Distribution of turbulence energy dissipation rates in a Rushton turbine stirred mixer

An approximate method of measuring the turbulence energy dissipation rate () in mixers by use of laser-Doppler measurements of the velocity autocorrelation and turbulence energy was successful in yielding remarkably consistent values. The necessary corrections for periodic, non-dissipative velocity fluctuations were made by an autocorrelation method. Two modes of periodic fluctuation were found to be significant. Transformation of the corrected autocorrelations yielded completely normal turbulence energy spectra.List of symbols c fluctuating concentration, C–C - D impeller diameter - D molecular diffusivity - f() autocorrelation function - E ₁ (n) one-dimensional energy spectrum function - k turbulence energy (=q) - L _s macroscale of segregation - L _x integral velocity scale - N impeller rotation rate - N _Sc Schmidt number (v/D) - q turbulence energy (=k) - r radial distance from impeller shaft - R impeller radius - T tank diameter - U, V, W velocity in x, y, z directions - u, v, w velocity fluctuations - u _r , u , u _z fluctuating velocities in radial, tangential, and axial (shaft) directions - U _r , U , U _z velocities - z axial distance from impeller disk - Z tank height - turbulence energy dissipation rate - viscosity - time delay     

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     

Flow transitions in a rotating magnetic field

Critical Rayleigh numbers have been measured in a liquid metal cylinder of finite height in the presence of a rotating magnetic field. Several different stability regimes were observed, which were determined by the values of the Rayleigh and Hartmann numbers. For weak rotating magnetic fields and small Rayleigh numbers, the experimental observations can be explained by the existence of a single non-axisymmetric meridional roll rotating around the cylinder, driven by the azimuthal component of the magnetic field. The measured dependence of rotational velocity on magnetic field strength is consistent with the existence of laminar flow in this regime.List of symbols B ₀ magnitude of magnetic induction - B_r, B radial and azimuthal magnetic induction components - C wall admittance - d cell diameter - d _w wall thickness - g gravity at earth's surface - Ha Hartmann number - h cell height - k _f thermal conductivity of fluid - k _w thermal conductivity of wall - L1, L2, L3, L4 thermistor temperatures - Ra Rayleigh number - Ra ^c critical Rayleigh number for the transition from no flow to laminar flow - Ra ^t critical Rayleigh number for the transition from time-independent to time-dependent flow - r radial coordinate - T _a temperature at top of cell - T _b temperature at bottom of cell - T temperature difference between cell bottom and cell top - T_c critical temperature difference between cell bottom and top time - t time - U1, U2, U3, U4 thermistor temperatures - z vertical coordinate - volumetric thermal expansion coefficient - skin depth - k thermal diffusivity - magnetic permeability - kinematic viscosity - density - electrical conductivity - azimuthal coordinate - angular frequency of magnetic induction This work was supported by the Microgravity Science and Applications Division of the National Aeronautics and Space Administration.     

小型高速离心压气机级内部的三维流场分析

利用三维数值模拟技术对微型燃气轮机中的离心压气机部分进行了数值分析,得到了离心压气机设计转速下的级特性曲线和各通流部件中的流动情况。数值分析表明:设计转速下压气机的级特性非常陡峭;整个特性线范围内离心叶轮基本在亚音速情况下工作,而径向扩压器是在跨音速条件下工作,离心压气机整机的最大流量是由径向扩压器的喉部面积决定的;离心压气机级内部各通流部件之间流动的相互干扰是引起流动分离的重要原因,各通流部件之间流动的相互匹配和协调将决定了离心压气机整机的性能和稳定性。     

Effect of angular inertia of lubricant in MHD hydrostatic thrust bearings

Summary Circumferential motion of a conducting lubricant in a hydrostatic thrust bearing is caused either by the angular motion of a rotating disk or by the interaction of a radial electric field and an axial magnetic field. Under the assumption that the fluid inertia due to radial motion is negligibly small in comparison with that due to angular motion, it is found analytically that the rotor causes an increase in flow rate and a decrease in load capacity, while both are increased by the application of an electric field in the presence of an axial magnetic field. The critical angular speed of the rotor at which the bearing can no longer support any load is obtained, and the possibility of flow separation in the lubricant is discussed.Nomenclature a recess radius - b outside disk radius - B ₀ magnetic induction of uniform axial magnetic field - E ₀ radial electric field at r=a - E _r radial electric field - h half of lubricant film thickness - M Hartmann number = (B ₀ ² h ²/)^1/2 - P pressure - P ₀ pressure at r=a - P _e pressure at r=b - Q volume flow rate of lubricant - Q ₀ flow rate of a nonrotating bearing without magnetic field - r radial coordinate - r _s position of flow separation on stationary disk - u, v fluid velocity components in radial and circumferential directions, respectively - W load carrying capacity of bearing - W ₀ load capacity of a nonrotating bearing without magnetic field - z axial coordinate - coefficient of viscosity - _e magnetic permeability - fluid density - electrical conductivity - electric potential - angular speed of rotating disk - _c critical rotor speed at which W=0     

A cylindrical coaxial MHD generator

A MHD generator with a novel geometry is analyzed as a possible dc power source. The generator channel consists of two coaxial cylinders with a smooth annular space between them through which pressure driven ionized gas flows axially. Magnetic poles and electrodes separated by insulators are embedded in both the inner and outer cylinders. A one-dimensional steady state analysis is presented. It is shown that the internal impedance of the generator is a very sensitive function of the ratio of areas of the charge collecting electrodes to that of the magnetic poles. The generator efficiency analysis, on the other hand, indicates that there is an optimum area ratio corresponding to the maximum conversion efficiency. A comparison of the performance characteristics of this generator with those of a generator of rectangular cross section is presented. The average gas temperature and velocity, the magnetic flux density at the poles, and the volume displacement rate, etc., are assumed identical for the two cases in comparison. It is inferred that the novel channel analyzed herein is, in general, superior to the simple rectangular channel in the energy conversion scheme.Nomenclature a _n - 2a width of the rectangular channel - a _1n , a _2n , b _1n , b _2n constants - B magnetic flux density, both induced and applied - B _r0 maximum value of radial component of B at r=r _i - B ₀ applied magnetic field in the rectangular generator = B _r0 - 2b height of the rectangular channel - C _n r _i r _o ⁿ +r _o r _i ⁿ - C _–n r _i r _o ⁿ +r _o r _i ^–n - c integration constant - D _n - E electric field strength - maximum value of azimuthal component of E at r=r _i - G _n C _–n r _n +C _n r ^– _n - G _–n C _–n r _n –C _n r ^– _n - H _n G _n r ^–1 - H _–n G _–n r ^–1 - I _r total radial current between a pair of opposite electrodes - j electric current density - p pressure of the ionized gas - P number of magnetic poles in each cylinder of the generator - P _HT power loss due to heat transfer to the walls - P _i power input - P _o power output - R _ic internal impedance of the coaxial channel MHD generator consisting of an opposite pair of electrodes associated with the magnetic poles, insulators, and the channel in between, for a unit length of the channel - R _ir internal impedance of the rectangular generator for a unit length of the channel = a/b - R ₀ external load connected to the MHD generator - r radial coordinate of the cylindrical coordinate system - r _i, r _o radii of the inner and outer cylinders, respectively - V fluid velocity - z axial coordinate of the cylindrical coordinate system - _n nP/2 - azimuthal coordinate of the cylindrical coordinate system - _e electrode angular width - _pi pole-insulator angular width - electrical conductivity of the ionized gas - permeability of the medium - _v coefficient of viscosity - (r, ) electric potential - (r _i, )–(r _o, ) potential difference between an opposite pair of electrodes - conversion efficiency of a MHD generator A paper based on some of this material was presented at the International Electron Devices Meeting, Washington (D.C.) October 1967.     

Time-resolved particle imaging velocimetry for the investigation of rotating stall in a radial pump

The operating range of turbomachines is limited in terms of the low flow rate by instabilities appearing in flow-leading parts of the machinery resulting in the creation of vortices. If the flow is further throttled, stall cells can start to propagate in the impeller at a fraction of the rotor speed. This article presents an investigation of rotating stall at different flow rates in a radial pump using time-resolved particle imaging velocimetry (PIV). This technique was used to investigate the flow field at the same position in every channel of the impeller during several revolutions. Frequency analysis was applied to the measured velocities to calculate the angular speed of the rotating stall in the impeller. The interest of time-resolved PIV to understand rotating stall is demonstrated, as it allows measurement of transient, irregularly appearing flow fields.     

Interaction of zones of flow separation in a centrifugal impeller-stationary vane system

In a radial flow pump operating in off-design conditions, regions of stall can exist on the rotating impeller blade and on the downstream diffuser blade, vane or tongue. Interaction of these stall zones can generate complex patterns of vorticity concentrations. In turn, these vorticity concentrations are related to sources of unsteady stagnation enthalpy. The form of these patterns is strongly dependent on the instantaneous location of the impeller trailing-edge relative to the leading-edge of the vane.Comparison of instantaneous with ensemble-averaged images shows that the flow structure in the gap region between the impeller and the vane is highly repetitive. Away from this region, in particular in the separated shear layer from the vane, the nonrepetitive nature of the vorticity field is manifested in substantial reduction of peak levels of vorticity in the ensemble-averaged image, relative to the instantaneous image.The three-dimensional flow structure resulting from these separation zone interactions was characterized via end views of the flow patterns. Particularly pronounced concentrations of vorticity can occur in this plane. They tend to be located in the shear layer at the outer edge of the large-scale separation zone. These vorticity concentrations are, however, highly non-stationary for successive passages of the impeller blade. Ensemble-averaging reveals that they persist primarily on the endwalls of the diffuser.The authors are grateful to the Office of Naval Research for support of this research program     

Dynamic stall experiments on the NACA 23012 aerofoil

An experimental investigation was conducted to examine the dynamic stall characteristics of a NACA 23012 aerofoil section at a Reynolds number of 1.5 million. Time-dependent data were obtained from thirty miniature pressure transducers and three hot film gauges situated at the mid-span of the wing. The static stall mechanism of the NACA 23012 was determined to be via abrupt upstream movement of trailing edge separation. Under dynamic conditions, stall was found to occur via leading edge separation, followed by a strong suction wave that moved across the aerofoil. This suction wave is characteristic of a strong moving vortex disturbance. Evidence of strong secondary vortex shedding was also found to occur, and this appears symptomatic of dynamic stall only at low Mach numbers. Some evidence of flow reversals over the trailing edge of the aerofoil were indicated prior to the development of leading edge separation and dynamic stall.List of symbols c aerofoil chord - C _L sectional lift coefficient - C _M sectional pitching moment coefficient measured about the quarter-chord location - C _p pressure-coefficient - k reduced frequency, c/2V - M Mach number - P pressure - R _c Reynolds number based on chordc - t time - V free stream velocity - x distance along chord line - y distance along span - angle of attack - _a oscillation amplitude - _M Mean angle of oscillation - shear stress - circular frequency     

The Reissner–Sagoci Problem for a Non-homogeneous Half-space with a Surface Constraint

This paper deals with the problem of twisting a non-homogeneous, isotropic, half-space by rotating a circular part of its boundary surface (0 r < a, z = 0) through a given angle. A ring (a < r < b, z = 0) outside the circle is stress-free and the remaining part (r > b, z = 0) is rigidly clamped. The shear modulus is assumed to vary with the cylindrical coordinates, r, z by the relation (z) = ₁(c + z), c 0 where ₁, c and are real constants. Expressions for some quantities of physical importance, such as torque applied at the surface of the disk and stress intensity factors, are obtained. The effects of non-homogeneity on torque and stress intensity factor are illustrated graphically.     

Effect of interference on the pressure distribution on the ground plane around two low-rise prismatic bodies in tandem arrangement

In this paper the pressure distribution on the ground plane along the centre line of prismatic bodies with square cross section and height to width (h/b) ratio of 1 and 3 are presented when the bodies are in tandem arrangement. The dimensions used are typical of low-rise buildings. For bodies with h/b=1, the results indicate that when the gap between the bodies is small, it is the front body which influences the gap pressures. But for bodies with h/b=3, even at small gaps both the front and rear bodies influence the gap pressure distribution. An increase in the base pressure of the rear body compared to the no-interference case is observed. The interference effects are in general, stronger for the body with h/b=3 than for h/b=1. Flow visualisation results are presented which reveal the changes in the flow patterns that occur with interference. There is good correspondence between the pressure distribution results and the flow visualisation studies. The results presented are applicable to low-rise building aerodynamic problems with some limitations.Nomenclature b width of the body - C _pw (p–p _r)/1/2U _r ² , pressure coefficient on the ground - g gap between the bodies in tandem arrangement - h height of the body - I length of the body along the flow direction - p pressure at any point on the surface plate with the body - p _r pressure at the same point on the surface plate without the body - s distance from the leading edge of the surface plate to the front face of the body - U local velocity at a distance y from the ground - U _r free stream reference velocity - x coordinate along the flow direction - x coordinate along the centre line measured from the front face of the rear body - y vertical distance from the ground - density of air     

Near-wall velocity distributions within a straight conical diffuser

The measured mean velocity profiles at the various stations along a conical diffuser (8° total divergence angle) were found to consist of log regions, half-power law regions and linear regions. The describing coefficients for the inner half-power law region (which followed a rather narrow log region) differed from the standard values due to the axi-symmetric geometry and lack of moving equilibrium of the flow as it attempted to adjust to a varying adverse pressure gradient. However, these coefficients (like those for the linear region) correlated with the local wall shear stress and the kinematic pressure gradient.List of symbols A, B coefficients in logarithmic law velocity distribution (Eq. (1)) - C, D coefficients in half-power law velocity distribution (Eq. (5)) - D_i inside diameter of feed pipe (10.16 cm) - d _p outer diameter of Preston tube - E, F coefficients in linear law velocity distribution (Eq. (10)) - P _s local static pressure - R local radius of diffuser, (D _i /2) + x _w sin 4° - Re Reynolds number, D _i U _b /v - U local mean velocity in the x _w direction - U _b cross-sectional average mean velocity (x-direction) in feed pipe - U _c mean velocity at the diffuser centerline - u ^* local friction velocity - u ⁺ dimensionless local mean velocity, U/u ^* - axial distance along diffuser centerline (measured from inlet to diffuser) Fig. (2) - _w distance along diffuser wall (measured from inlet to difusser (Fig. 2) - y _w distance from wall in direction orthogonal to wall (Fig. 2) - y ⁺ dimensionless position, y _w u ^*/v - kinematic (axial static) pressure gradient, (1/g9) dP _s/dx - ^* displacement thickness (Eq. (4)) - dimensionless pressure gradient parameter, x v/(u*) ³ - Von Karman constant (0.41) - density - kinematic viscosity - shear stress     

Crack repair using an elastic filler

The effect of repairing a crack in an elastic body using an elastic filler is examined in terms of the stress intensity levels generated at the crack tip. The effect of the filler is to change the stress field singularity from order 1/r^1/2 to 1/r^(1-λ) where r is the distance from the crack tip, and λ is the solution to a simple transcendental equation. The singularity power (1-λ) varies from (the unfilled crack limit) to 1 (the fully repaired crack), depending primarily on the scaled shear modulus ratio γ_r defined by G₂/G₁=γ_rε, where 2πε is the (small) crack angle, and the indices (1, 2) refer to base and filler material properties, respectively. The fully repaired limit is effectively reached for γ_r≈10, so that fillers with surprisingly small shear modulus ratios can be effectively used to repair cracks. This fits in with observations in the mining industry, where materials with G₂/G₁ of the order of 10^-3 have been found to be effective for stabilizing the walls of tunnels. The results are also relevant for the repair of cracks in thin elastic sheets.     

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     

Investigations of mean and turbulent flow characteristics of a two dimensional plane diffuser

This paper reports the investigation of mean and turbulent flow characteristics of a two-dimensional plane diffuser. Both experimental and theoretical details are considered. The experimental investigation consists of the measurement of mean velocity profiles, wall static pressure and turbulence stresses. Theoretical study involves the prediction of downstream velocity profiles and the distribution of turbulence kinetic energy using a well tested finite difference procedure. Two models, viz., Prandtl's mixing length hypothesis and k- model of turbulence, have been used and compared. The nondimensional static pressure distribution, the longitudinal pressure gradient, the pressure recovery coefficient, percentage recovery of static pressure, the variation of U _max/U _bar along the length of the diffuser and the blockage factor have been valuated from the predicted results and compared with the experimental data. Further, the predicted and the measured value of kinetic energy of turbulence have also been compared. It is seen that for the prediction of mean flow characteristics and to evaluate the performance of the diffuser, a simple turbulence model like Prandtl's mixing length hypothesis is quite adequate.List of symbols C ₁ , C ₂ ,C turbulence model constants - F _x body force - k kinetic energy of turbulence - l _m mixing length - L length of the diffuser - u, v, w rms value of the fluctuating velocity - u, v, w turbulent component of the velocity - mean velocity in the x direction - _A average velocity at inlet - U _bar average velocity in any cross section - U _max maximum velocity in any cross section - V mean velocity in the y direction - W local width of the diffuser at any cross section - x, y coordinates - dissipation rate of turbulence - _m eddy diffusivity - Von Karman constant - mixing length constant - _l laminar viscosity - _eff effective viscosity - v kinematic viscosity - density - _k effective Schmidt number for k - effective Schmidt number for - stream function - non dimensional stream function     

Three-segment electrodiffusion probes for measuring velocity fields close to a wall

Three-segment electrodiffusion probes embedded in a wall allow to determine simultaneously the three kinematic parameters of flow close to the probe surface: the flow direction, the wall shear rateq, and the normal velocity coefficientA,v _z = –A z ². A well-controlled three-dimensional flow, generated by a rotating disk, was used to demonstrate the capabilities of this new kind of electrodiffusion probes by comparing experimental results with the prediction based on the well-known hydrodynamical theory.List of symbols A normal flow coefficient, Eq. (1) - A axis of the adjustment rod, Fig. 2 - c ₀ concentration of depolarizer (mol/m³) - D diffusivity of depolarizer (m²/s) - E correction of total current on normal flow effect - e _x reference direction of the probe, Figs. 1 and 3 - F Faraday constant (F = 96,464 C/mol) - F _s normalized directional characteristic fors-th segment - f _sm ,g _sm Fourier coefficients of directional characteristics, Eq. (4) and Table 3 - h _m corrections of Fourier coefficients on normal flow effect, Eqs. (4) and (7) - i _s limiting diffusion current throughs-th segment (A) - i _tot (r) total current through the probe in dependence on its eccentricity (A) - K transport coefficient, Eqs. (3) and (5) - n number of electrons involved in redox reaction - O axis of the rotating disk, Fig. 2 - P centre of the probe, Fig. 2 - q magnitude of vectorial wall shear rate (s^-1) - q _x ,q _y components of vectorial wall shear rate - Q ratio of the currents in an eccentric and the central position of the probe, Eq. (15) - r radial coordinate, eccentricity of the probe - r _A eccentricity of the adjustment rod (r _A = , Fig. 2) - r, , z polar coordinates on the rotating disk - R effective radius of the probe (R = 0.337 mm) - S macroscopic area of the probe (S = 0.357 mm²) - x, y, z Cartesian coordinates moving with the probe - adjustment angle, Figs. 2 and 3 - angle included between local radius-vector ¯P of the probe and local direction of flow, Fig. 3 - angle included between reference directione _x of the probe and local direction of flow, Fig. 3 - ₀ theoretical prediction of, Eq. (11) - x ₀ theoretical prediction ofx, Eq. (14) - x _exp x calculated from experimental data using Eq. (4) - v kinematic viscosity (m²/s) - angle implied between gradient ofq and direction of flow, Eq. (8) - angular speed of the rotating disk (rad/s)     

Influence of wall riblets on diffuser flow

Diffuser flows have been investigated with different riblet combinations both experimentally and numerically. Wall pressures and velocity profiles have been measured throughout a large working domain. In addition numerical calculations have been performed by accounting for the strong coupling between the inviscid core flow and the viscous boundary layer flows by using a simple model based on the negative roughness for the riblet walls.Nomenclature c _f skin friction coefficient - p local static pressure - x streamwise distance - y distance normal to the wall - u friction velocity - A diffuser cross section - H shape factor - U _e fluid velocity in inviscid flow - W ₁ inlet height of the diffuser - * boundary layer displacement thickness - density of working fluid Institut de Machines Hydrauliques et de Mecanique des Fluides     

Investigation of diffuser characteristics in an aerodynamic shock tube

An investigation has been made in an aerodynamic shock tube, with M=8, of the diffuser starting with variable Reynolds numbers, and its throttle characteristics have been recorded. The results obtained enable conclusions to be made regarding the possibility of investigating diffusers in such types of tubes. In [1] the possibility of, in principle, determining the throat starting of a diffuser in an aerodynamic shock tube, and the time to establish flow in the diffuser channel, which was 300 sec, was measured. This paper is devoted to the further investigation of diffuser operation with variable Reynolds numbers, and to determining the throttle characteristics, in particular, the total pressure reduction coefficient.Moscow. Translated from Izvestiya Akademii Nauk SSSR, mekhanika Zhidkosti i Gaza, No. 1, pp. 156–161, January–February, 1972.     

Homogeneous diffusion in ℝ with power-like nonlinear diffusivity

We study the nonnegative solutions of the initial-value problem u_t=(u^r|u_x|^p-1u_x)_x,u(x, 0)L ¹(), where p>0, r+p>0. The local velocity of propagation of the solutions is identified as V = -v_x| v_x|^p-1 where v =cu (with r +p - 1)/p and c (r +p/(r +p- 1)) is the nonlinear potential. Our main result is the a priori estimate (v_x|v_x|^p-1)_x-