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.     

An analytical solution for the temperature profiles during injection molding,including dissipation effects

An analytical solution is obtained for the stationary temperature profile in a polymeric melt flowing into a cold cavity, which also takes into account viscous heating effects. The solution is valid for the injection stage of the molding process. Although the analytical solution is only possible after making several (at first sight) rather stringent assumptions, the calculated temperature field turns out to give a fair agreement with a numerical, more realistic approach. Approximate functions were derived for both the dissipation-independent and the dissipation-dependent parts which greatly facilitate the temperature calculations. In particular, a closed-form expression is derived for the position where the maximum temperature occurs and for the thickness of the solidified layer.The expression for the temperature field is a special case of the solution of the diffusion equation with variable coefficients and a source term.Nomenclature a thermal diffusivity [m²/s] - c specific heat [J/kg K] - D channel half-height [m] - L channel length [m] - m 1/ - P pressure [Pa] - T temperature [°C] - T _W wall temperature [°C] - T _i injection temperature [°C] - T _A Br independent part of T - T _B Br dependent part of T - T ^core asymptotic temperature - v _z() axial velocity [m/s] - W channel width [m] - x cross-channel direction [m] - z axial coordinate [m] - (x) gamma function - (a, x) incomplete gamma function - M(a, b, x) Kummer function - small parameter - () temperature function - thermal conductivity [W/mK] - viscosity [Pa · s] - ₀ consistency index - power-law exponent - density [kg/m] - similarity variable Dimensionless variables Br Brinkman number - Gz Graetz number -      

On surface waves in generalized thermoelasticity

Knowles' representation theorem for harmonically time-dependent free surface waves on a homogeneous, isotropic elastic half-space is extended to include harmonically time-dependent free processes for thermoelastic surface waves in generalized thermoelasticity of Lord and Shulman and of Green and Lindsay.r , , r , , .This work was done when author was unemployed.     

Experimental study of stability of strongly underexpanded laminar free and impact jets

The growth rates of density fluctuations in the mixing layer of strongly underexpanded lowdensity jets are measured by the electronbeam technique within the range of Reynolds numbers Re_L = 50 – 230. Regimes of selfinduced oscillations are determined for these flow conditions in the case of a jet incident normally onto a finitesize target. Results on the frequency of pressure oscillations on the target are obtained and compared with the spectra of the growth rates of density fluctuations in the mixing layer of free jets. It is shown that selfinduced oscillations cannot be sustained because of the development of instability in the mixing layer of the jet.     

Transport in ordered and disordered porous media II: Generalized volume averaging

In this paper we develop the averaged form of the Stokes equations in terms of weighting functions. The analysis clearly indicates at what point one must choose a media-specific weighting function in order to achieve spatially smoothed transport equations. The form of the weighting function that produces the cellular average is derived, and some important geometrical theorems are presented.Roman Letters A interfacial area of the- interface associated with the local closure problem, m² - A _e area of entrances and exits for the-phase contained within the averaging system, m² - A _p surface area of a particle, m² - d _p 6V _p/A_p, effective particle diameter, m - g gravity vector, m/s² - I unit tensor - K _m permeability tensor for the weighted average form of Darcy's law, m² - L general characteristic length for volume averaged quantities, m - L _p general characteristic length for volume averaged pressure, m - L characteristic length for the porosity, m - L _v characteristic length for the volume averaged velocity, m - l characteristic length (pore scale) for the-phase - l _i i=1, 2, 3 lattice vectors, m - (y) weighting function - m(–y) (y), convolution product weighting function - _v special weighting function associated with the traditional averaging volume - m _v special convolution product weighting function associated with the traditional averaging volume - m _g general convolution product weighting function - m _V unit cell convolution product weighting function - m _C special convolution product weighting function for ordered media which produces the cellular average - m _D special convolution product weighting function for disordered media - m _M master convolution product weighting function for ordered and disordered media - n unit normal vector pointing from the-phase toward the-phase - p pressure in the-phase, N/m² - p_m superficial weighted average pressure, N/m² - p _m intrinsic weighted average pressure, N/m² - p traditional intrinsic volume averaged pressure, N/m² - p p p _m , spatial deviation pressure, N/m² - r ₀ radius of a spherical averaging volume, m - r _m support of the convolution product weighting function, m - r position vector, m - r position vector locating points in the-phase, m - V averaging volume, m³ - V volume of the-phase contained in the averaging volume, m³ - V _cell volume of a unit cell, m³ - V velocity vector in the-phase, m/s - vm superficial weighted average velocity, m/s - v _m intrinsic weighted average velocity, m/s - V volume of the-phase contained in the averaging volume, m³ - V _p volume of a particle, m³ - v traditional superficial volume averaged velocity, m/s - v v p _m spatial deviation velocity, m/s - x position vector locating the centroid of the averaging volume or the convolution product weighting function, m - y position vector relative to the centroid, m - y position vector locating points in the-phase relative to the centroid, m Greek Letters indicator function for the-phase - Dirac distribution associated with the- interface - V /V, volume average porosity - _m m * . weighted average porosity - mass density of the-phase, kg/m³ - viscosity of the-phase, Ns/m² - V /V, volume fraction of the-phase     

Explanation of the “anomalous” dispersal of the discrete particles in a two-phase turbulent jet with allowance for their inertia

The anomalous dispersal of the discrete particles in a two-phase turbulent jet (growth of their concentration in the region of the axis, rapid damping of the concentration in the initial sections of the jet, and wavelike axial distribution) is studied with allowance for the initial conditions and the Magnus effect.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 36–41, January–February, 1984.I thank G. N. Abramovich and A. N. Sekundov for discussing the results.     

The effect of background particulates on the delayed transition of a heated 9:1 ellipsoid

An experimental study was done to quantify the effects of a variety of background particulates on the delayed laminar-turbulent transition of a thermally stabilized boundary layer in water. A Laser-Doppler Velocimeter system was used to measure the location of boundary layer transition on a 50 mm diameter, 9:1 fineness ratio ellipsoid. The ellipsoid had a 0.15 m RMS surface finish. Boundary layer transition locations were determined for length Reynolds numbers ranging from 3.0 × 10⁶ to 7.5 × 106. The ellipsoid was tested in three different heating conditions in water seeded with particles of four distinct size ranges. For each level of boundary layer heating, measurements of transition were made for clean water and subsequently, water seeded with 12.5 m, 38.9 m, 85.5 m and 123.2 m particles, alternately. The three surface heating conditions tested were no heating, T = 10°C and T = 15°C where T is the difference between the inlet model heating water temperature, T _i, and free stream water temperature, T . The effects of particle concentration were studied for 85.5 m and 123.2 m particulates.The results of the study can be summarized as follows. The 12.5 m and 38.9 m particles has no measurable effect on transition for any of the test conditions. However, transition was significantly affected by the 85.5 m and 123.2 m particles. Above a length Reynolds number of 4 × 10⁶ the boundary layer transition location moved forward on the body due to the effect of the 85.5 m particles for all heating conditions. The largest percentage changes in transition location from clean water, were observed for 85.5 m particles seeded water.Transition measurements made with varied concentrations of background particulates indicated that the effect of the 85.5 m particles on the transition of the model reached a plateau between 2.65 particulates/ml concentration and 4.2 particles/ml. Measurements made with 123.3 m particles at concentrations up to 0.3 part/ml indicated no similar plateau.     

Effect of blowing on the resistance of a sphere in laminar flow of viscous fluid

Using the method of matching asymptotic expansions [1–3], a stationary field of velocities is obtained in the vicinity of a sphere for Reynolds numbers R and R computed from the blowing velocity and the fluid flow, respectively; they satisfy the relations R² 1 and R 1. It is also shown that for intensive blowing (R 1), the resistive force is considerably smaller than that found by using the Stokes formula. For weak blowing the results are in good agreement with the solution of Oseen.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskai Fiziki, No. 3, pp. 110–114, May–June, 1972.     

Filament stretching equation in Lagrangian coordinates

The general integral of the very simple equation ^21/n/() was found to describe the cross sectional area of filaments of isothermal power law fluids while in transient stretching where is time and is the initial location of fluid molecules at time = 0 given as the distance from a reference point fixed in space. Any such stretching transient given as a solution of the above equation is physically realizable subject to the restrictions > 0 and/ < 0.     

Effect of the Streamwise Component of the Vorticity Formed in a Turbulent Jet Source on the Acoustic Characteristics of the Jet

The results of an experimental study of the effects of different nozzle heads on turbulent jet noise are analyzed. A configuration of four cylindrical heads, tabbed heads, and chevron nozzles are considered and the decreases in the acoustic-mechanical efficiency of the jet (acoustic power reduction) for jets exposed to different modes of action are compared.It is shown that the effects of tabbed and cylindrical heads, as well as of chevrons, share a common property which is associated with the occurrence of vorticity in the jet source and can be described on the basis of a unified criterion characterizing the action on both the jet flow structure and the jet noise.     

Flow in vicinity of blunt body stagnation point in diverging hypersonic stream

In the hypersonic thin shock layer approximation for a small ratio k of the densities before and after the normal shock wave the solution of [1] for the vicinity of the stagnation point of a smooth blunt body is extended to the case of nonuniform outer flow. It is shown that the effect of this nonuniformity can be taken into account with the aid of the effective shock wave radius of curvature R_*, whose introduction makes it possible to reduce to universal relations the data for different nonuniform outer flows with practically the same similarity criterion k. The results of the study are compared with numerical calculations of highly underexpanded jet flow past a sphere.Notations x, y a curvilinear coordinate system with axes directed respectively along and normal to the body surface with origin at the forward stagnation point - R radius of curvature of the meridional plane of the body surface - uV, vV., , p V ² respectively the velocity projections on the x, y axes, density, and pressure - and V freestream density and velocity The indices =0 and=1 apply to plane and axisymmetric flows Izv. AN SSSR, Mekhanika Zhidkosti i Gaza, Vol. 5, No. 3, pp. 102–105, 1970.     

Structure of a supersonic jet with varied geometry of the nozzle entrance

The structure of a supersonic underexpanded lowpressureratio jet exhausted from a nozzle with variable geometry of the entrance section is experimentally studied. Total pressure distributions in the initial cross sections of the examined jets are obtained. Based on these distributions, the coordinates of the mixingregion boundaries are found. The curvature of streamlines in the mixing layer within the first two barrels of an underexpanded jet is determined. A dependence generalizing the measurement results on the curvature of streamlines in the first barrel of a weakly underexpanded jet is obtained in dimensionless coordinates.     

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.     

Helium jets discharging normally into a swirling air flow

The characteristics of helium jets injected normally to a swirling air flow are investigated experimentally using laser Doppler and hot-wire anemometers. Two jets with jet-to-crossflow momentum flux ratios of 0.28 and 12.6 are examined. The jets follow a spiral path similar to that found in the swirling air flow alone. Swirl acts to decrease jet penetration, but this is being counteracted by the lighter jet fluid density which is being pressed towards the tube center by the inward pressure gradient. Consequently, in spite of the large variation in momentum flux ratio, jet penetration into the main flow for the two jets investigated is about the same. The presence of the jet is felt only along the spiral path and none at all outside this region. Upstream of the jet, the oncoming swirling flow is essentially unaffected. These characteristics are quite different from jets discharging into a uniform crossflow at about the same momentum flux ratios, and can be attributed to the combined effects of swirl and density difference between the jet fluid and the air stream. Finally, the jets lose their identity in about fifteen jet diameters.List of symbols C mean volume concentration of helium - C _j mean volume concentration of helium at jet exit - c fluctuating volume concentration of helium - instantaneous volume concentration of helium - c RMS volume concentration of helium - D _j jet nozzle diameter - D _T diameter of tube - F flatness factor of c - J = _j U _j ² / _a U _a ^{gn 2} jet-to-crossflow momentum flux ratio - P(c) probability density function of c - r radial coordinate measured from tube centerline - R = D _T /2 radius of tube - Re _j = D _j U _j / _j jet Reynolds number - S = = tan swirl number - Sk skewness of c - instantaneous axial velocity - u RMS axial velocity - U mean axial velocity - local average mean axial velocity across tube - U _j jet exit velocity - U _a overall average mean axial velocity across tube - instantaneous circumferential velocity - w RMS circumferential velocity - W mean circumferential velocity - x axial coordinate measured from exit plane of swirler - x ₁ axial coordinate measured from centerplane of normal jet - y normal distance measured from tube wall - _j jet fluid kinematic viscosity - _a air density - _j jet fluid density - vane angle (constant)     

On laminar flow through a uniformly porous pipe

Numerous investigations ([1] and [4–9]) have been made of laminar flow in a uniformly porous circular pipe with constant suction or injection applied at the wall. The object of this paper is to give a complete analysis of the numerical and theoretical solutions of this problem. It is shown that two solutions exist for all values of injection as well as the dual solutions for suction which had been noted by previous investigators. Analytical solutions are derived for large suction and injection; for large suction a viscous layer occurs at the wall while for large injection one solution has a viscous layer at the centre of the channel and the other has no viscous layer anywhere. Approximate analytic solutions are also given for small values of suction and injection.

Nomenclature

General r distance measured radially - z distance measured along axis of pipe - u velocity component in direction of z increasing - v velocity component in direction of r increasing - p pressure - density - coefficient of kinematic viscosity - a radius of pipe - V velocity of suction at the wall - r ²/a ² - R wall or suction Reynolds number, Va/ - f() similarity function defined in (6) - u ₀() eigensolution - U(0) a velocity at z=0 - K an arbitrary constant - B _K Bernoulli numbers Particular Section 5 perturbation parameter, –2/R - ² a constant, –K - x / - g(x) f()/ Section 6 perturbation parameter, –R/2 - ² a constant, –K - g() f() - g _c ()=g() near centre of pipe - * point where g()=0 Section 7 2/R - ² K - t (1–)/ - w(t, ) [1–f(t)]/ - ₀, ₁ constants - g() f()– ₀ - ₀/ - ₀ a constant - * point where f()=0     

Effect of Dynamic Prehistory on Aerodynamics of a Laminar Separated Flow in a Channel Behind a Rectangular Backward‐Facing Step

The effect of dynamic prehistory of the flow and the channelexpansion ratio on aerodynamics of a steady separated laminar flow behind a rectangular backwardfacing step located in a planeparallel channel is numerically studied. It is shown that the boundary layer upstream of the flow separation exerts a strong effect on flow characteristics behind the step. A decrease in the boundarylayer thickness in the cross section of the step leads to a decrease in the separationregion length, and an increase in the channelexpansion ratio with a fixed initial boundarylayer thickness and Reynolds number leads to an increase in the separationregion length.     

Pressure pulsations during interaction between a subsonic jet and the subsonic section of a supersonic jeton a flat obstacle

Pressure pulsations were measured during in-leakage of a subsonic jet and the subsonic section of a heated supersonic jet on a flat obstacle. Data have been obtained on the total and spectrum levels of the pressure pulsations at different spacings X of the obstacle from the nozzle exit. It is shown that when the obstacle is disposed at the section of the jet where the local velocity is subsonic, the pulsation levels outside the dependence on the conditions at the nozzle exit (Mach number Maxa 0 a 3.0; stagnation temperature T₀=280–1200K) vary in direct proportion to the local velocity head q. The ratio between the total level and q is (/g)=0.2–0.3. It is established that for a subsonic velocity ahead of the obstacle, all the spectra obtained for different values of M _a , T₀, d _a and X in the coordinates Sh=f(d/V) and (₁*/q)(V/d) will lie on a single generalized spectrum. Here ₁* is the pulsation level in a 1-Hz band, and d and V are, respectively, the jet diameter and velocity directly in front of the obstacle.Translated from Izvestiya Akademiya Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 172–174, September–October, 1975.     

Mean and turbulent velocity measurements of supersonic mixing layers

The behavior of supersonic mixing layers under three conditions has been examined by schlieren photography and laser Doppler velocimetry. In the schlieren photographs, some large-scale, repetitive patterns were observed within the mixing layer; however, these structures do not appear to dominate the mixing layer character under the present flow conditions. It was found that higher levels of secondary freestream turbulence did not increase the peak turbulence intensity observed within the mixing layer, but slightly increased the growth rate. Higher levels of freestream turbulence also reduced the axial distance required for development of the mean velocity. At higher convective Mach numbers, the mixing layer growth rate was found to be smaller than that of an incompressible mixing layer at the same velocity and freestream density ratio. The increase in convective Mach number also caused a decrease in the turbulence intensity ( _u/U).List of symbols a speed of sound - b total mixing layer thickness between U ₁ – 0.1 U and U ₂ + 0.1 U - f normalized third moment of u-velocity, f u³/(U)³ - g normalized triple product of u² , g u²/(U)³ - h normalized triple product of u ², h u ²/(U)³ - l _u axial distance for similarity in the mean velocity - l _u axial distance for similarity in the turbulence intensity - M Mach number - M _c convective Mach number (for ₁ = ₂), M _c (U ₁ – U ₂)/(a ₁ + a ₂) - P static pressure - r freestream velocity ratio, r U ₂/U ₁ - Re unit Reynolds number, Re U/ - s freestream density ratio, s ₂/₁ - T _t total temperature - u instantaneous streamwise velocity - u deviation of u-velocity, uu – U - U local mean streamwise velocity - U ₁ primary freestream velocity - U ₂ secondary freestream velocity - average of freestream velocities, (U ₁ + U ₂)/2 - U freestream velocity difference, U U ₁ – U ₂ - instantaneous transverse velocity - v deviation of -velocity, – V - V local mean transverse velocity - x streamwise coordinate - y transverse coordinate - y ₀ transverse location of the mixing layer centerline - ensemble average - ratio of specific heats - boundary layer thickness (y-location at 99.5% of free-stream velocity) - similarity coordinate, (y – y ₀)/b - compressible boundary layer momentum thickness - viscosity - density - standard deviation - dimensionless velocity, (U – U ₂)/U - 1 primary stream - 2 secondary stream A version of this paper was presented at the 11th Symposium on Turbulence, October 17–19, 1988, University of Missouri-Rolla     

Normal forms for random diffeomorphisms

Given a dynamical system (,, ,) and a random diffeomorphism (): ^{d d} with fixed point at x=0. The normal form problem is to construct a smooth near-identity nonlinear random coordinate transformation h() to make the random diffeomorphism ()=h()^–1() h() as simple as possible, preferably linear. The linearization D(, 0)=:A() generates a matrix cocycle for which the multiplicative ergodic theorem holds, providing us with stochastic analogues of eigenvalues (Lyapunov exponents) and eigenspaces. Now the development runs pretty much parallel to the deterministic one, the difference being that the appearance of turns all problems into infinite-dimensional ones. In particular, the range of the homological operator is in general not closed, making the conceptof-normal form necessary. The stochastic versions of resonance and averaging are developed. The case of simple Lyapunov spectrum is treated in detail.     

Free convection heat transfer from an isothermal plate with arbitrary inclination

This paper presents a new formulation for the laminar free convection from an arbitrarily inclined isothermal plate to fluids of any Prandtl number between 0.001 and infinity. A novel inclination parameter is proposed such that all cases of the horizontal, inclined and vertical plates can be described by a single set of transformed equations. Moreover, the self-similar equations for the limiting cases of the horizontal and vertical plates are recovered from the transformed equations by setting=0 and=1, respectively. Heated upward-facing plates with positive and negative inclination angles are investigated. A very accurate correlation equation of the local Nusselt number is developed for arbitrary inclination angle and for 0.001 Pr .

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Wärmeübertragung bei freier Konvektion an einer isothermen Platte mit beliebiger Neigung

Zusammenfasssung Diese Untersuchung stellt eine neue Formulierung der laminaren freien Konvektion von Flüssigkeiten mit einer Prandtl-Zahl zwischen 0,001 und unendlich an einer beliebig schräggestellten isothermen Platte dar. Ein neuer Neigungsparameter wird eingeführt, so daß alle Fälle der horizontalen, geneigten oder vertikalen Platte von einem einzigen Satz transformierter Gleichungen beschrieben werden können. Die unabhängigen Gleichungen für die beiden Fälle der horizontalen and vertikalen Platte wurden für=0 und=1 aus den transformierten Gleichungen wieder abgeleitet. Es wurden erwärmte aufwärtsgerichtete Platten mit positiven und negativen Neigungswinkeln untersucht. Eine sehr genaue Gleichung wurde für die lokale Nusselt-Zahl bei beliebigen Neigungswinkeln und für 0,001 Pr entwickelt.

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Nomenclature C _p specific heat - f reduced stream function - g gravitational acceleration - Gr local Grashof number,g(T _w –T _w ) x³/v² - h local heat transfer coefficient - k thermal conductivity - n constant exponent - Nu local Nusselt number,hx/k - p pressure - Pr Prandtl number, v/ - Ra local Rayleigh number,g(T _w –T )J x³/v - T fluid temperature - T _w wall temperature - T temperature of ambient fluid - u velocity component in x-direction - v velocity component in y-direction - x coordinate parallel to the plate - y coordinate normal to the plate Greek symbols thermal diffusivity - thermal expansion coefficient - (Ra¦sin¦)^1/4/( Ra cos()^1/5 - pseudo-similarity variable, (y/) - dimensionless temperature, (T–T )/(T _w–T ) - ( Ra cos)^1/5+(Rasin)^1/4 - v kinematic viscosity - 1/[1 +(Ra cos)^1/5/( Ra¦sin)^1/4] - density of fluid - Pr/(1+Pr) - _w wall shear stress - angle of plate inclination measured from the horizontal - stream function - dimensionless dynamic pressure