Effect of surface layers on the constriction resistance of an isothermal spot. Part II: Analytical results for thick layers

Solution of a non-homogeneous Fredholm integral equation of the second kind [1], which forms the basis for the evaluation of the constriction resistance of an isothermal circular spot on a half-space covered with a surface layer of different material, is considered for the case when the ratio, , of layer thickness to spot radius is larger than unity. The kernel of the integral equation is expanded into an infinite series in ascending odd-powers of (1/) and an approximate kernel accurate to (^–(2M+1)) is derived therefrom by terminating the series after an arbitrary but finite number of terms, M. The approximate kernel is rearranged into a degenerate form and the integral equation with this approximate kernel is reduced to a system of M linear equations. An explicit analytical solution is obtained for a four-term approximation of the kernel and the resulting constriction resistance is shown to be accurate to (^–9). Solutions of lower orders of accuracy with respect to (1/) are deduced from the four-term solution. The analytical approximations are compared with very accurate numerical solutions and it is shown that the (^–9)-approximation predicts the constriction resistance exceedingly well for any 1 over a four orders of magnitude variation of layer-to-substrate conductivity ratio for both conducting and insulating layers. It is further shown that, for all practical purposes, an (^–3)-approximation gives results of adequate accuracy for > 2.     

Analysis of turbulent boundary layer interaction with an external supersonic flow behind a step

An integral method of analyzing turbulent flow behind plane and axisymmetric steps is proposed, which will permit calculation of the pressure distribution, the displacement thickness, the momentum-loss thickness, and the friction in the zone of boundary layer interaction with an external ideal flow. The characteristics of an incompressible turbulent equilibrium boundary layer are used to analyze the flow behind the step, and the parameters of the compressible boundary layer flow are connected with the parameters of the incompressible boundary layer flow by using the Cowles-Crocco transformation.A large number of theoretical and experimental papers devoted to this topic can be mentioned. Let us consider just two [1, 2], which are similar to the method proposed herein, wherein the parameter distribution of the flow of a plane nearby turbulent wake is analyzed. The flow behind the body in these papers is separated into a zone of isobaric flow and a zone of boundary layer interaction with an external ideal flow. The jet boundary layer in the interaction zone is analyzed by the method of integral relations.The flow behind plane and axisymmetric steps is analyzed on the basis of a scheme of boundary layer interaction with an external ideal supersonic stream. The results of the analysis by the method proposed are compared with known experimental data.Notation x, y longitudinal and transverse coordinates - X, Y transformed longitudinal and transverse coordinates - , ^*, ^** boundary layer thickness, displacement thickness, momentum-loss thickness of a boundary layer - , ^*, ^** layer thickness, displacement thickness, momentum-loss thickness of an incompressible boundary layer - u, velocity and density of a compressible boundary layer - U, velocity and density of the incompressible boundary layer - , stream function of the compressible and incompressible boundary layers - , dynamic coefficient of viscosity of the compressible and incompressible boundary layers - r₁ radius of the base part of an axisymmetric body - r radius - R transformed radius - M Mach number - friction stress - p pressure - a speed of sound - s enthalpy - v Prandtl-Mayer angle - P Prandtl number - P_t turbulent Prandtl number - r₂ radius of the base sting - b step depth - =0 for plane flow - =1 for axisymmetric flow Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 33–40, May–June, 1971.In conclusion, the authors are grateful to M. Ya. Yudelovich and E. N. Bondarev for useful comments and discussions.     

An analytical solution for the velocity and shear rate distribution of non-ideal Bingham fluids in concentric cylinder viscometers

An analytical solution is presented for the calculation of the flow field in a concentric cylinder viscometer of non-ideal Bingham-fluids, described by the Worrall-Tuliani rheological model. The obtained shear rate distribution is a function of the a priori unknown rheological parameters. It is shown that by applying an iterative procedure experimental data can be processed in order to obtain the proper shear rate correction and the four rheological parameters of the Worrall-Tuliani model as well as the yield surface radius. A comparison with Krieger's correction method is made. Rheometrical data for dense cohesive sediment suspensions have been reviewed in the light of this new method. For these suspensions velocity profiles over the gap are computed and the shear layer thicknesses were found to be comparable to visual observations. It can be concluded that at low rotation speeds the actually sheared layer is too narrow to fullfill the gap width requirement for granular suspensions and slip appears to be unavoidable, even when the material is sheared within itself. The only way to obtain meaningfull measurements in a concentric cylinder viscometer at low shear rates seems to be by increasing the radii of the viscometer. Some dimensioning criteria are presented.Notation A, B Integration constants - C Dimensionless rotation speed = µ/_y - c = 2µ - d = ₀ ²–2c_y - f() = (–₀)²+2c(–_y) - r Radius - r _b Bob radius - r _c Cup radius - r _y Yield radius - r ₀ Stationary surface radius - r Rotating Stationary radius - Y ₀ Shear rate parameter = /µ Greek letters Shear rate - = (r _y /r _b )²– 1 - µ Bingham viscosity - µ₀ Initial differential viscosity - µ µ₀-µ - Rotation speed - Angular velocity - Shear stress - _b Bob shear stress - _B Bingham stress - _y (True) yield stress - ₀ Stress parameter = _B +µ Y ₀ - _B - _y      

Viscoelastic properties of semidilute poly(methyl methacrylate) solutions

Viscoelastic properties were examined for semidilute solutions of poly(methyl methacrylate) (PMMA) and polystyrene (PS) in chlorinated biphenyl. The number of entanglement per molecule, N, was evaluated from the plateau modulus, G _N . Two time constants, _s and ₁, respectively, characterizing the glass-to-rubber transition and terminal flow regions, were evaluated from the complex modulus and the relaxation modulus. A time constant _k supposedly characterizing the shrink of an extended chain, was evaluated from the relaxation modulus at finite strains. The ratios ₁/ _s and _k / _s were determined solely by N for each polymer species. The ratio ₁/ _s was proportional to N ^4.5 and N ^3.5 for PMMA and PS solutions, respectively. The ratio _k / _s was approximately proportional to N ^2.0 in accord with the prediction of the tube model theory, for either of the polymers. However, the values for PMMA were about four times as large as those for PS. The result is contrary to the expectation from the tube model theory that the viscoelasticity of a polymeric system, with given molecular weight and concentration, is determined if two material constants _s and G _N are known.     

The eddy viscosity of turbulent equilibrium layers

Summary Similarity laws for the mean flow and scaling laws for the turbulent motion are used in an attempt to obtain a general expression for the eddy viscosity of equilibrium layers. It is found that =0.09 w ² /w*, in which w ² is a Reynolds stress representative for the region of overlap between the law of the wall and the velocity-defect law, while w* is the logarithmic slope of the mean velocity profile in that region. The distinction between w and w* is related to the strong inhomogeneity of the mean rate of strain in the inner layer. The results of the theory agree with experimental evidence obtained from transpired equilibrium layers.     

Thermodynamic activation functions of viscous flow of non-polar liquids

The thermodynamic activation function of viscous flow may be determined from the expression for the pre-exponential factor in the Eyring relationship (the viscosity coefficient), which is a function of density and relative permittivity, together with the thermal dependence of the viscosity coefficient. This method of determination is demonstrated for a series of n-alkanes C₆–C₂₀.List of symbols A, B, C parameters in empirical viscosity-temperature dependence - E molecular bond energies of liquid calculated from heat of evaporation in vacuum - E _v viscous flow activation energy - G activation Gibbs function of viscous flow - H _v heat of evaporation - H _vb heat of evaporation at boiling point - H viscous flow activation enthalpy - k Boltzmann constant - m weight of molecule - M relative molecular weight - N _A Avogadro constant - q total number of bonds in one mole of liquid - R gas constant - S _id entropy of one mole of compound in ideal gas state - S _sat entropy of one mole of compound in saturated vapour state - S activation entropy of viscous flow - T absolute temperature - T _c critical temperature - T _r reduced temperature - T _c critical volume - T _f free volume of liquid per molecule - V ₀ molar volume of molecules - V _m molar volume of liquid - relative permittivity - viscosity - density - _r reduced density     

High Reynolds number thick axisymmetric turbulent boundary layer measurements

Experimental measurements of the wall shear stress and momentum thickness for thick axisymmetric turbulent boundary layers are presented. The use of a full-scale towing tank allowed zero pressure gradient turbulent boundary layers to be developed on cylinders with diameters of 0.61, 0.89, and 2.5 mm and lengths ranging from 30 m to 150 m. Moderate to high Reynolds numbers (10⁴<Re <10⁵, 10⁸<Re _L<10⁹) are considered. The relationship between the mean wall shear stress, cylinder diameter, cylinder length, and speed was investigated, and the spatial growth of the momentum thickness was determined. The wall shear stress is significantly higher, and the spatial growth of the boundary layers is shown to be lower than for a comparable flat-plate case. The mean wall shear stress exhibits variations with length that are not seen in zero pressure gradient flat plate turbulent boundary layers. The ratio of outer to inner boundary layer length scales is found to vary linearly with Re , which is qualitatively similar to a flat plate turbulent boundary layer. The quantitative effect of a riblet cylindrical cross-sectional geometry scaled for drag reduction based on flat plate criteria was also measured. The flat plate criteria do not lead to drag reduction for this class of boundary layer shear flows.List of symbols a cylinder radius, mm - A _s total cylindrical surface area, m² - C _d tangential drag coefficient - D drag force, Newtons - boundary layer thickness, mm - * displacement thickness, mm - h riblet height, mm - L cylinder length, m - kinematic viscosity, m²/s - momentum thickness, mm - fluid density, kg/m³ - r radial coordinate, mm - Re _L Reynolds number based on length= - Re Reynolds number based on momentum thickness= - s riblet spacing, mm - _w mean wall shear stress, N/m² - u(r) mean streamwise velocity, m/s - u friction velocity= - U _o tow speed, m/s - x streamwise coordinate, m     

Layered petroleum reservoirs with cross-flows

Studied is a cylindrical reservoir consisting of three layers: a water-containing bottom layer, and two oil-containing top layers from whose upper layer oil is produced. For its solution, a corrected version of the finite Hankel transform for Neumann-Neumann boundary conditions was used together with numerical inversion of the Laplace transform. The effects of the water zone on the unsteady state pressure in the reservoir were evaluated at distances away from the well and at the well-bore itself. We found that the vertical pressure drop increases gradually with time and is more significant in the vicinity of the well-bore. For constant production and at any time t, smaller reservoirs experience higher pressure drops than larger ones. For the reservoir investigated, we found that for nondimensional time t _Dw <10⁴ the presence of a second fluid (water) has no effect on the pressure drop. Of all the formation and fluid properties investigated, porosity has the largest effect on pressure.Nomenclature c ₁, c ₂ Oil and water compressibilities, vol/vol/atm, vol/vol/psi - h Height of water zone from bottom of reservoir, cm, ft - h _D h/r _w , non-dimensional - H Height of reservoir, cm, ft - H _D H/r _w, non-dimensional - J ₀, J ₁ Bessel functions of the first kind, zero and first-order - K _r2, K _r1 Oil and water zones, horizontal permeabilities, darcies, md - K _z2, K _z1 Oil and water zones, vertical permeabilities, darcies, md - k ₁ n=1, 2, 3... - k ₂ n=1, 2, 3... - k _1,0 - k _2,0 - p(r, z, t) P(r, z, 0)–P(r, z, t), atm, psi - P(r, z, t) Pressure at any layer in the reservoir, atm, psi - P(r, z, 0) Initial pressure at any layer in the reservoir, atm, psi - P _D , non-dimensional - q Constant production rate of well, cc/sec, barrels/day - r Radius of reservoir, cm, ft - r _D r/r _w , non-dimensional - r _e Drainage radius, cm, ft - r _eD re/r _w , non-dimensional - r _w Well-bore radius, cm, ft - t Time, sec, hr - _Dw (k _r2 t)/( _{2 2} c ₂ r _w ² ), non-dimensional - Y ₀, Y ₁ Bessel functions of the second kind, zero and first-order - z Distance z measured vertically upward from bottom of reservoir, cm, ft - Z _D z/r _w , non-dimensional - z ₁ Height of the bottom of the producing layer, cm, ft - z _1D z ₁/r _w , non-dimensional - z ₂ Height of the top of the producing layer, cm, ft - z _2,D z ₂/r _w , non-dimensional - _n nth positive root of equation (18) - ₁ k _z1/k _r1, non-dimensional - ₂ k _z2/k _r2, non-dimensional - _{1 1 1} c ₁/k _r1, hydraulic diffusivity of layer I - _{2 2 2} c ₂/k _r2, hydraulic diffusivity of layers II and III - ₂, ₁ Viscosity of oil and water, cp, cp - _{n n} /r _w , l/cm, l/ft - ₂, ₁ Porosity of oil and water-filled zones, fraction - ( ₁/ ₂) (k _z2/k _z1), non-dimensional     

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     

Heat transport in an infinitely long Non-Newtonian liquid bridge due to marangoni convection

An infinitely long liquid bridge, observing a Non-Newtonian flow law (Ostwald-de-Waal and Bingham), is subjected under weightless condition to a linear temperature field, which yields due to the local change of the surface tension a Marangoni convection. The velocity- and temperature distribution inside the liquid bridge is determined analytically.

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Wärmetransport in einer unendlich langen nicht-newtonischen Flüssigkeitsbrücke bezüglich Marangoni-Konvektion

Zusammenfassung Eine unendlich lange Flüssigkeitsbrücke, die einem Nicht-Newtonischen Fließgesetz (Ostwald-de-Waal und Bingham) folgt, ist im schwerelosen Raum einem linearen Temperaturfeld ausgesetzt, das in der Flüssigkeitssäule eine Marangoni-Konvektion hervorruft. Es wird die Geschwindigkeits- und Temperaturverteilung in der Flüssigkeitsbrücke analytisch bestimmt.

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Nomenclature a radius of liquid column - k thermal conductivity - K material constant of Ostwald-de-Waal liquid - n material constant of Ostwald-de-Waal liquid - p liquid pressure - r radial coordinate - T temperature - ¯T₁ temperature gradient along z-axis - w velocity distribution of liquid in axial direction - z axial coordinate - x diffusivity - liquid surface tension - tangential stress - ₀ flow shear stress for Bingham plastic - dynamic viscosity of Newtonian liquid(k=, n=1)     

Flow stability of a Grad-model liquid layer on an inclined plane

A study is presented of the flow of stability of a Grad-model liquid layer [1, 2] flowing over an inclined plane under the influence of the gravity force.It is assumed that at every point of the considered material continuum, along with the conventional velocity vector v, there is defined an angular velocity vector , the internal moment stresses are negligibly small, and in the general case the force stress tensor _kj is asymmetric. The model is characterized by the usual Newtonian viscosity , the Newtonian rolling viscosity _r, and the relaxation time = J/4 _r, where J is a scalar constant of the medium with dimensions of moment of inertia per unit mass, is the density. It is assumed that the medium is incompressible, the coefficients , _r, J are constant [2].The exact solution of the equations of motion, corresponding to flow of a layer with a plane surface, coincides with the solution of the Navier-Stokes equations in the case of flow of a layer of Newtonian fluid. The equations for three-dimensional periodic disturbances differ considerably from the corresponding equations for the problem of the flow stability of a layer of a Newtonian medium. It is shown that the Squire theorem is valid for parallel flows of a Grad liquid.The flow stability of the layer with respect to long-wave disturbances is studied using the method of sequential approximations suggested in [3, 4].     

Thermal wave propagation within a medium with the inert heat source

A heat conduction equation of a new type is derived which takes into account the finite velocity of heat flux propagation and the relaxation of heat source capacity. The equation is solved for a semi-infinite body and a step change in temperature at the surface. The analysis shows that as the time increases the obtained solution moves from the solution of the classical hyperbolic equation without energy generation towards the solution of the classical hyperbolic equation with energy generation.

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Ausbreitung thermischer Wellen in einem Medium mit träger Wärmequelle

Zusammenfassung Es wird eine neuartige Wärmeleitungsgleichung abgeleitet, welche die endliche Geschwindigkeit der Ausbreitung des Wärmestromes und die Relaxation der Kapazität der Wärmequelle berücksichtigt. Die Gleichung wird für einen halbunendlichen Körper und eine schrittweise Temperaturänderung an der Oberfläche gelöst. Die Analyse zeigt, daß mit zunehmender Zeit sich die Lösung der klassischen hyperbolischen Gleichung ohne Wärmeerzeugung in eine solche mit ebenfalls klassischer hyperbolischer Gleichung mit Wärmeerzeugung wandelt.

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Nomenclature a thermal diffusivity,k/( c _p - c _p specific heat at constant pressure - C speed of heat propagation - C ₁,C ₂ constants - k thermal conductivity - q _v steady capacity of internal heat source - q _vd transient capacity of internal heat source - r ₁,r ₂ roots of characterisitc equation - t time - t _k relaxation time of heat flux - t _q relaxation time of internal heat source capacity - T temperature - T ₀ surface temperature - u() unit step function - x, y, z Cartesian coordinates - X dimensionless coordinate - , constant coefficients - dimensionless temperature - density - dimensionless time - _r-t_qt_k ratio of relaxation times - dimensionless steady capacity of internal heat source - _d dimensionless transient capacity of internal heat source     

Outer layer similarity in fully rough turbulent boundary layers

Turbulent boundary layer measurements were made on a flat plate covered with uniform spheres and also on the same surface with the addition of a finer-scale grit roughness. The measurements were carried out in a closed return water tunnel, over a momentum thickness Reynolds number (Re) range of 3,000–15,000, using a two-component, laser Doppler velocimeter (LDV). The results show that the mean profiles for all the surfaces collapse well in velocity defect form. Using the maximum peak to trough height (R_t) as the roughness length scale (k), the roughness functions (U⁺) for both surfaces collapse, indicating that roughness texture has no effect on U⁺. The Reynolds stresses for the two rough surfaces also show good agreement throughout the entire boundary layer and collapse with smooth wall results outside of the roughness sublayer. Quadrant analysis and the velocity triple products show changes in the rough wall boundary layers that are confined to y<8k_s, where k_s is the equivalent sand roughness height. The present results provide support for Townsends wall similarity hypothesis for uniform three-dimensional roughness. However, departures from wall similarity may be observed for rough surfaces where 5k_s is large compared to the thickness of the inner layer.     

Consecutive chemical reactions in an annular reactor with non-newtonian flow

The purpose of this paper is to analyze the homogeneous consecutive chemical reactions carried out in an annular reactor with non-Newtonian laminar flow. The fluids are assumed to be characterized by a Ostwald-de Waele (powerlaw) model and the reaction kinetics is considered of general order. Effects of flow pseudoplasticity, dimensionless reaction rate constants, order of reaction kinetics and ratio of inner to outer radii of reactor on the reactor performances are examined in detail.Nomenclature c _A concentration of reactant A, g.mole/cm³ - c _B concentration of reactant B, g.mole/cm³ - c _A0 inlet concentration of reactant A, g.mole/cm³ - C ₁ dimensionless concentration of A, c _A/c _A0 - C ₂ dimensionless concentration of B, c _B/c _A0 - C ₁ dimensionless bulk concentration of A - C ₂ dimensionless bulk concentration of B - D _A molecular diffusivity of A, cm²/sec - D _B molecular diffusivity of B, cm²/sec - k _A first reaction rate constant, (g.mole/cm³)^1–m /sec - k _B second reaction rate constant, (g.mole/cm³)^1–n /sec - K ₁ dimensionless first reaction rate constant, k _A r ₀ ² c _A0 ^m–1 /D _A - K ₂ dimensionless second reaction rate constant, k _B r ₀ ² c _A0 ^n–1 /D _B - K apparent viscosity, dyne(sec) ^m /cm² - m order of reaction kinetics - n order of reaction kinetics - P pressure, dyne/cm² - r radial coordinate, cm - r _i radius of inner tube, cm - r _max radius at maximum velocity, cm - r _o radius of outer tube, cm - R dimensionless radial coordinate, r/r _o - s reciprocal of rheological parameter for power-law model - u local velocity, cm/sec - u _max maximum velocity, cm/sec - u bulk velocity, cm/sec - U dimensionless velocity, u/u - z axial coordinate, cm - Z dimensionless axial coordinate, zD _A/r ₀ ² /u - ratio of molecular diffusivity, D _B/D _A - ratio of inner to outer radius of reactor, r _i/r _o - ratio of radius at maximum velocity to outer radius, r _max/r _o     

Die scheinbare Wärmedurchgangszahl von Plattenwärmeaustauschern

Zusammenfassung Um bei Platten-Wärmeaustauschern die übertragene Wärme richtig zu erhalten, muß das Produkt aus der Wärmedurchgangszahl k=1/(1/a ₁ + / + 1/a ₂), der gesamten Heizfläche A und der mittleren logarithmischen Temperatur differenz _m im allgemeinen noch mit einem Korrekturfaktor Z multipliziert werden, der von der Fließweganzahl n und der dimensionslosen Größe =(k A₀)/(M c) abhängt: Q=Z (kA_m) =k_sA_m (k_s=scheinbare Wärmedurchgangszahl). Unter der Voraussetzung, daß Gegenstrom herrscht und das Umwälzverhältnis U=M₁c₁/M₂c₂=1 ist, konnte diese Funktion Z=Z (n, )=k_s/k jetzt im Anschluß an eine frühere Rechnung für die reine Hintereinanderschaltung beliebig vieler Fließwege bestimmt werden. Die gefundenen Formeln weisen für gerade und ungerade Fließweganzahlen kleine Unterschiede auf. Doch verlaufen beide Kurvenscharen so gleichartig, daß sie sich gegenseitig ergänzend sehr gut ineinander fügen. — Eine wichtige Folgerung aus den Rechnungsergebnissen ist, daß die scheinbare Wärmedurchgangszahl k_s bei einem gegebenem Austauscher für jeden Massenstrom einen absoluten Höchstwert hat.

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The apparent overall heat transfer coefficient of plate heat exchangers

In order to get the correct value of the transfered heat Q of plate heat exchangers one must multiply the product of the overall heat transfer coefficient k=(1/a ₁ + / + 1/a ₂), the total heating area A and the logarithmic mean temperature difference _m with a correction factor Z: Q=Z · k · A · _m =k_s · A_m, where k_s means the so called apparent overall heat transfer coefficient. Z is, as was shown in a previous paper, a function of the numbers of flow channels and the dimensionless quantity =(k · A₀)/M · c. In this paper, assuming counter flow and the validity of the relation U=M₁c₁/M₂c₂=1, the correction factor Z is determined for the pure series con nexion of any desired number of flow channels. — An important conclusion drawn from our results is, that for a given heat exanger, k_s has an absolute maximum value for every mass flow rate.

     

A numerical study of supersonic turbulence

We report on simulations of nonstationary supersonic isotropic turbulent flow using the Piecewise-Parabolic Method on uniform grids of 2048² in two dimensions and 256³ in three dimensions. Intersecting shock waves initiate the transfer of energy from long to short wavelengths. Weak shocks survive for many acoustic times _ac. In two dimensions eddies merge over many _ac. In three dimensions vortex sheets break-up into short vortex filaments within two _ac. Entropy fluctuations, produced by strong shocks, are stretched into filaments over several _ac. These filaments persist and are mirrored in the density. We observe three temporal phases: onset, with the initial formation of shocks; quasi-supersonic, with strong density contrasts; and post-supersonic, with a slowly decaying root mean square Mach number. Compressive modes quickly establish a k ^–2 velocity power spectrum. In three dimensions solenoidal modes build-up during the supersonic phase delineating through time a Kolmogorov k ^–5/3 envelope and leaving a self-similarly decaying k ^–0.9 spectrum at lower wave numbers.This work was supported at the University of Minnesota by grants DE-FG02-87ER25035 from the Office of Energy Research of the Department of Energy, AST-8611404 from the National Science Foundation, and by equipment grants from Sun Microsystems, Gould Electronics, Seagate Technology, and the Air Force Office of Scientific Research (AFOSR-86-0239). Partial support for this work has also come from the Army Research Office Contract Number DAAL03-89-C-0038 funding the Army High Performance Computing Research Center (AHPCRC) at the University of Minnesota. At Nice this work was supported under DRET Contract 500-276, under the GdR CNRS-SPI Mécanique des Fluides, and under two special grants from the Observatoire de la Côte d'Azur.     

Response of an elastic Bingham fluid to oscillatory shear

The response of an elastic Bingham fluid to oscillatory strain has been modeled and compared with experiments on an oil-in-water emulsion. The newly developed model includes elastic solid deformation below the yield stress (or strain), and Newtonian flow above the yield stress. In sinusoidal oscillatory deformations at low strain amplitudes the stress response is sinusoidal and in phase with the strain. At large strain amplitudes, above the yield stress, the stress response is non-linear and is out of phase with strain because of the storage and release of elastic recoverable strain. In oscillatory deformation between parallel disks the non-uniform strain in the radial direction causes the location of the yield surface to move in-and-out during each oscillation. The radial location of the yield surface is calculated and the resulting torque on the stationary disk is determined. Torque waveforms are calculated for various strains and frequencies and compared to experiments on a model oil-in-water emulsion. Model parameters are evaluated independently: the elastic modulus of the emulsion is determined from data at low strains, the yield strain is determined from the phase shift between torque and strain, and the Bingham viscosity is determined from the frequency dependence of the torque at high strains. Using these parameters the torque waveforms are predicted quantitatively for all strains and frequencies. In accord with the model predictions the phase shift is found to depend on strain but to be independent of frequency.Notation A plate strain amplitude (parallel plates) - A _R plate strain amplitude at disk edge (parallel disks) - G elastic modulus - m torque (parallel disks) - M normalized torque (parallel disks) = 2m/R ³₀ - N ratio of viscous to elastic stresses (parallel plates) =µ A/ ₀ ratio of viscous to elastic stresses (parallel disks) =µ A _R/₀ - r normalized radial position (parallel disks) =r/R - r radial position (parallel disks) - R disk radius (parallel disks) - t normalized time = t — /2 - t time - _E elastic strain - _P plate strain (displacement of top plate or disk divided by distance between plates or disks) - _PR plate strain at disk edge (parallel disks) - ₀ yield strain - _E normalized elastic strain = _E/₀ - _P normalized plate strain = _P/₀ - _PR normalized plate strain at disk edge (parallel disks) = _PR/₀ - ₀ normalized plate strain amplitude (parallel plates) =A/ ₀ — normalized plate strain amplitude at disk edge (parallel disks) =A _R/₀ - phase shift between _P andT (parallel plates) — phase shift between _PR andM (parallel disks) - µ Bingham viscosity - stress - ₀ yield stress - T normalized stress =/ ₀ - frequency     

Turbulent boundary layer over a smooth wall with widely separated transverse square cavities

Laser-Doppler velocimetry (LDV) measurements and flow visualizations are used to study a turbulent boundary layer over a smooth wall with transverse square cavities at two values of the momentum thickness Reynolds number (R =400 and 1300). The cavities are spaced 20 element widths apart in the streamwise direction. Flow visualizations reveal a significant communication between the cavities and the overlying shear layer, with frequent inflows and ejections of fluid to and from cavities. There is evidence to suggest that quasi-streamwise near-wall vortices are responsible for the ejections of fluid out of the cavities. The wall shear stress, which is measured accurately, increases sharply immediately downstream of the cavity. This increase is followed by a sudden decrease and a slower return to the smooth wall value. Integration of the wall shear stress in the streamwise direction indicates that there is an increase in drag of 3.4% at bothR .Nomenclature C _f skin friction coefficient - C _fsw friction coefficient for a continuous smooth wall - k height of the cavity - k ⁺ ku / - R Reynolds number based on momentum thickness (U ₁ /v) - Rx Reynolds number based on streamwise distance (U ₁ x/) - s streamwise distance between two cavities - t time - t ⁺ tu ₂ / - U ₁ freestream velocity - mean velocity inx direction - u,v,w rms turbulent intensities inx,y andz directions - u local friction velocity - u _sw friction velocity for a continuous smooth wall - w width of the cavity - x streamwise co-ordinate measured from the downstream edge of the cavity - y co-ordinate normal to the wall - z spanwise co-ordinate - y ⁺ yu / - boundary layer thickness - ₀ boundary layer thickness near the upstream edge of the cavity - _i thickness of internal layer - kinematic viscosity of water - ⁺ zu / - momentum thickness     

Analysis of suddenly started laminar flow in the entrance region of a circular tube

Suddenly started laminar flow in the entrance region of a circular tube, with constant inlet velocity, is investigated analytically by using integral momentum approach. A closed form solution to the integral momentum equation is obtained by the method of characteristics to determine boundary layer thickness, entrance length, velocity profile, and pressure gradient.Nomenclature M(, , ) a function - N(, , ) a function - p pressure - p* p/1/2U ², dimensionless pressure - Q(, , ) a function - R radius of the tube - r radial distance - Re 2RU/, Reynolds number - t time - U inlet velocity, constant for all time, uniform over the cross section - u velocity in the boundary layer - u* u/U, dimensionless velocity - u ₁ velocity in the inviscid core - x axial distance - y distance perpendicular to the axis of the tube - y* y/R, dimensionless distance perpendicular to the axis - boundary layer thickness - * displacement thickness - /R, dimensionless boundary layer thickness - momentum thickness - absolute viscosity of the fluid - /, kinematic viscosity of the fluid - x/(R Re), dimensionless axial distance - density of the fluid - tU/(R Re), dimensionless time - _w wall shear stress