Investigation of plasma flux interaction with an axisymmetric magnetic field

The motion of the plasma flux in an axisymmetric magnetic field is examined for a magnetic Reynolds number R_m 10, magnetohydrodynamic interaction parameter N 1, and Hall parameter 1. Flux deceleration in a circular channel is studied at the entrance to the magnetic field because of the formation of azimuthal electrical current eddies.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 3, pp. 37–39, May–June, 1972.     

Similarity in the flow of a magnetized plasma around a plate and a cylinder

The article is devoted to a verification of the law of similarity in the flow of a rarefied magnetized plasma around a body under conditions which simulate the conditions of flow around artificial earth satellites in the ionosphere. The law of similarity for flow around plates and cylinders of different sizes (R₀/_i0.5–1, V₀/V_i1.5–2) is confirmed experimentally. It is shown that the patterns of flow around a plate and a cylinder coincide at small values of the parameter (=R₀/z_H). The effect of the potential of the bodies on their flow patterns is studied.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 2, pp. 3–12, March–April, 1973.The authors thank A. V. Gurevich for constant cooperation and valuable advice, and K. Tinchurin for help in the measurements.     

Nonlinear analysis of the forced response of a beam with three mode interaction

An analysis is presented for the primary resonance of a clamped-hinged beam, which occurs when the frequency of excitation is near one of the natural frequencies,_n . Three mode interaction (₂ 3₁ and _{3 1} + 2₂) is considered and its influence on the response is studied. The case of two mode interaction (₂ 3₁) is also considered to compare it with the case of three mode interaction. The straight beam experiencing mid-plane stretching is governed by a nonlinear partial differential equation. By using Galerkin's method the governing equation is reduced to a system of nonautonomous ordinary differential equations. The method of multiple scales is applied to solve the system. Steady-state responses and their stability are examined. Results of numerical investigations show that there exists no significant difference between both modal interactions' influences on the responses.     

Some remarks on differential equations of quadratic type

In this paper we study differential equations of the formx(t) + x(t)=f(x(t)), x(0)=x ₀ C HereC is a closed, bounded convex subset of a Banach spaceX,f(C) C, and it is often assumed thatf(x) is a quadratic map. We study the differential equation by using the general theory of nonexpansive maps and nonexpansive, non-linear semigroups, and we obtain sharp results in a number of cases of interest. We give a formula for the Lipschitz constant off: C C, and we derive a precise explicit formula for the Lipschitz constant whenf is quadratic,C is the unit simplex inR ⁿ, and thel ₁ norm is used. We give a new proof of a theorem about nonexpansive semigroups; and we show that if the Lipschitz constant off: CC is less than or equal to one, then lim_tf(x(t))–x(t)=0 and, if {x(t):t 0} is precompact, then lim_tx(t) exists. Iff¦C=L¦C, whereL is a bounded linear operator, we apply the nonlinear theory to prove that (under mild further conditions on C) lim_t f(x(t))–x(t)=0 and that lim_t x(t) exists if {x(t):t 0} is precompact. However, forn 3 we give examples of quadratic mapsf of the unit simplex ofR ⁿ into itself such that lim_t x(t) fails to exist for mostx ₀ C andx(t) may be periodic. Our theorems answer several questions recently raised by J. Herod in connection with so-called model Boltzmann equations.     

Infinite Memory Expectations in a Dynamic Model with Hyperbolic Demand

In this study we will research the dynamics shown by a cobweb-type model with hyperbolic demand, sigmoidal supply and with backward-looking mechanism of expectation creation, whereby the new state of the system is obtained from all the previous states observed by weighted arithmetical mean with exponentially decreasing weights in the region. The study herewith presented aims at confirming the existence of a stabilising effect due to the presence of infinite memory since, with all the other conditions begin the same, a memory rate > exists at which market equilibrium is a sink. An unstable system, therefore, becomes stable in the presence of sufficiently resistant expectations with infinite historical memory, although this transition to stability is accompanied by the onset of chaos. The resulting effect, therefore is one of qualitative destabilisation, that is with reference to the qualitative dynamic performance produced, associated to a quantitative stabilisation, that is to say with reference to the decreasing width of the invariant sets within which relevant dynamics occur.     

Pressure pulsations on the surface of a sphere in a subsonic stream

Many data are available on the drag C_x and the distribution of the static pressure over the surface of a sphere [1, 2]. However, there are virtually no data on pulsations of the pressure over the surface of a sphere. In the present paper, the results are given of an investigation of the total and spectral levels of the pressure pulsations at different points of the surface of a sphere at M 0.5–1.0 and Re (1.7–2.7)·.10⁶. It was found that the strongest pressure pulsations occur on the side in the region of the angle 90°. In this region at M 0.6–0.8 the relative total level o/q where q is the velocity head in the oncoming stream, reaches values 0.18–0.22. It was established that at M = 0.7–0.9 narrow-band maxima occur in the spectra of the pressure pulsations at frequencies Sh fD/V = 0.2–0.3. Data are also presented on the pulsations of the base pressure behind a spherical segment with short cylindrical and conical trailing edges.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 164–168, September–October, 1981.     

An application of nonsymmetric Lax-Milgram lemma to nonassociated plasticity

Based on a general assumption for plastic potential and yield surface, some properties of the nonassociated plasticity are studied, and the existence and uniqueness of the distribution of incremental stress and displacement for work-hardening materials are proved by using nonsymmetric Lax-Milgram lemma, when the work-hardening parameter A>F/Q/–F/, Q/.     

Spherical couette flow: Transitions and onset of chaos

The results are given of an experimental investigation of the transition to chaos and of the properties of the chaotic regimes in a wide range of Reynolds numbers: 460 Re 3200 7Re₀. Estimates of the probability dimension of the attractors and Lyapunov exponents and the exponential damping of the highest-frequency part of the spectrum indicate a deterministic nature of the chaos in the considered range of Re. It is noted that in one and the same range of Re values the route to chaos is not unique and can depend on the prehistory of the flow development; the simultaneous existence of chaotic and regular regimes for fixed values of the parameters is also noted.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 10–18, January–February, 1991.     

Reflection, transmission and excitation of SH-surface waves by a discontinuity in mass-loading on a semi-infinite elastic medium

The scattering of an SH-wave by a discontinuity in mass-loading on a semi-infinite elastic medium is investigated theoretically. The incident wave is either a plane body wave or a plane SH-surface wave. The problem is reduced to a Wiener-Hopf problem for the scattered wave. In this problem the amplitude spectral density of the particle displacement occurs as unknown function. Special attention is given to the numerical values of the surface wave contributions to the scattered field.Nomenclature x ₁, x ₂, x ₃ Cartesian coordinates - , polar coordinates in x ₁, x ₃-plane - volume mass density - surface mass density of mass-loading - , Lamé constants - U scalar wave function, defined by (2.1) - c _S speed of propagation of uniform shear waves in bulk medium (c _S=(/)^1/2) - angular frequency - t time - k _S wave number of uniform shear waves (k _S=/c _S) - reduced specific acoustic impedance of mass-loading (=k _S /) - k _m wave number of SH-surface wave (k _m=k _S(1+ ²)^1/2) - _1,2,3 partial differentiation with respect to x _1,2,3 - ⁱ angle between x ₃-axis and direction of propagation of incident body wave - ⁱ wave number in horizontal direction of incident body wave ( ⁱ=k _S sin( ⁱ)) - ⁱ wave number in vertical direction of incident body wave ( ⁱ=k _S cos( ⁱ)) - C _1,2 complex amplitude of surface wave excited by a body wave - R reflection factor of surface wave, when surface wave is incident - T transmission factor of surface wave, when surface wave is incident - S particle displacement vector The research presented in this paper has been carried out with partial financial support from the Delfts Hogeschoolfonds.     

Rheologische Messungen an Polypropylenschmelzen

Zusammenfassung An einer Anzahl von Polypropylenproben wurden die Grenzviskositäten [] und dieNewtonschen Schmelzviskositäten ₀ bei 190 °C gemessen; die meisten Proben wurden fraktioniert und die Molekulargewichtsverteilung festgestellt. Die Messung der Schmelzviskosität erfolgte in einem Extrusionsrheometer mit drei Düsen verschiedener Länge. Es wurden Endkorrekturen bestimmt und die Schubspannung entsprechend korrigiert. Für die-M-Beziehung ergibt sich: log ₀=3,69 log¯M _w –11,90.Proben mit sehrenger bzw. weiter Molekulargewichtsverteilung weichen von dieser Beziehung deutlich ab. Eine für die Molekulargewichtsverteilungsbreite besonders empfindliche Größe ist die Steigungskonstante (1/ vs.), die gleichzeitig ein Maß für den Grad des Nicht-Newtonschen Verhaltens der Schmelze darstellt, bzw. ₀ ; diese letzte Größe eignet sich ganz besonders zur Kennzeichnung der Verteilung sowohl für Proben mit besonders enger Verteilung (Fraktionen, Abbauprodukte, Luparen) als auch für den anderen Extremfall (Mischungen).Vorgetragen auf der Arbeitstagung der Sektion Rheologie des Vereins Österreichischer Chemiker am 28. September 1965 in Graz.Die Messungen und experimentellen Arbeiten wurden von FräuleinHeather Furley und HerrnHelmuth Lehner durchgeführt.     

Hydrodynamic instability of the contact zone between accelerated gases

An experimental apparatus for investigating Rayleigh-Taylor instability in the transition layer between two gases at accelerations g 10⁵g₀ (g₀ is the acceleration of gravity) is described. The constantly acting acceleration is communicated to the contact zone by the compression wave formed ahead of a flame front. The linear stage of development is investigated together with the effect of the thickness of the contact zone. It is shown that on the interval 0.3 < <- ( is the wavelength of the disturbance at the edge of the contact zone) the rate of growth of the perturbation amplitude 0.5₀, where ₀ is the amplitude growth rate for media separated by an interface with a discontinuous change of density.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 15–21, November–December, 1991.     

Shock heating of plasma in a rapidly increasing magnetic field

The experimental excitation of intense collisionless shock waves (M 5) with subsequent plasma compression by the magnetic field of a shock coil is described. A magnetic plug > 20 kOe is produced in 100 × 10^–9 sec by a current generator, a long line with 250-kV water insulation and a characteristic impedance of l At an initial deuterium-plasma density of 2 × 10¹⁴ cm^–3, shock waves with a front width of 20c/₀₃and a velocity of 5 × 10₇ cm/sec are recorded. The ion energy after the accumulation, determined from the neutron yield, turns out to be 2 ke V. Axial shock waves excited by the plasma flow beneath the shock coil are observed.Translated from Zhurnal Prikladnoi Mekhaniki i Teknicheskoi Fiziki, Vol. 11, No. 2, pp. 28–38, March–April, 1970.The authors thank G. I. Budker and R. Z. Sagdeev for formulating the problem, R. I. Soloukhin for interest in the study, and S. P. Shalamov for construction of the apparatus.     

Similar solutions and the asymptotics of filtration equations

In this paper we consider the asymptotic behavior of solutions of the quasilinear equation of filtration as t. We prove that similar solutions of the equation u _t = (u )_xx asymptotically represent solutions of the Cauchy problem for the full equation u _t = [(u)]_xx if (u) is close to u for small u.     

Modeling pressure-strain rate correlations in turbulence theory

On the basis of a spectral representation of the rapid part _ij,2 of the correlation tensor p(u _i /x _j ) using Cramer's theorem the inequality _ij,2(U _j /x _i )0 is obtained. As distinct from the realizability conditions, it can serve as a direct and very rigorous test of the adequacy of model expressions for _ij,2. In particular, it is shown that the best known of such expressions do not satisfy this test.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.2, pp. 42–46, March–April, 1992.     

Two-phase flow in heterogeneous porous media II: Numerical experiments for flow perpendicular to a stratified system

In this work, we make use of numerical experiments to explore our original theoretical analysis of two-phase flow in heterogeneous porous media (Quintard and Whitaker, 1988). The calculations were carried out with a two-region model of a stratified system, and the parameters were chosen be consistent with practical problems associated with groundwater flows and petroleum reservoir recovery processes. The comparison between theory (the large-scaled averaged equations) and experiment (numerical solution of the local volume averaged equations) has allowed us to identify conditions for which the quasi-static theory is acceptable and conditions for which a dynamic theory must be used. Byquasi-static we mean the following: (1) The local capillary pressure,everywhere in the averaging volume, can be set equal to the large-scale capillary pressure evaluated at the centroid of the averaging volume and (2) the large-scale capillary pressure is given by the difference between the large-scale pressures in the two immiscible phases, and is therefore independent of gravitational effects, flow effects and transient effects. Bydynamic, we simply mean a significant departure from the quasi-static condition, thus dynamic effects can be associated with gravitational effects, flow effects and transient effects. To be more precise about the quasi-static condition we need to refer to the relation between the local capillary pressure and the large-scale capillary pressure derived in Part I (Quintard and Whitaker, 1990). Herep _c ¦_y represents the local capillary pressure evaluated at a positiony relative to the centroid of the large-scale averaging volume, and {p _c }¦_x represents the large-scale capillary pressure evaluated at the centroid.In addition to{p _c } ^c being evaluated at the centroid, all averaged terms on the right-hand side of Equation (1) are evaluated at the centroid. We can now write the equations describing the quasi-static condition as , , This means that the fluids within an averaging volume are distributed according to the capillary pressure-saturation relationwith the capillary pressure held constant. It also means that the large-scale capillary pressure is devoid of any dynamic effects. Both of these conditions represent approximations (see Section 6 in Part I) and one of our main objectives in this paper is to learn something about the efficacy of these approximations. As a secondary objective we want to explore the influence of dynamic effects in terms of our original theory. In that development only the first four terms on the right hand side of Equation (1) appeared in the representation for the local capillary pressure. However, those terms will provide an indication of the influence of dynamic effects on the large-scale capillary pressure and the large-scale permeability tensor, and that information provides valuable guidance for future studies based on the theory presented in Part I.Roman Letters A scalar that maps {}*/t onto - A scalar that maps {}*/t onto - A interfacial area between the -region and the -region contained within, m² - A interfacial area between the -region and the -region contained within, m² - A interfacial area between the -region and the -region contained within, m² - a vector that maps ({}*/t) onto , m - a vector that maps ({}*/t) onto , m - b vector that maps ({p}– g) onto , m - b vector that maps ({p}– g) onto , m - B second order tensor that maps ({p}– g) onto , m² - B second order tensor that maps ({p}– g) onto , m² - c vector that maps ({}*/t) onto , m - c vector that maps ({}*/t) onto , m - C second order tensor that maps ({}*/t) onto , m² - C second order tensor that maps ({}*/t) onto . m² - D third order tensor that maps ( ) onto , m - D third order tensor that maps ( ) onto , m - D second order tensor that maps ( ) onto , m² - D second order tensor that maps ( ) onto , m² - E third order tensor that maps () onto , m - E third order tensor that maps () onto , m - E second order tensor that maps () onto - E second order tensor that maps () onto - p _c =(), capillary pressure relationship in the-region - p _c =(), capillary pressure relationship in the-region - g gravitational vector, m/s² - largest of either or - - - i unit base vector in thex-direction - I unit tensor - K local volume-averaged-phase permeability, m² - K local volume-averaged-phase permeability in the-region, m² - K local volume-averaged-phase permeability in the-region, m² - {K } large-scale intrinsic phase average permeability for the-phase, m² - K –{K }, large-scale spatial deviation for the-phase permeability, m² - K –{K }, large-scale spatial deviation for the-phase permeability in the-region, m² - K –{K }, large-scale spatial deviation for the-phase permeability in the-region, m² - K ^* large-scale permeability for the-phase, m² - L characteristic length associated with local volume-averaged quantities, m - characteristic length associated with large-scale averaged quantities, m - I _i i = 1, 2, 3, lattice vectors for a unit cell, m - l characteristic length associated with the-region, m - ; characteristic length associated with the-region, m - l _H characteristic length associated with a local heterogeneity, m - - n unit normal vector pointing from the-region toward the-region (n =–n ) - n unit normal vector pointing from the-region toward the-region (n =–n ) - p pressure in the-phase, N/m² - p local volume-averaged intrinsic phase average pressure in the-phase, N/m² - {p } large-scale intrinsic phase average pressure in the capillary region of the-phase, N/m² - p local volume-averaged intrinsic phase average pressure for the-phase in the-region, N/m² - p local volume-averaged intrinsic phase average pressure for the-phase in the-region, N/m² - p –{p }, large scale spatial deviation for the-phase pressure, N/m² - p –{p }, large scale spatial deviation for the-phase pressure in the-region, N/m² - p –{p }, large scale spatial deviation for the-phase pressure in the-region, N/m² - P _c p –{p }, capillary pressure, N/m² - {p_c}^c large-scale capillary pressure, N/m² - r ₀ radius of the local averaging volume, m - R ₀ radius of the large-scale averaging volume, m - r position vector, m - , m - S /, local volume-averaged saturation for the-phase - S ^* {}*{}*, large-scale average saturation for the-phaset time, s - t time, s - u , m - U , m² - v -phase velocity vector, m/s - v local volume-averaged phase average velocity for the-phase in the-region, m/s - v local volume-averaged phase average velocity for the-phase in the-region, m/s - {v } large-scale intrinsic phase average velocity for the-phase in the capillary region of the-phase, m/s - {v } large-scale phase average velocity for the-phase in the capillary region of the-phase, m/s - v –{v }, large-scale spatial deviation for the-phase velocity, m/s - v –{v }, large-scale spatial deviation for the-phase velocity in the-region, m/s - v –{v }, large-scale spatial deviation for the-phase velocity in the-region, m/s - V local averaging volume, m³ - V volume of the-phase in, m³ - V large-scale averaging volume, m³ - V capillary region for the-phase within, m³ - V capillary region for the-phase within, m³ - V _c intersection of m³ - V volume of the-region within, m³ - V volume of the-region within, m³ - V () capillary region for the-phase within the-region, m³ - V () capillary region for the-phase within the-region, m³ - V () , region in which the-phase is trapped at the irreducible saturation, m³ - y position vector relative to the centroid of the large-scale averaging volume, m Greek Letters local volume-averaged porosity - local volume-averaged volume fraction for the-phase - local volume-averaged volume fraction for the-phase in the-region - local volume-averaged volume fraction for the-phase in the-region - local volume-averaged volume fraction for the-phase in the-region (This is directly related to the irreducible saturation.) - {} large-scale intrinsic phase average volume fraction for the-phase - {} large-scale phase average volume fraction for the-phase - {}* large-scale spatial average volume fraction for the-phase - –{}, large-scale spatial deviation for the-phase volume fraction - –{}, large-scale spatial deviation for the-phase volume fraction in the-region - –{}, large-scale spatial deviation for the-phase volume fraction in the-region - a generic local volume-averaged quantity associated with the-phase - mass density of the-phase, kg/m³ - mass density of the-phase, kg/m³ - viscosity of the-phase, N s/m² - viscosity of the-phase, N s/m² - interfacial tension of the - phase system, N/m - , N/m - , volume fraction of the-phase capillary (active) region - , volume fraction of the-phase capillary (active) region - , volume fraction of the-region ( + =1) - , volume fraction of the-region ( + =1) - {p }– g, N/m³ - {p }– g, N/m³     

Experimental investigation of the distribution of the pressure on a triangular airfoil with blunt edges at small angles of attack

The article gives the results of an experimental investigation of the pressure on a triangular airfoil with blunt edges with a half aperture angle =45° under angles of attack =0,5,10°, with M=11.6 and Re1.5×10⁶. It has been observed that in a region adjacent to the axis of symmetry, at a certain distance from the apex, there is observed a considerable lowering of the pressure.Moscow. Translated from Izvestiya Akademii Nauk SSSR. Mekhanika Zhidkosti i Gaza, No. 2, pp. 166–169, March–April, 1972.     

Messung und Berechnung des örtlichen und mittleren Stoffübergangs an stumpf angeströmten Kreisscheiben bei unterschiedlicher Turbulenz

Zusammenfassung In einer vergleichenden Literaturübersicht zu Umströmung, Druck- bzw. Geschwindigkeitsverteilung sowie Wärme- und Stoffübergang werden bislang vorliegende Angaben zu stumpf angeströmten Kreisscheiben und -Zylindern zusammengefaßt. Wenige und zudem divergierende Ergebnisse zum Wärme- und Stoffübergang machen grundlegende experimentelle und theoretische Untersuchungen notwendig, wie sie in [l, 2] für die Eichung von Stoffübergangsmeßmethoden benötigt werden.Unter Einbeziehung des quer angeströmten Kreiszylinders wird gezeigt, daß genaue Angaben zum Wärme- und Stoffübergang bei zwei- wie dreidimensionalen Staupunktströmungen bislang nur über die Messung möglich sind. Über gemessene Geschwindigkeitsverteilungen berechnete Stoffübergangskoeffizienten werden von der Messung nicht bestätigt. Sie liegen gegenüber dem Experiment zu niedrig.Die Messungen wurden bei Turbulenzintensiten 0,8%Tu6%, Reynolds-Zahlen 2·10³5 und Scheibendurchmessern 9,3mmd73,7mm durchgeführt. Der Einfluß der Turbulenz auf den Stoffübergang im Staupunkt von Kreisscheiben kann nur näherungsweise über den Smith-Kuethe-Parameter Tu · Re/100 erfaßt werden. Differenzen zwischen Theorie nach Smith und Kuethe für Tu· Re<5 und Messung lassen sich über die Stabilitätstheorie erklären. Für eine genauere Erfassung des Stoffübergangs muß den unterschiedlichen Transportvorgängen über Turbulenzballen oder Längswirbeln sowie der Struktur der Turbulenz Rechnung getragen werden.

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Measuring and computation of local and average mass transfer to disks in cross flow at different turbulence intensities

The results of different publications concerning the flow, pressure and velocity distributions as well as the heat and mass transfer of disks and cylinders in cross flow are compared by a literature review. A few diverging results for heat and mass transfer require new experimental and theoretical approaches. The calibration of recently developed techniques for the determination of mass transfer rates as published in [1, 2] make these investigations expecially necessary. Including the cylinder in cross flow the authors show, that up to now exact data of heat and mass transfer for two- or three-dimensional flow at a forward stagnation region can be obtained by direct measuring only.Mass transfer coefficients computed from measured velocity distributions are not confirmed by the experimental results. Compared to the experimental data they are too low. The measurements were accomplished for turbulence intensities 0.8%Tu6%, Reynolds-numbers 2· 10³5 and disk diameters 9.3 mm d 73.7 mm.The influence of the turbulence on the stagnation point mass transfer of disks can be obtained only approximately by the Smith-Kuethe-parameter Tu·Re/100. Differences between theoretical results of Smith and Kuethe and experimental ones for Tu·Re/100<5 may be explained by the stability theory. For a more accurate determination of the mass transfer the different transport mechanisms of the scale of turbulence or the tree-dimensional flow pattern like Taylor-Görtler-vortices as well as the structure of the turbulence itself have to be regarded.

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Bezeichnungen a Temperaturleitkoeffizient - C_p Beiwert für den statischen Druck - C₂, C₃ Gradient der bezogenen Geschwindigkeit U⁺ am Staupunkt bei ebener, räumlicher Strömung - D_A Diffusionskoeffizient von Ammoniak in Luft - d Durchmesser - Fr=Sh/Re Frössling-Zahl für den Stoffübergang - Fr=Nu/Re Frössling-Zahl für den Wärmeübergang - Le=a/D_A Lewis-Zahl - L Bezugslänge - M Maschenweite von Turbulenzgittern - Nu=·d/ Nußbelt-Zahl - n Exponent der Prandtl-bzw. Schmidt-Zahl - Pr=/a Prandtl-Zahl - p Druck, Partialdruck - p_x statischer Druck an der Stelle x am Rand der Grenzschicht - Re=U · d/ Reynolds-Zahl - r Radius - r(x) radiale Distanz von der Rotationsachse eines Körpers zu einem Oberflächenelement - Sc=/D_A Schmidt-Zahl - Sh= _A ·d/D_A Sherwood-Zahl - T absolute Temperatur - Tu Turbulenzintensität (Turbulenzgrad) in% - U Strömungsgeschwindigkeit in x-Richtung am Rand der Grenzschicht - U Hauptströmungsgeschwindigkeit im freien Kanalquerschnitt - U⁺=U/U bezogene Geschwindigkeit in x-Richtung am Rand der Grenzschicht - u Strömungsgeschwindigkeit in x-Richtung, tangential zur Oberfläche - mittlere turbulente Geschwindig-keitsschwankung in x-Richtung - v Strömungsgeschwindigkeit in y-Richtung, normal zur Oberfläche - x Koordinate in Strömungsrichtung, tangential zur Oberfläche - x_G Entfernung vom Turbulenzgitter in Strömungsrichtung - x⁺ bezogene Länge x/r - y Koordinate normal zur Oberfläche - Wärmeübergangskoeffizient - _A Stoffübergangskoeffizient (Ammoniak) - dimensionsloses Temperaturgefälle an der Wand - Keilvariable - Wärmeleitkoeffizient - Wirbelweilenlänge (mm) - kinematische Zähigkeit - transformierte bezogene Länge - _A Partialdichte von Ammoniak Indices B mit Korrektur aufgrund der Verengung - m mittel - S bezogen auf die Kreisscheibe - Z bezogen auf den Kreiszylinder Herrn Prof. Dr.-Ing. habil. Josef Ipfelkofer zum 70. Geburtstag am 7. April 1977 gewidmet.     

The Hele-Shaw problem and area-preserving curve-shortening motions

We prove existence, locally in time, of a solution of the following Hele-Shaw problem: Given a simply connected curve contained in a smooth bounded domain, find the motion of the curve such that its normal velocity equals the jump of the normal derivatives of a function which is harmonic in the complement of the curve in and whose boundary value on the curve equals its curvature. We show that this motion is a curve-shortening motion which does not change the area of the region enclosed by the curve. In case is the whole plane ², we also show that if the initial curve is close to an equilibrium curve, i.e., to a circle, then there exists a global solution and the global solution tends to some circle exponentially fast as time tends to infinity.     

Digital phase-stepping holographic interferometry in measuring 2-D density fields

This paper presents a holographic interferometer technique for measuring transparent (2-D or quasi 2-D) density fields. To be able to study the realization of such a field at a certain moment of time, the field is frozen on a holographic plate. During the reconstruction of the density field from the hologram the length of the path traversed by the reconstruction beam is diminished in equal steps by applying a computer controlled voltage to a piezo-electric crystal that translates a mirror. Four phase-stepped interferograms resulting from this pathlength variation are digitized and serve as input to an algorithm for computing the phase surface. The method is illustrated by measuring the basically 2-D density field existing around a heated horizontal cylinder in free convection.List of symbols wavelength - x, y cartesian coordinate system - phase - phase step - K Gladstone-Dale constant - L width of 2-D density field - density - ₀ density at reference conditions - I ₀, I ₁, I ₂, I ₃ recorded interferograms - I _mod modulation intensity - I _bias bias intensity - N numerator determining tan { (x, y)} - D denominator determining tan { (x, y)} - dimensionless temperature - T temperature - T _c temperature of cylinder - T temperature of environment - p pressure - R gas constant     

Asymptotic theory for calculating heat fluxes near the corner of a body

A method is presented for calculating the distribution of the thermal fluxes, friction stresses, and pressure near the corner point of a body contour in whose vicinity the outer supersonic flow passes through an expansion wave. The method is based on a study of the asymptotic solutions of the Navier-Stokes equations as the Reynolds number R approaches infinity for the flow region in which the longitudinal gradients of the flow functions are large, invalidating conventional boundary layer theory. This problem was examined in part in [1], in which the distribution of the friction and pressure in a region with length on the order of a few thicknesses of the approaching boundary layer was obtained in the first approximation. The leading term of the expansion for the thermal flux to the surface of the body vanishes for a value of the Prandtl number equal to unity and for other values of the Prandtl number does not match directly with its value in the undisturbed boundary layer.The thermal-flux distribution is obtained for values of the Prandtl number approaching unity. For this purpose it was necessary to consider a more general double passage to the limit as 1 and 0 for a finite value of the parameter B=[(–1)/] [–ln ^1/4/]^1/4 characterizing the ratio of the effects of thermal conduction, viscous dissipation, and convection. The solution obtained previously [1] corresponds to the particular case B and therefore for actual values of R=10⁴–10⁶, ~ 0.7 overestimates considerably the effect of the dissipative term on heat transfer, although even in first approximation it describes the pressure distribution well and the friction distribution satisfactorily. For smooth matching of the solutions with the corresponding flow functions in the undisturbed boundary layer it was necessary to introduce a flow region with free interaction for the expansion flow. Equations and boundary conditions which describe the flow as a whole are presented. Examples are given of numerical calculations and comparison with experiment.