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.     

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     

Digital image plane holography (DIPH) for two-phase flow diagnostics in multiple planes

A technique for measuring the size and displacement of the disperse phase in two planes of a two-phase flow is presented. Digital image plane holography (DIPH) is used for the simultaneous recording and independent reconstruction of both planes. Each fluid plane is illuminated with two laser sheets propagating in opposite directions. The defocused image fields are holographically recorded at 90°, and can be reconstructed either in a defocused or in the best-focused plane. The analysis of the images in a defocused plane provides the sizes, while the cross-correlation of the focused images provides the velocity field, as in a regular particle image velocimetry (PIV) experiment. For air bubbles freely drifting in glycerine, diameters from 50 m to 400 m and displacements of up to 300 m have been measured.     

Turbulent dispersion of droplets for air flow in a pipe

An optical technique was used to study the dispersion of 50 m, 90 m and 150 m droplets downstream of a source located in the center of a vertical pipe through which turbulent air is flowing. A turbulent dispersion coefficient, _P, and a mean-square of the fluctuations in the turbulent velocity, v _p ² , are determined from the change of the measured mean-square displacement of the droplets over the pipe cross section with time. The interesting aspect of the experiments is that they explored conditions where the inertia of the particles is believed to be a much more important effect than that of the crossing of trajectories associated with the inequality of the average velocities of the particles and the fluid.     

Laminar assisting mixed convection heat transfer from a vertical isothermal plate to water with variable physical properties

The steady laminar boundary layer flow, with an external force, along a vertical isothermal plate is studied in this paper. The external force may be produced either by the motion of the plate or by a free stream. The fluid is water whose density-temperature relationship is non-linear at low temperatures and viscosity and thermal conductivity are functions of temperature. The results are obtained with the numerical solution of the boundary layer equations with , k and variable across the boundary layer. Both upward and downward flow is considered. It was found that the variation of , k and with temperature has a strong influence on mixed convection characteristics.Nomenclature c_p water specific heat - f dimensionless stream function - g gravitational acceleration - Gr_x local Grashof number - k thermal conductivity - Nu_x local Nusselt number - Pr Prandtl number - Pr_a ambient Prandtl number - Re_x local Reynolds number - s salinity - T water temperature - T_a ambient water temperature - T_o plate temperature - u vertical velocity - u_a free stream velocity - u_o plate velocity - v horizontal velocity - x vertical coordinate - y horizontal coordinate - pseudo-similarity variable - nondimensional temperature - dynamic viscosity - _f film dynamic viscosity - _o dynamic viscosity at plate surface - kinematic viscosity - buoyancy parameter - water density - _a ambient water density - _f film water density - _o water density at plate surface - physical stream function     

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     

Performance prediction of dual cylindrical wave laser devices for flow measurements

The dual cylindrical wave system is a variant of laser Doppler velocimetry, in which two cylindrical waves of laser light are used to illuminate a moving particle. This instrument is being used for local measurement of the unsteady skin friction in turbulent boundary layers, as well as droplet sizing in spray flows. In the present work, performance of these new devices is examined using the electromagnetic theory of light. Various requirements for the design and operation of these instruments have been further elaborated and extended. The accuracy of the previous experimental results has also been considered. The optics-related errors are shown to be negligible in the measurements of streamwise as well as spanwise wall velocity gradients. However, rigorous simulations appear to be essential for proper calibration of the particle sizing device.List of symbols A, B, C three particle positions - a half-width of an optical slit - a _gm amplitude of a plane wave in the spectrum of a cylindrical wave - d _f fringe spacing - d _p particle diameter - E amplitude of the electric oscillation in the optical field - E _c combined electric field of two cylindrical waves - E _o maximum strength of the electric field at the source of a cylindrical wave - E _s electric field of a scattered wave - E _y time-dependent electric field in the case of electric polarization - f characteristic length for the phase of the scattering amplitude - f _a anisotropic frequency - f _D Doppler frequency - F _DCW transfer function of DCW system for particle sizing - F _pDA Phase Doppler transfer function - g wall velocity gradient - g _m measured wall velocity gradient - I ₀, I₂ integrals in the asymptotic expansion of the scattering amplitude - I _s intensity of the scattered light - k wave number of laser light in the fluid medium - m refractive index of the particle relative to the surrounding medium - N ₀ nominal number of fringes resulting from interference of two cylindrical waves - P phase of a plane wave - P ₁, P₂ phases of plane waves from downstream and upstream cylindrical waves respectively - P _s scattered light power at a receiving aperture - r unit vector in the direction of light scattering - r _D distance of the signal detector from the particle center - S scattering amplitude of a cylindrical wave - S ₁, S₂ Scattering amplitudes of the cylindrical waves emanating from S₁ and S₂ respectively - magnitude of the scattering amplitude for a plane wave - S _c combined scattering amplitude of two cylindrical waves - S₁, S₂ downstream and upstream sources of cylindrical waves, respectively - S scattering amplitude of a plane wave - s half-spacing between sources of the cylindrical waves - t time - u velocity along x-axis - w ₀ 1/e half-width of the field distribution at the waist of a laser sheet - X ₀ nominal width of the fringe volume along the particle path - X particle position in the measuring volume - x, y, z Cartesian coordinates Greek symbols angle of the direction of wave propagation from x-axis - coefficient of the second-order term in the phase function of a cylindrical wave - angular size of the signal receiving aperture - incremental used for numerical differentiation the ratio of to _p - — ₀ integration parameter for I₀ and I₂ - half-angle between the directions of propagation of two waves - wavelength of laser in the fluid medium - ₀ wavelength of laser in vacuum - parameter defining the direction of propagation of a plane wave - 1/e 1/e half-width of the function A - ₀ direction of propagation of the dominant plane wave in the spectrum - _0s the direction of propagation of the plane wave that contributes predominantly to scattering in a particular direction - _p the value of . corresponding to one cycle of P - _s change in corresponding to a lobe of the scattering amplitude - a dimensional form of that determines lobes in the scattered field - signal phase, ₀ + _a - _a anisotropic phase shift - ₀ phase difference between two indicent waves - off-axis angle - elevation angle - circular frequency of laser light     

Second order phase equilibrium for classical bodies

Summary In this note the Author, recalling a previous work[15], gives a new formulation of second order phase equilibria for classical bodies such as those defined by Truesdell and Toupin in[8].The Author arrives at three equivalent systems of partial differential equations (generalized Ehrenfest equations), the conditions for whose integration are shown to be always satisfied.Finally, as particular cases, the equations ruling the phase equilibria for classical fluids and for n component-classical fluid mixtures are given.

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Sommario In questa nota l'Autore, rifacendosi ad un lavoro precedente[15], presenta una nuova formulazione degli equilibri di fase del secondo ordine per corpi classici come quelli definiti da Truesdell e Toupin in[8].L'Autore perviene a tre sistemi equivalenti di equazioni alle derivate parziali (equazioni di Ehrenfest generalizzate) dei quali viene dimostrata la integrabilità.Infine, come casi particolari, si ottengono le equazioni che governano l'equilibrio di fase per fluidi classici e per miscele fluide classiche ad n componenti.

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This work was supported by the Gruppo Nazionale per la Fisica Matematica of C.N.R.     

Deformed particle image pattern matching in Particle Image Velocimetry

The deformation of particle image patterns due to velocity gradients causes errors of velocity measurements and false velocity detections in PIV (Particle Image Velocimetry). A novel technique to overcome those limitations inherent in the conventional PIV by correcting the particle image pattern according to the local velocity gradients in two dimensional flows, i.e. u/x, u/y, v/x and v/y, is proposed and successfully applied to a water flow downstream of a backward facing step.     

The stokes problems for a conducting fluid with a suspension of particles

The Stokes problems of an incompressible, viscous, conducting fluid with embedded small spherical particles over an infinite plate, set into motion in its plane by impulse and by oscillation, in the presence of a transverse magnetic field, are studied. The velocities of the fluid and of the particles and the wall shear stress are obtained. The stress is found to increase due to the particles and the magnetic field, with the effect of the particles diminishing as the field strength is increased.Nomenclature H ₀ strength of the imposed magnetic field - k density ratio of particles to fluid (per unit volume of flow field) - m _e ² H ₀ ² / - t time - y co-ordinate normal to the plate - u fluid velocity - v particle velocity - _e magnetic permeability of the fluid - kinematic viscosity of the fluid - electric conductivity of the fluid - fluid density - particle relaxation time - frequency of oscillation of the plate     

Laser velocimetry measurements in a horizontal gas-solid pipe flow

This paper presents laser measurements of particle velocities in a horizontal turbulent two-phase pipe flow. A phase Doppler particle analyzer, (PDPA), was used to obtain particle size, velocity, and rms values of velocity fluctuations. The particulate phase consisted of glass spheres 50 m in diameter with the volume fraction of the suspension in the range _p=10^-4 to _p=10^-3. The results show that the turbulence increases with particle loading.List of symbols a particle diameter - C _va velocity diameter cross-correlation - d pipe diameter - Fr ² Froude number - g gravitational constant - p(a) Probability density of the particle diameter - Re pipe Reynolds number based on the friction velocity - T characteristic time scale of the energy containing eddies - T _L integral scale of the turbulence sampled along the particle path - u, U, u characteristic fluid velocities: fluctuating, mean and friction - v characteristic velocity of the paricle fluctuations - f expected value of any random variable f - f¦g expected value of f given a value of the random variable g - _p particle volume fraction - _p particle response time - absolute fluid viscosity - v kinematic fluid viscosity - _p, _f densities, particle and fluid - _a ² particle diameter variance - _va ² velocity variance due to the particle diameter variance - _vT ² total particle velocity variance - _vt ² particle velocity variance due to the response to the turbulent field     

Bifurcations and complex instability in a 4-dimensional symplectic mapping

Summary We study the main periodic solutions of a 4-dimensional symplectic mapping composed of two coupled 2-dimensional mappings. Their bifurcations were calculated numerically and also theoretically for small values of the coupling parameter . Most bifurcating families of period 2ⁿ (n0) have complex unstable regions that extend from =0 to the maximum allowed value of for each family. These complex unstable regions do not allow the transmisssion of the stability of the corresponding families to families of higher order. We found only one family with a complex unstable region not extending to the maximum , but in this case also the transmission of the stability is stopped at an inverse bifurcation. Thus although there are infinite sequences of bifurcations (of the Feigenbaum type) in the limiting 2-dimensional case =0, all such sequences are interrupted when the system is 4-dimensional (i.e. for 0). The appearance of complex instability for =0 can be predicted by studying the cases =0 and applying the Krein-Moser theorem.

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Sommario Si svolge uuno studio dettagliato delle soluzioni periodiche principali di due mappe simplettiche bidimensionali accoppiate, calcolandone sia analiticamente che numericamente le biforcazioni per piccoli valori del parametro di accoppiamento . Quasi tutte le famiglie di periodo 2ⁿ (n0) prodotte dalle biforcazioni presentano regioni di instabilità complessa che si estendono da =0 fino al massimo valore di considerato. Queste regioni di instabilità complessa impediscono il trasferimento della stabilità di una famiglia a famiglie di ordine più elevato. In un solo caso si osserva una famiglia la cui regione di instabilità complessa non arriva ad estendersi fino al valore massimo di ; in questo caso però il trasferimento della stabilità viene interrotto da una biforcazione inversa. Se ne conclude che, nonostante I'esistenza di una famiglia di infinite biforcazioni di tipo Feigenbaum nel caso limite bidimensionale (=0), tutte le sequenze si interrompono se il sistema è a quattro dimensioni. Il formarsi di regioni di instabilità complessa per 0 può essere previsto studiando il caso =0 ed appplicando il teorema di Krein-Moser.

     

Asymmetric snap-buckling of a column restrained by a stiff wire

Summary A relatively simple example of asymmetric snap-through buckling in a continuous structure is the nonlinear problem of a cantilevered column restrained at its tip by a stiff wire, which is inclined at an acute angle to the column centerline, and loaded at its tip by a force perpendicular to the centerline. A parameter called , which is the nondimensional ratio of the flexural rigidity of the column to the combined extensional stiffness of the wire and the column centerline, determines the essential features of the buckling. If is zero, or is small compared to unity, the bending of the column is small enough to justify the use of linear bending theory for the column. Hence, even though the constraint is nonlinear, the solution to this problem is obtained in closed form. The critical point for the structure is found to be an asymmetric branching point for =0, while for positive, the critical point is a snap-through type. The effect of is similar to that induced by initial imperfections in more complex structures. For very small , the critical load is markedly decreased from the value for =0. Moreover, the graph of the load vs. tip deflection has the appearance of having an acute discontinuity in slope at the critical point for very small, although it is actually found that the graph has a horizontal tangent there.

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Sommario Un esempio relativamente semplice di collasso asimmetrico per carico di punta in una struttura continua è il problema non lineare di una colonna incastrata ad una estremità e vincolata all'altra estremità da un filo rigido che forma un angolo acuto con l'asse della colonna e caricata a quella estremità con una forza perpendicolare allo stesso asse. Un parametro chiamato , che è il rapporto adimensionale fra la rigidità flessionale della colonna e la rigidezza longitudinale del filo e dell'asse della colonna, determina le caratteristiche essenziali del cedimento. Se è zero o è piccolo rispetto all'unità, l'inflessione della colonna è sufficientemente piccola per giustificare l'uso della teoria lineare di inflessione per la colonna. Di conseguenza anche se il vincolo non è lineare la soluzione del problema è ottenuta in forma compatta. Il punto critico della struttura si trova nel punto di biforcazione asimmetrica per =0, mentre per positivo, il punto critico rappresenta un punto di collasso. L'effetto di è simile a quello prodotto da imperfezioni iniziali in strutture più complesse. Per molto piccolo il carico critico è notevolmente ridotto rispetto al valore per =0. Inoltre il grafico del carico in funzione della curvatura all'estremità sembra avere una netta discontinuità nella pendenza dal punto di biforcazione per molto piccolo benchè, in realtà si trovi che il grafico ha lì una tangente orizzontale.

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This research was supported by the Advanced Research Projects Agency (Project DEFENDER) and was monitored by the U.S. Army Research Office, Durham, under Contract DA-31-124-ARO-D-257.     

Motion of large gas bubbles ascending in a liquid

An equation is derived for the ascent velocity of large gas bubbles in a liquid. This velocity is assumed to be governed by the propagation of a wavelike perturbation caused by the bubble in the liquid.Notation w bubble (or drop) velocity - specific gravity - dynamic viscosity - kinematic viscosity - r bubble (or drop) radius - surface tension - coefficient of friction - g gravitational acceleration - D bubble (or drop) diameter - p pressure - c propagation velocity of the wavelike perturbation - wavelength     

Some problems of nonisothermal steady flow of a viscous fluid

The paper presents solutions to the problems of plane Couette flow, axial flow in an annulus between two infinite cylinders, and flow between two rotating cylinders. Taking into account energy dissipation and the temperature dependence of viscosity, as given by Reynolds's relation =₀ exp (–T) (₀, =const). Two types of boundary conditions are considered: a) the two surfaces are held at constant (but in general not equal) temperatures; b) one surface is held at a constant temperature, the other surface is insulated.Nonisothermal steady flow in simple conduits with dissipation of energy and temperature-dependent viscosity has been studied by several authors [1–11]. In most of these papers [1–6] viscosity was assumed to be a hyperbolic function of temperature, viz. =_m 1/1+²(T–T_m.Under this assumption the energy equation is linear in temperature and can he easily integrated. Couette flow with an exponential viscosity-temperature relation. =₀ e ^–T (₀, =const), (0.1) was studied in [7, 8]. Couette flow with a general (T) relation was studied in (9).Forced flow in a plane conduit and in a circular tube with a general (T) relation was studied in [10]. In particular, it has been shown in [10] that in the case of sufficiently strong dependence of viscosity on temperature there can exist a critical value of the pressure gradient, such that a steady flow is possible only for pressure gradients below this critical value.In a previous work [11] the authors studied Polseuille flow in a circular tube with an exponential (T) relation. This thermohydrodynamic problem was reduced to the problem of a thermal explosion in a cylindrical domain, which led to the existence of a critical regime. The critical conditions for the hydrodynamic thermal explosion and the temperature and velocity profiles were calculated.In this paper we treat the problems of Couette flow, pressureless axial flow in an annulus, and flow between two rotating cylinders taking into account dissipation and the variation of viscosity with temperature according to Reynolds's law (0.1). The treatment of the Couette flow problem differs from that given in [8] in that the constants of integration are found by elementary methods, whereas in [8] this step involved considerable difficulties. The solution to the two other problems is then based on the Couette problem.     

Analysis of suddenly started laminar flow in the entrance region of a circular tube

Suddenly started laminar flow in the entrance region of a circular tube, with constant inlet velocity, is investigated analytically by using integral momentum approach. A closed form solution to the integral momentum equation is obtained by the method of characteristics to determine boundary layer thickness, entrance length, velocity profile, and pressure gradient.Nomenclature M(, , ) a function - N(, , ) a function - p pressure - p* p/1/2U ², dimensionless pressure - Q(, , ) a function - R radius of the tube - r radial distance - Re 2RU/, Reynolds number - t time - U inlet velocity, constant for all time, uniform over the cross section - u velocity in the boundary layer - u* u/U, dimensionless velocity - u ₁ velocity in the inviscid core - x axial distance - y distance perpendicular to the axis of the tube - y* y/R, dimensionless distance perpendicular to the axis - boundary layer thickness - * displacement thickness - /R, dimensionless boundary layer thickness - momentum thickness - absolute viscosity of the fluid - /, kinematic viscosity of the fluid - x/(R Re), dimensionless axial distance - density of the fluid - tU/(R Re), dimensionless time - _w wall shear stress     

Shock velocity measurements using microwave Doppler shift

A commercially available 10.587 GHz microwave Doppler module is used for the measurement of shock velocity in a conventional shock tube. With proper electronic circuits the Doppler frequency obtained is found to be quite noisefree and consistent for shock velocities in the range of 1.8 mm/sec to 2.0 mm/sec.     

Heat transfer and equilibrium temperature of a sphere in a supersonic rarefied gas flow

Some results are presented of experimental studies of the equilibrium temperature and heat transfer of a sphere in a supersonic rarefied air flow.The notations D sphere diameter - u, , T,,l, freestream parameters (u is velocity, density, T the thermodynamic temperature,l the molecular mean free path, the viscosity coefficient, the thermal conductivity) - T₀ temperature of the adiabatically stagnated stream - T_e mean equilibrium temperature of the sphere - T_w surface temperature of the cold sphere (T_we) - mean heat transfer coefficient - _e air thermal conductivity at the temperature T_e - P Prandtl number - M Mach number     

Asymptotics of the homogenized moduli for the elastic chess-board composite

We find the asymptotic behavior of the homogenized coefficients of elasticity for the chess-board structure. In the chess board white and black cells are isotropic and have Lamé constants (, ,) and (, ) respectively. We assume that the black cells are soft, so 0. It turns out that the Poisson ratio for this composite tends to zero with .     

Reynolds number dependence of the drag coefficient for laminar flow through fine-scale photoetched screens

The laminar steady flow downstream of fine-mesh screens is studied. Instead of woven-wire screens, high-uniformity screens are fabricated by photoetching holes into 50.8 m thick Inconel sheets. The resulting screens have minimum wire widths of 50.8 m and inter-wire separations of 254 m and 318 m for the two screens examined. A flow facility has been constructed for experiments with these screens. Air is passed through the screens at upstream velocities yielding wire width Reynolds numbers from 2 to 35. To determine the drag coefficient, pressure drops across the screens are measured using pressure transducers and manometers. Threedimensional flow simulations are also performed. The computational drag coefficients consistently overpredict the experimental values. However, the computational results exhibit sensitivity to the assumed wire cross section, indicating that detailed knowledge of the wire cross section is essential for unambiguous interpretation of experiments using photoetched screens. Standard semi-empirical drag correlations for woven-wire screens do not predict the present experimental results with consistent accuracy.List of symbols A ₁, A ₂ screen aspect ratios - c _d screen drag coefficient - d woven-wire diameter - D photoetched minimum wire width (spanwise) - f woven-wire screen drag function - M distance between adjacent wires - N spectral-element order - o woven-wire open area fraction - O photoetched open area fraction - _p pressure drop across screen - Re _d woven-wire diameter Reynolds number - Re _D photoetched wire width Reynolds number - U fluid velocity upstream of screen - W photoetched sheet thickness (streamwise) - x, y, z spatial coordinates - fluid density - fluid viscosity