall region, so planar averages must be considered in context with the distributions. The mass-flow ratios and orifices investigated were often very large (mass-flow ratio greater than 1 and ratio of orifice area-to-mainstream cross-sectional area up to 0.5), and the axial planes of interest were sometimes near the orifice trailing edge. Three-dimensional flow was a key part of efficient mixing and was observed for all configurations. The results shown also seem to indicate that non-reacting dimensionless scalar profiles can emulate the reacting flow equivalence ratio distribution reasonably well. The results cited suggest that further study may not necessarily lead to a universal 'rule of thumb' for mixer design for lowest emissions, because optimization will likely require an assessment for a specific application.

Holdeman, James D.↗
NASA NTRS2016

Pre-Test Assessment of the Use Envelope of the Normal Force of a Wind Tunnel Strain-Gage Balance

The relationship between the aerodynamic lift force generated by a wind tunnel model, the model weight, and the measured normal force of a strain-gage balance is investigated to better understand the expected use envelope of the normal force during a wind tunnel test. First, the fundamental relationship between normal force, model weight, lift curve slope, model reference area, dynamic pressure, and angle of attack is derived. Then, based on this fundamental relationship, the use envelope of a balance is examined for four typical wind tunnel test cases. The first case looks at the use envelope of the normal force during the test of a light wind tunnel model at high subsonic Mach numbers. The second case examines the use envelope of the normal force during the test of a heavy wind tunnel model in an atmospheric low-speed facility. The third case reviews the use envelope of the normal force during the test of a floor-mounted semi-span model. The fourth case discusses the normal force characteristics during the test of a rotated full-span model. The wind tunnel model's lift-to-weight ratio is introduced as a new parameter that may be used for a quick pre-test assessment of the use envelope of the normal force of a balance. The parameter is derived as a function of the lift coefficient, the dimensionless dynamic pressure, and the dimensionless model weight. Lower and upper bounds of the use envelope of a balance are defined using the model's lift-to-weight ratio. Finally, data from a pressurized wind tunnel is used to illustrate both application and interpretation of the model's lift-to-weight ratio.

strain-gage balance↗
NASA NTRS2000

Currents and Flows in Distant Magnetospheres

Space scientists have explored, described, and explained the terrestrial magnetosphere for four decades. Rarely do they point out that the planetary and solar wind parameters controlling the size, shape, and activity of Earth's magnetosphere map out only a small portion of the space of dimensionless parameters that govern magnetospheric properties. With the discovery of Ganymede's magnetosphere, the range of parameters relevant to magnetospheric studies has grown by orders of magnitude. Consider the extremes of Ganymede's and Jupiter's magnetospheres. Jupiter's magnetosphere forms within a plasma flowing at super-Alfvenic speed, whereas Ganymede's forms in a sub-Alfvenic flow. The scale sizes of these magnetospheres, characterized by distances to the magnetopause of order 7x10(exp 6) km and 5x10(exp 3) km, respectively, differ by three orders of magnitude, ranging from 100 to 0.1 times the scale of Earth's magnetosphere. The current systems that control the structure and dynamics of a magnetosphere depend on specific plasma and field properties. Magnetopause currents at Ganymede differ greatly from the forms familiar for Earth and Jupiter, principally because the Mach number of the ambient plasma flow greatly influences the shape of the magnetosphere. A magnetodisk current, present at Jupiter because of its rapid rotation, is absent at Earth and Ganymede. The ring current, extensively investigated at Earth, is probably unimportant at Ganymede because the dynamical variations of the external flow are slow. The ring current is subsumed within the magnetodisk current at Jupiter. This paper describes and contrasts aspects of these and other current systems for the three bodies.

Kivelson, Margaret Galland↗
NASA NTRS1986

An experimental investigation of the flap-lag-torsion aeroelastic stability of a small-scale hingeless helicopter rotor in hover

A small scale, 1.92 m diam, torsionally soft, hingeless helicopter rotor was investigated in hover to determine isolated rotor stability characteristics. The two-bladed, untwisted rotor was tested on a rigid test stand at tip speeds up to 101 m/sec. The rotor mode of interest is the lightly damped lead-lag mode. The dimensionless lead-lag frequency of the mode is approximately 1.5 at the highest tip speed. The hub was designed to allow variation in precone, blade droop, pitch control stiffness, and blade pitch angle. Measurements of modal frequency and damping were obtained for several combinations of these hub parameters at several values of rotor speed. Steady blade bending moments were also measured. The lead-lag damping measurements were found to agree well with theoretical predictions for low values of blade pitch angle. The test data confirmed the predicted effects of precone, droop, and pitch control stiffness parameters on lead-lag damping. The correlation between theory and experiment was found to be poor for the mid-to-high range of pitch angles where the theory substantially overpredicted the experimental lead-lag damping. The poor correlation in the mid-to-high blade pitch angle range is attributed to low Reynolds number nonlinear aerodynamics effects not included in the theory. The experimental results also revealed an asymmetry in lead-lag damping between positive and negative thrust conditions.

Sharpe, David L.↗
NASA NTRS1996

Turbulence Scales, Rise Times, Caustics, and the Simulation of Sonic Boom Propagation

The general topic of atmospheric turbulence effects on sonic boom propagation is addressed with especial emphasis on taking proper and efficient account of the contributions of the portion oi the turbulence that is associated with extremely high wavenumber components. The recent work reported by Bart Lipkens in his doctoral thesis is reexamined to determine whether the good agreement between his measured rise times with the 1971 theory of the author is fortuitous. It is argued that Lipken's estimate of the distance to the first caustic was a gross overestimate because of the use of a sound speed correlation function shaped like a gaussian curve. In particular, it is argued that the expected distance to the first caustic varies with the kinematic viscosity nu and the energy epsilon dissipated per unit mass per unit time, and the sound speed c as : d(sub first caustic) = nu(exp 7/12) c(exp 2/3)/ epsilon(exp 5/12)(nu x epsilon/c(exp 4))(exp a), where the exponent a is greater than -7/12 and can be argued to be either O or 1/24. In any event, the surprising aspect of the relationship is that it actually goes to zero as the viscosity goes to zero with s held constant. It is argued that the apparent overabundance of caustics can be grossly reduced by a general computational and analytical perspective that partitions the turbulence into two parts, divided by a wavenumber k(sub c). Wavenumbers higher than kc correspond to small-scale turbulence, and the associated turbulence can be taken into account by a renormalization of the ambient sound speed so that the result has a small frequency dependence that results from a spatial averaging over of the smaller-scale turbulent fluctuations. Selection of k(sub c). can be made so large that only a very small number of caustics are encountered if one adopts the premise that the frequency dispersion of pulses is caused by that part of the turbulence spectrum which lies in the inertial range originally predicted by Kolmogoroff. The acoustic propagating wave's dispersion relation has the acoustic wavenumber being of the form k = (omega/c) + F(omega), where c is a spatially averaged sound speed and where, for mechanical turbulence, the extra term F(omega) must depend on only the angular frequency omega, the sound speed c, and the turbulent energy dissipation epsilon per unit fluid mass and per unit time. If the turbulence is weak, then the quantity F(omega) has to be of second order in the portions of the turbulent fluid velocity in the inertial range, so, following Kolmogoroff's reasoning, it must vary with epsilon as epsilon(exp 2/3). Simple dimensional analysis then reveals that F(omega) is K epsilon(exp 2/3) c(exp -7/3) omega(exp l/3), K being a universal dimensionless complex constant.

Pierce, Allan D.↗
NASA NTRS1989

Cosmic strings and ultra-high energy cosmic rays

The flux is calculated of ultrahigh energy protons due to the process of cusp evaporation from cosmic string loops. For the standard value of the dimensionless cosmic string parameter epsilon is identical to G(sub mu) approx. = 10(exp -6), the flux is several orders of magnitude below the observed cosmic ray flux of ultrahigh energy protons. However, the flux at any energy initially increases as the value of epsilon is decreased. This at first suggests that there may be a lower limit on the value of epsilon, which would imply a lower limit on the temperature of a cosmic string forming phase transition in the early universe. However, the calculation shows that this is not the case -- the particle flux at any energy reaches its highest value at epsilon approx. = 10(exp -15) and it then decreases for further decrease of the value of epsilon. This is due to the fact that for too small values of epsilon (less than 10(exp -15)), the energy loss of the loops through the cusp evaporation process itself (rather than gravitational energy loss of the loops) becomes the dominant factor that controls the behavior of the number density of the loops at the relevant times of emission of the particles. The highest flux at any energy remains at least four orders of magnitude below the observed flux. There is thus no lower limit on epsilon.

Bhattacharjee, Pijushpani↗
NASA NTRS1990

A new continuum model for suspensions of gyrotactic micro-organisms

A new continuum model is formulated for dilute suspensions of swimming micro-organisms with asymmetric mass distributions. Account is taken of randomness in a cell's swimming direction, p, by postulating that the probability density function for p satisfies a Fokker-Planck equation analogous to that obtained for colloid suspensions in the presence of rotational Brownian motion. The deterministic torques on a cell, viscous and gravitational, are balanced by diffusion, represented by an isotropic rotary diffusivity Dr, which is unknown a priori, but presumably reflects stochastic influences on the cell's internal workings. When the Fokker-Planck equation is solved, macroscopic quantities such as the average cell velocity Vc, the particle diffusivity tensor D and the effective stress tensor sigma can be computed; Vc and D are required in the cell conservation equation, and sigma in the momentum equation. The Fokker-Planck equation contains two dimensionless parameters, lambda and epsilon; lambda is the ratio of the rotary diffusion time Dr-1 to the torque relaxation time B (balancing gravitational and viscous torques), while epsilon is a scale for the local vorticity or strain rate made dimensionless with B. In this paper we solve the Fokker-Planck equation exactly for epsilon = 0 (lambda arbitrary) and also obtain the first-order solution for small epsilon. Using experimental data on Vc and D obtained with the swimming alga, Chlamydomonas nivalis, in the absence of bulk flow, the epsilon = 0 results can be used to estimate the value of lambda for that species (lambda approximately 2.2; Dr approximately 0.13 s-1). The continuum model for small epsilon is then used to reanalyse the instability of a uniform suspension, previously investigated by Pedley, Hill & Kessler (1988). The only qualitatively different result is that there no longer seem to be circumstances in which disturbances with a non-zero vertical wavenumber are more unstable than purely horizontal disturbances. On the way, it is demonstrated that the only significant contribution to sigma, other than the basic Newtonian stress, is that derived from the stresslets associated with the cells' intrinsic swimming motions.

NASA Discipline Number 28-20↗
NASA NTRS2000

Thermocapillary Flow and Aggregation of Bubbles on a Solid Wall

During the electrolytic evolution of oxygen bubbles forming on a vertically oriented transparent tin oxide electrode, bubbles were found to be mutually attractive. The mechanism of the aggregation had never been explained satisfactorily until Guelcher et al. attributed it to thermocapillary flow. The gradient of surface tension of the liquid at the bubble's surface, which was established because of reaction heat and ohmic heat loss at the electrode wall, drives flow of the liquid adjacent to each bubble; the bubble "pumps" fluid along its surface away from the wall. Fluid flows toward the bubble to conserve mass and entrains nearby bubbles in the flow pattern. The same logic would apply when two bubbles of equal size are adjacent to each other on a warm wall. Each bubble drives thermocapillary flow and hence entrains the other in its flow pattern, which drives the aggregation. Our objective here is to perform experiments where the temperature gradient at the wall is well known and controlled. The theory can be quantitatively tested by studying aggregation of bubble pairs of equal size, and by varying system parameters such as temperature gradient, bubble size and fluid viscosity. The results are then compared with the theory in a quantitatively rigorous manner. We demonstrate that the theory without adjustable parameters is capable of quantitatively modeling the rate of aggregation of two bubbles. The equations governing the thermocapillary flow around a single stationary bubble on a heated or cooled wall in a semi-infinite domain were solved. Both Reynolds number and Marangoni number were much less than unity. The critical result is that liquid in the vicinity of a warm wall flows toward a stationary collector bubble. Consequently the thermocapillary flow around the stationary bubble entrains another bubble toward itself. The bubbles undergo hindered translation parallel to the wall with velocity U while the fluid flow field is described with u. Two velocities were equated by using a wall hindrance parameter q: U = qu [1] which shows the velocity of bubble is proportional to the entraining velocity. The hindrance parameter q can experimentally be measured independently. q can also be calculated by solving the equations of motion for a bubble translating parallel to a solid wall. The experimental cell is cylindrical with an ID of 10 cm and consists of a 1 cm deep main cell filled with silicone oil and flanked by two thermal reservoirs. The upper thermal reservoir was heated and the lower thermal reservoir was cooled so that the bubbles aggregate. Two types of silicone oil (eta = 0.02 and 0.50 Pa s) were used. Two equal sized air bubbles were injected into the cell with a syringe. The center-to-center distance of bubbles was observed through a microscope. Bubble radius ranged from 0.40 mm to 0.65 mm and the temperature gradients along with the cell ranged from 1400 to 5000 K/m. The bubbles aggregated when heat flows from the wall to the fluid. The velocities of bubbles were in the range of 1 - 10 microns/s. The separation r decreased more quickly when the temperature gradient was higher, bubble size was larger, and the oil viscosity was lower. r decreased more rapidly as the bubbles approached each other. Dimensionless time was arbitrarily set to be zero when the dimensionless center-to-center distance between the bubbles was 4. All the bubble trajectories fall onto one line, especially in the range of dimensionless distance from 4 to 3. This means the relative movement of the bubble pair is proportional to the temperature gradient and bubble size and it is inversely proportional to the viscosity of the oil. This result strongly suggests that the thermocapillary flow-based aggregation mechanism is correct. A value of q can be estimated by fitting the scaled data to Eq. [1]. A best fit value of q was obtained as q = 0.26 with a standard deviation of 0.03. Independent experimental results for q for a 0.5 mm radius bubble, give values of q in the range 0.11 to 0.23. The value of q obtained from solving the equations of motion reveals q has values in the range 0.23 - 0.30. Since the full scale of possible values of q is zero to one, the maximum deviation of independently determined values of q from the best fit value was 15% of this full scale. Thus reasonable quantitative agreement between theory and experiment has been obtained.

Kasumi, Hiroki↗