Streamwise evolution of a high-Schmidt-number passive scalar in a turbulent plane wake

The concentration fluctuation c of diluted fluorescein dye, a high-Schmidt-number passive scalar (Sc=ν/D ≈ 2000, ν and D are the fluid momentum and dye diffusivities, respectively), is measured in the wake of a circular cylinder using a single-point laser-induced fluorescence (SPLIF) technique. The streamwise decay rate of the mean and rms values of c is slow in comparison to that of θ, the temperature fluctuation for which the molecular Prandtl number Pr=ν/κ is about 0.7 (κ is the thermal diffusivity). The comparison between mean and rms distributions of c and θ highlights the combined role the Reynolds and Schmidt numbers play in terms of dispersing the scalar. The streamwise evolution of the probability density functions (pdfs) of c and θ suggest that while p(θ) is approximately Gaussian in the intermediate wake (x/d ≈ 80), p(c) is strongly non-Gaussian, and depends on both x/d and Re. The skewness of c is larger than that of θ along the wake centreline. Arguably, the asymmetry of p(c) reflects the relatively strong organisation of the large-scale motion in the far-wake. Received: 27 July 2000/Accepted: 22 December 2000     

Computation of conjugate heat transfer in the turbulent mixed convection regime in a vertical channel with multiple heat sources

This paper presents the results of a comprehensive numerical study to analyze conjugate, turbulent mixed convection heat transfer from a vertical channel with four heat sources, uniformly flush-mounted to one of the channel walls. The results are presented to study the effect of various parameters like thermal conductivity of wall material (k _s), thermal conductivity of flush-mounted discrete heat source (k _c), Reynolds number of fluid flow (Re _s), modified Richardson number (Ri ⁺) and aspect ratio (AR) of the channel. The standard k-ε turbulence model, modified by including buoyancy effects with physical boundary conditions, i.e. without wall functions, has been used for the analysis. Semi-staggered, non-uniform grids are used to discretise the two dimensional governing equations, using finite volume method. A correlation, encompassing a wide range of parameters, is developed for the non-dimensional maximum temperature (T ^*) using the asymptotic computational fluid dynamics (ACFD) technique.     

Density Maximum Effect on Double-diffusive Natural Convection in a Porous Cavity with Variable Wall Temperature

The effect of density maximum of water on double-diffusive natural convection in a two-dimensioned cavity filled with a water saturated isotropic porous medium is studied numerically. The horizontal walls of the cavity are insulated. The opposing vertical walls are kept at different temperatures θ _h (linearly varies with height) and θ _c (θ _c ≤ θ _h ). The concentration levels at cold wall and hot wall are, respectively, c ₁ and c ₂ with c ₁ > c ₂. Brinkman-Forchheimer extended Darcy model is used to investigate the average heat and mass transfer rates. The non-dimensional equations for momentum, energy, and concentration are solved by finite volume method with power law scheme for convection and diffusion terms. The results are presented in the form of streamlines, isotherms, and isoconcentration lines for various values of Grashof numbers, Schmidt number, porosity, and Darcy numbers. It is observed that the density maximum of water has profound effect on the thermosolutal convection. The effects of different parameters on the velocity, temperature, and species concentrations are also shown graphically.     

Onset of Buoyancy-Driven Convection in Porous Media Saturated with Cold Water Cooled from Above

When porous media saturated with initially stagnant cold water around the density maximum temperature are cooled from above, convection may be induced in an unstable lower layer. In this study, the onset of buoyancy-driven convection during time-dependent cooling is investigated using the propagation theory, which transforms disturbance equations similarly, and also considering the density inversion effect. The critical Darcy–Rayleigh number Ra _D,c is found as a function of the dimensionless density maximum temperature θ _max. For Ra _D > Ra _D,c the dimensionless critical time τ _c to mark the onset of instability is presented as a function of Ra _D and θ _max. These critical conditions are compared with previous theoretical results.     

Radiation-conduction interaction on mixed convection from a horizontal circular cylinder

A mixed convection flow of an optically dense viscous incompressible fluid along a horizontal circular cylinder has been studied with the effect of radiation when the surface temperature is uniform. Using appropriate transformations, the boundary layer equations governing the flow are reduced to local nonsimilarity form. Solutions of the governing equations are obtained employing the implicit finite difference method. Effects of varying the pertinent parameters, such as, the Planck number, R _w the surface temperature parameter, θ_w and the buoyancy parameter, α on the local skin-friction and local heat transfer coefficients are shown graphically as well as in tabular form against the curvature parameter ξ, while taking Prandtl number Pr = 1.0. It is found that an increase of R _d,θ_w or α leads to increases in the values of the local skin-friction and the local rate of heat transfer coefficients. At the stagnation point asymptotic solutions for large value of α are also obtained and the effect of the other pertinent parameters on the formation of the flow separation are studied. Received on 28 July 1998     

Simulation of the thermal method for reducing turbulent friction

The results of calculating a supersonic turbulent boundary layer on a heated surface on the basis of the algebraic two-parameter (k-ε) and four-parameter (k-ε-θ ²-ε ₆) models of turbulence are compared with experimental data. Emphasis is placed on the ability of the models to predict the behavior of the friction and heat-transfer coefficients on a heated surface. The optimal model of turbulence is chosen. The possibility of improving the efficiency of viscous drag reduction by localizing the regions of heat addition to the boundary layer is demonstrated on the basis of numerical calculations. Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, pp. 59–68, January–February, 1998. This research was carried out with financial support from the International Scientific and Technological Center (project No. 199).     

Steady mixed convection boundary-layer flow over a vertical flat surface in a porous medium filled with water at 4°C: variable surface heat flux

The steady mixed convection boundary-layer flow over a vertical impermeable surface in a porous medium saturated with water at 4°C (maximum density) when the surface heat flux varies as x ^m and the velocity outside the boundary layer varies as x ^(1+2m)/2, where x measures the distance from the leading edge, is discussed. Assisting and opposing flows are considered with numerical solutions of the governing equations being obtained for general values of the flow parameters. For opposing flows, there are dual solutions when the mixed convection parameter λ is greater than some critical value λ _c (dependent on the power-law index m). For assisting flows, solutions are possible for all values of λ. A lower bound on m is found, m > −1 being required for solutions. The nature of the critical point λ _c is considered as well as various limiting forms; the forced convection limit (λ = 0), the free convection limit (λ → ∞) and the limits as m → ∞ and as m → −1.     

Turbulence Modeling Effects on the Prediction of Equilibrium States of Buoyant Shear Flows

The effects of turbulence modeling on the prediction of equilibrium states of turbulent buoyant shear flows were investigated. The velocity field models used include a two-equation closure, a Reynolds-stress closure assuming two different pressure-strain models and three different dissipation rate tensor models. As for the thermal field closure models, two different pressure-scrambling models and nine different temperature variance dissipation rate ɛ_τ) equations were considered. The emphasis of this paper is focused on the effects of the ɛ_τ-equation, of the dissipation rate models, of the pressure-strain models and of the pressure-scrambling models on the prediction of the approach to equilibrium turbulence. Equilibrium turbulence is defined by the time rate of change of the scaled Reynolds stress anisotropic tensor and heat flux vector becoming zero. These conditions lead to the equilibrium state parameters, given by /ɛ, _τ/ɛ_τ, , Sk/ɛ and G/ɛ, becoming constant. Here, and _τ are the production of turbulent kinetic energy k and temperature variance , respectively, ɛ and ɛ_τ are their respective dissipation rates, R is the mixed time scale ratio, G is the buoyant production of k and S is the mean shear gradient. Calculations show that the ɛ_τ-equation has a significant effect on the prediction of the approach to equilibrium turbulence. For a particular ɛ_τ-equation, all velocity closure models considered give an equilibrium state if anisotropic dissipation is accounted for in one form or another in the dissipation rate tensor or in the ɛ-equation. It is further found that the models considered for the pressure-strain tensor and the pressure-scrambling vector have little or no effect on the prediction of the approach to equilibrium turbulence. Received 21 April 2000 and accepted 21 February 2001     

Heat and mass transfer for micropolar flow with radiation effect past a nonlinearly stretching sheet

In this study, an analysis has been performed for heat and mass transfer with radiation effect of a steady laminar boundary-layer flow of a micropolar flow past a nonlinearly stretching sheet. Parameters n, K, k ₀, Pr, Ec, and Sc represent the dominance of the nonlinearly effect, material effect, radiation effect, heat and mass transfer effects which have presented in governing equations, respectively. The similar transformation, the finite-difference method and Runge–Kutta method have been used to analyze the present problem. The numerical solutions of the flow velocity distributions, temperature profiles, the wall unknown values of θ′(0) and ϕ′(0) for calculating the heat and mass transfer of the similar boundary-layer flow are carried out as functions of n, Ec, k ₀, Pr, Sc. The value of n, k ₀, Pr and Sc parameters are important factors in this study. It will produce greater heat transfer efficiency with a larger value of those parameters, but the viscous dissipation parameter Ec and material parameter K may reduce the heat transfer efficiency. On the other hand, for mass transfer, the value of Sc parameter is important factor in this study. It will produce greater heat transfer efficiency with a larger value of Sc.     

The compressible mixing layer: an LES study

This article employs LES to simulate temporal mixing layers with Mach numbers ranging from M _c  = 0.3 to M _c  = 1.2. A form of approximate deconvolution together with a dynamic Smagorinsky subgrid model are employed as subgrid models. A large computational domain is used along with relatively good resolution. The LES results regarding growth rate, turbulence levels, turbulence anisotropy, and pressure–strain correlation show excellent agreement with those available from previous experimental and DNS results of the same flow configuration, underlining the effectiveness and accuracy of properly conducted LES. Coherent structures during the transitional stage change from spanwise aligned rollers to streamwise-aligned thinner vortices at high Mach number. In the quasi-self-similar turbulent stage, the resolved-scale vorticity is more isotropic at higher M _c , and its vertical correlation length scale is smaller. The ratio of the vertical integral length scale of streamwise velocity fluctuation to a characteristic isotropic estimate is found to decrease with increasing M _c . Thus, compressibility leads to increased spatial decorrelation of turbulence which is one reason for the reduction in pressure–strain correlation with increasing M _c . The balance of the resolved-scale fluctuating vorticity is examined, and it is observed that the linear production by mean shear becomes less important compared to nonlinear vortex stretching at high M _c . A spectral decomposition of the pressure fluctuations into low- and intermediate-to-high-wave numbers is performed. The low-wave number part of the pressure field is found not to correlate with the strain field, although it does have a significant contribution to the r.m.s of the fluctuating pressure. As a consequence, the pressure–strain correlation can be analyzed using a simplified Green’s function for the Poisson equation as is demonstrated here using the LES data.     

Finite difference analysis of laminar free convection flow past a non isothermal vertical cone

A finite-difference analysis for the transient free convection flow of an incompressible viscous fluid past a vertical cone with variable wall surface temperature T _w^′ (x) = T _∞^′ + a x ⁿ varying as power function of distance from the apex (x = 0) is presented here. The dimensionless governing equations of the flow that are unsteady, coupled and non-linear partial differential equations are solved by an efficient, accurate and unconditionally stable finite difference scheme of Crank-Nicolson type. The velocity and temperature fields have been studied for various parameters such as Prandtl number and n (exponent in power law variation in surface temperature). The local as well as average skin-friction and Nusselt number are also presented and analyzed graphically. The present results are compared with available results in literature and are found to be in good agreement.     

Two‐dimensional prediction of time dependent,turbulent flow around a square cylinder confined in a channel

This paper presents two‐dimensional and unsteady RANS computations of time dependent, periodic, turbulent flow around a square block. Two turbulence models are used: the Launder–Sharma low‐Reynolds number k–ε model and a non‐linear extension sensitive to the anisotropy of turbulence. The Reynolds number based on the free stream velocity and obstacle side is Re=2.2×10⁴. The present numerical results have been obtained using a finite volume code that solves the governing equations in a vertical plane, located at the lateral mid‐point of the channel. The pressure field is obtained with the SIMPLE algorithm. A bounded version of the third‐order QUICK scheme is used for the convective terms. Comparisons of the numerical results with the experimental data indicate that a preliminary steady solution of the governing equations using the linear k–ε does not lead to correct flow field predictions in the wake region downstream of the square cylinder. Consequently, the time derivatives of dependent variables are included in the transport equations and are discretized using the second‐order Crank–Nicolson scheme. The unsteady computations using the linear and non‐linear k–ε models significantly improve the velocity field predictions. However, the linear k–ε shows a number of predictive deficiencies, even in unsteady flow computations, especially in the prediction of the turbulence field. The introduction of a non‐linear k–ε model brings the two‐dimensional unsteady predictions of the time‐averaged velocity and turbulence fields and also the predicted values of the global parameters such as the Strouhal number and the drag coefficient to close agreement with the data. Copyright © 2009 John Wiley & Sons, Ltd.     

Numerical solution of the turbulent flow of an incompressible fluid in curved planar and axially symmetrical channels

The governing equations for axially symmetric flow, where the Reynolds stresses are expressed by scalar turbulent viscosity, are the Reynolds equations. The turbulence model k, ? is used in the well-known form for fully developed turbulent flow.The numerical method, a continuation of the MAC system¹, is adapted so that even for high Reynolds cell numbers precision (δx²) can be achieved for the steady flow. Irregular cells join the rectangular network on the curved surface. Von Neumann's stability condition of the linearised numerical system is investigated. Special problems concerning the numerical solution of the turbulence model equations are stated and a special procedure is worked out to ensure that the fields k, ? do not converge to physically meaningless values. The program for the computer is universal in that the boundary problems can be assigned by input data.As an example, an axially symmetrical diffuser with an area ratio of widening 1.40 is computed. Fields of velocity and pressure at the wall as well as fields v_T and k are assessed. The results are compared with an experiment. The conclusion is that this method is suitable for the problems mentioned in this study as well as for unsteady flow.     

Thermal radiation effects on the natural convection flow over an isothermal horizontal plate

The natural convection boundary layer flow with conduction-radiation interaction of a viscous incompressible fluid along an isothermal horizontal surface has been studied. The equations valid in the upstream, downstream as well as in the entire regime are obtained. Solutions of the non-similar equations governing the flow for the entire regime and the downstream regime are obtained by employing an efficient implicit finite difference approximation together with the Keller box method, for a Prandtl number of 0.73. Also, the effects of the pertinent parameters, R _d, the radiation-conduction parameter and θ_w, the surface heating parameter are shown graphically in terms of the local skin-friction and the local rate of heat transfer. Comparison of the results obtained for the upstream and the downstream regimes shows good agreement over the entire regime. Effects of R _d and θ_w are also shown on the streamlines and the isotherms. Received on 15 December 1998     

Analysis of 2D infinite anisotropic elastic strips and semi-strips

Summary  Using Stroh's formalism and the theory of analytic functions, simple and explicit solutions for a line dislocation in an infinite anisotropic elastic strip are obtained. The two boundaries of the strip are free of traction. The problem of a dislocation in an anisotropic elastic semi-infinite strip with traction-free boundaries is also studied. A set of singular integral equations governing the unknown functions is derived. When the medium is orthogonal anisotropic and the coordinate axes x ₁ x ₂ x ₃ are coincident with the material principal axes, all the eigenvalues of the material coefficient matrix are pure imaginary. Explicit expressions of the unknown functions are given for this case. The results obtained are valid not only for plane and anti-plane problems but also for coupled problems between in-plane and out-of-plane deformations. Received 30 October 2000; accepted for publication 28 March 2001     

Effect of different sized transverse square grooves on a turbulent boundary layer

The effect of three different sized transverse square grooves (5, 10, and 20 mm) on a turbulent boundary layer was investigated at two values of momentum thickness Reynolds numbers (R _θ =1,000 and 3,000) using hot-wire anemometry. The ratios of the groove depth to the boundary layer thickness (d/δ ₀) are approximately 0.07, 0.13, and 0.27. Wall shear stress (τ _w), mean velocity (U), and turbulence intensity downstream of the grooves are compared to those on a corresponding smooth wall The effects of the grooves are more significant at the higher R _θ , with the most pronounced effects caused by the largest size groove. There is an increase in mean velocity (U), streamwise (u′/U ₀), and wall-normal (ν′/U ₀) turbulence intensities in the near-wall region immediately downstream of the grooves. The increase propagates outwards in the layer as the streamwise distance increases downstream of the grooves. The increase in ν′/U ₀ is much more significant than that of u′/U ₀, which is also evident in the spectra of u′ and ν′. There is an increase in τ _w over the smooth wall value immediately downstream of the grooves at R _θ =1,000, with the increase being more pronounced as the groove size increases. The growth of the internal layer downstream of the grooves is found to scale with the groove size, and is more rapid at R _θ =3,000. Published online: 23 November 2002     

High-resolution imaging of dissipative structures in a turbulent jet flame with laser Rayleigh scattering

High-resolution 2-D imaging of laser Rayleigh scattering is used to measure the detailed structure of the thermal dissipation field in a turbulent non-premixed CH₄/H₂/N₂ jet flame. Measurements are performed in the near field (x/d = 5–20) of the flame where the primary combustion reactions interact with the turbulent flow. The contributions of both the axial and radial gradients to the mean thermal dissipation are determined from the 2-D dissipation measurements. The relative contributions of the two components vary significantly with radial position. The dissipation field exhibits thin layers of high dissipation. Noise suppression by adaptive smoothing enables accurate determination of the dissipation-layer widths from single-shot measurements. Probability density functions (PDF) of the dissipation-layer widths conditioned on temperature are approximately log-normal distributions. The conditional layer width PDFs are self-similar functions with the layer widths scaling with temperature to the 0.75 power. The high signal-to-noise ratio of the Rayleigh scattering images coupled with an interlacing technique for noise suppression enable fully resolved measurements of the mean power spectral density (PSD) of the temperature gradients. These spectra are used to determine the turbulence microscales by measuring a cutoff wavelength, λ _C , at 2% of the peak PSD. The Batchelor scale is estimated from λ _C , and the results are compared with estimates from scaling laws in non-reacting flows. At x/d = 20, the different approaches to determining the Batchelor scale are comparable on the jet centerline. However, the estimates from non-reacting flow scaling laws are significantly less accurate in off-centerline regions and at locations closer to the nozzle exit. Throughout the near field of the jet flame, the measured ratio of a characteristic dissipation-layer width to the local Batchelor scale is larger than values previously reported for the far field of non-reacting flows.     

Complete Orbits for Twist Maps on the Plane: Extensions and Applications

Twist maps (θ ₁, r ₁) = f (θ, r) on the plane are considered which do not exhibit any kind of periodicity in their dependence on θ. Some general results are obtained which typically yield the existence of infinitely many complete and bounded orbits. Examples that can be treated with this theory include oscillators of the type [(x)\ddot]+V￠(x)=p(t){\ddot{x}+V'(x)=p(t)} under appropriate hypotheses, the bouncing ball system, and the standard map.