Heat transfer to an isothermal flat plate in turbulent flow of a liquid over a wide range of prandtl and reynolds numbers

The study of heat transfer in turbulent flow over a flat plate is very important, not only because this situation frequently arises in practice, but also in that data for an isothermal flat plate are used to calculate heat transfer in more complex cases. In particular, such data are necessary when one uses the limiting relative laws which allow calculation of the effect of compressibility, pressure gradient, blowing, and other perturbing factors [1]. Most papers dealing with heat transfer for an isothermal flat plate refer to comparatively low Re values, when the velocity distribution in the boundary layer over almost its entire thickness can be described by the universal law of the wall. However, as Re increases there is an increasing layer adjacent to the outer boundary in which the velocity distribution cannot be described by the law of the wall, and therefore the results obtained for low Re are inapplicable. In the present paper coefficients of heat transfer from a turbulent flow to an isothermal flat plate have been obtained by numerical integration of the thermal boundary-layer equations over a wide range of the parameters 3 · 10⁵ Re 2.5·10¹², 10² Pr 10³.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 4, pp. 94–100, July–August, 1976.     

On the Viscous Drag of a Plate with Spherical Recesses

The drag of flat plates with spherical recesses was measured using the direct balance method. The experiments were run in a low-turbulence wind tunnel with a cross-section measuring 1000 × 1000 mm and a length of 400 mm. Three surfaces with recesses 7.0, 3.9, and 1.3 mm in diameter and 0.5, 0.3, and 0.2 mm deep, respectively, were tested. It is shown that on the Reynolds and Mach number ranges Re= (3–9)· 10⁶ and M 0.3 the spherical recesses add to the drag of a flat plate in turbulent flow. The recesses have almost no effect on the location of laminar-turbulent transition, which occurs at Re_t 3· 10⁶.     

Effect of longitudinal riblets on the aerodynamic characteristics of a straight wing

The results of balance aerodynamic tests on model straight wings with smooth and ribbed surfaces at an angle of attack =–4°–12°, Mach number M=0.15–0.63, and Reynolds number Re=2.4·10⁶–3.5·10⁶ are discussed. The nondimensional riblet spacings ⁺, which determines the effect of the riblets on the turbulent friction drag, and the effect of riblets on the upper and/or lower surface of a straight wing on its drag, lift, and moment characteristics are estimated.Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 2, pp. 33–38, March–April, 1995.     

Spectral and correlation characteristics of the turbulent boundary layer on an extended flexible cylinder

In marine geophysical seismological prospecting extensive use is made of towed receiving systems consisting of extended flexible cylinders containing acoustic sensors over which the water flows in the longitudinal direction. The boundary layer pressure fluctuations on the cylinder surface are picked up by the sensors as hydrodynamic noise. This paper is concerned with the study of the energy and spacetime characteristics of the pressure fluctuations in the turbulent boundary layer on an extended flexible cylinder in a longitudinal flow. The pressure fluctuations on the cylinder surface have been investigated experimentally for Re_X=(2–4)·10⁷, a dimensionless diameter of the pressure fluctuation sensors d _p ⁺ =d_pu^*/=500, where d_p is the sensor diameter, u^* the dynamic viscosity, and the viscosity coefficient, and frequencies 0.02311.259 (=^*/U). The spectral and correlation characteristics of the pressure fluctuations on the surface of the flexible cylinder are found to differ from the corresponding characteristics for a rigid cylinder [1–4].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i aza, No, 5, pp. 49–54, September–October, 1989.     

Analysis of slip and temperature jump coefficients in a binary gas mixture

The possibility of simplifying the formulas obtained by the Maxwell-Loyalka method for the velocity _u, temperature _T and diffusion _d slip coefficients and the temperature jump coefficient in a binary gas mixture with frozen internal degrees of freedom of the molecules is considered. Special attention is paid to gases not having sharply different physicochemical properties. The formulas are written in a form convenient for use without linearization in the thermal diffusion coefficient. They are systematically analyzed for mixtures of inert gases, N₂, O₂, CO₂, and H₂ at temperatures extending from room temperature to 2500°K. It is shown that for the molecular weight ratios m^* = m₂/m₁ considered the expressions for _u and can be radically simplified. With an error acceptable for practical purposes (up to 10%) it is possible to employ expressions of the same structural form as for a single-component gas: for _u if 1 m^* 6, and for if 1 m^* 3. When 1 m^* 2 the expression for _T can be simplified with a maximum error of 5%. Within the limits of accuracy of the method the expression for _t can be linearized in the thermal diffusion coefficient. An approximate expression convenient for practical calculations is proposed for _d Finally, the , _u, and _T for a single-component polyatomic gas with easy excitation of the internal degrees of freedom of the molecules are similarly analyzed; it is shown that these expressions can be considerably simplified.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 152–159, November–December, 1990.     

Interpolation formulas for maxwell viscosity of certain metals as a function of shear-strain intensity and temperature

The purpose of this study is the construction of interpolation formulas for the dependence of Maxwell viscosity, a quantity which is the reciprocal of shear-strain relaxation time , on shear-strain intensity and temperature for several metals: iron, aluminum, copper, and lead. This function was interpolated in various temperature and deformation velocity ranges in accordance with available experimental data for iron (0 10⁷ sec^–1, 200 ° T 1500 °); aluminum (0 10⁷ sec^–1, 300 ° T 900 °); copper (0 10⁵ sec^–1, 300 ° T 1300 °); lead (0 10⁶ sec^–1, 90 ° T 400 °); temperatures in °K.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 4, pp. 114–118, July–August, 1974.     

Possible Schemes of Cavity Closure

Two new schemes of incompressible inviscid steady-state symmetric (circulationless) cavitation flow are considered. In these schemes the cavity is closed on a flow domain bounded by a closed streamline encircling either a dipole or twin point vortices with circulations equal in magnitude and opposite in sign. A comparative analysis of the efficiency of applying these schemes is performed for the case of transverse flow past a flat plate. On the cavitation number range 0.01 0.15 the drag coefficient and the relative cavity diameter and length obtained in this study are compared with their values calculated from the Riabouchinsky and Efros schemes. For a particular value of the cavitation number, the shapes of the cavities are also compared. The solutions obtained possess the properties of univalence and closure and satisfy the Brillouin condition.     

Rheological properties of reactive polymer melts.

A new method for describing the rheological properties of reactive polymer melts, which was presented in an earlier paper, is developed in more detail. In particular, a detailed derivation of the equation of a first-order rheometrical flow surface is given and a procedure for determining parameters and functions occurring in this equation is proposed. The experimental verification of the presented approach was carried out using our data for polyamide-6.Notation E Dimensionless reduced viscosity, eq. (34) - E ₀ Newtonian asymptote of the function (36) - E power-law asymptote of the function (36) - E _{= 1} the value ofE at = 1 - k degradation reaction rate constant, s^–1 - k ₁ rate constant of function (t), eq. (26), s^–1 - k ₂ rate constant of function (t), eq. (29), s^–1 - K(t) residence-time-dependent consistency factor, eq. (22) - M _w weight-average molecular weight - M _x x-th moment of the molecular weight distribution - R gas constant - S _x M _x /M _w - t residence time in molten state, s - t _j thej-th value oft, s - T temperature, K - % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xd9vqpe0x% c9q8qqaqFn0dXdir-xcvk9pIe9q8qqaq-xir-f0-yqaqVeLsFr0-vr% 0-vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaaieGaceWFZo% Gbaiaaaaa!3B4E!\[\dot \gamma \] shear rate, s^–1 - _i thei-th value of , s^–1 - _r =1 the value of at = 1, s^–1 - ^* reduced shear rate, eq. (44), s^–1 - dimensionless reduced shear rate, eq. (35) - viscosity, Pa · s - shear-rate and residence-time dependent viscosity, Pa · s - zero-shear-rate degradation curve - degradation curve at - _{t0 (t)} zero-residence-time flow curve - Newtonian asymptote of the RFS - instantaneous flow curve - power-law asymptote of the RFS - _0,0 zero-shear-rate and zero-residence-time viscosity, Pa · s - _E=1 value of viscosity atE=1, Pa · s - ^* reduced viscosity, eq. (43), Pa · s - zero-residence-time rheological time constant, s - density, kg/m³ - (t),(t) residence time functions     

Calculation of intense blowing on a blunt body and an airfoil

Various aspects of the problem of intense blowing through the surface of bodies have, been theoretically studied by a number of authors, within the framework of inviscid flow theory. A detailed bibliography on this topic is given, e.g., in [1, 2]. The well-known approaches to solution of this problem have a limited area of application. For example, asymptotic methods can be used for hypersonic flow regimes only at relatively low levels of the blown gas momentum ( = ² = _ov_o ²/ V ² 1). The same limitation applies to the numerical method of straight lines [2]. The forward Eulerian calculation schemes [3, 4] smear the contact discontinuity severely, and cannot handle the case where the blown gas and the gas in the incident flow have different thermodynamic properties (_o ). This paper presents results of a numerical investigation of supersonic flow over two-dimensional and axisymmetric bodies with intense blowing on the forward surface, performed using a time-dependent finite-difference method [5] with an explicit definition of the contact interface between the two cases. The calculations encompass a family of elliptic cylinders with semiaxis ratio 0.5 4, a flat-face cylinder, and a flat plate with rounding near the midsection, with variations in the blowing law, the incident flow Mach number M (3 M 10), the adiabatic indices, and the blowing parameter 0 0.5.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 117–124, January–February, 1977.In conclusion, the authors thank T. S. Novikov and I. D. Sandomirskii, who took part In the present calculations.     

Jet methods of gas injection into fluid boundary layer for drag reduction

Two jet methods for saturating the fluid boundary layer with microbubbles for drag reduction in contrust with gas injection through porous materials are considered. The first method is the gas injection through the slot under a special fluid wall jet. The second method is the saturation of boundary layer by microbubbles via the gas-water mixture injection through the slot. Experimental data, reflecting the skin friction drag reduction on the flat plate and total drag reduction of axisymmetric bodies, are presented. The comparison between a jet methods of gas injection and gas injection through porous materials is made.Nomenclature v free-stream velocity - v _j mean velocity of a water through slot - v _g mean velocity of a gas through slot - h width of slot for realizing water jet - h ₁ width of slot for gas injection - incidence angle - Q volume airflow rate - C _Q airflow rate coefficient (v _g/v ) - C _f skin friction coefficient - v _j/v - C _f0 C _f ifQ=0 andv _j=0 - _f C _f/C _{f 0} - d diameter of an axisymmetric body - L length of body - C _Q 4 · ·Q/d ² v - C _D 4 ·D/1/2v ² ·d ² - C _Q 4 ·Q/d ² v - Q _j volume flow rate of water jet - C 8 ·Q _jv_j/d ² v ² - 1 fluid density of main flow - 2 fluid density of wall jet - B ₁ main stream total pressure - B ₂ wall jet total pressure - v ₁ main stream velocity - Be (B ₂ –B ₁)/1/2₁ v ₁ ² = Bernoulli number - ₂ v ₂/₁ v ₁ - p _st static pressure - p _at atmospheric pressure - p _st/p _at - D hydrodynamic drag of body     

Drag measurements on V-grooved surfaces on a body of revolution in axial flow

In water flows with velocities of up to 9 m/s the friction drag of a body of revolution in axial flow was investigated for dependence on the body surface structure. This was done for different types of riblet film fixed on the surface with the riblet direction aligned with the flow. The lateral spacing between the triangular shaped riblets varied between 0.033 mm and 0.152 mm. In all cases the riblet spacing was equal to the riblet height. For comparison a smooth reference film was used.Depending on the Reynolds number and the non-dimensional riblet spacings ⁺, a turbulent drag reduction of up to 9% could be achieved with riblets in comparison with the flow over a smooth surface.In the region of transition to turbulent flow and with non-dimensional riblet spacings ofs ⁺10–15 drag reductions of up to 13% were obtained. It is therefore conjectured, that in addition to hampering the near wall momentum exchange, the riblets can delay the development of initial turbulent structures in time and space.     

Einfluß von Anströmprofil und Turbulenzintensität auf die Umströmung längsangeströmter Platten endlicher Dicke

Zusammenfassung An längsangeströmten dicken Platten löst sich die Strömung im Bereich des Anströmprofils ab und legt sich nach einer gewissen Entfernung wieder an die Platte an. Dies führt gegenüber der dünnen Platte zu signifikanten Veränderungen von Umströmung, Wärmeübergang und Druckverlust.Die Aussagen zur Umströmung werden aus örtlichen Stoffübergangsmessungen (Sc=0,616) mit Hilfe einer remissionsfotometrischen Stoffübergangsmeßmethode auf der Basis von Absorption, chemischer Reaktion und gekoppelter Farbreaktion gewonnen. Als wesentliche Einflußgrößen auf Umströmung und zugeordneten Stoffübergang sind das Anströmprofil, die Anström-Reynolds-Zahl Re_sB sowie insbesondere die Turbulenzintensität Tu zu nennen. Die Untersuchungen erstrecken sich auf Platten mit Dicken 0,8 mms91, 3 mm mit stumpfem, halbrundem und keilförmigen Anströmprofilen sowie auf längsüberströmte Kreiszylinder mit stumpfem, halbkugelförmigem und kegelförmigen Anströmprofilen. Die mit der Plattendicke s gebildete Re_sB -Zahl wurde in weiten Grenzen 10² < Re_sB < 2 · 10⁵, die Turbulenzintensität zwischen 0,8 Tu 6% variiert.Die Ergebnisse zeigen, daß sich die Umströmung dicker Platten generell in drei Hauptströmungsformen Plattengrenzschicht, Ablöseblase und Querwirbelablösung untergliedern läßt. Der Übergang von einer Strömungsform zur anderen wird durch kritische Re_sB -Zahlen erfaßt und die Abhängigkeit vom Anströmprofil über einen Profilfaktor beschrieben. Bei Ablöseblasen konnten erstmals an Platten Längswirbel nachgewiesen werden mit ihren charakteristischen Auswirkungen auf Umströmung und Stoffübergang.Dicke Platten mit Strömungsablösung lassen sich in drei Abschnitte unterteilen: 1. Profilbereich, 2 Bereich abgelöster Strömung, 3. Plattengrenzschicht ab Wiederanlegen der Strömung. Im Hinblick auf eine Berechnung des örtlichen Stoffübergangs wird die Lage des Stoffübergangsmaximums beim Wiederanlegen der Strömung — als Trennlinie der beiden letzten Plattenabschnitte — unter Einbeziehung der wesentlichen Parameter Anströmprofil, Re_sB-Zahl und Turbulenzintensität erfaßt und als Berechnungsgleichung angegeben.

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The influence of nose section and turbulence intensity on the flow around thick plates in parallel flow

The flow parallel to thick plates separates in the nose section and reattaches on the flat plate after a certain distance. Compared with the thin flat plate the flow separation with reattachement causes significant changes of flow, heat transfer and pressure drop.The informations about the flow are recieved from measurements of local mass transfer coefficients (Sc=0,616), which are fotometrically determined by light-remission. The fundamentals of the measuring technique are based on absorbed ammonia and subsequent chemical reaction with immediate color reaction. The main parameters of flow and related mass transfer are the nose of the plate, the Reynolds-number Re_sB and especially the turbulence intensity Tu. The measurements were accomplished with plates of a thickness 0.8 mm s 91.3 mm of truncated, hemicylindrical and wedge-shaped noses and with cylinders in parallel flow of truncated, hemispherical and conical noses. The Re_sB -number with the plate thickness as characteristic length was varied in a wide range between 10² < Re_sB < 2 · 10⁵, the turbulence intensity Tu between 0,8% Tu 6%.The measured results indicate, that the flow around thick plates may generally be subdivided into three main forms of the flow: boundary layer of flat plate, separation bubble and vortex shedding with reattachement. The critical Reynolds-numbers for the transition of one form to another were determined as a function of the different shapes of noses. For the first time longitudinal vortices can be observed for separation bubbles with significant influence on the flow and on the mass transfer.Thick plates with flow separation and reattachement may be subdivided into three sections: 1. the nose section, 2. the section of separated flow, 3. the section of boundary layer of flat plate downstream of reattachement. For the computation of local mass transfer rates, the position of the maximum mass transfer at the point of reattachement — the dividing line of the 2nd and 3rd section — are determined as a function of the main parameters plate nose, Re_sB -number and turbulence intensity.

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Bezeichnungen d Zylinderdurchmesser - D_A Diffusionskoeffizient von Ammoniak in Luft - K_P Koeffizient in Gl. (7) - K_Tu Koeffizient in Gl. (6) - Re_dB=U_B · d/ Reynolds-Zahl - Re_sB=U_B · s/ Reynolds-@#@ Zahl - Re_skrit1=U_B · s/ Reynolds für den Übergang von Plattengrenzschicht zu Strömungsablösung - Re_skrit2=U_B · s/ Reynolds-Zahl für den Übergang von Ablöseblase zu Querwirbelablösung - Re_xkrit=U_B · x_krit/ Reynolds-Zahl für den Übergang von laminarer zu turbulenter Plattengrenzschicht - s Plattendicke einschließlich Trägerfolie - Sh_s=_A · s/D_A Sherwood-Zahl - Tu Turbulenzintensität in % - U_B Hauptstromgeschwindigkeit im verengten Kanalquerschnitt - U Hauptstromgeschwindigkeit im freien Kanal querschnitt - mittlere turbulente Geschwindigkeitsschwankung in x-Richtung - x Koordinate in Strömungsrichtung, tangential zur Oberfläche - X_P Profillänge des Anströmprofils - x Koordinate in x-Richtung ab Profilende (Plattenbeginn) - x_max' Entfernung von Plattenbeginn bis zum Wiederanlegen der Strömung nach Strömungsablösung (Stoffübergangs-maximum) - x_min Entfernung von Plattenbeginn bis zur Stelle minimalen Stoffübergangs - x^* Koordinate in x-Richtung ab Wiederanlegen der Strömung nach Strömungsablösung - _A Stoffübergangskoeffizient (Ammoniak) - _Am mittlerer Stoffübergangskoeffizient - kinematische Zähigkeit Unter dem Titel Einfluß des Anströmprofils auf die Umströmung von Platten endlicher Dicke gekürzt vorgetragen auf der Sitzung des GVC-Fachausschusses Wärme- und Stoffübertragung am 5./6. April 1976 in Schliersee.Herrn Prof. Dr. phil. Dr.-Ing. E.h. Peter Grassmann zum 70. Geburtstag am 13. August 1977.     

Comparison of Calculated and Experimental Data on Supersonic Flow past a Circular Cylinder

The supersonic perfect-gas flow past a circular cylinder is studied on the basis of a numerical analysis of the time-dependent two-dimensional Reynolds equations using a differential q– turbulence model with reference to the experimental conditions. The calculations are carried out at Reynolds and Mach numbers Re=2× 10⁵ and M=1.1, 1.3, and 1.7 and the experimental investigations at Re=1.62×10⁵–2×10⁵ and Mach numbers on the interval 0.7 M 1.7. The calculated and experimental data on the pressure coefficient distribution over the cylinder surface, the location of the separation point on the surface, and the pressure drag coefficient are compared.     

Internal waves generated by the turbulent wake of a sphere

Internal waves generated by the turbulent wake of a sphere travelling horizontally through a linearly stratified fluid were studied using shadowgraph and particle-streak photography. The Reynolds and internal Froude number ranges considered were 2,000 Re 12,900 and 2.0 Fi 28.0, respectively. Two quite distinct flow regimes based on the structure of the turbulent wake were identified. In one, the wake is characterized by large-scale coherent structures. In the other, the wake, as viewed on a side-view shadowgraph, grows in a roughly symmetric fashion to a maximum height and then collapses slowly; such flows are termed the smallscale structures regime.Wave lengths and maximum wave heights of the internal waves were measured as functions of Nt and Fi, where N is the Brunt-Väisälä frequency and t the time. It was found that the wave lengths scale well with the streamwise dimension of the spiralling coherent structures. The maximum amplitude of the internal waves were found to scale with the vertical dimension of the turbulent wake, upon varying the internal Froude number.     

Investigation of pre-separation boundary layer flows on the basis of various algebraic models of turbulence

The problem of the three-dimensional incompressible turbulent boundary layer developing ahead of a circular cylinder mounted at right angles on a flat plate is considered. The direction of the uniform approach stream is normal to the leading edge of the plate. The turbulence is simulated by means of five different isotropic algebraic models of eddy viscosity. The boundary layer equations are solved numerically by means of a second-order-accurate implicit finite-difference method. The principal characteristics of the flow obtained on the basis of the turbulence models selected are compared for a free-stream Reynolds number Re = 10⁷.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 36–43, November–December, 1991.     

Numerical Investigation of the Natural Convection of Water in the Neighborhood of the Density Inversion Point For Grashof Numbers up to 106

The natural convection of fresh water in a square cell is considered at a temperature close to the density inversion temperature for Grashof numbers 2.9 · 10⁴ Gr 10⁶. As a result of the numerical investigation, one steady-state and three self-oscillating regimes are found in addition to the three steady-state flows previously detected earlier and described for low Grashof numbers ( 0 Gr 2 · 10⁵). The basic characteristics of the unsteady flows are analyzed by means of the Fourier method, the fundamental oscillation frequencies are found, and the flow evolution and the variation of the oscillation characteristics with increase in the Gr number are considered.     

Study of initial segment of turbulent jets of various gases in a parallel air stream

Results are presented of a study of the gasdynamic parameters and the geometric characteristics of the mixing zone of axisymmetric jets of gases of differing density (Freon-12, air, and helium) propagating in a parallel air stream, within the limits of the initial segment (0x/R3–30). Experimental data are presented on the effect of different densities (0. 27 n8.2) and velocities (0m1.7) of the gas jet and the parallel stream on the mixing process.     

Turbulent boundary layer on a permeable surface

Several theoretical [1–4] and experimental [5–7] studies have been devoted to the study of the effect of distributed injection of a gaseous substance on the characteristics of the turbulent boundary layer. The primary study has been made of flow past a flat plate with gas injection. The theoretical methods are based primarily on the semiempirical theories of Prandtl [1] and Karman [2].In contrast with the previous studies, the present paper proposes a power law for the mixing length; this makes it possible to obtain velocity profiles which degenerate to the known power profiles [8] in the case of flow without blowing and heat transfer. This approach yields analytic results for flows with moderate pressure gradient.Notation x, y coordinates - U, V velocity components - density - T temperature - h enthalpy - H total enthalpy - c mass concentration - , , D coefficients of molecular viscosity, thermal conductivity, diffusion - c_p specific heat - adiabatic exponent - r distance from axis of symmetry to surface - boundary layer thickness - U velocity in stream core - friction - c_f friction coefficient - P Prandtl number - S Schmidt number - S_t Stanton number - M Mach number - j=0 plane case - j=1 axisymmetric case The indices 1 injected gas - 2 mainstream gas - w quantities at the wall - core of boundary layer - 0 flow of incompressible gas without injection - v=0 flow of compressible gas without injection - ^* quantities at the edge of the laminar sublayer - quantities at the initial section - turbulent transport coefficients     

Experimental investigation of the resistance to the flow of a conducting fluid in flat insulated ducts in the presence of a transverse magnetic field with allowance for fringe effects and wall roughness

The distribution of pressure, velocity, and electrical potential has been investigated for a mercury flow in insulated rectangular ducts with a large side ratio (Hartmann-type flow). The ranges of variation of the Reynolds, Hartmann, and Stewart numbers were 7·10²R5·10⁵, 0H490, and 0N24, respectively. Special attention is given to the sections of the channel where the flow enters and leaves the magnetic field. In these zones the pressure is sharply nonuniform and the velocity profiles in a plane perpendicular to the field acquire an M shape. A relation is established between the length of the entrance section, where the flow is three-dimensional, and the MHD similarity criteria. It is shown that ducts which are hydraulically smooth in the absence of a magnetic field become increasingly rough as the field grows stronger. Data are obtained on the resistance coefficient for a stabilized flow measured in a magnetic field and on the dependence of the critical Reynolds number on the Hartmann number.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 10–21, July–August, 1971.     

Flow of a viscous gas in a shock layer with equilibrium chemical reactions

Steady flow of supersonic air over a sphere is examined, allowing for viscosity, heat conduction, and actual physical and chemical processes. Flow in the shock layer at flight speeds in the range 3 km/sec V10 km/sec (10⁴R10⁶) is investigated, under the assumption of local thermodynamic equilibrium. The flow is described by simplified Navier-Stokes equations, which are solved by a finite difference method. The case of a cooled surface is examined. The distribution of gasdynamic parameters is obtained in different flow regimes. The distribution of heat flux and friction coefficient is investigated as a function of the oncoming-stream parameters and the sphere radius. The shape and position of the shock wave are determined, and the stream lines and sonic lines are constructed.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 4, pp. 150–153, July–August, 1970.The authors thank Yu. P. Lun'kin and F. D. Popov for their help in formulating the problem and their constant interest.