Investigation of the microstructure a turbulent jet in a wake

The study of the characteristics of the turbulence in the boundary layer and in free jets is one of the most important problems of the aerodynamics of viscous fluids. The accumulation of information on the pulsation characteristics of jet flows and the establishment of the corresponding governing laws may serve to verify the basic hypotheses of the semiempirical theories of turbulence, and also for the development of more advanced computational methods. In many cases the measurement of the pulsation characteristics of turbulent jets is of practical interest.The studies made up till now [1–5] of the microstructure of turbulent flow in the primary region of submerged axisymmetric jets have made it possible to obtain several interesting results. In particular, in addition to the average velocity profiles, hot-wire anemometric equipment has been used to measure the normal and tangential Reynolds stresses and also the intermittency factor in cross sections of the jet, the distribution of the intensity of the longitudinal and lateral velocity pulsations along the axis, the correlation coefficients and the corresponding integral turbulence scales, etc. These measurements have made it possible to draw several important conclusions on the mechanism of turbulent exchange, on the order of the terms omitted in the equation of motion, and on the semiempirical theories of turbulence [6–9].The common deficiency of the studies mentioned above is that near the boundary of a submerged jet, where the average velocity is practically equal to zero, the intensity of the pulsations is so great that it makes the reliability of the results obtained by means of the hotwire anemometer questionable. In this connection Townsend [6] indicated the advisability of studying the microstructure of a turbulent jet issuing into a low-velocity ambient flow.The present study had as its objective the investigation of the microstructure of the primary region of an axisymmetric jet in a wake flow over quite a broad range of the flow ratio parameter m=u/u₀;here u₀ is the average velocity at the nozzle exit, u is the velocity of the ambient stream. For various values of the parameter m in the primary region of the jet measurements were made of the profiles of the three components of the pulsation velocity and the Reynolds shear stresses, and also the values of the average velocity and two components of the pulsation velocity at a large number of points on the jet axis. The measured profiles of the Reynolds shear stresses were compared with the corresponding profiles calculated on the basis of the boundary layer equations from the experimentally determined average velocity profiles. For two values of the parameter m, in one of the sections of the jet measurements were made of the correlation coefficients of the longitudinal components of the pulsation velocity and the variation across the jet of the integral turbulence scale was determined.The results obtained give an idea of the influence of the parameter m on the characteristics of the turbulent jet in an ambient stream.     

Study of heat transfer in the separation zones formed in front of supersonic jets in a subsonic carrier flow

The heat transfer taking place between the gas and the surface of the plate in the zone of three-dimensional separation of the turbulent boundary layer in front of a set of supersonic jets injected perpendicularly to a subsonic carrier flow is considered. The aim of this investigation is to establish the main physical characteristics of heat transfer in the separation zones in front of jet obstacles and to obtain the distributions of the local heat-transfer coefficients and the temperature of the thermally insulating wall as functions of the parameters of the carrier flow and the injected jets. Analysis of the experimental results yields certain approximating relationships for the distribution of the local heat-transfer coefficients as functions of the Mach number of the carrier flow M, the Mach number of the jet M_j, the relative boundary-layer displacement thickness _s= _s ^* /d, and the degree of jet superheating T_ojT_o relative to the separation zones in front of supersonic jet obstacles.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 68–72, July–August, 1975.     

Experimental investigation of a turbulent jet carrying heavy particles of a disperse phase

The results are given of an experimental investigation of an immersed jet carrying heavy spherical particles in the case of uniform velocity fields of the particles and the gas and uniform concentration of the particles at the nozzle exit. The laser-optical method was used to measure the velocity fields of the gas and the particles and the concentration of the latter in the flow-rate concentration range of the additive from 0 to 1.5 and particle diameters from 35 to 67 m. It was found that the flow in the jet is essentially nonequilibrium, the disequilibrium and intensity of the mixing in the jet depending on the concentration of the additive and the diameter of the particles. The investigation of the initial section revealed a somewhat anomalous behavior of the jet at a comparatively small (₀=0.5) flow-rate concentration of the additive, and this indicates a need to take into account the prehistory of the flow in the construction of a method for calculating the initial section of a two-phase jet.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 26–31, September–October, 1981.     

Calculation of the transition region of a turbulent jet

The presently known methods for calculating plane and axisymmetric turbulent jets in a wake flow are based on dividing the flow region into two segments, initial and basic [1–3], Here the matching of the parameters of the initial and basic segments is of an artificial nature, since it permits the existence of a physically impossible discontinuity of the curves of the velocity distribution and the jet width along the axis.The aerodynamic characteristics of the transition segment, extending from the point of convergence of the boundary layers at the end of the initial segment to the section corresponding to the point of inflection of the curve u_m(x), differ significantly from the characteristics of the initial and basic segments. This difference is due not only to the sharp increase of the velocity pulsations, but also the marked deformation of the average longitudinal velocity component profile. Consequently, the calculation of the transition segment, in contrast to the initial and basic segments, cannot be based on the single-parameter method.Generally speaking, the flow development in the transition segment may be calculated with the aid of the method [4], which reduces the solution of the problem to an equation of the heat conduction type and assumes the use of an experimental curve of the velocity distribution along the jet axis. Abramovich has carried out the calculation of the transition segment of a plane submerged jet on the basis of certain assumptions which are based on the results of experimental studies [1].Below is presented an approximate method of calculating the transition segment of plane and axisymmetric turbulent jets in a wake flow in which the velocity profiles obtained for the extreme sections of this segment are used for calculating the flow parameters in the initial and basic segments.     

Sensitivity of three-segment electrodiffusion probes to eddy shedding

The influence of eddy shedding on the instantaneous readings of a three-segment cylindrical electrodiffusion velocity probe was investigated in an immersed jet with a very low turbulence intensity, = 1.2%. The velocity fluctuations measured by the three-segment probe were smaller than 2.6%, and the maximum error in the flow angle estimation was 2. Vortices with the Strouhal frequency were detected by a simple electrodiffusion probe placed downstream of the three-segment probe, but no peaks with this frequency were found on the frequency spectra of the three-segment probe. From the probe response to a stepwise change of the polarization voltage the characteristic times of the transient process were estimated. List of symbols a parameter in Eq. (1) [A s^b m^-b] - A amplitude gain - b parameter in Eq. (1) - c parameter in Eq. (3) [A s^–1/2] - d probe diameter [m] - f frequency [s^–1] - f _s recording frequency [s^–1] - G power spectrum - I _k relative current through k-th segment, Eq. (2) - i total current [A] - i _k current through k-th segment [A] - N number of data samples - Re Reynolds number, - Sr Strouhal number, - t time [s] - t ₀ characteristic transient time [s] - v jet velocity [m s^-1] - v time mean value of velocity [m s^-1] - v _x, y velocity components measured by probe [m s^-1] - var variance, var - dynamic viscosity [Pa s] - density [kg m^-3] - relative deviation, [%] - flow angle, see Fig. 1 - dimensionless frequency For the financial support of this work we express our thanks to the DFG, Bonn. The assistance of Dr. Ondra Wein and Dr. Pavel Mitschka is greatly appreciated.     

Development characteristics of drag-reducing surfactant solution flow in a duct

Development characteristics of dilute cationic surfactant solution flow have been studied through the measurements of the time characteristics of surfactant solution by birefringence experiments and of the streamwise mean velocity profiles of surfactant solution duct flow by a laser Doppler velocimetry system. For both experiments, the concentration of cationic surfactant (oleylbishydroxymethylethylammonium chloride: Ethoquad O/12) was kept constant at 1000 ppm and the molar ratio of the counter ion of sodium salicylate to the surfactants was at 1.5. From the birefringence experiments, dilute surfactant solution shows very long retardation time corresponding to micellar shear induced structure formation. This causes very slow flow development of surfactant solution in a duct. Even at the end of the test section with the distance of 112 times of hydraulic diameter form the inlet, the flow is not fully developed but still has the developing boundary layer characteristics on the duct wall. From the time characteristics and the boundary layer development, it is concluded that the entry length of 1000 to 2000 times hydraulic diameter is required for fully developed surfactant solution flow.List of abbreviations and symbols A₁, A₂ Coefficients for time constant fitting [-] - B Breadth of the test duct [m] - C₁, C₂ Coefficients for time constant fitting [-] - D Pipe diameter [m] - D_H Hydraulic diameter [m] - g Impulse response function [Pa] - H Width of the test duct [m] - n Index of Bird-Carreau model [-] - Re Reynolds number (=U_mD_H/) - Re_D Pipe Reynolds number (=U_mD/) - Re_x Streamwise distance Reynolds number (=U₀x/) - T Absolute temperature [K] - t Time [s] - t_a Retardation time [s] - t_b Build-up time [s] - t_x Relaxation time [s] - t_x1, t_x2 Relaxation time for double time constant fitting [s] - t Time constant in Bird-Carreau model [s] - U Time mean velocity [m/s] - U_m Bulk mean velocity [m/s] - U_max Maximum velocity in a pipe [m/s] - U₀ Main flow velocity [m/s] - u Friction velocity [m/s] - x, y Coordinates [m] - Shear rate [s^–1] - Mean shear rate [s^–1] - n Birefringence [-] - 99% boundary layer thickness [m] - Solution viscosity [Pa·s] - _P, _S Surfactant and solvent viscosity [Pa·s] - ₀, Zero and infinite viscosity of Bird-Carreau model [Pa·s] - Characteristic time in Maxwell model [s] - Water kinematic viscosity [m²/s] - Density [kg/m³] - Solution shear stress [Pa] - _P, _S Surfactant and solvent shear stress [Pa] - Time in convolution [s]     

Axisymmetric impingement of a hot jet on a flat plate: equilibrium flow analysis of high-temperature air

The flow structure of an underexpanded supersonic jet with high reservoir temperature impinging on a flat plate has been numerically investigated using a Total Variation Diminishing (TVD) scheme. When the temperature of the flow field is high enough to cause chemical reaction, the specific heat ratio,, is no longer equal to 1.4, nor constant. This explains the difference found in the literature between the flow properties of the calorically perfect gas and that of the chemically reacting flow. Under the equilibrium flow assumption the effect of high temperature gas on the impinging jet has been taken into account in the present paper by using specific heat ratio and speed of sound given by correlation polynomials of thermodynamic variables. The limiting case of cold jet calculation in the present numerical results agreed well with the existing experimental data. For the equilibrium jet with high reservoir temperature,T _o=1000K, qualitative support of the present result has been provided by means of the approximation theory.This article was processed using Springer-Verlag T_EX Shock Waves macro package 1.0 and the AMS fonts, developed by the American Mathematical Society.     

A new method for measuring time constants of a thermocouple wire in varying flow states

A new measuring method is suggested for determining the time constant of a thermocouple wire to be applied for the measurement of the true fluid temperatures in varying flow states. Based on the techniques of internal heating which are commonly used to measure mean time constants, we extend the existing method to measure instantaneous time constants continuously. A method of measurement and analysis is presented and verified experimentally.List of Symbols A _s surface area [m²] - c specific heat [J/kg K] - D diameter [m] - h heat transfer coefficient [W/m² K] - I current [A] - k thermal conductivity [W/m K] - L length [m] - r resistance per unit length [/m] - T temperature [°C] - t time [s] - t _c characteristic time to reach uniform state [s] - u velocity of stream [m/s] - V volume [m³] - x axial coordinate [m] - thermal diffusivity [m²/s] - normalized temperature (TT )/(T _RT )) - density [kg/m³] - time constant [s] - angular velocity [rad/s] - a amplitude - i initial condition - j junction of thermocouple - R reference point - surrounding The work was supported by Turbo and Power Machinery Research Center at Seoul National University and the authors are grateful to Mr. M. H. Yang for his assistance in the experiment.     

Couette flow of a multicomponent ionized gas

We consider equilibrium flow of a multicomponent ionized gas between two catalytic plates of infinite length, one of which moves parallel to the other with constant velocity. The results of [1] are generalized for ionized gaseous mixtures which are in local thermodynamic equilibrium. Formulas are presented for calculating the thermal flux and the effective thermal conductivity for ambipolar diffusion.Then a special ionization case is discussed.Notation A_i chemical symbol of the i-th component - W_i projection of the molar diffusive flux vector of the i-th component on the y-axis - x_i molar concentration - H_i enthalpy - m_i molecular weight - Q_s heat of the s-th reaction - K_ps(T) equilibrium constant of the s-th reaction - W_i mass formation rate of the i-th component per unit volume - Z_i charge number - e unit charge (electron charge) - E electric field intensity - distance between the plates - N number of components - v _sl stoichiometric coefficients - density - T temperature - p pressure - u projection of average velocity on y-axis - viscosity - thermal conductivity - D_ij binary diffusion coefficient - R universal gas constant - k Boltzmann constant In conclusion, the author wishes to thank G. A. Tirskii for proposing the study and for suggestions made in the course of the investigation.     

The effect of ablation products on heat exchange in the disintegration of graphite in a radiating air plasma

In this article we consider the problem of the stationary hypersonic flow of a viscous, heat-conducting stream of radiating high-temperature air past the forward critical point of a blunt body made of graphite in the region between a passing shock wave and the surface of the body.We investigated the radiative and convective heat exchange taking place at an impenetrable surface, as well as such heat exchange when air is injected.We obtained the characteristics of the graphite ablation on the assumption that there is radiative transfer of heat by the graphite-disintegration products.The diffusion was calculated on the basis of a binary model, i.e., it was assumed that the mixture consists of two components: the advancing air and the disintegration products; the chemical reactions in the boundary layer were assumed to be frozen, and on the outer edge of the boundary layer, reaching out as far as the shock wave, they were assumed to be equilibrium reactions.The condition of the gas on the disintegrating surface was also determined on the assumption that there was chemical equilibrium, and the saturation pressure of the vapors was assumed to be equal to the stagnation pressure.It should be noted that the effect of ablation on heat exchange has been considered in [1–3]. Anfimov and Shari [1] assumed that pure air was injected and their calculations were carried out up to stagnation temperatures of 15·10³° K. Hoshisaki and Zasher [2] gave an analysis of a small number of variants, where the values of the absorption cross sections of the injected components used in their calculations were independent of temperature; and the discussion in [3] was limited to the case of low injection velocities.Notation c_p specific heat at constant pressure - c_i mass concentration of the i-th component - D₁₂ coefficient of binary diffusion - f dimensionless stream function - h static enthalpy of the mixture - h_i enthalpy per unit mass of the i-th component - k thermal conductivity - m_w total rate of mass loss through the surface - p stagnation pressure - q heat flux - T absolute temperature - u velocity component along the x axis - v velocity component along the y axis - optical thickness - coefficient of absorption - Lees-Dorodnitsyn self-similar variable - viscosity coefficient - absorption cross section - density - Stefan-Boltzmann constant - wavelength - angle between the normal to the surface and the incident intensity ray Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 23–31, January–February, 1971.     

Interaction of a turbulent jet and a cocurrent confined incompressible flow

The effects induced in a coaxial circular channel flow by an axisymmetric turbulent jet are investigated for various values of the velocity and radius ratios 0.16m<1 and 2.5f30.9. The problem is solved by means of an e-L model of turbulence [1, 2]. The calculation scheme differs from the usual one for boundary layers, jets and wakes in that the pressure p is assumed to be unknown and is determined by assigning the boundary conditions for the radial velocity component and the transverse gradient of the longitudinal velocity component on both boundaries. On the basis of the calculations and the experimental data of [3, 4] generalized relations are obtained. These make it possible to estimate the turbulence characteristics of an axisymmetric jet in a confined cocurrent flow when the pressure is variable along the flow.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 14–19, September–October, 1986.In conclusion, the author wishes to thank G. S. Glushko for constructive discussion of the results and useful advice.     

Instationäre eindimensionale Wärmeleitung beim Erstarren einer Flüssigkeit mit konstanter Außentemperatur

Zusammenfassung Die Temperaturverteilung in der festen und flüssigen Phase einer erstarrenden Flüssigkeit mit eindimensionaler Wärmeleitung und konstanter Außentemperatur der festen Phase wurde mit Hilfe von Laplace-Transformationen abgeleitet und mit der Neumannschen Losung des gleichen Problems verglichen.Die Übereinstimmung zwischen der Losung im vorliegen Beitrag und der Neumannschen Lösung ist recht gut.

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Transient one dimensional heat transfer in a liquid in solidification and with constant outer surface temperature of the solid phase

The temperature distribution in the solid and liquid phase of a liquid in solidification with one dimensional heat transfer and constant outer surface temperature of the solid phase is laid down by using Laplace transforms.The agreement between the present solution and Neumanns solution of the same problem is very good.

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Formelzeichen h Schmelz- oder Erstarrungswärme [J/kg] - k Erstarrungskoeffizient [m/s^0,5] - k₁ Temperaturleitfähigkeit der festen Phase [m²/s] - k₂ Temperaturleitfähigkeit der fl. Phase [m²/s] - T_o Außentemperatur der festen Phase [K] - T_s Schmelz- Oder Erstarrungstemperatur [K] - T Temperatur der fl. Phase zur Zeit t=0[K] - t Zeit [t] - x Entfernung von der Außenfläche der festen Phase [t] - ₁ Wärmeleitzahl der festen Phase [W/m·k] - ₂ Wärmeleitzahl der fl. Phase [W/m·k] - Dicke der festen Phase, Entfernung der Erstarrungsfront von der Außenfläche der festen Phase [m] - Dichte der fl. Phase [Kg/m³]     

Solution of the temperature jump (Knudsen layer flow) and linear heat-transfer problems for two parallel plates in a rarefied gas

The Monte Carlo method [1, 2] is used to solve the linearized Boltzmann equation for the problem of heat transfer between parallel plates with a wall temperature jump (Knudsen layer flow). The linear Couette problem can be separated into two problems: the problem of pure shear and the problem of heat transfer between two parallel plates. The Knudsen layer problem is also linear [3] and, like the Couette problem, can be separated into the velocity slip and temperature jump problems. The problems of pure shear and velocity slip have been examined in [2].The temperature jump problem was examined in [4] for a model Boltzmann equation. For the linearized Boltzmann equation the problems noted above have been solved either by expanding the distribution function in orthogonal polynomials [5–7], which yields satisfactory results for small Knudsen numbers, or by the method of moments, with an approximation for the distribution function selected from physical considerations in the form of polynomials [8–10]. The solution presented below does not require any assumptions on the form of the distribution function.The concrete calculations were made for a molecular model that we call the Maxwell sphere model. It is assumed that the molecules collide like hard elastic spheres whose sections are inversely proportional to the relative velocity of the colliding molecules. A gas of these molecules is close to Maxwellian or to a gas consisting of pseudo-Maxwell molecules [3].     

Nonlinear problem of a thin jet impinging on the surface of a heavy fluid

In a previous note by the author [1] the problem of symmetric forms of contact with oblique incidence of a free jet on a liquid was posed as a problem of the eigenfunctions of a nonlinear integral equation.Here we consider a more general flow scheme-a model of the jet curtain of an air cushion vehicle above the water surface (Fig. 1); the jet of inviscid, incompressible, weightless fluid of density ₁ impinges from a nozzle on the surface of a stationary liquid of density ₂, where, generally speaking, the pressures p₀ and p₁ are different. The problem is two-dimensional. We derive nonlinear integral equations, one of which is analogous to the Nekrasov equation for exact wave theory [2], In the limiting case of a thin jet we obtain a simple differential equation and exact solutions of the problem are constructed.Some data from the numerical calculations for the nonlinear problem of a thin jet curtain are presented in [3]; the problem has been solved in linearized form in [4],The author wishes to thank M. I. Gurevich and G. Yu. Stepanov, to whom he is indebted for his interest in the problem on jet impingement on a liquid and whose advice has been of assistance in improving the present note.     

Optimum rib size to enhance forced convection in a horizontal triangular duct with ribbed internal surfaces

The optimum rib size to enhance heat transfer had been proposed through an experimental investigation on the forced convection of a fully developed turbulent flow in an air-cooled horizontal equilateral triangular duct fabricated on its internal surfaces with uniformly spaced square ribs. Five different rib sizes (B) of 5 mm, 6 mm, 7 mm, 7.9 mm and 9 mm, respectively, were used in the present investigation, while the separation (S) between the center lines of two adjacent ribs was kept at a constant of 57 mm. The experimental triangular ducts were of the same axial length (L) of 1050 mm and the same hydraulic diameter (D) of 44 mm. Both the ducts and the ribs were fabricated with duralumin. For every experimental set-up, the entire inner wall of the duct was heated uniformly while the outer wall was thermally insulated. From the experimental results, a maximum average Nusselt number of the triangular duct was observed at the rib size of 7.9 mm (i.e. relative rib size ). Considering the pressure drop along the triangular duct, it was found to increase almost linearly with the rib size. Non-dimensional expressions had been developed for the determination of the average Nusselt number and the average friction factor of the equilateral triangular ducts with ribbed internal surfaces. The developed equations were valid for a wide range of Reynolds numbers of 4,000 < Re _D < 23,000 and relative rib sizes of under steady-state condition. A Inner surface area of the triangular duct [m²] - A _C Cross-sectional area of the triangular duct [m²] - B Side length of the square rib [mm] - C _P Specific heat at constant pressure [kJ·kg^–1·K^–1] - C ₁, C ₂, C ₃ Constant coefficients in Equations (10), (12) and (13), respectively - D Hydraulic diameter of the triangular duct [mm] - Electric power supplied to heat the triangular duct [W] - f Average friction factor - F View factor for thermal radiation from the duct ends to its surroundings - h Average convection heat transfer coefficient at the air/duct interface [W·m^–2 ·K^–1] - k Thermal conductivity of the air [W·m^–1 ·K^–1] - L Axial length of the triangular duct [mm] - Mass flow rate [kg·s^–1] - n ₁, n ₂, n ₃ Power indices in Equations (10), (12) and (13), respectively - Nu _D Average Nusselt number based on hydraulic diameter - P Fluid pressure [Pa] - Pr Prandtl number of the airflow - _c Steady-state forced convection from the triangular duct to the airflow [W] - _l Heat loss from external surfaces of the triangular duct assembly to the surroundings [W] - _r Radiation heat loss from both ends of the triangular duct to the surroundings [W] - Re _D Reynolds number of the airflow based on hydraulic diameter - S Uniform separation between the centre lines of two consecutive ribs [mm] - T Fluid temperature [K] - T _a Mean temperature of the airflow [K] - T _ai Inlet mean temperature of the airflow [K] - T _ao Outlet mean temperature of the airflow [K] - T _s Mean surface temperature of the triangular duct [K] - T Ambient temperature [K] - U Mean air velocity in the triangular duct [m·s^–1] - _r Mean surface-emissivity with respect to thermal radiation - Dynamic viscosity of the fluid [kg·m^–1·s^–1] - Kinematic viscosity of the airflow [m²·s^–1] - Density of the airflow [kg·m^–3] - Stefan-Boltzmann constant [W·m^–2·K^–4]     

Heat transfer enhancement in a channel with a built-in rectangular cylinder

A two dimensional numerical investigation of the unsteady laminar flow pattern and forced convective heat transfer in a channel with a built-in rectangular cylinder is presented. The channel in the entrance region has a length to plate spacing of ten. The computations were made for several Reynolds number and two rectangular cylinder aspect ratios. Hydrodynamic behavior and heat transfer results are obtained by solution of the complete Navier-Stokes and energy equation. The results show that these flow exhibits laminar self-sustained oscillations for Reynolds numbers above the critical one. This study show that oscillatory separated flows result in a significant heat transfer enhancement but also in a significant pressure drop increase.

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Erhöhung des Wärmeübergangs in einem Spaltkanal mit quer eingebautem Rechteckprisma

Zusammenfassung Es wird eine zweidimensionale numerische Untersuchung des instationären Wärmeübergangs und Druckverlustes im laminar durchströmten Spaltkanal mit quer eingebautem Rechteckprisma dargelegt und zwar für verschiedene Reynoldszahlen und zwei Prismenabmessungen. Als Lösung der Navier-Stokes- und der Energiegleichung resultieren selbsterregt oszillieren de Strömungs- und Temperaturfelder, verbunden mit starker Erhöhung des Wärmeübergangs und des Druckverlustes.

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List of symbols C _f skin friction coefficient, Eq. (11) - C _D drag coefficient, Eq. (11) - D drag [N/m] - f _app apparent friction factor, Eq. (10) - h cylinder height [m] - H channel height [m] - k thermal conductivity of cylinder [W/mK] - k ₀ thermal conductivity of air [W/mK] - l cylinder length [m] - L channel length [m] - Nu Nusselt number, Eq. (7) - P dimensionless pressure - Pr Prandtl number of air - Re Reynolds number, Eq. (6) - t time [s] - T temperature [K] - T _b bulk temperature [K], Eq. (8) - U, V dimensionless velocity components - X, Y dimensionless coordinates Greek symbols thermal diffusivity [m²/s] - velocity factor, Eq. (11) - dimensionless temperature, Eq. (5) - fluid density [kg/m³] - kinematic viscosity [m²/s] - dimensionless time, Eq. (5) - difference     

Velocity and temperature distributions on a semi-infinite flat plate embedded in a saturated porous medium

In this paper the velocity and temperature distributions on a semi-infinite flat plate embedded in a saturated porous medium are obtained for the governing equations (Kaviany [7]) following the technique adopted by Chandrashekara [2] which are concerned with the interesting situations of the existence of transverse, velocity and thermal boundary layers. Here the pressure gradient is just balanced by the first and second order solid matrix resistances for small permeability and observed that by increasing of the flow resistance the asymptotic value for the heat transfer rate increases. Further we concluded that the transverse boundary layers are thicker than that of axial boundary layers. Hence we evaluated the expressions for the boundary layer thickness, the shear stress at the semi-infinite plate and _T (the ratio of the thicknesses of the thermal boundary layer and momentum boundary layer). The variations of these quantities for different values of the porous parameterB and the flow resistanceF have been discussed in detail with the help of tables. The curves for velocity and temperature distributions have been plotted for different values ofB andF.Lastly we have evaluated the heat fluxq(x) and found that it depends entirely upon the Reynolds numberRe, Prandtl numberPr,B andF.     

Simulation by a source of a thin jet from a body

An exact analytic solution is found to the following plane hydrodynamic problem. An unbounded flow of an ideal incompressible fluid flows around a plate BB' placed at right angles to the velocity vector of the flow at infinity. The pressure on the free boundary P is equal to the pressure in the flow. From an opening in the center of the plate, a jet with flow rate Q from a cavity with pressure P₀ encounters the flow head-on. As a result of the solution, it is found that for fixed width of the opening the values of Q allowed by the scheme are limited. In the limiting case Q = 0 Chaplygin's flow is obtained with stagnation region at the front [1], and in the limiting case Q = Q_max a jet out of a cavity with pressure P₀ into a cavity with pressure P. As Q varies in this interval, the total drag, regarded as the drag of the plate and the chamber from which the jet emerges, takes a minimal value at a certain point. If the width of the opening tends to the length of the slab, the problem of the collision of two jets is obtained; if the width of the opening tends to zero (Q o), the problem of jet flow past a slab with a source is obtained. It is shown that the replacement of the jet by the source gives a good approximation in both the sense of the force characteristics and in the sense of the behavior of the free streamlines.Translated from Izvestiya Akademii Nauk SSSR, Hekhanika Shidkosti i Gaza, No. 5, pp. 47–54, September–October, 1979.We thank L. I. Sedov for his interest in the work and G. Yu. Stepanov for proposing the method of solution and for a helpful discussion.     

Effect of a magnetic field on the flow in the vicinity of the stagnation point of a blunt body with ablation of a protective layer

During hypersonic flow around a blunt-nosed body, the gas which passes through the bow shock is heated to high temperatures, where dissociation, ionization, and inverse phenomena (recombination) take place in the gas. If an ionized gas moves in a magnetic field, the ponderomotive force which is set up changes the nature of its motion close to the stagnation point, decreasing the frictional stress and heat transfer at the wall (at the contact surface of the gas and the body about which the gas flows). In this case, the intense heat fluxes from the strongly heated gas to the body about which the gas flows cause phase changes in the surface of the body (melting, sublimation, etc.). These processes, in turn, affect the flow in the vicinity of the stagnation point due to realization of the heat of phase transition, the conduction of heat from the entrained mass, and the diffusion of evaporating material into the boundary layer. References [1, 2] are devoted to a study of the joint influence of the magneto-gasdynamic and ablation effects. The magnetogasdynamic layers and the wall profile of the external velocity (flow around wedges) are discussed in [1], and special cases of such boundary layers-flow close to the stagnation line (the two-dimensional case) and close to the stagnation point (the axisymmetric case) of a blunt body are considered in [2]. Melting and evaporation are taken into account by setting the longitudinal and the transverse velocity components at the wall not equal to zero-the first taking into account the flow of the molten material and the second pyrolysis of the vapor of the surface material into the gaseous boundary layer. However, the values of these components, also the enthalpy on the wall h_w(in[1, 2] h_W 0), are not known beforehand and must be determined from the boundary conditions at the wall which express the mass and heat balances. The general formulation of the problem given in the gasdynamics case by G. A. Triskii in [3, 4], and elsewhere includes a consideration of the boundary-layer equations in the gas, the boundary-layer equations in the melted zone, and the heat conductivity equations in the solid with boundary conditions at the outer edge of the boundary layer, on the gas-molten zone interface, on the molten zone-solid interface, and inside the solid. This approach to the problem can also be utilized in the magnetogasdynamic case, as it is in this article with certain simplifying assumptions as compared with [3, 4]. In this sense, the present article is an extension of the results of [3, 4] to the field of magnetogasdynamics.In conclusion, the author thanks K. A. Lur'e for proposing the subject and useful discussions.     

The effect of a nonlinear temperature dependent Prandtl-number on heat transfer of fully developed flow of liquids in a straight tube

If Nu_o is the Nusselt Number for a temperature-independent Prandtl number Pr, and Nu the Nusselt number for a temperature dependent Prandtl number, it is usual to define the correction factor Nu/Nu_o=C. A correction factor which has been defined in this form has, up to now, only been expressed as a function of two characteristic Pr numbers. Thus it was simply assumed that the Pr number was a linear function of the temperature. Fluids with very large Pr numbers show a (T;Pr) relationship which deviates considerably from a linear one. This leads to a very large difference (up to 70%) between the calculated and the measured values of the Nusselt number. In the following study we shall determine a so-called curvature parameter of the (T;Pr) curve and obtain a semi-empirical formula for C. This formula has a deviation from the actual relationship many times smaller than that of the formulae for a linear (T;Pr) relationship.

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Zusammenfassung Ist Nu_o die Nusseltzahl bei temperaturunabhangiger und Nu die Nusseltzahl bei temperaturabhangiger Prandtlzahl Pr, so ist es üblich, mit Nu/Nu_o=C den Korrekturfaktor zu bezeichnen. Ein in dieser Form definierter Faktor C ist bisher als Funktion nur zweier charakteristischer Pr-Zahlen ausgedrückt worden. Es wurde somit nur eine lineare Abhängigkeit von der Pr-Zahl von der Temperatur T vorausgesetzt. Flüssigkeiten mit großen Pr-Zahlen weisen eine (T;Pr)-Charakteristik auf, die sehr stark von der linearen abweicht. Sehr große Abweichungen (bis — 70%) der gerechneten von den gemessenen Nu-Zahlen sind eine Folge davon. In vorliegender Arbeit bilden wir mit einer dritten charakteristischen Pr-Zahl einen sogenannten Krümmungsparameter der Kurve (T;Pr) und leiten eine semiempirische Formel für C ab, die um ein großes Vielfaches kleinere Fehler aufweist, als die Formeln für den linearen (T;Pr)-Verlauf.

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Nomenclature

Material constants c_p specific heat at constant pressure [J/kgK] - k heat conductivity [W/mK] - density [kg/m³] - a temperature diffusivity, a=k/c_p [m²/s] - dynamic viscosity [Ns/m²] - kinematic viscosity [m²/s] Fluid dynamics D inner diameter of the tube [m] - L length of tube [m] - w mean speed of fluid [m/s] Heat transfer h coefficient of heat transfer [W/m²K] - T absolute temperature [K] - T_b bulk temperature (corresponding to the adiabatic mixing temperature) [K] - T_w tube wall temperature [K] - T_f=(T_b+T_w)/2 film temperature [K] - T=T_b-T_w temperature forcing difference of heat transfer [K] Characteristic quantities without dimensions Re=wD/ Reynolds number - Pr=/a Prandtl number - Nu=hD/k Nusselt number - related temperature - related Prandtl number - curvature parameter of the Prandtl number Various - C=Nu_b/Nu_o correction factor according to Eq.(5) - p exponent in Eq.(6), (a), (8) and (16) Indices o corresponding to the quasi-isothermal heat transfer - b,w,f with reference to quantities, including physical properties which correspond to the temperatures T_b, T_wor T_f - Pr,k,, for quantities calculated corresponding to the Prandtl number, the thermal conductivity coefficient, the density or the dynamic viscosity - H,C for heating or cooling exp for quantities calculated from experimental data