Showing papers on "K-epsilon turbulence model published in 1971"

On the nature of turbulence

Abstract: A mechanism for the generation of turbulence and related phenomena in dissipative systems is proposed.

The production of turbulence near a smooth wall in a turbulent boundary layer

Abstract: The structure of the flat plate incompressible smooth-surface boundary layer in a low-speed water flow is examined using hydrogen-bubble measurements and also hot-wire measurements with dye visualization. Particular emphasis is placed on the details of the process of turbulence production near the wall. In the zone 0 < y+ < 100, the data show that essentially all turbulence production occurs during intermittent ‘bursting’ periods. ‘Bursts’ are described in some detail.The uncertainties in the bubble data are large, but they have the distinct advantage of providing velocity profiles as a function of time and the time sequences of events. These data show that the velocity profiles during bursting periods assume a shape which is qualitatively distinct from the well-known mean profiles. The observations are also used as the basis for a discussion of possible appropriate mathematical models for turbulence production.

Turbulence in an atmosphere with a non-uniform temperature

Abstract: In this work the effect of static stability on the development of atmospheric turbulence is investigated. This influence is considered quantitatively by generalizing Prandtl's semi-empirical theory, i.e., by using a correcting factor in the form of a universal function of the Richardson number. An evaluation of the thickness of the layer of dynamic turbulence under various external conditions is successfully achieved quantitatively.

Atmospheric Predictability and Two-Dimensional Turbulence

Abstract: Recent observations indicate that the planetary-scale motions of the atmosphere obey some of the laws of two-dimensional turbulence. The eddy-damped Markovian approximation to two-dimensional turbulence is applied to these motions to predict for an observed energy spectrum the nonlinear transfer rates, characteristic error spectra, and the rate of error growth. In this way estimates are derived of the predictability of the atmosphere and of the errors inherent in numerical models. The use of stochastic models for turbulence approximations is described in an Appendix.

An almost-Markovian Galilean-invariant turbulence model

Abstract: A model equation of Langevin type for the turbulent velocity field is constructed, in which the non-linear terms of the Navier–Stokes equation are replaced by a dynamical damping term and a random forcing term, with strength parameters determined by the past history of the energy spectrum The model leads to a closed set of first-order differential equations in time for the evolution of two functions: the energy spectrum and the effective memory times for the interaction of mode triads Invariance of the energy transfer to random Galilean transformation is achieved by using the interaction between solenoidal and compressive parts of a convected test field to determine the memory–time functions The model equation is developed from the direct-interaction approximation as starting-point At an intermediate stage, before the Galilean invariance is introduced, a model representation of Edwards's (1964) theory is obtained which extends the latter to statistically non-stationary states

Experiments on internal intermittency and fine-structure distribution functions in fully turbulent fluid

Abstract: Spatial ‘intermittency’ in the velocity field fine-structure of fully turbulent flow regions, first observed by Batchelor & Townsend (1949), is studied further here in grid-generated nearly isotropic turbulence and on the axis of a round jet. At large enough Reynolds numbers, appropriately filtered hot-wire anemometer signals appear intermittent as the turbulent patterns are convected past the hot wire by the mean flow. Measurements show that there is a decrease in the relative fluid volume (equal to the ‘intermittency factor’) occupied by fine-structure of given size as the turbulence Reynolds number is increased. They show also that, for a fixed Reynolds number, the relative volume is smaller for smaller fine-structure. The average linear dimension of the fine-structure regions turns out to be much larger than the sizes of fine-structure therein. At Rλ, = 110, for example, the ratio ranges from 15 to 30, decreasing with decreasing ‘eddy’ size. It appears to be approaching an asymptote with increasing Rλ.The flatness factors and probability distributions of the first derivative, the second derivative, band-passed and high-passed velocity fluctuation signals were also measured. The turbulence Reynolds numbers Rλ ranged from 12 to 830. The flatness factors of the first and the second derivatives increase monotonically with Rλ. Those of the second derivative vary with Rλ0.25 for Rλ 300. No indication of asymptotic constant values was observed for Rλ up to the order of one thousand.The probability distributions of velocity fluctuations and large-scale signals are nearly normal, while the small-scale signals are not. The flatness factor of the filtered band-pass velocity signal increases with increasing frequency.At the larger Reynolds numbers, the square of the signal associated with large wave-numbers may be approximated by a log-normal probability distribution for amplitudes when probabilities fall between 0·3 and 0·95, in limited agreement with the theory of Kolmogorov (1962), Oboukhov (1962), Gurvich & Yaglom (1967).

Structure of Turbulence in the Zeta Plasma

Abstract: The structure of turbulent electric and magnetic field fluctuations is described, and the internal dynamics of the associated velocity fluctuations are compared with those of fluid turbulence. It is suggested that the plasma turbulence is qualitatively similar to fluid turbulence but with the turbulent elements elongated along the mean magnetic field to form rolls. This suggests that an appropriate comparison might be with the hypothetical two‐dimensional limit of fluid turbulence, in which energy is expected to be transferred toward long wavelengths rather than to short wavelengths as for isotropic turbulence. In the plasma case, the direction of energy flow is inferred to be toward short wavelengths but the measured form of the triple correlation is close to that expected for two‐dimensional turbulence; we cannot, therefore, make a clear choice between the two alternatives. Effects arising from the particle structure of the plasma do not appear to be important, except that at low pressures an increased damping occurs which may be ion Landau damping. The source of the turbulent energy is not primarily convection due to the pressure gradient but involves some mechanism of direct coupling with the plasma current associated with the “excess resistance” of the discharge.

Flow in a Turbulent Trailing Vortex

Abstract: The structure of turbulent line vortices is examined. A general argument is constructed to show that the vortex must develop an overshoot of circulation if it entrains fluid at a rate greater than that due to molecular diffusion. A weak hypothesis on the distribution of Reynolds stress leads to the logarithmic profile of Hoffman and Joubert and an estimate of the maximum Reynolds stress. The results of a turbulence model due to Saffman are presented and shown to be poor.

Turbulent flow in wavy pipes.

Abstract: A primarily experimental investigation was undertaken to determine the internal structure of steady, quasi-uniform, non-separated, axisymmetric flows in circular pipes with sinusoidal wall profiles. The quantities measured include radial and longitudinal distributions of mean velocity, pressure, and total head; the Reynolds shear stress and all three components of turbulence velocity; and boundary shear stress and pressure. Two different wall-wave steepnesses were investigated, and a constant Reynolds number of 1·13 × 105 (based on the average pipe diameter) was maintained in most experiments. The boundary shear stress was found to be shifted upstream relative to the boundary wave, whereas the wall pressure is shifted slightly downstream. The turbulence measurements revealed that there is a central core extending over some 60% of the pipe radius in which the turbulence quantities are constant along the pipe. Near the boundary, however, the turbulence velocities and stress vary periodically along the boundary waves. The longitudinal component of mean velocity was found to be distributed radially according to the power law, but with an exponent that varies along each wave; a simple analytical model is constructed to predict the variation of the exponent. It was not found possible to relate the local boundary shear stress to just the local flow characteristics, since the convective or ‘history’ effects play a significant role in its determination. An empirical formula is derived relating the local boundary shear stress to the local velocity distribution and the first two derivatives of the boundary profile.

Flow in the wake of bluff-body flame stabilizers

Abstract: Experimental results for the flow around bluff bodies situated in a nozzle are reported in two parts: o (a) The influence of initial flow-boundary conditions on the characteristics of the mean flow field are given The spatial distributions of reverse mass-flow rate and of the normalized wake boundary (σ/σ o =0) are presented as functions of bluff body blockage ratios in the range 0 to 054 and of forebody geometry (b) The turbulence characteristics of the flow in the immediate wake of a “disc-in-nozzle” arrangement are given; for example, the spatial distributions of turbulent shear stress, turbulence intensities, and turbulence kinetic energy These were determined using a new method of interpreting anemometer responses, developed for use in highly turbulent flows When correlated with the mean flow characteristics, the turbulence measurements show that simple models of turbulence are inadequate for the evaluation of momentum exchange coefficients in highly turbulent flows The correlation between the spatial distributions of mean stream function and local kinetic energy of turbulence, both calculated from measurement data, shows good agreement with that predicted for a similar flow system using the Prandtl-Kolmogorov model of turbulence

Generalizations of the Richardson criterion for the onset of atmospheric turbulence

Abstract: The energetics of air-parcel interchange are examined for the case in which the atmospheric potential temperature θ and background wind U are continuous but otherwise arbitrarily varying in space, and in which the direction of U is free from constraint. Interchange along a single s-axis, inclined at angle α to the vertical, is shown to be energetically feasible if where g is the gravitational acceleration. This condition, which is in general far less stringent than that imposed by the standard Richardson criterion, is suggested as a condition portending the onset of anisotropic turbulence. An isotropic interchange throughout a sphere is shown to be energetically feasible if where the z co-ordinate of the Cartesian x, y, z system is directed upward. This condition, which is in general somewhat less stringent than that imposed by the elementary Richardson criterion, may well portend the occurrence of isotropic turbulence. An outer time scale for the turbulence is estimated in each case, and its relevance to the occurrence of turbulence is discussed for the case when the background parameters of the system change with time.

The effect of velocity sensitivity on temperature derivative statistics in isotropic turbulence

Abstract: The velocity sensitivity of a resistance-wire temperature sensor is expressed in terms of sensor parameters, and the resulting errors in temperature derivative moments in isotropic turbulence are evaluated. It is shown that velocity sensitivity of a degree completely negligible for most purposes causes severe contamination of the measured third moment. The contamination terms are shown to be production rates of the mean square temperature gradient and vorticity, respectively, and therefore create positive values of measured derivative skewness. The dominant contamination term is related to the temperature spectrum through the balance equation for the mean-square temperature gradient, and calculations based on an assumed spectral form show that under typical conditions the measured skewness is large. This mechanism could provide an alternative to anisotropy as an explanation of the positive skewnesses recently measured in the atmosphere.

The langevin model for turbulent diffusion

Abstract: The Lagrangian approach to turbulent diffusion is considered through an improved Langevin model approach. The improved model partially accounts for the interaction between pressure and viscous stresses, an interaction not included in previous studies. The model is used to obtain one and two particle dispersion relations and it permits calculation of Eulerian-Lagrangian scale relations. Turbulence Reynolds number and time span limits for Richardson's law of diffusion are obtained. Non-stationarity of the acceleration between two particles is discussed as it relates to Richardson's law and the conclusions are considered in view of both laboratory and geophysical fluid dynamical data.

Stochastic Properties of Turbulence Excited Rotor Blade Vibrations

Abstract: The developed analytical methods are applicable to the linearized equations of blade motion up to high rotor advance ratios. Rigid flapping blades with elastic flapping restraint are stipulated, only vertical turbulence components are considered and the ratio of turbulence scale length over rotor radius L/R is assumed to be large. On the basis of computed threshold crossing expectations, the blade response can be described as a quasi-coherent narrowband random process. The limiting case of infinite L/R is easy to compute and to interpret and yields conservative values for the mean square blade response. Numerical analysis for a slowed unloaded rotor based on low-altitude turbulence data indicates in the absence of gust alleviating device an appreciable probability of excessive blade flapping or flap-bending. w L cor , (S) -^subscripts

Sphere Drag in Solid Rockets—Non-Continuum and Turbulence Effects

Abstract: Momentum transfer between gas and condensed phase in metallized solid propellant rocket motors, measuring noncontinuum and turbulence effects on sphere drag

Kinematic-dynamo action under incompressible, isotropic velocity turbulence.

Abstract: Regenerative kinematic-dynamo action under incompressible isotropic velocity turbulence, noting turbulent Lorentz force role

An Implicit Numerical Solution of the Turbulent Three-Dimensional Incompressible Boundary Layer Equations.

Abstract: A method of solving the three-dimensional, incompressible turbulent boundary-layer equations was developed using a Crank-Nicholson implicit finite-difference technique, with the turbulent stress terms modeled with an eddy-viscosity model obtained from mixing length theory. The method was applied to two three-dimensional flow geometries for which experimental data exists and a comparison with this data showed excellent agreement. A complete computer program was sufficiently generalized for application to two-dimensional laminar and turbulent flows with arbitrary pressure gradients. The method was applied to several such test cases and the solutions agreed well with both theory and experiment. An analysis was presented to determine the conditions for which the finite difference equations were stable and convergent. (Author)

Burgers turbulence model for a channel flow

Abstract: The truncated Burgers models have a unique equilibrium state which is defined continuously for all the Reynolds numbers and attainable from a realizable class of initial disturbances. Hence, they represent a sequence of convergent approximations to the original (untruncated) Burgers problem. We have pointed out that consideration of certain degenerate equilibrium states can lead to the successive turbulence-turbulence transitions and finite-jump transitions that were suggested by Case & Chiu. As a prototype of the Navier–Stokes equations, Burgers model can simulate the initial-value type of numerical integration of the Fourier amplitude equations for a turbulent channel flow. Thus, the Burgers model dynamics display certain idiosyncrasies of the actual channel flow problem described by a truncated set of Fourier amplitude equations, which includes only a modest number of modes due to the limited capability of the computer at hand.

Decay of Homogeneous Turbulence from a Given Initial State

Abstract: The homogeneous turbulence problem is formulated by first specifying the multipoint velocity correlations or their spectral equivalents at an initial time. Those quantities, together with the correlation or spectral equations are then used to calculate initial time derivatives of correlations or spectra. The derivatives are in turn used in time series to calculate the evolution of turbulence quantities with time. When the problem is treated in this way, the correlation equations are closed by the initial specification of the turbulence, and no assumption is necessary for closing those equations. An exponential series, which is an iterative solution of the Navier‐Stokes equations, gave much better results than a Taylor power series when used with the limited available initial data. In general, the agreement between theory and experiment was good.

Inclusion of turbulence effects in numerical fluid dynamics

Abstract: To show this distinction, consider the flow of water past a circular cylinder For very slow speeds, the flow development is uniquely determined by the magnitude of the Reynolds number, Re E ud/9, in which u is the input flow speed, d is the diameter of the cylinder, and 9 is the kinematic viscosity of the water Since these are macroscopic parameters, this flow is called laminar As the speed increases, a double vortex develops behind the cylinder, and when the Reynolds number exceeds about 40, the double vortex begins to oscillate Except for an arbitrary phase, however, the oscillations are uniquely determined by the macroscopic parameters, u, d, and ~, and the flow is still completely laminar As the Reynolds number continues to increase, the changing pattern maintains its laminar property until a stage is reached in which noticeable effeots are observed that do not depend upon the macroscopic parameters These spurious features are the result of microscopic perturbations, which for lower Reynolds numbers could never amplify to significance They depend upon the details of cylinder roughness , slight mechanical vibrations and minor fluctuations of input water speed, and signify the onset of turbulence

Tests on Cylinders in Turbulent Flow

Abstract: The note describes the experimental investigation under way to study the effect of turbulence on the drag coefficient of cylinders at Lehigh University.