Combined effect of external turbulence and strong acceleration on boundary layer transition in a nozzle

The effect of external turbulence on the boundary layer flow in a convergent-divergent nozzle with a high expansion ratio has been studied numerically. The external turbulence was simulated by the turbulent viscosity _e, for which we used the partial differential equation that serves to close the system of boundary layer equations [1–4]. It was found that there exists a critical value _cr such that for all _e< _cr the flow regime in the nozzle remains perfectly laminar, whereas for _ecr a laminar-turbulent transition takes place and the boundary layer in the supersonic part of the nozzle becomes turbulent. For postcritical values of _e the heat fluxes and friction losses are approximately an order greater than for precritical values. With increase in the Reynolds number, determined from the parameters in the nozzle throat, the value of _cr decreases; as the coordinate of the onset of boundary layer formation is displaced in the direction of flow the value of _cr increases.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 34–37, January–February, 1906.The authors are grateful to L. V. Gogish for participating in the discussion of the results.     

Asymptotic analysis of equilibrium states for rotating turbulent flows

The equilibrium states of homogeneous turbulence simultaneously subjected to a mean velocity gradient and a rotation are examined by using asymptotic analysis. The present work is concerned with the asymptotic behavior of quantities such as the turbulent kinetic energy and its dissipation rate associated with the fixed point (/kS)=0, whereS is the shear rate. The classical form of the model transport equation for (Hanjalic and Launder, 1972) is used. The present analysis shows that, asymptotically, the turbulent kinetic energy (a) undergoes a power-law decay with time for (P/)<1, (b) is independent of time for (P/)=1, (c) undergoes a power-law growth with time for 1<(P/)<(C ₂–1), and (d) is represented by an exponential law versus time for (P/)=(C ₂–1)/(C ₁–1) and (/kS)>0 whereP is the production rate. For the commonly used second-order models the equilibrium solutions forP/,II, andIII (whereII andIII are respectively the second and third invariants of the anisotropy tensor) depend on the rotation number when (P/kS)=(/kS)=0. The variation of (P/kS) andII versusR given by the second-order model of Yakhot and Orzag are compared with results of Rapid Distortion Theory corrected for decay (Townsend, 1970).     

Heteroclinic orbits in singular systems: A unifying approach

We consider singularly perturbed systems , such that=f(, _o, 0). _o ^m , has a heteroclinic orbitu(t). We construct a bifurcation functionG(, ) such that the singular system has a heteroclinic orbit if and only ifG(, )=0 has a solution=(). We also apply this result to recover some theorems that have been proved using different approaches.     

An experimental study on effects of solid concentration and ζ-potential on pseudoplastic flow of suspensions

The pseudoplastic flow of suspensions, alumina or styrene-acrylamide copolymer particles in water or an aqueous solution of glycerin has been studied by the step-shear-rate method. The relation between the shear rate,D, and the shear stress,, in the step-shear-rate measurements, where the state of dispersion was considered to be constant, was expressed as = AD ^1/2 +CD. The effective solid volume fraction,ø _F, andA were dependent on the shear rate and expressed byø _F =aD ^b andA = D . Combining the above relations, the steady flow curve was expressed by = D ^{1/2 +} + ₀ (1 – a D ^b/0.74)^–1.85 D, where ₀ is the viscosity of the medium.With an increase in solid volume fraction and a decreases in the absolute value of the-potential, the flow behavior of the suspensions changed from Newtonian ( = = b = 0), slightly pseudoplastic ( = b = 0), pseudoplastic ( = 0) to a Bingham-like behavior.The change in viscosity of the medium had an effect on the change in the effective volume fraction.     

Über den Einfluß von Querkrümmung und Dichte bei inhomogenen turbulenten Freistrahlen

Zusammenfassung Zur Berechnung turbulenter Strömungen wird das k--Modell im Ansatz für die turbulente Scheinzähigkeit erweitert, so daß es den Querkrümmungs- und Dichteeinfluß auf den turbulenten Transportaustausch erfaßt. Die dabei zu bestimmenden Konstanten werden derart festgelegt, daß die bestmögliche Übereinstimmung zwischen Berechnung und Messung erzielt wird. Die numerische Integration der Grenzschichtgleichungen erfolgt unter Verwendung einer Transformation mit dem Differenzenverfahren vom Hermiteschen Typ. Das erweiterte Modell wird auf rotationssymmetrische Freistrahlen veränderlicher Dichte angewendet und zeigt Übereinstimmung zwischen Rechnung und Experiment.

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On the influence of transvers-curvature and density in inhomogeneous turbulent free jets

The prediction of turbulent flows based on the k- model is extended to include the influence of transverse-curvature and density on the turbulent transport mechanisms. The empirical constants involved are adjusted such that the best agreement between predictions and experimental results is obtained. Using a transformation the boundary layer equations are solved numerically by means of a finite difference method of Hermitian type. The extended model is applied to predict the axisymmetric jet with variable density. The results of the calculations are in agreement with measurements.

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Bezeichnungen Wirbelabsorptionskoeffizient - c_i Massenkonzentration der Komponente i - c_D, c_L, c, c₁, c₂ Konstanten des Turbulenzmodells - d Düsendurchmesser - E bezogene Dissipationsrate - f bezogene Stromfunktion - f Korrekturfunktion für die turbulente Scheinzähigkeit - j turbulenter Diffusionsstrom - k Turbulenzenergie - k_i Schrittweite in -Richtung - K dimensionslose Turbulenzenergie - L turbulentes Längenmaß - M_i Molmasse der Komponente i - p Druck - allgemeine Gaskonstante - r Querkoordinate - r_0,5 Halbwertsbreite der Geschwindigkeit - r_0,5c Halbwertsbreite der Konzentration - T Temperatur - u Geschwindigkeitskomponente in x-Richtung - v Geschwindigkeitskomponente in r-Richtung - x Längskoordinate - y allgemeine Funktion - Y_i diskreter Wert der Funktion y - Relaxationsfaktor für Iteration - turbulente Dissipationsrate - transformierte r-Koordinate - kinematische Zähigkeit - Exponent - transformierte x-Koordinate - Dichte - _k, Konstanten des Turbulenzmodells - Schubspannung - allgemeine Variable - Stromfunktion - Turbulente Transportgröße Indizes 0 Strahlanfang - m auf der Achse - r mit Berücksichtigung der Krümmung - t turbulent - mit Berücksichtigung der Dichte - im Unendlichen - Schwankungswert oder Ableitung einer Funktion - – Mittelwert Herrn Professor Dr.-Ing. R. Günther zum 70. Geburtstag gewidmet     

The Gibbs-Thompson relation within the gradient theory of phase transitions

This paper discusses the asymptotic behavior as 0+ of the chemical potentials associated with solutions of variational problems within the Van der Waals-Cahn-Hilliard theory of phase transitions in a fluid with free energy, per unit volume, given by ²¦¦²+ W(), where is the density. The main result is that is asymptotically equal to E/d+o(), with E the interfacial energy, per unit surface area, of the interface between phases, the (constant) sum of principal curvatures of the interface, and d the density jump across the interface. This result is in agreement with a formula conjectured by M. Gurtin and corresponds to the Gibbs-Thompson relation for surface tension, proved by G. Caginalp within the context of the phase field model of free boundaries arising from phase transitions.     

On the shape of a slender axisymmetric nonstationary cavity

Very few studies have been made of three-dimensional nonstationary cavitation flows. In [1, 2], differential equations were obtained for the shape of a nonstationary cavity by means of a method of sources and sinks distributed along the axis of thin axisymmetric body and the cavity. In the integro-differential equation obtained in the present paper, allowance is made for a number of additional terms, and this makes it possible to dispense with the requirement ¦ In ¦ 1 adopted in [1, 2]. The obtained equation is valid under the weaker restriction 1. In [3], the problem of determining the cavity shape is reduced to a system of integral equations. Examples of calculation of the cavity shape in accordance with the non-stationary equations of [1–3] are unknown. In [4], an equation is obtained for the shape of a thin axisymmetric nonstationary cavity on the basis of a semiempirical approach. In the present paper, an integro-differential equation for the shape of a thin axisymmetric nonstationary cavity is obtained to order ² ( is a small constant parameter which has the order of the transverse-to-longitudinal dimension ratio of the system consisting of the cavity-forming body, the cavity, and the closing body). A boundary-value problem is formulated and an analytic solution to the corresponding differential equation is obtained in the first approximation (to terms of order ² In ), A number of concrete examples is considered.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 38–47, July–August, 1980.I thank V. P. Karlikov and Yu. L. Yakimov for interesting discussions of the work.     

Discrete time semigroup transformations with random perturbations

Let (X, ) and (Y,C) be two measurable spaces withX being a linear space. A system is determined by two functionsf(X): X X and:X×YX, a (small) positive parameter and a homogeneous Markov chain {y _n } in (Y,C) which describes random perturbations. States of the system, say {x _n X, n=0, 1,}, are determined by the iteration relations:x _n+1 =f(x _n )+(x _n ,Y_n+1) forn0, wherex ₀ =x ₀ is given. Here we study the asymptotic behavior of the solutionx _n as 0 andn under various assumptions on the data. General results are applied to some problems in epidemics, genetics and demographics.Supported in part by NSF Grant DMS92-06677.Supported in part by NSF Grant DMS93-12255.     

Turbulent heat and mass transfer for Pr ≫ 1 and law of turbulent diffusion damping at a solid wall

The question of determining the law of damping for the turbulent diffusion coefficient at a smooth wall according to data on mass and heat transfer for Pr 1 is discussed. It is proved that the hypothesis that this law is determined by the first member of the Taylor series expansion of , namely, / = yⁿ ₊ is valid in the Pr range from 10³ to 10⁵ only under the assumption that the subsequent terms in the expansion have smaller coefficients. A statistical analysis of electrochemical and other experiments devoted to this problem shows that apparently n = 3, but singularities in the experimental results do not permit making a final conclusion. Requirements on a conclusive experiment are formulated on the basis of the analysis made.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 172–175, March–April, 1977.     

Effect of blowing on the resistance of a sphere in laminar flow of viscous fluid

Using the method of matching asymptotic expansions [1–3], a stationary field of velocities is obtained in the vicinity of a sphere for Reynolds numbers R and R computed from the blowing velocity and the fluid flow, respectively; they satisfy the relations R² 1 and R 1. It is also shown that for intensive blowing (R 1), the resistive force is considerably smaller than that found by using the Stokes formula. For weak blowing the results are in good agreement with the solution of Oseen.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskai Fiziki, No. 3, pp. 110–114, May–June, 1972.     

Modellvorstellung zur turbulenten Filmkondensation strömenden Dampfes

Zusammenfassung Der Wärmeübergang bei turbulenter Film kondensation strömenden Dampfes an einer waagerechten ebenen Platte wurde mit Hilfe der Analogie zwischen Impuls-und Wärmeaustausch untersucht. Zur Beschreibung des Impulsaustausches im Film wurde ein Vierbereichmodell vorgestellt. Nach diesem Modell wird die wellige Phasengrenze als starre rauhe Wand angesehen. Die Abhängigkeit einer Schubspannungs-Nusseltzahl von der Film-Reynoldszahl und Prandtlzahl wurde berechnet und dargestellt.

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A model for turbulent film condensation of flowing vapour

The heat transfer in turbulent film condensation of flowing vapour on a horizontal flat plate was investigated by means of the analogy between momentum and heat transfer. To describe the momentum transfer in the film a four-region model was presented. With this model the wavy interfacial surface is treated as a stiff rough wall. A shear Nusselt number has been calculated and represented as a function of film Reynolds number and Prandtl number.

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Formelzeichen a Temperaturleitkoeffizient - k Mischungswegkonstante - k _s äquivalente Sandkornrauhigkeit - Nu _x lokale Schubspannungs-Nusseltzahl,Nu _x=x_xv/uw - Pr Prandtlzahl,Pr=v/a - Pr _t turbulente Prandtlzahl,Pr _t =_m/_q - q Wärmestromdichte _q - R Wärmeübergangswiderstand - R_f Wärmeübergangswiderstand des Films - Re _F Reynoldszahl der Filmströmung - T Temperatur - U, V Geschwindigkeitskomponenten des Dampfes in waagerechter und senkrechter Richtung - u, Geschwindigkeitskomponenten des Kondensats in waagerechter und senkrechter Richtung - V Querschwankungsgeschwindigkeit des Kondensats und des Dampfes - u _/gtD Schubspannungsgeschwindigkeit an der Phasengrenze für die Dampfgrenzschicht, uD =(/)^1/2 - u _F Schubspannungsgeschwindigkeit an der Phasengrenze für den Kondensatfilm,u _F =(/)^1/2 - u _w Schubspannungsgeschwindigkeit an der Wand der Kühlplatte,u _w =(w/)^1/2 - y Wandabstand - x Wärmeübergangskoeffizient - gemittelte Kondensatfilmdicke - _s Dicke der zähen Schicht der Filmströmung an der welligen Phasengrenze - ₄ Dicke der zähen Schicht der Filmströmung an der gemittelten glatten Phasengrenze - Wärmeleitzahl - dynamische Viskosität - v kinematische Viskosität - Dichte - Oberflächenspannung - _w Wandschubspannung - Schubspannung an der Phasengrenzfläche - _m turbulente Impulsaustauschgröße - _q turbulente Wärmeaustauschgröße Indizes d Wert des Dampfes - w Wert an der Wand - x lokaler Wert inx - Wert an der Phasengrenze Stoffgrößen ohne Index gelten für das Kondensat     

An absence of a certain class of periodic solutions in the Navier-Stokes equations

In the case of the 3D Navier-Stokes equations, it is proved that there exists a constant>0 with the following property: Every time-periodic solution with a period smaller than is necessarily a stationary solution. An explicit formula for is also provided.     

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

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

Dirichlet problem of higher order elliptic equation with perturbation both in differential operator and in boundary

This paper is the continuation of article [7]. It gives further results about the asymptotic expression for the solution of higher order elliptic equation in the case of boundary perturbation combined with operator perturbation. When unperturbed problemA ₀ is not on the spectrum, the asymptotic expression for the solution of perturbation problemA may be expanded with respect to the small parameter . WhileA ₀ is on the spectrum, the asymptotic expression of the solution contains negative powers of the small parameter . The approximation of arbitrary order to the solution is considered and the recursive formula for the general term and the estimation of remainder term are given.     

Ein Rechenmodell für reagierende turbulente Scherströmungen im chemischen Nichtgleichgewicht

Zusammenfassung Zur Berechnung turbulenter Strömungen mit chemischen Reaktionen wird ein Schlie\ungsmodell 2. Ordnung vorgeschlagen, das auch die Berücksichtigung von chemischem Nichtgleichgewicht erlaubt. Es besteht aus dem k- Modell zur Schlie\ung der gemittelten Impulsgleichungen, einem thermodynamischen Modell zur Schlie\ung der Zustandsgieichungen und der Energiegleichung und einem Mischungsmodell, das den Grad der Vermischung der Komponenten beschreibt und damit die Schlie\ung der gemittelten Stofferhaltungsgleichungen erlaubt.Für die Behandlung der gemittelten reaktionskinetischen Quellterme der Stofferhaltungsgleichungen wird eine Modifikation des Reihenansatzes von Borghi [7] vorgeschlagen, der die AnnÄherung an den Gleichgewichtszustand besser beschreibt. Das Modell wird auf die von Batt [11, 12] vermessene ebene Scherströmung angewendet und zeigt gute übereinstimmung zwischen Rechnung und Experiment.

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A method for predicting reacting turbulent shear flows with chemical non-equilibriums

A prediction model based on second order closure for the calculation of reacting turbulent flows including chemical non-equilibrium is put forward. It consists of the k- model for the closure of the mean momentum equations, the thermodynamic model for the closure of the mean equations of state and the mean energy-equation and the mixing model that describes the degree of mixedness of the components and consequently leads to the closure of the mean mass transport equations. A modification of the series truncation method of Borghi [7] is suggested that improves the representation of the mean chemical source terms as equilibrium is approached. The results of the calculations are compared with the measurements of Batt [11, 12] in a turbulent plane shear layer with and without reaction and show good agreement.

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Bezeichnungen C_i,j Turbulenzkonstante - D Divergenz der Geschwindigkeit (m/s) - D_b, D_f Vorexponentielle Faktoren im Arrheniusansatz (m³/kmol) - E_b, E_f Exponenten im Arrheniusansatz - F_b, F_f Aktivierungsenergie (K) - H_i Bezugsenthalpie der i-ten Komponente (kJ/kg) - K_i,j Turbulenzkonstante - KON Anzahl der im System vorkommenden Komponenten - M_i Molmasse (kg/kmol) - NR Anzahl der im System vorkommenden Elementargleichungen - Q_ij Konzentrationskorrelation zwischen der i-ten und der j-ten Komponente (kg²/m⁶) - R Restglied der Taylorentwicklung für die Geschwindigkeitskoeffizienten - T Temperatur (K) - a_b, a_f,b_b, b_f Koeffizienten der Taylorreihenentwicklung der Geschwindigkeitskoeffizienten - c_i Massenkonzentration der i-ten Komponente - c₁ Turbulenzkonstante der -Gleichung - c₂ Turbulenzkonstante der -Gleichung - c₃ Turbulenzkonstante der -Gleichung - c_pi spezifische WÄrmekapazitÄt der i-ten Komponente (kJ/kg/K) - h_i spezifische Enthalpie des Gesamtsystems (kJ/kg/K) - k Turbulenzenergie (m²/s²) - k_fj, k_bj Geschwindigkeit der j-ten Elementarreaktion (m³/kmol) - p Druck (N/m²) - v, (u,v,w) Geschwindigkeit (m/s) - x, (x,y,z) Raumkoordinate (m) - Molekularer Diffusionskoeffizient (m²/s) - Allgemeine Gaskonstante (kJ/kmol K) - w_i Quellterm der Konzentrationserhaltungsgleichungen (kg/m³/s) - _ijij stöchiometrische Koeffizienten deriten Komponente in der j-ten Elementarreaktion (VorwÄrtsreaktion ,RückwÄrtsreaktion) - _ij Kronecker-Symbol - Turbulente Dissipation (m²/s³) - Molekulare WÄrmeleitfÄhigkeit (KJ/m/s/K) - Dynamische ZÄhigkeit (kg/m/s) - _t Turbulente dynamische ZÄhigkeit(kg/ m/s) - kinematische ZÄhigkeit (m²/s) - _t Turbulente kinematische ZÄhigkeit (m²/s) - Turbulente Prandtlzahl Mittelwert und Schwankungsgrö\en Unbewichteter Mittelwert - · Unbewichtete Schwankungsgrö\e - Bewichteter (Favre-) Mittelwert - · Bewichtete (Favre-)Schwankungsgrö\e Indizes i,j Komponentenindex - , Summations-oder Vektorindex (,=1,2,3) - b RückwÄrtsreaktion - f VorwÄrtsreaktion     

Effect of free-stream turbulence on surface friction in a turbulent boundary layer

The combined effect of the turbulence intensity , the turbulence scaleL, and the Reynolds number Re^** on the surface friction coefficientc _f in a turbulent boundary layer is studied. The dependence of the relative friction increment on the equivalent turbulence level _cq, which takes into account the simultaneous variation in ,L and Re^**, is determined. The threshold value _cq ^* below which the value ofc _f does not depend on _cq is found.Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 2, pp. 65–75, March–April, 1995.     

Viscous properties of soil

Dynamic problems connected with the wave propagation in soils not saturated with water and with wave interaction with obstacles and structural elements at the present time are solved on the basis of models in which plastic but not viscous soil properties are taken into account [1–5]. An analysis of experimental data and their comparison with the calculated results [4, 5] confirms that it is permissible to apply the model of an elasticplastic medium to soils in problems concerning the interaction of waves and structures. At the same time plane-wave damping in soils takes place more intensively than would follow from calculations carried out on the basis of models of an elastic-plastic medium. For example, if in a section of a poured sandy soil, taken as the initial section, the maximum stress in the wave is _m=ll kgf/cm² and its duration is 6=8 msec, then at a distance of 25 cm the calculations give _m=9.5 kgf/cm², while the experiment gives _m= 5 kgf/cm². If in the initial section _m= 20 kgf/cm² and =6 msec, then at a distance of 35 cm the calculation gives _m= l7 kgf/cm², while the experiment gives _m= 9 kgf/cm². In the calculations it was assumed that unloading takes place with a constant strain. This deviation of the calculated results from the experiment can be explained, in the first place, by the dependence of the () on the strain rate , which is not taken into account in the model of an elastic-plastic medium. The viscous properties cause additional energy losses and a more intensive damping of the waves. Experimentally the dependence of the () curves on the strain rate has been investigated for many soils [5–8]. The dynamic load on the test sample was produced by a body falling from a height or being accelerated by some method. Below we present test results of viscous soil properties when the test sample is compressed by an air shock wave. Compression curves and approximate numerical values of the coefficient of viscosity are obtained.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, Vol. 9, No. 4, pp. 68–71, July–August, 1968.The author thanks A. I. Shishikin for his participation in the experiments.     

Attractors for the Two-Dimensional Navier–Stokes-α Model: An α-Dependence Study

The two-dimensional Navier–Stokes- model is considered on the torus and on the sphere. Upper and lower bounds for the dimension of the global attractors are given. The dependence of the dimension of the global attractors on is studied. Special attention is given for the limiting cases when 0, that is, when the Navier–Stokes- model tends to the Navier–Stokes equations, and to the case when .     

Application of dispersion theory to time domain reflectometry in soils

With time domain reflectometry (TDR) two dispersive parameters, the dielectric constant, , and the electrical conductivity, can be measured. Both parameters are nonlinear functions of the volume fractions in soil. Because the volume function of water ( _w) can change widely in the same soil, empirical equations have been derived to describe these relations. In this paper, a theoretical model is proposed based upon the theory of dispersive behaviour. This is compared with the empirical equations. The agreement between the empirical and theoretical aproaches was highly significant: the ( _w) relation of Topp et al. had a coefficient of determination r ² = 0.996 and the (_u) relation of Smith and Tice, for the unfrozen water content, _u, at temperatures below 0°C, had an r ² = 0.997. To obtain ( _w) relations, calibration measurements were performed on two soils: Caledon sand and Guelph silt loam. For both soils, an r ² = 0.983 was obtained between the theoretical model and the measured values. The correct relations are especially important at low water contents, where the interaction between water molecules and soil particles is strong.     

The problems of nonlinear bending for orthotropic rectangular plate with four clamped edges

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