Power spectra of fluid velocities measured by laser Doppler velocimetry

The power spectrum and the correlation of the laser Doppler velocimeter velocity signal obtained by sampling and holding the velocity at each new Doppler burst are studied. Theory valid for low fluctuation intensity flows shows that the measured spectrum is filtered at the mean sample rate and that it contains a filtered white noise spectrum caused by the steps in the sample and hold signal. In the limit of high data density, the step noise vanishes and the sample and hold signal is statistically unbiased for any turbulence intensity.List of symbols A cross-section of the LDV measurement volume, m² - A empirical constant - B bandwidth of velocity spectrum, Hz - C concentration of particles that produce valid signals, number/m³ - d _m diameter of LDV measurement volume, m - f(₁, ₂ | u) probability density of t _i; and t _j given (t) for all t, Hz² - probability density for t _j-t_i, Hz - n (t, t) number of valid bursts in (t, t) = N + n - N (t, t) mean number of valid bursts in (t, t) - N _e mean number of particles in LDV measurement volume - valid signal arrival rate, Hz - mean valid signal arrival rate, Hz - R _uu time delayed autocorrelation of velocity, m²/s² - S _u power spectrum of velocity, m²/s²/Hz - t ₁, t ₂ times at which velocity is correlated, s - t _i, t _j arrival times of the bursts that immediately precede t ₁ and t ₂, respectively, s - t _ij t _j–t _i s - T averaging time for spectral estimator, s - T _u integral time scale of u (t), s - T Taylor's microscale for u (t), s - u velocity vector = U + u, m/s - u fluctuating component of velocity, m/s - U mean velocity, m/s - u _m sampled and held signal, m/s Greek symbols (t) noise signal, m/s - _m (t) sampled and held noise signal, m/s - bandwidth of spectral estimator window, radians/s - time between arrivals in pdf, s - Taylor's microscale of length = UT m - kinematic viscosity - ₁, ₂ arrival times in pdf, s - root mean square of noise signal, m/s - _u root mean square of u, m/s - delay time = t ₂ - t ₁ s - B duration of a Doppler burst, s - circular frequency, radians/s - _c low pass frequency of signal spectrum radians/s Other symbols ensemble average - conditional average - ^ estimate     

Spectral analysis of hydrodynamic tracer dispersion with an electrochemical measurement technique

We report tracer dispersion measurements in a capillary tube performed in the frequency domain using an electrochemical technique. Tracer Fe⁺⁺⁺ ions are produced by oxidizing Fe⁺⁺ ions at an emission anode; the inverse reaction allows to detect the tracer on a measurement electrode at the outlet of the sample. The amplitude and phase of the steady state signal detected at the outlet of the sample are measured as a function of the frequency of a sinusoidal concentration modulation induced at the inlet of the tube. Measurement results at two flow velocities are in agreement with predictions of the Taylor-Aris model.List of symbols A(f) output signal modulation amplitude at a modulation frequency f - a capillary tube radius - C _ox concentration of the solution ferricyanide ions - C _red concentration of the solution ferrocyanide ions - D _ox diffusion coefficient of tracer ions - D _m molecular diffusion coefficient - D longitudinal dispersion coefficient - E _e potential of emission electrode - E _d potential of detection electrode - F Faraday constant - J _m number of ions-g of tracer - I electrical current intensity on emitter electrode - I limiting current on detection electrode - k _x , k _y real and imaginary part of tracer concentration modulation wavevector - L total capillary tube length - Pe _L (= UL/D _m ) global Peclet number - S flow section - T _f characteristic exchange time with low velocity regions and dead zones - T ₀ mean transit time through the capillary - U mean fluid velocity - boundary layer thickness on detection electrode - phase shift between tracer concentration modulations at the inlet and the outlet of the sample - tracer concentration modulations spatial wavelength along the capillary tube - _a (= a ²/D _m ) characteristic diffusion time across the capillary section - tracer concentration modulation pulsation - _c cut-off frequency for concentration modulations at the capillary outlet     

Lattice gas simulation of Darcy flow with memory-free collisions

A lattice gas algorithm is proposed for the simulation of water flow in the unsaturated zone. Microscopic dynamics of a two-dimensional model system are defined. Up to four fluid particles occupy the sites of a square lattice. At each time step, the particles are sent to neighbouring sites according to probabilistic rules which depend on the permeability and the potential but not on the input velocities of the particles. On the macroscopic scale, the flow is described by a diffusion term and a Darcy term. Several extensions including higher dimension are discussed.List of Symbols c ⁽ⁿ⁾ constant in the definition of the rejection probabilityP forn = 1,2,3 particles at a site 0 c ⁽ⁿ⁾ 1 - D diffusion constant - D vertical extent of the system, measured in cells - E _i vector connecting a site to its neighbour in directioni - i direction of a nearest neighbour site,i = 1,..., 4 - j direction of a nearest neighbour site,j = 1,..., 4 - j mass transport (fluid flow),j = v - j ^x x-component of the flowj - k(x) spatial dependence of the permeability, user defined under the constraint 0 k 1 - k () the part of the permeability which depends on the degree of saturation (seek) - k ⁽ⁿ⁾ (x) effective permeability at a sitex that holdsn particles - L horizontal extent of the system, measured in cells - l _mac macroscopic length scale, e.g. one meter - l _mic microscopic length scale (one lattice constant) - m integer number of time steps - n (x) number of particles at the lattice sitex - N _A total number of particles on all A-sites - P probability for rejection of a randomly selected direction or set of directions - p arithmetic mean of the probability for a site to receive a particle from a particular neighbour (the average is taken over the four neighbours) - p _i ⁽ⁿ⁾ probability that one out ofn particles at a site is sent in directioni - p _ij ⁽²⁾ probability that the two particles at a site are sent in directionsi andj - t time - t _mac macroscopic time scale, e.g. one day - t _mic microscopic time scale (one time step) - v fluid velocity - x space vector, mostly two-dimensional:x = (x, y) - x horizontal component ofx - y vertical component ofx - quotient of microscopic and macroscopic time scales,t _mic /t _mac - quotient of microscopic and macroscopic length scales,l _mic /l _mac - _i p + _i is the probability that a particle is received from the neighbour atx +E _i - K(X, ) effective permeability,k =k(x)k () - correlation length - degree of saturation, used synonymously with density (homogeneous porosity) - ₀ value of a homogeneous particle density - ø(x) external potential (user defined), ø = ^gr + ^mat - ø(x) arithmetic mean of the external potential at the four sites surroundingx - ø _i external potential at the sitex +E _i - total potential, = ø + ^den - ^gr(x) gravitational potential - ^mat(x) matrix potential - ^den() density-dependent potential - _n potential depending on the occupation number - ⁽ⁿ⁾ (x) probability that sitex is occupied byn particles - ₀ ^{(n) (n)} in a system with homogeneous particle density - mac macroscopic - mic microscopic     

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     

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     

Time-dependent viscous flow in a straight channel

An infinite straight channel, filled with an incompressible viscous fluid, is closed at one end by a piston. This is set in motion with finite acceleration and then maintained at constant velocity until the flow pattern in the fluid reaches a steady state. The development of velocity profiles, stream lines, and streak lines is investigated by direct numerical solution of the complete Navier Stokes equations. It is found that the nonconvex velocity profiles found in previous work on steady-state problems appear from the beginning, and their development is studied. In the downstream region alternative methods can be used which serve as a check on the accuracy of the numerical procedures. The asymptotic behaviour downstream is studied in some detail.Nomenclature a acceleration of piston - f(t) nondimensional distance travelled by piston up to time t - 2l width of channel - p pressure (in units of u ₀ ² ) - R Reynolds number, lu ₀/ - t ₀ time during which piston is accelerated - u ₀ final velocity of piston - (u, v) (x, y) components of fluid velocity (relative to piston, in units of u ₀) - x distance measured downstream from piston (in units of l) - y distance from central axis of channel (in units of l) - vorticity - density of fluid - kinematic viscosity - stream function     

Estimation of three-phase relative permeabilities

Starting from the statistical structural model of Alemánet al. (1988), we have developed an alternative to Stone's (1970, 1973; Aziz and Settari, 1979) methods for estimating steady-state, three-phase relative permeabilities from two sets of steady-state, two-phase relative permeabilities. Our result reduces to Stone's (1970; Aziz and Settari, 1979) first method, when the steady-state, two-phase relative permeability of the intermediate-wetting phase with respect to either the wetting phase or the nonwetting phase is a linear function of the saturation of the intermediate-wetting phase. As the curvature of either of these relative permeability functions increases, the deviation of our result from Stone's (1970; Aziz and Settari, 1979) first method increases. Currently, there are no data available that are sufficiently complete to form the basis of a comparison between our result and either of the methods of Stone (1970, 1973; Aziz and Settari, 1979).Notation a free parameter in Equation (19) - B(m, n) Beta function defined by Equation (17) - F ^(w), F^(nw) defined by Equations (31) and (27), respectively - G ⁽ⁱ⁾ defined by Equations (37) and (39) - H ⁽ⁱ⁾ defined by Equations (38) and (40) - k ⁽ⁱ⁾ three-phase relative permeability fo phasei - k ^(i)* defined by Equations (34) through (36) - k ^(i,j) relative permeability to phasei during a two-phase flow with phasej, possibly in the presence of an immobile phase - k ^(i,j)* defined by analogy with Equations (41) and (42) - k ^(i,j)** defined by Equations (49), (50), (53), and (54) - k _max ⁽ⁱ⁾ defined by Equation (11) - k ₁₉₇₀ ^(iw) defined by Equation (10) - k ₁₉₇₃ ^(iw)* defined by Equation (58) - k ₁₉₇₃ ^(iw) defined by Equation (13) - L length and diameter of cylindrical averaging surfaceS - L _t length of an individual capillary tube enclosed byS - L _t ^* defined by Equation (19) - L _t,min length of pore whose radius isR _max - N total number of pores contained within the averaging surfaceS - p ₁ ⁽ⁱ⁾ ,p ₂ ⁽ⁱ⁾ pressure of phasei at entrance and exit of averaging surfaceS, respectively - p defined by Equation (21) - p _c ^(i,j) capillary pressure function - p _c ^(i,j)* defined by Equations (23), (29), and (32) - p ⁽ⁱ⁾ intrinsic average of pressure within phasei defined by Alemánet al. (1988) - R pore radius - R ^* defined by Equation (18) - R _max maximum pore radius that occurs withinS - s ⁽ⁱ⁾ local saturation of phasei - s ^(i)* defined by Equation (7) - s _min ⁽ⁱ⁾ minimum or immobile saturation of phasei - S averaging surface introduced in local volume averaging - V ⁽ⁱ⁾ volume of phasei occupying the pore space enclosed byS Greek Letters , parameters in the Beta distribution defined by Equation (16) - ^(w), ^(nw) functions of only the wetting phase saturation and the non-wetting phase saturation, respectively. Introduced in Equation (6) - ^(i,j) interfacial tension between phasesi andj - (x) Gamma function - defined by Equation (57) - , spherical coordinates in system centered upon the axis of the averaging surfaceS - _max maximum value of , 45 °, in view of assumption (9) - ^(i,j) contact angle between phasesi andj measured through the displacing phase - ^(w),^(nw) functions of only the wetting phase saturation and the non-wetting phase saturation, respectively. Introduced in Equation (12) Other gradient operator Amoco Production Company, PO Box 591 Tulsa, OK 74102, U.S.A.     

Two-phase brine mixtures in the geothermal context and the polymer flood model

Two-phase mixtures of hot brine and steam are important in geothermal reservoirs under exploitation. In a simple model, the flows are described by a parabolic equation for the pressure with a derivative coupling to a pair of wave equations for saturation and salt concentration. We show that the wave speed matrix for the hyperbolic part of the coupled system is formally identical to the corresponding matrix in the polymer flood model for oil recovery. For the class ofstrongly diffusive hot brine models, the identification is more than formal, so that the wave phenomena predicted for the polymer flood model will also be observed in geothermal reservoirs.Roman Symbols A,B coefficient matrices (5) - c(x,t) salt concentration (primary dependent variable) - C(p, s, c, q _t) wave speed matrix (6) - f source term (5) - g acceleration due to gravity (constant) - h _b(p, c) brine specific enthalpy - h _v(p) vapour specific enthalpy - j conservation flux (1) - k absolute permeability (constant) - k _b(s), k_v(s) relative permeabilities of the brine and vapour phases - K conductivity - p(x,t) pressure (primary dependent variable) - q volume flux (Darcy velocity) (3) - s(x,t) brine saturation (primary dependent variable) - t time (primary independent variable) - T=T _sat(p) saturation temperature - u _b(p, c) brine specific internal energy - u _m T rock matrix specific internal energy - u _v(p) vapour specific internal energy - U(x, t) shock velocity - x space (primary independent variable) Greek Symbols porosity (constant) - _b(p, c) brine dynamic viscosity - _v(p) vapour dynamic viscosity - (p, s, c) conservation density (1) - _b(p, c) brine density - _v(p) vapour density Suffixes b brine - m rock matrix - t total - v vapour - S salt - M mass - E energy     

A Discussion of the Effect of Tortuosity on the Capillary Imbibition in Porous Media

In the past decades, there was considerable controversy over the Lucas–Washburn (LW) equation widely applied in capillary imbibition kinetics. Many experimental results showed that the time exponent of the LW equation is less than 0.5. Based on the tortuous capillary model and fractal geometry, the effect of tortuosity on the capillary imbibition in wetting porous media is discussed in this article. The average height growth of wetting liquid in porous media driven by capillary force following the [`(L)] _s(t) ~ t^1/2D_T{\overline L _{\rm {s}}(t)\sim t^{1/{2D_{\rm {T}}}}} law is obtained (here D _T is the fractal dimension for tortuosity, which represents the heterogeneity of flow in porous media). The LW law turns out to be the special case when the straight capillary tube (D _T = 1) is assumed. The predictions by the present model for the time exponent for capillary imbibition in porous media are compared with available experimental data, and the present model can reproduce approximately the global trend of variation of the time exponent with porosity changing.     

The equations of miscible flow with negligible molecular diffusion

A set of equations with generalized permeability functions has been proposed by de la Cruz and Spanos, Whitaker, and Kalaydjian to describe three-dimensional immiscible two-phase flow. We have employed the zero interfacial tension limit of these equations to model two phase miscible flow with negligible molecular diffusion. A solution to these equations is found; we find the generalized permeabilities to depend upon two empirically determined functions of saturation which we denote asA andB. This solution is also used to analyze how dispersion arises in miscible flow; in particular we show that the dispersion evolves at a constant rate. In turn this permits us to predict and understand the asymmetry and long tailing in breakthrough curves, and the scale and fluid velocity dependence of the longitudinal dispersion coefficient. Finally, we illustrate how an experimental breakthrough curve can be used to infer the saturation dependence of the underlying functionsA andB.Roman Letters A a surface area; cross-sectional area of a slim tube or core - A _1s pore scale area of interface between solid and fluid 1 - A ₁₂ pore scale area of interface between fluid 1 and fluid 2 - A(S ₁) fluid flow weighting function defined by Equation (3.21) - a _i ,b _a ,c _a ,d _i macro scale parameters,i=1...2 (Section 3); polynomial coefficients,i=1...N (Section 7) - B(S ₁) fluid flow weighting function defined by Equation (3.16) - c _e effluent concentration - c _i mass concentration fluidi=1...2 - c _fi fractional mass concentration of fluidi=1...2 - D dispersion tensor - D _m mechanical dispersion tensor - D ₀ molecular dispersion tensor - D _L longitudinal dispersion coefficient - D _T transverse dispersion coefficient - D _L ⁰ defined by Equation (6.21) - F(c _f2) defined by Equation (5.17) - f ₁(S ₁) fractional flow - g acceleration of gravity - j ₂ deviation mass flux of fluid 2 - K permeability of porous medium - K _ij generalized relative permeability function,i=1...2,j=1...2 - K _ri relative permeability functions,i=1...2 - L length of a slim tube or core - M _i total mass of fluidi=1...2 in volumeV - N number of points used to generate numerical curves - n unit normal to a surface - P pressure - P _i pressure in fluidi=1...2 - P _c capillary pressure - P ₁₂ macroscopic capillary pressure parameter - P(x) normal distribution function - q Darcy velocity of total fluid - q _i Darcy velocity of fluidi=1...2 - S _i saturation of fluidi=1...2 - S _L a low saturation value forS ₁ - S _H a high saturation value forS ₁ - u average intersitial fluid velocity - u _S isosaturation velocity - V volume used for volume averaging - V(c _f2) function defined by Equation (6.28) - V _e effluent volume - V _f fluid volume - V _i volume of fluidi=1...2 (Section 2); injected fluid volume - V _p pore volume of a slim tube or core - v macro scale fluid velocity - v _i macro scale velocity of fluidi=1...2 - _q (S ₁) isosaturation speed - _g (S ₁) component of isosaturation velocity due to gravity - w(S _L,S _H,t) width of a displacement front - w(t) overall width of a displacement front Greek Letters static interfacial tension - _ME macroscopic dispersivity - divergence operator - porosity - _i fraction of pore space occupied by fluidi=1...2 - (S ₁) effective viscosity of the fluid - _i viscosity of fluidi=1...2 - ₁₂ macroscopic fluid viscosity coupling parameter - macro scale fluid density - _i density of fluid i=1...2 - _q effective gravitational fluid density     

Thermodynamic analysis of the one-dimensional heat conduction with inhomogeneities in the heat flux direction

Starting from the results of Li, Prigogine and others about the one-dimensional heat conduction with constant temperature boundary conditions, the aim of this paper is to study, according to the methods and the purposes of the generalized thermodynamics, the more general case of one-dimensional heat conduction in systems, whose conductivity is function of both temperature and the coordinate in the heat flux direction and presents a finite number of discontinuities.

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Thermodynamische Analyse für eindimensionale Wärmeströmung mit Ungleichartigkeiten in der Wärmeflußleitung

Zusammenfassung Ausgehend von den Ergebnissen von Li, Prigogine und anderen, für eindimensionale Wärmeströmung mit konstanten Temperaturen an den Grenzen, versucht diese Arbeit, gemäß den Methoden und den Zielen der verallgemeinerten Thermodynamik, den allgemeineren Fall der eindimensionalen Wärmeströmung mit Ungleichartigkeiten zu examinieren.

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Nomenclature c volumetric specific heat - G discriminating parameter [G (t) =k ₂ (t)t ² orG _i (t)=k_i(t) t², when the separation of the variables for thermal conductivity can be done, or in general: G(x, t)=k(x,t)t ²] - J heat transfer rate (generalized flux) [W] - J ₀ heat flux [W/m²] - k thermal conductivity - k ₁ component of thermal conductivity depending upon the coordinate in the heat flux direction - k ₂ component of thermal conductivity depending upon temperature - k _i component of the thermal conductivity of a homogeneous layer (i) depending upon temperature - L ₁,L ₂ extreme coordinates - Lip Lipschitz's function - P entropy production rate - (P(T))_min temperature distribution in a system corresponding to the minimum of entropy production rate - p ₁ thermokinetic potential - (P ₁ (T))_min temperature distribution in a system corresponding to the minimum of thermokinetic potential - P ₂ generalized force potential - Q differential form - dQ total differential form - ¯R set of all real numbers - S(x) area of the isothermal surface corresponding to the coordinatex - T system temperature distribution - t absolute temperature - ¯T set of all the possible system temperature distributions - x symbol of Cartesian product - x coordinate in the heat flux direction - X local generalized force - B,Y ₀,Y ₀ ^*,Z sets (defined in this paper) - function representing the time evolution of the temperature distribution in a system - time - ₀ reference time interval     

Limiting analytical solutions for complex saturated and unsaturated transport problems

Non-linear diffusion and velocity-dependent dispersion problems are under consideration. The necessary and sufficient conditions allowing the comparison of solutions to the two dimensional convection-dispersion equations with different coefficients are obtained. These conditions provide a framework within which solutions to the complex non-linear problems mentioned above can be estimated by solutions to the problems possessing analytical solvability.Nomenclature c(x, y, t) concentration of solute in solution,ML ^–3 - C(h)=d/dh moisture capacity function - D,D _ij hydrodynamic dispersion coefficient, a second order tensor,L ² T ^–1 - D _L longitudinal hydrodynamic dispersion coefficient,L ² T ^–1 - D _m molecular diffusion coefficient,L ² T ^–1 - D _T transverse hydrodynamic coefficient,L ² T ^–1 - G flow domain for the unsaturated flow problem - G _z , G _w flow domain and complex potential domain, respectively, for the hydrodynamic dispersion problem - h piezometric head,L - I _n given mass flux normal to the boundary,MLT ^–1 - k hydraulic conductivity,LT ^–1 - K(h) unsaturated hydraulic conductivity,LT ^–1 - L continuously differentiable function with respect to all arguments - m porosity - n(x,t) outer normal vector to the boundary - t time,T - V(x, y, t) seepage velocity vector withV=V,LT ^–1 - x Cartesian coordinate system - x horizontal coordinate,L - y vertical coordinate (elevation),L - (x),(x,t) given functions in initial and boundary conditions (3), (4) - ₁(,) angle between vectors ₁c andV - boundary of the flow domain - _L , _T longitudinal and transverse dispersivities, respectively,L - water mass density,ML ^–3 - v _i components of a unit vector in the direction of the outward normal to the boundary - =–kh velocity potential - =/m - stream function defined such thatw=+i is the complex potential - =/m     

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)     

On the use of a nickel-tube resonator for measuring the complex shear modulus of liquids in the kHz-range

Summary An apparatus for the measurement of liquid-shear impedance in the frequency range 4–200 kHz with the aid of a thin-walled Ni-tube resonator is described. A magnetostrictive mechanism is used for setting the tube into torsional oscillation. Real and imaginary parts of the liquid-shear impedance are found from the change in the 3 dB band-width of the resonance curve and the shift of the resonance frequency, respectively, when the tube is immersed from the air into the liquid. The amount of liquid required is 20 ml. The necessary theory is given and some preliminary results are presented.

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Zusammenfassung Es wird über einen Apparat zur Messung der Scherimpedanz von Flüssigkeiten im Frequenzbereich von 4 bis 200 kHz mit Hilfe eines Resonators berichtet. Der Resonator, ein Nickelrohr mit geringer Wandstärke, wird mittels eines magnetostriktiven Mechanismus in Torsionsschwingungen versetzt. Real- und Imaginärteil der Scherimpedanz der Flüssigkeit werden aus der Änderung der Bandbreite der Resonanzkurve und der Verschiebung der Resonanzfrequenz, wenn das Rohr aus der Luft in die Flüssigkeit eingetaucht wird, berechnet. Die benötigte Flüssigkeitsmenge beträgt 20 ml. Die zugehörige Theorie wird mitgeteilt, und einige vorläufige Meßergebnisse werden vorgestellt.

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a,b outer and inner radii of the tube - 3 dB band-width of resonance curve of the loaded and unloaded tube - c,c ₀ phase velocities of torsional waves in the loaded and unloaded tube - c ^*,c ₀ ^* complex velocities of propagation of torsional waves in the loaded and unloaded tube - c ^* =c ₁ + ic₂ - c _L ^* complex velocity of propagation of shear waves in the fluid - f frequency - f _n, f _n ⁰ resonance frequency ofn-th overtone of the loaded and unloaded tube - G ^* complex shear modulus of the tube material - G _L ^* complex shear modulus of the fluid - i - I ₀ moment of inertia per cm of unloaded tube - I effective moment of inertia per cm of loaded tube - K instrument constant - l length of the tube - m =b/a - M angular momentum - n positive integer - R _L, X_L real and imaginary parts of the characteristic plane shear impedance of the fluid - Z _L characteristic plane shear impedance of the fluid (=R _L + iX_L) - Z _cyl characteristic cylindrical shear impedance of the fluid - , ₀ damping factors of torsional waves in the loaded and unloaded tube - , ₀ phase factors of torsional waves in the loaded and unloaded tube - propagation constant (= + i) - loss angle - tan cos^–1 (/ _max) - viscosity - angular displacement - wave length - displacement amplitude - , ₀ densities of the fluid and the tube material - angular frequency (= 2f) - _n ⁰ = 2f _n ⁰ - _R reduced frequency With 4 figures and 2 tables     

Magnetohydrodynamic laminar source flow between two parallel disks

The steady axisymmetrical laminar source flow of an incompressible conducting fluid between two circular parallel disks in the presence of a transverse magnetic field is analytically investigated. A solution is obtained by expanding the velocity and the pressure distribution in terms of a power series of 1/r. Velocity, induced magnetic field, pressure and shear stress distributions are determined and compared with the case of the hydrodynamic solution. Pressure is found to be a function of both r and z in the general case and the flow is not parallel. At high magnetic fields, the velocity distribution degenerates to a uniform core surrounded by a boundary layer near the disks.Nomenclature C _f skin friction coefficient - H ₀ impressed magnetic field - H _r induced magnetic field in the radial direction, H _r /H ₀ - M Hartmann number, H ₀ t(/)^1/2 - P dimensionless static pressure, P*t ⁴/Q - P* static pressure - P ₀ reference dimensionless pressure - Q source discharge - R outer radius of disks - Rm magnetic Reynolds number, Q/t - Re Reynolds number, Q/t - 2t channel width - u dimensionless radial component of the velocity, u*t ²/Q - u* radial component of the velocity - w dimensionless axial component of the velocity, w*t ²/Q - w* axial component of the velocity - z, r dimensionless axial and radial directions, z*/t and r*/t, respectively - z*, r* axial and radial direction, respectively - magnetic permeability - coefficient of kinematic viscosity - density - electrical conductivity - ² LaPlacian operator in axisymmetrical cylindrical coordinates     

Transient heat transfer in the laminar thermal entry region of a pipe: an analytical solution

Attention is directed toward the problem of unsteady convective heat transfer to a fluid flowing inside a pipe in a laminar, fully developed fashion when suddenly, an ambient fluid outside the pipe undergoes a step change in temperature. For the fastest portion of the resultant transient, time domain I, an analytical solution of the governing partial differential thermal energy equation is effected via the Laplace transformation. From this solution, response functions are found for the pipe wall temperature, surface heat flux, and fluid bulk mean temperature as a function of non-dimensional time for a range of values of a parameter which characterizes the heat transfer between the ambient and the pipe.Comparison of results is made with a recent finite difference solution in the literature and with the standard quasi-steady type of analysis. It is found that the analytical solution presented herein extends and complements the finite difference solution and that the quasi-steady solution can be severely in error in this part of the transient.Nomenclature â c _p R/_wc_pwb Ratio of thermal energy storage capacity of fluid to wall material - b pipe wall thickness - C _n defined by equation (24) - c _p , c _pw specific heat capacity of fluid and pipe wall, respectively - D _n functions defined by equation (23) - erf, erfc error function and complimentary error function, respectively - F t/R ² Fourier number - g 1–2S - h local surface coefficient of heat transfer between inside of pipe wall and inside flowing fluid - i ⁿ erfc n ^th repeated integral of the error function - k thermal conductivity of the inside fluid - N h(2R)/k Nusselt number - p Laplace transform parameter - q _w local, instantaneous surface heat flux at inside of pipe wall - Q _w 2Rq _w /k(T _L –T _i ) nondimensional surface heat flux - R pipe inside radius - S UR/k - t time - T local instantaneous fluid temperature - T _B , T _L , T _i bulk mean, ambient, and initial, as well as inlet, temperature, respectively - u, u _m local and mass average, fluid velocity, respectively - U overall heat transmission coefficient between ambient fluid outside of pipe and inside pipe wall - X, Y x/R, y/R nondimensional space coordinates along, and radially inward from, the pipe wall, respectively - k/c _p thermal diffusivity of inside fluid - , _w mass density of inside fluid and wall, respectively - (T(x, y, t)–T _i )/(T _L –T _i ) - _w , _B wall, bulk mean value of , respectively     

A distributed electrical analog for transient diffusion-techniques of model fabrication and calibration

Simulation of transient two-dimensional diffusion by means of a distributed electrical analog is discussed. After present techniques of analog model construction and calibration are reviewed, an improved calibration technique is presented and a convenient method of analog fabrication, not previously reported, is described. The proposed new method allows complete access to any point on the analog model during a test. Frequency response and step response measurements indicate that an adequate simulation is provided by this particular type of analog model.

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Zusammenfassung Die Simulierung eines nichtstationären, zweidimensionalen Diffusionsvorganges mittels eines kontinuierlichen, elektro-thermischen Analogapparates wird besprochen. Eine Übersicht der gegenwärtigen Methoden für die Konstruktion und Kalibrierung von elektro-thermischen Analogmodellen wird gegeben. Ein verbessertes Verfahren für die Kalibrierung und eine handliche Fertigungsmethode von Analogmodellen, die noch nicht in der Literatur beschrieben wurden, werden dargestellt. Das vorgeschlagene neue Verfahren gestattet vollständigen Zugang zu jedem Punkt im Analogmodell während des Experiments. Meßbeobachtungen des periodischen Frequenzverhaltens und des nichtstationären Verhaltens zeigen, daß dieses spezielle Analogmodell eine ausreichende Simulierung des Diffusionsvorganges gestattet.

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Nomenclature A ₀ Potential amplitude atx=0 (see Fig. 5) - A _x Potential amplitude at locationx (see Fig. 5) - A _L Potential amplitude atx=L (see Fig. 5) - C Capacitance per unit area - j Frequency - L Characteristic length - R Resistance per square - t Time - x Coordinate - X Dimensionless distance,x/L - y Coordinate - Y Dimensionless distance,y/L - Diffusivity - _x Phase angle at locationx (see Fig. 5) - _L Phase angle atx=L (see Fig. 5) - _d Thickness of dielectric sheet - _r Thickness of resistance sheet - Dielectric constant - Resistivity - Dimensionless time,t/L ² - Potential - ₁ Reference potential - ₂ Reference potential - Dimensionless potential, ( – ₁)/( ₂ – ₁) - Angular frequency, 2f - Dimensionless frequency,L ²/ The investigation was performed while the first author was Visiting Associate Professor at Purdue University during 1967/68.     

On the derivation of the Forchheimer equation by means of the averaging theorem

The averaging theorem is applied to the microscopic momentum equation to obtain the macroscopic flow equation. By examining some very simple tube models of flow in porous media, it is demonstrated that the averaged microscopic inertial terms cannot lead to a meaningful representation of non-Darcian (Forchheimer) effects. These effects are shown to be due to microscopic inertial effects distorting the velocity and pressure fields, hence leading to changes in the area integrals that result from the averaging process. It is recommended that the non-Darcian flow regime be described by a Forchheimer number, not a Reynolds number, and that the Forchheimer coefficient be more closely examined as it may contain information on tortuosity.English a _i gravitational acceleration (m/s²) - A _fs interfacial area between the fluid and solid phases (m²) - Fo Forchheimer number - k permeability (m²) - k ₀ permeability at zero velocity (m²) - p thermodynamic pressure (Pa) - r _i coordinate on the microscopic scale (m) - Re Reynolds number - t time (s) - u _i ,u bulk velocity (m/s) - V volume (m³) - V _f fluid volume (m³) - w _i ,w microscopic velocity (m/s) - x _i ,x coordinate on the macroscopic scale (m) Greek the Forchheimer coefficient (1/m) - _ij extra (viscous) stress tensor (Pa) - _ij stress tensor (Pa) - Viscosity (Pa. s) - density (kg/m³) - porosity - a general variable Symbols < > phase average - < > ^f intrinsic phase average - the fluctuating part of a variable     

Study of the non-isothermal glass fibre drawing process

The glass fibre drawing process is simulated using a finite-element method. The two-dimensional energy and momentum equations are solved in their fully non-linear forms. These are coupled via the temperature-sensitive viscosity function. Both convective and radiative cooling mechanisms are taken into account on the filament surface. An effective emissivity of about 0.2 is found to be applicable to the drawing conditions in this paper. Even at this fairly low effective emissivity, radiation is found to be the dominant mode of cooling. The material thermal conductivity is found to have a small but definite influence on the filament profiles. Two-dimensionsl effects of the kinematic field are only significant up to a distance of about two orifice radii from the nozzle exit.The symbols in the square brackets show the dimensions of the parameters;M Mass,L Length,T Temperature,t Time. a Constant radius of a uniform cylinder [L] - A Local cross-sectional area of the filament [L ² ] - b _i Total tension applied on the filament boundary surface in thei ^th direction [ML/t ² ] - c Specific heat [L ² /t ² T] - D Local filament diameter [L] - f _i i ^th component of the body-force vector [L/t ² ] - h Surface convective heat transfer coefficient of the filament [M/t ³ T] - H Total equivalent heat transfer coefficient due to both convection and radiation [M/t ³ T] - k Thermal conductivity [ML/t ³ T] - M Mass-flow rate [M/t] - n Coordinate normal to the local filament surface [L] - Nu Local Nusselt number [–] - Average Nusselt number [–] - Q Rate of heat transfer [ML ² /t ³ ] - Volume-flow rate [ ³ /t] - r Radial coordinate [L] - R Local radius of the filament [L] - Re _x Reynolds number based on characteristic length scalex [–] - s Coordinate along the filament surface [L] - T Temperature [T] - u Radial component of the velocity [T/t] - U Free-stream velocity of a uniform flow [L/t] - v Local speed of a fluid particle defined by v = ;[L/t] - V Volume [L ³ ] - v _f Constant velocity of a filament with a uniform radius [L/t] - w Axial component of the velocity [L/t] - Average axial velocity of the fluid inside the tube [L/t] - z Axial coordinate, i.e. axial distance from the orifice exit [L] - Exponential coefficient of the viscosity function [T ^–1 ] - _ij Kronecker delta [–] - Emissivity or total hemispherical emissivity [–] - µ Viscosity [M/Lt] - µ ₀ Reference viscosity defined byµ = µ ₀ e ^–T [M/Lt] - Fluid density [M/L ³ ] - Stefan-Boltzmann constant [M/t ³ T ⁴ ] - Viscous dissipation function [M/Lt ³ ] - a Of air - a Based on the (constant) filament radius - C.L. Referred to the centre line of the filament - conv Referred to convection - D Dased on the diameter - f Referred to the filament local condition - g Referred to glass - i,j Species in multi-component systems - o Quantity evaluated at the orifice exit - R Based on the radius - rad Referred to radiation - s Evaluated at the filament surface - tot Referred to the total heat transfer from the filament surface - w Evaluated at the tube wall - Ambient condition - * Refers to non-dimensional quantities - — Indicating quantities averaged over the filament cross-section     

Entropy generation due to laser step input pulse heating

Laser heating of surfaces results in thermal expansion of the substrate material in the region irradiated by a laser beam. In this case, the thermodynamic irreversibility associated with the thermal process is involved with temperature and thermal stress fields. In the present study, entropy analysis is carried out to quantify the thermodynamic irreversibility pertinent to laser pulse heating process. The formulation of entropy generation due to temperature and stress fields is presented and entropy generation is simulated for steel substrate. It is found that the rapid rise of surface displacement in the early heating period results in high rate of entropy generation due to stress field in the surface region while entropy generation due to temperature field increases steadily with increasing depth from the surface. c ₁ Wave speed in the solid (m/s) - c ₁* Dimensionless wave speed - c ₂ Constant - C _p Specific heat (J/kg.K) - E Elastic modules (Pa) - I Power intensity (W/m²) - I ₁ Power intensity after surface reflection (W/m²) - I _o Laser peak power intensity (W/m²) - k Thermal conductivity (W/m.K) - r _f Reflection coefficient - s Laplace variable - S Entropy generation rate (W/m³K) - S* Dimensionless entropy generation rate - T(x, t) Temperature (K) - T*(x*, t*) Dimensionless temperature - Temperature in Laplace domain (K) - Dimensionless reference temperature - t Time (s) - t* Dimensionless time - U Displacement (m) - U* Dimensionless displacement (U) - W* _lost Dimensionless lost work - x Spatial coordinate (m) - x* Dimensionless distance (x) - Thermal diffusivity (m²/s) - _T Thermal expansion coefficient (1/K) - Poissons ratio - Absorption coefficient (1/m) - Density (kg/m³) - _x Thermal stress (Pa) - _x * Dimensionless thermal stress