The linearization of certain class of nonlinear simultaneous equation set containing amplitude and phase frequency characteristics and application

A class of complex function of rational fraction type is frequently used to describe the dynamical properties of systems. It is however quite difficult to establish a mathematical model of this type on the basis of amplitude and phase frequency data collected from experiments conducted on the related physical system. Since the erection of mathematical model G(j) would involve the solution of a set of nonlinear simultaneous equations with the unknown coefficients a_is and b_is(i=0, 1, ..., m, ..., n) in G(j). Up to now, these nonlinear equations have been considered to be very difficult to solve directly. In spite of the fact there are special computer programmes in certain software packages available to tackle this problem, it is by no means an easy task due to the complex procedures involved in picking up a set of initial values that should be close enough to the exact solutions. This paper proposes a simplified method of linearizing these nonlinear equations set so that direct solution is possible. The method can also be applied to systems with factors of (j) andej0 in G(j). An illustration by a workable example is furnished at the end of this paper to show its versatility.     

A new statistical approach to study and to predict wall turbulence

A new analysis method is developed to study the double- and triple-correlations of velocity fluctuations inside a stationary three-dimensional turbulent boundary layer (3D-TBL). Experimental eigenvalues and eigenvectors of measured Reynolds stress-tensors are obtained by diagonalization; a set of semi-empirical relationships is derived and these are interpreted (qualitatively) in terms of statistics of gas dynamics. Sample-averaged double- and triple-correlations are Monte Carlp (MC-) simulated, simultaneously, with 3 independent perturbed centered-Gaussians (trial probability density functions) along experimental eigenvectors. Comparisons with corresponding time-averaged measurements show excellent agreement for the double-correlations and qualitative agreement for the triple-correlations. Also, a statistical model for the double-correlations is presented: it can predict the -profiles inside the S-shaped wind tunnel at EPFL, given .     

Three-dimensional vorticity in a turbulent cylinder wake

The three components of the vorticity vector in the intermediate region of a turbulent cylinder wake were measured simultaneously using a multi-hot-wire probe. This probe has an improved spatial resolution compared with those reported in the literature. The behavior of the instantaneous velocity and vorticity signals is examined. Both coherent and incoherent vorticity fields are investigated using a phase-averaged technique. The iso-contours of the phase-averaged longitudinal and lateral vorticity variances, and , wrap around the spanwise structures of opposite sign and run through the saddle point along the diverging separatrix. The observation conforms to the previous reports of the occurrence of the longitudinal structures based on flow visualizations and numerical simulations. The magnitude of these contours is about the same as that of the maximum coherent spanwise vorticity at the vortex center, indicating that the strength of the longitudinal structures is comparable to that of the spanwise vortices. Furthermore, and exhibit maximum concentration away from the vortex center, probably because of a combined effect of the large-scale spanwise vortices and the intermediate-scale longitudinal structures. Coherent structures contribute about 36% to the spanwise vorticity variance at x/d=10. The contribution decreases rapidly to about 5% at x/d=40. The present results suggest that vorticity largely reside in relatively small-scale structures.     

Certain yawed magnetogasdynamic flows

Certain steady yawed magnetogasdynamic flows, in which the magnetic field is everywhere parallel to the velocity field, are related to certain reduced three-dimensional compressible gas flows having zero magnetic field. Under a restriction, the reduced flows are linked, by certain reciprocal relations, to a four parameter class of plane gas flows. In the instance of constant entropy an approximation method is suggested for obtaining magnetogasdynamic flows from the corresponding plane, irrotational gasdynamic flows and examples are given.

Nomenclature

magnetogasdynamic flow variables H magnetic intensity - q fluid velocity - fluid density - p pressure - s entropy - Q _t, H _t component of q, H in the x–y plane - w , h component of q, H perpendicular to the x–y plane reduced gasdynamic flow factor of proportionality - q* fluid velocity - * fluid density - p* pressure - Q _t ^* =u*î+v*, w* components of q* - l arbitrary constant - A _v Alfvén speed - Q _t, , p fluid velocity, density, pressure of the reciprocal gas dynamic flow - L, n, k, arbitrary constants - , velocity potential, stream function - angle made by Q _t, Q _t ^* , and V with the x-axis - adiabatic gas constant - a ²=(–1)/2 constant - M Mach number - W constant value of w* - E approximate constant value of g(p) - * modified potential function - modified velocity coordinate - +i - complex potential of the irrotational flow - B arbitrary constant - V incompressible flow velocity - V modified fluid velocity - X _p, Y _p points on the profile     

The investigation on the properties and structures of starting vortex flow past a backward-facing step by WBIV technique

An experimental investigation of a starting vortex flow around a backward-facing step was conducted in a water channel. The properties and structures of the flow were investigated by qualitative flow visualization using the hydrogen bubble method and by quantitative velocity and vorticity measurements using White-light Bubble Image Velocimetry (WBIV) — a newly developed PIV method. Some invariant properties and 4-stage structures of starting vortex flow were observed.List of symbols a flow acceleration during starting stage - h height of backward-facing step - d _v dimensionless vortex size - t time - t dimensionless time - U free uniform velocity - u, v streamwise and spanwise velocity components respectively - Re Reynolds number based on a and h - x, y streamwise and spanwise coordinates respectively in flow field - x _c , y _c dimensionless vortex center position - vorticity - ov dimensionless vorticity - _max maximum vorticity - ov _max dimensionless maximum vorticity - circulation - dimensionless circulation - kinematic viscosity This work was supported by the CNSF Grant 1939 100-1-3     

The unsteady aerodynamics of a first stage stator vane row

The fundamental unsteady aerodynamics on a vane row of an axial flow research compressor stage are experimentally investigated, demonstrating the effects of airfoil camber and steady loading. In particular, the rotor wake generated unsteady surface pressure distributions on the first stage vane row are quantified over a range of operating conditions. These cambered airfoil unsteady data are correlated with predictions from a flat plate cascade inviscid flow model. At the design point, the unsteady pressure difference coefficient data exhibit good correlation with the nonseparated predictions, with the aerodynamic phase lag data exhibiting fair trendwise correlation. The quantitative phase lag differences are associated with the camber of the airfoil. An aft suction surface flow separation region is indicated by the steady state surface static pressure data as the aerodynamic loading is increased. This separation affects the increased incidence angle unsteady pressure data.List of symbols b airfoil semi-chord - C airfoil chord - C _p dynamic pressure coefficient, - _p static pressure coefficient, - i incidence angle - k reduced frequency, - N number of rotor revolutions - p dynamic pressure difference - static pressure difference, - S stator vane circumferential spacing - U _t rotor blade tip speed - u longitudinal perturbation velocity - V absolute velocity - V _axial absolute axial velocity - v transverse perturbation velocity - x _sep location of separation point - inlet angle - inlet air density - blade passing angular frequency     

An analogue study for flame flickering

An analogue experiment is proposed to simulate flame flickering comprising a free ascending column fed on its side with a light gas (helium) emerging from a vertical slot in ambient air. The convective motion of the helium jet is considered to represent the motion of burnt gases of buoyant jet flames. The helium jet is accelerated by buoyancy effects and the flow field is similar to that of burnt gases observed for real buoyant flames. The vertical velocity profile of the steady helium jet is measured at different vertical distances. The unsteady helium jet is also studied by measuring the instability frequency as a function of ambient pressure at different injection flow rates, and by analyzing the tomography images of the helium jet. The instability morphology is the same as that observed on real buoyant flames. We conclude that this type of instability can be approximately characterized by the maximum vertical velocityu _max, and the distance betweenu _max in the helium ascending column andu = o in the ambient air. For this type of instability the local vorticity is proportional to which can be influenced by gravity and ambient pressure. Theoretical prediction of the instability frequency as a function of gravity and ambient pressure has been obtained, and is in good agreement with the experimental results.List of symbols C ₁,C ₂ constants - F instability frequency - F _c critical frequency - F _m the most amplified frequency - F (K, ) function defined in (11) - g gravitational acceleration - g reduced gravity acceleration g(₀-^*)/^* - k real wave number of the disturbance - K reduced wave numberK=2k - K _c reduced wave number of the critical instability mode - K _m nondimensional wavenumber of the most amplified mode - L vertical characteristic length (in x direction) - P ambient pressure - u local vertical buoyant velocity (inx direction) - u _max local maximum vertical velocity - v local velocity component iny direction (horizontal) - V ₀ injection velocity of helium (iny direction) - x vertical distance measured from the leading edge of boundary layer - y horizontal distance measured from the exit plane of the vertical slot - Z(K, ) function defined in equation (11) Greek symbols distance betweenu _max in the helium ascending column andu = o in the ambient air - - wavelength of instability - _c critical wavelength - _m the most amplified wavelength - ^* helium density at slot exit - ₀ ambient air density - ^* helium dynamic viscosity at slot exit - v ^* helium kinematic viscosity at slot exit - complex number presented in disturbancee ^i(kx+t) - _i imaginary part of , representing the amplification rate of disturbance - _r real part of , where ( _r /k) represents the group velocity - reduced complex number of , defined      

Transient Free Convection from a Horizontal Surface in a Porous Medium Subjected to a Sudden Change in Surface Heat Flux

This paper presents an analytical and numerical study of transient free convection from a horizontal surface that is embedded in a fluid-saturated porous medium. It is assumed that for time steady state velocity and temperature fields are obtained in the boundary-layer which occurs due to a uniform flux dissipation rate q ₁ on the surface. Then, at the heat flux on the surface is suddenly changed to q ₂ and maintained at this value for . Firstly, solutions which are valid for small and large are obtained. The full boundary-layer equations are then integrated step-by-step for the transient regime from the initial unsteady state ( ) until such times at which this forward marching approach is no longer well posed. Beyond this time no valid solutions could be obtained which matched the final solution from the forward integration to the steady state profiles at large times .     

On the universality of the velocity profiles of a turbulent flow in an axially rotating pipe

If a fluid enters an axially rotating pipe, it receives a tangential component of velocity from the moving wall, and the flow pattern change according to the rotational speed. A flow relaminarization is set up by an increase in the rotational speed of the pipe. It will be shown that the tangential- and the axial velocity distribution adopt a quite universal shape in the case of fully developed flow for a fixed value of a new defined rotation parameter. By taking into account the universal character of the velocity profiles, a formula is derived for describing the velocity distribution in an axially rotating pipe. The resulting velocity profiles are compared with measurements of Reich [10] and generally good agreement is found.Nomenclature b constant, equation (34) - D pipe diameter - l mixing length - l ₀ mixing length in a non-rotating pipe - N rotation rate,N=Re /Re _D - p pressure - R pipe radius - Re _D flow-rate Reynolds number, - Re rotational Reynolds number, Re =v _w D/ - Re* Reynolds number based on the friction velocity, Re*=v*R/ - (Re*)₀ Reynolds number based on the friction velocity in a non-rotating pipe - Ri Richardson number, equation (10) - r coordinate in radial direction - dimensionless coordinate in radial direction, - v _r ,v ,v _z time mean velocity components - v _r ,v ,v _z velocity fluctations - v _w tangential velocity of the pipe wall - v* friction velocity, - axial mean velocity - v _ZM maximum axial velocity - dimensionless radial distance from pipe wall, - y ⁺ dimensionless radial distance from pipe wall - y ₁ ⁺ constant - Z rotation parameter,Z =v _w/v _* =N Re _D /2Re* - _m eddy viscosity - ( _m )₀ eddy viscosity in a non-rotating pipe - coefficient of friction loss - von Karman constant - ₁ constant, equation (31) - density - dynamic viscosity - kinematic viscosity     

Fluctuating flow in a rotating fluid

Summary Fluctuating flow of a viscous fluid rotating over a disk whose angular velocity oscillates about a nonzero mean is investigated. Initially the disk and the fluid rotate in the same sense with different angular velocities ₁ and ₂ ( ₂> ₁) and at a particular instant of time, the angular velocity of the disk becomes ₁[1+ sin( )]. The problem is solved as an initial boundary value problem and it is found that for small values of the results of analytical and numerical methods are in excellent agreement. The effect of frequency parameter on surface skin frictions has been analysed for various values of angular velocity ratio s and amplitude parameter .

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Fluktuierende Strömung in einer rotierenden Flüssigkeit

Übersicht Untersucht wird die fluktuierende Strömung einer viskosen Flüssigkeit, die über einer Scheibe, deren Winkelgeschwindigkeit um einen von Null verschiedenen Mittelwert schwankt, rotiert. Anfangs drehen sich die Scheibe und die Flüssigkeit gleichsinnig, aber mit verschiedenen Winkelgeschwindigkeiten ₁ und ₂ ( ₂> ₁). Zu einem Anfangszeitpunkt geht die Winkelgeschwindigkeit der Scheibe über in ₁[1+ sin ( )]. Die Aufgabe wird als Anfangs-/Randwertproblem gelöst. Für kleine Werte stimmen die analytischen und numerischen Ergebnisse hervorragend überein. Für verschiedene Werte des Winkelgeschwindigkeitsverhältnisses und des Amplitudenparameters wurde der Einfluß des Frequenzparameters auf die Reibspannungen an der Scheibe untersucht.

     

Wärmeübertragung in einem axial rotierenden,durchströmten Rohr im Bereich des thermischen Einlaufs

Zusammenfassung Der Einfluß der Rotation auf das Temperaturprofil und die Wärmeübergangszahl einer turbulenten Rohrströmung im Bereich des thermischen Einlaufs wird theoretisch untersucht und mit Meßwerten verglichen. Es wird angenommen, daß das Geschwindigkeitsprofil voll ausgebildet ist. Die Rotation hat aufgrund der radial ansteigenden Zentrifugalkräfte einen ausgeprägten Einfluß auf die Unterdrückung der turbulenten Bewegung. Dadurch verschlechtert sich die Wärmeübertragung mit steigender Rotations-Reynoldszahl und die thermische Einlauflänge nimmt beträchtlich zu.

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Heat transfer in an axially rotating pipe in the thermal entrance region. Part 1: Effect of rotation on turbulent pipe flow

The effects of rotation on the temperature distribution and the heat transfer to a fluid flowing inside a tube are examined by analysis in the thermal entrance region. The theoretical results are compared with experimental findings. The flow is assumed to have a fully developed velocity profile. Rotation was found to have a very marked influence on the suppression of the turbulent motion because of radially growing centrifugal forces. Therefore, a remarkable decrease in heat transfer with increasing rotational Reynolds number can be observed. The thermal entrance length increases remarkably with growing rotational Reynolds number.

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Formelzeichen a Temperaturleitzahl - C _n , ,C ₁,C ₃ Konstanten - c _p spezifische Wärme bei konstantem Druck - D Rohrdurchmesser - E Funktion nach Gl. (30) - H _n Eigenfunktionen - l hydrodynamischer Mischungsweg - l _q thermischer Mischungsweg - Massenstrom - N=Re /Re Reynoldszahlenverhältnis - Nu Nusseltzahl - Nu Nusseltzahl für die thermisch voll ausgebildete Strömung - Pr Prandtlzahl - Pr _t turbulente Prandtlzahl - Wärmestromdichte - Re _* Schubspannungsreynoldszahl - R _n Eigenfunktionen - Durchfluß-Reynoldszahl - Re v =D/ Rotations-Reynoldszahl - Ri Richardsonzahl - R Rohrradius - r Koordinate in radialer Richtung - dimensionslose Koordinate in radialer Richtung - T Temperatur - T Temperaturschwankung - T _b bulk temperature - mittlere Axialgeschwindigkeit - v Geschwindigkeit - v Geschwindigkeitsschwankung - turbulenter Wärmestrom - dimensionsloser Wandabstand - =1/6 Konstante - Integrationsvariable - Integrationsvariable - , ₁, ₂, dimensionslose Temperaturen - Wärmeleitzahl - _n Eigenwerte - kinematische Viskosität - Dichte - tangentiale Koordinate - , Hilfsfunktionen Indizes m in der Rohrmitte - r radial - w an der Rohrwand - z axial - 0 am Rohreintritt - 0 ohne Rotation - tangential     

Vector potential–vorticity relationship for the Stokes flows: application to the Stokes eigenmodes in 2D/3D closed domain

The unsteady dynamics of the Stokes flows, where , is shown to verify the vector potential–vorticity ( ) correlation , where the field is the pressure-gradient vector potential defined by . This correlation is analyzed for the Stokes eigenmodes, , subjected to no-slip boundary conditions on any two-dimensional (2D) closed contour or three-dimensional (3D) surface. It is established that an asymptotic linear relationship appears, verified in the core part of the domain, between the vector potential and vorticity, , where is a constant offset field, possibly zero.     

On the Asymptotic Properties of Solutions of Linear Functional Differential Equations with Constant Coefficients and Linearly Transformed Argument

We establish new properties of C ¹(0, +)-solutions of the linear functional differential equation in the neighborhood of the singular point t = +.     

Apparent thickening behavior of dilute polystyrene solutions in extensional flows

An experimental investigation was undertaken to study the apparent thickening behavior of dilute polystyrene solutions in extensional flow. Among the parameters investigated were molecular weight, molecular weight distribution, concentration, thermodynamic solvent quality, and solvent viscosity. Apparent relative viscosity was measured as a function of wall shear rate for solutions flowing from a reservoir through a 0.1 mm I.D. tube. As increased, slight shear thinning behavior was observed up until a critical wall shear rate was exceeded, whereupon either a large increase in or small-scale thickening was observed depending on the particular solution under study. As molecular weight or concentration increased, decreased and, the jump in above , increased. increased as thermodynamic solvent quality improved. These results have been interpreted in terms of the polymer chains undergoing a coil-stretch transition at . The observation of a drop-off in at high (above ) was shown to be associated with inertial effects and not with chain fracture due to high extensional rates.     

Stoffaustausch zwischen zwei fluiden Phasen,von denen sich eine als Strahl in der anderen bewegt

Zusammenfassung Der Übergang eines Stoffes zwischen zwei fluiden Phasen wird betrachtet, von denen sich einer als Strahl in der anderen bewegt. Die Geschwindigkeit der laminar strömenden Phase wird durch eine Gleichung ausgedrückt, die Geschwindigkeitsprofile zwischen der Kolben- und der Rohrströmung kontinuierlich beschreibt. Der Transport des Stoffes im Strahl durch Diffusion in radialer und durch Konvektion in axialer Richtung wird für den isothermen, stationären Fall untersucht. Die das Problem beschreibende Differentialgleichung wird anscheinend erstmals geschlossen gelöst. Die Lösungen beinhalten konfluente hypergeometrische Funktionen. Berechnet werden Eigenwerte, Koeffizienten, örtliche und mittlere Konzentrationsfelder sowie Stoffübergangszahlen.

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Mass transfer between two fluids, one of the two fluids is moving as jet within the other

The mass transfer between two fluids is calculated, one of the two fluids is moving as a jet within the other. The velocity of the laminar flowing phase is expressed by an equation, which describes continously the velocity profiles from plug flow to tubular flow. For the isothermal, stationary state the transport of substance i by radial diffusion and by axial convection is investigated. It appears to be that the differential equations describing the problem are solved rigorously for the first time. The solutions contain confluent hypergeometrical functions. Results include eigenvalues, coefficients, local and mean concentration fields, mass transfer numbers.

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Verwendete Zeichen und ihre Bedeutung a - A, A_n Koeffizienten - B, B_n Koeffizienten - c Konzentration, Konstante im Anhang - C_r=0 Mittenkonzentration - c₀ Konzentration in Phase I bis z=0 - c_II Konzentration in Phase II - ¯c mittlere Konzentration, definiert in Gl. (35) - C Koeffizient, definiert in Gl. (A 21) - D Diffusionskoeffizient - Da Damköhlerzahl - E Funktion, gegeben durch Gl. (A 12) - f, f(R) Funktion f von R - f_n, f_n (R) Funktionswerte - g, g(Z) Funktion g von Z - g_n, g_n (Z) Funktionswerte - h(Z) Funktion h von z - H_q Koeffizienten, gegeben durch Gl. (A 10) - j Massenstromdichte - J _k , J_q Besselfunktion der Ordnungk, q - k definiert durch Gl. (A 9) - n laufende Zahl - m laufende Zahl - p laufende Zahl - Pe=Re·Sc Pecletzahl - q laufende Zahl - Q_n Koeffizienten, definiert in Gl. (31) - r radiale Koordinate - r₀ Radius - R r/r₀ - Re=u₀r₀/ Reynoldszahl - S=2r₀z Zylinderfläche - Sc=/D Schmidtzahl - Sh=2r₀ /D Sherwoodzahl - Sherwoodzahl, definiert in Gl. (52) - Sh_u Sherwoodzahl, definiert in Gl. (54) - Sh_z Sherwoodzahl, definiert in Gl. (40) - Sherwoodzahl, definiert in Gl. (45) - t R² - u Geschwindigkeit - u₀ maximale Geschwindigkeit - v - Volumenstrom - w Variable - x Variable - y abhängige Variable - z axiale Koordinate, Lauflänge - Z z/r₀ - Z_Pe dimensionslose Lauflänge, definiert durch Gl. (34) - a_n Koeffizienten, definiert durch Gl. (A 19) - Stoffübergangskoeffizient - Stoffübergangskoeffizient, definiert in Gl. (48) - _u Stoffübergangskoeffizient, definiert in Gl. (49) - _z Stoffübergangskoeffizient, definiert in Gl. (38) - Stoffübergangskoeffizient, definiert in Gl. (44) - definiert in Gl. (A 21) - Gammafunktion - c Konzentrationsdifferenz - m Stoffmenge - Zahl zwischen Null und Eins - laufende Zahl - kinematische Zähigkeit - (v) (t) - konfluente hypergeometrische Funktion - (t) - konfluente hypergeometrische Funktion - , _n Eigenwerte Hochzeichen - * kennzeichnet asymptotische Lösungen     

Torsion of a hemisphere embedded in an elastic half-space

This paper deals with the stress distribution in a homogeneous isotropic elastic hemisphere embedded in a semi-infinite homogeneous isotropic elastic medium when a rigid circular disc is clamped to the plane face of the hemisphere and the stresses are caused by the rotation of the disc through an angle . The problem is reduced to the solution of a Fredholm integral equation of the second kind in the auxiliary function (t). An analytical expression for the torque T required to rotate the die through an angle is obtained in terms of (t). The Fredholm integral equation is solved numerically, and the numerical values of T are graphed.This work has been supported by the National Research Council of Canada through NRC-Research Grant No. A4177.     

Solution for plane strain forward and backward extrusions with a fractional reductionR=0.5 by the integartion depending on a parameter

A parameter t is introduced to boundary slip line of rigid regions for plane strain and indirect extrusions with a fractional reduction R=0.5. Integration by substitution has been used along the boundary slip line in order to obtain the extrusion pressure. By the integration depending on a parameter, the following results are obtained, and die pressure is 5.14k for backward extrusion; and pad average pressure is 2.57k for forward extrusion. All the results from this method are the same as those of the conventional solution.     

Local Bifurcation Analysis of a Second Gradient Model for Deformations of a Rectangular Slab

In this paper we carry out a derivation of the equilibrium equations of nonlinear elasticity with an added second-gradient term proportional to a small parameter . These equations are given by a fourth order semilinear system of pdes. We discuss different types of possible boundary conditions for these equations. We then specialize the equations to a rectangular slab and study the linearized problem about a homogenous deformation. We show that these equations admit solutions representable as Fourier series in one of the independent variables. Furthermore, we obtain the characteristic equation for the eigenvalues (possible bifurcation points) for the linear problem and derive asymptotic representations for this equation for small . We used these expressions to show that in the limit as the characteristic equation for converges uniformly (in certain regions of the parameter space) to the corresponding characteristic equation for . When the base material () is that of a Blatz–Ko type, we get conditions for the existence of eigenvalues of the linear problem with and small. Our numerical results in this case indicate that the number of bifurcation points is finite when and that this number monotonically increases as . For the problem with we get conditions for the existence of local branches of non-trivial solutions.      

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     

Motion of discrete particles in a turbulent fluid

Summary Various approximations to Basset's equation for the motion of a particle in a viscous fluid have been applied to the complex phenomenon of dispersion in a turbulent fluid. The deviations of the particle motion from the fluid motion, as predicted by the various approximations, is explored, and the frequencies for which this deviation is large are described. The approximations are found to be invalid for such cases as sediment transport and motion of gas bubbles in liquids. For small, 7 micron, liquid or solid particles in air, however, all approximations are shown to be valid for turbulent frequencies below 812 cps.Nomenclature a parameter in equation (2.3) - b parameter in equation (2.3) - c parameter in equation (2.3) - d diameter of sphere - E _f energy spectrum of the fluid - E _p energy spectrum of the particle - F frequency of oscillation - f ₁ parameter defined by equation (2.10) - f ₂ parameter defined by equation (2.10) - g acceleration of gravity - N _S , Stokes number - s density ratio - t time - t ₀ initial time - u _f fluid velocity - u _p particle velocity - V velocity of sphere - phase angle - parameter in equation (2.8) - amplitude ratio - parameter in equation (2.8) - dynamic viscosity - kinematic viscosity - f density of the fluid - p density of the particle - parameter in equation (2.8) - parameter in equation (2.8) - circular frequency of the motion