Mixing Length Distributions Research Articles

Mixing-Length Distributions in the Near-Wall Region of Transpired Turbulent Boundary Layers

Mixing-Length Distributions in the Near-Wall Region of Transpired Turbulent Boundary Layers

Compressible Flow Ejectors: Part I—Development of a Finite-Difference Flow Model

A general method of calculating two-dimensional (plane and axisymmetric) mixing of a compressible jet in variable-area ducts has been developed. The method incorporates finite-difference approximations to the conservation equations, and is applicable to a wide range of Mach number, mass flow ratio, and initial conditions. The model was based on mixing length approximations deduced from boundary-layer and free-jet mixing for the upstream portion of the flow, and on a new mixing length distribution for the downstream zone which is entirely occupied by shear flow. The method has been tested and found satisfactory with existing data on boundary layer, constant-diameter duct entrance flow, constant-pressure jet mixing, and jet mixing in variable-area ducts. Part II of the paper describes tests of the method with new data from an experimental ejector study.

Prediction of Transpired Turbulent Boundary Layers

A model for the mixing length distribution near the wall in turbulent boundary layer flow with transpiration is presented. The model is based on a new formulation of the exponential damping function originally suggested by Van Driest. The analysis used to evaluate the damping function employs the same set of assumptions successfully used by several investigators in the past to develop the law of the wall with blowing. This mixing length model is then used with the calculation method previously developed by the author to solve the governing conservation equations of mass, momentum, and energy in partial differential form. Predicted velocity profiles, skin friction coefficients, and Stanton numbers are compared with experimental results taken over a wide range of transpiration for both incompressible and compressible flows.

A Unified View of the Law of the Wall Using Mixing-Length Theory

SummaryIt is shown that, if the well-known mixing-length formula is regarded simply as a relationship between the velocity and the stress distributions in the wall region of a turbulent flow, then a truly universal distribution of mixing length is sufficient to describe the experimentally observed departures of the velocity distribution from the usual law of the wall as a result of severe pressure gradients and transverse surface curvature. Comparisons have been made with a wide variety of experimental data to demonstrate the general validity of the mixing-length model in describing the flow close to a smooth wall.An extension of the re-laminarisation criterion of Patel and Head, and some experimental evidence, suggest that the thick axisymmetric boundary layer on a slender cylinder placed axially in a uniform stream cannot be maintained in a fully turbulent state for values of the Reynolds number, based on friction velocity and cylinder radius, below a certain critical value.

Eddy Viscosity Distributions in Compressible Turbulent Boundary Layers with Injection

SummaryShear stress, eddy viscosity and mixing length distributions have been obtained from measured boundary-layer developments over porous surfaces with air and carbon dioxide injection at Mach numbers up to M=3·6. It is found that, if the eddy viscosity is non-dimensionalised by dividing by the product of the free-stream velocity and the kinematic displacement thickness then this non-dimensional ratio is almost independent of injection ratio, but decreases slightly with Mach number.

Observational studies in the atmospheric boundary layer

AbstractWinds and temperatures in the boundary layer measured during two Australian expeditions are analysed according to the similarity scheme, with the use of four stability classifications.Under conditions of deep convection there is a minimum potential temperature and a maximum velocity component in the direction of the surface wind, at a height of about 0.12 u*/f. In the very stable cases, the temperature gradient follows rather closely a z−2 law for a considerable height range from 0.08 u*/f upwards. An expression closely fitting the mean data in stable conditions is suggested for the vertical temperature structure at all levels in the boundary layer.Wind data processed in this way show, for all four stability classes, a rudimentary Ekman spiral. With deep convection the spiral is found to be reversed in sense, but the flow in the convecting layer is down the gradient of pressure. If the convective limit lies broadly within the Ekman layer, a spiral of the expected sense is found. The upper limit of the Ekman layer (as defined by the ‘spiral’) is found to lie at a height of 0.17 to 0.24 (increasing with stability) in units of u*/f. Stress and heat‐flux are apparently considerable above this level, with sub‐geostrophic wind, when deep convection is occurring.Approximations to the universal distributions of stress, eddy coefficients, mixing length and rate of degradation of mean flow kinetic energy are computed for the various stabilities. The mixing length in unstable conditions increases almost as height up to a level of about 0.08 u*/f, and then decreases, but in general appears not to vanish in stable layers above the boundary layer. In the unstable boundary layer with deep convection, the eddy transfer coefficient for heat exceeds that for momentum up to 0.12 u*/f, where it becomes infinite, and is negative at higher levels. In stable conditions the transfer coefficients for a small sample of soundings were estimated to be closely similar.The universal functions of stability, A, B and C, which enable one to compute free atmosphere wind vector and temperature, given surface conditions, have been evaluated with moderate success, although B, which essentially describes the change of wind direction with height, exhibits excessive variability. A method is suggested for computing horizontal advection in the boundary layer when this is to be ‘parameterized’ in mathematical models.The drag coefficient, in terms of free atmosphere wind, has almost a 50‐fold range, due to stability variation only. Most of the variation occurs relatively close to neutral, so warning against too ready an assumption of neutrality in practical applications.It is suggested that, for modelling purposes, it is preferable to adopt boundary layer formulations which are not too sensitive to departures from ideal conditions, and eddy coefficients, perhaps based on mixing lengths, may well provide the best approach currently available.

On a finite-difference solution for the constant-property turbulent boundary layer.

A finite-differe nce method has been developed for the solutions to the governing partial differential equations for the constant-property turbulent boundary layer. Prandtl's mixinglength concept was used to express the apparent turbulent shearing stress according to a hypothesized mixing-length distribution through the boundary layer. The model for the mixing-length distribution is in general agreement with experimental data for a variety of flow conditions. The method, an always stable explicit finite-difference formulation that requires no iterative procedures, is numerically more direct than other methods recently proposed and is unique in its evaluation of the apparent shearing stress by using an assumed mixing-length distribution as the only empirical input. The predicted skin-friction coefficients and velocity profiles agree well with experimental data for the several comparisons made, which included flows in both favorable and adverse pressure gradients as well as flat plate cases with and without blowing. The numerical method presented is not restricted to use with the particular mixing-length model tentatively proposed in this work, but can be used to compare various models and help establish their properties and range of applicability, thereby serving to further understanding of the most fundamental aspects of turbulent flow. a c Cf gc I

Equilibrium-Turbulent Boundary Layer Prediction for a Proposed Prandtl Mixing Length Distribution

The momentum equation is solved numerically for a suggested ramp variation of the Prandtl mixing length across an equilibrium-turbulent boundary layer. The predictions of several important boundary-layer functions are compared with the equilibrium experimental data. Comparisons are also made with some recent universal recommendations for turbulent boundary layers since the equilibrium experimental data are limited. Good agreement is found between the predictions, the experimental data, and the recommendations.

Mixing length and kinematic eddy viscosity in a compressible boundary layer.

The effect of Mach number on the mixing lengths and kinematic eddy viscosities in the turbulent, flat-plate boundary layer has been evaluated. Using the generalized velocities suggested by Van Driest, it was possible to correlate measured boundary-layer profiles, for a range of Reynolds numbers and Mach numbers from 0 to 5, on a single curve. From this correlation, velocity profiles at desired Reynolds and Mach numbers were generated, and the boundary-layer equations of motion were then integrated, using these profiles, to obtain the local turbulent shear stress. The local mixing lengths and kinematic eddy viscosities were subsequently evaluated from the computed shear-stress distribution. The results of this study show that within the Mach number range of 0 to 5 the effect of compressibility on the normalized, turbulent shear stress and mixing length distribution is quite small, in keeping with Morkovin's hypothesis concerning the structure of compressible turbulent shear flow. The kinematic eddy viscosity displays a more pronounced sensitivity to Mach number, but a normalizing length scale can be found such that this sensitivity is all but eliminated.

Distribution of eddy viscosity and mixing length in smooth tubes

AbstractProfiles of eddy viscosity and Prandtl mixing length in fluids flowing steadily and isothermally in smooth tubes have been calculated from the velocity data of several investigators for Reynolds numbers between 1.2 × 103 and 3.2 × 106. In the transition range unusually high values of eddy viscosity and mixing length are obtained in some portions of the stream. In the fully turbulent range the effect of Reynolds number is small and the mixing length tends toward zero at the center of the tube. The parameters for turbulent flow between parallel plates have been correlated through the concept of an equivalent tube. The results are of importance in designing equipment for heat and mass transfer and mixing.

On the Velocity Distribution of Turbulent Flow in Pipes and Channels of Constant Cross Section

Abstract This paper follows the Prandtl conception of momentum transport and gives a critical examination of the so-called Prandtl-Nikuradse formula and the von Kármán formula for the velocity distribution of the turbulent flow in tubes or channels at large Reynolds number. It shows that both formulas would not give a good picture of the turbulent flow near the center of the conduit, and indeed they actually do not. A new formula for the velocity distribution is developed from a study of the mixing-length distribution across the section. This new formula checks quite well with the experiments and yields the same skin-friction formula as derived by von Kármán and Prandtl, which itself is in very good agreement with experiments.