Influence of cavitation on blade characteristics

An experimental study of flow around a blade with a modified NACA 4418 profile was conducted in a water tunnel that also enables control of the cavitation conditions within it. Pressure, lift force, drag force and pitching moment acting on the blade were measured for different blade angles and cavitation numbers, respectively. Relationships between these parameters were elaborated and some of them are presented here in dimensionless form. The analysis of results confirmed that cavitation changes the pressure distribution significantly. As a consequence, lift force and pitching moment are reduced, and the drag force is increased. When the cavitation cloud covers one side of the blade and the flow becomes more and more vaporous, the drag force also begins to decrease. The cavity length is increased by increasing the blade angle and by decreasing thé cavitation number.List of symbols A (m²) blade area,B ·L - B (m) blade width - C _D (–) drag coefficient,F _D /(p _d ·A) - C _L (–) lift coefficient,F _L /(P _d ·A) - C _M (–) pitching moment coefficient,M/(P _d ·A ·L) - C _p (–) pressure coefficient, (p-p _r )/p _d - F (N) force - L (m) blade length - M (Nm) pitching moment - p (Pa) local pressure on blade surface - p _d (Pa) dynamic pressure, ·V ²/2 - p _r (Pa) reference wall pressure at blade nose position if there would be no blade in the tunnel - p _v (Pa) vapor pressure - p ₁ (Pa) wall pressure 350 mm in front of thé blade axis - Re (–) Reynolds number,V ·L/v - V (m/s) mean velocity of flow in the tunnel - x (m) Cartesian coordinate along thé blade profile cord - x _c (m) cavity length,x-coordinate of cavity end - (°) blade angle - v (m²/s²) kinematic viscosity - (kg/m³) fluid density - (–) cavitation number, (p _r –p _v )/p _d - (°) angle of tangent to thé blade profile contour     

Dynamic interaction of various beams with the underlying soil – finite and infinite, half-space and Winkler models

Various beams lying on the elastic half-space and subjected to a harmonic load are analyzed by a double numerical integration in wavenumber domain. The compliances of the beam–soil systems are presented for a wide frequency range and for a number of realistic parameter sets. Generally, the soil stiffness G has a strong influence on the low-frequency beam compliance whereas the beam parameters EI and m^′ are more important for the high-frequency compliance. An important parameter is the elastic length l=(EI/G)^1/4 of the beam–soil system. Around the corresponding frequency ω_l=v_S/l, the wave velocity of the combined beam–soil system changes from the Rayleigh wave v_R≈v_S to the bending wave velocity v_B and the combined beam–soil wave has typically a strong damping. The interaction frequency ω_l is found not far from the characteristic frequency ω₀=(G/m^′)^1/2 where an amplification compared to the static compliance is observed for special parameter constellations. In contrast, real foundation beams show no resonance effects as they are highly damped by the radiation into the soil. At medium and high frequencies, asymptotes for the compliance of the beam–soil system are found, u/P(ρv_Paiω)^−3/4 in case of the dominating damping and u/P(−m^′ω²)^−3/4 for high frequencies. The low-frequency compliance of the coupled beam–soil system can be approximated by u/P1/Gl, but it also depends weakly on the width a of the foundation. All numerical results of different beam–soil systems are evaluated to yield a unique relation u/P₀=f(a/l). The integral transform method is also applied to ballasted and slab tracks of railway lines, showing the influence of train speed on the deformation of the track beam. The presented results of infinite beams on half-space are compared with results of finite beams and with infinite beams on a Winkler support. Approximating Winkler parameters are given for realistic foundation-soil systems which are useful when vehicle-track interaction is analyzed for the prediction of railway induced vibration.     

Landslide generated impulse waves.

Landslide generated impulse waves were investigated in a two-dimensional physical laboratory model based on the generalized Froude similarity. Digital particle image velocimetry (PIV) was applied to the landslide impact and wave generation. Areas of interest up to 0.8 m by 0.8 m were investigated. The challenges posed to the measurement system in an extremely unsteady three-phase flow consisting of granular matter, air, and water were considered. The complex flow phenomena in the first stage of impulse wave initiation are: high-speed granular slide impact, slide deformation and penetration into the fluid, flow separation, hydrodynamic impact crater formation, and wave generation. During this first stage the three phases are separated along sharp interfaces changing significantly within time and space. Digital masking techniques are applied to distinguish between phases thereafter allowing phase separated image processing. PIV provided instantaneous velocity vector fields in a large area of interest and gave insight into the kinematics of the wave generation process. Differential estimates such as vorticity, divergence, elongational, and shear strain were extracted from the velocity vector fields. The fundamental assumption of irrotational flow in the Laplace equation was confirmed experimentally for these non-linear waves. Applicability of PIV at large scale as well as to flows with large velocity gradients is highlighted.List of symbols a wave amplitude (L) - c wave celerity (LT^–1) - d_diff diffraction limited minimum particle image diameter (L) - d_e diffracted particle image diameter (L) - d_g granulate grain diameter (L) - d_p seeding particle diameter (L) - d recorded particle image diameter (L) - f focal length (L) - f_# f number (-) - F slide Froude number (-) - g gravitational acceleration (LT^–2) - h still-water depth (L) - H wave height (L) - l_s slide length (L) - L wavelength (L) - M magnification (-) - m_s slide mass (M) - n refractive index (-) - n_por slide porosity (-) - N_iw number of seeding particles in an interrogation window (-) - N_pair number of detected particle image pairs in window (-) - p interrogation window size p×p pixels; 1 pixel=9 m (L) - P probability (-) - P_il probability of in-plane loss of particle (-) - P_ol probability of out-of-plane loss of particle (-) - s slide thickness (L) - S relative slide thickness (-) - t time after impact (T) - T wave period (T) - v velocity (LT^–1) - v_p particle velocity (LT^–1) - v_px streamwise horizontal component of particle velocity (LT^–1) - v_py crosswise horizontal component of particle velocity (LT^–1) - v_pz vertical component of particle velocity (LT^–1) - v_s slide centroid velocity at impact (LT^–1) - V dimensionless slide volume (-) - V_iw interrogation volume (L³) - V_s slide volume (L³) - x streamwise coordinate (L) - x_ip area of view x dimension in image plane (L) - z vertical coordinate (L) - slide impact angle (°) - bed friction angle (°) - y depth of field (L) - t laser pulse separation (T) - x mean particle image x displacement in interrogation window (L) - _x random displacement x error (L) - _v random velocity v error (LT^–1) - _tot total random velocity v error (LT^–1) - _bias velocity v error due to biased correlation analysis (LT^–1) - _optics velocity v error due to optical imaging errors (LT^–1) - _track velocity v error due to particle flow tracking error (LT^–1) - _xx streamwise horizontal elongational strain component (1/T) - _xz shear strain component (1/T) - _zx shear strain component (1/T) - _zz vertical elongational strain component (1/T) - water surface displacement (L) - wavelength (L) - dynamic viscosity (ML^–1T^–1) - density (ML^–3) - _g granulate density (ML^–3) - _p particle density (ML^–3) - _s mean slide density (ML^–3) - _w water density (ML^–3) - granulate internal friction angle (°) - _y vorticity vector component (out-of-plane) (1/T)     

Numerical Study of Hydrodynamics and Heat and Mass Transfer of a Ducted Gas–Vapor‐Droplet Flow

A computational model has been developed to predict heat and mass transfer and hydrodynamic characteristics of a turbulent gas–vapor–droplet flow. Turbulent characteristics of the gas phase are computed using the k– model of turbulence. It is shown that, with increasing inlet droplet diameter, the rate of heat transfer between the duct surface and the vapor–gas mixture decreases appreciably, whereas the wall friction increases only insignificantly. The predicted values agree fairly well with available experimental and numerical data     

On Asymptotics of Solutions of Nonlinear Differential Equations of the Second Order

We study the problem of asymptotics of unbounded solutions of differential equations of the form y″ = α₀ p(t)ϕ(y), where α₀ ∈ {−1, 1}, p: [a, ω[→]0, +∞[, −∞ < a < ω ≤ +∞, is a continuous function, and ϕ: [y ₀, +∞[→]0, +∞[ is a twice continuously differentiable function close to a power function in a certain sense.__________Translated from Neliniini Kolyvannya, Vol. 8, No. 1, pp. 18–28, January–March, 2005.     

Investigation of chemically reacting and radiating supersonic flow in channels

The two-dimensional time dependent Navier-Stokes equations are used to investigate supersonic flows undergoing finite rate chemical reaction and radiation interaction for a hydrogen-air system. The explicit multi-stage finite volume technique of Jameson is used to advance the governing equations in time until convergence is achieved. The chemistry source term in the species equation is treated implicitly to alleviate the stiffness associated with fast reactions. The multidimensional radiative transfer equations for a nongray model are provided for general configuration, and then reduced for a planar geometry. Both pseudo-gray and nongray models are used to represent the absorption-emission characteristics of the participating species.The supersonic inviscid and viscous, nonreacting flows are solved by employing the finite volume technique of Jameson and the unsplit finite difference scheme of MacCormack to determine a convenient numerical procedure for the present study. The specific problem considered is of the flow in a channel with a 10° compression-expansion ramp. The calculated results are compared with the results of an upwind scheme and no significant differences are noted. The problem of chemically reacting and radiating flows are solved for the flow of premixed hydrogen-air through a channel with parallel boundaries, and a channel with a compression corner. Results obtained for specific conditions indicate that the radiative interaction can have a significant influence on the entire flowfield.Nomenclature A band absorptance (m^–1) - A _o band width parameter (m^–1) - C _j concentration of thejth species (kg mol/m³) - C _o correlation parameter ((N/m²)^–1m^–1) - C _p constant pressure specific heat (J/kgK) - e Planck's function (J/m²S) - E total internal energy (J/kg) - f _j mass fraction of thejth species - h static enthalpy of mixture (J/kg) - H total enthalpy (J/kg) - I identity matrix - I _v spectral intensity (J/m s) - I _bv spectral Planck function - k thermal conductivity (J/m sK) - K _b backward rate constant - K _f forward rate constant - I unit vector in the direction of - M _j molecular weight of thejth species (kg/kg mol) - P pressure (N/m²) - P _j partial pressure of thejth species (N/m²) - P _e equivalent broadening pressure ratio - Pr Prandtl number - P _w a point on the wall - q _R total radiative heat flux (J/m² s) - spectral radiative heat flux (J/m³ s) - R gas constant (J/KgK) - r _w distance between the pointsP andP _w(m) - S integrated band intensity ((N/m²)^–1/m^–2) - S integrated band intensity ((N/m²)^–1 m^–2) - T temperature (K) - u, v velocity inx andy direction (m/s) - production rate of thejth species (kg/m³ s) - x, y physical coordinate - z dummy variable in they direction Greek symbols ratio of specific heats - t _ch chemistry time step (s) - t _f fluid-dynamic time step (s) - absorption coefficient (m^–1) - ,_v spectral absorption coefficient (m^–1) - _p Planck mean absorption coefficient - second coefficient of viscosity, wavelength (m) - dynamic viscosity (laminar flow) (kg/m s) - , computational coordinates - density (kg/m³) - Stefan-Boltzmann constant (erg/s cm² K³) - shear stress (W/m²) - equivalence ratio - wave number (m^–1) - _c frequency at the band center     

Landslide generated impulse waves. 2. Hydrodynamic impact craters

Landslide generated impulse waves were investigated in a two-dimensional physical laboratory model based on the generalized Froude similarity. Digital particle image velocimetry (PIV) was applied to the landslide impact and wave generation. Areas of interest up to 0.8 m by 0.8 m were investigated. PIV provided instantaneous velocity vector fields in a large area of interest and gave insight into the kinematics of the wave generation process. Differential estimates such as vorticity, divergence, and elongational and shear strain were extracted from the velocity vector fields. At high impact velocities flow separation occurred on the slide shoulder resulting in a hydrodynamic impact crater, whereas at low impact velocities no flow detachment was observed. The hydrodynamic impact craters may be distinguished into outward and backward collapsing impact craters. The maximum crater volume, which corresponds to the water displacement volume, exceeded the landslide volume by up to an order of magnitude. The water displacement caused by the landslide generated the first wave crest and the collapse of the air cavity followed by a run-up along the slide ramp issued the second wave crest. The extracted water displacement curves may replace the complex wave generation process in numerical models. The water displacement and displacement rate were described by multiple regressions of the following three dimensionless quantities: the slide Froude number, the relative slide volume, and the relative slide thickness. The slide Froude number was identified as the dominant parameter.List of symbols a wave amplitude (L) - b slide width (L) - c wave celerity (LT^–1) - d _g granulate grain diameter (L) - d _p seeding particle diameter (L) - F slide Froude number - g gravitational acceleration (LT^–2) - h stillwater depth (L) - H wave height (L) - l _s slide length (L) - L wave length (L) - M magnification - m _s slide mass (M) - n _por slide porosity - Q _d water displacement rate (L³) - Q _D maximum water displacement rate (L³) - Q _s maximum slide displacement rate - s slide thickness (L) - S relative slide thickness - t time after impact (T) - t _D time of maximum water displacement volume (L³) - t _qD time of maximum water displacement rate (L³) - t _si slide impact duration (T) - t _sd duration of subaqueous slide motion (T) - T wave period (T) - v velocity (LT^–1) - v _p particle velocity (LT^–1) - v _px streamwise horizontal component of particle velocity (LT^–1) - v _pz vertical component of particle velocity (LT^–1) - v _s slide centroid velocity at impact (LT^–1) - V dimensionless slide volume - V _d water displacement volume (L³) - V _D maximum water displacement volume (L³) - V _s slide volume (L³) - x streamwise coordinate (L) - z vertical coordinate (L) - slide impact angle (°) - bed friction angle (°) - x mean particle image x-displacement in interrogation window (L) - _x random displacement x error (L) - _tot total random velocity v error (LT^–1) - _xx streamwise horizontal elongational strain component (1/T) - _xz shear strain component (1/T) - _zx shear strain component (1/T) - _zz vertical elongational strain component (1/T) - water surface displacement (L) - density (ML^–3) - _g granulate density (ML^–3) - _p particle density (ML^–3) - _s mean slide density (ML^–3) - _w water density (ML^–3) - granulate internal friction angle (°) - _y vorticity vector component (out-of-plane) (1/T)     

The wake flow control behind a circular cylinder using ion wind

Active and passive flow control methods have been studied for decades, but there have been only a few studies of flow control methods using ion wind, which is the bulk motion of neutral molecules driven by locally ionized air from a corona discharge. This paper describes an experimental study of ion wind wake control behind a circular cylinder. The experimental conditions consisted of a range of electrohydrodynamic numbers—the ratio of an electrical body force to a fluid inertial force—from 0 to 2 and a range of Reynolds numbers from 4×10³ to 8×10³. Pressure distributions over the cylinder surface were measured and flow visualizations were carried out using a smoke-wire method. The flow visualizations confirmed that ion wind significantly affects the wake structure behind a circular cylinder, and that the pressure drag can be dramatically reduced by superimposing ion wind.List of symbols BR blockage ratio - C _d coefficient of the pressure drag - C _p coefficient of the surface pressure, 2(p–p ₀)/(U ₀ ²) - C _pb coefficient of the base surface pressure, 2(p _b–p ₀)/(U ₀ ²) - D diameter of the cylinder - D _P pressure drag - d _p diameter of particle - E the electric field - F _e Coulombian force (qE) - F _v viscous force - H wire-to-cylinder spacing - I total electric current (A) - L the axial length of cylinder (m) - N _EHD electrohydrodynamic number - p _b base pressure of cylinder at =180° - p ₀ reference static pressure at 10D upstream - q the charge on the particle - R radius of the cylinder - V applied voltage (kV) - U ₀ mean flow velocity (m/s) - ion mobility in air (m²/(s V)) - ₀ permittivity of free space - viscosity of fluid (kg/ms) - density of fluid (kg/m³) - installation angle of a wire electrode (°)     

The viscous air flow pattern in the Stefan diffusion tube

In this paper, we consider the problem of the binary viscous diffusion of vapour through a Stefan tube, which is the model of an elementary capillary. While some preceding results in particular cases supposed parabolic velocity profiles and showed air recirculation, we treat here the general problem of a tube of finite length, submitted to a double viscous diffusion of vapour and air from a liquid surface. The movement of gas is expressed with conservation equations and ideal gas equations. The following added restrictions: constant temperature, no buyoancy effect, no inertial forces, are compatible with a capillary. A numerical solution based on the control volume method is obtained at every point in the tube. The results give the vapour and air flux, describe the circulation pattern and show that the vapour profile of concentration is level. In the lower part of the cylindrical tube space, over a distance of the length of a radius an important radial movement occurs, due to the recirculation of air which changes direction once it reaches the liquid surface. The velocity profile of the gas flow then becomes parabolic in the upper part of the tube.In order to easily obtain a numerical solution, the system of dimensionless equations is expanded to a series and transformed into a set of sub-systems. The little parameter used for this expansion is tied to the vapour concentration on the liquid surface. The solution of the sub-system of order zero, which is easier to compute, represents a good approximation of the complete solution. These solutions are situated in comparison with the Stefan diffusion and show that the influence of the viscous effect on the vapour flux is limited to a few percents.In order to apply the results to porous media where the pores are not so regular, we consider at last the diffusion in a tube including a contracted section in the middle of the tube. Since the diffusion paths are longer, the vapour flux is reduced, while the viscosity effect becomes more considerable. The reduction of the air flux is more significant than that of the vapour. This part of the study provides a better understanding of the diffusion through the pits at the wall fiber, and gives data for the air flux which permeates into the oak wood and produces tannin oxidation and thus discolouration.Nomenclature m _v/ vapour concentration - m _l vapour concentration in equilibrium with its liquid phase - D coefficient of molecular diffusion of a vapour in air (m²/s) - J _a vector density of mass flux of dry air (kg/m²s) - J _v vector density of mass flux of vapour (kg/m²s) - L Capillary length (m) - M _a dry air molar mass (kg/mole) - M _v Vapour molar mass (kg/mole) - P _atm atmospheric pressure - P gas mixture total pressure (Pascal) - R Ideal gas constant (J/mole K) - r _a= R/M _a,r _v=R/M _v - r ₀ tube or capillary radius (m) - T Temperature (K) - u axial component ofV - V gas mixture velocity vector (m/s) - v radial component ofV Greek Letters density of gas (kg/m³) - gas mixture dynamic viscosity (kg/ms) Numerical Values of Parameters D 3×10^–5m²/s (water vapour in air) - 2×10^–5kg/ms - M _a 29×10^–3kg/mole - M _v 18×10^–3kg/mole - T 323K - R 8.31 J/moleK - P _atm 10⁵Pa     

Transients in flow and local heat transfer due to a pressure wave in pipe flow

The effect of a pressure wave on the turbulent flow and heat transfer in a rectangular air flow channel has been experimentally studied for fast transients, occurring due to a sudden increase of the main flow by an injection of air through the wall. A fast response measuring technique using a hot film sensor for the heat flux, a hot wire for the velocities and a pressure transducer have been developed. It was found that in the initial part of the transient the heat transfer change is independent of the Reynolds number. For the second part the change in heat transfer depends on thermal boundary layer thickness and thus on the Reynolds number. Results have been compared with a simple numerical turbulent flow and heat transfer model. The main effect on the flow could be well predicted. For the heat transfer a deviation in the initial part of the transient heat transfer has been found. From the turbulence measurements it has been found that a pressure wave does not influence the absolute value of the local turbulent velocity fluctuations. They could be considered to be frozen.Nomenclature A surface area (m²) - D diameter (m) - h heat transfer coefficient (Wm^–2 K^–1) - p pressure drop (Pa) - P pressure (Pa) - Q heat flow (W) - R tube radius (m) - T bulk temperature (K) - T _s surface temperature (K) - t time (s) - u velocity (m/s) - V voltage (V) - y distance from wall (m) - viscosity (N s m^–2) - kinematic viscosity (m^–2 s^–1) - density (kg m^–3) - _w wall shear stress (N m^–2) - Nu Nusselt number - Re Reynolds number     

Effect of solids concentration in dense suspensions of silica-iron(III) oxide and water

The rheological properties of dense suspensions, of silica, iron (III) oxide and water, were studied over a range of solids concentrations using a viscometer, which was modified so as to prevent settling of the solid components. Over the conditions studied, the material behaved according to power—law flow relationships. As the concentrations of silica and iron(III) oxide were increased, an entropy term in the flow equation was identified which had a silica dependent and an iron (III) oxide dependent component. This was attributed to a tendency to order into some form of structural regularity. A, A, B, C pre-exponential functions (K Pa^–n s^–1) - C _ox volume fraction iron (III) oxide - Q activation energy (kJ mol^–1) - R gas constant (kJ mol^–1 K^–1) - R _v silica/water volume ratio - T temperature (K) - n power-law index - H enthalpy (kJ mol^–1) - S entropy change (kJ mol^–1 K^–1) - shear strain rate (s^–1) - shear stress (Pa)     

A specific slit die rheometer for extruded starchy products. Design,validation and application to maize starch

A novel in-line rheometer, called Rheopac, has been designed and built in order to study the rheological behaviour of starchy products or, more generally, of products sensitive to a thermomechanical treatment. It is based on the principle of a twin channel, using a balance of feed rate between each of them, in order to make local shear rate vary in the measuring section without changing the flow conditions into the extruder. A wide range of shear rate could be reached and measurements were performed more swiftly than with a classical slit die. The viscous behaviour of maize starch was studied by taking into account the influence of the thermomechanical history, which modified the starch degradation and thus led to important variations in the viscosity. Experimental results were satisfactorily compared to previously published models.Nomenclature E activation energy (J · mol^–1) - h channel depth (m) - h ₁ depth under the piston valve in channel 1 (m) - h ₂ depth under the piston valve in channel 2 (m) - K consistency (Pa·s ⁿ ) - K ₀ reference consistency (Pa·s ⁿ ) - L total channel length (m) - L _p length of the piston valve (m) - MC moisture content (wet basis) - n power law index - N screw rotation speed (rpm) - P ₀ entrance pressure (Pa) - P _e pressure at the entry of the piston valve (Pa) - Q ₁ flow rate in channel 1 (m3 · s^–1) - Q ₂ flow rate in channel 2 m³·s^–1) - Q _T total flow rate (m3 · s^–1) - R constant of perfect gas (8.314 J·mol^–1·K^–1) - SME specific mechanical energy (kWh · t^–1) - T temperature (°C) - T _a absolute temperature (K) - T _b barrel temperature (°C) - T _d die temperature (°C) - T _p product temperature (°C) - w channel width (m) - W energetical term (J·m^–3) - viscosity (Pa · s) - [gh ₀] intrinsic viscosity of native starch (ml·g^–1) - [] intrinsic viscosity (ml·g^–1) - shear rate (s^–1) - shear rate in measuring section (s^–1) - maximum shear rate (s^–1)     

ANALYSIS OF FINANCIAL DERIVATIVES BY MECHANICAL METHOD (Ⅱ)-BASIC EQUATION OF MARKET PRICE OF OPTION

The basic equation of market price of option is formulated by taking assumptions based on the characteristics of option and similar method for formulating basic equations in solid mechanics: hv ₀(t) = m ₁ v ₀ ^–1(t) – n ₁ v ₀(t) + F, where h, m ₁, n ₁, F are constants. The main assumptions are: the ups and downs of market price v ₀(t) are determined by supply and demand of the market; the factors, such as the strike price, tenor, volatility, etc. that affect on v ₀(t) are demonstrated by using proportion or inverse proportion relation; opposite rules are used for purchasing and selling respectively. The solutions of the basic equation under various conditions are found and are compared with the solution v _f (t) of the basic equation of market price of futures. Furthermore the one-one correspondence between v _f and v ₀(t) is proved by implicit function theorem, which forms the theoretic base for study of v _f affecting on the market price of option v ₀(t).     

Simulation method in the kinetic theory of gases

We develop a method that is based on use of the simulation equation vf/v=vl ^–1(f ₀–f), where v is the modulus of the molecular velocity andl is the mean free-path length. A number of general properties of the model are clarified and the transition to the limit of a continuous medium is discussed in detail.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 110–115, November–December, 1973.     

Wall visualization of swirling decaying flow using a dot-paint method

This paper deals with the visualization of swirling decaying flow in an annular cell fitted with a tangential inlet. A wall visualization method, the so-called dot-paint method, allows the determination of the flow direction on both cylinders of the cell. This study showed the complex structure of the flow field just downstream of the inlet, where a recirculation zone exists, the effects of which are more sensitive on the inner cylinder. The flow structure can be considered as three-dimensional in the whole entrance section. The swirl number and the entrance length were estimated using the measured angle of the streamlines. Experimental correlations of these two parameters, taking into account the Reynolds number and the axial distance from the tangential inlet, are given.List of symbols e = R ₂ – R ₁ thickness of the annular gap (m) - L _ax entrance length of axial flow on the outer cylinder (m) - L _ti length of the three-dimensional flow region on the inner cylinder (m) - L _to length of the three-dimensional flow region on the outer cylinder (m) - Q _v volumetric flowrate in the annular cell (m³s^–) - r radial position (m) - R ₁ external radius of the inner cylinder (m) - R ₂ internal radius of the outer cylinder (m) - Re=2eU _m /v Reynolds number - Sn swirl number - T time average resulting velocity (m s^–) - u time average axial velocity component (ms ^–) - average velocity in the annulus (m s^–) - w time average tangential velocity component (m s^–) - x axial location from the tangential inlet (m) - _e diameter of the tangential inlet (m) - streak angle with respect to the horizontal (degree) - angle with respect to the tangential inlet axis (degree) - gn kinematic viscosity of the working liquid (m²s^–)     

Pulsating slurry flow in pipelines

An experimental study on pulsating turbulent flow of sand-water suspension was carried out. The objective was to investigate the effect of pulsating flow parameters, such as, frequency and amplitude on the critical velocity, the pressure drop per unit length of pipeline and hence the energy requirements for hydraulic transportation of a unit mass of solids. The apparatus was constructed as a closed loop of 11.4 m length and 3.3 cm inner diameter of steel tubing. Solid volumetric concentrations of up to 20% were used in turbulent flow at a mean Reynolds number of 33,000–82,000. Pulsation was generated using compressed air in a controlled pulsation unit. Frequencies of 0.1–1.0 Hz and amplitude ratios of up to 30% were used. Instantaneous pressure drop and flow rate curves were digitized to calculate the energy dissipation associated with pulsation. The critical velocity in pulsating flow was found to be less than that for the corresponding steady flow at the same volumetric concentration. Energy dissipation for pulsating flow was found to be a function of both frequency and amplitude of pulsation. A possible energy saving was indicated at frequencies of 0.4–0.8 Hz and moderate amplitudes ratios of less than 25%.List of symbols A cross-section area of the tube (m²) - C _D drag coefficient of sand particles - C _v volumetric concentration (%) - D inner diameter of test-section pipe (m) - F frequency (Hz) - f friction factor - g gravitational constant (m/s²) - J energy dissipation of suspension (W/m)/(kg/s) - J _p energy dissipation of pulsating suspension (W/m)/(kg/s) - J _s energy dissipation of steady component of suspension (W/m)/(kg/s) - J _w energy dissipation of pure water (W/m)/(kg/s) - L length of test-section (m) - m mass flow rate (kg/s) - P pressure drop in test-section (N/m²) - S specific gravity of sand - V instantaneous flow velocity (m/s) - V _c steady flow critical velocity (m/s) - V _cp pulsating flow critical velocity (m/s) - V _F settling velocity of particles (m/s) - V _s steady component of mean flow velocity (m/s) - dynamic viscosity (g/cm sec) - _m mean density of suspension (kg/m³) - angular velocity (rad/sec) - amplitude ratio (V — V _s)/V - nondimentional factor equal to - nondimentional factor equal to (V–V _s/V - NI nondimentional factor equal to (V ²C _d/g D(S – 1)) - Re Reynolds number (V ²C _d/C _v g D(S – 1))     

Jet methods of gas injection into fluid boundary layer for drag reduction

Two jet methods for saturating the fluid boundary layer with microbubbles for drag reduction in contrust with gas injection through porous materials are considered. The first method is the gas injection through the slot under a special fluid wall jet. The second method is the saturation of boundary layer by microbubbles via the gas-water mixture injection through the slot. Experimental data, reflecting the skin friction drag reduction on the flat plate and total drag reduction of axisymmetric bodies, are presented. The comparison between a jet methods of gas injection and gas injection through porous materials is made.Nomenclature v free-stream velocity - v _j mean velocity of a water through slot - v _g mean velocity of a gas through slot - h width of slot for realizing water jet - h ₁ width of slot for gas injection - incidence angle - Q volume airflow rate - C _Q airflow rate coefficient (v _g/v ) - C _f skin friction coefficient - v _j/v - C _f0 C _f ifQ=0 andv _j=0 - _f C _f/C _{f 0} - d diameter of an axisymmetric body - L length of body - C _Q 4 · ·Q/d ² v - C _D 4 ·D/1/2v ² ·d ² - C _Q 4 ·Q/d ² v - Q _j volume flow rate of water jet - C 8 ·Q _jv_j/d ² v ² - 1 fluid density of main flow - 2 fluid density of wall jet - B ₁ main stream total pressure - B ₂ wall jet total pressure - v ₁ main stream velocity - Be (B ₂ –B ₁)/1/2₁ v ₁ ² = Bernoulli number - ₂ v ₂/₁ v ₁ - p _st static pressure - p _at atmospheric pressure - p _st/p _at - D hydrodynamic drag of body     

Shear stress and normal stress measurements of aqueous polymer solutions in a cone-and-plate rheogoniometer

Summary The rheological behaviour of aqueous solutions of Separan AP-30 and Polyox WSR-301 in a concentration range of 10–10000 wppm is investigated by means of a cone-and-plate rheogoniometer. The relation between the shear stress and the shear rate is for lower shear rates characterized by a timet ₀, which is concentration dependent. Both polymers show for 4000 s^–1 < < 10000 s^–1 a behaviour similar to that of a Bingham material, characterized by a dynamic viscosity ₀ and an apparent yield stress ₀, which also depend on the concentration. The inertial forces are measured for water and some other Newtonian liquids. An explanation is given why the theoretical model developed for these forces does not match the experimental values; the shape of the liquid surface is shear rate dependent. To obtain the first normal stress difference, we have to correct for these inertial forces, the surface tension and the buoyancy. The normal forces, measured for Separan AP-30, appear to be a linear function of the shear rate for 350 s^–1 < < 3300 s^–1.

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Zusammenfassung Das rheologische Verhalten wäßriger Polymerlösungen von Separan AP-30 und Polyox WSR-301 wird in einem Konzentrationsgebiet von 10–10000 wppm in einem Kegel-Platte-Rheogoniometer untersucht. Der Zusammenhang zwischen Schubspannung und Schergeschwindigkeit wird für niedrige Schergeschwindigkeiten durch eine konzentrationsabhängige Zeitt ₀ gekennzeichnet. Für Schergeschwindigkeiten 4000 s^–1 < < 10000 s^–1 zeigen beide Polymere ein genähert binghamsches Verhalten, gekennzeichnet durch eine dynamische Viskosität ₀ und eine scheinbare Fließgrenze ₀, welche ebenfalls konzentrationsabhängig sind. Die Trägheitskräfte werden für Wasser und einige newtonsche Öle bestimmt. Die Abweichung der experimentellen Ergebnisse vom theoretischen Modell wird durch die Abhängigkeit der Gestalt der Flüssigkeitsoberfläche von der Schergeschwindigkeit erklärt. Um die Werte der ersten Normalspannungsdifferenz zu erhalten, muß man bezüglich der Trägheitskräfte, der Oberflächenspannung und der Auftriebskräfte korrigieren. Die Normalspannungen für Separan AP-30, gemessen für 350 s^–1 < < 3300 s^–1, zeigen eine lineare Abhängigkeit von der Schergeschwindigkeit.

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c concentration (wppm) - g acceleration of gravity (ms^–2) - K force (N) - K _b buoyant force (N) - K _c force, acting on the cone (N) - K ₀ dimensional constant def. by eq. [24] (N) - K _s force, def. by eq. [22] (N) - M dimensional constant def. by eq. [24] (Ns) - P _s pressure def. by eq. [17] (Nm^–2) - P ₀ average pressure in the liquid atr = 0 (Nm^–2) - P _R average pressure in the liquid atr = R (Nm^–2) - r ₁,r ₂ radii of curved liquid surface (m) - R platen radius (m) - R _w radius of wetted platen area (m) - S _x standard deviation ofx - t ₀ characteristic time def. by eq. [1] (s) - T temperature (°C) - V volume of the submerged part of the cone (m³) - v tangential velocity of liquid (ms^–1) - x distance (m) - angle (rad) - ₀ cone angle (rad) - calibration constant (Nm^–3) - shear rate (s^–1) - dynamic viscosity (mPa · s) - ₀ viscosity def. by eq. [1] (mPa · s) - contact angle (rad) - density (kgm^–3) - static surface tension (Nm^–1) - shear stress (Nm^–2) - ₀ yield stress def. by eq. [1] (Nm^–2) - _c, _p angular velocity (c = cone,p = plate) (s^–1) With 8 figures and 3 tables     

The L1-Stability of Boundary Layers for Scalar Viscous Conservation Laws

Let v=v(x) be a non-trivial bounded steady solution of a viscous scalar conservation law u _t+f(u) _x =u _xx on a half-line R⁺, with a Dirichlet boundary condition. The semi-group of this IBVP is known to be contractive for the distance d(u, u)u–u₁ induced by L ¹(R⁺). We prove here that v is asymptotically stable with respect to d: if u ₀–vL ¹, then u(t)–v₁0 as t+. When v is a constant, we show that this property holds if and only if f(v)0. These results complement our study of the Cauchy problem [2].