Numerical simulation of turbulent free-surface flow in curved channel

This paper presents first results of numerical simulation of turbulent free-surface flow. Simple implementation of surface capturing method is based on the variable density approach. The flow is treated as if there is only one fluid, but with variable material properties (density, viscosity). The switch in these values is done by a function resulting from the mass conservation principle. This approach simplifies the implementation of turbulence model. In this case the SST k−ω model was chosen in modification given by Hellsten.Numerical solution was carried out by finite-volume method with explicit Runge-Kutta time-integration. The artificial compressibility method was used for time-marching search for steady state solution. The whole model was tested on horizontally placed square-sectioned 90^∘ bend, which was partially filled by the water. The main goal of this study was to demonstrate the applicability of this model and solution method for capturing the water-air interface as well as for predicting the turbulent effects in both fluids.     

Study of Density Effects in Turbulent Buoyant Jets Using Large-Eddy Simulation

Large-Eddy simulations (LES) of spatially evolving turbulent buoyant round jets have been carried out with two different density ratios. The numerical method used is based on a low-Mach-number version of the Navier–Stokes equations for weakly compressible flow using a second-order centre-difference scheme for spatial discretization in Cartesian coordinates and an Adams–Bashforth scheme for temporal discretization. The simulations reproduce the typical temporal and spatial development of turbulent buoyant jets. The near-field dynamic phenomenon of puffing associated with the formation of large vortex structures near the plume base with a varicose mode of instability and the far-field random motions of small-scale eddies are well captured. The pulsation frequencies of the buoyant plumes compare reasonably well with the experimental results of Cetegen (1997) under different density ratios, and the underlying mechanism of the pulsation instability is analysed by examining the vorticity transport equation where it is found that the baroclinic torque, buoyancy force and volumetric expansion are the dominant terms. The roll-up of the vortices is broken down by a secondary instability mechanism which leads to strong turbulent mixing and a subsequent jet spreading. The transition from laminar to turbulence occurs at around four diameters when random disturbances with a 5% level of forcing are imposed to a top-hat velocity profile at the inflow plane and the transition from jet-like to plume-like behaviour occurs further downstream. The energy-spectrum for the temperature fluctuations show both −5/3 and −3 power laws, characteristic of buoyancy-dominated flows. Comparisons are conducted between LES results and experimental measurements, and good agreement has been achieved for the mean and turbulence quantities. The decay of the centreline mean velocity is proportional to x ^−1/3 in the plume-like region consistent with the experimental observation, but is different from the x ⁻¹ law for a non-buoyant jet, where x is the streamwise location. The distributions of the mean velocity, temperature and their fluctuations in the near-field strongly depend upon the ratio of the ambient density to plume density ρ_a/ρ₀. The increase of ρ_a/ρ₀ under buoyancy forcing causes an increase in the self-similar turbulent intensities and turbulent fluxes and an increase in the spatial growth rate. Budgets of the mean momentum, energy, temperature variance and turbulent kinetic energy are analysed and it is found that the production of turbulence kinetic energy by buoyancy relative to the production by shear is increased with the increase of ρ_a/ρ₀. Received 16 June 2000 and accepted 26 June 2001     

A technique to determine total shear stress and polymer stress profiles in drag reduced boundary layer flows

Two methods of recovering the entire total shear stress profile from incomplete velocity data in turbulent boundary layers are presented and validated for both DNS simulations and experimental measurements. The first method, an exponential–polynomial curve fit, recovers the whole total shear stress profile using the data from the outer part of the boundary layer (y/δ>0.3). However, while performing well, this curve fit is sensitive to the quality of the data. The second method, a new (1−y/δ) weighted straight line fit, which is very simple and accurate, has been applied to current experiments of drag reduction in zero pressure gradient turbulent boundary layers with and without polymer injection. The total shear stress profile obtained from this fit is used to estimate the contribution of the polymer stress to the total shear stress. It shows that the polymer stress is significant only in the inner part of the boundary layer and the magnitude of the polymer stress is not always proportional to the drag reduction.     

Local measurements of turbulent angular momentum transport in circular Couette flow

We report on velocity fluctuations and the fluctuation-driven radial transport of angular momentum in turbulent circular Couette flow. Our apparatus is short (cylinder height to gap width ratio Γ ~ 2) and of relatively high wall curvature (ratio of cylinder radii η ~ 0.35). Fluctuation levels and the mean specific angular momentum are found to be roughly constant over radius, in accordance with previous studies featuring narrower gaps. Synchronized dual beam Laser Doppler Velocimetry (2D LDV) is used to directly measure the r − θ Reynolds stress component as a function of Reynolds number (Re), revealing approximate scalings in the non-dimensional angular momentum transport that confirm previous measurements of torque in similar flows. 2D LDV further allows for a decomposition of the turbulent transport to assess the relative roles of fluctuation intensity and r − θ cross-correlation. We find that the increasing angular momentum transport with Re is due to intensifying absolute fluctuation levels accompanied by a slightly weakening cross-correlation.     

Wake visualization of a heaving and pitching foil in a soap film

Many fish depend primarily on their tail beat for propulsion. Such a tail is commonly modeled as a two-dimensional flapping foil. Here we demonstrate a novel experimental setup of such a foil that heaves and pitches in a soap film. The vortical flow field generated by the foil correlates with thickness variations in the soap film, which appear as interference fringes when the film is illuminated with a monochromatic light source (we used a high-frequency SOX lamp). These interference fringes are subsequently captured with high-speed video (500 Hz) and this allows us to study the unsteady vortical field of a flapping foil. The main advantage of our approach is that the flow fields are time and space resolved and can be obtained time-efficiently. The foil is driven by a flapping mechanism that is optimized for studying both fish swimming and insect flight inside and outside the behavioral envelope. The mechanism generates sinusoidal heave and pitch kinematics, pre-described by the non-dimensional heave amplitude (0–6), the pitch amplitude (0°–90°), the phase difference between pitch and heave (0°–360°), and the dimensionless wavelength of the foil (3–18). We obtained this wide range of wavelengths for a foil 4 mm long by minimizing the soap film speed (0.25 m s⁻¹) and maximizing the flapping frequency range (4–25 Hz). The Reynolds number of the foil is of order 1,000 throughout this range. The resulting setup enables an effective assessment of vortex wake topology as a function of flapping kinematics. The efficiency of the method is further improved by carefully eliminating background noise in the visualization (e.g., reflections of the mechanism). This is done by placing mirrors at an angle behind the translucent film such that the camera views the much more distant and out-of-focus reflections of the black laboratory wall. The resulting high-quality flow visualizations require minimal image processing for flow interpretation. Finally, we demonstrate the effectiveness of our setup by visualizing the vortex dynamics of the flapping foil as a function of pitch amplitude by assessing the symmetry of the vortical wake.     

Numerical investigation of turbulent flow,heat transfer and entropy generation in a helical coiled tube with larger curvature ratio

Three-dimensional turbulent forced convective heat transfer and flow characteristics, and the non-dimensional entropy generation number in a helical coiled tube subjected to uniform wall temperature are simulated using the k–ε standard turbulence model. A finite volume method is employed to solve the governing equations. The effects of Reynolds number, curvature ratio, and coil pitch on the average friction factor and Nusselt number are discussed. The results presented in this paper cover a Reynolds number range of 2 × 10⁴ to 6 × 10⁴, a pitch range of 0.1–0.2 and a curvature ratio range of 0.1–0.3. The results show that the coil pitch, curvature ratio and Reynolds number have different effects on the average friction factor and Nusselt number at different cross-sections. In addition, the flow and heat transfer characteristics in a helical coiled tube with a larger curvature ratio for turbulent flow are different from that of smaller curvature ratio for laminar and turbulent flow in certain ways. Some new features that are not obtained in previous researches are revealed. Moreover, the effects of Reynolds number, curvature ratio, and coil pitch on the non-dimensional entropy generation number of turbulent forced convection in a helical coiled tube are also discussed.     

A carbon nanotube sensor for wall shear stress measurement

A novel carbon nanotube (CNT) sensor is being developed to measure the mean and fluctuating wall shear stress (WSS) in a turbulent boundary layer. The CNT WSS sensor is based on the thermal principle and featured by high spatial and temporal resolutions (in the order of nm and kHz, respectively), low power consumption (in the order of μW), and a compact fabrication process compared with traditional WSS measurement sensors. The CNT WSS-sensing element was characterized in detail before its calibration. The CNT sensor was operated under a constant temperature (CT) operation mode and an overheat ratio range of −0.15 to −0.19 and calibrated in a fully developed turbulent channel flow. It has been observed for the first time in a macroscopic flow that the sensor output power is approximately proportional to the 1/3 powered WSS, as expected for a thermal-principle-based WSS sensor, and the wall shear stress measurement is demonstrated for a low Reynolds number flow.     

Speckle technique for dynamic drop profile measurement on rough surfaces

A technique has been developed for measuring three-dimensional instantaneous drop profiles on rough surfaces. The surface is illuminated using a laser and images are captured of the resulting speckle pattern with and without the drop in place. The analysis consists of finding the contact line, measuring the deformation of the speckle field caused by refraction of light at the drop surface, then reconstructing the drop using simulated annealing optimization to find the drop shape whose shift vector field best matches the one measured. An error analysis of the technique was performed using a Monte Carlo technique and comparisons to sideview drop images for a large sample of drops. Mean contact angle measurement error was found to be −1.6° with a 1 − σ error bound of −6.9°, +2.0°.     

The concentration field in a turbulent channel flow with polymer injection at the wall

Instantaneous concentration profiles have been measured in turbulent water channel flows at 5 axial locations immediately downstream of a line, wall injection of a dyed 700 ppm polymer solution and for comparison, dyed water. Concentration was deduced from a line of fluoresced radiation that was stimulated by a laser beam directed through the dyed injectant and normal to the channel wall. Both statistical and time-resolved results show how the turbulent mixing is modified and damped when the injectant is a polymer solution. A version of this paper was presented at the 11th Symposium on Turbulence, University of Missouri-Rolla, Oct. 17–19, 1988     

Optical technique for direct measurement of power spectra in compressible turbulence

We present exploratory tests of a new optical method which measures directly two components of the 3-D power spectrum of the refractive-index fluctuation. The method was applied to a Mach 2 turbulent shear layer. Length scales of 0.1 mm to 1.0 mm were resolved within a shear-layer thickness ranging from 5 mm to 10 mm. At low-to-moderate wavenumbers, the spectral behavior is roughlyκ ⁻³, while at higher wavenumbers it isκ ^−3.7. These spectral slopes are in agreement with prevailing theoretical predictions for anisotropic (low-wavenumber) and isotropic (high-wavenumber) turbulence. ]This study was funded by a grant from Lawrence Livermore National Laboratory and by a grant from the California Space Institute. DP is indebted to HFR and Dr. Alfred C. Buckingham of LLNL for their support.     

Effects of the extensional rate on two-dimensional turbulence of semi-dilute polymer solution flows

An experimental study has been performed to investigate the relationship between the extensional viscosity of polymers and the turbulent drag reduction. In order to obtain the flow which is mostly dominated by extensional flow, two-dimensional (2D) grid turbulence made by flowing soap films was used. Extensional rate added to the flow was controlled by changing the conformation of the grid. Polyethylene oxide, as a flexible polymer, and hydroxypropyl cellulose, as a rigid rod-like polymer were added to the flow. Several extensional rates affect polymer behaviors, which induce different effects. Drag reduction effects of polymers under several extensional rates were visualized and analyzed by image processing. Rheological properties of the polymer solutions were also measured by a rheometer. The results indicated that the mechanisms of energy transfer are different in the streamwise and normal directions. The critical concentration to observe drag reduction effects in 2D turbulence was changed by the extensional rate. When extensional rate is higher, the effects were started to observe from lower concentration. These results were confirmed to correspond to the drag reduction effects obtained by classical pressure drop experiments in a pipe flow.     

Semi-analytic solution of the two-dimensional turbulent energy equation in round tubes using an analytic velocity profile and its experimental validation

 A semi-analytic solution of the temperature development of single-phase, turbulent viscous flows inside smooth round tubes is performed. The special feature of the theoretical analysis revolves around two single universal functions of analytic form for the accurate characterization of the turbulent diffusivity of momentum and the turbulent velocity profile in the entire cross-section of a round tube. Using this valuable information that emanates from the analytic solution of the one-dimensional momentum balance equation, the two-dimensional energy balance equation was reformulated into an adjoint system of ordinary differential equations of first–order with constant coefficients. Each equation in the system of differential equations governs the axial variation of the average temperature of a finite volume of fluid of annular shape. Exploiting the linearity of the system of differential equations, an analytic solution of it was obtained via the matrix eigenvalue method with LAPACK, a library of Fortran 77 subroutines for numerical linear algebra. Reliable series have been determined for the axial variation of the two thermal quantities of importance: (a) the time-mean bulk temperature and (b) the local Nusselt number. The semi-analytic nature of the local Nusselt number distribution is advantageous because it may be viewed as an analytic-based correlation equation. Prediction of the local Nusselt numbers for turbulent air flows compare satisfactorily with the comprehensive correlation equations and the abundant experimental data that are accessible from the literature. The air flows are regulated by a wide spectrum of turbulent Reynolds numbers. Received on 4 June 2001 RID="★" ID="★" Current address Mechanical Engineering Dept. The University of Vermont Burlington, VT 05405, USA     

A study of turbulent mixing in a turbine-agitated tank using a fluorescence technique

Two-dimensional images of (Plane) Laser Induced Fluorescence (PLIF) have been used to study the turbulent mixing process in a model stirred tank. A calibration procedure is presented and discussed in terms of its accuracy. Data from the literature are used for comparison. A pattern-recognition algorithm has been designed to identify and quantitatively describe large-scale structures in the flow. This methodology, called “structural analysis”, is based on a conditional analysis of the PLIF data and requires the definition of an appropriate structure-detector function which is calculated locally. The mathematical tools developed have been used to study the mixing in a Rushton turbine-agitated reactor. Particular attention is paid to two specific regions of the tank; namely the bulk and the impeller stream regions, at two measured power input (0.3 and 0.7 W kg⁻¹). The averaged concentration fields show a common two-dimensional steady circulation pattern. Concentration probability density functions reflect well the instability of the flow in the two regions investigated. The data reveal the non-isotropic distribution of these instabilities around a reference point when the feed port is situated in the bulk region only. In this case, the structural analysis quantitatively shows the presence of a folding of the concentration field. It was found that this phenomenon can last several seconds. Received: 16 June 1998/Accepted: 14 April 1999     

A general one-equation turbulence model for free shear and wall-bounded flows

The purpose of thiswork is to introduce a complete and general one-equation model capable of correctly predicting a wide class of fundamental turbulent flows like boundary layer, wake, jet, and vortical flows. The starting point is the mature and validated two-equation k−ω turbulence model of Wilcox. The newly derived one-equation model has several advantages and yields better predictions than the Spalart-Allmaras model for jet and vortical flows while retaining the same efficiency and quality of the results for near-wall turbulent flows without using a wall distance. The derivation and validation of the new model using findings computed by the Spalart-Allmaras and the k−ω models are presented and discussed for several free shear and wall-bounded flows.     

Temperature Dependent Viscoelastic Properties of Polymers Investigated by Small-Scale Dynamic Mechanical Analysis

The temperature-dependent viscoelastic properties of polymers were investigated by small-scale dynamic mechanical analysis in the range of −100°C to 200°C. The polymers tested included glassy polymer (atactic polystyrene), semicrystalline polymer (high-density polyethylene) and rubbery polymer (polyisobutylene). The small-scale dynamic mechanical analyses were performed by using a flat-tip indenter with an oscillating displacement of amplitude 36 nm. The force amplitude and phase angle were measured, from which the storage modulus E′ and loss tangent tanδ were calculated. The results obtained from indentation experiments are consistent with those obtained from conventional dynamic mechanical analyzer (DMA). It is thus demonstrated that the indentation technique can quantitatively measure the temperature-dependent viscoelastic properties of polymers at small dimensions.     

Acoustic Lagrangian velocity measurement in a turbulent air jet

Measuring Lagrangian velocities in a turbulent flow is of a great interest for turbulence modeling. We report measurements made in an axisymmetric turbulent air jet at Reynolds number R _λ ≃ 320, using acoustical Doppler scattering. Helium-filled soap bubbles are used as Lagrangian tracers. We describe an experimental setup which allows the simultaneous measurement of the full three-component Lagrangian velocity and the longitudinal Eulerian one. Lagrangian velocity probability density functions (PDF) are found Gaussian, close to Eulerian ones. Velocity correlations are analysed as well as the statistical dependence between components.     

A soap film shock tube to study two-dimensional compressible flows

 A new experimental approach to the study of the two-dimensional compressible flow phenomena is presented. In this technique, a variety of compressible flows were generated by bursting plane vertical soap films. An aureole and a “shock wave” preceding the rim of the expanding hole were clearly observed using traditional high-speed flash photography and a fast line-scan charge coupled device (CCD) camera. The moving shock wave images obtained from the line-scan CCD camera were similar to the x–t diagrams in gas dynamics. The moving shock waves cause thickness jumps and induce supersonic flows. Photographs of the supersonic flows over a cylinder and a wedge are presented. The results suggest clearly the feasibility of the “soap film shock tube”. Received: 11 May 2000/Accepted: 2 November 2000     

A Variational Model of Disjoining Pressure: Liquid Film on a Nonplanar Surface

Variational methods have been successfully used in modelling thin liquid films in numerous theoretical studies of wettability. In this article, the variational model of the disjoining pressure is extended to the general case of a two-dimensional solid surface. The Helmholtz free energy functional depends both on the disjoining pressure isotherm and on the shape of the solid surface. The augmented Young–Laplace equation (AYLE) is a nonlinear second-order partial differential equation. A number of solutions describing wetting films on spherical grains have been obtained. In the case of cylindrical films, the phase portrait technique describes the entire variety of mathematically feasible solutions. It turns out that a periodic solution, which would describe wave-like wetting films, does not satisfy Jacobi’s condition of the classical calculus of variations. Therefore, such a solution is nonphysical. The roughness of the solid surface significantly affects liquid film stability. AYLE solutions suggest that film rupture is more likely at a location where the pore-wall surface is most exposed into the pore space, and the curvature is positive.     

Evaluation of transient turbulent flow fields using digital cinematographic particle image velocimetry

This paper describes an experimental development for temporal and spatial reconstruction of continuously varying flow fields by means of digital cinematographic particle image velocimetry (PIV). The system uses a copper-vapor laser illumination synchronized with a high-speed camera, and continuously samples at 250 fps to measure transient and non-periodic turbulent flows with relatively low frequencies, i.e., the surf zone turbulence produced by depth-limited wave break in a long laboratory flume. The use of the developed PIV system comprehensively records the temporal development of both phase-averaged and instantaneous turbulent vortex flows descended from the breaking waves to the bottom. Also, the measured power spectra show harmonic frequencies, ranging from the orbital frequency of 0.5 Hz up to the order of 5 Hz, and the well-known −5/3 dependence upon the turbulence fluctuation frequencies thereafter. Received: 2 December 1999/Accepted: 6 September 2000     

Mixing Layers in Sink Flow: Effect of Length of Flight on Mixing in a Channel Downstream

We have experimentally and analytically studied transport of a passive scalar in the wake of a thin flat plate located at the centerline of a planar contraction with flat walls. The constant Launder parameter in the contraction, K = 6.25 ×10^− 6, was twice the value required for a turbulent boundary layer to relaminarize. In addition to the mixing analysis inside the contraction, layer mixing is also investigated downstream, where the flow continues inside a constant cross-section channel. In order to generate the passive scalar, the airflow above the plate was heated and the temperature stratification in the wake was traced by measuring the temperature field using constant current anemometry. Using different plate lengths, we found that the degree of mixing, obtained at a given position in the straight channel, is a function of the distance from the plate trailing edge to the contraction outlet. For a plate which does not protrude into the straight channel, we demonstrate the existence of an optimal trailing edge-contraction outlet distance that results in the lowest possible degree of mixing at a given downstream position in the straight channel. This finding is also supported by a semi-empirical relationship based on our developed self-similar solution for mixing layers in planar contractions.