Experimental investigation of mean flow characteristics of slot jet impingement on a cylinder

An experimental investigation is made to study the flow characteristics of slot jet impingement on a cylinder. The velocity profiles and pressure distribution around the cylinder are reported for various parameters namely, the flow rate, width of the nozzle, distance of the cylinder from the jet exit and eccentricity of the cylinder to the jet axis.

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Experimentelle Untersuchung über die Strömungseigenschaften eines Düsenstrahls, der auf einen Zylinder aufprallt

Zusammenfassung Es wurde eine experimentelle Untersuchung gemacht, um die Strömungseigenschaften eines Düsenstrahls zu unterschen, der auf einen Zylinder prallt. Die Geschwindigkeitsprofile und die Druckverteilungen an dem Zylinder wurden für unterschiedliche Parameter dokumentiert. Die Parameter sind die Strömungsgeschwindigkeit, Düsengröße, Abstand zwischen Zylinder und Strahlaustritt und die Exzentrizität von Zylinder und Strahlachse.

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Nomenclature B breadth of the nozzle at the exit - D diameter of the cylinder - C _p pressure coefficient - g acceleration due to gravity - L distance of the cylinder from jet exit - P _a atmospheric pressure - P _c static pressure along the jet center-line - P ₀ stagnation pressure - P _W wall static pressure - Re _D Reynolds numberu _j D/ _a - Re _W Reynolds numberu _j W/ _a - r distance measured from cylinder surface in radial direction - r _m position of maximum velocity from cylinder surface - r _0.5 half width of the jet - u mean velocity - u _j mean velocity at the jet exit - u _m maximum velocity - W width of the nozzle - _a density of air - _m density of mercury - _w density of water - absolute viscosity - kinematic viscosity     

An experimental study on the two-dimensional opposed wall jet in a turbulent boundary layer: Change in the flow pattern with velocity ratio

Results of the measurement of flow properties in a two-dimensional turbulent wall jet which is injected into the turbulent boundary layer in the direction opposite to that of the boundary layer flow are presented by varying the ratio of the jet issuing velocity to the mainstream velocity . This flow includes the flow separation and the recirculating flow, and so it requires the magnitude and direction of instantaneous velocity be measured. In the present work, a tandem hot-wire probe is manufactured and its characteristics are examined experimentally. With the use of this probe the change in the penetration length of the jet with the velocity ratio is investigated. It is clarified that two regimes of flow patterns exist: in the low velocity ratio the penetration length of the jet changes little with , and in the high velocity ratio it goes far from the nozzle with increasing . Streamlines, turbulence intensity contours and static pressure contours are presented in the two typical velocity ratios corresponding to each of two flow patterns, and they are compared.List of symbols b ₀ nozzle width (Fig. 1) - , e mean and fluctuating output voltages of hot-wire anemometer - p, p mean static pressure, p = p – p _o - p ₀ static pressure in undisturbed mainstream - p _w , p _w mean wall pressure, p _w = p _w – p _o - U ₀ mainstream velocity - U _j jet velocity at the nozzle exit - , u mean and fluctuating velocity components in x-direction - u _e effective cooling velocity - x distance along the wall from nozzle exit - x _pmax, x _pmin positions where the wall pressure has maximum and minimum values respectively - x _s penetration length of jet - y distance from the wall - forward flow fraction - ₁, ₂ yaw and pitch angles of flow direction (Fig. 4) - velocity ratio, = U _j /U _o - density of the fluid - nondimensional stream function The authors wish to express their appreciation to Prof. Toshio Tanaka of Gifu University for his advice in the course of the experiment. They also would like to thank the Research Foundation for the Electrotechnology of Chubu which partly supported this work.     

Two-phase flow in heterogeneous porous media III: Laboratory experiments for flow parallel to a stratified system

Two-phase flow in stratified porous media is a problem of central importance in the study of oil recovery processes. In general, these flows are parallel to the stratifications, and it is this type of flow that we have investigated experimentally and theoretically in this study. The experiments were performed with a two-layer model of a stratified porous medium. The individual strata were composed of Aerolith-10, an artificial: sintered porous medium, and Berea sandstone, a natural porous medium reputed to be relatively homogeneous. Waterflooding experiments were performed in which the saturation field was measured by gamma-ray absorption. Data were obtained at 150 points distributed evenly over a flow domain of 0.1 × 0.6 m. The slabs of Aerolith-10 and Berea sandstone were of equal thickness, i.e. 5 centimeters thick. An intensive experimental study was carried out in order to accurately characterize the individual strata; however, this effort was hampered by both local heterogeneities and large-scale heterogeneities.The theoretical analysis of the waterflooding experiments was based on the method of large-scale averaging and the large-scale closure problem. The latter provides a precise method of discussing the crossflow phenomena, and it illustrates exactly how the crossflow influences the theoretical prediction of the large-scale permeability tensor. The theoretical analysis was restricted to the quasi-static theory of Quintard and Whitaker (1988), however, the dynamic effects described in Part I (Quintard and Whitaker 1990a) are discussed in terms of their influence on the crossflow.Roman Letters A interfacial area between the -region and the -region contained within V, m² - a vector that maps onto , m - b vector that maps onto , m - b vector that maps onto , m - B second order tensor that maps onto , m² - C second order tensor that maps onto , m² - E energy of the gamma emitter, keV - f fractional flow of the -phase - g gravitational vector, m/s² - h characteristic length of the large-scale averaging volume, m - H height of the stratified porous medium , m - i unit base vector in the x-direction - K local volume-averaged single-phase permeability, m² - K - {K}, large-scale spatial deviation permeability - { K} large-scale volume-averaged single-phase permeability, m² - K ^* large-scale single-phase permeability, m² - K ^** equivalent large-scale single-phase permeability, m² - K local volume-averaged -phase permeability in the -region, m² - K local volume-averaged -phase permeability in the -region, m² - K - {K } , large-scale spatial deviation for the -phase permeability, m² - K ^* large-scale permeability for the -phase, m² - l thickness of the porous medium, m - l characteristic length for the -region, m - l characteristic length for the -region, m - L length of the experimental porous medium, m - characteristic length for large-scale averaged quantities, m - n outward unit normal vector for the -region - n outward unit normal vector for the -region - n unit normal vector pointing from the -region toward the -region (n = - n ) - N number of photons - p pressure in the -phase, N/m² - p ₀ reference pressure in the -phase, N/m² - local volume-averaged intrinsic phase average pressure in the -phase, N/m² - large-scale volume-averaged pressure of the -phase, N/m² - large-scale intrinsic phase average pressure in the capillary region of the -phase, N/m² - - , large-scale spatial deviation for the -phase pressure, N/m² - p_c , capillary pressure, N/m² - p _c capillary pressure in the -region, N/m² - p capillary pressure in the -region, N/m² - {p _c } ^c large-scale capillary pressure, N/m² - q -phase velocity at the entrance of the porous medium, m/s - q -phase velocity at the entrance of the porous medium, m/s - S_wi irreducible water saturation - S /, local volume-averaged saturation for the -phase - S _i initial saturation for the -phase - S _r residual saturation for the -phase - S ^* { }^*/}^*, large-scale average saturation for the -phase - S saturation for the -phase in the -region - S saturation for the -phase in the -region - t time, s - v -phase velocity vector, m/s - v local volume-averaged phase average velocity for the -phase, m/s - {v } large-scale averaged velocity for the -phase, m/s - v local volume-averaged phase average velocity for the -phase in the -region, m/s - v local volume-averaged phase average velocity for the -phase in the -region, m/s - v -{v } , large-scale spatial deviation for the -phase velocity, m/s - v -{v } , large-scale spatial deviation for the -phase velocity in the -region, m/s - v -{v } , large-scale spatial deviation for the -phase velocity in the -region, m/s - V large-scale averaging volume, m³ - y position vector relative to the centroid of the large-scale averaging volume, m - {y}^c large-scale average of y over the capillary region, m Greek Letters local porosity - local porosity in the -region - local porosity in the -region - local volume fraction for the -phase - local volume fraction for the -phase in the -region - local volume fraction for the -phase in the -region - {}^* { }^*+{ }^*, large-scale spatial average volume fraction - { }^* large-scale spatial average volume fraction for the -phase - mass density of the -phase, kg/m³ - mass density of the -phase, kg/m³ - viscosity of the -phase, N s/m² - viscosity of the -phase, Ns/m² - V /V , volume fraction of the -region ( + =1) - V /V , volume fraction of the -region ( + =1) - attenuation coefficient to gamma-rays, m^-1 - -      

Liquid metal flow in a finite-length cylinder with a rotating magnetic field

A liquid metal flow induced by a rotating magnetic field in a cylindrical container of finite height was investigated experimentally. It was demonstrated that the flow in a rotating magnetic field is similar to geophysical flows: the fluid rotates uniformly with depth and the Ekman layer exists at the container bottom. Near the vertical wall the flow is depicted in the form of a confined jet whose thickness determines the instability onset in a rotating magnetic field. It was shown that the critical Reynolds number can be found by using the jet velocity u ₀ for Re _cr =u ² ₀/ u/ r. The effect of frequency of a magnetic field on the fluid flow was also studied. An approximate theoretical model is presented for describing the fluid flow in a uniform rotating magnetic field.List of Symbols U _r , U , U _z radial, azimuthal and vertical velocity components, respectively - B _r , U , B _z radial, azimuthal and vertical magnetic induction components - A vector potential of magnetic field - j induced electric current density - electrical conductivity of fluid - electrical potential - kinematic viscosity - tf electromagnetic volume force - angular velocity of fluid rotation - R container radius - H container height - aspect ratio - E Ekman number - Re _cr critical Reynolds number - r, z radial and axial coordinates     

Laminar source flow between parallel porous disks with equal suction and injection

An analysis is presented for laminar source flow between parallel stationary porous disks with suction at one of the disks and equal injection at the other. The solution is in the form of an infinite series expansion about the solution at infinite radius, and is valid for all suction and injection rates. Expressions for the velocity, pressure, and shear stress are presented and the effect of the cross flow is discussed.Nomenclature a distance between disks - A, B, ..., J functions of R _w only - F static pressure - p dimensionless static pressure, p(a ²/ ²) - Q volumetric flow rate of the source - r radial coordinate - r dimensionless radial coordinate, r/a - R radial coordinate of a point in the flow region - R dimensionless radial coordinate of a point in the flow region, R - Re source Reynolds number, Q/2a - R _w wall Reynolds number, Va/ - reduced Reynolds number, Re/r ² - critical Reynolds number - velocity component in radial direction - u dimensionless velocity component in radial direction, a/ - average radial velocity, Q/2a - u dimensionless average radial velocity, Re/r - ratio of radial velocity to average radial velocity, u/u - velocity component in axial direction - v dimensionless velocity component in axial direction, v - V magnitude of suction or injection velocity - z axial coordinate - z dimensionless axial coordinate, z a - viscosity - density - kinematic viscosity, / - shear stress at lower disk - shear stress at upper disk - ₀ dimensionless shear stress at lower disk, - ₁ dimensionless shear stress at upper disk, - dimensionless stream function     

Measurements of turbulent inclined plane dual jets

Measurements of mean velocities, flow direction, velocity fluctuations and Reynolds shear stress were made with a split film probe of hot wire anemometer to investigate the interactions created by two air jets issuing from two identical plane inclined nozzles. The reverse flow was detected by using the split film probe and observed by flow visualization. Experimental results with an inclined angle of 9° are presented in the paper. Some experimental results with an inclined angle of 27° are presented to investigate the effect of inclination on the flow field.Mean velocities approach self-preservation in both the converging region and the combining region. Velocity fluctuations and Reynolds shear stress approach self-preservation in the combining region only. The spreads of jet and the square of the decay of maximum mean velocity increase linearly as the distance from the nozzle exit increases.List of symbols D nozzle width - h nozzle height - J momentum of jet - J ₀ momentum of jet at nozzle exit - M mass flow rate - M ₀ mass flow rate at nozzle exit - S nozzle spacing - U, V mean velocities in the X and Y axis respectively - U _m maximum axial velocity - U ₀ axial velocity at nozzle exit - u, v velocity fluctuations in the X and Y axis respectively - u, v r.m.s. of u and v - Reynolds shear stress - X, Y Cartesian coordinates - X _m , Y _m coordinates at the location of maximum axial velocity - y _0.5 distance from the location of maximum axial velocity to the location where the velocity is half of maximum axial velocity - inclined angle - yY/S - Reynolds stress correlation coefficient - C.P combining point - max maximum value - M.P merging point - o nozzle exit plane - V.C vortex center     

Effect of the flow regime in an underexpanded jet issuing into a subsonic counterstream on the pressure distribution on a body

The pressure distribution on a cone with a half-angle =75°, from which a single central underexpanded jet issues into a subsonic counterstream, has been experimentally investigated. The effect of the flow regime in the jet on the pressure distribution is demonstrated. Generalized relations for the pressure on the body are obtained for various jet-flow momentum ratios J and flow Mach numbers M = 0.35–0.9; the Mach number M_a at the exit of the conical nozzle with half-angle _a=10° was equal to 2.9. The working medium of the jet and the flow was air with stagnation temperatures T_0a = T₀ 260–265°K. The ratio of the nozzle outlet radius to the radius of the maximum cross section of the cone _a/R_M=0.1.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 182–185, January-February, 1988.     

Prediction of stagnation point heat transfer for a single round jet impinging on a concave hemispherical surface

This paper deals with a systematic procedure for assessment of fluid flow and heat transfer parameters for a single round jet impinging on a concave hemispherical surface. Based on Scholkemeier's modifications of the Karman-Pohlhausen integral method, expressions are derived for evaluation of the momentum thickness, boundary layer thickness and the displacement thickness at the stagnation point. This is followed by the estimation of thermal boundary layer thickness and local heat transfer coefficients. A correlation is presented for the Nusselt number at the stagnation point as a function of the Reynolds number for different non-dimensional distances from the exit plane of the jet to the impingement surface.

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Bestimmung des Staupunktes bei der Wärmeübertragung für einen einzelnen Strahl, der auf eine konkave halbkugelige Oberfläche trifft

Zusammenfassung Diese Arbeit beschäftigt sich mit dem systematischen Verfahren der Bewertung von Fluidströmungen und Wärmeübertragungsparametern für einen einzelnen runden Strahl, der auf eine konkave halbkugelförmige Oberfläche trifft. Das Verfahren beruht auf Scholkemeiers Modifikation des Karman-Pohlhausen Integrationsverfahrens. Ausdrücke sind für die Berechnung der Impuls-Dicke, der Grenzschichtdicke und der Verschiebungsdicke am Staupunkt hergeleitet worden. Dies ist aus der Berechnung der thermischen Grenzschichtdicke und des lokalen Wärmeübertragungskoeffizienten abgeleitet worden. Es wird eine Gleichung für die Nusselt-Zahl am Staupunkt als Funktion der Reynolds-Zahl für verschiedene dimensionslose Abstände vom Strahlaustrittspunkt bis zum Auftreffpunkt auf die Oberfläche vorgestellt.

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Nomenclature c _p specific heat at constant pressure - d diameter of single round nozzle - h ₀ heat transfer coefficient at the stagnation point - H distance from the exit plane of the jet to the impingement surface - k thermal conductivity - Nu _0.5 Nusselt number based on impinging jet quantities=h _0.50/k - Nu _{0.5, 0} stagnation point Nusselt number=h _{0 0,50}/k - p pressure - p _a ambient pressure - p ₀ maximum pressure or stagnation pressure - p(x) static pressure at a distancex from the stagnation point - R radius of curvature of the hemisphere - Re _J jet Reynolds number=U _Jd/ - Re _0.5 Reynolds number based on impinging jet quantities=u _{m0 0.50}/ - T temperature - T _a room temperature - T _J jet temperature - T _W wall temperature - u velocity component inx andx directions (Fig. 1) - u _m jet centerline (or maximum) free jet velocity: external (or maximum) boundary layer velocity aty= _m - u _m0 arrival velocity defined as the maximum velocity the free jet would have at the plane of impingement if the plane were not there - U _J jet exit velocity - x* non-dimensional coordinate starting at the stagnation point=x/2 _0.50 - x, y rectangular Cartesian coordinates - y coordinate normal to the wall starting at the wall - ratio of thermal to velocity boundary layer thickness= _T/_m - ₀ ratio of thermal to velocity boundary layer thickness at the stagnation point - * inner layer displacement thickness - _0.50 jet half width at the plane of impingement if the plate were not there - _m inner boundary layer thickness atu=u _m - Pohlhausen's form parameter - dynamic viscosity - kinematic viscosity=/ - fluid density - momentum thickness - ₀ momentum thickness at the stagnation point     

Simultaneous measurements of velocity and temperature fluctuations in thermal boundary layer in a drag-reducing surfactant solution flow

The mechanism of turbulent heat transfer in the thermal boundary layer developing in the channel flow of a drag-reducing surfactant solution was studied experimentally. A two-component laser Doppler velocimetry and a fine-wire thermocouple probe were used to measure the velocity and temperature fluctuations simultaneously. Two layers of thermal field were found: a high heat resistance layer with a high temperature gradient, and a layer with a small or even zero temperature gradient. The peak value of was larger for the flow with the drag-reducing additives than for the Newtonian flow, and the peak location was away from the wall. The profile of was depressed in a similar manner to the depression of the profile of in the flow of the surfactant solution, i.e., decorrelation between v and compared with decorrelation between u and v. The depression of the Reynolds shear stress resulted in drag reduction; similarly, it was conjectured that the heat transfer reduction is due to the decrease in the turbulent heat flux in the wall-normal direction for a flow with drag-reducing surfactant additives.List of symbols ensemble averaged value - (·)⁺ normalized by the inner wall variables - (·) root-mean-square value - C concentration of cetyltrimethyl ammonium chloride (CTAC) solution - c _p heat capacity - D hydraulic diameter - f friction factor - H channel height - h heat transfer coefficient - j _H Colburn factor - l length - Nu Nusselt number, h - Pr Prandtl number, c _p/ - q _w wall heated flux - Re Reynolds number, U _b/ - T temperature - T _b bulk temperature - T _i inlet temperature - T _w wall temperature - T friction temperature, q _w /c _p u - U local time-mean streamwise velocity - U ₁ velocity signals from BSA1 - U ₂ velocity signals from BSA2 - U _b bulk velocity - u streamwise velocity fluctuation - u1 velocity in abscissa direction in transformed coordinates - u friction velocity, - v wall-normal velocity fluctuation - v1 velocity in ordinate direction in transformed coordinates - var(·) variance - x streamwise direction - y wall-normal direction - z spanwise direction - _j junction diameter of fine-wire TC - _w wire diameter of fine-wire TC - angle of principal axis of joint probability function p(u,v) - _f heat conduction of fluid - _w heat conduction of wire of fine-wire TC - kinematic viscosity - local time-mean temperature difference, T _w –T - temperature fluctuation - standard deviation - density - _w wall shear stress     

Pulsatile flow in a tube with wall injection and suction

This paper studies similarity solutions for pulsatile flow in a tube with wall injection and suction. The Navier-Stokes equations are reduced to a system of three ordinary differential equations. Two of the equations represent the effects of suction and injection on the steady flow while the third represents the effects of suction and injection on pulsatile flow. Since the equations for steady flow have been studied previously, the analysis centers on the third equation. This equation is solved numerically and by the method of matched asymptotic expansions. The exact numerical solutions compare well with the asymptotic solutions.The effects of suction and injection on pulsatile flow are the following: a) Small values of suction can cause a resonance-like effect for low frequency pulsatile flow. b) The annular effect still occurs but for large injection or suction the frequency at which this effect becomes dominant depends on the cross-flow Reynolds number. c) The maximum shear stress at the wall is decreased by injection, but may be increased or decreased by suction.Nomenclature a radius of the tube - a ₀ ² i ² - A₀, B₀, C₀, D₀, E₀ constant coefficients appearing in the expression for pressure - b a non-dimensionalized length - b ₀ ² i ^{2 2} - b _k complex coefficients of a power series - B - C ₁, C ₂, D complex constants - d - D _1,2 - f() F(a ^1/2)/aV - f ₀,f ₁,... functions of order one used in asymptotic expansions of f() - F(r) rv _r - g() - G(r) a steady component of velocity in axial direction - h() 4/C₀ a ² H(a ^1/2) - h ₀,h ₁,h ₂,...;l ₀,l ₁,l ₂,... functions of order one used in asymptotic expansions for h() in outer regions - H(r) complex valued function giving unsteady component of velocity - H ₀, H ₁, H ₂, ... K ₀, K ₁, K ₂, ...; L ₀, L ₁, L ₂, ... functions of order one used in asymptotic expansions for h() in inner regions - i - J ₀, J ₁, Y ₀, Y ₁ Bessel functions of first and second kind - k - K Rk/2b ² - O order symbol - p pressure - p ₁(z, t) arbitrary function related to pressure - r radial coordinate - r ₀ (1+16 ^{4 4})^1/4 - R Va/, the crossflow Reynolds number - t time - u() G(r)/V - v _r radial velocity - v _z axial velocity - V constant velocity at which fluid is injected or extracted - z axial coordinate - ² a ²/4 - 4.196 - small parameter; =–2/R (Sect. 4); =–R/2 (Sect. 5); =2/R(Sect. 6) - r ²/a ² - * 0.262 - Arctan (4 ^{2 2}) - , inner variables - kinematic viscosity - b - * zero of g() - density - (r, t) arbitrary function related to axial velocity - frequency     

Mean and turbulent velocity measurements of supersonic mixing layers

The behavior of supersonic mixing layers under three conditions has been examined by schlieren photography and laser Doppler velocimetry. In the schlieren photographs, some large-scale, repetitive patterns were observed within the mixing layer; however, these structures do not appear to dominate the mixing layer character under the present flow conditions. It was found that higher levels of secondary freestream turbulence did not increase the peak turbulence intensity observed within the mixing layer, but slightly increased the growth rate. Higher levels of freestream turbulence also reduced the axial distance required for development of the mean velocity. At higher convective Mach numbers, the mixing layer growth rate was found to be smaller than that of an incompressible mixing layer at the same velocity and freestream density ratio. The increase in convective Mach number also caused a decrease in the turbulence intensity ( _u/U).List of symbols a speed of sound - b total mixing layer thickness between U ₁ – 0.1 U and U ₂ + 0.1 U - f normalized third moment of u-velocity, f u³/(U)³ - g normalized triple product of u² , g u²/(U)³ - h normalized triple product of u ², h u ²/(U)³ - l _u axial distance for similarity in the mean velocity - l _u axial distance for similarity in the turbulence intensity - M Mach number - M _c convective Mach number (for ₁ = ₂), M _c (U ₁ – U ₂)/(a ₁ + a ₂) - P static pressure - r freestream velocity ratio, r U ₂/U ₁ - Re unit Reynolds number, Re U/ - s freestream density ratio, s ₂/₁ - T _t total temperature - u instantaneous streamwise velocity - u deviation of u-velocity, uu – U - U local mean streamwise velocity - U ₁ primary freestream velocity - U ₂ secondary freestream velocity - average of freestream velocities, (U ₁ + U ₂)/2 - U freestream velocity difference, U U ₁ – U ₂ - instantaneous transverse velocity - v deviation of -velocity, – V - V local mean transverse velocity - x streamwise coordinate - y transverse coordinate - y ₀ transverse location of the mixing layer centerline - ensemble average - ratio of specific heats - boundary layer thickness (y-location at 99.5% of free-stream velocity) - similarity coordinate, (y – y ₀)/b - compressible boundary layer momentum thickness - viscosity - density - standard deviation - dimensionless velocity, (U – U ₂)/U - 1 primary stream - 2 secondary stream A version of this paper was presented at the 11th Symposium on Turbulence, October 17–19, 1988, University of Missouri-Rolla     

Heat transfer and equilibrium temperature of a sphere in a supersonic rarefied gas flow

Some results are presented of experimental studies of the equilibrium temperature and heat transfer of a sphere in a supersonic rarefied air flow.The notations D sphere diameter - u, , T,,l, freestream parameters (u is velocity, density, T the thermodynamic temperature,l the molecular mean free path, the viscosity coefficient, the thermal conductivity) - T₀ temperature of the adiabatically stagnated stream - T_e mean equilibrium temperature of the sphere - T_w surface temperature of the cold sphere (T_we) - mean heat transfer coefficient - _e air thermal conductivity at the temperature T_e - P Prandtl number - M Mach number     

A crossed beam laser shadow technique for spatially resolved measurements in turbulent flow

Optical methods such as the shadowgraph and schlieren techniques do not directly allow three-dimensional spatial resolution because of the inherent integration along the line of sight. For the particular case of the laser shadow technique, it is attempted to circumvent this restriction by employing a pair of crossed laser beams and cross-correlating the optical signals obtained from the beams. The information obtained is supposed to pertain only to the common volume of both beams. It is theoretically shown that this concept will work if the turbulence spectrum does not decay faster than K ^–5 in the range of wave numbers corresponding to the inverse diameter of the laser beam. Experiments on a round jet confirm that local values of the mean flow velocity can be obtained.List of symbols A electromagnetic amplitude at x = 0 - B ₁₂ correlation of light intensity fluctuations - C ₁₂ normalized correlation (Eq. (1)) - I (I) light intensity (fluctuating part) - k wavenumber of light - K wavenumber of turbulent eddies - L ₀ outer scale of turbulence - n refractive index - R ₀ radius of curvature of the wavefront of the laser beam at x = 0 - u (u ₀) (undisturbed) electromagnetic field of the laser beam - drift velocity of the laser shadow pattern - W spatial weighting function (Eq. (11)) - W ₀ diameter of the laser beam at x = 0 - x, y, z spatial coordinates (Figs. 1, 2) - see Eq. (6) - see Eq. (A3) spectrum of turbulent refractive index fluctuations exponent of _n - = (y, z) - ₁ see Eq. (7) - , spatial coordinates (Fig. 2) - , spatial and temporal delay variables in a correlation function     

Rheological behaviour of two-phase flow of solid particles in a gas

The present investigation was concerned with the rheological behaviour of dilute suspensions of solid particles in a gas in a vertical cocurrent flow moving upwards. Starting from the experimentally determined dependence of the pressure drop on the concentration of solid particles and the Reynolds number of the carrier medium in the steady flow region, the rheological parameters were estimated using pseudo-shear diagrams. Air was the carrier medium and the dispersed phase was one of six fractions of polypropylene powder and five fractions of glass ballotini. The results show that the investigated two-phase systems have pseudoplastic character which becomes more pronounced with increases in concentration, equivalent diameter and density of solid particles in the flowing suspension. C _d coefficient of particle resistance - d _e equivalent diameter of particles - D column diameter - Fr Froude number - g gravitational acceleration - K rheological parameter - L length - n rheological parameter - p _t pressure drop due to friction - p _m total pressure drop - p _ag pressure drop due to acceleration of the gas phase - p _as pressure drop due to acceleration of the solid phase - p _g hydrostatic pressure of the gas phase - p _s specific effective weight of the dispersed phase - r radius - Re Reynolds number - Re _p Reynolds number of a particle - Re _G generalized Reynolds number - Re _G1 generalized Reynolds number relating to the end of the laminar flow region - Re _G2 generalized Reynolds number relating to the beginning of the turbulent flow region - w _z axial component of velocity - u _t steady free-fall velocity of a single particle - w average velocity - w _g average velocity of the gas phase - w _s average velocity of the dispersed phase of solid particles - relative mass fraction of solid particles - x _s volume fraction of solid particles - _g coefficient of pressure drop due to friction - µ dynamic viscosity - _g density of the gas phase - _m density of the suspension - _s density of solid particles - _ds density of the dispersed phase - _w shear stress at the wall     

An experimental investigation of separating flow on a convex surface

The mean and turbulent characteristics of an incompressible turbulent boundary layer developing on a convex surface under the influence of an adverse pressure gradient are presented in this paper.The turbulence quantities measured include all the components of Reynolds stresses, auto-correlation functions and power spectra of the three components of turbulence. The results indicate the comparative influence of the convex curvature and adverse pressure gradient which are simultaneously acting on the flow. The investigation provides extensive experimental information which is much needed for a better understanding of turbulent shear flows.Nomenclature a, b constants in equation for velocity defect profile (Fig. 6) - c _f skin-friction coefficient (= _w/F _1/2 U ₁ ² ) - E(k ₁) one-dimensional wave number spectra - f frequency in Hz - G Clauser's equilibrium parameter = (H–1)/H(c _f /2) - H shape parameter (= ₁/ ₂) - k ₁ wave number (=2f/U) - L _u, L _v, L _w length scales of u, v and w fluctuations - p _s static pressure on the measurement surface - p _w reference tunnel wall static pressure - q ² total turbulent kinetic energy - R radius of curvature of the convex surface - R() auto-correlation function - T _u, T _v, T _w time scales of u, v and w fluctuations - U local mean velocity - U ₁ local free stream velocity - U _* friction velocity - u, v, w velocity fluctuations in x, y and z directions respectively - X streamwise coordinate measured along the surface from A (Fig. 1b) - x streamwise coordinate measured along the surface reckoned from station 9 - y coordinate normal to the surface - z spanwise coordinate - ₁/ _w · dp/dx - - boundary layer thickness - ₁ displacement thickness - ₂ momentum thickness - ₃ energy thickness - kinematic viscosity - density - time delay - _w wall shear stress     

Outflow of an unexpanded jet into counter supersonic and subsonic flows

The article gives the results of an experimental investigation of the geometric structure of an opposing unexpanded jet. It discusses flow conditions with interaction between the jet and sub- and supersonic flows. It is shown that, with the outflow of an unexpanded jet counter to a supersonic flow, there are unstable flow conditions. For stable flow conditions with one roll, dependences are proposed determining the form of a jet in a supersonic opposing flow. A generalized dependence is obtained for the distribution of the pressure at the surface of a body with a jet, flowing out counter to a subsonic flow. The range of change in the determining parameters are the following: Mach numbers at outlet cross section of nozzle, M _a = 1 and 3; Mach numbers of opposing flow, M = 0.6–0.9 and 2.9; degree of effectiveness of jet, n = p _a /p = 0.5–800 (p _a and p are the static pressures at the outlet cross section of the nozzle and in the opposing flow); the ratios of the specific heat capacities, _a = = 1.4; the drag temperatures of the jet and the flow, T_o = T_oa = 290°K.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 89–96, January–February, 1977.     

Momentum and energy balances for dispersed two-phase flow

Summary The equations of motion and the mechanical energy balances for two-phase flow systems are derived by integration over a volume containing a large number of elements of the dispersed phase.List of symbols A, A boundary of volumes V, V - dA, dA surface element of A, A - A _s boundary of particles in V - dA _s surface element of A _s - F force per unit volume of the system - g –gz=gravity vector - g acceleration by gravity - I unit tensor - p pressure - Q dissipation in the continuous phase - Q _s dissipation in the dispersed phase - R compression work in the continuous phase - R _s compression work in the dispersed phase - t time - u velocity of continuous phase - u _s velocity of dispersed phase - u magnitude of u - u _s magnitude of u _s - V volume in the two-phase system - V part of V occupied by the continuous phase - W work done by F - z vertical coordinate - local volume fraction of the dispersed phase - pI–=stress tensor of the continuous phase - _s turbulent particle stress tensor - density of the continuous phase - _s density of the dispersed phase - shearing-stress tensor of the continuous phase - _s turbulent particle shearing-stress tensor - nabla operator - u, u _s velocity gradient tensor - substantial derivative (Shell Internationale Research Maatschappij N.V.)(Bataafse Internationale Petroleum Maatschappij N.V.)     

A three-parameter model describing the experimental relation between shear stress and shear rate for laminar flow of aqueous polymer solutions

Summary A three-parameter model is introduced to describe the shear rate — shear stress relation for dilute aqueous solutions of polyacrylamide (Separan AP-30) or polyethylenoxide (Polyox WSR-301) in the concentration range 50 wppm – 10,000 wppm. Solutions of both polymers show for a similar rheological behaviour. This behaviour can be described by an equation having three parameters i.e. zero-shear viscosity ₀, infinite-shear viscosity , and yield stress ₀, each depending on the polymer concentration. A good agreement is found between the values calculated with this three-parameter model and the experimental results obtained with a cone-and-plate rheogoniometer and those determined with a capillary-tube rheometer.

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Zusammenfassung Der Zusammenhang zwischen Schubspannung und Schergeschwindigkeit von strukturviskosen Flüssigkeiten wird durch ein Modell mit drei Parametern beschrieben. Mit verdünnten wäßrigen Polyacrylamid-(Separan AP-30) sowie Polyäthylenoxidlösungen (Polyox WSR-301) wird das Modell experimentell geprüft. Beide Polymerlösungen zeigen im untersuchten Schergeschwindigkeitsbereich von ein ähnliches rheologisches Verhalten. Dieses Verhalten kann mit drei konzentrationsabhängigen Größen, nämlich einer Null-Viskosität ₀, einer Grenz-Viskosität und einer Fließgrenze ₀ beschrieben werden. Die Ergebnisse von Experimenten mit einem Kegel-Platte-Rheogoniometer sowie einem Kapillarviskosimeter sind in guter Übereinstimmung mit den Werten, die mit dem Drei-Parameter-Modell berechnet worden sind.

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a Pa^–1 physical quantity defined by:a = {1 – ( / ₀)}/ ₀ - c l concentration (wppm) - D m capillary diameter - L m length of capillary tube - P Pa pressure drop - R m radius of capillary tube - u m s^–1 average velocity - v _r m s^–1 local axial velocity at a distancer from the axis of the tube - shear rate (–dv _r /dr) - local shear rate in capillary flow - s^–1 wall shear rate in capillary flow - Pa s dynamic viscosity - _a Pa s apparent viscosity defined by eq. [2] - ( _a ) Pa s apparent viscosity in capillary tube at a distanceR from the axis - ₀ Pa s zero-shear viscosity defined by eq. [4] - Pa s infinite-shear viscosity defined by eq. [5] - l ratior/R - kg m^– density - Pa shear stress - ₀ Pa yield stress - _r Pa local shear stress in capillary flow - _R Pa wall shear stress in capillary flow _R = (PR/2L) - _v m³ s^–1 volume rate of flow With 8 figures and 1 table     

Distribution of eddy viscosity and mixing length in a non-Newtonian flow of a solid particle-gas suspension

The present work concerns the study of the radial distribution of eddy viscosity and mixing length and their dependence upon the Reynolds number and the concentration of the solid phase in a non-Newtonian flow of a suspension of solid particles in a gas. The investigated systems have a pseudoplastic character and the deviation from Newtonian behaviour increases with an increase in the concentration of the dispersed phase. Relations are presented for the eddy viscosity and mixing length in the flow of pseudoplastic fluids. From the analysis of results it follows that the mixing length and eddy viscosity increases with an increase in the Reynolds number. In contrast, an increase in the concentration of the solid phase and consequently of the pseudoplasticity causes a decrease in the investigated quantities. The radial distribution of the mixing length and the eddy viscosity is characterized by a maximum, after which the investigated quantities vary only slightly. This enables the area of the core of the turbulent flow to be defined. E Non-dimensional eddy viscosity - K fluid consistency defined by Ostwald-De Waele's formula (power law) - K fluid consistency, eq. (12) - L mixing length - L _t non-dimensional mixing length - N ⁺ position parameter, eq. (3) - n power-law index - n ⁺ Reichardt's position parameter, eq. (4) - n slope of the dependence ln _w = f[ln (8w/D)] - R pipe radius - r radial distance from the pipe axis - Re=D W /µ Reynolds number - U ⁺ non-dimensional local mean axial velocity, eq. (2) - u ^* = w (/8)^0.5 friction velocity - non-dimensional local mean axial velocity - local mean axial velocity - u turbulence velocity component - mean axial velocity at pipe axis - w average velocity over cross-section of pipe - loading ratio of solid to air, i.e. ratio of mass flow rates of solids to air - Y _m ⁺ =Rⁿ u^2–n /K 8^n–1 non-dimensional distance of pipe centre from the wall - y distance from pipe wall - y ⁺ non-dimensional distance from pipe wall, eq. (5) - friction factor - µ _L laminar part of viscosity - µ _t eddy viscosity - density - shear stress - _w shear stress on the wall     

Transient flow towards a well in an aquifer including the effect of fluid inertia

Transient propagation of weak pressure perturbations in a homogeneous, isotropic, fluid saturated aquifer has been studied. A damped wave equation for the pressure in the aquifer is derived using the macroscopic, volume averaged, mass conservation and momentum equations. The equation is applied to the case of a well in a closed aquifer and analytical solutions are obtained to two different flow cases. It is shown that the radius of influence propagates with a finite velocity. The results show that the effect of fluid inertia could be of importance where transient flow in porous media is studied.List of symbols b Thickness of the aquifer, m - c ₀ Wave velocity, m/s - k Permeability of the porous medium, m² - n Porosity of the porous medium - p( ,t) Pressure, N/m² - Q Volume flux, m³/s - r Radial coordinate, m - r _w Radius of the well, m - s Transform variable - S Storativity of the aquifer - S _d(r, t) Drawdown, m - t Time, s - T Transmissivity of the aquifer, m²/s - ( ,t) Velocity of the fluid, m/s - Coordinate vector, m - z Vertical coordinate, m - Coefficient of compressibility, m²/N - Coefficient of fluid compressibility, m²/N - Relaxation time, s - (r, t) Hydraulic potential, m - Dynamic viscosity of the fluid, Ns/m² - Dimensionless radius - Density of the fluid, Ns²/m⁴ - (, ) Dimensionless drawdown - Dimensionless time - , x Dummy variables - ₀, ₁ Auxilary functions