The interaction of two opposing,asymmetric curved wall jets

An experimental investigation was made of a two dimensional flow formed by the interaction of two asymmetric turbulent curved wall jets past a circular cylinder. Measurements were made of velocity and turbulence intensity profiles of the two curved wall jets before the interaction, and those of the merged jet after the interaction. The location of the interaction region of the two opposing curved wall jets and the flow direction of the merged jet were found to depend primarily on the ratio of initial momentum fluxes. The velocity and turbulence intensity profiles of the merged jet were similar to those of the plane turbulent jet. However, the growth rate of the merged jet was approximately 1.5 times larger than that of the plane jet. The influence of the momentum flux ratio on the growth rate appeared to be insignificant.List of symbols C _f friction coefficient - h slot height - J _p, J _c initial momentum flux of a power jet and of a control jet, respectively - P, Pa wall static and atmospheric pressure, respectively - Re Reynolds number based on slot height - Re _m local Reynolds number U _m y _m /v - U local mean velocity - U _c velocity along the center line of the merged jet - U _m local maximum velocity of the curved wall jet - u r.m.s. value of velocity fluctuations - u _u friction velocity - U ⁺ U/u_t - x distance along the cylinder surface - x distance along the center line of the merged jet - y _1/2, y _1/2 position of y and y where U = U _m /2 and U = U _c /2, respectively - y ⁺ yu _t/V - deflection angle of the merged jet (Fig. 4) - interaction angle (Fig. 4) - merged jet angle (Fig. 4) - angle measured from the center line of the cylinder (Fig. 4) - interception angle (Fig. 8) - , normalized coordinates, y/y _1/2 and y/y _1/2, respectively     

Experimental study of turbulent axisymmetric cavity flow

An experimental study is made of turbulent axisymmetric cavity flow. The flow configuration consists of a sudden expansion and contraction pipe joint. In using the LDV system, in an effort to minimize refraction of laser beams at the curved interface, a refraction correction formula for the Reynolds shear stress is devised. Three values of the cavity length (L = 300, 600 and 900 mm) are chosen, and the cavity height (H) is fixed at 55 mm. Both open and closed cavities are considered. Special attention is given to the critical case L = 600 mm, where the cavity length L is nearly equal to the reattachment length of the flow. The Reynolds number, based on the inlet diameter (D = 110 mm) is 73,000. Measurement data are presented for the static wall pressure, mean velocity profiles, vorticity thickness distributions, and turbulence quantities.List of symbols C _f velocity correction factor - C _p static wall pressure coefficient - D diameter of inlet pipe = 110 mm - H step height or difference in radii of two pipes = 55 mm - L cavity length = 300, 600 and 900 mm - n _a , n _w , n _f refraction indices of the medium between the transmitting lens and window, the window itself, and the working fluid - signal validation rate in LDV, Hz - P wall static pressure, Pa - P _ref wall static pressure at x = -70 mm, Pa - r radial distance from centreline, m - r _a radial position of the virtual intersection, m - r _d radial location of the dividing streamline, m - r _f radial position of the real beam intersection, m - Re Reynolds number based on the inlet diameter - R _i inner radius of the cylindrical cavity=110 mm - t thickness of the window, m - T ₁ integral time scale, s - U streamwise mean velocity, m/s - U _c centreline mean velocity, m/s - U _ref maximum upstream velocity at x= -70 mm, m/s - r.m.s. intensity of streamwise, radial and circumferential velocity fluctuations respectively, m/s - Reynolds shear stress, m²/s² - x distance in the streamwise direction, m - x _a streamwise position of virtual intersection, m - x _f streamwise position of real beam intersection, m - x _r mean reattachment length, m - x nondimensional streamwise distance - y distance normal to the wall=R–r, m Greek symbols vorticity thickness - stream function of dividing streamline      

Effect of a Short Roughness Strip on a Turbulent Boundary Layer

The response of a turbulent boundary layer to a short roughness strip is investigated using laser Doppler velocimetry (LDV) and laser induced fluorescence (LIF). Skin friction coefficients are inferred from accurate near-wall measurements. There is an undershoot in , where is the undisturbed smooth wall skin friction coefficient, immediately after the strip. Downstream of the strip, overshoots before relaxing back to unity in an oscillatory manner. The roughness strip has a major effect on the turbulent stresses ; these quantities increase, relative to the undisturbed smooth wall, in the region between the two internal layers originating at the upstream and downstream edges of the strip. The increase in the ratio suggests a decrease in near-wall anisotropy. From the flow visualizations, it is inferred that streamwise vortical structures are weakened immediately downstream of the strip. Consistently, streamwise length scales are also reduced; direct support for this is provided by measured two-point velocity correlations.     

Mean flow and turbulence measurements of the impingement wall jet on a semi-circular convex surface

The detailed mean flow and turbulence measurements of a turbulent air slot jet impinging on two different semi-circular convex surfaces were investigated in both free jet and impingement wall jet regions at a jet Reynolds number Re_w=12,000, using a hot-wire X-probe anemometer. The parametric effects of dimensionless circumferential distance, S/W=2.79-7.74, slot jet-to-impingement surface distance Y/W=1-13, and surface curvature D/W=10.7 and 16 on the impingement wall jet flow development along a semi-circular convex surface were examined. The results show that the effect of surface curvature D/W increases with increasing S/W. Compared with transverse Reynolds normal stress, [`(v<sup>2</sup> )] /U<sub>m</sub><sup>2</sup> \overline {v^2 } /U_{\rm m}^2 , the streamwise Reynolds normal stress, [`(u² )] /U_m² \overline {u^2 } /U_{\rm m}^2 , is strongly affected by the examined dimensionless parameters of D/W, Y/W and S/W in the near-wall region. It is also evidenced that the Reynolds shear stress, -[`(uv)] /U_m² - \overline {uv} /U_{\rm m}^2 is much more sensitive to surface curvature, D/W.     

Generating high freestream turbulence levels

This paper presents a method of generating a highly turbulent freestream flow, up to levels of 20% with a relatively uniform mean velocity field. This method was developed as a result of a combined water channel and wind tunnel study. The method for generating these high turbulence levels includes using high-velocity jets issuing into a mainstream cross-flow. A range of turbulence levels can be generated, using this same flow geometry, by adjusting the jet-to-mainstream velocity ratio or the Reynolds number of the flow.List of symbols b Grid bar width - D Turbulence generator jet hole diameter - Eu (f) Spectral energy for streamwise velocity fluctuations - f Frequency - H Channel height - L _u Dissipation length scale, - m Exponent for length scale growth - M Grid mesh size - n Exponent for turbulence decay - Re _D Reynolds number based on jet hole diameter - Re _T Turbulent Reynolds numbers,u _g /V - S Lateral spacing between the jet holes - T Integral time scale of turbulence - Tu Streamwise turbulence intensity,u/U - u RMS velocity in streamwise direction - U Mean local velocity in streamwise direction - U Freestream velocity in streamwise direction - v RMS velocity in normal direction - x Streamwise distance measured from the turbulence generator jets - y Vertical distance from the wall - z Spanwise distance - Boundary layer thickness (U = 0.99U ) - _x Longitudinal integral length scale of turbulence This project was supported by Wright Laboratory and Allied-Signal. The authors would also like to thank Mr. David Dotson for his help in constructing the turbulence generator and Mr. Don Schmidt for his help in procuring the blower. The first author would also like to thank Professor Sigmar Wittig and the Institut für Thermische Strömungsmaschinen for support while writing this paper     

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     

Measurements of turbulent inclined plane dual jets

Measurements of mean velocities, flow direction, velocity fluctuations and Reynolds shear stress were made with a split film probe of hot wire anemometer to investigate the interactions created by two air jets issuing from two identical plane inclined nozzles. The reverse flow was detected by using the split film probe and observed by flow visualization. Experimental results with an inclined angle of 9° are presented in the paper. Some experimental results with an inclined angle of 27° are presented to investigate the effect of inclination on the flow field.Mean velocities approach self-preservation in both the converging region and the combining region. Velocity fluctuations and Reynolds shear stress approach self-preservation in the combining region only. The spreads of jet and the square of the decay of maximum mean velocity increase linearly as the distance from the nozzle exit increases.List of symbols D nozzle width - h nozzle height - J momentum of jet - J ₀ momentum of jet at nozzle exit - M mass flow rate - M ₀ mass flow rate at nozzle exit - S nozzle spacing - U, V mean velocities in the X and Y axis respectively - U _m maximum axial velocity - U ₀ axial velocity at nozzle exit - u, v velocity fluctuations in the X and Y axis respectively - u, v r.m.s. of u and v - Reynolds shear stress - X, Y Cartesian coordinates - X _m , Y _m coordinates at the location of maximum axial velocity - y _0.5 distance from the location of maximum axial velocity to the location where the velocity is half of maximum axial velocity - inclined angle - yY/S - Reynolds stress correlation coefficient - C.P combining point - max maximum value - M.P merging point - o nozzle exit plane - V.C vortex center     

Helium jets discharging normally into a swirling air flow

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

Experimental investigation of certain aspects of upward vertical bubble and slug flows

The paper describes an installation, and measurements performed with its aid, in which it was possible to observe bubble flows as well as slug flows. Measurements and observations were carried out in vertical upward flow of water, with air injected into it, flowing through a plexiglass tube of 20 mm i.d. and 1500 mm long. The purpose of the investigation was to identify the parameters and influences which determine the observed flow pattern. The results show that there exists a range of values of the superficial velocity U _LS and U _GS in which it is possible to observe both flow patterns depending on the method of air injection employed. The transition zone bubble-slug shows reasonable agreement with the data of Taitel et al., whereas that for the slug-froth transition is close to the data of Oshinowa and Charles, and Griffith and Wallis. The distributions of bubble diameters and plug and Taylor-bubble dimensions are acceptably Gaussian. It is surmised that considerable discrepancies in the delineation of flow-regime boundaries which exist between different investigators are due to hitherto unidentified influences and parameters.List of symbols a distance between aligned bubbles - C ₀ distribution parameter - D inner tube diameter - d _S bubble diamter - L _b length of a Taylor bubble - L _E entrance length - L _S length of liquid plug - U _G actual gas velocity - U _GS superficial gas velocity: - U _L actual liquid velocity - U _LS superficial liquid velocity: - U _S superficial velocity - t time - average void fraction     

Velocity and turbulence measurements in combustion systems

A laser-Doppler velocimeter is used in the measurement of high-temperature gas flows. A two-stage fluidization particle generator provides magnesium oxide particles to serve as optical scattering centers. The one-dimensional dual-beam system is frequency shifted to permit measurements of velocities up to 300 meters per second and turbulence intensities greater than 100 percent.Exiting flows from can-type gas turbine combustors and burners with pre-mixed oxy-acetylene flames are described in terms of the velocity, turbulence intensity, and temperature profiles.The results indicate the influence of the combustion process on turbulence.List of Symbols A exit area of combustor or burner (m²) - A/F mass air-fuel ratio - D exit diameter of combustor or burner (m) - M mass flow rate of gases (kg/s) - N _D number of Doppler bursts used in each velocity measurement - Q volumetric flow rate at T _r (m³/s) - R exit radius of combustor or burner (m) - R _1/2 distance from centerline to radius where the velocity is one-half of the local centerline velocity (m) - Re exit Reynolds number based on cold flow, QD/A - r distance from centerline of flow (m) - T temperature (°C) - T _CL centerline temperature (°C) - T _r inlet (cold) air temperature of combustor or burner (°C) - T.I. turbulence intensity, - mean velocity (m/s) - U _i instantaneous velocity individually realized by LDV (m/s) - mean velocity at centerline of flow (m/s) - mean square velocity fluctuation (m²/s² - x distance along centerline downstream of exit (m) - absolute viscosity at T _r (kg/(ms)) - density at T _r (kg/m³)     

Production of temperature fluctuations in grid turbulence: Wiskind's experiment revisited

Measurements have been made in nearly-isotropic grid turbulence on which is superimposed a linearly-varying transverse temperature distribution. The mean-square temperature fluctuations, , increase indefinitely with streamwise distance, in accordance with theoretical predictions, and consistent with an excess of production over dissipation some 50% greater than values recorded in previous experiments. This high level of production has the effect of reducing the ratio,r, of the time scales of the fluctuating velocity and temperature fields. The results have been used to estimate the coefficient,C, in Monin's return-to-isotropy model for the slow part of the pressure terms in the temperature-flux equations. An empirical expression by Shih and Lumley is consistent with the results of earlier experiments in whichr 1.5, C 3.0, but not with the present data where r 0.5, C 1.6. Monin's model is improved when it incorporates both time scales.List of symbols C coefficient in Monin model, Eq. (5) - M grid mesh length - m exponent in power law for temperature variance, x ^m - n turbulence-energy decay exponent,q ² x ^-n - p production rate of - p pressure - q ² - R microscale Reynolds number - r time-scale ratiot/t - T mean temperature - U mean velocity - mean-square velocity fluctuations (turbulent energy components) - turbulent temperature flux - x, y, z spatial coordinates - temperature gradient dT/dy - thermal diffusivity - dissipation rate ofq ²/2 - dissipation rate of - Taylor microscale (₂=5q₂/) - temperature microscale - _v temperature-flux correlation coefficient, /v - dimensionless distance from the grid,x/M     

A new statistical approach to study and to predict wall turbulence

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

LDV measurements of a turbulent air-solid two-phase flow in a 90° bend

Simultaneous measurements of the mean streamwise and radial velocities and the associated Reynolds stresses were made in an air-solid two-phase flow in a square sectioned (10×10 cm) 90° vertical to horizontal bend using laser Doppler velocimetry. The gas phase measurements were performed in the absence of solid particles. The radius ratio of the bend was 1.76. The results are presented for two different Reynolds numbers, 2.2×10⁵ and 3.47×10⁵, corresponding to mass ratios of 1.5×10^–4 and 9.5×10^–5, respectively. Glass spheres 50 and 100 m in diameter were employed to represent the solid phase. The measurements of the gas and solid phase were performed separately. The streamwise velocity profiles for the gas and the solids crossed over near the outer wall with the solids having the higher speed near the wall. The solid velocity profiles were quite flat. Higher negative slip velocities are observed for the 100 m particles than those for the 50 gm particles. At angular displacement =0°, the radial velocity is directed towards the inner wall for both the 50 and 100 m particles. At =30° and 45°, particle wall collisions cause a clear change in the radial velocity of the solids in the region close to the outer wall. The 100 m particle trajectories are very close to being straight lines. Most of the particle wall collisions occur between the =30° and 60° stations. The level of turbulence of the solids was higher than that of the air.List of symbols D hydraulic diameter (100 mm) - De Dean number,De = - mass flow rate - number of particles per second (detected by the probe volume) - r radial coordinate direction - r _i radius of curvature of the inner wall - r ₀ radius of curvature of the outer wall - r ^* normalized radial coordinate, - R mean radius of curvature - Re Reynolds number, - R _r radius ratio, - U ,U _z mean streamwise velocity - U _r ,U _y mean radial velocity - U _b bulk velocity - , _z rms fluctuating streamwise velocity - _r , _y rms fluctuating radial velocity - -r shear stress component - z-y shear stress component - x spanwise coordinate direction - x ^* normalized spanwise coordinate, - y radial coordinate direction in straight ducts - y ^* normalized radial coordinate in straight ducts, - z streamwise coordinate direction in straight ducts - z ^* normalized streamwise coordinate in straight ducts, Greek symbols streamwise coordinate direction - kinematic viscosity of air     

The flow regime in the turbulent near wake of a hemisphere

The work presented is a wind tunnel study of the near wake region behind a hemisphere immersed in three different turbulent boundary layers. In particular, the effect of different boundary layer profiles on the generation and distribution of near wake vorticity and on the mean recirculation region is examined. Visualization of the flow around a hemisphere has been undertaken, using models in a water channel, in order to obtain qualitative information concerning the wake structure.List of symbols C _p pressure coefficient, - D diameter of hemisphere - n vortex shedding frequency - p pressure on model surface - p ₀ static pressure - Re Reynolds number, - St Strouhal number, - U, V, W local mean velocity components - mean freestream velocity inX direction - U ^* shear velocity, - u, v, w velocity fluctuations inX, Y andZ directions - X Cartesian coordinate in longitudinal direction - Y Cartesian coordinate in lateral direction - Z Cartesian coordinate in direction perpendicular to the wall - it* boundary layer displacement thickness, - diameter of model surface roughness - elevation angleI - O boundary layer momentum thickness, - _w wall shearing stress - dynamic viscosity of fluid - density of fluid - streamfunction - _x longitudinal component of vorticity, - _y lateral component of vorticity, - _z vertical component of vorticity, This paper was presented at the Ninth symposium on turbulence, University of Missouri-Rolla, October 1–3, 1984     

On model constants and second order closure for curved shear flows

The pressure-strain correlation model of Hanjalic and Launder is extended to rotating curved shear flows. Stationary plane flow data together with the requirement that the critical Richardson number should be predicted correctly are used to determine the model constants. The value of the constants are found to be the same as those determined by Hanjalic and Launder through computer optimization. It is also found that the pressure-strain correlation model of Hanjalic and Launder has negligible effects on the shear stress relation for curved shear flows compared to the simple return to isotropy model of Rotta. In view of this, significant improvement on second order closure of the mean flow equations for curved shear flows cannot be obtained by improving the simple Rotta model.Nomenclature a ₁, a ₂, a ₃ Constants defined in (11a...c) - A ₁...A ₆ Constants defined in (11d...i) - b _ij Reynolds stress matrix defined in (12)–(15) - c ₁ Model constant introduced by Rotta [2] and defined in (6) - c ₂ Model constant introduced by Rotta [2] and also in (3) - c _r, c ₃ Model constants introduced in (3) - F _c Curvature parameter defined in (11j) - F _R Rotation parameter defined in (11k) - k Longitudinal surface curvature - l ₁ Length scale introduced in (3) - l _b Mixing length for corner flow defined by Gessner and Po [13] - Fluctuating pressure - Twice the turbulence kinetic energy - Ri Gradient Richardson number for curved shear flows - Ri _cr Critical gradient Richardson number - u _i i-th component of fluctuating velocity field - U _i i-th component of mean velocity field - U Mean flow in x-direction - u, v, w Fluctuating velocity components along x, y, z axes, respectively - x _i i-th component of coordinate system - x, y, z Coordinate axes attached to surface - _i Coefficients defined in (20) - Dissipation length scale introduced in (5) - Fluid kinematic viscosity - Fluid density - _i i-th component of rotation vector - Speed of rotation of surface about oz axis     

Momentum balance in highly distorted turbulent pipe flows

An in depth study into the development and decay of distorted turbulent pipe flows in incompressible flow has yielded a vast quantity of experimental data covering a wide range of initial conditions. Sufficient detail on the development of both mean flow and turbulence structure in these flows has been obtained to allow an implied radial static pressure distribution to be calculated. The static pressure distributions determined compare well both qualitatively and quantitatively with earlier experimental work. A strong correlation between static pressure coefficient Cp and axial turbulence intensity is demonstrated.List of symbols C _p static pressure coefficient = (pw-p)/1/2 - D pipe diameter - K turbulent kinetic energy - (r, , z) cylindrical polar co-ordinates. / 0 - R, y pipe radius, distance measured from the pipe wall - U, V axial and radial time mean velocity components - mean value of u - u, u/ , / - u, , w fluctuating velocity components - axial, radial turbulence intensity - turbulent shear stress - u friction velocity, (u ² = ₀/p) - ₀ wall shear stress - ^* boundary layer thickness A version of this paper was presented at the Ninth Symposium on Turbulence, University of Missouri-Rolla, October 1–3, 1984     

On integrals of Bessel functions related to Weber's second exponential integral

Two closely related integrals of Bessel functions are discussed: Whenn, m, l are integers (m andl=1,2, ...,n), both integrals can be written as closed expressions in modified Bessel functionsI _k (z). The results are interpreted in terms of hypergeometric series_p F _q. Series expansions inz and in 1/z are given.     

Slip flow in the hydrodynamic entrance region of a tube and a parallel plate channel

Summary The problem of slip flow in the entrance region of a tube and parallel plate channel is considered by solving a linearized momentum equation. The condition is imposed that the pressure drop from momentum considerations and from mechanical energy considerations should be equal. Results are obtained for Kn=0, 0.01, 0.03, 0.05, and 0.1 and the pressure drop in the entrance region is given in detail.Nomenclature A cross-sectional area of duct - c mean value of random molecular speed - d diameter of tube - f _p - f _t - h half height of parallel plate channel - Kn Knudsen number - L molecular mean free path - n directional normal of solid boundary - p pressure - p ₀ pressure at inlet - r radial co-ordinate - r _t radius of tube - R non-dimensional radial co-ordinate - Re _p 4hU/ - Re _t 2r _t U/ - s direction along solid boundary - T absolute temperature - u velocity in x direction - u* non-dimensional velocity - U bulk velocity = (1/A) _A u dA - v velocity in y direction - x axial co-ordinate - x* stretched axial co-ordinate - X non-dimensional axial co-ordinate - X* non-dimensional stretched axial co-ordinate - Y non-dimensional channel co-ordinate - eigenvalue in parallel plate channel - stretching factor - eigenvalue in tube - density - kinematic viscosity - i index - p parallel plate - t tube - v velocity vector - gradient operator - ² Laplacian operator     

Nonlinear Taylor stability of viscoelastic fluids

Using Stuart's energy method, the torque on the inner cylinder, for a second order fluid, in the supercritical regime is calculated. It is found that when the second normal stress difference is negative, the flow is more stable than for a Newtonian fluid and the torque is reduced. If the second normal stress difference is positive, then the flow is more stable and there is no torque reduction. Experimental data related to the present work are discussed.Nomenclature a amplitude of the fundamentals - A _ij ⁽¹⁾ , A _ij ⁽²⁾ first and second Rivlin-Ericksen tensors - d r ₂–r ₁ - D d/dx - E - F - g _ij metric tensor - G torque on the inner cylinder in the supercritical regime - h height of the cylinders - k ₀ /d ² - k ₁ /d ² - I ₁ - I ₂ - I ₃ - I ₄ - r ₁, r ₂ radii of inner and outer cylinders respectively - r ₀ 1/2(r ₁+r ₂) - R Reynolds number ₁ r ₁ d/ ₀ - R _c critical Reynolds number - T Taylor number r _{1 1} ² d ^{3 2}/ ₀ ² *) - T _c critical Taylor number - u ₁, v ₁, w ₁ Fundamentals of the disturbance - u _i , v _i , w _i , (i>1) harmonics - mean velocity (not laminar velocity) - u –u ₁/ar _{1 1} - v v ₁/Rar _{1 1} - x (r–r ₀)/d - , material constants - 0 viscosity - wave number d - density - ₁ angular velocity of inner cylinder - tilde denotes complex conjugate     

Shell-side distribution and the influence of inlet conditions in a model of a disc-and-doughnut heat exchanger

Measurements of wall pressure and of three orthogonal velocity components with their corresponding fluctuations are reported for two systems of alternating and equi-spaced doughnut and disc baffles axisymmetrically located in a water turbulent pipe flow, simulating the isothermal shell-side flow in shell and tube heat exchangers. The influence of inlet Reynolds number and of asymmetric inlet flow conditions was studied for two geometries. The velocity field was dominated by the pressure gradient and the flow around each individual baffle was influenced by the relative position of its neighbouring baffles.List of symbols C _p wall static-pressure coefficient - D internal diameter of upstream and downstream pipes (mm) - D _s internal diameter of test section (mm) - d _d disc diameter (mm) - d _c doughnut-hole diameter (mm) - l baffle-pitch (mm) - l _i entrance length in the model before first baffle (mm) - l ₀ exit length in the model after last baffle (mm) - mass flow rate (kg/s) - p local wall-static pressure (mm H₂O) - p density of water (1.006 kg/dm³ at 17°C) - Re _b Reynolds number based on bulk velocity - U _b bulk velocity - U _max maximum centre-line axial velocity (ms^–1) - x, y, z Cartesian coordinates - mean and turbulent velocity components along x, y, z respectively