Turbulent Wall Flows Research Articles

The law of the wall in turbulent flow

The ‘law of the wall’ for the inner part of a turbulent shear flow over a solid surface is one of the cornerstones of fluid dynamics, and one of the very few pieces of turbulence theory whose results include a simple analytic function for the mean velocity distribution, the logarithmic law. Various aspects of the law have recently been questioned, and this paper is a summary of the present position. Although the law of the wall for velocity has apparently been confirmed by experiment well outside its original range, the law of the wall for temperature seems to apply only to very simple flows. Since the two laws are derived by closely analogous arguments this throws suspicion on the law of the wall for velocity. Analysis of simulation data, for all the Reynolds stresses including the shear stress, shows that law-of-the-wall scaling fails spectacularly in the viscous wall region, even when the logarithmic law is relatively well behaved. Virtually all turbulence models are calibrated to reproduce the law of the wall in simple flows, and we discuss whether, in practice or in principle, their range of validity is larger than that of the law of the wall itself: the present answer is that it is not; so that when the law of the wall (or the mixing-length formula) fails, current Reynolds-averaged turbulence models are likely to fail too.

A model of transient oscillatory pressure-flow relationships of canine airways.

In a previous paper (27) we developed a lumped parameter model of canine pulmonary airway mechanics featuring airway wall elasticity, gas inertance, and laminar and turbulent gas flow. The model accurately accounted for the steady-state pressure-flow data we obtained during sinusoidal cycling of the lung following a period of apnea. In the present paper, we extend the model to account for the transient decrease in the amplitude of the trans-airway pressure swings that we observed immediately following the apnea, which we have shown to be due to a vagally mediated bronchodilatation reflex. The extended model accounts for this transient in terms of a sudden change in airway smooth muscle tone acting on the viscoelastic properties of the airway wall and tissues mechanically coupled to it. Consequently, this model is able to temporarily store a volume of gas in the conducting airway tree as its volume changes cyclically with that of the whole lung. This means that the flow entering the airway tree from the trachea at any instant (V) is not precisely equal to that entering the alveoli (Valv) even when the gas is considered incompressible. We found that assuming V to be equal to Valv can lead to errors in estimating respiratory tissue impedance of as much as 10%. However, tissue hysteresivity remained almost unaffected, suggesting that the hysteretic properties of respiratory system tissues and airway wall are well matched.

Heat transfer and associated energy dissipation for oscillatory flow in baffled tubes

We report experimental data on the heat transfer performance of a periodically baffled tube subject to both steady (net) flow and oscillatory flow. The data show that, in particular, at a low net flow Reynolds number, significant heat transfer enhancement can be achieved with the superposition of fluid oscillations. A general correlation is derived for the measured Nusselt number as a function of both net flow and oscillatory Reynolds number. Dynamic pressure drop data for oscillatory flow are also reported, and estimates of energy efficiency for obtaining heat transfer enhancement made from these measurements are compared with smooth wall turbulent flow equations. For large amplitudes of oscillation (equivalent to half the tube diameter) the overall power dissipation follows the quasi-steady theory. At smaller amplitudes of oscillation the power dissipation was larger than predicted by the quasi-steady theory, indicating an increased eddy interaction.

A Two-Equation Heat-Transfer Model Reflecting Second-Moment Closures for Wall and Free Turbulent Flows.

Abstract A new two-equation heat transfer model which incorporates some essential features of second-order modeling is proposed. The present model is applicable to heat transfer problems in both wall and free turbulent flows not unusually deviating from the equilibrium state. Furthermore, by introducing the Kolmogorov velocity scale, the model can appropriately express the low Reynolds number effects in the near-wall region and is also applicable to complex heat transfer fields with flow separation and reattachment. It is shown that the proposed model predicts quite successfully heat transfer in both wall and free turbulent flows; i.e., a homogeneous isotropic decaying flow, a homogeneous shear flow, a boundary-layer flow heated from the origin, and a boundary-layer flow subjected to a sudden change in the wall-heating condition; whereas, such predictions have been almost impossible with existing two-equation heat transfer models.

Critical Conditions for Upward Flame Spread and Comparison with Empirical Flammability Indices

ABSTRACT We use recent advances in material flame spread physics to emphasize that empirical flammability indices might be replaced by fundamentally derived parameters. Flame spread rates in upward wall fire situations can be described both for charring and non-charring materials by using a characteristics length scale, lm , and a characteristic time, tp . The length scale, lm , is related to the distribution and magnitude of the heat flux from the flames to the wall surface as well as to burning material properties and mixing processes in the turbulent wall flow. The same characteristic time, tp describes both the spread rate, by the determination of ignition time of yet unpyrolyzed material, and the transient pyrolysis of the solid for situations where such pyrolysis can be approximated by a thermal pyrolysis model.

Diffusion of Matter in the Wall Turbulent Flow. 1st Report. Similarity of a Diffusion Plume from a Wall Point Source in a Pipe Flow.

The aim of this study is to elucidate experimentally the spatial structure and the statistical properties of the plume diffusion field of matter from a wall point source in a turbulent pipe flow. The fully developed water turbulent flow of Reynolds number 20000 has been realized in the horizontal straight pipe of inner diameter 100 mm. The concentration field downstream from a continuous point source on the wall has been measured using a reflection-type fiber optic probe (sampling volume s 6.3×10-6cc). It is found that the vertical profiles of the mean concentration and concentration fluctuation r. m. s. value at the different downstream positions from the source show self-similarity on the symmetric plane of the diffusion field. In particular, the mean concentration profile can be described quite well by the universal function which was suggested for a point or a line source plume in the turbulent boundary layer on the flat plate.

Calculation of Turbulent Flows with Pressure Gradients Using a k-.EPSILON. Model.

A k-e turbulence model is developed to calculate wall turbulent shear flows under various pressure gradient conditions. In the present model, we set the dissipation rate of turbulent energy at a wall equal to zero, though the wall limiting behavior of velocity fluctuations is reproduced exactly. Thus, the model assures computational expediency and convergence. The proposed model is constructed to properly take into account the effect of pressure gradients on shear layers. It was found by Nagano, Tagawa and Tsuji (Turbulent Shear Flows 8, (1992), 7) that in adverse pressure gradient flows the Van Driest constant decreased with increasing dimensionless pressure gradient parameter P+. Therefore, the present model incorporates the modified Van Driest dumping function which is a function of P+. The validity of the proposed model was tested by application to a turbulent channel flow and boundary layers with P+ 0. The model predictions indicate that agreement with the experiment and the direct simulation data is very good over a wide range of pressure variations.

Advanced turbulence closure models: a view of current status and future prospects

This report presents the status and reviews some of the current activities in the development of advanced one-point turbulence closure models with special reference to the modeling and computation of turbulent wall flows. It gives a brief retrospective of the development of the one-point closure model and highlights major achievementss. It also recalls some of the pertinent issues that are still open and that pose challenges to modelers. Some pertaining deficiencies, which have been obvious since the very beginning of the development and are still present, are discussed, together with some marked differences in the views of various groups of modelers. The review is limited to conventional one-point two-equation Eddy Viscosity Models (EVMs) and Differential Second-Moment (Re-stress) Models (DSMs). It does not deal with any spectral approach, nor partial field models, such as subgrid modeling or similar methods used within the framework of other simulation techniques. Some new approaches, such as the Renormalization group theory (RNG), are mentioned briefly and only insofar as they may inspire modifications and improvements of conventional models. The report deals with general aspects of modeling complex turbulent flows, but is restricted to incompressible (or mildly compressible) fluid, with particular focus on computation of wall flows.

Multiple-Scale Turbulence Model for Wall and Free Turbulent Flows.

It is well known that turbulence models based on the eddy viscosity assumption have been quite successful in the numerical predictions of various kinds of turbulent flows. In particular, the k-εmodel is the most popular and has been proven to be effective in predicting wall shear layer turbulence. The k-ε models, however, show poor performance in the prediction of free shear flows. Such inconsistency is not caused by the crudeness of the eddy viscosity approximation, but by the inappropriate estimation of the relevant time scale of turbulence. Thus, in order to settle this problem, we propose a low-Reynolds-number-type "multiple-scale" turbulence model on the basis of the original model of Hanjalic et al. (Turbulent Shear Flows 2, 1980, Springer), in which multiple scales characterizing a turbulent flow are introduced by dividing the turbulence energy spectrum into two wave-number regions. In the present study, much emphasis is put on making a proposed model applicable to the prediction of near-wall turbulent flows without the controversial wall functions and reproduce the wall-limiting behavior of turbulence quantities. The proposed model has been tested in canonical wall and free shear flows and are found to work quite well regardless of the flow regimes.

Evaluation of ejection detection schemes in turbulent wall flows

Velocity techniques for detecting ejections have been systematically examined by separating the detection schemes into trigger and delimiter algorithms. A new technique for grouping ejections based on a period of quiescence between bursts was also developed. The combination of improved ejection detection and the new grouping technique have reduced the error in the time between bursts from 25% to 10%. There is a similar improvement in the uncertainty.

Reynolds stress and vorticity in turbulent wall flows

Reynolds stress and vorticity in turbulent wall flows

Calculations of Turbulent Flows with Pressure Gradients using a k-.EPSILON. Model.

A k-ε turbulence model is developed to calculate wall turbulent shear flows under various pressure gradient conditions. In the present model, we set the dissipation rate of turbulent energy at a wall equal to zero, though the wall limiting behavior of velocity fluctuations is reproduced exactly. Thus, the model assures computational expediency and convergence. The proposed model is constructed to properly take into account the effects of pressure gradients on shear layers. It was found by Nagano et al. (1992) that in adverse pressure gradient flows the Van Driest damping constant decreased with increasing dimensionless pressure gradient parameter P+. Therefore, the present model has introduced the modified Van Driest damping function which is a function of P+. The proposed model was tested by application to a turbulent channel flow and boundary layers with P+ < 0, P+=0 and P+ > 0. The model predictions indicate that agreement with the experiment and the direct simulation data is very good over a wide range of pressure variations.

A two-equation heat transfer model for predicting turbulent thermal fields under arbitrary wall thermal conditions

A new proposal for the P 2 ̄ -ε 2 heat transfer model is presented along with an accurate prediction of wall turbulent thermal fields. The proposed model reproduces the correct wall limiting behavior of velocity and temperature under arbitrary wall thermal conditions. Assessment of the model constants and functions are made to generalize applicability of the P 2 ̄ -ε 2 model. The proposed model is tested with five different typical thermal fields, which often occur in engineering applications, in wall turbulent shear flows. The predicted results are compared with the available experimental and full simulation data, together with the previous model predictions. It is shown that the present model works much better than the previous models.

Application of a new K-tau model to near wall turbulent flows

A recently developed K-tau model for near wall turbulent flows is applied to a variety of test cases. The turbulent flows considered include the incompressible flat plate boundary layer with adverse pressure gradients, incompressible flow past a backward facing step, and the supersonic flat plate boundary layer at zero pressure gradient. Calculations are performed for this two-equation model using an anisotropic as well as isotropic eddy-viscosity. The model predictions are shown to compare quite favorably with experimental data.

Critical evaluation of two-equation models for near-wall turbulence

A variety of two-equation turbulence models,including several versions of the K-epsilon model as well as the K-omega model, are analyzed critically for near wall turbulent flows from a theoretical and computational standpoint. It is shown that the K-epsilon model has two major problems associated with it: the lack of natural boundary conditions for the dissipation rate and the appearance of higher-order correlations in the balance of terms for the dissipation rate at the wall. In so far as the former problem is concerned, either physically inconsistent boundary conditions have been used or the boundary conditions for the dissipation rate have been tied to higher-order derivatives of the turbulent kinetic energy which leads to numerical stiffness. The K-omega model can alleviate these problems since the asymptotic behavior of omega is known in more detail and since its near wall balance involves only exact viscous terms. However, the modeled form of the omega equation that is used in the literature is incomplete-an exact viscous term is missing which causes the model to behave in an asymptotically inconsistent manner. By including this viscous term and by introducing new wall damping functions with improved asymptotic behavior, a new K-tau model (where tau is identical with 1/omega is turbulent time scale) is developed. It is demonstrated that this new model is computationally robust and yields improved predictions for turbulent boundary layers.

Distributions of Turbulence Quantities and Their Production, Diffusion and Dissipation in the Thermal Entrance Region of a Pipe.

Measurements have been made to investigate the mechanism of turbulent heat transport in the thermal entrance region of a pipe. To capture the interface between hot and ambient cold fluids, we have developed new detector functions based on the the time-derivative of temperature fluctuations. The instantaneous position of the interface fluctuates widely around the time-averaged position in conformity to a Gaussian probability distribution function. As a result, ambient cold fluids penetrate deeply into the near-wall hot region. From the analysis of both the developing processes and the energy budgets of temperature variance and radial turbulent heat flux, it is shown that peak values of the productions of these quantities become gradually larger as the thermal boundary layer develops, whereas their diffusive transports near the wall reach a fully developed state very rapidly. The distributions of the turbulent Prandtl number show no similarity throughout the entrance region ; furthermore, they are not constant within the thermal boundary layer. This means that the analogy between turbulent heat and momentum transfer, which is often assumed in conventional heat transfer analyses, does not hold in the thermal entrance region of a wall turbulent shear flow.

Unrolling of macromolecules under the conditions of wall turbulence

The results of polarization-optical studies of turbulent wall flow of a solution of polyethylene oxide are presented, showing the existence of strong deforming action of the hydrodynamic field on macromolecules in certain zones of the boundary layer.

Semiempirical theories of turbulent diffusion in an atmospheric surface layer

The equation for the mean concentration 〈c〉=C of a passive admixture in a turbulent flow is known not to be closed, since it includes, besides the unknown C and the mean velocity vector U=(U1, U2, U3), the turbulent fluxes of the admixture which are new unknowns. (These fluxes are given as Si=〈−u′ic′〉, i=1, 2, 3, where c′ and ui are the turbulent fluctuations of concentration and velocity components, and the angular brackets symbolize averaging.) Therefore, the equation for C cannot be solved without using some closure hypothesis. The simplest closure hypothesis is based on Boussinesq’s assumption of the proportionality of turbulent flux S=(S1, S2, S3) and the gradient of admixture concentration. According to Schmidt (1925) and Richardson (1926) this assumption can be written Si=Ki(∂C/∂xi), i=1, 2, 3. (1) Of course, (1) can be interpreted as the definition of new unknowns Ki instead of the turbulent fluxes Si. Therefore, without any supplementary hypotheses these equations are useless. Moreover, it is easy to see that Eqs. (1) contradict some known experimental facts which can be explained only if (1) are replaced by more general equations, Si=𝒥j=13 Kij (∂C)/(∂xij), i=1, 2, 3, (2) which include nine unknown eddy diffusivities. In the meteorological literature the off-diagonal components of the eddy diffusivity tensor Kij are usually neglected and the diagonal components are determined with the aid of crude, purely speculative hypotheses. A preferable approach that fits well with the modern development of semiempirical turbulence theory is based on the application of second-order closures for the equations of turbulence mechanics. Such closures supplement the exact equations for the mean fields by the equation for the second moments of turbulent fluctuations, where the new unknowns (third-order moments, correlations with pressure fluctuations) are approximated by simple functions of mean fields and second moments. It is easy to show that the so-called algebraic models of second-order closure, based on local equilibrium approximation (i.e., neglect of time variations and turbulent diffusion of the second moments), lead automatically to an equation of the form (2) and permit one to determine all the eddy diffusivities Kij. A number of different algebraic models of second-order closure can be found in the available literature and many of them require further modifications to become consistent with the experimental data for wall turbulent flows. However, the analysis of the algebraic models shows that different models consistent with the experimental data often lead to close values of turbulent diffusivities Kij. A survey of the results obtained from the evaluation of eddy diffusivities Kij for a nonstratified and a thermally stratified atmospheric surface layer with the aid of second-order closures will be reported. Some results of the study of turbulent diffusion by more general differential models of second-order closures will be briefly outlined. These results permit one to estimate approximately the accuracy of the assumptions (2).

Wave-number-frequency spectrum of turbulent wall pressure fluctuations

The study of the properties of turbulent wall flows is important to understanding the structural excitations that might induce flow noise. Models of the forcing functions of the structural excitations have been based on generalized properties of the spectral features of these flows. This paper will highlight the characteristics of turbulent spectral energy in wave number space, and present new experimental results. Discussion will address the wave number spectrum as a model for the forcing function of structural excitation. [Funded by ONR and IBM.]

A Treatment of wall boundaries for turbulent flows by the use of a transmission finite element method

A method to connect momentum Navier-Stokes equations with the universal law of the wall using the finite element method is developed for turbulent wall flows. This method is based on a domain decomposition of the fluid into subdomains near a solid boundary where the law of the wall is valid. A transmission formulation is introduced to match these regions and a new class of boundary finite element is used. This finite element takes into account the near-wall profile of the velocity and the transmission conditions. Computational results are presented for Poiseuille flow and flow over a backward-facing step.