Oscillatory flow in microchannels

Commonly used, lumped-parameter expressions for the impedance of an incompressible viscous fluid subjected to harmonic oscillations in a channel were compared with exact expressions, based on solutions of the Navier-Stokes equations for slots and channels of circular and rectangular cross-section, and were found to differ by as much as 30% in amplitude. These differences resulted in predicted discrepancies by as much as 400% in frequency response amplitude for simple second-order systems based on size scales and frequencies encountered in microfluidic devices. These predictions were verified experimentally for rectangular microchannels and indicate that underdamped fluidic systems operating near the corner frequency of any included flow channel should be modeled with exact expressions for impedance to avoid potentially large errors in predicted behavior.List of symbols A Channel cross-sectional area (m²) - A_c Membrane area (m²) - a Rectangular duct and slot half-width or radius (m) - b Rectangular duct half-depth and slot depth (m) - C Capacitance (m³/Pa) - C - D_h Channel hydraulic diameter (m) - E Voltage (V) - f Darcy friction factor - F Force (N) - I Channel inertance (Pa s²/m³) - i - Imaginary part of a complex number - J_k Bessel function of the first kind of order k - System transfer function - K Sum of minor loss factors - k Membrane stiffness (N/m) - L Channel length (m) - n Outward unit normal vector - P Fluid pressure (Pa) - p_n - Q Volumetric flow rate (m³/s) - R Channel resistance (Pa s/m³) - Real part of a complex number - Re Reynolds number, - V Velocity (m/s) - _V Volume (m³) - w Axial component of velocity (m/s) - Harmonic amplitude of membrane centerline displacement - Fluid impedance (kg/m⁴ s) - Duct aspect ratio, b/a - ² Nondimensional frequency parameter, - Nondimensional corner frequency, - Membrane shape factor - C/C - µ Fluid dynamic viscosity (Pa s) - Fluid kinematic viscosity (m²/s) - Mass density (kg/m³) - Radian frequency - _c R_s/I_s cutoff or corner frequency - _n Undamped natural frequency - Channel shape parameter in Eqs. 29 and 30 - Damping ratio - ( )_e Exact property - ( )_s Simplified property - (^–) Spatial average - Complex quantity     

The effect of cross flow on one-dimensional spectra measured using hot wires

Expressions were developed to estimate the cross-flow error that occurs in the one-dimensional velocity spectra determined by applying Taylors frozen field hypothesis to measurements with single- and cross-wire probes. The cross-flow error and the error caused by the unsteady convection of the small-scale motions were evaluated for typical measurements. It was found that the cross-flow error could be significant in inertial range of the measured one-dimensional spectra, and was much larger than the error caused by the unsteady convection of the small-scale motions in the one-dimensional spectra of the cross-stream velocity components, and . The results indicate that the one-dimensional spectra of the streamwise velocity component measured with a single-wire probe should be significantly more accurate than the spectra measured with a cross-wire probe. The cross-flow error in the one-dimensional spectra also becomes much less important in the dissipation range of the measured spectra.

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Pulsed-wire measurements in the near-wall layer in a reattaching separated flow

Pulsed-wire velocity measurements have been made in the near-wall layer, including the viscous sublayer, beneath a separated flow. A method for correcting the error caused by fluctuations in velocity gradient is given, extending the work of Schober et al. (1998). The measurements show that the r.m.s. of the streamwise velocity fluctuations scale closely in accordance with an inner-layer scaling, where the velocity scale, , is based on the r.m.s. of the wall shear stress fluctuations (measured by means of a pulsed-wire shear stress probe), rather than the mean wall shear stress. The effects of velocity gradient are only significant beneath of 10 or less.List of symbols C Calibration constant - f Function representing mean velocity - h_f Height of fence above splitter plate surface - L Length scale of outer-layer structures - s Distance between pulsed and sensor wires - u r.m.s. of U - Velocity scale based on r.m.s of wall shear stress fluctuation - U Instantaneous velocity in x-direction - U_m Instantaneous measured velocity in x-direction - U_r Free-stream reference velocity - x Streamwise direction from separation point - y Distance from splitter plate surface, in normal direction - X_r Length of separation bubble - ₀ Thickness scale in oscillating layer - Blasius laminar boundary layer parameter - Density - Wall shear stress - r.m.s. of wall shear stress fluctuation - Frequency of oscillating layer - Kinematic viscosity - Overbar denotes time average     

Control of unstable oscillations in a confined and bounded injecting wall channel

The analysis of a confined flow field generated through two separated injecting walls was carried out by studying the effect of a mass flow rate difference between the two walls. Such a parameter has been found to play a major role in unstable flow field behaviour. Concretely speaking, we have identified two vortex shedding phenomena, i.e. the main flow and the wall vortex shedding phenomena. Results clearly show that when the mass flow rate is increased at the second injecting wall, wall vortex coherence is enhanced and impinging of such structures forces a coupling phenomenon to develop between flow field dynamics and acoustics. On the other hand, only a 15% mass flow rate difference of the first injecting block is sufficient to prevent such coupling between acoustics and vortex shedding phenomenon. Consequently, the resonance phenomenon is pronouncedly weakened and significant oscillation reduction is achieved.Nomenclature sound velocity (m/s) - nth longitudinal acoustic mode (Hz) - h_t height of the nozzle throat (m) - h_c channel height (m) - l length between the edge of the second injecting block and the nozzle location (m) - L channel length (m) - P mean pressure at the front-head (Pa) - P fluctuating pressure (Pa) - q_m, q₁, q₂ total, first and second injecting block mass flow rate (kg/s) - S_x normalised power spectral density of the x fluctuations (Hz^-1) - s=w h_c characteristic surface area (m²) - T temperature of the flow (K) - u, v longitudinal and lateral velocity component (m/s) - u_X longitudinal mean velocity at X location (m/s) - u_m maximum longitudinal velocity (m/s) - u, v longitudinal and lateral fluctuating velocity (m/s) - characteristic acoustic velocity (m/s) - v_w wall injection velocity (m/s) - w channel width (m) - X, Y, Z non-dimensional axis normalised respectively by l, h_c and w - dynamic viscosity (kg/ms) - density (kg/m³) - time delay (s)Dimensionless parameters turbulence intensity - Mach number - Re_c= M a h_c/ Reynolds number - Re_w= v_w h_c/ wall injection Reynolds number - correlation coefficient of pressure and velocity fluctuations - normalised longitudinal velocity component - parameter of unbalanced mass flow rate between the two injecting blocks - specific heat ratio - coherence function     

A technique for evaluation of three-dimensional behavior in turbulent boundary layers using computer augmented hydrogen bubble-wire flow visualization

A system is described which allows the recreation of the three-dimensional motion and deformation of a single hydrogen bubble time-line in time and space. By digitally interfacing dualview video sequences of a bubble time-line with a computer-aided display system, the Lagrangian motion of the bubble-line can be displayed in any viewing perspective desired. The u and v velocity history of the bubble-line can be rapidly established and displayed for any spanwise location on the recreated pattern. The application of the system to the study of turbulent boundary layer structure in the near-wall region is demonstrated.List of Symbols Reynolds number based on momentum thickness u /v - t⁺ nondimensional time - u shear velocity - u local streamwise velocity, x-direction - u ⁺ nondimensional streamwise velocity - v local normal velocity, -direction - x ⁺ nondimensional coordinate in streamwise direction - ⁺ nondimensional coordinate normal to wall - ₊ ^wire wire nondimensional location of hydrogen bubble-wire normal to wall - z ⁺ nondimensional spanwise coordinate - momentum thickness - v kinematic viscosity - _W wall shear stress     

A phenomenological model of electrorheological fluids

The fundamental assumption of the paper is that the extra stress tensor of an electrorheological fluid is an isotropic tensor valued function of the rate of strain tensor D and the vector n (which characterizes the orientation and length N of the fibers formed by application of an electric field). The resulting constitutive equation for is supplemented by the solution of the previously studied time evolution equation for n. Plastic behavior for the shear and normal stresses is predicted. Anticipating that the action of increasing shear rate is i) to orient the fibers more and more in the direction of flow and ii) simultaneously to break up the fibers leads to the conclusion that for the same behavior is encountered as without an electric field. Using realistically possible approximation formulas for the dependence of and N on leads to the Bingham behavior for and power law behavior for large shear rates.

Two-point laser Doppler velocimetry measurements in a Mach 1.2 cold supersonic jet for statistical aeroacoustic source model

Single and multi-point laser Doppler velocimetry measurements performed in a cold Mach 1.2 jet flow are used to assess those properties of the aerodynamic field most relevant in the generation of turbulence mixing noise. Single point measurements yield mean velocity profiles, turbulence intensity profiles and power spectral densities of both the velocity and Reynolds stress fields at seven axial stations between the jet exit and the end of the potential core. The longitudinal components of the second-order and fourth-order two-point velocity correlation tensor are obtained from a series of multi-point LDV measurements, whence a cartography of integral space and time scales, convection velocities and acoustic compactness is effected. These results are used to examine differences between subsonic and supersonic jet aerodynamics in terms of their sound generating potential. Finally analytical expressions are proposed for the spatial and temporal parts of the longitudinal correlation coefficient function. These are scaled using the integral space and time scales of the velocity and Reynolds stress fields, and excellent agreement is found with experimentally determined functions.Nomenclature  c_o  Sound speed - D  Exit nozzle diameter - f, f  Spatial correlation function - g, g  Temporal correlation function - f_St  Frequency based Strouhal number - _i  2nd-order integral length scales in i-th direction -  4th-order integral length scales in i-th direction - M_c  Convective Mach number - M_j Jet exit Mach number - q Quantity q evaluated at location y into the flow -  Time average of the quantity q - r Radial distance from the jet exit - r_0.5  Radial location of the shear layer axis - r* Normalised radial coordinate - r_ij Second-order velocity correlation - r_ijkl Fourth-order velocity correlation - St Jet Strouhal number - U_c Convective velocity - U_e Subsonic coflow velocity - U_j Jet exit velocity - U_i Mean part of u_i - u_i Local velocity in i-th direction - u_ti Fluctuating part of u_i - x Distance from the exit nozzle - y Location test point - _i Variance of the velocity component u_i - _ij Variance of the velocity product u_iu_j -  Constant value - _ij Kronecker delta - _c Shear layer thickness - t_i Interarrival time between two ldv samples -  Separation distance in the moving pattern (components _i ) -  Polynomial function -  Separation distance in the fixed pattern (components _i ) -  Time delay -  2nd-order time scale in the fixed pattern -  2nd-order time scale in the moving pattern -  4th-order time scale in the fixed pattern -  4th-order time scale in the moving pattern -  Typical radian frequency where

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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.     

Investigation of the secondary corner vortex in a benchmark turbulent backward-facing step using cross-correlation particle imaging velocimetry

An experimental study of a turbulent backward-facing step (BFS) was undertaken to investigate the vortex structures behind the step. Attention was given to the secondary vortex because of its poor representation in literature and its potential for evaluating computational turbulence models. A 2D, cross-correlation particle image velocimeter (PIV) was developed, which allowed measurement of the highly turbulent, reversing step flow. Global, high resolution data was obtained for the cross-sectional plane of the BFS and for several other planes parallel to it. Measurement planes across the step revealed the 3D nature of the secondary vortex and an unexpected flow structure was identified. The secondary vortex was found to traverse across the flow, from the cross-sectional plane towards the step edge–sidewall corner.List of symbols AR aspect ratio - d particle displacement (m) - d error in particle displacement (m) - D expansion channel height (mm) - D₀ inlet channel height (mm) - ER expansion ratio - H step height (mm) - N number of samples - Re_H Reynolds number based on step height - Sp(x,y) centre coordinates of primary vortex (mm) - Ss(x,y) centre coordinates of secondary vortex (mm) - t laser pulse separation time (s) - t error in pulse separation time (s) - U horizontal velocity (m/s) - ̄ mean horizontal velocity (m/s) - horizontal velocity variance (m²/s²) - inlet centreline mean velocity (m/s) - inlet centreline velocity variance (m²/s²) - V vertical velocity (m/s) - VM velocity magnitude (m/s) - VM error in velocity magnitude (m/s) - W step width (mm) - x length dimension (mm) - y height dimension (mm) - z width dimension (mm) - Xr shear layer reattachment point (mm) - Xr reattachment point for infinite step width (mm) - Xs secondary vortex separation point (mm) - Ys secondary vortex reattachment point (mm) - U velocity error (m/s) - mean velocity error estimate (m/s) - velocity variance error estimate (m²/s²) - _bot bottom inlet boundary layer thickness (mm) - _top top inlet boundary layer thickness (mm) - ₉₉ 0.99 boundary layer thickness (mm)     

Counterflow flames of air and methane,propane and ethylene,with and without periodic forcing

The extinction of forced and unforced turbulent premixed counterflow flames has been quantified with lean mixtures of air and each of methane, propane and ethylene. Symmetric flames were produced with two streams of equal equivalence ratios between 0.6 and 1.0, and nozzle separations from 0.2 to 2.5 D, while acoustic drivers were used to force the flow at discrete frequencies. Photographs confirmed visual observation of unforced twin flames and their merging with increasing strain rate into one reaction zone at the stagnation plane before extinction. Propane flames merged at velocities closer to the extinction limit. At separations less than 0.4 D local quenching and extinction and relight occurred at equivalence ratios less than 0.7, independent of fuel type. Unforced extinction times were determined by igniting mixtures with equivalence ratios of 0.6 to 0.9 and bulk velocities above the extinction limit, and observing the extinction process with high-speed video: they were found to increase quasi-exponentially with reduction in strain rate, and were strongly dependent on equivalence ratio and fuel type. Forced extinction times also increased with decrease in strain rate and with reduction in forcing amplitude and instantaneous strain rates greater than the unforced limit were observed. Ethylene flames were more sensitive to the cyclic weakening with more rapid temperature decay rates and shorter extinction times.Abbreviations f Forcing frequency (Hz) - H Nozzle separation (m) - D Nozzle diameter (m) - Bulk strain rate, 2U _b/H, (s^-1) - Bulk strain rate at extinction (s^-1) - Maximum instantaneous forced strain rate (s^-1) - Maximum instantaneous unforced strain rate (s^-1) - Forcing time to extinction (s) - Time of one period of forcing oscillation (s) - Bulk velocity, flow rate/nozzle exit area (ms^-1) - Bulk velocity at extinction (ms^-1) - u Fluctuating component of turbulent velocity (ms^-1) - Fluctuating component of forced velocity (ms^-1) - Equivalence ratio (dimensionless)     

Simultaneous measurements of velocity and temperature fluctuations in thermal boundary layer in a drag-reducing surfactant solution flow

The mechanism of turbulent heat transfer in the thermal boundary layer developing in the channel flow of a drag-reducing surfactant solution was studied experimentally. A two-component laser Doppler velocimetry and a fine-wire thermocouple probe were used to measure the velocity and temperature fluctuations simultaneously. Two layers of thermal field were found: a high heat resistance layer with a high temperature gradient, and a layer with a small or even zero temperature gradient. The peak value of was larger for the flow with the drag-reducing additives than for the Newtonian flow, and the peak location was away from the wall. The profile of was depressed in a similar manner to the depression of the profile of in the flow of the surfactant solution, i.e., decorrelation between v and compared with decorrelation between u and v. The depression of the Reynolds shear stress resulted in drag reduction; similarly, it was conjectured that the heat transfer reduction is due to the decrease in the turbulent heat flux in the wall-normal direction for a flow with drag-reducing surfactant additives.List of symbols ensemble averaged value - (·)⁺ normalized by the inner wall variables - (·) root-mean-square value - C concentration of cetyltrimethyl ammonium chloride (CTAC) solution - c _p heat capacity - D hydraulic diameter - f friction factor - H channel height - h heat transfer coefficient - j _H Colburn factor - l length - Nu Nusselt number, h - Pr Prandtl number, c _p/ - q _w wall heated flux - Re Reynolds number, U _b/ - T temperature - T _b bulk temperature - T _i inlet temperature - T _w wall temperature - T friction temperature, q _w /c _p u - U local time-mean streamwise velocity - U ₁ velocity signals from BSA1 - U ₂ velocity signals from BSA2 - U _b bulk velocity - u streamwise velocity fluctuation - u1 velocity in abscissa direction in transformed coordinates - u friction velocity, - v wall-normal velocity fluctuation - v1 velocity in ordinate direction in transformed coordinates - var(·) variance - x streamwise direction - y wall-normal direction - z spanwise direction - _j junction diameter of fine-wire TC - _w wire diameter of fine-wire TC - angle of principal axis of joint probability function p(u,v) - _f heat conduction of fluid - _w heat conduction of wire of fine-wire TC - kinematic viscosity - local time-mean temperature difference, T _w –T - temperature fluctuation - standard deviation - density - _w wall shear stress     

Non-autonomous 2D Navier–Stokes System with Singularly Oscillating External Force and its Global Attractor

We study the global attractor of the non-autonomous 2D Navier–Stokes (N.–S.) system with singularly oscillating external force of the form . If the functions g ₀(x, t) and g ₁ (z, t) are translation bounded in the corresponding spaces, then it is known that the global attractor is bounded in the space H, however, its norm may be unbounded as since the magnitude of the external force is growing. Assuming that the function g ₁ (z, t) has a divergence representation of the form where the functions (see Section 3), we prove that the global attractors of the N.–S. equations are uniformly bounded with respect to for all . We also consider the “limiting” 2D N.–S. system with external force g ₀(x, t). We have found an estimate for the deviation of a solution of the original N.–S. system from a solution u ⁰(x, t) of the “limiting” N.–S. system with the same initial data. If the function g ₁ (z, t) admits the divergence representation, the functions g ₀(x, t) and g ₁ (z, t) are translation compact in the corresponding spaces, and , then we prove that the global attractors converges to the global attractor of the “limiting” system as in the norm of H. In the last section, we present an estimate for the Hausdorff deviation of from of the form: in the case, when the global attractor is exponential (the Grashof number of the “limiting” 2D N.–S. system is small).      

Existence of a Solution “in the Large” for Ocean Dynamics Equations

For the system of equations describing the large-scale ocean dynamics, an existence and uniqueness theorem is proved “in the large”. This system is obtained from the 3D Navier–Stokes equations by changing the equation for the vertical velocity component u ₃ under the assumption of smallness of a domain in z-direction, and a nonlinear equation for the density function ρ is added. More precisely, it is proved that for an arbitrary time interval [0, T], any viscosity coefficients and any initial conditions

Entropy generation due to laser step input pulse heating

Laser heating of surfaces results in thermal expansion of the substrate material in the region irradiated by a laser beam. In this case, the thermodynamic irreversibility associated with the thermal process is involved with temperature and thermal stress fields. In the present study, entropy analysis is carried out to quantify the thermodynamic irreversibility pertinent to laser pulse heating process. The formulation of entropy generation due to temperature and stress fields is presented and entropy generation is simulated for steel substrate. It is found that the rapid rise of surface displacement in the early heating period results in high rate of entropy generation due to stress field in the surface region while entropy generation due to temperature field increases steadily with increasing depth from the surface. c ₁ Wave speed in the solid (m/s) - c ₁* Dimensionless wave speed - c ₂ Constant - C _p Specific heat (J/kg.K) - E Elastic modules (Pa) - I Power intensity (W/m²) - I ₁ Power intensity after surface reflection (W/m²) - I _o Laser peak power intensity (W/m²) - k Thermal conductivity (W/m.K) - r _f Reflection coefficient - s Laplace variable - S Entropy generation rate (W/m³K) - S* Dimensionless entropy generation rate - T(x, t) Temperature (K) - T*(x*, t*) Dimensionless temperature - Temperature in Laplace domain (K) - Dimensionless reference temperature - t Time (s) - t* Dimensionless time - U Displacement (m) - U* Dimensionless displacement (U) - W* _lost Dimensionless lost work - x Spatial coordinate (m) - x* Dimensionless distance (x) - Thermal diffusivity (m²/s) - _T Thermal expansion coefficient (1/K) - Poissons ratio - Absorption coefficient (1/m) - Density (kg/m³) - _x Thermal stress (Pa) - _x * Dimensionless thermal stress      

Measurement of the turbulent diffusivity in the near field of variable density jets using conditional velocimetry

Pulsed laser Mie scattering and laser Doppler velocimetry (LDV), both conditioned on the origin of the seed particles, have been successively performed in turbulent jets with variable density. In the early stages of the jet developments, significant differences are measured between the ensemble average LDV data obtained by jet seeding and those obtained by seeding the ambient air. Careful analysis of the marker statistics shows that this difference is a quantitative measure of the turbulent mixing. The good agreement with gradient–diffusion modelling suggests the validity of a general diffusion equation where the velocities involved are expressed in terms of ensemble conditional Favre averages. This operator accounts for all events (including intermittent ones) and for variations in the density of the marked fluid whose velocity is still specified by the binary origin of the marker.List of symbols D_L laminar diffusivity, m²/s - D_T turbulent diffusivity, m²/s - d diameter of the jet nozzle, m - F_r Froude number - J diffusion vector, m/s - k global sensitivity of the detection system for one particle (signal level) - N_P number of seed particles in the probe volume - N_P,i number of seed particles in sample i - N_P⁽ⁱ⁾ value of N_P in channel i - N_B number of Doppler bursts - count rate of bursts, s^–1 - N_v number of validated Doppler bursts - count rate of validated bursts, s^–1 - N_id number of ideal particles - N_id* number of marked ideal particles - P* probability that an ideal particle be marked by a seed particle - P(z) probability density function for z, m³/kg - probability to have k seed particles in the probe volume - probability of having k seed particle conditioned on a given value of z - r radial coordinate, m - R =⁽¹⁾/⁽²⁾, density ratio - S₁ local signal level with jet seeding - S₁⁽¹⁾ reference signal level in pure stream 1 with jet seeding - s₁ = S_1/S₁⁽¹⁾, normalized signal - v_c volumic capacity of the probe volume, m³ - V velocity vector, m/s - V_x axial velocity component, m/s - V_r radial velocity component, m/s - V_P particulate velocity vector, m/s - V_Pj velocity vector of particle j, m/s - V_Pij velocity vector of the jth particle in sample i, m/s - V_i velocity vector of the marked flow for realization i, m/s - V_1,i velocity vector of the flow such it is marked in realization i by particles issuing only from stream 1, m/s - x axial coordinate, m - Y_i local mass fraction of species i - Z mixture fraction:local mass fraction of jet fluid - Z_i mixture fraction for realization iGreek local density, kg/m³ - _i local density for realization i, kg/m³ - ⁽¹⁾ density in stream 1 (density of the jet fluid), kg/m³ - ₁ time of flight of jet seed particles to reach the probe volume, s - _B duration of a Doppler burst, sAverages <A> ensemble average of A - Ā time average of A - Favre average, , ( ) the present notation is only due to printing problems - A Favre fluctuation,      

Existence of Solutions with the Prescribed Flux of the Navier–Stokes System in an Infinite Cylinder

The existence and uniqueness of a solution to the nonstationary Navier–Stokes system having a prescribed flux in an infinite cylinder is proved. We assume that the initial data and the external forces do not depend on x₃ and find the solution (u, p) having the following form

A study of stationary crack-tip deformation fields in thin sheets by computer vision

The in-plane deformation fields near a stationary crack tip for thin, single edge-notched (SEN) specimens, made from Plexiglas, 3003 aluminum alloy and 304 stainless steel, have been successfully obtained by using computer vision. Results from the study indicate that (a) in-plane deformations ranging from elastic to fully plastic can be obtained accurately by the method, (b) for U, and , the size of the HRR dominant zone is much smaller than forV and , respectively. Since these results are in agreement with recent analytical work, suggesting that higher order terms will be needed to accurately predict trends in the data, it is clear that the region where the first term in the asymptotic solution is dominant is dependent on the component of the deformation field being studied, (c) the HRR solution can be used to quantity only in regions where theplastic strains strongly dominate the elastic strain components (i.e., when ); forV, the HRR zone appears to extend somewhat beyond this region, (d) the displacement componentU does not have the HRR singularity anywhere within the measurement region for either 3003 aluminum or 304 SS. However, the displacement componentV agrees with the HRR slope up to the plastic-zone boundary in 3003 aluminum ( ) and over most of the region where measurements were obtained ( ) in 304 SS and (e) the effects of end conditions must be included in any finite-element model of typical SEN specimen geometries to accurately calculate theJ integral and the crack-tip fields.Paper was presented at the 1992 SEM Spring Conference on Experimental Mechanics held in Las Vegas, NV on June 8–11.     

Analysis of hysteresis and phase shifting phenomena in unsteady three-dimensional wakes

The aerodynamic characteristics of automobiles are greatly influenced by the unsteady change in the direction of relative airflow. The aim of this paper is to analyse how such a change influences vehicle wake flow patterns. An analysis was conducted on a simplified model capable of reproducing the typical structures encountered under the aerodynamic conditions of an automobile. The results were processed by mapping the steady and unsteady total pressure losses around the model. The findings should enable automobile development engineers inter alia to identify and analyse the physical phenomena that occur when a vehicle is subjected to a sudden gust of side wind.List of symbols B rod - l length of rod B (m) - angle of rod B (degrees) - D disk - P connection point between disk D and rod B - M _o drive motor of disk D - O centre of rotation of disk D - e radius of disk D (m) - angle of disk (degrees) - t moment of time t (s) - _o angle (degrees) at instant of time t=0 - d diameter of model (m) - C centre of rotation of model - x _c abscissa of centre of rotation C of model (m) - M connection point between model and rod B - pi value=3.14159 - incidence of model (degrees) - _M maximum value of incidence (degrees) - _m minimum value of incidence (degrees) - angular amplitude (degrees) - _c critical angle of incidence (degrees) for steady evolutions - critical angle of incidence (degrees) for the unsteady evolutions - ̄ mean angle of incidence (degrees) - angle (degrees) of the model such that =+ - pulse (rad s^–1) - T period (s) - f frequency (Hz) - R radius of model in meter (m) - velocity vector of incident airflow - V _o intensity of velocity vector (m s^–1) - P _io total pressure associated with upstream airflow velocity (Pa) - P _i local total pressure (Pa or J m^–3) - density (kg m^–3) - C _x drag coefficient - total pressure coefficient - (m, n) dimensions of grid: lines m, columns n - x X coordinate of sampling plane (m) - y _j Y coordinate of point of index j for j[1,n] - z _k Z coordinate of point of index k for k[1,m] - P _i (x,y _j ,z _k ,(t)) continuous data of unsteady total pressure (Pa) - discrete data of unsteady total pressure (Pa) - N number of tomographic images, from 1 upwards over an oscillation period T - maximum value of total pressure coefficients for steady evolutions - maximum value of total pressure coefficients for unsteady evolutions and increasing incidences - maximum value of total pressure coefficients for unsteady evolutions and decreasing incidences - differential between and - differential between and - phase shifting (degrees)