An integral approach to lifting wing theory at Mach one

Summary An approach to lifting wing theory at Mach one is presented that utilizes an integral method similar to the Karman-Pohlhausen method in boundary layer theory. As in any integral method the results obtained are approximate in nature. Nonetheless, comparison with experimental data shows good agreement in cases for which experimental data are available. The method can easily be used to determine the lift on wings of finite aspect ratio and also to solve transient lifting problems. The method is demonstrated by solving for the pressure distribution on a lifting airfoil of arbitrary symmetric cross-section, the lift on a wing of rectangular planform, and the transient lift on an airfoil due to a sudden change in angle of attack. These cases were chosen to illustrate the versatility of the method and are not meant to be exhaustive of all possibilities. The computational time required to obtain numerical results is very small in all cases considered.List of symbols A parameter associated with Guderley airfoil, defined in equation (28) - AR aspect ratio - AR reduced aspect ratio=AR ^1/3( + 1)^1/3 - c chord of airfoil - C _l sectional lift coefficient - C _L lift coefficient - C _p pressure coefficient - M Mach number - p Laplace transform variable - s span of wing (in units ofc) - t time (in units ofc/U) - U free stream velocity - x streamwise coordinate (in units ofc) - x ^* distance from leading edge to sonic point (in units ofc) - y spanwise coordinate (in units ofc) - z coordinate normal to plane of wing (in units ofc) - angle of attack - y/2s - ratio of specific heats (=1.4 in all calculations) Research sponsored by the Air Force Office of Scientific Research/AFSC, United States Air Force, under Contract No. F44620-72-C-0079. The United States Government is authorized to reproduce and distribute reprints for governmental purposes notwithstanding any copyright notation hereon.     

Analysis of the parameters of discrete particle motion in axisymmetric turbulent impinging jets

The parameters of discrete particle motion in axisymmetric turbulent impinging air jets are determined.Notation x, y coordinates (Fig. 1) - v_x jet velocity - V_o maximum jet velocity - r_o nozzle radius - l _i length of the initial jet section - L spacing between the nozzle and the collision plane - ¯x dimensionless coordinate referred to the nozzle radius - ¯x_i dimensionless length of the initial section referred to the nozzle radius - d particle diameter - ₁ jet density - particle density - c_x particle drag coefficient - v particle velocity - v₁ axial jet velocity - kinematic coefficient of the flow viscosity - ¯x_o dimensionless coordinate referred to the distance L - d_c cement particle diameter - d_s sand particle diameter - ¯v_i dimensionless velocity of particle insertion into the jet, referred to V_o Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 37, No. 5, pp. 813–817, November, 1979.     

Formation and properties of amorphous (Fe1−x Nb x ) l B100−l

Amorphous (Fe_1–x Nb _{x l} B_100–l alloys with 0 x 0.15 and 74 T _g, crystallization temperatureT _x, and microhardnessH _v, but to decrease the magnetization and Curie temperatureT _c. The effects of niobium onT _x,H _v, andT _c in iron-based amorphous alloys are similar to those of chromium, manganese, molybdenum, tungsten and vanadium.     

Temperature similarity of heat transfer processes

New relationships are presented which describe the temperature and humidity of the medium at the edges of the boundary layer in heat and mass transfer processes.Notation A thermal equivalent of mechanical work - g acceleration due to gravity - c_p specific heat of medium - T_f and T_w arithmetic mean temperatures of medium in flow core and at heat transfer surface - w_x and w_y, u_v and v_v projections of mixture and vapor velocities on the X and Y axes - and _T molecular and turbulent thermal conducitvities of medium - andT molecular and turbulent diffusion coefficients - P pressure - r latent heat of condensation - l characteristic geometrical dimension (tube diameter) - l ₀ arbitrary characteristic dimension taken as zero reading - x humidity of mixture - I heat content of mixture - and _v density of mixture and vapor - specific weight - a thermal diffusivity - k diffusion conductivity - Gr Grashof number - Pr Prandtl number     

Isentropic exponents of real gases and application for the air at temperatures from 150 K to 450 K

Summary Three real gas isentropic exponentsk _Tv,k _rv,k _pT are introduced, which when used in place of the classical isentropic exponentk=c _p/c in the ideal gas isentropic change equations, the latter may describe very accurately the isentropic change of real gases. The usual practice of employing exponentk may lead to considerably incorrect results even when the value ofk corresponds to the correct local value ofc _p/c _v of the real gas under examination. The numerical values of the new exponents are calculated in the case of real air for temperatures from 150 K to 450 K and pressures from 1 bar to 1000 bar. It is seen that at low temperatures and high pressures the values of the new exponents differ considerably from the values of the classical exponentk. Therefore, the error resulting by approximating, as is usually the case, the behaviour of real gases by the ideal gas isentropic change equations in a stepwise fashion with exponentk instead of the new exponents, is considerable. It follows that exponentk, which appears in various relations in thermodynamics, fluid mechanics, gasdynamics, heat transfer etc., should be suitably replaced by combinations of the three exponents. Related numerical examples, made in the case of real air, showed that the use ofk leads (in the temperature and pressure ranges examined) to a 5% error in the calculation of blowby rate in internal combustion enginers, high pressure compressors or steam turbines and to a 50% error in the calculation of the isentropic expansion or compression.Nomenclature A _ij,B _i,N _ij,O _ij,Q _ij Coefficients - c Velocity - c _p Specific heat under constant pressure - c _v Specific heat under constant volume - h Specific enthalpy - k Isentropic exponent,k=c _p/c _v - k _pT Real gas isentropic exponent corresponding to the pair of variablesp,T - k _pv Real gas isentropic exponent corresponding to the pair of variablesp, v - k _Tv Real gas isentropic exponent corresponding to the pair of variablesT,v - M Mach number,M=c/ - p Pressure - p _c Pressure at the critical point - R Constant of the air,R=287.22 J/kg K - s Specific entropy - T Temperature - T _c Temperature at the critical point - v Specific volume - v _c Specific volume at the critical point - z Compressibility factor - Sound velocity - T Temperature increment With 14 Figures     

Structure of flows in vortices

The structure of a gradient vortical flow was studied experimentally.Notation v_x, v_y, v_z flow velocity components in a rectangular coordinate system - v, v_r, v_z flow velocity components in a cylindrical coordinate system - v₁ tangential velocity at the boundary of solid revolution at r = r₁ - l length of the vortex - kinematic viscosity - R radius of the forming cylinder - circulation in the region of potential flow - second air flow rate through the eddy of ascending flows - Re=v₁r₁/ tangential Reynolds number - N=Q/r_o radial Reynolds number - a=l/r₀ configuration ratio for the vortex model - s=r_o/2Q effective exchange coefficient - a ^*=l/r configuration ratio for the vortex generator - s^*=R/Q constructive exchange coefficient - p=p–p pressure drop in the vortex relative to atmospheric pressure p - r^*= r/r₁ dimensionless radius of the vortex - v^*=v/v₁ dimensionless tangential velocity - a ^*/a gradient ratio Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 39, No. 4, pp. 611–618, October, 1980.     

Effect of a protective coating on the frequency characteristics of film transducers in thermoanemometers

An expression is derived for the thermal characteristic impedance. Equations are obtained for calculating the amplitude-frequency and the phase-frequency characteristics of planarfilm transducers with protective coating.Notation time constant - radian frequency - f frequency - T temperature - Q thermal flux - k thermal conductivity - heat transfer coefficient - a thermal diffusivity in solid state - _T thermal wavelength - l length of film on sensing element, normal to the direction of the stream - d width of film on sensing element, parallel to the direction of the stream - l _b thickness of substrate - _g kinematic viscosity of fluid - Z_t thermal characteristic impedance - R_T thermal characteristic resistance - phase-shift angle - T_j temperature of hot film (°C) - T_c temperature of protective coating, at distanceL _c from the film - T_b temperature of substrate, at distance l_b from the film - l_c=_T, l_c=_T/2, etc. thickness of protective coating, equal to thermal wavelength, equal to half the thermal wavelength, etc. - Q_fc thermal flux from hot film to protective coating - Q_fb thermal flux from hot film to substrate - Q_gc heat transferred from protective coating to fluid stream - Z characteristic impedance of transient heat transfer from film to coating (°C·sec/J) - Z characteristic impedance of transient heat transfer from film to substrate (°C·sec/J) - W relative sensitivity, dB Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 22, No. 6, pp. 1036–1041, June, 1972.     

Lift and drag characteristics of a delta-wing-half-cone configuration with subsonic leading edges,using slender-body theory

Summary Configurations, composed of a cone with a half-circular cross-section mounted above or below a delta wing of zero thickness with subsonic leading edges and placed in a supersonic flow, are studied using the slender-body theory in order to determine their lift and drag characteristics. These are compared to the lift and drag of configurations composed of the same wing and a symmetrically disposed circular cone with equal volume as the half-cone. The comparison is made to investigate whether it is possible to attain better lift efficiency by placing the body on one side of the wing.For configurations having a body diameter-wing span ratio larger than approximately 0.45, a disposition of a half-cone on one side of the delta wing shows a drag reduction at a given lift, and therefore a higher value of (C_L/C_D)_max, compared to the corresponding symmetrical combination. However, the high-wing combination is preferable to the low-wing, since lower angles of incidence are needed to attain a certain C_L.If the body diameter-wing span ratio is less than this value, the symmetrical system appears to be more favourable.The lift curve slope dC_L/d of the asymmetrical configurations studied is larger than that of the symmetrical configurations.     

A method of joint determination of the effective coefficients of horizontal mixing of large particles and of external heat exchange in an organized fluidized bed

A method is proposed for the joint determination of the coefficients of horizontal particle diffusion and external heat exchange in a stagnant fluidized bed.Notation c_f, c_s, c_n specific heat capacities of gas, particles, and nozzle material, respectively, at constant pressure - D effective coefficient of particle diffusion horizontally (coefficient of horizontal thermal diffusivity of the bed) - d equivalent particle diameter - d_t tube diameter - H₀, H heights of bed at gas filtration velocities u₀ and u, respectively - H_a height of active section - l width of bed - L tube length - l _o width of heating chamber - N number of partition intervals - p=H/H₀ expansion of bed - s_n surface area of nozzle per unit volume of bed - S_h, S_v horizontal and vertical spacings between tubes - t_c, t₀, t_s, t_n, t_w initial temperature of heating chamber, entrance temperature of gas, particle temperature, nozzle temperature, and temperature of apparatus walls, respectively - u₀, u velocity of start of fluidization and gas filtration velocity - y horizontal coordinate - ^*, coefficient of external heat exchange between bed and walls of apparatus and nozzle - ₁, ₁, ₂, ... coefficients in (4) - thickness of tube wall - _b bubble concentration in bed - ₀ porosity of emulsion phase of bed - _n porosity of nozzle - =(t_s – t₀)/(t_c – t₀) dimensionless relative temperature of particles - _n coefficient of thermal conductivity of nozzle material - f, _s, _n densities of gas, particles, and nozzle material, respectively - _be=_s(1 – ₀) (1 – _b) average density of bed - time - _max time of onset of temperature maximum at a selected point of the bed - R =l _o/l Fourier number - Pe = ₁ l ²/D Péclet number - Bi = /_n Biot number Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 41, No. 3, pp. 457–464, September, 1981.     

New equations for enthalpy of vaporization from the triple point to the critical point

Two new equations are proposed for the enthalpy of vaporization from the triple point to the critical point. One of these equations containing four parameters is exceptionally good for fitting the data. The other equation containing three parameters is quite adequate for fitting the data but it is exceptionally suited for interpolation when the data do not cover the entire range. These equations have been tested using the enthalpy of vaporization of water from the triple point to the critical point and are compared with other equations.Nomenclature T _c Critical temperature, K - T _t Triple point, K - T _x Any particular temperature, K - T _r Reduced temperature - P _r Reduced pressure - R Gas constant - P Vapor pressure - X (T _c–T)/T _c - Y (T _c–T)/T - X _x (T _c–T)/(T _c–T _x) - X _t (T _c–T)/(T _c–T _t) - H _vt Enthalpy of vaporization at the triple point, kJ · mol^–1 - H _vx Enthalpy of vaporization at any temperature x, kJ · mol^–1 - Z _v Compressibilty factor of the saturated vapor - Z ₁ Compressibilty factor of the saturated liquid Relative deviation = 100[H_v(obs)–H_v(cal)]/H_v(obsd) Standard deviation = { [H _v(obs)–H _v(cal)]²/(No. points — No. parameters)}^0.5     

Breakup of an anomolously viscous liquid film in a centrifugal force field

An equation is obtained for the breakup radius with consideration of tipping moments and Laplacian pressure forces acting on the liquid ridge at the critical point.Notation K, n rhenological constants - density - surface tension - r current cup radius - R maximum cup radius - r_c critical radius for film breakup - ¯r=¯r=r/R dimensionless current radius - ¯r_c=r_c/R dimensionless critical radius - ₀, _c actual and critical film thicknesses - current thickness - R_r ridge radius - h₀ ridge height - h current ridge height - ₀ limiting wetting angle - current angle of tangent to ridge surface - angle between axis of rotation and tangent to cup surface - angular velocity of rotation - q volume liquid flow rate - v₁ and v meridional and tangential velocities - =4vv _lm/r,=4v_m/r dimensionless velocities - M moments of surface and centrifugal forces - M_v moment from velocity head - p_r pressure within ridge - P_vm pressure from velocity head - p_m, p_pm pressures from centrifugal force components tangent and normal to cup surface - deviation range of breakup radius from calculated value - ¯r_max, ¯r_min limiting deviations of breakup radius - _c angle of tangent to curve _c/°₀=f(¯r) at critical point - t random oscillation of ratio _c/_c Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 39, No. 1, pp. 51–56, July, 1980.     

Heat transfer in the flow of low-density air in narrow channels

Results are given from measurements on air flow in narrow channels; relationships in dimensionless terms are derived for the heat transfer over a wide range in speed (1–120 m/sec) and in pressure (1 · 10⁵ > P > 1.33 · 10³) N/m².Notation V volume flow rate of air - N total number of buret divisions - P₀ pressure in measuring tank - l length of measuring section of buret - t time of oil column rise to the height h_i - n number of buret division corresponding to_i - _o, _m specific weights of oil and mercury - c scale division of buret - h₂ height of oil drop in measuring cylinder - v₀ total volume of system from needle throttle to heat exchanger inlet - P_p pressure at heat exchanger inlet - T_p, T₀ temperature at heat exchanger inlet and of surrounding air - G flow rate in mass terms - c_p mean specific heat of air - t temperature variation over measuring section - Nu, Re Nusselt and Reynolds numbers - l, d length and diameter of channel Translated from Inzhenerno-Fizicheskii Zhurnal, vol. 20, No. 5, pp. 879–883, May, 1971.     

Effect of swelling in the extrusion of an elastic liquid from a capillary

The article demonstrates that the profile of a jet extruded from a capillary can be constructed from the data of uniform stretching.Notation V₁ mean velocity of the liquid in the capillary - r₁ radius of the capillary - q flow rate - P pressure in the preinlet zone of the capillary - F₁ and F₀ extrusion force - ₁ and ₂ elastic deformations in parallel Maxwellian elements upon stretching - p, z radial and longitudinal coordinates, respectively - v_z(p) velocity of steady-state flow in the capillary - _i modulus of elasticity in the i-th Maxwellian element - swelling coefficient - t time - l length of the specimen at instant t - elastic deformation - x rate of deformation - ₀, ₁ deformation - L length from which onward the radius of the jet extruded from the capillary practically does not change - r radius of the jet - t₁ time from which onward the radius of the specimen in retardation practically does not change - d₀,l ₀ diameter and length, respectively, of the undeformed specimen - l _r length to which the extended specimen tends after removal of the load - b length of the capillary - ₁ tensile stress (for uniform stretching at t = 0, for extrusion from the capillary at z = 0) - l _H length of the stretched specimen at the instant of beginning reading of t Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 39, No. 2, pp. 343–350, August, 1980.     

A method of determining the thermal conductivity of powdered and fibrous insulation as a function of filler gas pressure

An examination is made of the theoretical basis and implementation of a nonstationary method of rapid measurement of the thermal conductivity of powdered and fibrous insulation under conditions of monotonic change of filler gas pressure.Notation t temperature - ,a thermal conductivity and diffusivity of test material - k, k_a relative temperature coefficients of anda - thickness of test layer - x variable layer coordinate reckoned from shell - =(x), _c excess temperature of material at section x and of core over shell - b_c, b_v rate of cooling of core and of variation of volume-mean temperature of layer - c_c, c total heat capacity of core and material - f_s, F_c area of working surfaces of shell and core - d diameter of particles of bulk material - p material porosity - volume density of material     

The temperature of an adiabatic surface at a turbulent boundary layer

A calculation of the temperature decrease of an adiabatic surface at a supersonic turbulent boundary layer is conducted. It is shown that the temperature decrease is a consequence of the appearance of a vortex chain in the flow near the walls. Comparison of calculated data with experimental gives qualitative agreement.Notation V incident flow velocity - V_v vortex velocity - v local velocity - u induced velocity - T thermodynamic temperature - T_w, T recovery temperature and undisturbed flow temperature - L length of depression - h₀ depth of depression - h distance from vortex center to wall - b relative vortex velocity - l _v vortex spacing - r recovery coefficient - R₀ recovery coefficient on smooth surface - c_p gas heat capacity at constant pressure - n vortex passage frequency - Re Reynolds number - M Mach number - velocity potential - time - vortex intensity Translated from Ihzhenerno-Fizicheskii Zhurnal, Vol. 20, No. 5, pp. 903–908, May, 1971.     

Heat transfer in plane film formation

A mathematical model is obtained for the process of cooling with formation of a planar film. The solution obtained is verified experimentally.Notation mean axial velocity gradient - v_x current axial velocity - v_o initial polymer velocity - v₁ sampling velocity - K draw ratio - deformation rate tensor - x, y, z spatial coordinates - X, Y dimensionless coordinates - L() differential operator - T temperature - T_o initial temperature - T_c temperature of surrounding medium - dimensionless temperature - dimensionless temperature averaged over film thickness - thermal-diffusivity coefficient - 2_o initial film thickness - thermal conductivity - heat-transfer coefficient - f(X) distance function - Bi Biot criterion, Bi_o, Biot criterion calculated for initial film thickness - Gz* modified Graetz criterion - V dimensionless velocity - ₁, ₂, ₃ heat-transfer coefficients produced by radiation, free convection, and forced convection - v_c, _c mean velocity and film half-thickness in formation zone - T₁ calculated temperature value - T₂ experimental temperature value - l formation zone length Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 37, No. 5, pp. 854–858, November, 1979.     

Construction of kernels G l, 0 l of the nonlinear collision integral in the Boltzmann equation for arbitrary l

It was shown in [1] that kernels L _l (v, v ₁) of linear collision integral and kernels G _{l, 0} ^l (v, v ₁, v ₂) of nonlinear collision integral are related by the Laplace transform. Here, analytical expressions are derived for nonlinear kernels G _{l, 0} ^+l (v, v ₁, v ₂) with arbitrary l for models of hard spheres and pseudo-Maxwellian molecules using the Laplace transform method.     

Complex moduli of aligned discontinuous fibre-reinforced polymer composites

This paper describes recent analytical and experimental efforts to determine the effects of fibre aspect ratio, fibre spacing, and the viscoelastic properties of constituent materials on the damping and stiffness of aligned discontinuous fibre-reinforced polymer matrix composites. This includes the analysis of trade-offs between damping and stiffness as the above parameters are varied. Two different analytical models show that there is an optimum fibre aspect ratio for maximum damping, and that the predicted optimum aspect ratios lie in the range of actual aspect ratios for whiskers and microfibres when the fibre damping is small. When the fibre damping is great enough, however, the optimum fibre aspect ratio corresponds to continuous fibre reinforcement. Experimental data for E-glass/epoxy specimens are presented for comparison with predictions.Nomenclature A _c,A_f,A_m Cross-sectional area of composite, fibre, and matrix, respectively - d Fibre diameter - E _c ^* ,E _f ^* ,E _m ^* Complex extensional modulus of composite, fibre, and matrix, respectively. - E_c,E_f,E_m Extensional storage modulus of composite, fibre, and matrix, respectively - E_c,E_f,E_m Extensional loss modulus of composite, fibre, and matrix, respectively - G_m Complex shear modulus of matrix - G_m Shear storage modulus of matrix - i –1^1/2 - K Defined in Equation A9 - K ₁ Defined in Equation A5 - l Fibre length - r Radial distance from centre of fibre - r ₀ Fibre radius - R Radius of representative volume element, or one-half of centre-to-centre fibre spacing - v _f,v _m Volume fraction of fibre and matrix, respectively - W _c Total strain energy stored in a unit volume of composite - W _f Strain energy stored in volumev _f of fibre - W _m Strain energy stored in a volumev _m of matrix - W _m Shear strain energy stored in a volumev _m of matrix - W _m Extensional strain energy stored in a volumev _m of matrix - w _rm Shear strain energy stored in the matrix inr ₀rR - w _f Extensional strain energy stored in a single fibre - x Distance along fibre from end of fibre - Defined in Equation 12 - Defined in Equation 2 - ^* Defined in Equation A2 - Extensional (longitudinal) strain - _c, _f, _m Extensional loss factor of composite, fibre, and matrix, respectively - _Gm Shear loss factor of matrix - Polar angle measured in a plane perpendicular to fibre axis - ¯gs _c,¯gs _f,¯gs _m Average longitudinal stress in composite, fibre, and matrix, respectively - _f Longitudinal stress in fibre - Shear stress in matrix - Defined in Equation 27     

On the analysis of the thermal diffusivity measurement method with modulated heat input

The present paper proposes a simplified way to analyze thermal diffusivity experiments in which the phase shift is measured between the modulations of the temperatures on either face of a disk-shaped sample. The direct application of complex numbers mathematics avoids the use of the cumbersome formulae which hitherto have hampered a wider confirmation of the method and which restricted the range of the phase lag to an angle of 180°. The algorithm exposed makes it more practical to refine the analysis, which may lead to a higher accuracy and a wider use of the method. The origins of some possible errors in the calculated results are briefly reviewed.Nomenclature a Thermal diffusivity, m² · s^–1 - c Index denoting a constant part, dimensionless - c _l, c ₀ Inverse extrapolation length, m^–1 - C _p Specific heat, J · kg^–1 · K^–1 - f Modulation frequency, Hz - l Thickness of disk-shaped sample, m - Q _c Equilibrium energy per unit surface deposited on surface x=l, W · m^–2 - Q _m(t) Energy of modulation per unit surface deposited on surface x=l, W · m^–2 - Q(t) Total energy per unit surface deposited on surface x=l, W · m^–2 - q Complex energy modulation amplitude, W · m^–2 - T _l Equilibrium temperature of heated surface, K - t ₀ Equilibrium temperature of nonheated surface, K - T(x, t) Total temperature of any plane at distance x and at time t, K - T _m(x, t) Modulation temperature at any distance x and at time t, K - t Time, s - x Distance perpendicular to the specimen's surface and with the nonheated surface as the reference, m - Thermal linear expansion coefficient, dimensionless - Intermediary parameter, m^–2 - Phase difference between heated and nonheated specimen face, radian - ₀ Phase difference between energy modulation and nonheated face, radian - _l Phase difference between energy modulation and heated face, radian - Total emissivity, dimensionless - _s Spectral emissivity, dimensionless - Temperature, amplitude of modulated part argument, K - Thermal conductivity, W · m^–1 · K^–1 - Density, kg · m^–3 - Stefan-Boltzmann constant, 5.66961×10^–8W · m^–2 · K^–4 - Angular frequency=2f, s^–1