Exact solutions and mathematical properties of boundary‐value problems for dynamic‐diffusion boundary layers

The paper studies boundaryvalue problems for dynamicdiffusion boundary layers occurring near a vertical wall at high Schmidt numbers and for dynamic boundary layers whose inner edge is adjacent to the dynamicdiffusion layers. Exact solutions for boundary layers at small and large times are derived. The wellposedness of the boundaryvalue problem for a steady dynamicdiffusion layer is studied.     

Heat transfer in magnetohydrodynamic flow near a stagnation point

Summary Steady, axisymmetric, magnetohydrodynamic flow with a stagnation point on an infinite plane wall is considered with a magnetic field applied normal to the wall. Solutions are obtained in the form of series for the velocity, magnetic field and temperature when the magnetic field parameter () and the ratio of viscosity to magnetic diffusivity () are small. The case=O(1) is considered briefly when solutions which Meyer³⁾ obtained by physical order-of-magnitude arguments are derived mathematically as expansions in. Some remarks are made on the consistency of extending the results to flow within the boundary layer near the nose of a bluff body.     

Analytical Modeling of Nonaqueous Phase Liquid Dissolution with Green's Functions

Equilibrium and bicontinuum nonequilibrium formulations of the advection–dispersion equation (ADE) have been widely used to describe subsurface solute transport. The Green's Function Method (GFM) is particularly attractive to solve the ADE because of its flexibility to deal with arbitrary initial and boundary conditions, and its relative simplicity to formulate solutions for multidimensional problems. The Green's functions that are presented can be used for a wide range of problems involving equilibrium and nonequilibrium transport in semiinfinite and infinite media. The GFM is applied to analytically model multidimensional transport from persistent solute sources typical of nonaqueous phase liquids (NAPLs). Specific solutions are derived for transport from a rectangular source (parallel to the flow direction) of persistent contamination using first, second, or thirdtype boundary or source input conditions. Away from the source, the first and thirdtype condition cannot be expected to represent the exact surface condition. The secondtype condition has the disadvantage that the diffusive flux from the source needs to be specified a priori. Near the source, the thirdtype condition appears most suitable to model NAPL dissolution into the medium. The solute flux from the pool, and hence the concentration in the medium, depends strongly on the mass transfer coefficient. For all conditions, the concentration profiles indicate that nonequilibrium conditions tend to reduce the maximum solute concentration and the total amount of solute that enters the porous medium from the source. On the other hand, during nonequilibrium transport the solute may spread over a larger area of the medium compared to equilibrium transport.     

Near-wall turbulence closure modeling without “damping functions”

An elliptic relaxation model is proposed for the strongly inhomogeneous region near the wall in wall-bounded turbulent shear flow. This model enables the correct kinematic boundary condition to be imposed on the normal component of turbulent intensity. Hence, wall blocking is represented. Means for enforcing the correct boundary conditions on the other components of intensity and on the k — equations are discussed. The present model agrees quite well with direct numerical simulation (DNS) data. The virtue of the present approach is that arbitrary damping functions are not required.     

Transient laminar free convection in horizontal cylinders

A numerical study of laminar natural convection inside uniformly heated, partially or fully filled horizontal cylinders is made. A coordinate transformation which simplifies the discretization of the equations of motion and energy is utilized. The resulting system of partial differential equations with their boundary conditions is solved using central differences for various Prandtl and Grashof numbers for two different grid sizes. The flow in completely filled cylinders for which experimental data are available is predicted. Close agreement between steady-state predictions and experiments is obtained for temperature and velocity profiles as well as for the streamline contours and isotherms. The technique is further demonstrated by solving the transient natural convection flow inside a partially filled horizontal cylinder with an adiabatic free surface and subjected to uniform wall heating.

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Laminare freie Konvektion in horizontalen Zylindern

Zusammenfassung Es wurde eine numerische Berechnung der laminaren, freien Konvektion in gleichmäßig beheizten, teilweise oder ganz gefüllten, horizontalen Zylindern durchgeführt. Dabei wird eine Koordinatentransformation benützt, welche die Diskretisierung der Bewegungs- und der Energiegleichung vereinfacht. Das so resultierende System von partiellen Differentialgleichungen wird, zusammen mit seinen Randbedingungen, unter Verwendung einer Differenzenmethode für verschiedene Prandtl und Grashof-Zahlen sowie für zwei verschiedene Gittergrößen gelöst. Für den vollständig gefüllten Zylinder, für den experimentelle Daten verfügbar sind, wird die Strömung vorhergesagt. Dabei wird für stationäre Zustände gute Übereinstimmung zwischen Rechnung und Experiment erzielt. Dies gilt sowohl für den Verlauf der Stromlinien als auch für den der Isothermen. Das Verfahren wird weiterhin am Beispiel der Berechnung instationärer, freier Konvektion in einem partiell gefüllten, horizontalen Zylinder demonstriert, wobei eine adiabate, freie Oberfläche und gleichmäßige Beheizung der Wand angenommen sind.

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Nomenclature g acceleration due to gravity, m/s² - Gr _R ^* modified Grashof number =gqR⁴/kv² - Gr _R Grashof number =gTR³/v² - H heat function vector, dimensionless - k thermal conductivity, W/mK - L(Y) cord length associated with coordinateY, dimensionless - Pr Prandtl number=v/ - q wall heat flux, W/m² - R radius, m - r(X, Y,Z) distance of a boundary point from the reference axis, dimensionless - S vector derived from the flow field solution, dimensionless - T temperature, K - T _w wall temperature, K - T reference temperature, K - t time, s - u, v velocity components inx, y directions, m/s - U, V dimensionless velocity components inX- and Y-direction normalized withU - U reference velocity=gqR²/k or gTR, m/s - V velocity vector, dimensionless - W vorticity vector, dimensionless - W vorticity, dimensionless - x, y, z cartesian coordinates, m - X, Y, Z cartesian coordinates normalized with a reference length, dimensionless Greek letters thermal diffusivity, m²/s - coefficient of thermal expansion, K^–1 - ,,, non-dimensional coordinates in the transformed domain - non-dimensional temperature =(T–T)_k/q^R or T–T/T_w–T - v kinematic viscosity, m²/s - non-dimensional time=v/R² Gr_Rt or v/R² G _R ^* t - angle measured from the bottom of the cylinder, rads - ^* angle measured from the axis on (– ) plane, rads - heat potential, dimensionless - angle of incidence of the heat flux vector, rads - non-dimensional stream function - vector potential, dimensionless - grid size, dimensionless - ² Laplacian operator - gradient vector     

Calculation of heat transfer accompanying hypersonic three-dimensional flow of air over slender blunt-nosed cones

The effective length method [1, 2] has been used to make systematic calculations of the heat transfer for laminar and turbulent boundary layers on slender blunt-nosed cones at small angles of attack ( + 5° in a separationless hypersonic air stream dissociating in equilibrium (half-angles of the cones 0 20°, angles of attack 0 15°, Mach numbers 5 M 25). The parameters of the gas at the outer edge of the boundary layer were taken equal to the inviscid parameters on the surface of the cones. Analysis of the results leads to simple approximate dependences for the heat transfer coefficients.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 173–177, September–October, 1981.     

Boundary layer stability on a rotating disk with corotation of the surrounding fluid

The stability of the steady self-similar flow in the boundary layer on a rotating disk of infinite radius with corotation of the surrounding fluid is analyzed by the normal mode method. The spectral problem for infinitesimal three-dimensional disturbances is solved by a collocation method with expansion of the amplitude functions in Chebyshev polynomials. It is established that for all values of the parameter 0, equal to the ratio of the angular velocities of the fluid and the disk, the lower critical Reynolds number is determined byA-type, waves, whose development is governed by the parallel instability mechanism typical of an Ekman layer. TheB-type instability, associated with the presence of an inflection point on the velocity profile, disappears when 4. The neutral surfaces are calculated for Karman flow (=0) and Bödewadt flow (). It is found that in Karman flowA-type waves may grow at values of the Reynolds number several times smaller than the critical Reynolds number for spiral vortices. The results of the analysis are compared with the available experimental data.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.5, pp. 69–77, September–October, 1992.     

Modelling of heat transfer by conduction in a transition region between a porous medium and an external fluid

We study the modelling of purely conductive heat transfer between a porous medium and an external fluid within the framework of the volume averaging method. When the temperature field for such a system is classically determined by coupling the macroscopic heat conduction equation in the porous medium domain to the heat conduction equation in the external fluid domain, it is shown that the phase average temperature cannot be predicted without a generally negligible error due to the fact that the boundary conditions at the interface between the two media are specified at the macroscopic level.Afterwards, it is presented an alternative modelling by means of a single equation involving an effective thermal conductivity which is a function of point inside the interfacial region.The theoretical results are illustrated by means of some numerical simulations for a model porous medium. In particular, temperature fields at the microscopic level are presented.Roman Letters _sf interfacial area of thes-f interface contained within the macroscopic system m² - A _sf interfacial area of thes-f interface contained within the averaging volume m² - C _p mass fraction weighted heat capacity, kcal/kg/K - g vector that maps to _s , m - h vector that maps to _f , m - K _eff effective thermal conductivity tensor, kcal/m s K - l _s,l _f microscopic characteristic length m - L macroscopic characteristic length, m - n _fs outwardly directed unit normal vector for thef-phase at thef-s interface - n outwardly directed unit normal vector at the dividing surface. - R ₀ REV characteristic length, m - T _i macroscopic temperature at the interface, K - error on the external fluid temperature due to the macroscopic boundary condition, K - T ^* macroscopic temperature field obtained by solving the macroscopic Equation (3), K - V averaging volume, m³ - V _s,V _f volume of the considered phase within the averaging volume, m³. - _mp volume of the porous medium domain, m³ - _ex volume of the external fluid domain, m³ - _s , _f volume of the considered phase within the volume of the macroscopic system, m³ - dividing surface, m² - x, z spatial coordinates Greek Letters _s, _f volume fraction - ratio of the effective thermal conductivity to the external fluid thermal conductivity - ^* macroscopic thermal conductivity (single equation model) kcal/m s K - _s, _f microscopic thermal conductivities, kcal/m s K - spatial average density, kg/m³ - microscopic temperature, K - ^* microscopic temperature corresponding toT ^*, K - spatial deviation temperature K - error in the temperature due to the macroscopic boundary conditions, K - ^* _i macroscopic temperature at the interface given by the single equation model, K - spatial average - ^s , ^f intrinsic phase average.     

Group Classification of Equations of the Form y″ = f(x,y)

The problem of classification of ordinary differential equations of the form y = f(x,y) by admissible local Lie groups of transformations is solved. Standard equations are listed on the basis of the equivalence concept. The classes of equations admitting a oneparameter group and obtained from the standard equations by invariant extension are described.     

Spatial numerical simulations of linear and weakly nonlinear wave instabilities in supersonic boundary layers

The spatial development of disturbances with small and moderate amplitudes in a two-dimensional (2-D) supersonic flat-plate boundary layer at Mach 4.8 is investigated using direct numerical simulations based on the compressible 3-D Navier-Stokes equations. Disturbances are introduced into the boundary layer by blowing and suction within a narrow disturbance strip at the wall. In response to the timewise periodic forcing, two types of disturbance waves are generated, a first-mode wave and a multiple-viscous-solution. The multiple-viscous-solution was described by Mack (1969, 1984) but was not seen before in a direct numerical simulation. The results of the simulations are compared with results of linear stability theory, and the agreement is very good. In simulations for larger amplitudes, fundamental resonance is observed, where both types of 3-D waves are nonlinearly amplified and synchronize their phase velocities with the 2-D disturbance waves. Subharmonic resonance is found for 3-D waves with large wave numbers, where the phase velocities of the linear 2-D and 3-D waves are nearly the same.This work was supported by the Deutsche Forschungsgemeinschaft (DFG), Bonn-Bad Godesberg, as part of SFB 259.     

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     

First order phase equilibrium for classical bodies

Summary In this note the Author gives a general and unified treatment of first order phase equilibria for classical bodies like those considered by Truesdell and Toupin in[3].The Author reaches a system of partial differential equations (generalized Clapeyron equations) the conditions of whose solution are shown always to be satisfied.In particular, the Author derives the equations governing the polarized phase equilibrium for a fluid.Besides the equations ruling the phase equilibrium for a two phase n-component fluid mixture are given and the equivalence with the statical Gibbs-Duhem relation is shown.

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Sommario In questa nota l'Autore presenta una trattazione generale ed unificata degli equilibri di fase del primo ordine per corpi classici come quelli definiti da Truesdell e Toupin in[3].L'Autore perviene ad un sistema di equazioni alle derivate parziali (equazioni di Clapeyron generalizzate) del quale si dimostra la integrabilità.In particolare, si deducono le equazioni che governano gli equilibri di fase polarizzati.Inoltre si ottengono le equazioni che regolano l'equilibrio di fase per una miscela fluida a n componenti; in questo caso si dimostra l'equivalenza delle equazioni con la relazione statica di Gibbs-Duhem.

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This work was supported by the Gruppo Nazionale per la Fisica Matematica of C.N.R.     

Theory of plane flows of a readily conducting plasma in a pipe

Simplified equations are obtained describing slowly changing plane flows of a readily conducting quasineutral inviscid plasma in a pipe. The practically interesting case of flow in a channel with solid metal ideally conducting walls (electrodes) is analyzed. When the gas pressure is large by comparison with the magnetic pressure ( 1), the field and current distribution is determined by gas dynamic factors, and the solid electrodes perturb the longitudinal electric field in a skin of the flow, symmetrically on the two sides of the flow, leading to attenuation of the longitudinal electric field near the input to the pipe; we also consider problems in the motion of the plasma under ideal and under poor conductivity. In the converse limiting case ( 1), it is shown that as the motion of the plasma in the pipe accelerates near the anode, there is observed an increase in the intensity of the electric field which is sharply inhomogeneous in the transverse direction. The possibility of the plasma breaking away from the anode (the limiting regime) is indicated, this being accompanied by a divergence between the electron velocity and the velocity of the ions. A criterion is obtained for the breakaway of the plasma, and its possible connection with the occurrence of pre-anode explosions is noted. It is shown that for 1, Joule losses are small by comparison with the power in the charge and the magnitude of the losses is independent of the conductivity of the plasma.Translated from Zhurnal Prikladnoi Mekhaniki i Teknicheskoi Fiziki, No. 4, pp. 9–19, July–August, 1970.     

Effect of Streamwise Streaky Structures on Turbulization of a Circular Jet

Experimental investigations of the influence of streamwise streaky structures on turbulization of a circular laminar jet are described. The qualitative characteristics of jet evolution are studied by smoke visualization of the flow pattern in the jet and by filming the transverse and longitudinal sections of the jet illuminated by the laser sheet with image stroboscopy. It is shown that the streaky structures can be generated directly at the nozzle exit, and their interaction with the Kelvin–Helmholtz ring vortices leads to emergence of azimuthal beams ( structures) by a mechanism similar to threedimensional distortion of the twodimensional Tollmien–Schlichting wave at the nonlinear stage of the classical transition in nearwall flows. The effect of the jetexhaustion velocity and acoustic action on jet turbulization is considered.     

Boundary conditions for contacts lines in coextrusion flows

A bicomponent coextrusion process is modelled using a 3-D finite element formulation. The layer uniformity problem in coextrusion is addressed by examining the effects of the polymer melt/polymer melt/die wall contact line boundary condition. It has been observed that the less viscous polymer layer will tend to displace the more viscous polymer layer near the die wall. The behaviour of the contact lisle is considered to be either a stick or slip boundary condition. In the stick boundary condition, the contact line does not move from its original position after the two polymer layers meet, A slip boundary condition allows the contact line to move along the die wall. The calculated interfaces which result from different contact line assumptions are determined. Results show that if a stick boundary condition is appropriate for a given fluid/fluid/solid contact line, then a very thin entrained layer of the more viscous polymer melt will be trapped between the less viscous polymer melt and the die wall. Slip boundary conditions would allow complete displacement of the contact line along the die wall. Both slip and stick boundary conditions produce similar interface profiles far away from the die wall for small viscosity ratios. In certain eases, the displacement of the more viscous material by the less viscous material will cease and a static interface structure is produced regardless of die length. Experimental work with polycarbonate melts is compared with the numerical simulations.A. Torres on leave from Investigación y Desarrollo,, C.A. (INDESCA), P.O. Box 10319, Complejo Petroquímico El Tablazo, Maracaibo, 4001, Venezuela.     

Direct numerical simulation of laminar breakdown in high-speed,axisymmetric boundary layers

The laminar breakdown of high-speed, axisymmetric boundary-layer flow is simulated numerically by solving the compressible Navier-Stokes equations using spectral collocation and high-order compact-difference techniques. Numerical test cases include Mach 4.5 flow along a hollow cylinder and Mach 6.8 flow along a sharp cone. From initial states perturbed by second-mode primary and subharmonic (H-type) secondary disturbances, the well-resolved (temporal) calculations proceed well into the laminar breakdown stages, characterized by saturation of the primary and secondary instability waves, explosive growth of higher harmonics, and rapid increase in the wall shear stress. The numerical results qualitatively replicate two previously unexplained phenomena which have been observed in high-speed transition experiments: the appearance of so-called rope-like waves and the precursor transition effect, in which transitional flow appears to originate near the critical layer well upstream of the transition location at the wall. The numerical results further reveal that neither of these effects can be explained, even qualitatively, by linear stability theory alone. Structures of rope-like appearance are shown to arise from secondary instability. Whereas certain features of the precursor transition effect also emerge from secondary instability theory, its nature is revealed to be fundamentally nonlinear.This research was accomplished largely while the first author was in residence as a National Research Council Associate at NASA Langley Research Center, Hampton, VA 23665, USA.     

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.     

Airfoil design with external flow suction

A method is proposed for the numerical-analytic design of a high-lift airfoil with slit air suction from the external flow within the framework of ideal fluid theory. The suction slit is represented by a channel with constant wall velocity. Examples of the design of wing profiles with non-detached flows having lift coefficientsC _y =2.68 and 4 and maximum relative velocities over the profile [_max|/=2 and 2.2 are given. A scheme for an aircraft of the flying wing type with internally located engines and pay load is briefly discussed.Kazan', Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 6, pp. 23–28, November–December, 1996.     

Boundary conditions for the equations of two-temperature gas dynamics of a binary mixture with strongly differing masses of the components

A study is made of the problem of the boundary conditions for the equations of two-temperature gas dynamics for a binary mixture with strongly differing masses when (m/M)^1/2 Kn 1 (Kn is the Knudsen number, m is the mass of the light molecules, and M the mass of the heavy molecules). The flow structure is established at velocities of the light and heavy components of the order of the velocity of sound of the heavy component. The formulation of the boundary conditions for the gas-dynamic equations is investigated. It is shown that the only closed boundary layer theory is Prandtl's theory taking into account terms of order Kn^1/2.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 93–98, September–October, 1982.I thank M. N. Kogan, N. K. Makashev, and E. S. Asmolov for assistance in the work and valuable discussions.     

Instability of axially symmetric MHD flows

The problem of linear stability of axially symmetric steady-state flows of an ideal incompressible fluid in a magnetic field is studied. A necessary and sufficient condition of stability of these flows with respect to perturbations of the same symmetry type is obtained by the direct Lyapunov method. This condition represents a generalization of the well-known Rayleigh criterion [3, 4] of centrifugal stability of rotating streams to the magnetohydrodynamic case. Two-sided exponential estimates of the perturbation growth are derived. A class of the most rapidly growing perturbations is identified and exact formulas for determining their growth rate are obtained. The corresponding exponents are calculated using the steady flow parameters and initial data for the perturbation field. From the mathematical point of view, the results of the present paper are preliminary in character, since the theorems of existence of the solutions of the problem in question have not been proved.Novosibirsk. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 6, pp. 19–25, November–December, 1995.