Unsteady Radiative–Convective Heat Transfer in a Flow Emitting‐Absorbing and Scattering Medium around an Ablating Plate

A conjugate problem of radiative–convective heat transfer in a turbulent hightemperature gasdisperse flow around a thermally thin ablating plate is considered. The plate experiences intense radiative heating by an external source, which is a blackbody. The temperature fields and the distributions of heat fluxes along the plate under unsteady conditions are calculated. The data gained make it possible to examine the effect of the Stark number and phasetransition heat in the plate material on the time evolution of the thermal state of the boundarylayer medium and the plate itself being heated by a hightemperature radiation source.     

Heat Transfer on the Surface on a Spherically Blunted Cone Exposed to a Supersonic Spatial Flow with Injection of a Coolant Gas

The heattransfer processes in a supersonic spatial flow around a spherically blunted cone with allowance for heat overflow along the longitudinal and circumferential coordinates and injection of a coolant gas are studied numerically. The prospects of using highly heatconducting materials and injection of a coolant gas for reduction of the maximum temperatures at the body surface are demonstrated. The solutions of the direct and inverse problems in one, two, and threedimensional formulations for different shell materials are compared. The error of the thinwall method in determining the heat flux on the heatloaded boundary of the body is estimated.     

Multiple Steady State Solutions Resulting From Coupling Between Mixed Convection And Radiation In An Inclined Channel

In this work, we present a numerical study of mixed convection coupled with radiation in an inclined channel with an aspect ratio B = L/H=10, and locally heated from one side. Convective, radiative and total Nusselt numbers, evaluated on the cold surface and at the exit of the channel, are presented for different combinations of the governing parameters namely, the surface emissivity (0 1), the Reynolds number (10 Re 50), the inclination of the channel with respect to the horizontal surface (0° 90°) and the Rayleigh number (Ra = 10⁵). The ratio, R = Q_C/Q_E, of the heat quantities, leaving the channel through the cold wall, Q_C, and through the exit, Q_E, is presented to identify the most favorable issue to the heat transfer in the studied configuration. The results obtained show that the flow structure is significantly altered by radiation which contributes to reduce or to enhance the number of the solutions obtained.     

Turbulent boundary layer on a permeable surface

Several theoretical [1–4] and experimental [5–7] studies have been devoted to the study of the effect of distributed injection of a gaseous substance on the characteristics of the turbulent boundary layer. The primary study has been made of flow past a flat plate with gas injection. The theoretical methods are based primarily on the semiempirical theories of Prandtl [1] and Karman [2].In contrast with the previous studies, the present paper proposes a power law for the mixing length; this makes it possible to obtain velocity profiles which degenerate to the known power profiles [8] in the case of flow without blowing and heat transfer. This approach yields analytic results for flows with moderate pressure gradient.Notation x, y coordinates - U, V velocity components - density - T temperature - h enthalpy - H total enthalpy - c mass concentration - , , D coefficients of molecular viscosity, thermal conductivity, diffusion - c_p specific heat - adiabatic exponent - r distance from axis of symmetry to surface - boundary layer thickness - U velocity in stream core - friction - c_f friction coefficient - P Prandtl number - S Schmidt number - S_t Stanton number - M Mach number - j=0 plane case - j=1 axisymmetric case The indices 1 injected gas - 2 mainstream gas - w quantities at the wall - core of boundary layer - 0 flow of incompressible gas without injection - v=0 flow of compressible gas without injection - ^* quantities at the edge of the laminar sublayer - quantities at the initial section - turbulent transport coefficients     

Unsteady forced convection heat/mass transfer from a flat plate

Numerical methods are used to investigate the transient, forced convection heat/mass transfer from a finite flat plate to a steady stream of viscous, incompressible fluid. The temperature/concentration inside the plate is considered uniform. The heat/mass balance equations were solved in elliptic cylindrical coordinates by a finite difference implicit ADI method. These solutions span the parameter ranges 10 Re 400 and 0.1 Pr 10. The computations were focused on the influence of the product (aspect ratio) × (volume heat capacity ratio/Henry number) on the heat/mass transfer rate. The occurrence on the plates surface of heat/mass wake phenomena was also studied.     

Free convection effects on the oscillatory flow of water at 4° C. past an infinite vertical,porous plate with constant suction

An analysis of a two-dimensional flow of water at 4 °C past an infinite vertical, porous plate is presented under the following conditions — i) suction velocity normal to the plate is constant, ii) the free stream oscillates in time about a constant mean, iii) the plate temperature is constant, iv) the difference between the temperature of the plate and the free stream is moderately large causing free convection currents. — Approximate solutions to coupled non-linear equations are derived for the mean velocity, the mean temperature, the mean skin-friction, the mean rate of heat transfer, the transient velocity and the transient temperature, the amplitude and the phase of the skin-friction and the rate of heat transfer. The mean flow of water at 4 °C is compared with that of water at 20 °C in a quantitative manner for both G >0 (cooling of the plate) and G < 0 (heating of the plate). — It is observed that owing to a fall in the temperature of the water from 20 °C to 4 ° C, there is a fall in the mean skin-friction when the plate is being cooled by the free convection currents, and a rise in the mean skin-friction when the plate is being heated by the free convection currents. The amplitude of the skin-friction, for water at 4°C, remains the same for both G > <0 whereas greater cooling of the plate causes a rise in the amplitude of the rate of heat transfer ¦Q ¦ /E and greater heating of the plate causes a fall in ¦ Q ¦ /E.

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Zusammenfassung Die zweidimensionale Stromung von Wasser bei 4 °C an einer unendlichen senkrechten Wand wird unter folgenden Bedingungen untersucht: 1) konstante Absauggeschwindigkeit normal zur Wand, 2) zeitliche Schwankungen der Freistromgeschwindigkeit um einen Mittelwert, 3) konstante Wandtemperatur, 4) mäßige Temperaturdifferenz zwischen Platte und Freistrom zur Erzeugung freier Konvektion. — Näherungslösungen der gekoppelten nichtlinearen Gleichungen sind abgeleitet für die mittlere Geschwindigkeit, die mittlere Temperatur, die mittlere Wandreibung, die mittlere Wärmeübertragung, die nichtstationäre Geschwindigkeit und Temperatur und die Amplitude und Phase der Wandreibung und der Warmeübertragung. Die Strömung von Wasser bei 4°C is quantitativ verglichen mit der bei 20°C für G > 0 (Kühlung der Platte) und G < 0 (Heizung der Platte). — Erniedrigung der Temperatur von 20°C auf 4°C ergibt geringere Wandreibung bei Kühlung und höhere Wandreibung bei Heizung der Platte. Für Wasser von 4°C bleibt die Amplitude der Wandreibung für G < 0 gleich; stärkere Kühlung ergibt einen Anstieg in der Amplitude der Warmeübertragung ¦Q¦/E, starkere Heizung einen Abfall in ¦q¦/E.

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Nomenclature ¦B¦ amplitude of the skin-friction - C_p specific heat at constant pressure - E Eckert numer {U ₀ ² /c_p(T'_w–T')} - g_x acceleration due to gravity - G Grashoff number {vg_x(T'_w–T')/u₀v ₀ ² } - k thermal conductivity - M_r, M_i fluctuating parts of the velocity profile - P Prandtl number,c _p /k - p pressure - q' rate of heat transfer - ¦Q¦ amplitude of the rate of heat transfer - t' time - T' temperature of fluid - T'_w temperature of the plate - T' temperature of the fluid in the free-stream - T_r,T_i fluctuating parts of the temperature profile - u',v' velocity components in the X8,y' directions - U' free stream velocity - U₀ amplitude of free stream fluctuations - u₀ mean velocity - v₀ suction velocity - x', y' coordinate system - ' frequency of free stream oscillations - non-dimensional frequency,'/vsk0/2 - ' skin-friction - ₀ mean tempeature - ₁ amplitude of the temperature fluctuations - phase angle of the skin-friction - ₁ coefficient of volume expansion - ' density of fluid in the boundary layer - ' density of fluid in the free-stream - viscosity     

Application of Rayleigh-Ritz method to forced convection turbulent heat transfer

An analytical model to predict heat transfer rates to an incompressible fluid in turbulent flow, with fully developed velocity profile, between a heated plate and a parallel, insulated plate is developed. The model employs van Driest's mixing length expression near the wall, a constant eddy diffusivitiy in the core and a constant turbulent Prandtl number. An approximate solution obtained by employing Rayleigh-Ritz method is shown to compare well with the exact solution obtained by numerical integration of the differential equations. The results are compared with the available experimental data and analytical solutions.

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Anwendung der Rayleigh-Ritz-Methode auf die Wärmeübertragung bei erzwungener turbulenter Strömung

Zusammenfassung Es wird ein analytisches Modell zur Berechnung der Wärmeübertragung an ein inkompressibles Fluid in turbulenter Strömung mit voll ausgebildetem Geschwindigkeitsprofil zwischen einer beheizten Platte und einer dazu parallelen isolierten Platte angegeben. Das Modell verwendet van Driest's Ausdruck für die wandnahe Mischungslänge, eine konstante Wirbeldiffusivität im Kern und eine konstante turbulente PrandtlZahl. Eine Näherungslösung nach der Rayleigh-Ritz-Methode läßt sich gut mit der exakten Lösung vergleichen, die durch numerische Integration der Differentialgleichungen erhalten wurde. Die Ergebnisse werden mit verfügbaren Versuchswerten und analytischen Lösungen verglichen.

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Nomenclature A⁺ dimensionless constant in van Driest formula - a⁺ dimensionless distance from the wall after which the eddy diffusivity of momentum is constant - b half-gap of passage - b⁺ dimensionless half-gap=bu^*/ - C_f skin friction coefficient - C_p constant pressure specific heat - d hydraulic mean diameter defined as 4xarea/perimeter=4b - h convective heat transfer coefficient - K⁺ dimensionless constant in van Driest formula - k fluid thermal conductivity - m mass flow rate of fluid - Nu Nusselt number hd/k - P pressure - Pr Prandtl number=/ - Pr_t turbulent Prandtl number=_m/ - q_w heat flux at wall - Re Reynolds number=v_md/ - T Temperature - u⁺ dimensionless velocity=V_x/u^* - u^* friction velocity= - V_x axial velocity - x axial distance from the entrance - x⁺ dimensionless distance=x/d - y distance from the heated wall - y⁺ dimensionless distance=yu^*/ Greek Symbols thermal molecular diffusivity - function equal to (_H+)/ - boundary layer thickness - _H eddy diffusivity of heat - _m eddy diffusivity of momentum - _m0 uniform eddy diffusivity of momentum in the core - dimensionless temperature - T-T_i/q_wd/k uniform heat flux - T-T_w/T_i-T_w uniform temperature - fluid kinematic viscosity - fluid density - fluid shearing stress - bulk mean temperature—fully developed region - fully developed transverse temperature profile Suffixes 1 fully developed - 2 in the entrance region - i at the inlet - m bulk mean value - w at the heated wall     

Heat transfer in laminar flow in a uniformly porous channel with an applied transverse magnetic field

Summary The steady laminar flow of an incompressible, viscous, and electrically conducting fluid between two parallel porous plates with equal permeability has been discussed by Terrill and Shrestha [6]. In this paper, using the solution of [6] for the velocity field, the heat transfer problems of (i) uniform wall temperature and (ii) uniform heat flux at wall are solved.For small suction Reynolds numbers we find that the Nusselt number, with increasing Reynolds number, increases for case (i) and decreases for (ii).Nomenclature stream function - 2h channel width - x, y distances measured parallel, perpendicular to the channel walls - U velocity of fluid in the x direction at x=0 - V constant velocity of suction at the wall - nondimensional distance, y/h - nondimensional distance, x/h - f() function defined in (1) - density - coefficient of kinematic viscosity - R suction Reynolds number, V h/ - Re channel Reynolds number, 4U h/ - B ₀ magnetic induction - electrical conductivity - M Hartmann number, B ₀ h(/)^1/2 - K constant defined in (3) - A constant defined in (5) - 4R/Re - q local heat flux per unit area at the wall - k thermal conductivity - T temperature of the fluid - X –1/ ln(1–) - C _p specific heat at constant pressure - j current density - Pr Prandtl number, C _p/k - P mass transfer Péclet number, R Pr - Pe mass transfer Péclet number, P/ - T ₀ temperature at x=0 - T _H() temperature in the fully developed region - T _h(X, ) temperature in the entrance region - Y _n () eigenfunctions, uniform wall temperature - _n eigenvalues - e() function defined by (24) - B _n 2/3 _n ² - A _n constants defined by (28) - a _2m constants defined by (30) - F _n () eigenfunctions, uniform wall heat flux - a _n , b _n , c _n , d _n , e _n constants defined by (45) and (48) - S a parameter, U ²/q - h ₁ heat transfer coefficient - T _m mean temperature - Nu Nusselt number - Nu _T Nusselt number, uniform wall temperature - Nu _q Nusselt number, uniform wall heat flux     

Mixed convection flow over a vertical plate with localized heating (cooling), magnetic field and suction (injection)

An analysis is carried out to study the effects of localized heating (cooling), suction (injection), buoyancy forces and magnetic field for the mixed convection flow on a heated vertical plate. The localized heating or cooling introduces a finite discontinuity in the mathematical formulation of the problem and increases its complexity. In order to overcome this difficulty, a non-uniform distribution of wall temperature is taken at finite sections of the plate. The nonlinear coupled parabolic partial differential equations governing the flow have been solved by using an implicit finite-difference scheme. The effect of the localized heating or cooling is found to be very significant on the heat transfer, but its effect on the skin friction is comparatively small. The buoyancy, magnetic and suction parameters increase the skin friction and heat transfer. The positive buoyancy force (beyond a certain value) causes an overshoot in the velocity profiles.A mass transfer constant - B magnetic field - C_fx skin friction coefficient in the x-direction - C_p specific heat at constant pressure, kJ.kg^–1.K - C_v specific heat at constant volume, kJ.kg^–1.K^–1 - E electric field - g acceleration due to gravity, 9.81 m.s^–2 - Gr Grashof number - h heat transfer coefficient, W.m².K^–1 - Ha Hartmann number - k thermal conductivity, W.m^–1.K - L characteristic length, m - M magnetic parameter - Nu_x local Nusselt number - p pressure, Pa, N.m^–2 - Pr Prandtl number - q heat flux, W.m^–2 - Re Reynolds number - Re_m magnetic Reynolds number - T temperature, K - T_o constant plate temperature, K - u,v velocity components, m.s^–1 - V characteristic velocity, m.s^–1 - x,y Cartesian coordinates - thermal diffusivity, m².s^–1 - coefficient of thermal expansion, K^–1 - , transformed similarity variables - dynamic viscosity, kg.m^–1.s^–1 - ₀ magnetic permeability - kinematic viscosity, m².s^–1 - density, kg.m^–3 - buoyancy parameter - electrical conductivity - stream function, m².s^–1 - dimensionless constant - dimensionless temperature, K - w, conditions at the wall and at infinity     

Droplet motion in an electrohydrodynamic fine spray

The results of a combined experimental and numerical study on droplet behavior within an electrohydrodynamic fine spray are presented. The fine spray exists in the transition region between the multiple cone-jet and rim emission spray modes. Experiments were conducted specifically to characterize the motion of droplets within the spray. Light-sheet visualizations and measurements of droplet speed and velocity using laser-based, single-particle counters were obtained. Additionally, a numerical simulation of the droplet motion within the spray was made and compared to the experimental results. The electrohydrodynamic fine spray of ethanol droplets ( 1 to 40 m diameter) was generated using a typical capillary-plate configuration, with a capillary tip electric field intensity of 10⁶ V/m and a spray charge density of 70 C/m³. Acquired images of the spray revealed a zone of rapid expansion near the capillary followed by a more gradual expansion farther from the capillary. In situ laser-diagnostic measurements confirmed these observations. Measured droplet speeds decreased rapidly with increasing axial distance from the capillary, but then increased beyond the spray's axial mid-plane as a result of a change in the sign of the axial internal electric field. Droplet axial velocity components behaved similarly. The radial velocity components exhibited a maximum value off of the spray's centerline in the near-capillary region. Farther away from the capillary, they increased monotonically with increasing radial position. These trends identified the significant role that the radial internal electric field plays in spray expansion. The numerical simulation of the normal spray verified the inferred change in sign of the axial internal field and underscored the dominant contribution of the external electric field in the near-capillary region and of the internal electric field farther away.     

Nonsimilar Solutions for Mixed Convection in Non‐Newtonian Fluids Along a Vertical Plate in a Porous Medium

A nonsimilar boundary layer analysis is presented for the problem of mixed convection in powerlaw type nonNewtonian fluids along a vertical plate with powerlaw wall temperature distribution. The mixed convection regime is divided into two regions, namely,the forced convection dominated regime and the free convection dominated regime. The two solutions are matched. Numerical results are presented for the details of the velocity and temperature fields. A discussion is provided for the effect of viscosity index on the surface heat transfer rate.     

The problems of nonlinear bending for orthotropic rectangular plate with four clamped edges

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Water Impingement on a Vertical Wall due to Discontinuity Decay Above a Drop

The paper reports experimental data on the height of water impingement on a vertical wall during wave reflection due to discontinuity decay above an even bottom and a bottom drop. It is shown that for a bore type wave with a roller in its head, propagating over finitedepth water, the impingement height is proportional to the initial difference in freesurface level. In the case of a dry bottom in the tail water, this is also true for other types of waves formed during discontinuity decay.     

Stress–strain state of a composite anisotropic plate with curvilinear cracks and thin rigid inclusions

A complexpotential solution of a mixed problem of the linear theory of elasticity is given for an infinite plate composed of two anisotropic halfplanes. The plate contains cuts and thin undeformable inclusions shaped like arbitrary open smooth curves that do not intersect each other and the interface between the halfplanes.     

Effects of magnetic field and variable viscosity on forced non‐darcy flow about a flat plate with variable wall temperature in porous media in the presence of suction and blowing

This paper presents a study of the effect of a magnetic field and variable viscosity on steady twodimensional laminar nonDarcy forced convection flow over a flat plate with variable wall temperature in a porous medium in the presence of blowing (suction). The fluid viscosity is assumed to vary as an inverse linear function of temperature. The derived fundamental equations on the assumption of small magnetic Reynolds number are solved numerically by using the finite difference method. The effects of variable viscosity, magnetic and suction (or injection) parameters on the velocity and temperature profiles as well as on the skinfriction and heattransfer coefficients were studied. It is shown that the magnetic field increases the wall skin friction while the heattransfer rate decreases.     

Whose Thoughts Are They, Anyway? Dimensionally Exploding Bion's “Double-Headed Arrow” into Coadapting, Transitional Space

The physics and biology that found psychoanalysis account for discontinuous experience only in the presence of nonmeasurable, metaphysical operators; these include the ego and its subsystems as well as biological experience inherited through Lamarckian principles. Complex, self-organizing systems, however, can link biology to experience without metaphysics. They can also account for psychoanalytically relevant behaviors without appealing to stable internal representations. These behaviors include what W. R. Bion called transformation in O and its corollary, the appearance of the selected fact. By dimensionally exploding the double-headed arrow that he used to link the states Ps and D in his model for thinking (Ps D), we can generate a space that is, at once, psychoanalytically imaginal and dynamically coadapting. Isomorphic to D. W. Winnicott's transitional space, it is self-organizing. It is describable according to dynamics formulated by W J. Freeman, S. Kauffman and C. Langton and it can generate instantaneous conscious contents by way of a selective process analogous to spatio-temporal binding. As a whole, this model supports a clinical stance advanced by D. W. Winnicott as play, within transitional space.     

Experimental investigation of impinging jet arrays

We report on measurements of the velocity field and turbulence fluctuations in a hexagonal array of circular jets, impinging normally on a plane wall, using particle image velocimetry (PIV). Results for mean velocity and turbulent stresses are presented in various horizontal and vertical planes. From the measurements, we have identified some major features of impinging jet arrays and we discuss their mutual interaction, collision on the plate, and consequent backwash, which generate recirculating motion between the jets. The length of the jet core, the production of turbulence kinetic energy, and the model of the exhaust mechanisms for spent fluid are also discussed. The measurements indicated that the interaction between the self-induced cross flow and the wall jets resulted in the formation of a system of horseshoe-type vortices that circumscribe the outer jets of the array. The instantaneous snapshots of the velocity field reveal some interesting features of the flow dynamics, indicating a breakdown of some of the jets before reaching the plate, which may have consequences on the distribution of the instantaneous heat transfer.List of symbols D_m Nozzle diameter in multiple jet array nozzle plate (m) - D_s Pipe diameter in single jet rig (m) - H Distance between nozzle and impingement plate (m) - k Turbulent kinetic energy (m²/s²) - L Pipe length (m) - P_k Production of turbulent kinetic energy (m²/s³) - P_uu , P_vv Normal components of P_k (m²/s³) - P_uv Shear component of P_k (m²/s³) - s Pitch (m) - U_bulk Surface-averaged exit velocity (single jet) (m/s) - U_CL Center line jet exit velocity (jet array), m/s - u, v Mean velocity components in x and y directions (m/s) - u, v, w Instantaneous velocity in x, y, and z directions (m/s) - u, v, w Velocity fluctuation in x, y, and z directions (m/s) - u², v², w² Reynolds normal stress components (m²/s²) - uv Reynolds shear stress component (m²/s²) - x, z Coordinates parallel to impingement plate (m) - y Coordinate perpendicular to impingement plate (m)     

Heat transfer between gas-liquid spray stream flowing perpendicularly to the row of the cylinders

Experimental investigation and analysis of heat transfer process between a gas-liquid spray flow and the row of smooth cylinders placed in the surface perpendicular to the flow has been performed. Among others, there was taken into account in the analysis the phenomenon of droplets bouncing and omitting the cylinder as well as the phenomenon of the evaporation process from the liquid film surface.In the experiments test cylinders were used, which were placed between two other cylinders standing in the row.From the experiments and the analysis the conclusion can be drawn that the heat transfer coefficients values for a row of the cylinders are higher than for a single cylinder placed in the gasliquid spray flow.

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Wärmeübergang an eine senkrecht anf eine Zylinderreihe auftreffende Gas-Flüssigkeits-Sprüh-Strömung

Zusammenfassung Es wurden Messungen und theoretische Analysen des Wärmeübergangs zwischen einer Gas-FlüssigkeitsSprüh-Strömung und den glatten Oberflächen einer Zylinderreihe durchgeführt, die senkrecht zum Sprühstrahl angeordnet waren. Dabei wurde in der Analyse unter anderem das Phänomen betrachtet, daß die Tropfen die Zylinderwand treffen und verfehlen können und daß sich ein Verdampfungsprozeß aus dem flüssigen Film an der Zylinderoberfläche einstellt.Gemessen wurde an einem zwischen zwei Randzylindern befindlichen Zylinder.Die Experimente und die Analyse gestatten die Schlußfolgerung, daß der Wärmeübergangskoeffizient für eine Zylinderreihe höher ist als für einen einzelnen Zylinder in der Sprühströmung.

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Nomenclature a distance between axes of cylinders, m - c _l specific heat capacity of liquid, J/kg K - c _g specific heat capacity of gas, J/kg K - D cylinder diameter, m - g _l mass velocity of liquid, kg/m²s - ¯k average volume ratio of liquid entering film to amount of liquid directed at the cylinder in gas-liquid spray flow, dimensionless - k() local volume ratio of liquid entering film to amount of liquid directed at the cylinder in gas-liquid spray flow, dimensionless - L specific latent heat of vaporisation, J/kg - m mass fraction of water in gas-liquid spray flow, dimensionless - M constant in Eq. (9) - p pressure, Pa - p _g statical pressure of gas, Pa - p _w pressure of gas on the cylinder surface, Pa - p external pressure on the liquid film surface, Pa - r cylindrical coordinate, m - R radius of cylinder, m - T temperature, K, °C - T _l, t_l liquid temperature in the gas-liquid spray, K, °C - T _w,t_w temperature of cylinder surface, K, °C - T temperature of gas-liquid film interface, K - U liquid film velocity, m/s - w gas velocity on cylinder surface, m/s - w _g gas velocity in free stream, m/s - W _l liquid vapour mass ratio in free stream, dimensionless - W liquid vapour mass ratio at the edge of a liquid film, dimensionless - x coordinate, m - y coordinate, m - z complex variable, dimensionless - average heat transfer coefficient, W/m²K - local heat transfer coefficient, W/m² K - average heat transfer coefficient between cylinder surface and gas, W/m² K - _g, local heat transfer coefficient between cylinder surface and gas, W/m² K - mass transfer coefficient, kg/m²s - liquid film thickness, m - _lg dynamic diffusion coefficient of liquid vapour in gas, kg/m s - pressure distribution function on a cylinder surface - function defined by Eq. (3) - _l liquid dynamic viscosity, kg/m s - _g gas dynamic viscosity, kg/m s - cylindrical coordinate, rad, deg - _l thermal conductivity of liquid, W/m K - _g thermal conductivity of gas, W/m K - mass transfer driving force, dimensionless - _l density of liquid, kg/m³ - _g density of gas, kg/m³ - _w shear stress on the cylinder surface, N/m² - _w shear stress exerted by gas at the liquid film surface, N/m² - air relative humidity, dimensionless - T -T _w - _w =T _wT_l Dimensionless parameters I= enhancement factor of heat transfer - m ^*=M _l/M_g molar mass of liquid to the molar mass of gas ratio - Nu _g= D/ _g gas Nusselt number - Pr _g=c _{g g}/_g gas Prandtl number - Pr _l=c_lll liquid Prandtl number - ¯r=(r–R)/ dimensionless coordinate - Re _g=w_gD _g/_g gas Reynolds number - Re _g,max=w_g,max D _g/_g gas Reynolds number calculated for the maximal gas velocity between the cylinders - Sc=m ^* _g/_l–g Schmidt number =/R dimensionless film thickness     

Heat transfer to an isothermal flat plate in turbulent flow of a liquid over a wide range of prandtl and reynolds numbers

The study of heat transfer in turbulent flow over a flat plate is very important, not only because this situation frequently arises in practice, but also in that data for an isothermal flat plate are used to calculate heat transfer in more complex cases. In particular, such data are necessary when one uses the limiting relative laws which allow calculation of the effect of compressibility, pressure gradient, blowing, and other perturbing factors [1]. Most papers dealing with heat transfer for an isothermal flat plate refer to comparatively low Re values, when the velocity distribution in the boundary layer over almost its entire thickness can be described by the universal law of the wall. However, as Re increases there is an increasing layer adjacent to the outer boundary in which the velocity distribution cannot be described by the law of the wall, and therefore the results obtained for low Re are inapplicable. In the present paper coefficients of heat transfer from a turbulent flow to an isothermal flat plate have been obtained by numerical integration of the thermal boundary-layer equations over a wide range of the parameters 3 · 10⁵ Re 2.5·10¹², 10² Pr 10³.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 4, pp. 94–100, July–August, 1976.     

Shock Waves in a Fluid with Bubbles Containing Evaporating Drops of a Liquefied Gas

The propagation and reflection of one-dimensional plane unsteady waves and pulses in a mixture of a fluid with two-phase bubbles containing evaporating drops is investigated. A significant effect of unsteady evaporation of the drops in the zone ahead of the shock wave on the wave propagation is demonstrated. The evaporation of the drops results in a pressure increase ahead of the wave and the shock wave as it were climbs to increasing pressure level. In contrast to bubbly fluids with single-phase bubbles, in a fluid with two-phase bubbles, at a fixed phase volume fraction, a decrease in bubble size results in an increase rather than a decrease of the oscillation amplitude. The wave reflection from a solid wall is essentially nonlinear and the maximum pressure attained at the wall is several times greater than the incident-wave intensity.