Showing papers on "Reynolds number published in 1987"

Viscous-inviscid analysis of transonic and low Reynolds number airfoils

Abstract: A method of accurately calculating transonic and low Reynolds number airfoil flows, implemented in the viscous-inviscid design/analysis code ISES, is presented. The Euler equations are discretized on a conservative streamline grid and are strongly coupled to a two-equation integral boundary-layer formulation, using the displacement thickness concept. A transition prediction formulation of the e type is derived and incorporated into the viscous formulation. The entire discrete equation set, including the viscous and transition formulations, is solved as a fully coupled nonlinear system by a global Newton method. This is a rapid and reliable method for dealing with strong viscous-inviscid interactions, which invariably occur in transonic and low Reynolds number airfoil flows. The results presented demonstrate the ability of the ISES code to predict transitioning separation bubbles and their associated losses. The rapid airfoil performance degradation with decreasing Reynolds number is thus accurately predicted. Also presented is a transonic airfoil calculation involving shock-induced separation, showing the robustness of the global Newton solution procedure. Good agreement with experiment is obtained, further demonstrating the performance of the present integral boundary-layer formulation.

Alignment of vorticity and scalar gradient with strain rate in simulated Navier-Stokes turbulence

Abstract: The alignment between vorticity and eigenvectors of the strain‐rate tensor in numerical solutions of Navier–Stokes turbulence is studied. Solutions for isotropic flow and homogeneous shear flow from pseudospectral calculations using 1283 grid points have been examined. The Taylor Reynolds number is 83 or greater. In both flows there is an increased probability for the vorticity to point in the intermediate strain direction and at three‐fourths of the sample points this strain is positive (extensive). This propensity for vorticity alignment with a positive intermediate strain is a consequence of angular momentum conservation, as shown by a restricted Euler model of the coupling between strain and vorticity. Probability distributions for intermediate strain, conditioned on total strain, change from a symmetric triangular form at small strain to an asymmetric one for large strain. The most probable value of the asymmetric distribution gives strains in the ratios of 3:1: −4. The evolution of the distribution from a symmetric to an asymmetric form as the strain magnitude increases is essentially the same in both flows, indicating a generic structure of intense turbulence. The alignment between the gradient of a passive scalar and eigenvectors of the strain‐rate tensor for Prandtl numbers of 0.1, 0.2, 0.5, and 1.0 has also been studied. There is an increased probability for the scalar gradient to align in the most compressive strain direction, and the average gradient is larger when it is pointing in that direction. Estimates for the scalar dissipation from the turbulent kinetic energy, its dissipation, and the root‐mean‐square scalar value are in reasonable agreement with calculated scalar dissipation if no explicit Prandtl number dependence is used in the estimate. Statistical analysis of scalar dissipation conditioned on energy dissipation yields a power‐law relation between conditioned mean values. Both simulated flows are found to obey the qualitative predictions of the Gurvich–Yaglom (lognormal) intermittency model. Energy and scalar intermittency exponents are estimated and compared to measured values.

Oscillations and concentrations in weak solutions of the incompressible fluid equations

Abstract: The authors introduce a new concept of measure-valued solution for the 3-D incompressible Euler equations in order to incorporate the complex phenomena present in limits of approximate solutions of these equations. One application of the concepts developed here is the following important result: a sequence of Leray-Hopf weak solutions of the Navier-Stokes equations converges in the high Reynolds number limit to a measure-valued solution of 3-D Euler defined for all positive times. The authors present several explicit examples of solution sequences for 3-D incompressible Euler with uniformly bounded local kinetic energy which exhibit complex phenomena involving both persistence of oscillations and development of concentrations. An extensions of the concept of Young measure is developed to incorporate these complex phenomena in the measure-valued solutions constructed here.

Turbulent burning velocities: a general correlation in terms of straining rates

Abstract: All known experimental values of turbulent burning velocity have been scrutinized. These number 1650, a significant proportion of which at the higher turbulent Reynolds numbers we measured in a fan-stirred bomb. Dimensionless correlations which have a theoretical basis are presented. These are in terms of flame straining rates and the effective r.m.s. turbulent velocity, as well as the laminar burning velocity of the mixture. When a flame develops from an ignition source it is not initially exposed to the lower frequencies of the turbulent spectrum. As the kernel grows the flame is affected by ever-lower frequencies and the turbulent burning velocity increases towards a fully developed value. An experimental dimensionless power spectral density function is presented, and used to show how both effective r.m.s. turbulent velocity and flame straining rate develop in an explosion. The results are relevant to a variety of practical devices, including gasoline engines, as well as atmospheric explosions.

Turbulent oscillatory flow over rough beds

Abstract: Velocity measurements are presented for turbulent oscillatory flow over rough beds. Two components of velocity were measured with a laser-Doppler anemometer and the rough beds consisted of a single layer of sand, gravel or pebbles on a flat surface. Turbulence intensities showed significant variation during the course of the cycle. Maximum turbulence intensity propagated out from the bed at a more or less constant velocity for all beds. Variation of time-mean turbulence intensity with height was qualitatively similar to that observed in steady flows. Reynolds stress showed several interesting features. Near the bed, maximum Reynolds stress was in phase with one of the two peaks of turbulence intensity but further out it was in phase with the other, i.e. the phase of maximum Reynolds stress showed a 180° phase shift at a certain height above the bed. A related effect was seen in the time-mean eddy viscosity which was negative near the bed but positive further out. It is suggested that these effects are caused by the jets of fluid associated with vortex formation and ejection in oscillatory flow over rough beds. Maximum Reynolds stress was also significantly less than the horizontal force per unit area of bed obtained from the momentum integral. Eddy viscosity and mixing length were found to vary significantly during the course of the cycle. Variation with height of time-mean values of these variables showed similar trends, except in the near-bed region, to those observed in steady flow but derived values of the Karman constant were significantly lower. Non-dimensional defect velocity appeared to show dependence on a/ks as well as on y/δ in the outer layer away from the bed, even at high Reynolds numbers.

Laminar separation bubble characteristics on an airfoil at low Reynolds numbers

Abstract: An experimental investigation was conducted in order to document the structure and behavior of laminar separation bubbles at low Reynolds numbers. Data of this type are necessary if the currently insufficient analytical and numerical models are to be improved. The laminar separation bubble that forms on a NACA 663-018 airfoil model was surveyed at chord Reynolds numbers of 50,000-200,000 at angles of attack of 8-12 deg. The effects of the various testing conditions on the separation bubble were isolated and the data were analyzed in relation to existing separation bubble correlations in order to test their low Reynolds number applicability. This analysis indicated that the chord Reynolds number and the disturbance environment strongly influence the experimental pressure distributions. These effects must be included in any analytic prediction technique applied to the low Reynolds number flight regime.

An experimental study of the steady and unsteady flow characteristics of stirred reactors

Abstract: Liquid flow in a baffled stirred reactor vessel driven by a six-blade disk impeller has been investigated experimentally. Laser-slit photography provided an overview of the flows which were quantified by measurements of velocity characteristics, obtained with a laser-Doppler anemometer, for an impeller rotational speed of 300 r.p.m. and for three impeller clearances from the bottom of the vessel. The mean flow results show an inclination of the impeller stream and the formation of ring vortices above and below the impeller, which depend on the clearance; the flow was strongly three-dimensional with large regions having circumferential velocities in a direction opposite to that of the impeller rotation. Impeller-induced torque measurements show that the Power number is invariant with clearance for turbulent-flow Reynolds numbers ([ges ] 40000) and increases with impeller diameter. The flow structure was controlled mainly by convection and pressure forces with turbulent mixing important in the impeller region.

Experimental Study of Particle Deposition in Bends of Circular Cross Section

Abstract: The deposition efficiency of liquid particles in tube bends of circular cross section has been measured for flow Reynolds numbers of 100, 1000, 6000, and 10,000. The particle Reynolds number, Re p, was in the range 0.6–3.9 for the laminar flow cases (i.e., Re = 100 and 1000), whereas for the turbulent flow cases (i.e., Re = 6000 and 10,000) Re p was in the range 1.3–12.7. Bends constructed of stainless steel and glass tubes of different diameters were used. The experiments were performed using monodisperse aerosols generated by the vibrating orifice aerosol generator. The results were in good agreement with the theory of Cheng and Wang for Re = 1000, but differed from theory for Re = 100. For the turbulent cases, no dependence was found on the flow Reynolds number and an exponential curve of deposition efficiency versus Stokes number was fitted to the experimental results. A theoretical justification of the form of the curve is given in the paper.

The resistance to motion of a solid sphere in a fluid

Abstract: The published correlations for the drag force on a sphere moving in a fluid have been critically examined and their limitations have been identified. A series of new correlations has been proposed, each covering a wide range of particle Reynolds number Re(10−2< Re < 3 × 105) by choice of the appropriate form iterative calculations can be avoided.

The structure of a turbulent shear layer bounding a separation region

Abstract: Detailed measurements within the separated shear layer behind a flat plate normal to an airflow are reported. A long, central splitter plate in the wake prevented vortex shedding and led to an extensive region of separated flow with mean reattachment some ten plate heights downstream. The Reynolds number based on plate height was in excess of 2 × 1044.Extensive use of pulsed-wire anemometry allowed measurements of all the Reynolds stresses throughout the flow, along with some velocity autocorrelations and integral timescale data. The latter help to substantiate the results of other workers obtained in separated flows of related geometry, particularly in the identification of a very low-frequency motion with a timescale much longer than that associated with the large eddies in the shear layer. Wall-skin-friction measurements are consistent with the few similar data previously published and indicate that the thin boundary layer developing beneath the separated region has some ‘laminar-like’ features.The Reynolds-stress measurements demonstrate that the turbulence structure of the separated shear layer differs from that of a plane mixing layer between two streams in a number of ways. In particular, the normal stresses all rise monotonically as reattachment is approached, are always considerably higher than the plane layer values and develop in quite different ways. Flow similarity is not a useful concept. A major conclusion is that any effects of stabilizing streamline curvature are weak compared with the effects of the re-entrainment at the low-velocity edge of the shear layer of turbulent fluid returned around reattachment. It is argued that the general features of the flow are likely to be similar to those that occur in a wide range of complex turbulent flows dominated by a shear layer bounding a large-scale recirculating region.

The three‐dimensional instability of strained vortices in a viscous fluid

Abstract: The recent theory describing 3‐D exact solutions of the Navier–Stokes equations is applied to the problem of stability of 2‐D viscous flow with elliptical streamlines. An intrinsically inviscid instability mechanism persists in all such flows provided the length scale of the disturbance is sufficiently large. Evidence is presented that this mechanism may be responsible for 3‐D instabilities in high Reynolds number flows whose vortex structures can be locally described by elliptical streamlines.

Near-surface turbulence in a grid-stirred tank

Abstract: In order to elucidate the turbulent structure below a shear-free gas-liquid interface, turbulence measurements were made in a 50 cm square by 40 cm deep tank stirred by a vertically oscillating grid well below the surface, using a split-film anemometer probe rotating in a horizontal circle. This instrument is able to measure both vertical and horizontal velocity fluctuations to within 0.4 mm of the surface, from which spatial spectra and profiles of r.m.s. velocity fluctuations and integral lengthscales can be calculated. The turbulent structure is affected by the presence of the surface within a ‘surface-influenced layer’ roughly one integral scale, or ten per cent of the distance from the surface to the centre of the grid stroke, in thickness. The shapes of the spectra and profiles within the surface-influenced layer are predicted to a good first approximation by the source theory of Hunt & Graham (1978), which treats the turbulent structure as the superposition of homogeneous turbulence with an irrotational velocity field driven by a source distribution at the surface which cancels the vertical velocity fluctuations there. The magnitudes (as opposed to the shapes) of the profiles scale according to the values that would otherwise occur in the vicinity of the surface-influenced layer were the surface not present. These magnitudes are adequately predicted by the bulk relations determined by Hopfinger & Toly (1976) and Thompson & Turner (1975), with no apparent dependence on turbulent Reynolds number. There are some minor discrepancies between the measured profiles and those of Hunt & Graham. A thin layer of reduced velocity fluctuations below what would be expected from the theory was observed near the surface. Also, anisotropy in the velocity spectra at depths within the surface-influenced layer extended well into the inertial subrange, whereas the Hunt & Graham theory predicts no anisotropy at high wavenumbers.

Strange attractors in weakly turbulent Couette-Taylor flow.

Abstract: An experiment is conducted on the transition from quasi-periodic to weakly turbulent flow of a fluid contained between concentric cylinders with the inner cylinder rotating and the outer cylinder at rest. Power spectra, phase-space portraits, and circle maps obtained from velocity time-series data indicate that the nonperiodic behavior observed is deterministic, that is, it is described by strange attractors. Various problems that arise in computing the dimension of strange attractors constructed from experimental data are discussed and it is shown that these problems impose severe requirements on the quantity and accuracy of data necessary for determining dimensions greater than about 5. In the present experiment the attractor dimension increases from 2 at the onset of turbulence to about 4 at a Reynolds number 50-percent above the onset of turbulence.

Self-Propulsion at Low Reynolds Number

Abstract: We formulate the problem of self-propulsion at low Reynolds number in terms of a gauge field over the space of shapes. The computation of this field is discussed, and carried out in some examples. We apply our results to determine maximally efficient infinitesimal swimming motions of spheres and circular cylinders.

Stability of viscous flow past a circular cylinder

Abstract: A spectral method which employs trigonometric functions and Chebyshev polynomials is used to compute the steady, incompressible laminar flow past a circular cylinder. Linear stability methods are used to formulate a pair of decoupled generalized eigenvalue problems for the growth of symmetric and asymmetric (about the dividing streamline) perturbations. We show that, while the symmetric disturbances are stable, the asymmetric perturbations become unstable at a Reynolds number about 40 with a Strouhal number about 0.12. The critical conditions are found to depend on the size of the computational domain in a manner similar to that observed in the laboratory.

Rough-wall turbulent boundary layers in the transition regime

Abstract: An experimental investigation of turbulent boundary layers over two-dimensional spanwise groove and three-dimensional sandgrain roughnesses in the transition regime between hydraulically smooth and fully rough conditions is presented. It is found that a self-preserving state can be reached in boundary layers developing over both d-type groove and sandgrain roughnesses, and that the drag of a k-type rough wall can be reduced by lowering the spanwise aspect ratio of the roughness elements. The two roughness Reynolds numbers defining the boundaries of the transition regime of the k-type roughnesses are shown to decrease with increasing roughness-element spanwise aspect ratio, and the upper critical transition Reynolds number is shown to determine the roughness behavior in both the transition and fully rough regime.

Turbulent energy dissipation in a wake

Abstract: The average turbulent energy dissipation is often estimated by assuming isotropy and measuring the temporal derivative of the longitudinal velocity fluctuation. In this paper, the nine major terms that make up the total dissipation have been measured in the self-preserving region of a cylinder wake for a small turbulence Reynolds number. The results indicate that local isotropy is not satisfied; the isotropic dissipation, computed by assuming isotropic relations, being smaller than the total dissipation by about 45% on the wake centreline and by about 80% near the wake edge. Indirect verification of the dissipation measurements is provided by the budget of the turbulent kinetic energy. This budget leads to a plausible distribution for the pressure diffusion term, obtained by difference.

A second-moment closure study of rotating channel flow

Abstract: The second-moment closure applied by Gibson & Launder (1978) to buoyant turbulent flows is here employed without modification to compute the effects of Coriolis forces on fully-developed flow in a rotating channel. The augmentation of turbulent transport on the pressure surface of the channel and its damping on the suction surface seem to be well captured by the computations, provided the flow near the suction surface remains turbulent. The rather striking alteration in shape of the mean velocity profile that occurs as the Rossby number is increased from 0.06 to 0.2 is shown to be explicable in terms of the modification to the intensity of the turbulent velocity fluctuations normal to the plate; for the larger value of Rossby number these fluctuations become larger than those in the flow direction causing what at low spin rates is a source of shear stress to become a sink.

Numerical Simulation of Supersonic Flow Over a Three-Dimensional Cavity

Abstract: A numerical solution is presented for the unsteady flow over a three-dimensio nal cavity at a freestream Mach number of 1.5 and Reynolds number of 1.09 x 10 6. The self-sustained oscillatory motion within the cavity is generated numerically by integration of the time-dependent compressible three-dimensional Reynolds averaged Navier-Stokes equations. Effects of fine-scale turbulence are simulated via a simple algebraic closure model. Details of the flowfield structure are elucidated, and it is verified that the fundamental behavior of the unsteady phenomena is two dimensional. Comparison with experimental data is made in terms of the mean static pressure and overall acoustic sound pressure levels within the cavity, as well as with the acoustic frequency spectra of the oscillation along the cavity floor and rear bulkhead.

Coherent Motion in Excited Free Shear Flows

Abstract: The application of the inviscid instability approach to externally excited turbulent free shear flows at high Reynolds numbers is explored. Attention is given to the cases of a small-deficit plane turbulent wake, a plane turbulent jet, an axisymmetric jet, the nonlinear evolution of instabilities in free shear flows, the concept of the 'preferred mode', vortex pairing in turbulent mixing layers, and experimental results for the control of free turbulent shear layers. The special features often attributed to pairing or to the preferred mode are found to be difficult to comprehend; the concept of feedback requires further substantiation in the case of incompressible flow.

Fast dynamo action in a steady flow

Abstract: The existence of fast dynamos caused by steady motion of an electrically conducting fluid is established by consideration of a two-dimensional spatially periodic flow: the velocity, which is independent of the vertical coordinate z, is finite and continuous everywhere but the vorticity is infinite at the X-type stagnation points. A mean-field model is developed using boundary-layer methods valid in the limit of large magnetic Reynolds number R. The magnetic field is confined to sheets, width of order R−½. The mean magnetic field lies and is uniform on horizontal planes: its direction is independent of time but rotates once about the vertical axis over a short distance 2πl, where l−1 = R½β and β is a vertical stretched wavenumber independent of R. Its alternating direction gives it a rope-like structure within the sheets. An α-effect is calculated for the model, whose strength for a given flow is a function of β and R. Two sources of α-effect are isolated whose relative importance depends critically on the size of β. When the vorticity is finite everywhere and β [Lt ] 1, the dynamo is ‘almost’ fast with growth rates of order (ln R)−1. The maximum growth rate ln (ln R)/ln R occurs when, correct to leading order, β is (ln R)−½. The asymptotic results valid for large R compare excellently with Roberts (1972) modal analysis for finite R.

Axisymmetric vortex breakdown with and without temperature effects in a container with a rotating lid

Abstract: A flow circulation in a closed circular-cylindrical container is produced by a rotating lid. After a transient phase from an initial state at rest a steady-flow situation is reached for a certain parameter range. In a subspace of this parameter range an undulating meridional flow occurs that may exhibit at the axis of rotation one or several separation bubbles which are interpreted as vortex breakdown. Numerical calculations on the basis of the Navier-Stokes equations for incompressible homogeneous and Boussinesq fluids enable the study of the influence of various flow parameters on the properties of these separation bubbles. The parameters varied are the Reynolds, Prandtl, Rayleigh, and Eckert numbers together with the ratio of height to radius of the container. The numerical results are in good agreement with experiments performed by Vogel, Ronnenberg, and Escudier. The stability of the fluid motions in these experiments with respect to non-axisymmetric disturbances strongly suggests that the corresponding axisymmetric solutions of the Navier-Stokes equations are stable configurations.

Navier-Stokes computations of prolate spheroids at angle of attack

Abstract: Three-dimensional viscous flow calculations are made for a 6:1 prolate spheroid at conditions for which detailed experimental data are available. The computations are made with two finite-volume algorithms for the compressible Navier-Stokes equations, one using central differencing for the convective and pressure terms and the other using an upwind-biased flux-difference-splitting approach. The effects of grid density and artificial dissipation on the accuracy of the numerical results are included. Generally good agreement of the computations with the experimental results is obtained over a range of Reynolds numbers and angles-of-attack, up to 30 deg, although the results at lower Reynolds numbers are sensitive to the assumed transition location.