On natural convection from a vertical plate with a prescribed surface heat flux in porous media

This paper presents a theoretical and numerical investigation of the natural convection boundary-layer along a vertical surface, which is embedded in a porous medium, when the surface heat flux varies as (1 +x ²)), where is a constant andx is the distance along the surface. It is shown that for > -1/2 the solution develops from a similarity solution which is valid for small values ofx to one which is valid for large values ofx. However, when -1/2 no similarity solutions exist for large values ofx and it is found that there are two cases to consider, namely < -1/2 and = -1/2. The wall temperature and the velocity at large distances along the plate are determined for a range of values of .Notation g Gravitational acceleration - k Thermal conductivity of the saturated porous medium - K Permeability of the porous medium - l Typical streamwise length - q _w Uniform heat flux on the wall - Ra Rayleigh number, =gK(q _w /k)l/(v) - T Temperature - Too Temperature far from the plate - u, v Components of seepage velocity in the x and y directions - x, y Cartesian coordinates - Thermal diffusivity of the fluid saturated porous medium - The coefficient of thermal expansion - An undetermined constant - Porosity of the porous medium - Similarity variable, =y(1+x ) ^/3/x ^1/3 - A preassigned constant - Kinematic viscosity - Nondimensional temperature, =(T – T )Ra^1/3 k/qw - Similarity variable, = =y(log_e x)^1/3/x ^2/3 - Similarity variable, =y/x ^2/3 - Stream function     

Binary laminar boundary layer on a vertical surface in the presence of combined free and forced convection

Heat and mass transfer at a vertical surface is examined in the case of combined free and forced convection. The boundary layer equations, transformed to ordinary differential equations, contain a parameter that determines the effect of free convection on the forced motion. Criteria are offered for differentiating the free-convection, forced-convection, and combined regimes.Notation x, y coordinates - u, v velocity components - g acceleration of gravity - T temperature - kinematic viscosity - coefficient of thermal expansion - a thermal diffusivity - ₁ partial vapor density - D diffusion coefficient - W₂ mass velocity of air - independent variable - _w shear stress at wall - thermal conductivity - r latent heat of phase transition - , dimensionless temperature and partial vapor density - m^* the complex (m ₁–m _1w )/(1–m(_1w ) - c_p specific heat at constant pressure - G Grashof number - R Reynolds number - P Prandtl number - S Schmidt number     

Laminar flow in the entrance region of a porous-wall channel

Summary The effect of fluid injection at the walls of a two-dimensional channel on the development of flow in the entrance region of the channel has been investigated. The integral forms of the boundary layer equations for flow in the channel were set up for an injection velocity uniformly distributed along the channel walls.With an assumed polynomial of the n-th degree for the one-parameter velocity profile a solution of the above boundary layer equations was obtained by an iteration method. A closed form solution was also obtained for the case when a similar velocity profile was assumed. The agreement between the entrance region velocity profiles of the present analysis for an impermeable-walled channel and of Schlichting¹) and Bodoia and Osterle²) is found to be very good.The results of the analysis show that fluid injection at the channel walls increases the rate of the growth of the boundary layer thickness, and hence reduces considerably the entrance length required for a fully developed flow.Nomenclature h half channel thickness - L entrance length with wall-injection - L ₀ entrance length without wall-injection - p static pressure - p=p/U ₀ ² dimensionless pressure - Re=U ₀ h/ Reynolds number at inlet cross-section - u velocity in the x direction at any point in the channel - =u/U ₀ dimensionless velocity in the x direction at any point in the channel - U _av average velocity at a channel cross-section - U _c center line velocity - U ₀ inlet cross-section velocity - _c =U _c /U ₀ dimensionless center line velocity - v velocity in the y direction at any point in the channel - v ₀ constant injection velocity of fluid at the wall - v=v/v ₀ dimensionless velocity in the y direction at any point in the channel - x distance along the channel wall measured from the inlet cross-section - x=x/hRe dimensionless distance in the x direction - y distance perpendicular to the channel wall - y=y/h dimensionless distance in the y direction - thickness of the boundary layer - =/h dimensionless boundary layer thickness - =/ dimensionless distance within the boundary layer region - =v ₀ h/ injection parameter or injection Reynolds number - kinematic viscosity - 1+ie - mass density of the fluid - parameter defined in (14)     

On the effects of a periodic, spanwise variation of suction upon asymptotic suction flow

Summary The effects of superposing streamwise vorticity, periodic in the lateral direction, upon two-dimensional asymptotic suction flow are analyzed. Such vorticity, generated by prescribing a spanwise variation in the suction velocity, is known to play an important role in unstable and turbulent boundary layers. The flow induced by the variation has been obtained for a freestream velocity which (i) is steady, (ii) oscillates periodically in time, (iii) changes impulsively from rest. For the oscillatory case it is shown that a frequency can exist which maximizes the induced, unsteady wall shear stress for a given spanwise period. For steady flow the heat transfer to, or from a wall at constant temperature has also been computed.Nomenclature (x, y, z) spatial coordinates - (u, v, w) corresponding components of velocity - (, , ) corresponding components of vorticity - t time - stream function for v and w - v _w mean wall suction velocity - nondimensional amplitude of variation in wall suction velocity - characteristic wavenumber for variation in direction of z - T temperature - P pressure - density - coefficient of kinematic viscosity - coefficient of thermal diffusivity - (/v _w)² - frequency of oscillation of freestream velocity - nondimensional amplitude of freestream oscillation - /v _w ² - z z - y –v _w y/ - v _w ² t/4 - /v _w - U ₀ characteristic freestream velocity - u/U ₀ - coefficient of viscosity - _w wall shear stress - Prandtl number (/) - q heat transfer to wall - T _w wall temperature - T (T _w–T)/(T _w–)     

Convection velocity measurements in a cylinder wake

The convection velocity of vortices in the wake of a circular cylinder has been obtained by two different approaches. The first, implemented in a wind tunnel using an array of X-wires, consists in determining the velocity at the location of maximum spanwise vorticity. Four variants of the second method, which estimates the transit time of vortices tagged by heat or dye, were used in wind and water tunnels over a relatively large Reynolds number range. Results from the two methods are in good agreement with each other. Along the most probable vortex trajectory, there is only a small streamwise increase in the convection velocity for laminar conditions and a more substantial variation when the wake is turbulent. The convection velocity is generally greater than the local mean velocity and does not depend significantly on the Reynolds number.Nomenclature d diameter of circular cylinder - f frequency in spectrum analysis - f _v average vortex frequency - r _v vortex radius - Re Reynolds number U _o d/v - t time - Th , Th , Th _r thresholds for _zp, , and r _v respectively - U _o free stream velocity - U ₁ maximum value of (U _o – U) - U _c convection velocity of the vortex, as obtained either by Eq. (1) or Eq. (2) - U _co convection velocity used in Eq. (3) U _cd, U _cu average convection velocities of downstream and up-stream regions respectively of the vortex - U _cv the value of U _c at y = 0.5 - u, v the velocity fluctuations in x and y directions respectively - U, V mean velocity components in x and y directions respectively - U,V U = U + u, V = V + v - x, y, z co-ordinate axes, defined in Fig. 1 Greek Symbols circulation - mean velocity half-width - x spacing between two cold wires or grid spacing - ₁, ₂ temperature signals from upstream and downstream cold wires respectively - v kinematic viscosity - _c transit time for a vortex to travel a distance x - phase in the cross-spectrum of ₁ and ₂ - _z instantaneous spanwise vorticity - _zc cut-off vorticity used in determining the vortex size - _zp peak value of _z - a denotes conditional average, defined in Eq. (12) - a prime denoting rms value     

Effect of measuring volume length on two-component laser velocimeter measurements in a turbulent boundary layer

The effects of finite measuring volume length on laser velocimetry measurements of turbulent boundary layers were studied. Four different effective measuring volume lengths, ranging in spanwise extent from 7 to 44 viscous units, were used in a low Reynolds number (Re=1440) turbulent boundary layer with high data density. Reynolds shear stress profiles in the near-wall region show that u v strongly depends on the measuring volume length; at a given y-position, u v decreases with increasing measuring volume length. This dependence was attributed to simultaneous validations on the U and V channels of Doppler bursts coming from different particles within the measuring volume. Moments of the streamwise velocity showed a slight dependence on measuring volume length, indicating that spatial averaging effects well known for hot-films and hot-wires can occur in laser velocimetry measurements when the data density is high.List of symbols time-averaged quantity - u wall friction velocity, ( _w /)^1/2 - v kinematic viscosity - d _p pinhole diameter - l _eff spanwise extent of LDV measuring volume viewed by photomultiplier - l ⁺ non-dimensional length of measuring volume, l _eff u /v - y ⁺ non-dimensional coordinate in spanwise direction, y u /v - z ⁺ non-dimensional coordinate in spanwise direction, z u /v - U ⁺ non-dimensional mean velocity, /u - u instantaneous streamwise velocity fluctuation, U &#x2329;U - v instantaneous normal velocity fluctuation, V–V - u RMS streamwise velocity fluctuation, u ^21/2 - v RMS normal velocity fluctuation, v ^21/2 - Re Reynolds number based on momentum thickness, U 0/v - R _uv cross-correlation coefficient, u v/u v - R₁₂(0, 0, z) two point correlation between u and v with z-separation, <u(0, 0, 0) v (0, 0, z)>/<u(0, 0, 0) v (0, 0, 0)> - N rate at which bursts are validated by counter processor - T Taylor time microscale, u (dv/dt²)^–1/2     

Mixed convection on vertical plate for fluids of any prandtl number

A mixed convection parameter=(Ra) ^1/4/(Re)^1/2, with=Pr/(1+Pr) and=Pr/(1 +Pr)^1/2, is proposed to replace the conventional Richardson number, Gr/Re², for combined forced and free convection flow on an isothermal vertical plate. This parameter can readily be reduced to the controlling parameters for the relative importance of the forced and the free convection,Ra ^1/4/(Re ^1/2 Pr ^1/3) forPr 1, and (RaPr)^1/2/(RePr ^1/2 forPr 1. Furthermore, new coordinates and dependent variables are properly defined in terms of, so that the transformed nonsimilar boundary-layer equations give numerical solutions that are uniformly valid over the entire range of mixed convection intensity from forced convection limit to free convection limit for fluids of any Prandtl number from 0.001 to 10,000. The effects of mixed convection intensity and the Prandtl number on the velocity profiles, the temperature profiles, the wall friction, and the heat transfer rate are illustrated for both cases of buoyancy assisting and opposing flow conditions.

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Mischkonvektion an einer vertikalen Platte für Fluide beliebiger Prandtl-Zahl

Zusammenfassung Für die kombinierte Zwangs- und freie Konvektion an einer isothermen senkrechten Platte wird ein Mischkonvektions-Parameter=( Ra) ^1/4 (Re)^1/2, mit=Pr/(1 +Pr) und=Pr/(1 +Pr)^1/2 vorgeschlagen, den die gebräuchliche Richardson-Zahl, Gr/Re², ersetzen soll. Dieser Parameter kann ohne weiteres auf die maßgebenden Kennzahlen für den relativen Einfluß der erzwungenen und der freien Konvektion reduziert werden,Ra ^1/4/(Re ^1/2 Pr ^1/3) fürPr 1 und (RaPr)^1/4/(RePr)^1/2 fürPr 1. Weiterhin werden neue Koordinaten und abhängige Variablen als Funktion von definiert, so daß für die transformierten Grenzschichtgleichungen numerische Lösungen erstellt werden können, die über den gesamten Bereich der Mischkonvektion, von der freien Konvektion bis zur Zwangskonvektion, für Fluide jeglicher Prandtl-Zahl von 0.001 bis 10.000 gleichmäßig gültig sind. Der Einfluß der Intensität der Mischkonvektion und der Prandtl-Zahl auf die Geschwindigkeitsprofile, die Temperaturprofile, die Wandreibung und den Wärmeübergangskoeffizienten werden für die beiden Fälle der Strömung in und entgegengesetzt zur Schwerkraftrichtung dargestellt.

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Nomenclature C _f local friction coefficient - C _p specific heat capacity - f reduced stream function - g gravitational acceleration - Gr local Grashoff number,g T _w –T )x³/v² - Nu local Nusselt number - Pr Prandtl number,v/ - Ra local Rayleigh number,g T _w –T x ³/( v) - Re local Reynolds number,u x/v - Ri Richardson number,Gr/Re ² - T fluid temperature - T _w wall temperature - T free stream temperature - u velocity component in thex direction - u free stream velocity - v velocity component in they direction - x vertical coordinate measuring from the leading edge - y horizontal coordinate Greek symbols thermal diffusivity - thermal expansion coefficient - mixed convection parameter (Ra)1/4/Re)^1/2 - pseudo-similarity variable,(y/x) - ₀ conventional similarity variable,(y/x)Re ^1/2 - dimensionless temperature, (T–T T _W –T - unified mixed-flow parameter, [(Re) ^1/2 + (Ra)^1/4] - dynamic viscosity - kinematic viscosity - stretched streamwise coordinate or mixed convection parameter, [1 + (Re)^1/2/(Ra) ^1/4]^–1=/(1 +) - density - Pr/(1 + Pr) _w wall shear stress - stream function - Pr/(l+Pr)^1/3 This research was supported by a grand from the National Science Council of ROC     

Finite difference analysis of plane Poiseuille and Couette flow developments

Summary The problem of flow development from an initially flat velocity profile in the plane Poiseuille and Couette flow geometry is investigated for a viscous fluid. The basic governing momentum and continuity equations are expressed in finite difference form and solved numerically on a high speed digital computer for a mesh network superimposed on the flow field. Results are obtained for the variations of velocity, pressure and resistance coefficient throughout the development region. A characteristic development length is defined and evaluated for both types of flow.Nomenclature h width of channel - L ratio of development length to channel width - p fluid pressure - p ₀ pressure at channel mouth - P dimensionless pressure, p/ ² - P ₀ dimensionless pressure at channel mouth - P pressure defect, P ₀–P - (P)₀ pressure defect neglecting inertia - Re Reynolds number, uh/ - u fluid velocity in x-direction - mean u velocity across channel - u ₀ wall velocity - U dimensionles u velocity u/ - U _c dimensionless centreline velocity - U ₀ dimensionless wall velocity - v fluid velocity in y-direction - V dimensionless v velocity, hv/ - x coordinate along channel - X dimensionless x-coordinate, x/h ² - y coordinate across channel - Y dimensionless y-coordinate, y/h - resistance coefficient, - ₀ resistance coefficient neglecting inertia - fluid density - fluid viscosity     

A method for the analysis of recirculating turbulent flows in a cavity

The complex fluid-dynamic aspects of a turbulent recirculating flow in a cavity with axial throughflow, and a rotating wall, were investigated by adopting a simple procedure for evaluating the turbulent stresses. The flow field was divided into two regions, a core and a wall region respectively. A wall function was adopted in the zones near to the solid boundaries, while a constant eddy diffusivity was assumed, in the core, following the indications of computed heat transfer coefficients in comparison with existing experimental data. The distributions of the stream function and of the tangential velocity are presented for a range of the rotational Reynolds number of the rotating wall and of the Reynolds number of the throughflow.

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Turbulente Rezirkulationsströmung in einem Hohlraum

Zusammenfassung Die komplizierten fluiddynamischen Aspekte einer turbulenten Rezirkulationsströmung in einem Hohlraum mit axialem Durchfluß und einer rotierenden Wand werden unter Verwendung einer vereinfachten Methode zur Berechnung der turbulenten Spannungen betrachtet. Das Strömungsfeld wird in einen Kern und einen Wandbereich aufgeteilt. Für die wandnahen Zonen wird eine Wandfunktion angenommen, während im Kern mit konstanter Wirbeldiffusivität gerechnet wird, was durch den Vergleich berechneter mit gemessenen Wärmeübergangskoeffizienten gerechtfertigt erscheint. Verteilungen der Stromfunktion und der tangentialen Geschwindigkeit sind für einen bestimmten Bereich der Reynoldszahlen für die Wandrotation und der für den Durchfluß angegeben.

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Nomenclature L axial length of enclosure - P dimensionless pressure, p^*2 - p static pressure - R dimensionless radial coordinate, r/r^* - r radial coordinate - r^* reference length, equal to r_O for enclosure - r_i radii of inlet and exit apertures - Re Reynolds number, v^*r^*/ - Re_i pipe Reynolds number, ¯v_zi(2r_i)/ - Re_t turbulent Reynolds number, Re(/) - Re rotational Reynolds number, r ₀ ² / - t dimensionless time,t/(r^*/v^*) - t time - V_r, V, Vz dimensionless velocity components, V_r/v^*, v, v_z/v^* - v_i turbulent fluctuation of the i-component of velocity - v_r, v, v_z velocity components - v^* reference velocity, equal to ¯v_zi for enclosure - X coordinate along a wall, x/r^* - Y coordinate normal to a wall, y/r^* - Z dimensionless axial coordinate, z/r^* - z axial coordinate - eddy diffusivity for momentum - dynamic viscosity - kinematic viscosity - density - shear stress - dimensionless shear stress, /v^*2 - dimensionless stream function, /r*²v*² - stream function - angular velocity - tangential vorticity component - ()_eff effective - ()_l laminar - ()_t turbulent - mean over the time     

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     

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     

Determination of permeability tensors for two-phase flow in homogeneous porous media: Theory

In this paper we continue previous studies of the closure problem for two-phase flow in homogeneous porous media, and we show how the closure problem can be transformed to a pair of Stokes-like boundary-value problems in terms of pressures that have units of length and velocities that have units of length squared. These are essentially geometrical boundary value problems that are used to calculate the four permeability tensors that appear in the volume averaged Stokes' equations. To determine the geometry associated with the closure problem, one needs to solve the physical problem; however, the closure problem can be solved using the same algorithm used to solve the physical problem, thus the entire procedure can be accomplished with a single numerical code.Nomenclature a a vector that maps V onto , m^-1. - A a tensor that maps V onto . - A area of the - interface contained within the macroscopic region, m². - A area of the -phase entrances and exits contained within the macroscopic region, m². - A area of the - interface contained within the averaging volume, m². - A area of the -phase entrances and exits contained within the averaging volume, m². - Bo Bond number (= (=(–)g²/). - Ca capillary number (= v/). - g gravitational acceleration, m/s². - H mean curvature, m^-1. - I unit tensor. - permeability tensor for the -phase, m². - viscous drag tensor that maps V onto V. - ^* dominant permeability tensor that maps onto v , m². - ^* coupling permeability tensor that maps onto v , m². - characteristic length scale for the -phase, m. - l characteristic length scale representing both and , m. - L characteristic length scale for volume averaged quantities, m. - n unit normal vector directed from the -phase toward the -phase. - n unit normal vector representing both n and n . - n unit normal vector representing both n and n . - P pressure in the -phase, N/m². - p superficial average pressure in the -phase, N/m². - p intrinsic average pressure in the -phase, N/m². - p –p , spatial deviation pressure for the -phase, N/m². - r ₀ radius of the averaging volume, m. - r position vector, m. - t time, s. - v fluid velocity in the -phase, m/s. - v superficial average velocity in the -phase, m/s. - v intrinsic average velocity in the -phase, m/s. - v –v , spatial deviation velocity in the -phase, m/s. - V volume of the -phase contained within the averaging volmue, m³. - averaging volume, m³. Greek Symbols V /, volume fraction of the -phase. - viscosity of the -phase, Ns/m². - density of the -phase, kg/m³. - surface tension, N/m. - (v +v ^T ), viscous stress tensor for the -phase, N/m².     

An exact solution for non-darcy free convection from a horizontal line source of heat

In this paper we consider the free convection from a horizontal line source of heat which is embedded in an unbounded porous medium saturated with a fluid at rest under gravity. The convective fluid and the porous medium are in local thermal equilibrium.

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Eine exakte Lösung der nicht-darcy'schen freien Konvektion von einer horizontalen, linienförmigen Wärmequelle

Zusammenfassung In dem Aufsatz wird die freie Konvektion von einer horizontalen, linienförmigen Wärmequelle untersucht, die in ein unbegrenztes poröses Medium eingebettet ist. Die Poren des porösen Mediums sind mit einem Fluid gefüllt, das unter Schwerkrafteinfluß ruht. Das strömende Fluid und das poröse Medium sind örtlich im thermischen Gleichgewicht.

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Nomenclature c _p specific heat of convective fluid - F _o parameter,=/(vl>g - g acceleration due to gravity - k thermal conductivity of the saturated porous medium - l typical length scale of body - Q heat flux per unit length of a line source - Ra Rayleigh number, =gQl/²c_p - Ra _x local Rayleigh number, =xg Qx/ a²c_p - T temperature - T temperature of ambient fluid - u, x andy components of velocity - x, y coordinates vertically upwards and normal to axis of plume - X, Y non-dimensional coordinates vertically upwards and normal to axis of plume Greek symbols equivalent themal diffusivity - coefficient of thermal expansion - similarity variable - non-dimensional temperature - x permeability of porous medium - viscosity of convective fluid - v kinematic viscosity of convective fluid - density of convective fluid - stream function - non-dimensional stream function - the Forchheimer's coefficient     

The finite element solutions of laminar flow and combined convection of air in a staggered or an in-line tube-bank

A finite element method is used to solve the full Navier-Stokes and energy equations for the problems of laminar combined convection from three isothermal heat horizontal cylinders in staggered tube-bank and four isothermal heat horizontal cylinders in in-line tube-bank. The variations of surface shear stress, pressure and Nusselt number are obtained over the entire cylinder surface including the zone beyond the separation point. The predicted values of total, pressure and friction drag coefficients, average Nusselt number and the plots of velocity flow fields and isotherms are also presented.

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Die Finite-Elemente-Lösung von laminarer Strömung und kombinierter Konvektion von Luft in einer versetzten oder fluchtenden Rohranordnung

Zusammenfassung Eine Methode der finiten Elemente wird zur Lösung der vollständigen Navier-Stokes- und der Energiegleichung für die Probleme der laminaren kombinierten Konvektion an drei isothermen geheizten horizontalen Zylindern in versetzter Rohranordnung sowie für vier isotherme geheizte horizontale Zylinder in fluchtender Anordnung verwendet.Die Veränderung der Wandschubspannung, des Druckes und der Nusselt-Zahl werden für die gesamte Zylinderoberfläche, einschließlich des Bereiches nach dem Ablösepunkt, bestimmt. Die Werte des gesamten Widerstandsbeiwertes aufgrund von Druck und Reibung, die durchschnittliche Nusselt-Zahl und die Diagramme des Geschwindigkeitsfeldes und der Isothermen werden ebenfalls aufgezeigt.

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Nomenclature C specifie heat - C _D total drag coefficient - C _f friction drag coefficient - C _p pressure drag coefficient - D diameter of cylinder,L=2R ₀ - G, g gravitational acceleration - Gr Grashof number, g(T_w–T )D ³/v ² - h local heat transfer coefficient - K thermal conductivity - L spacing between the centers of cylinder - M _l shape function - N _i shape function - Nu, local and average Nusselt numbers - P dimensionless pressure, p^*/u ² - p ^*,p pressure, free stream pressure - Pe Peclet number,RePr - Pr Prandtl number, c/K - Ra Rayleigh number,Gr Pr - Re Reynolds number,Du /v - R ₀ radius of cylinder - T temperature - T _w temperature on cylinder surface with fixed value - T free stream temperature - v dimensionless x-direction component of velocity,v ^*/u - u ^* x-direction component of velocity - u free stream velocity - v dimensionless Y-direction component of velocity,v ^*/u - v ^* Y-direction component of velocity - X x-direction axis - x dimensionless x-direction coordinate,x ^*/D - x^* x-direction coordinate - Y Y-direction axis - y dimensionless Y-direction coordinate,y ^*/D - y ^* Y-direction coordinate Greek symbols coefficient of volumetric thermal expansion - plane angle - dynamic viscosity - kinematic viscosity, / - density of fluid - _w dimensionless surface shear stress, _* ^w /u ² - skw/* surface shear stress - dimensionless temperature,      

Constituency measurements in the mixing region of a cross flow jet using a laser velocimeter

The velocities in the mixing region of a cross flow jet injected into a freestream were studied in detail with a laser velocimeter. Three jet to freestream momentum ratios were used (3.1, 8.1, 16.2). By purposely seeding the jet and freestream separately (as well as both simultaneously), marking the fluid was feasible. Thus, determining the velocities that emanated from the different streams was possible. By methodically analyzing the three sets of dependent data, the size and location of the mixing region was determined. The mixing regions for the three momentum ratios were found to be of different sizes and at different locations. By proper scaling, however, the regions for the three momentum ratios were found to collapse to one scaled region. Because of the intermittent behavior of the mixing, conventional turbulence models for such mixing may not be applicable; however, detailed velocities and turbulence quantities are included for benchmarking predictions.List of symbols B slot width - H channel height - MR momentum ratio, jet to free stream = _j V _j ²/ U ² - Re _H Reynolds number, U H/v - U free stream velocity - u axial velocity - u rms of axial velocity fluctuation - v transverse velocity - v rms of transverse velocity fluctuation - V _j slot exit transverse velocity - x axial direction (Fig. 3) - x _c x-center of mixing region - scaled value of x, = x/B - y transverse direction (Fig. 3) - y _c y-center of mixing region - scaled value of y, = y/ MRB - _x mixing region width in x-direction - _y mixing region width in y-direction - scaled mixing region width in x-direction, = _x /B - scaled mixing region width in y-direction, = _y / MRB - free stream density - _j slot exit density - v kinematic viscosity of freestream This research was sponsored in part by the Fulbright Commission (Bonn, Germany), the Institut für Thermische Strömungsmaschinen, Universität Karlsruhe (Karlsruhe, Germany), and the Rotating Machinery and Controls Industrial Research Program, University of Virginia (Charlottesville, VA, USA)     

Flow in porous media I: A theoretical derivation of Darcy's law

Stokes flow through a rigid porous medium is analyzed in terms of the method of volume averaging. The traditional averaging procedure leads to an equation of motion and a continuity equation expressed in terms of the volume-averaged pressure and velocity. The equation of motion contains integrals involving spatial deviations of the pressure and velocity, the Brinkman correction, and other lower-order terms. The analysis clearly indicates why the Brinkman correction should not be used to accommodate ano slip condition at an interface between a porous medium and a bounding solid surface.The presence of spatial deviations of the pressure and velocity in the volume-averaged equations of motion gives rise to aclosure problem, and representations for the spatial deviations are derived that lead to Darcy's law. The theoretical development is not restricted to either homogeneous or spatially periodic porous media; however, the problem ofabrupt changes in the structure of a porous medium is not considered.Roman Letters A interfacial area of the - interface contained within the macroscopic system, m² - A _e area of entrances and exits for the -phase contained within the macroscopic system, m² - A interfacial area of the - interface contained within the averaging volume, m² - A ^* interfacial area of the - interface contained within a unit cell, m² - Ae area of entrances and exits for the -phase contained within a unit cell, m² - B second order tensor used to represent the velocity deviation (see Equation (3.30)) - b vector used to represent the pressure deviation (see Equation (3.31)), m^–1 - d distance between two points at which the pressure is measured, m - g gravity vector, m/s² - K Darcy's law permeability tensor, m² - L characteristic length scale for volume averaged quantities, m - characteristic length scale for the -phase (see Figure 2), m - characteristic length scale for the -phase (see Figure 2), m - n unit normal vector pointing from the -phase toward the -phase (n =–n ) - n _e unit normal vector for the entrances and exits of the -phase contained within a unit cell - p pressure in the -phase, N/m² - p intrinsic phase average pressure for the -phase, N/m² - p –p , spatial deviation of the pressure in the -phase, N/m² - r ₀ radius of the averaging volume and radius of a capillary tube, m - v velocity vector for the -phase, m/s - v phase average velocity vector for the -phase, m/s - v intrinsic phase average velocity vector for the -phase, m/s - v –v , spatial deviation of the velocity vector for the -phase, m/s - V averaging volume, m³ - V volume of the -phase contained within the averaging volume, m³ Greek Letters V/V, volume fraction of the -phase - mass density of the -phase, kg/m³ - viscosity of the -phase, Nt/m² - arbitrary function used in the representation of the velocity deviation (see Equations (3.11) and (B1)), m/s - arbitrary function used in the representation of the pressure deviation (see Equations (3.12) and (B2)), s^–1     

A three-parameter model describing the behaviour of a viscoelastic liquid in a tangential annular flow

A three-parameter model describing the shear rate-shear stress relation of viscoelastic liquids and in which each parameter has a physical significance, is applied to a tangential annular flow in order to calculate the velocity profile and the shear rate distribution. Experiments were carried out with a 5000 wppm aqueous solution of polyacrylamide and different types of rheometers. In a shear-rate range of seven decades (5 10^–3 s^–1 < < 1.2 10⁵ s^–1) a good agreement is obtained between apparent viscosities calculated with our model and those measured with three different types of rheometers, i.e. Couette rheometers, a cone-and-plate rheogoniometer and a capillary tube rheometer. a physical quantity defined by:a = {1 – ( / ₀)}/ ₀ (Pa^–1) - C constant of integration (1) - r distancer from the center (m) - r ₁,r ₂ radius of the inner and outer cylinder (m) - v _r local tangential velocity at a distancer from the center (v _r = _r r) (m s^–1) - v ₂ local tangential velocity at a distancer ₂ from the center (m s^–1) - shear rate (s^–1) - local shear rate (s^–1) - ₁ wall shear rate at the inner cylinder (s^–1) - dynamic viscosity (Pa s) - _a apparent viscosity (_a = / ) (Pa s) - _a1 apparent viscosity at the inner cylinder (Pa s) - ₀ zero-shear viscosity (Pa s) - infinite-shear viscosity (Pa s) - shear stress (Pa) - _r local shear stress at a distancer from the center (Pa) - ₀ yield stress (Pa) - ₁, ₂ wall shear-stress at the inner and outer cylinder (Pa) - _r local angular velocity (s^–1) - ₂ angular velocity of the outer cylinder (s^–1)     

Correlation of experimental data on the pulsation velocity intensity for turbulent fluid flow in channels of different form

An analysis is made of experimental data on the intensity of the velocity pulsations in turbulent fluid flow in channels of different shape. Correlating relations are constructed for the intensity of the velocity pulsation components as a function of the flow regime and coordinates.Notation x, y, z coordinates in the flow direction, along the normal and parallel to the channel wall, respectively - a normal distance from the channel center to the wall - b distance from the channel corner to the point of intersection with the wall of the normal from the channel center - ₁ dimensionless distance along the normal from the channel wall - ₂ dimensionless distance in the direction parallel to the channel wall - U local fluid velocity - U_m maximal fluid velocity - U average fluid velocity across the section - _i intensity (mean square value) of the i-th component of the velocity pulsations (u, v, w are the indices in the directions x, y, z, respectively) - _i0 value of the intensity of the velocity pulsation components at the center of the channel - U_L velocity difference within the limits of the hydrcdynamic macroscale - q² total turbulence energy at a fixed point of the flow     

On the closure problem for Darcy's law

In a previous derivation of Darcy's law, the closure problem was presented in terms of an integro-differential equation for a second-order tensor. In this paper, we show that the closure problem can be transformed to a set of Stokes-like equations and we compare solutions of these equations with experimental data. The computational advantages of the transformed closure problem are considerable.Roman Letters A interfacial area of the- interface contained within the macroscopic system, m² - A _e area of entrances and exits for the-phase contained within the macroscopic system, m² - A interfacial area of the- interface contained within the averaging volume, m² - A _e area of entrances and exits for the-phase contained within the averaging volume, m² - B second-order tensor used to respresent the velocity deviation - b vector used to represent the pressure deviation, m^–1 - C second-order tensor related to the permeability tensor, m^–2 - D second-order tensor used to represent the velocity deviation, m² - d vector used to represent the pressure deviation, m - g gravity vector, m/s² - I unit tensor - K C ^–1,–D, Darcy's law permeability tensor, m² - L characteristic length scale for volume averaged quantities, m - characteristic length scale for the-phase, m - l _i i=1, 2, 3, lattice vectors, m - n unit normal vector pointing from the-phase toward the-phase - n _e outwardly directed unit normal vector at the entrances and exits of the-phase - p pressure in the-phase, N/m ² - p intrinsic phase average pressure, N/m² - p –p , spatial deviation of the pressure in the-phase, N/m² - r position vector locating points in the-phase, m - r ₀ radius of the averaging volume, m - t time, s - v velocity vector in the-phase, m/s - v intrinsic phase average velocity in the-phase, m/s - v phase average or Darcy velocity in the \-phase, m/s - v –v , spatial deviation of the velocity in the-phase m/s - V averaging volume, m³ - V volume of the-phase contained in the averaging volume, m³ Greek Letters V /V volume fraction of the-phase - mass density of the-phase, kg/m³ - viscosity of the-phase, Nt/m²     

Translatory accelerating motion of a circular disk in a viscous fluid

The exact solution of the equation of motion of a circular disk accelerated along its axis of symmetry due to an arbitrarily applied force in an otherwise still, incompressible, viscous fluid of infinite extent is obtained. The fluid resistance considered in this paper is the Stokes-flow drag which consists of the added mass effect, steady state drag, and the effect of the history of the motion. The solutions for the velocity and displacement of the circular disk are presented in explicit forms for the cases of constant and impulsive forcing functions. The importance of the effect of the history of the motion is discussed.Nomenclature a radius of the circular disk - b one half of the thickness of the circular disk - C dimensionless form of C ₁ - C ₁ magnitude of the constant force - D fluid drag force - f(t) externally applied force - F() dimensionaless form of applied force - F ₀ initial value of F - g gravitational acceleration - H() Heaviside step function - k magnitude of impulsive force - K dimensionless form of k - M a dimensionless parameter equals to (1+37#x03C0;_s/4_f) - S displacement of disk - t time - t ₁ time of application of impulsive force - u velocity of the disk - V dimensionless velocity - V ₀ initial velocity of V - V _t terminal velocity - parameter in (13) - parameter in (13) - (t) Dirac delta function - ratio of b/a - () function given in (5) - dynamical viscosity of the fluid - kinematic viscosity of the fluid - _f fluid density - _s mass density of the circular disk - dimensionless time - _i dimensionless form of t _i - dummy variable - dummy variable