Determination of specific thrust in unchoked regimes and design of a separationless convergent nozzle contour

The variation of the specific thrust R_Y on the angle of inclination of the wall is analyzed within the framework of the ideal gas model using the results of specific impulse and flow rate calculations for conical convergent nozzles. It is shown that in unchoked regimes nozzles with different have almost the same values of R_Y for both subcritical and supercritical pressure ratios _c. On the interval _C < 6 typical of convergent nozzles conical convergent nozzles with =30–90° have almost the same value of the specific thrust, maximal relative to the R_Y of nozzles with < 30°. In the presence of viscosity forces local boundary layer separation may occur in the neighborhood of the entrance section of the convergent nozzle. A method of constructing a separationless convergent nozzle contour with enhanced thrust is developed on the basis of a boundary layer separation criterion. The separationless contour is determined for given values of the flow rate, specific heat ratio, Reynolds number, wall temperature and initial boundary layer displacement thickness.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 158–164, January–February, 1990.     

On forced vibration of anisotropic cylinders

Summary The dynamic response of a circular cylinder with thick walls of transverse curvilinear isotropy subjected to a uniformly distributed pressure varying periodically with time is analyzed by means of the Laplace transformation, and the exact solution is obtained in closed form. The previously obtained solutions for forced vibrations with isotropy, and free vibrations with transverse curvilinear isotropy are included as special cases of the general results reported here.Nomenclature t time - r, , z cylindrical coordinates - _ii components of normal strain - _ii components of normal stress - u radial displacement - c _ij elastic constant - mass density - c ² c ₁₁/ - ² c ₂₂/c ₁₁ - a, b inner, outer radius of the cylinder - , A, B constants - forced angular frequency - function defined by (9) - p, real, complex variables - constant defined by (14) - real number - , Lamé elastic constants - J (x) Bessel function of first kind - Y (x) Bessel function of second kind - I (x) modified Bessel function of first kind - K (x) modified Bessel function of second kind     

Hypersonic flow over a narrow delta plate at large angles of attack

The problem of hypersonic flow over a flat delta plate with a high sweepback anglex at angles of attack close to /2 is solved using a numerical algorithm based on transition to the conical solution. The existence of conical flow at /2 with the velocity vector directed towards the apex of the plate is established. Values ofC _p/sin² and the thickness of the shock layer in the plane of symmetry of the plate are given as functions of the hypersonic similarity parameterk=tan tanx. A comparison of the calculated and experimental data shows that they are in good agreement.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.5, pp. 183–185, September–October, 1992.     

Transcendentally small transversality in the rapidly forced pendulum

The rapidly forced pendulum equation with forcing sin((t/), where =<₀^p,p = 5, for ₀, sufficiently small, is considered. We prove that stable and unstable manifolds split and that the splitting distanced(t) in the ( ,t) plane satisfiesd(t) = sin(t/) sech(/2) +O( ₀ exp(–/2)) (2.3a) and the angle of transversal intersection,, in thet = 0 section satisfies 2 tan/2 = 2S _s = (/2) sech(/2) +O(( ₀ /) exp(–/2)) (2.3b) It follows that the Melnikov term correctly predicts the exponentially small splitting and angle of transversality. Our method improves a previous result of Holmes, Marsden, and Scheuerle. Our proof is elementary and self-contained, includes a stable manifold theorem, and emphasizes the phase space geometry.     

Irrotational outflow of liquid from a stream through a lateral hole of finite depth

An algorithm is constructed for numerical determination of the flow parameters and coefficient of contraction of a jet in the case of irrotational lateral outflow of liquid from a semiinfinite stream through a nozzle of finite depth situated at an arbitrary angle to the mainstream flow. The solution is based on the use of N. E. Zhukovskii's method and the Schwarz-Christoffel formula. The results of calculations for a nozzle situated at an angle = /2 ± , where = /6, are given.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 162–164, January–February, 1977.     

Impacts of local dispersion and first-order decay on solute transport in randomly heterogeneous porous media

Stochastic subsurface transport theories either disregard local dispersion or take it to be constant. We offer an alternative Eulerian-Lagrangian formalism to account for both local dispersion and first-order mass removal (due to radioactive decay or biodegradation). It rests on a decomposition of the velocityv into a field-scale componentv , which is defined on the scale of measurement support, and a zero mean sub-field-scale componentv _s , which fluctuates randomly on scales smaller than. Without loss of generality, we work formally with unconditional statistics ofv _s and conditional statistics ofv . We then require that, within this (or other selected) working framework,v _s andv be mutually uncorrelated. This holds whenever the correlation scale ofv is large in comparison to that ofv _s . The formalism leads to an integro-differential equation for the conditional mean total concentration c which includes two dispersion terms, one field-scale and one sub-field-scale. It also leads to explicit expressions for conditional second moments of concentration cc. We solve the former, and evaluate the latter, for mildly fluctuatingv by means of an analytical-numerical method developed earlier by Zhang and Neuman. We present results in two-dimensional flow fields of unconditional (prior) mean uniformv . These show that the relative effect of local dispersion on first and second moments of concentration dies out locally as the corresponding dispersion tensor tends to zero. The effect also diminishes with time and source size. Our results thus do not support claims in the literature that local dispersion must always be accounted for, no matter how small it is. First-order decay reduces dispersion. This effect increases with time. However, these concentration moments c and cc of total concentrationc, which are associated with the scale below, cannot be used to estimate the field-scale concentrationc directly. To do so, a spatial average over the field measurement scale is needed. Nevertheless, our numerical results show that differences between the ensemble moments ofc and those ofc are negligible, especially for nonpoint sources, because the ensemble moments ofc are already smooth enough.     

Geometric properties of two phase flow in geothermal reservoirs

The two phase flow equations frequently used in geothermal engineering ignore capillary pressure, which results in a singular system of equations. Analysis of these equations reveals three mechanisms for altering saturation: local boiling, the spatial dependence of flowing enthalpy due to the convective transport of fluid, and counterflow. A scalar function is associated with each of these three mechanims. At each point in space, flows are essentially two dimensional, with gravity establishing a vertical hierarchy, in that volumetric, energy and mass fluxes can never point below a lower member in this triple. With increasing liquid saturation, the characteristics associated with the saturation equation move up from below this grouping of directions, and eventually may even point above volumetric fluxes. Finally, weak shocks and the associated entropy condition are considered. The characteristics of the saturation equation coincide with the velocity of extremely weak shocks, and saturation increases with the passage of a weak shock, provided the magnitude of the characteristic speed increases with saturation.Notation C_l liquid heat capacity - C_m rock heat capacity - C_v vapour heat capacity - G counterflow energy flux - h flowing enthalpy - hl liquid enthalpy - h_v vapour enthalpy - k permeability - k downward vector - P pressure - S liquid saturation - T temperature - dT/dP derivative at saturation - z vertical coordinate - l liquid viscosity - v vapour viscosity - Pl liquid density - _m rock density - _v vapour density - porosity     

Stability characteristics of condensate films

The linear stability theory is used to study stability characteristics of laminar condensate film flow down an arbitrarily inclined wall. A critical Reynolds number exists above which disturbances will be amplified. The magnitude of the critical Reynolds number is in all practical situations so small that a laminar gravity-induced condensate film can be expected to be unstable. Several stabilizing effects are acting on the film flow; at an inclined wall these effects are due to surface tension, gravity and condensation mass transfer.

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Zusammenfassung Mit Hilfe der linearen Stabilitätstheorie werden die Stabilitätseigenschaften laminarer Kondensatfilme an einer geneigten Wand untersucht. Es zeigt sich, daß Kondensatfilme in jedem praktischen Fall ein unstabiles Verhalten aufweisen. Der stabilisierende Einfluß von Oberflächenspannung, Schwerkraft und Stoffübertragung durch Kondensation bewkkt jedoch, daß Störungen in bestimmten Wellenlängenbereichen gedämpft werden.

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Nomenclature c=c^*/u₀ complex wave velocity, celerity, dimensionless - c^*=c _r ^* + i c _i ^* complex wave velocity, celerity, dimensional - c_p specific heat at constant pressure - g gravitational acceleration - h_fg latent heat - k thermal conductivity of liquid - p^* pressure - p=p^*/u₀ ² dimensionless pressure - Pe=Pr Re^* Peclet number - Pr Prandtl number - Re^*=u₀ / Reynolds number (defined with surface velocity) - S temperature perturbation amplitude - t^* time - t=t^* u₀/ dimensionless time - T temperature - T_s saturation temperature - T_w wall temperature - T=T_s-T_w temperature drop across liquid film - u^*, v^* velocity components - u=u^*/u₀ dimensionless velocity components - v=v^*/u₀ dimensionless velocity components - u₀ surface velocity of undisturbed film flow - v _g ^* vapor velocity - x^*, y^* coordinates - x=x^*/ dimensionless coordinates - y=y^*/ dimensionless coordinates Greek Symbols =^* wave number, dimensionless - ^*=2 /^* wave number dimensional - ^* wave length, dimensional - =^*/ wave length, dimensionless - local thickness of undisturbed condensate film - kinematic viscosity, liquid - density, liquid - _g density vapor - surface tension - = (1 +) film thickness of disturbed film, Fig. 1 - stream function perturbation amplitude - angle of inclination Base flow quantities are denoted by^–, disturbance quantities are denoted by.     

Identification of the Physical Parameters Used in the Thermo-Hygro-Mechanical Model

This paper describes different methods used to identify a large number of physical parameters of the thermo-hygro-mechanical coupling model. This model is developed on the basis of mechanics of porous media and deals with the prediction of response of a structure submitted to thermal, hygrometric and mechanical loading. The aim of this work is mainly to propose some experimental methods for the determination of physical parameters used previously in the model such as hygrometric parameters (liquid Biot's coefficient b _l , vapour and liquid permeability _v, _l and tangent capillary modulus N _ll). Thermal parameters such as thermal conductivity (), specific heat (C) and the thermo-hydrous expansion coefficient ( _i ^p ) have been identified using some works published previously. The different physical parameters were identified in the case of cement mortar without taking into account the influence of hysteresis.     

Interaction of a vortex couple with a free surface

The characteristics of three-dimensional flow structures (scars and striations) resulting from the interaction between a heterostrophic vortex pair in vertical ascent and a clean free surface are described. The flow features at the scar-striation interface (a constellation of whirls or coherent vortical structures) are investigated through the use of flow visualization, a motion analysis system, and the vortex-element method. The results suggest that the striations are a consequence of the short wavelength instability of the vortex pair and the helical instability of the tightly spiralled regions of vorticity. The whirls result from the interaction of striations with the surface vorticity. The whirl-merging is responsible for the reverse energy cascade leading to the formation and longevity of larger vortical structures amidst a rapidly decaying turbulent field.List of symbols A _c Area of a vortex core (Fig. 6b) - AR Aspect ratio of the delta wing model - B base width of delta wing - b ₀ initial separation of the vortex couple - d ₀ depth at which the vortex pair is generated - c average whirl spacing in the x-direction - E energy density - Fr Froude number ( ) - g gravitational acceleration - L length of the scar band - L _ko length of the Kelvin oval - N _w number of whirls in each scar band - P _c Perimeter of a vortex core - q surface velocity vector - r _c core size of the whirl ( = 2A _c/P _c) - Re Reynolds number ( = ) - Rnd a random number - s inboard edge of the scar front (Fig. 6 a) - t time - u velocity in the x-direction - velocity in the y-direction - V _b velocity imposed on a scar by the vortex couple (Fig. 6 a) - V ₀ initial mutual-induction velocity of the vortex couple (=₀/2b ₀) - V _t tangential velocity at the edge of the whirl core - w width of the scar front (Fig. 6 a) - z complex variable - z _k position of the whirl center - half included angle of V-shaped scar band - wave number - _m initial mean circulation of the whirls - ₀ initial circulation of the vortex pair - _w circulation of a whirl - _min minimum survival strength of a whirl - t time step - gDz increment of z - gD change in vorticity - cut-off distance in velocity calculations - critical merging distance - curvature of the surface - wavelength - kinematic viscosity - angular velocity of a whirl core     

Wärmestrahlung staubhaltiger Gase

Zusammenfassung Nach einem mehr qualitativen theoretischen Überblick über Absorption und Streuung der Strahlung an kleinen Partikeln wird gezeigt, daß sich bei kleinen optischen Dichten als analytischer Ausdruck für die Emissionszahl der Staubstrahlung ein der Gasstrahlung analoger Ausdruck ergibt. Dieses Ergebnis wird durch Messungen bestätigt. Insgesamt werden die Emissionsdaten von 20 untersuchten Kesselstäuben angegeben und interpretiert. Es werden Durchschnittswerte empfohlen, um bei Stäuben mit unbekannten Strahlungsdaten näherungsweise Austauschrechnungen durchzuführen. Die Untersuchungen gelten für Strahlungsräume von annähernd konstanter Temperatur.

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Radiation of gases containing dust particles

Having presented a more qualitative short survey about absorption and scattering on small particles, it is shown that in the range of small optical thicknesses expressions for the emissivity of dust clouds are analogue to those of gases. Measurements confirm this. The emissivities of twenty different dust materials are measured and interpreted. For calculations with unknown materials average emissivity data are recommended.

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Bezeichnungen A Koeffizient für Absorption bzw. Emission - B Staubbeladung, kg m^–3 - d Durchmesser, m - E Koeffizient für Extinktion - E _sn Intensität der schwarzen Strahlung, Watt/m² Raumwinkel - f Querschnittfläche kugelförmiger Teilchen,d ²/4, m² - F Spezifische Projektionsfläche 3/2 ^St d, m²kg^–1 - I Intensität, W/m² Raumwinkel - k Stoffkonstante, m^–1/3 - k _g Absorptionskoeffizient des Gases, m^–1 - L Schichtstärke, m - L _ä Äquivalente Schichtstärke, m - N Teilchenzahl pro Volumeneinheit, m^–3 - p Größenparameter d/ - S Streukoeffizient - S _V Streukoeffizient in Vorwärtsrichtung - S _R Streukoeffizient in Rückwärtsrichtung - T absolute Temperature, °K - _G Absorptionszahl des Gases - _St Absorptionszahl des Staubes, _St=_St - _G Emissionszahl des Gases - _St Emissionszahl des Staubes - _W Emissionszahl der Wand - _0,5 Bezugswellenlänge, , m - Wellenlänge, , m - _St Staubdichte, kg m^–3 - Optische DichteA F B L, A f N L     

An investigation of meterial properties of non-homogeneous cylindrically orthotropic circular plates

It has been been reported that the reduced stiffness of non-homogeneous cylindricallyorthotrpic circular plate varing exponentially with radius r is obtained by using thebending theory of a simple beamThe aim of this paper is to verify the effect of radius on the materal properties According to the flat stress-strain relation the values of material properties E_r E_a and v_θr which are the functions of radius r are obtained.Compared with the experimentalvalues the analytical values of the material properties are in essential agreement withtem.     

The chaotic response of a fluttering panel: The influence of maneuvering

The influence of maneuvering on the chaotic response of a fluttering buckled plate on an aircraft has been studied. The governing equations, derived using Lagrangian mechanics, include geometric non-linearities associated with the occurrence of tensile stresses, as well as coupling between the angular velocity of the maneuver and the elastic degrees of freedom. Numerical simulation for periodic and chaotic responses are conducted in order to analyze the influence of the pull-up maneuver on the dynamic behavior of the panel. Long-time histories phase-plane plots, and power spectra of the responses are presented. As the maneuver (load factor) increases, the system exhibits complicated dynamic behavior including a direct and inverse cascade of subharmonic bifurcations, intermittency, and chaos. Beside these classical routes of transition from a periodic state to chaos, our calculations suggest amplitude modulation as a possible new mode of transition to chaos. Consequently this research contributes to the understanding of the mechanisms through which the transition between periodic and strange attractors occurs in, dissipative mechanical systems. In the case of a prescribed time dependent maneuver, a remarkable transition between the different types of limit cycles is presented.Nomenclature a plate length - a _r u _r /h - D plate bending stiffness - E modulus of elasticity - g acceleration due to gravity - h plate thickness - j₁,j₂,j₃ base vectors of the body frame of reference - K spring constant - M Mach number - n 1 + ₀/g - N ₁ applied in-plane force - p–p aerodynamic pressure - P pa ⁴/Dh - q ₀/2 - Q _r generalized Lagrangian forces - R rotation matrix - R ₄ N, a ²/D - t time - kinetic energy - u plate deflection - u displacement of the structure - u _r modal amplitude - v₀ velocity - x coordinates in the inertial frame of reference - z coordinates in the body frame of reference - Ka/(Ka+Eh) - - elastic energy - 2qa ³/D - a/_mh - Poisson's ratio - material coordinates - air density - _m plate density - - _r prescribed functions - _r sin(r z/a) - angular velocity - a/v₀ - skew-symmetric matrix form of the angular velocity     

Shear-induced aggregation and break up of fibril clusters close to the percolation concentration

To probe the behaviour of fibrillar assemblies of ovalbumin under oscillatory shear, close to the percolation concentration, c_p (7.5%), rheo-optical measurements and Fourier transform rheology were performed. Different results were found close to c_p (7.3%), compared to slightly further away from c_p (6.9 and 7.1%). For 6.9 and 7.1%, a decrease in complex viscosity, and a linear increase in birefringence, n, with increasing strain was observed, indicating deformation and orientation of the fibril clusters. For 7.3%, a decrease in complex viscosity was followed by an increase in complex viscosity with increasing strain, which coincided with a strong increase in n, dichroism, n, and the intensity of the normalized third harmonic (I₃/I₁). This regime was followed by a second decrease in complex viscosity, where n,n and I₃/I₁ decreased. In the first regime where the viscosity was decreasing with increasing strain, deformation and orientation of existing clusters takes place. At higher oscillatory shear, a larger deformation occurs and larger structures are formed, which is most likely aggregation of the clusters. Finally, at even higher strains, the clusters break up again. An increase in complex viscosity, n, n and I₃/I₁ was observed when a second strain sweep was performed 30 min after the first. This indicates that the shear-induced cluster formation and break up are not completely reversible, and the initial cluster size distribution is not recovered after cessation of flow.     

An experimental investigation of separating flow on a convex surface

The mean and turbulent characteristics of an incompressible turbulent boundary layer developing on a convex surface under the influence of an adverse pressure gradient are presented in this paper.The turbulence quantities measured include all the components of Reynolds stresses, auto-correlation functions and power spectra of the three components of turbulence. The results indicate the comparative influence of the convex curvature and adverse pressure gradient which are simultaneously acting on the flow. The investigation provides extensive experimental information which is much needed for a better understanding of turbulent shear flows.Nomenclature a, b constants in equation for velocity defect profile (Fig. 6) - c _f skin-friction coefficient (= _w/F _1/2 U ₁ ² ) - E(k ₁) one-dimensional wave number spectra - f frequency in Hz - G Clauser's equilibrium parameter = (H–1)/H(c _f /2) - H shape parameter (= ₁/ ₂) - k ₁ wave number (=2f/U) - L _u, L _v, L _w length scales of u, v and w fluctuations - p _s static pressure on the measurement surface - p _w reference tunnel wall static pressure - q ² total turbulent kinetic energy - R radius of curvature of the convex surface - R() auto-correlation function - T _u, T _v, T _w time scales of u, v and w fluctuations - U local mean velocity - U ₁ local free stream velocity - U _* friction velocity - u, v, w velocity fluctuations in x, y and z directions respectively - X streamwise coordinate measured along the surface from A (Fig. 1b) - x streamwise coordinate measured along the surface reckoned from station 9 - y coordinate normal to the surface - z spanwise coordinate - ₁/ _w · dp/dx - - boundary layer thickness - ₁ displacement thickness - ₂ momentum thickness - ₃ energy thickness - kinematic viscosity - density - time delay - _w wall shear stress     

Dynamic behaviour of the constant temperature anemometer due to thermal inertia of the wire

The time dependent differential equation for the local wire temperature of a constant temperature anemometer is solved by a perturbation method in case of a harmonically changing heat transfer coefficient. The time dependent power supply to the wire follows from the condition of constant mean temperature imposed by the anemometer circuit. The influence of thin supporting wires, or copper-plated wire ends, is evaluated also. Numerical results are given for a number of cases that are of practical interest.Nomenclature c specific heat - D diameter of the wire - D _u diameter of the copper-plated ends of the wire - f D - g I ² r ₀ - I electric current - L length of the wire - P 1/4D ² c - q 1/4D ² - r resistance of the wire per unit length at temperature T' - r ₀ resistance of the wire per unit length at temperature T - T T' – T - T' local temperature of the wire - T ambient temperature - T _w constant mean temperature imposed by the anemometer circuit - T _u local difference between the temperature of the supporting wire and the ambient temperature - t time - x axial coordinate with the origin in the middle of the wire - heat transfer coefficient - temperature coefficient of the resistance - small parameter - time constant = cD ²/4D - _u time constant of the copper-plated ends cD _u ² /4D _u - thermal conductivity of wire material - _u thermal conductivity of the copper-plated wire ends - density - circular frequency     

Thermal stresses in rectangular plates

Summary A method of determining the thermal stresses in a flat rectangular isotropic plate of constant thickness with arbitrary temperature distribution in the plane of the plate and with no variation in temperature through the thickness is presented. The thermal stress have been obtained in terms of Fourier series and integrals that satisfy the differential equation and the boundary conditions. Several examples have been presented to show the application of the method.Nomenclature x, y rectangular coordinates - _x, _y direct stresses - _xy shear stress - ø Airy's stress function - E Young's modulus of elasticity - coefficient of thermal expansion - T temperature - ² Laplace operator: - ⁴ biharmonic operator - 2a length of the plate - 2b width of the plate - a/b aspect ratio - a _mr, b_ms, c_nr, d_ns Fourier coefficients defined in equation (6) - _m=m/a m=1, 2, 3, ... _n=n/2a n=1, 3, 5, ... - _r=r/b r=1, 2, 3, ... _s=s/2b s=1, 3, 5, ... - A _m, B_m, C_n, D_n, E_r, F_r, G_s, H_s Fourier coefficients - K _rand L _s Fourier coefficients defined in equation (20) - direct stress at infinity - T ₁(x, y) temperature distribution symmetrical in x and y - T ₂(x, y) temperature distribution symmetrical in x and antisymmetrical in y - T ₃(x, y) temperature distribution antisymmetrical in x and symmetrical in y - T ₄(x, y) temperature distribution antisymmetrical in x and y     

Natural convection heat transfer through an enclosed horizontal layer of supercritical carbon dioxide

In natural convection heat transfer through a thin horizontal layer of carbon dioxide, maxima in the equivalent thermal conductivities are obtained in the vicinity of the respective pseudocritical temperatures at pressures of 75.8, 89.6 and 103.4 bar. The maxima are the more pronounced, the closer the critical point is approached.Comparison of experimental results with Nusselt equations shows good agreement except for the immediate vicinity of the pseudocritical temperature.In visual observations a distinct change in flow structure appears in the immediate vicinity of the pseudocritical temperature. A steady state polygon pattern and a boiling-like action could not be observed in this geometry.

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Zusammenfassung Beim Wärmetransport durch freie Konvektion in einer dünnen waagerechten Schicht von Kohlendioxid ergaben sich Maxima der scheinbaren Wärmeleitfähigkeit in der Nähe der pseudokritischen Temperaturen bei Drükken von 75,8, 89,6 und 103,4 bar. Die Maxima sind um so ausgeprägter, je mehr man sich dem kritischen Punkt nähert.Ein Vergleich der Versuchsergebnisse mit Nusseltbeziehungen ergibt gute Übereinstimmung außer in unmittelbarer Umgebung der pseudokritischen Temperatur. Direkte Beobachtungen der Konvektionsmuster zeigen in unmittelbarer Umgebung der pseudokritischen Temperatur eine deutliche Strukturänderung. Ein stationäres Zellmuster und siedeähnliche Vorgänge konnten in dieser Anordnung nicht beobachtet werden.

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Nomenclature A area of the heating or cooling plate - C constant in the correlation - g acceleration of gravity - h heat transfer coefficient - k thermal conductivity of fluid in the gap - k _e equivalent thermal conductivity - m, n exponents of dimensionless numbers - q heat flux - T _C,PC absolute temperature; critical C, pseudocritical PC - Gr Grashof numberg ( _h– _c) ³/ ² - Nu Nusselt numberh/k - Pr Prandtl number/ - thermal diffusivity - coefficient of volume expansion - width of gap - _c,h temperature of cooling (c)-, heating (h)-plate - _m arithmetic mean temperature ( _c+ _h)/2 - kinematic viscosity - _c,h fluid density at the temperature of the cooling (c)- or heating (h)-plate - heat flow rate through the gap     

A visualization study of the interaction of a free vortex with the wake behind an airfoil

The two-dimensional interaction of a single vortex with a thin symmetrical airfoil and its vortex wake has been investigated in a low turbulence wind tunnel having velocity of about 2 m/s in the measuring section. The flow Reynolds number based on the airfoil chord length was 4.5 × 10³. The investigation was carried out using a smoke-wire visualization technique with some support of standard hot-wire measurements. The experiment has proved that under certain conditions the vortex-airfoil-wake interaction leads to the formation of new vortices from the part of the wake positioned closely to the vortex. After the formation, the vortices rotate in the direction opposite to that of the incident vortex.List of symbols c test airfoil chord - C vortex generator airfoil chord - TA test airfoil - TE test airfoil trailing edge - TE _G vortex generator airfoil trailing edge - t dimensionless time-interval measured from the vortex passage by the test airfoil trailing edge: gDt=(T-T- _TE)·U/c - T time-interval measured from the start of VGA rotation - U free stream velocity - U vortex induced velocity fluctuation - VGA vortex generator airfoil - y distance in which the vortex passes the test airfoil - Z vortex circulation coefficient: Z=/(U · c/2) - vortex generator airfoil inclination angle - vortex circulation - vortex strength: =/2     

Flow visualization of compressible vortex structures using density gradient techniques

Mathematical results are derived for the schlieren and shadowgraph contrast variation due to the refraction of light rays passing through two-dimensional compressible vortices with viscous cores. Both standard and small-disturbance solutions are obtained. It is shown that schlieren and shadowgraph produce substantially different contrast profiles. Further, the shadowgraph contrast variation is shown to be very sensitive to the vortex velocity profile and is also dependent on the location of the peak peripheral velocity (viscous core radius). The computed results are compared to actual contrast measurements made for rotor tip vortices using the shadowgraph flow visualization technique. The work helps to clarify the relationships between the observed contrast and the structure of vortical structures in density gradient based flow visualization experiments.Nomenclature a Unobstructed height of schlieren light source in cutoff plane, m - c Blade chord, m - f Focal length of schlieren focusing mirror, m - C _T Rotor thrust coefficient, T/( ² R ⁴) - I Image screen illumination, Lm/m ² - l Distance from vortex to shadowgraph screen, m - n _b Number of blades - p Pressure,N/m ² - p Ambient pressure, N/m ² - r, , z Cylindrical coordinate system - r _c Vortex core radius, m - Non-dimensional radial coordinate, (r/r _c ) - R Rotor radius, m - Tangential velocity, m/s - Specific heat ratio of air - Circulation (strength of vortex), m ²/s - Non-dimensional quantity, ² 8²p r _c ² - Refractive index of fluid medium - ₀ Refractive index of fluid medium at reference conditions - Gladstone-Dale constant, m ³/kg - Density, kg/m ³ - Density at ambient conditions, kg/m ³ - Non-dimensional density, (/ ) - Rotor solidity, (n _b c/ R) - Rotor rotational frequency, rad/s