Scalar transport in plane mixing layers

This paper describes the application of the Eulerian, single-point, single-time joint-scalar probability density function (PDF) equation for predicting the scalar transport in mixing layer with a high-speed and a low-speed stream. A finite-volume procedure is applied to obtain the velocity field with the k-ε closure being used to describe turbulent transport. The scalar field is represented through the modelled evolution equation for the scalar PDF and is solved using a Monte Carlo simulation. The PDF equation employs gradient transport modelling to represent the turbulent diffusion, and the molecular mixing term is modelled by the LMSE closure. There is no source term for chemical reaction as only an inert mixing layer is considered here. The experimental shear layer data published by Batt is used to validate the computational results despite the fact that comparisons between experiments and computational results are difficult because of the high sensitivity of the shear layer to initial conditions and free stream turbulence phenomena. However, the bimodal shape of the RMS scalar fluctuation as was measured by Batt can be reproduced with this model, whereas standard gradient diffusion calculations do not predict the dip in this profile. In this work for the first time an explanation is given for this phenomenon and the importance of a micromixing model is stressed. Also it is shown that the prediction of the PDF shape by the LMSE model is very satisfactory. Received on 27 October 1998     

Turbulent thermal boundary layer on a plate. Reynolds analogy and heat transfer law over the entire range of prandtl numbers

Arational asymptotic theory is proposed,which describes the turbulent dynamic and thermal boundary layer on a flat plate under zero pressure gradient. The fact that the flow depends on a finite number of governing parameters makes it possible to formulate algebraic closure conditions relating the turbulent shear stress and heat flux with the gradients of the averaged velocity and temperature. As a result of constructing an exact asymptotic solution of the boundary layer equations, the known laws of the wall for velocity and temperature, the velocity and temperature defect laws, and the expressions for the skin friction coefficient, Stanton number, and Reynolds analogy factor are obtained. The latter makes it possible to give two new formulations of the temperature defect law, one of which is identical to the velocity defect law and contains neither the Stanton number nor the turbulent Prandtl number, and the second formulation does not contain the skin friction coefficient. The heat transfer law is first obtained in the form of a universal functional relationship between three parameters: the Stanton number, the Reynolds number, and the molecular Prandtl number. The conclusions of the theory agree well with the known experimental data.     

Statistical characteristics of the diffusion of a chemically active additive in a turbulent mixing zone

In the article a numerical solution of the connected system of the equations of turbulent transfer for the fields of the velocity and concentration of a chemically active additive is used to calculate a number of the second moments of the concentration field in a flat mixing zone. The system of transfer equations is derived from the equations for a common function of the distribution of the fields of the pulsations of the velocity and the concentration [1] and is simplified in the approximation of the boundary layer. A closed form of the transfer equations is obtained on the level of three moments, using the hypothesis of four moments [2] and its generalized form for mixed moments of the field of the velocity and the field of a passive scalar. The differential operator of the closed system of the equations of turbulent transfer for the fields of the velocity and the concentration is found by a method of closure not of the parabolic type but of a weakly hyperbolic type [3]. An implicit difference scheme proposed in [4] is used for the numerical solution. The results of the numerical solution are compared with the experimental data of [5].     

A two-scale second-moment turbulence closure based on weighted spectrum integration

A two-scale second-moment turbulence closure has been derived based on the weighted integration of the dynamic equation for the covariance spectrum. The goal is to close the Reynolds stress equations with two additional scalar equations that provide separately the scales of the spectral energy transfer and of the turbulence energy dissipation rate. Such a model should provide better prediction of nonequilibrium turbulent flows. The derivation consists of analytical integration of the wave-number-weighted covariance spectrum using a model of the spectral equations with an assumed simple representation of the shape of the energy spectrum. The resulting closure consists of a set of three tensorial equations, one for the Reynolds stress and two for length scale tensors, the latter representing the energy containing- and dissipative eddies respectively. The trace of the two tensor-scale equations leads to a set of two scalar scale parameters. In the equilibrium limit, the model reduces to the standard second-moment single-scale closure. The approach makes it also possible to derive the scale equations in a more systematic manner as compared with the common single-scale and other multi-scale models. The performance of the model in capturing the scale dynamics is illustrated by predictions of several generic homogeneous and inhomogeneous unsteady flows, demonstrating the expected response of the two scale equations. PACS 03.50.De, 04.20-q, 42.65-k     

Multipoint correlations of concentration fluctuations in a turbulent passive scalar field

This study experimentally analyzes a turbulent passive scalar field using two-point and three-point correlation functions of the concentration fluctuations. The scalar field was created by the iso-kinetic release of a high Schmidt number dye into a fully developed turbulent boundary layer of an open channel flow. Concentration data at spatially separated measurement points were collected via the planar laser induced fluorescence (PLIF) technique. The current study complements previous research efforts by examining three-point correlations for several configurations in a turbulent shear flow. In the case of two streamwise-aligned points combined with one transversely separated point, contours of the three-point correlation function exhibit the symmetric properties reported in an earlier study of non-shear flow. In a second set of three-point configurations consisting of isosceles and collinear geometries, the influence of the orientation angle between the three-point configuration and the mean concentration gradient varies depending on the specified three-point geometry. The results also suggest that the scaling exponent in the inertial-convective regime is dependent on the injection length scale with weak dependence on the Reynolds number.     

Influence of turbulent Prandtl number on heat transfer of a flat plate

In computations involving heat transfer in turbulent flow past bodies it is necessary to assume turbulent Prandtl number distribution across the boundary layer. A review and comparison of results obtained by different authors are given, e.g., in [1–5]. Unfortunately, the existing data are so contradictory that, at present, it does not appear to be possible to establish reliably a function that determines turbulent Prandtl number distribution across the boundary layer. The absence of sufficiently reliable and general results on the distribution of turbulent Prandtl number led to the result that in the majority of studies conducted in earlier years its value was assumed a constant and either close to or equal to one. The effect of turbulent Prandtl number on the intensity of heat transfer from a flat plate is numerically investigated in the present paper. The thermal, turbulent boundary layer equation is integrated for this purpose at different values of turbulent Prandtl number and results are compared with experimental data. Results from [6], where the thermal boundary layer was numerically integrated with Pr_t=1 and compared with experimental data, were used for comparison in the present paper. The same numerical integration procedure as in [6] was used here.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 4, pp. 81–85, July–August, 1984.     

Presumed Mapping Functions for Eulerian Modelling of Turbulent Mixing

The mapping closure of Chen et al. [Phys. Rev. Lett., 63, 1989] is a transported probability density function (PDF) method that has proven very efficient for modelling of turbulent mixing in homogeneous turbulence. By utilizing a Gaussian reference field, the solution to the mapping function (in homogeneous turbulence) can be found analytically for a range of initial conditions common for turbulent combustion applications, e.g. for binary or trinary mixing. The purpose of this paper is to investigate the possibility of making this solution a presumed mapping function (PMF) for inhomogeneous flows. The PMF in turn will imply a presumed mixture fraction PDF that can be used for a wide range of models in turbulent combustion, e.g. flamelet models, the conditional moment closure (CMC) or large eddy simulations. The true novelty of the paper, though, is in the derivation of highly efficient, closed algebraic expressions for several existing models of conditional statistics, e.g. for the conditional scalar dissipation/diffusion rate or the conditional mean velocity. The closed form expressions nearly eliminates the overhead computational cost that usually is associated with nonlinear models for conditional statistics. In this respect it is argued that the PMF is particularly well suited for CMC that relies heavily on manipulations of the PDF for consistency. The accuracy of the PMF approach is shown with comparison to DNS of a single scalar mixing layer to be better than for the β-PDF. Not only in the shape of the PDF itself, but also for all conditional statistics models computed from the PDF.     

周期性扰动在湍流边界层中沿法向的衰减

在开口式循环水槽底部湍流边界层外区中引入周期性扰动,利用X型热膜探针对下游扰动进行测量。研究了湍流边界层中周期性人工扰动对湍流结构的影响,获得了人工扰动在湍流边界层中沿法向的衰减趋势。     

Investigation of scalar measurement error in diffusion and mixing processes

In variable density, multi-fluid and reacting flows, the degree of molecular mixing is a critical component of turbulent transfer and mixing models. Also, in many microflows and low Reynolds number flows, scalar diffusion length- and time-scales play a significant role in the mixing dynamics. Characterization of such molecular mixing processes requires scalar measurement devices with a small probe volume size. Spatial averaging, which occurs due to finite probe volume size, can lead to errors in resolving the density or scalar gradients between pockets of unmixed fluids. Given a probe volume size and a priori knowledge of the functional profile of the diffusion layer being measured, we obtain an estimate for the measurement error due to spatial averaging and make the corrections accordingly. An analytical model for the measure of scalar mixing is developed as a predictor for the growth of scalar gradients in a variable scalar flow. The model is applied to a buoyancy-driven mixing layer with a Prandtl number of 7. Measurements within the mixing layer have shown that initial entrainment of unmixed fluid causes a decrease in the measured amount of molecular mixing at the centerplane. Following this period of initial entrainment, the fluids within the mixing layer exhibit an increase in the degree of molecular mixing.     

Statistical modeling of the turbulent transition in the boundary layer

A method of statistical modeling the flow in the boundary-layer transition region is proposed on the basis of experimental data on kinematics and dynamics of turbulent spots (Emmons spots) on a flat plate in an incompressible fluid. This method allows one to determine the intermittency with allowance for overlapping of the spots, the forces on the plate surface, and the flow field in the vicinity of the transition region if the field of the streamwise component of the mean velocity in the developed turbulent boundary layer is known as a function of the Reynolds number. In contrast to multi-parameter models of the transition, this approach makes it possible to avoid the use of physically meaningless parameter values.     

Diffusion of suspended particles in an isotropic turbulence field

A model for turbulent motion is proposed which makes it possible to evaluate the pulsation characteristics and the diffusion coefficients of the dispersed phase and also makes it possible to describe the effect of the suspended particles on the turbulence of the dispersing medium. Specific calculations are made for the situation when the undisturbed turbulent field is isotropic.The diffusion of an admisture having inertia in a turbulent stream has been studied previously on the assumption that the three-dimensional turbulence characteristics have practically no effect on the behavior of the suspended particles, so that the random motion of the latter is described by ordinary differential equations containing the natural independent variable the motion travel time [1–4]. In many cases this assumption is incorrect and the corresponding theory is obviously deficient. For example, a fundamental result of this theory, asserting that the turbulent diffusion coefficients of the particles and of the fluid moles are equal for a long diffusion time, is obviously incorrect if the relative motion of the particles is significant [5].     

Scale similarity based models and their application to subgrid scale scalar flux modelling in the context of turbulent premixed flames

The performance of a variety of scale similarity (SS) type models for closure of sub-grid scalar flux in the context of Large Eddy Simulations (LES) of premixed turbulent combustion has been assessed. In addition to the well-known SS models, a more recent development by Anderson and Domaradzki (2012) is included in the analysis and also further model extensions and improvements are discussed. The work is based on a priori analysis of two Direct Numerical Simulation (DNS) databases of freely propagating turbulent premixed flames with a range of different Lewis and turbulent Reynolds numbers. Depending on the balance between the effects of flame normal acceleration due to heat release and the effects of turbulent velocity fluctuations, as well as the filter size, the subgrid-scalar flux exhibits both local gradient and counter-gradient transport which presents a considerable modelling challenge. The assessment is based on a correlation analysis and on the magnitude of the model expressions conditional on the Favre averaged reaction progress variable in comparison to the value obtained from DNS. Despite the fact that most of the models have been developed in the context of momentum transport in non-reactive flows they show either comparable or better performance in comparison to more conventional models used for reactive scalar flux closure. It is found that some models are sensitive to the test filter width and recommendations are provided in this regard. Further it is observed that the use of a Favre test filter substantially increases the correlation strength in direction of mean flame propagation where effects of heat release are most pronounced.     

Analyzing the influence of compressibility on the rapid pressure–strain rate correlation in turbulent shear flows

The influence of compressibility on the rapid pressure–strain rate tensor is investigated using the Green’s function for the wave equation governing pressure fluctuations in compressible homogeneous shear flow. The solution for the Green’s function is obtained as a combination of parabolic cylinder functions; it is oscillatory with monotonically increasing frequency and decreasing amplitude at large times, and anisotropic in wave-vector space. The Green’s function depends explicitly on the turbulent Mach number M _t , given by the root mean square turbulent velocity fluctuations divided by the speed of sound, and the gradient Mach number M _g , which is the mean shear rate times the transverse integral scale of the turbulence divided by the speed of sound. Assuming a form for the temporal decorrelation of velocity fluctuations brought about by the turbulence, the rapid pressure–strain rate tensor is expressed exactly in terms of the energy (or Reynolds stress) spectrum tensor and the time integral of the Green’s function times a decaying exponential. A model for the energy spectrum tensor linear in Reynolds stress anisotropies and in mean shear is assumed for closure. The expression for the rapid pressure–strain correlation is evaluated using parameters applicable to a mixing layer and a boundary layer. It is found that for the same range of M _t there is a large reduction of the pressure–strain correlation in the mixing layer but not in the boundary layer. Implications for compressible turbulence modeling are also explored.      

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.     

Dissipation rate correction methods

Dissipation rates of the turbulent kinetic energy and of the scalar variance are underestimated when the measurement resolution of the small scales of a turbulent flow field are insufficient. Results are presented of experiments conducted in a salt-stratified water tunnel (Schmidt number ∼700). Dissipation rates are determined to be underestimated, and thus correction techniques based on velocity structure functions and mixed-moment functions are proposed. Dissipation rates in laboratory experiments of shear-free, grid-generated turbulence are determined from balance calculations of the kinetic energy and scalar variance evolution equations. Comparisons between the structure function and balance estimates of dissipation show that the corrections are O(1) for the kinetic energy dissipation rate, and are O(100) for the scalar variance dissipation rate. This difference is due to the lack of resolution down to the Batchelor scales that is required for a high Schmidt number flow. Simple correction functions based on microscale Reynolds numbers are developed for both turbulent kinetic energy and scalar variance dissipation rates. Application of the technique to the results of laboratory experiments of density stratified turbulence, sheared turbulence, and sheared density stratified turbulence yields successful corrections. It is also demonstrated that the Karman–Howarth equality (and the analogous Yaglom equation) that relates second and third-order structure functions to dissipation rates is valid for both unstrained (decaying grid-generated turbulence) and density stratified and sheared turbulence at least up to the magnitudes of strains of the current experiments Nt∼10, St∼10, respectively. This is helpful for it allows the use of these equations in the analysis of turbulence even when the large scale background profiles of velocity and scalar are unknown.     

可压缩各向同性衰减湍流直接数值模拟研究

采用五阶有限差分WENO格式直接模拟了高初始湍流Mach数的可压缩均匀各向同性湍流,主要分析了湍流的统计特性 和压缩性的影响,包括能谱特征、激波串、耗散率、标度律等. 研究表明,湍动能主要来自于速度场螺旋分量的贡献;各向同性湍流的小尺度脉动对压缩性更为敏感,并且压缩性的增强加快了湍流大 尺度脉动向小尺度脉动的湍动能输运;随着湍流Mach数的升高,胀量(压缩)耗散率所占比率也显著增长. 标度律分析表明,强可压缩湍流的横向速度结构函数仍然具有扩展自相似性;当阶数较高(p ≥ 5)时,纵向速度结构函数的扩展自相似性则不再成立. 对于压缩性较弱的湍流,与不可压缩湍流一致,横向湍流脉动的间歇性要强于纵向湍流脉动;而对于强可压缩湍流,纵向湍流脉动的 间歇性要强于横向湍流脉动.     

Mass and momentum transport in the near field of swirling turbulent jets. Effect of swirl rate

Local transport of the flow momentum and scalar admixture in the near-field of turbulent swirling jets (Re = 5,000) has been investigated by using a combination of the particle image velocimetry and planar laser-induced fluorescence methods. Advection and turbulent and molecular diffusions are evaluated based on the measured distributions of the mean velocity and concentration and the Reynolds stresses and fluxes. As has been quantified from the data, the flow swirl intensifies the entrainment of the surrounding fluid and promotes mass and momentum exchange in the outer mixing layer. A superimposed swirl results in the appearance of a wake/recirculation region at the jet axis and, consequently, the formation of an inner shear layer. In contrast to the scalar admixture, the momentum exchange in the inner shear layer is found to be strongly intensified by the swirl. For the jet with the highest considered swirl rate, a substantial portion of the surrounding fluid is found to enter the unsteady central recirculation zone, where it mixes with the jet that is issued from the nozzle. The contribution of the coherent velocity fluctuations, which are induced by large-scale vortex structures, to the turbulent transport has been evaluated based on triple decomposition, which was based on proper orthogonal decomposition analysis of the velocity data sets. For the considered domain of the jet with the highest swirl rate and vortex breakdown, the contributions of detected helical vortex structures, inducing pressing vortex core, to the radial fluxes of the flow momentum and the scalar admixture are found to locally exceed 65% and 80%, respectively.     

Use of caged fluorescent dyes for the study of turbulent passive scalar mixing

The non-intrusive initialization of a flow field with distinct and spatially segregated scalar components represents a significant experimental difficulty. Here a new technique is described which makes possible the non-intrusive initialization of a spatially binary passive scalar field in a laminar or turbulent flow field. This technique uses photoactivatable (caged) fluorescent dyes dissolved in the flow medium. The scalar field within the flow field is tagged or initialized by uncaging the appropriate regions with an ultraviolet excimer laser. Mixing between the tagged and untagged regions is quantified using standard laser induced fluorescence techniques. The method is currently being used to study mixing in a turbulent pipe flow.     

Heat transfer enhancement in a turbulent natural convection boundary layer along a vertical flat plate

An experimental study on heat transfer enhancement for a turbulent natural convection boundary layer in air along a vertical flat plate has been performed by inserting a long flat plate in the spanwise direction (simple heat transfer promoter) and short flat plates aligned in the spanwise direction (split heat transfer promoter) with clearances into the near-wall region of the boundary layer. For a simple heat transfer promoter, the heat transfer coefficients increase by a peak value of approximately 37% in the downstream region of the promoter compared with those in the usual turbulent natural convection boundary layer. It is found from flow visualization and simultaneous measurements of the flow and thermal fields with hot- and cold-wires that such increase of heat transfer coefficients is mainly caused by the deflection of flows toward the outer region of the boundary layer and the invasion of low-temperature fluids from the outer region to the near-wall region with large-scale vortex motions riding out the promoter. However, heat transfer coefficients for a split heat transfer promoter exhibit an increase in peak value of approximately 60% in the downstream region of the promoter. Flow visualization and PIV measurements show that such remarkable heat transfer enhancement is attributed to longitudinal vortices generated by flows passing through the clearances of the promoter in addition to large-scale vortex motions riding out the promoter. Consequently, it is concluded that heat transfer enhancement of the turbulent natural convection boundary layer can be substantially achieved in a wide area of the turbulent natural convection boundary layer by employing multiple column split heat transfer promoters. It may be expected that the heat transfer enhancement in excess of approximately 40% can be accomplished by inserting such promoters.