Three-parameter Model Of Turbulence Research Articles

Three-Parameter Model of Turbulence for High-Velocity Flows

A formulation of a modernized version of the three-parameter model of turbulence developed earlier by the present authors for calculating high-velocity flows is presented in the paper. The modernized model of turbulence allows one to calculate both free and near-wall flows. The distinctive features of the model are the application of a separate equation for velocity fluctuations directed normally to streamlines, as well as of special functions of the gradient Mach number to account for compressibility. The model described was verified on a number of test problems. The results obtained agree well with experimental data.

The wind-field structure in a stably stratified atmospheric boundary layer over a rough surface

Both wind turning with height and ageostrophic flow in a stably stratified atmospheric boundary layer are analyzed using a three-parameter turbulence model. For a quasi-steady state of the boundary layer, the cross-isobaric flow is determined only by turbulent stress at the surface in the direction of geostrophic wind. The “operative” prediction models, in which the first-order turbulence closure schemes are used, tend to overestimate the boundary-layer depth and underestimate the angle between the surface and geostrophic winds when compared to “research” models (schemes of high-level turbulence closure). The true value of the angle between the surface and geostrophic winds is significant for the presentation of a large-scale flow. A nocturnal low-level jet is a mesoscale phenomenon reflected in data obtained from measurements in a stably stratified atmospheric boundary layer. It is found that such jets are of great importance in transporting humidity, momentum, and air pollution. In this study, the difference between jet flows over a homogeneous underlying surface and over a spatially localized large-scale aerodynamic roughness is shown.

Modeling Of A Stable Boundary Layer With A Low-Level Jet

An improved three-parameter model of turbulence is applied to study the characteristics of a flow of a stably stratified boundary layer with formation of a jet in its lower part. The results obtained show that the turbulent Prandtl number increases with increasing thermal stability, which agrees with the results of laboratory experiments and actual observations in the atmosphere. It is also shown that the gradient Richardson number is nonmonotonic in time in the range below the maximum jet velocity.

On the turbulent prandtl number in a stably stratified atmospheric boundary layer

The results obtained from both atmospheric and laboratory measurements and from LES data show that, in the stably stratified flows of the atmospheric boundary layer, turbulent mixing occurs at gradient Richardson numbers Ri g that significantly exceed one: the inverse turbulent Prandtl number Pr −1 decreases with an increase in the thermal flow stability. The decreasing trend of the inverse turbulent Ptandtl number is reproduced in a stably stratified atmospheric boundary layer in agreement with measurement data with the aid of an improved three-parameter turbulence model. In this model, a modified model that takes into account the effect of stratification in the expression for the time scale of the scalar field is used for the pressure-scalar correlation.

Modeling of the vertical structure of the nocturnal boundary layer over a rough surface

The results of numerical simulation of some features of the structure of the nocturnal atmospheric boundary layer over the urbanized surface are presented. The mesoscale model includes improved nonlocal parametrization of turbulence developed based on the three-parameter model of turbulence. The turbulent Prandtl number appears to be stably dependent on the gradient Richardson number, and the vertical turbulent heat transport is contragradient. It was shown that the structural features of an “upside-down” boundary layer are detected in the stable atmospheric boundary layer.

Compressible turbulent boundary layer on a permeable plate with injection of foreign gas

A numerical solution is obtained for equations of motion, diffusion, and energy using a three-parameter model of turbulence for boundary layer with different thermal boundary conditions on a permeable plate with injection of foreign gas. For an ideal gas with constant values of Prandtl and Schmidt numbers, the numerical solution is compared with limiting dependences for dimensionless temperature and enthalpy of gas on the wall. For helium/xenon mixtures with a low value of Prandtl number, the effect of intensity of injection and of the values of Reynolds and Mach numbers on the integral and local characteristics of flow and heat and mass transfer is investigated.

Three-parameter model of turbulence for the atmospheric boundary layer over an urbanized surface

A modified three-parameter model of turbulence for a thermally stratified atmospheric boundary layer (ABL) is presented. The model is based on tensor-invariant parametrizations for the pressure-strain and pressure-temperature correlations that are more complete than the parametrizations used in the Mellor-Yamada model of level 3.0. The turbulent momentum and heat fluxes are calculated with explicit algebraic models obtained with the aid of symbol algebra from the transport equations for momentum and heat fluxes in the approximation of weakly equilibrium turbulence. The turbulent transport of heat and momentum fluxes is assumed to be negligibly small in this approximation. The three-parameter E − e − model of thermally stratified turbulence is employed to obtain closed-form algebraic expressions for the fluxes. A computational test of a 24-h ABL evolution is implemented for an idealized two-dimensional region. Comparison of the computed results with the available observational data and other numerical models shows that the proposed model is able to reproduce both the most important structural features of the turbulence in an urban canopy layer near the urbanized ABL surface and the effect of urban roughness on a global structure of the fields of wind and temperature over a city. The results of the computational test for the new model indicate that the motion of air in the urban canopy layer is strongly influenced by mechanical factors (buildings) and thermal stratification.

Momentum and Heat Transfer in a Buoyant Turbulent Swirling Jet

Solutions are presented for a three-parameter model of turbulence for axisymmetric turbulent buoyant swirling jet flows in uniform and stagnant surroundings. The exit swirl number has a significant effect on the flow characteristics. The predicted results are in good agreement with experimental data. A discussion is provided for the effects of swirl and buoyancy on the growth of the jet width, the axial decay of mean flow properties, and the mass flow rates.

Transition to Turbulence in the Boundary Layer on a Plate in the Presence of a Negative Free-Stream Pressure Gradient

Transition to turbulence in the boundary layer on a flat plate is investigated numerically for an incompressible fluid flow with a given negative free-stream pressure gradient. The transition is investigated using the three-parameter turbulence model developed by the authors. The calculation results are compared with the available experimental data.

Numerical Study of Mixed Convection in Vertical Tubes under Conditions of Unstable Stratification

The present numerical study was performed using a three-parameter turbulence model supplemented by the thermogravitational terms and the transport equation for the transverse turbulent heat flux. The results obtained are compared with the few available experimental data for downward water and air flows through vertical heated tubes and with calculations for upward flow.

Loss of Specific Impulse Due to Friction in Spike Nozzles

The class of nozzles with a central body, so-called spike nozzles, is considered for axisymmetric and plane central body geometries. A method of constructing the nozzle contour is outlined. The boundary layer is calculated using a three-parameter turbulence model and the loss of specific impulse due to friction in both spike nozzles and a Laval nozzle with the same expansion ratio are determined. A comparative analysis of the calculation results obtained, which makes it possible to determine the advantages and limitations of the nozzles considered, is carried out.

Turbulent Transition in the Inlet Region of a Circular Pipe

Calculated and experimental data on turbulent transition in a circular pipe are analyzed. The calculations were performed using the three-parameter turbulence model. The dependence of the distance from the inlet to the point of minimum friction during transition on the Reynolds number for fixed inlet conditions and the distribution of the turbulence parameters over the pipe length and radius are obtained. The dependence of the maximum (critical) Reynolds number, Re*, for which there is no transition in the pipe, on the inlet intensity and scale of turbulence is found. It is suggested that Re* depends on the inlet perturbation parameters up to Re* = 1000, where the friction coefficients for laminar and turbulent flows coincide.

Tangential Slot Film Cooling in a Nozzle. Comparison of Calculation and Experiment

To test a method of calculating the boundary layer in liquid rocket engine (LRE) nozzles, developed by the authors and based on a differential three-parameter model of turbulence, nozzle flow with tangential-slot coolant injection in subsonic part of the nozzle is studied numerically. The effect of nozzle flow acceleration on the turbulence parameters and the dependence of the gas film effectiveness on the nozzle-inlet turbulence intensity are investigated. The numerical results are compared with available experimental data.

Turbulent flow in a channel with permeable walls. Direct numerical simulation and results of three-parameter model

Steady turbulent viscous incompressible fluid flow in a plane channel is calculated for the case of uniform blowing and suction through opposite walls. There are no experimental data for flows of this type. The flows were calculated by two methods: a direct numerical simulation method and using a three-parameter turbulence model. Direct numerical simulation was carried out using the same (apart from the boundary conditions) algorithm for numerical solution of the Navier-Stokes equations as that used earlier for calculating flows in pipes and channels with impermeable walls. In the second group of calculations the version of the model published in 1978 was used. The results obtained by the two methods are in good agreement. The difference is within the spread of the experimental data used for determining the parameters of the model. The agreement obtained makes it possible to assert that the turbulence direct numerical simulation algorithm developed can be used for the analysis of flows with quite different boundary conditions, including cases where there are no corresponding experimental data.

Supersonic boundary layer on a plate. Comparison of calculation and experiment

For verifying the method of calculating the boundary layer in liquid rocket engine (LRE) nozzles developed by the authors on the basis of a differential three-parameter turbulence model, the boundary layer on a plate in a zero-gradient flow is calculated. Over a wide range of variation of the free-stream Mach number, the temperature factor, and the Reynolds number, based on the momentum thickness of the boundary layer, the calculation agrees satisfactorily with the known experimental data, with respect to both integral and local flow and heat transfer characteristics.

Tangential Slot Gas Film Cooling. Comparison of Calculation and Experiment

A method of calculating of the boundary layer in liquid rocket engine (LRE) nozzles, developed by the authors and based on a differential three-parameter turbulence model, is tested using the known experimental data for the boundary layer on a plate in zero-pressure-gradient flow with tangential gas injection. Over a wide range of both bulk flow and injected gas parameters and for variable slot geometry, the calculations for both integral and local flow and heat transfer characteristics satisfactorily agree with experiment. The study has made it possible to obtain the governing parameters which significantly affect the gas film effectiveness.

Concerning the existence of wall terms in a three-parameter model of turbulence

We show the existence of a wall term in the equation for «u1u2» in a three-parameter model of turbulence by analogy with the known terms in an “E-e” model. We suggest a convenient system of differential equations for the boundary layer on a flat plate in fluid flow at small Reynolds numberse. We find the limiting calculated value of the turbulent Reynolds number at which wall terms must be taken into consideration.

Differential model of turbulence: A numerical study of mixed convection in vertical pipes

A three-parameter model of turbulence with the transport equation for the transverse turbulent heat flux is complemented by terms taking thermogravity effects into account. The results of a numerical study carried out without using the Boussinesq approximation are compared with known experimental data on rising air flows in vertical heated pipes.

Three-parameter turbulence model: Numerical investigation of the boundary layer in a nozzle with gas-film wall cooling

One of the principal problems in designing powerful liquid-fuel rocket engines (LRE) is how to protect the nozzle walls from heat fluxes, which in the region of the throat may reach tens of MW/m 2. The investigation of the processes in the nozzle wall zone is a problem of turbulent boundary layer theory under extreme conditions (with respect to temperature, velocity and pressure) in the external flow. The methods of calculating the turbulent boundary layer in a supersonic flow, as they existed in 1978, are reviewed in [1]. In[2] methods of calculating the boundary layer in LRE nozzles are described. Of these the best known among engineers are the methods of Avduevskii [3] and levlev [4]. These are integral methods, i.e., they are based on the use of integral momentum and energy relations closed by the algebraic relations for the friction coefficient, the form parameter and the Stanton number. Differential methods of calculating the supersonic boundary layer are also reviewed in [1]. Study [5] is a concrete example of nozzle boundary layer calculations using the transport equation for the turbulent viscosity. The above-mentioned studies relate to the case of so-called external cooling of the LRE nozzle walls. To reduce the high heat fluxes the walls are film-cooled by feeding the fuel through tangential slits [2]. Integral methods of calculating the film cooling of nozzle walls are reviewed in [6]. Differential models of turbulence have been used for calculating plane wall jets in supersonic [7] and subsonic [81 flows. These calculations were made in order to check the applicability of the models by comparison with experiment. Our aim is to calculate the boundary layer in a LRE nozzle using the three-parameter differential model of turbulence developed in [9] and extended to flow with heat transfer in [10], which has been comprehensively tested over a broad range of problems of flow and heat transfer in boundary layers and channels [11].

Three-parameter model of turbulence: Heat transfer calculations

Three-parameter model of turbulence: Heat transfer calculations