Showing papers on "K-epsilon turbulence model published in 1986"

Renormalization group analysis of turbulence I. Basic theory

Abstract: We develop the dynamic renormalization group (RNG) method for hydrodynamic turbulence. This procedure, which uses dynamic scaling and invariance together with iterated perturbation methods, allows us to evaluate transport coefficients and transport equations for the large-scale (slow) modes. The RNG theory, which does not include any experimentally adjustable parameters, gives the following numerical values for important constants of turbulent flows: Kolmogorov constant for the inertial-range spectrumCK=1.617; turbulent Prandtl number for high-Reynolds-number heat transferPt=0.7179; Batchelor constantBa=1.161; and skewness factor¯S3=0.4878. A differentialK-\(\bar \varepsilon \) model is derived, which, in the high-Reynolds-number regions of the flow, gives the algebraic relationv=0.0837 K2/\(\bar \varepsilon \), decay of isotropic turbulence asK=O(t−1.3307), and the von Karman constantκ=0.372. A differential transport model, based on differential relations betweenK,\(\bar \varepsilon \), andν, is derived that is not divergent whenK→ 0 and\(\bar \varepsilon \) is finite. This latter model is particularly useful near walls.

Renormalization-group analysis of turbulence.

Abstract: Using renormalization-group methods and the postulated equivalence between the inertial-range structures of turbulent flows satisfying initial and boundary conditions and of flows driven by a random force, we evaluate the Kolmogorov constant (1.617) and Batchelor constant (1.161), skewness factor (0.4878), power-law exponent (1.3307) for the decay of homogeneous turbulence, turbulent Prandtl number (0.7179), and von K\'arm\'an constant (0.372). This renormalization-group technique has also been used to derive turbulent transport models.

Self-accelerating turbidity currents

Abstract: Approximate layer-averaged equations describing the mechanics of turbid underflows are derived. Closure of the equations describing the balance of fluid mass, sediment mass, and mean flow momentum provides for the delineation of a three-equation model. A description of sediment exchange with the bed allows for the possibility of a self-accelerating turbidity current in which sediment entrainment from the bed is linked to flow velocity. A consideration of the balance of the mean energy of the turbulence yields a constraint on physically realistic solutions to the three-equation model. It is shown that the self-acceleration predicted by the three-equation model is so strong that the energy constraint fails to be satisfied. In particular, the turbulent energy consumed in entraining new bed sediment exceeds the supply of energy to the turbulence, so that the turbulence, and thus the turbidity current, must die. The problem is rectified by the formulation of a four-equation model, in which an explicit accounting is made of the mean energy of the turbulence. Sediment entrainment from the bed is linked to the level of turbulence in the four-equation model. Self-acceleration is again predicted, although it is somewhat subdued compared with that predicted by the three-equation model. The predictions of both models are summarized over a wide range of conditions.

The fractal facets of turbulence

Abstract: Speculations abound that several facets of fully developed turbulent flows are fractals. Although the earlier leading work of Mandelbrot (1974, 1975) suggests that these speculations, initiated largely by himself, are plausible, no effort has yet been made to put them on firmer ground by, resorting to actual measurements in turbulent shear flows. This work is an attempt at filling this gap. In particular, we examine the following questions: (a) Is the turbulent/non-turbulent interface a self-similar fractal, and (if so) what is its fractal dimension ? Does this quantity differ from one class of flows to another? (b) Are constant-property surfaces (such as the iso-velocity and iso-concentration surfaces) in fully developed flows fractals? What are their fractal dimensions? (c) Do dissipative structures in fully developed turbulence form a fractal set? What is the fractal dimension of this set? Answers to these questions (and others to be less fully discussed here) are interesting because they bring the theory of fractals closer to application to turbulence and shed new light on some classical problems in turbulence - for example, the growth of material lines in a turbulent environment. The other feature of this work is that it tries to quantify the seemingly complicated geometric aspects of turbulent flows, a feature that has not received its proper share of attention. The overwhelming conclusion of this work is that several aspects of turbulence can be described roughly by fractals, and that their fractal dimensions can be measured. However, it is not clear how (or whether), given the dimensions for several of its facets, one can solve (up to a useful accuracy) the inverse problem of reconstructing the original set (that is, the turbulent flow itself).

Multiscale model for turbulent flows

Abstract: A model is devised for computing general turbulent flows. The model is an improvement over two-equation turbulence models in a critically important manner, that is, the new model accounts for disalignment of the Reynolds-stress-tensor and the mean-strain-rate-tensor principal axes. The improved representation of the Reynolds-stress tensor has been accomplished through the introduction of a multiscale description of the turbulence, i.e., two energy scales are used corresponding to upper and lower partitions of the turbulence energy spectrum. A novel feature of the formulation is that the differential equation for the Reynolds-stress tensor is of first order, which in effect corresponds to what can be termed an "algebraic stress model" with convective terms. As a consequence of its mathematical simplicity, the model is very efficient and easy to implement computationally. The model is applied to a wide range of turbulent flows including homogeneous turbulence, compressible and incompressible two-dimensional boundary layers, and unsteady boundary layers including periodic separation and reattachment. Comparisons with corresponding experimental data show that the model reproducesall salient features of the flows considered/Perturbation analysis of the viscous sublayer shows that integration through the sublayer can be accomplished with no special viscous modifications to the closure coefficients appearing in the model.

A generalized Langevin model for turbulent flows

Abstract: A Langevin model appropriate to constant property turbulent flows is developed from the general equation for the fluid particle velocity increment proposed by Pope in an earlier paper [Phys. Fluids 26, 404 (1983)]. This model can be viewed as an analogy between the turbulent velocity of a fluid particle and the velocity of a particle undergoing Brownian motion. It is consistent with Kolmogorov’s inertial range scaling, satisfies realizability, and is consistent with second‐order closure models. The objective of the present work is to determine the form of a second‐order tensor appearing in the general model equation as a function of local mean quantities. While the model is not restricted to homogeneous turbulence, the second‐order tensor is evaluated by considering the evolution of the Reynolds stresses in homogeneous flows. A functional form for the tensor is chosen that is linear in the normalized anisotropy tensor and in the mean velocity gradients. The resulting coefficients are evaluated by matching the modeled Reynolds stress evolution to experimental data in homogeneous flows. Constraints are applied to ensure consistency with rapid distortion theory and to satisfy a consistency condition in the limit of two‐dimensional turbulence. A set of coefficients is presented for which the model yields good agreement with available data in homogeneous flows.

Scrutinizing the k-ε Turbulence Model Under Adverse Pressure Gradient Conditions

Abstract: The k-e model and a one-equation model have been used to predict adverse pressure gradient boundary layers. While the one-equation model gives generally good results, the k-e model reveals systematic discrepancies, e.g. too high skin friction coefficients, for these relatively simple flows. These shortcomings are examined and it is shown by an analytical analysis for the log-law region that the generation term of the e-equation has to be increased to conform with experimental evidence under adverse pressure gradient conditions. A corresponding modification to the e-equation emphasizing the generation rate due to deceleration was employed in the present investigation and resulted in improved predictions for both moderately and strongly decelerated flows.

Transonic, turbulent boundary-layer separation generated on an axisymmetric flow model

Abstract: Experimental data describing the transonic, turbulent, separated flow generated by an axisymmetric flow model are presented. The model consisted of a circular-arc bump affixed to a straight, circular cylinder aligned with the flow direction. Measurements of the mean velocity, turbulence intensity, and Reynolds shear-stress profiles were made in the separated flow. These data revealed dramatic changes in the shear-stress levels as the flow passed through the interaction to reattachment. Behavior of the turbulence reaction to the imposed pressure gradients was examined in terms of the mixing length and the excursions of the turbulence from equilibrium.

Calculation of swirling jets with a Reynolds stress closure

Abstract: The transport equations for the Reynolds stresses are closed by modeling the turbulence and mean‐strain parts of the pressure‐strain‐rate correlation. The model constants are determined from simple relationships deduced from measurements in rectilinear and longitudinally curved shear flows. It is found that the effects of complex strain fields are more correctly predicted when the influence of the mean‐strain part is reduced from levels indicated by rapid distortion theory, and the turbulence part is adjusted to conform approximately with the measured rates of return to isotropy. The case of the swirling jet is used to illustrate the improved performance of the model.

Turbulence and random processes in fluid mechanics

Abstract: Fluid flow turbulence is a phenomenon of great importance in many fields of engineering and science. Turbulence and related areas have continued to be subjects of intensive research over the last century. In this second edition of their successful textbook Professors Landahl and Mollo-Christensen have taken the opportunity to include recent developments in the field of chaos and its applications to turbulent flow. This timely update continues the original theme of the book: presenting the fundamental concepts and basic methods of fluid flow turbulence which enable the reader to follow the literature and understand current research. The emphasis upon the dynamic processes that create and maintain turbulent flows gives this book an original approach. This book should be useful to graduate students and researchers in fluid dynamics and, in particular, turbulence and related fields.

The limiting behaviour of turbulence near a wall

Abstract: Three different Navier-Stokes computational models of incompressible viscoussublayer turbulence have been developed. Comparison of computed turbulence quantities with experiment is made for the mean streamwise velocity, Reynolds stress, correlation coefficient and dissipation; for the r.m.s. fluctuation intensities of streamwise vorticity, Reynolds stress and three velocity components; and for the skewness and flatness of fluctuating streamwise velocity and Reynolds stress. The comparison is good for the first three of these quantities, and reasonably good for most of the remainder.Special computer runs with a very fine mesh and small Courant number were made to define the limiting power-law behaviour of turbulence near a wall. Such behaviour was found to be confined to about 0.3 wall units from the wall, and to be: linear for streamwise turbulence, spanwise turbulence, vorticity normal to the wall, and for the departures from their respective wall values of dissipation, streamwise vorticity and spanwise vorticity; second power for turbulence normal to the wall; third power for Reynolds stress; and a constant value of the correlation coefficient for Reynolds stress. A simple physical explanation is given for the third-power variation of Reynolds stress and for the broad generality of this limiting variation.Applications are made to Reynolds-average turbulence modelling: damping functions for Reynolds stress in eddy-viscosity models are derived that are compatible with the near-wall limiting behaviour; and new wall boundary conditions for dissipation in k-e models are developed that are similarly compatible.

Statistical Predictability of Decaying Turbulence.

Abstract: We use statistical models of turbulence with “eddy damping” (EDQNM) in order to study the problem of predictability of freely evolving two- and three-dimensional isotropic turbulent flows. The application of statistical theories to this problem necessitates taking into account long-range interactions between very different scales (“nonlocal” interactions) intervening in the evolution of the error spectrum. We have therefore developed an analytical and numerical modeling of the nonlocal interactions enabling us to ensure the “realizability” of the error spectrum. First, we validate our numerical codes by retrieving, in the case of a stationary turbulence, the results of Leith and Kraichnan. Second, the calculations carried out in the case of freely evolving three-dimensional and two-dimensional turbulence allow for the determination of temporal evolution laws of quantities characterizing the inverse error cascade. Various inertial ranges are displayed for the spectra and some analogies with the pa...

Turbulence Modeling of Flood Plain Flows

Abstract: A numerical model capable of predicting flow characteristics in a compound channel is described. The model solves the continuity and momentum equations along with the transport equations of kinetic energy of turbulence and the dissipation rate. Closure is achieved with the aid of algebraic relations for turbulence stresses. The model is capable of treating compound channels formed by regular geometrical sections of main channel and flood plain segments. The width of the main channel, the width of the total section, the depth of flow in flood plain, the total depth, channel slope and boundary roughness of main channel section and flood plain section can all be varied. The model predictions of total flow rate, shear stress distributions around the wetted perimeter, and the percentage of flow and shear force carried by the different sections were compared with published experimental data. Reasonable agreement between data and predictions was obtained.

Effects of a radial electric field on tokamak edge turbulence

Abstract: Turbulence associated with sheared radial electric fields such as those arising in tokamak edge plasmas is investigated analytically Two driving mechanisms are considered: in the region of maximum vorticity (maximum electric field shear), the electric field is the dominant driving mechanism Away from the maximum, turbulence is driven by the density gradient In the latter case, previous work is extended to include the effects of the electric field on the spatial scales of density correlation in the frequency‐Doppler‐shifted, density‐gradient‐driven turbulence For radial‐electric‐field‐driven turbulence, the effects of magnetic shear on linear instability and on fully developed turbulence are examined In the case of weak magnetic shear, saturation occurs through an enstrophy cascade process which couples regions of driving and dissipation in wavenumber space For stronger magnetic shear, such that the width of the dissipation region resulting from parallel resistivity is comparable to the radial electric field scale length, saturation occurs through nonlinear broadening of the mode structure, which pushes enstrophy into the region of dissipation Estimates of mode widths, fluctuation levels, and scalings are obtained for both mechanisms Comparison is made with the results of fluctuation measurements in the TEXT tokamak [Phys Fluids 27, 2956 (1984)]

Measurements of mean velocity and turbulent intensities in a free isothermal swirling jet

Abstract: The objectives of the present measurements are to provide data against which results of numerical prediction procedures can be compared, and quantitative information on the behavior of all turbulent stresses and their correlation to spatial distribution of mean velocity gradients, with a view to improving current understanding of relevant transport processes and to guiding turbulence modeling and prediction efforts of such flows. A laser velocimeter, operating in a constant time interval sampling mode, was used to obtain the measurements. Measured values are presented for the case of strong swirling velocities (with recirculating region). The location and extent of the recirculation region is established. Contours of turbulent kinetic energy, derived from measured data, locate the high-turbulence intensity zones (i.e., zones of intense mixing). Experimental data indicate a strong dependence of the turbulent stresses on the local strain of the mean flow in most regions of the flow, which suggests that an eddy viscosity type of turbulence model, e.g., the k-epsilon model, rather than a Reynolds stress model could be acceptable for the prediction of such flows. 22 references.

Near-field turbulence properties of single- and two-stream plane mixing layers

Abstract: Detailed mean flow and turbulence measurements have been made in the near-field of two plane mixing layers in air with a maximum velocity of 21 m/s. The experimental rig enabled mixing layers of velocity ratios 0 and 0.46 to be generated simultaneously. Cases with both tripped and untripped initial boundary layers were studied. In all cases, it was found that the two-stream layer developed to the self-preserving state in a distance much shorter than the single-stream layer, which followed accepted criteria for the development distance. The asymptotic levels of the turbulence quantities in the two-stream layer and the development of the single-stream layer showed agreement with existing data. The results suggest that the two-stream mixing layer should provide a better test case for the development of turbulence models and calculation methods than the single-stream mixing layer.

Turbulent flow in stirred tanks. Part II: A two‐scale model of turbulence

Abstract: The two-scale turbulence concept is recommended for modeling the turbulence in a baffled vessel equipped with a Rushton-type turbine impeller. A three-equation isotropic turbulence model is proposed that employs the balance equations for: the kinetic energy of the large scale vortices; the kinetic energy of the inertial subrange eddies; and the dissipation rate of the small-scale turbulence. The energy transfer rate from the large-scale vortices is prescribed algebraically. Flow patterns are modeled by solving the transport equations for vorticity, stream function, and tangential momentum. The Reynolds stresses are modeled by means of the effective viscosity, based on the three-equation model of turbulence. The calculated profiles of the mean velocity at the tank wall agree with experimental data obtained in the same system by means of a Pitot tube.

Navier-Stokes computations of transonic flows with a two-equation turbulence model

Abstract: A thin-layer Navier-Stokes code has been used to compute the turbulent flow over two axisymmetric bodies at transonic speeds. A critical element of calculating such flows is the turbulence model. Numerical computations have been made with an algebraic eddy viscosity model and the k — e two-equation model. The k — e equations are developed in a general spatial coordinate system and incorporated into the thin-layer, compressible, time-dependent Navier-Stokes code. The same implicit algorithm that simultaneously solves the Reynolds-averaged mean flow equations is extended to solve the turbulence field equations using block tridiagonal matrix inversions. Calculations with the k — e model are extended up to the wall. Numerical solutions have been obtained for two transonic flow situations to determine the accuracy of the k — e model. First, the attached flow over an axisymmetric projectile has been investigated. The second flow situation considered is the transonic separated flow over an axisymmetric bump model. Furthermore, the accuracy and applicability of the k — e model are determined by comparing the computed results with experimental data.

Is Navier-Stokes turbulence chaotic?

Abstract: Whether turbulent solutions of the Navier–Stokes equations are chaotic is considered. Initially neighboring solutions for a low‐Reynolds‐number fully developed turbulence are compared. The turbulence is sustained by a nonrandom time‐independent external force. The solutions separate exponentially with time, having a positive Liapunov characteristic exponent. Thus the turbulence is characterized as chaotic.

Numerical simulation of turbulent heat flows

Abstract: Previous work on simulating turbulence in finite beta tokamak plasmas with isothermal models is extended to include temperature perturbations and temperature gradients in a collisional plasma. Turbulent heat as well as plasma flows are treated. The toroidal ion temperature gradient mode is the dominant source of turbulence in the model. Electron temperature gradients can stabilize the electron collisional drift mode and lead to reverse plasma flows (or thermal pinches). Magnetic electron heat conduction is found to be very small when correlations between temperature and magnetic field perturbations are considered. Quasilinear theory and mixing length rules are found to give a good representation of the results.

Analysis of the Relation Between Doppler Spectral Width and Thunderstorm Turbulence

Abstract: A technique to separate ordered flow in a tornadic thunderstorm from the random velocities associated with turbulence is described. The relative importance of the ordered flow shear and turbulence in the broadening of the Doppler velocity spectrum is evaluated by least-squares fitting an assumed linear model of radial velocities to measured ones over an angular analysis domain (about 3° in azimuth and 3° in elevation). Fields and cumulative probabilities of Doppler spectral widths associated with turbulence and velocity shear, of root-mean-square (rms) velocity residuals, and of the turbulent kinetic energy dissipation rates ϵ are presented. In order to estimate ϵ from measurements of Doppler spectral width, the outer scale of the inertial subrange of turbulence must be at least three times larger than the size of the radar's resolution volume. Wind fields synthesized from the Doppler data of two radars are related to the turbulent kinetic energy dissipation rate and rms velocity residual fields....

Limitations of the near-wall k-epsilon turbulence model

Abstract: Comparisons are given at two different Reynolds numbers between the measured turbulent kinetic energy in channel flow and the predictions of several near-wall variants of the k-e closure. Included among these are new calculations based on a form of the e equation that is consistent with the physically correct boundary condition. It is observed that all of the approaches fail to account for the large peak k value in the wall region that is evident in the experimental data. By contrasting the computed energy budget with its measured values it is shown that this defect may be attributed to a fundamental inconsistency in the commonly used model for the pressure diffu- sion term in the k equation near the boundary. HE k-e closure is widely utilized today in the prediction of the mean properties of turbulent flowfields. In recent years, extensions of its basic form have been developed to permit treatment of complex fluid motions, such as those found in curved pipes,1 compressible boundary layers,2 com- bustion chambers,3 etc. A strong incentive for the continued development and application of the k-e model has been the remarkable success it has achieved in the prediction of mean velocity fields in wall-bounded shear flows, such as occur in channels, pipes, and boundary layers.4"6 The success of the k-e closure in these relatively simple flows is not complete, however, since the high accuracy obtained for mean velocity is not duplicated in the case of the turbulent kinetic energy, k. A notable defect4'7"9 is the 25% or more underprediction of the large peak k value shown by Clark 10 and Kreplin and Eckelmann11 to occur at y+ «15 in a channel flow. For ap- plications where predictions of turbulence levels are impor- tant, such as in internal combustion engine simulations,12 this failure can be of great practical significance. It may be expected that major errors in the prediction of k stem from deficiencies in the low turbulent Reynolds number forms of the k-e equations which have been developed for the region adjacent to boundaries. Jones and Launder4 for- mulated the first such model that permitted k and e to be calculated down to a solid wall without the use of wall func- tions. Several more recent studies, including those of Lam and Bremhorst,7 Hassid and Poreh,8 and Chien,9 have at- tempted to improve upon the Jones and Launder approach, but these authors report only modest gains in the accuracy of k. The object of the present study is to elucidate the cause of this discrepancy by making direct comparisons between the terms in the exact and modeled energy equations. To ac- complish this, a set of new calculations was performed in which e was required to satisfy the physically correct bound- ary condition at the wall. It was revealed that the failure of the k-e model to predict the correct peak k value is most likely due to reliance on a flawed model for the pressure- velocity correlation in the k equation. This relation will be shown to be inconsistent with the experimentally measured values of the exact term as found by Kreplin and Eckelmann,11 and may be directly implicated in suppressing the peak k value. Removal of this limitation should allow for better predictions of k near solid boundaries.

Numerical Simulation of the Effects of Coriolis Force on the Structure of Turbulence : 2nd Report, Structure of Turbulence

Abstract: Characteristics of the structure of a turbulent flow in a rotating channel have been investigated numerically using Large-eddy Simulation (LES). The authors previous paper reported successful prediction of the global effect of Coriolis force by simulation. In this paper, attention is focused on the behaviour of large scale eddies and the mechanism of production of turbulence under the action of Coriolis force is discussed. LES is confirmed to be useful to get experimentally. It is suggested that, in the Coriolis force field, the so-called sweep and ejection, which make a positive contribution to the production of Reynolds stress and turbulent energy, are intensified on the pressure side. Contrarily, on the stabilized side, the streak structure, which is closely associated with the production of turbulence, is observed to disappear.

Modelling a recirculating density-driven turbulent flow

Abstract: A mathematical model of turbulent density-driven flows is presented and is solved numerically. A form of the k–ϵ turbulence model is used to characterize the turbulent transport, and both this non-linear model and a sediment transport equation are coupled with the mean-flow fluid motion equations. A partitioned, Newton–Raphson-based solution scheme is used to effect a solution. The model is applied to the study of flow through a circular secondary sedimentation basin.

Turbulence Effect on Flow Around Circular Cylinder

Abstract: Free-stream turbulence has significant effects on the flow around a circular cylinder. Wind tunnel experiments were conducted on a smooth circular cylinder to investigate the effect of free-stream turbulence, particularly small-scale turbulence, on the flow past the cylinder. Tests were carried out in smooth flow and in a rod-generated turbulent flow. The results showed that increase in free-stream turbulence promotes early transition from laminar to turbulent boundary layer flow and a delay in flow separation. These changes resulted in a modification of the mean pressure distribution and a reduction in drag. It is suggested that only the small-scale turbulence, of an appropriate length, near the stagnation line of the cylinder is required to produce these effects of free-stream turbulence.