Turbulent Viscosity Coefficient Research Articles

Calculations of the flow resistance and heat emission of a sphere in the laminar and high-turbulent gas flows

An early drag crisis can occur at high turbulence of incoming gas flow to a sphere. To study the influence of a crisis on heat transfer from a sphere to gas, a numerical experiment was carried out in which the free gas flow around a sphere with a temperature lower than the sphere temperature was simulated for two cases. The flow was laminar in the first case and highly turbulent in the second case. To take into account turbulence, the kinematic coefficient of turbulent viscosity with a value, which is much higher (up to 2000 times) than that for physical viscosity, was introduced. The results of calculations show that the early drag crisis occurs at Reynolds numbers of about 100 and results in considerable (by four to seven times) decrease in the hydrodynamic force and sphere drag coefficient C d . The early drag crisis is also accompanied by the crisis of heat transfer from a sphere to gas with a decrease in Nusselt numbers Nu by three to six times.

AN ORIGIN OF MULTIPLE RING STRUCTURE AND HIDDEN PLANETS IN HL TAU: A UNIFIED PICTURE BY SECULAR GRAVITATIONAL INSTABILITY

ABSTRACT Recent ALMA observation has revealed multiple ring structures formed in a protoplanetary disk around HL Tau. Prior to the ALMA observation of HL Tau, theoretical analysis of secular gravitational instability (GI) described a possible formation of multiple ring structures with separations of 13 au around a radius of 100 au in protoplanetary disks under certain conditions. In this article, we reanalyze the viability of secular GI by adopting the physical values inferred from the observations. We derive the radial distributions of the most unstable wavelength and the growth timescale of secular GI and verify that secular GI can form the ring structures observed in HL Tau. When a turbulent viscosity coefficient α remains small in the inner region of the disk, secular GI grows in the whole disk. Thus, the formation of planetary mass objects should occur first in the inner region as a result of gravitational fragmentation after the nonlinear growth of secular GI. In this case, the resulting objects are expected to create gaps at r ∼ 10 au and ∼30 au. As a result, all ring structures in HL Tau can be created by secular GI. If this scenario is realized in HL Tau, the outer region corresponds to the earlier growth phase of the most unstable mode of secular GI, and the inner region corresponds to the outcome of the nonlinear growth of secular GI. Therefore, this interpretation suggests that we are possibly witnessing both the beginning and the end of planet formation in HL Tau.

Scaling flame height of fully turbulent pool fires based on the turbulent transport properties

Previous studies on turbulent pool fires were almost entirely based on the air entrainment concept, while the characteristics of turbulent mixing between fuel gas and air following entrainment have not been elucidated. This paper presents scaling analysis on turbulent viscosity (νt) and diffusivity (Dt) by using literature data of 2-D velocity and mass fraction in large scale turbulent pool fires and strong He plume, thereby the flame height correlations are developed based on the turbulent mixing concept. It is found that the radial position of the maximum shear stress is close to that of the maximum radial gradient of axial velocity at the same level. The combustion reaction can affect the radial profile of turbulent viscosity. It is found that the turbulent viscosity (νt) increases first and then decreases with radial distance in turbulent pool fires, while decreases steadily with radius in the near-field He plume and far-field plume. The turbulent mass diffusivity (Dt) is proven to increase slightly with radius in the near-field He plume. The radial-mean turbulent Schmidt number is about 0.75 in the near field of strong He plume. The mean turbulent viscosities (νt=) within the mean flame height (H) are scaled by F1/3H2/3 for the fully turbulent pool fires, where F is the buoyancy flux. The classical two-fifths law of H*(H/d, d is the pool size) regarding the dimensionless heat release rate (Q˙*) is developed based on the turbulent transport properties. The coefficient of the power law is related to known ambient and fuel parameters, the turbulent Schmidt number and turbulent viscosity coefficient. The predicted coefficient agrees well with the empirical values from experimental data of flame height in literature.

Дослідження та математичне моделювання турбулентного примежевого шару при додатному градієнті тиску

Мета: Математичне моделювання складних турбулентних пристінних течій, які виникають при обтіканні аеродинамічних профілів, неможливе без розуміння природи розвитку течії у примежевому шарі. З математичної точки зору розрахунок таких течій складає серйозну проблему і по сьогоднішній день, оскільки в практичних завданнях їх необхідно розглядати як турбулентні, а характеристики турбулентності значною мірою залежать від геометрії профіля поздовжньої складової осередненої швидкості пристінного струменя. Виходячи з цього, метою цієї роботи є дослідження та математичне моделювання турбулентних пристінних течій при взаємодії з реальною обтічною поверхнею, яка має свої певні особливості такі як кривизна, шорсткість тощо, а також вивчення й дослідження впливу градієнта тиску на емпіричні коефіцієнти, параметри течії, профілі швидкості і напруження тертя. Методи: Розрахунки виконано числовим скінченно-різнецевим маршевим методом з використанням алгебраїчної моделі коефіцієнта турбулентної в’язкості. Результати: У цій роботі викладено деякі результати числового дослідження впливу додатного градієнта тиску на емпіричні коефіцієнти перехідної зони і закону стінки пристінної та зовнішньої областей. Обговорення: Порівняння отриманих розрахункових результатів з експериментальними даними показує, що запропоновані підходи дають змогу моделювати течії максимально наближено їх фізичним властивостям. Запропонована математична модель для розрахунку турбулентних примежевих шарів і пристінних струменів дає можливість розраховувати і такий складний та цінний з практичної точки зору вид течії, як аеродинамічний слід за обтічним тілом, що вселяє надію на поширення даних підходів на більш складні види течій.

DUST AND GAS IN THE DISK OF HL TAURI: SURFACE DENSITY, DUST SETTLING, AND DUST-TO-GAS RATIO

ABSTRACT The recent ALMA observations of the disk surrounding HL Tau reveal a very complex dust spatial distribution. We present a radiative transfer model accounting for the observed gaps and bright rings as well as radial changes of the emissivity index. We find that the dust density is depleted by at least a factor of 10 in the main gaps compared to the surrounding rings. Ring masses range from 10–100 M ⊕ in dust, and we find that each of the deepest gaps is consistent with the removal of up to 40 M ⊕ of dust. If this material has accumulated into rocky bodies, these would be close to the point of runaway gas accretion. Our model indicates that the outermost ring is depleted in millimeter grains compared to the central rings. This suggests faster grain growth in the central regions and/or radial migration of the larger grains. The morphology of the gaps observed by ALMA—well separated and showing a high degree of contrast with the bright rings over all azimuths—indicates that the millimeter dust disk is geometrically thin (scale height ≈1 AU at 100 AU) and that a large amount of settling of large grains has already occurred. Assuming a standard dust settling model, we find that the observations are consistent with a turbulent viscosity coefficient of a few 10−4. We estimate the gas/dust ratio in this thin layer to be of the order of 5 if the initial ratio is 100. The HCO+ and CO emission is consistent with gas in Keplerian motion around a 1.7 M ⊙ star at radii from ≤10–120 AU.

NUMERICAL INVESTIGATION OF THE UNSTART SUPPRESSION IN A SUPERSONIC AIR INTAKE

The starting behavior of a supersonic axisymmetric mixed compression air intake was numerically investigated. The code solves Reynolds-averaged Navier–Stokes equations using an explicit finite volume method in a structured grid by the Roe flux difference splitting scheme. Further, it uses the Baldwin–Lomax algebraic model to compute the turbulent viscosity coefficient. The correct method of surveying the intake starting problem and effects of several geometrical parameters such as: intake throat area, cowl lip roundness and spike surface curvature upstream of the throat on the starting and performance of the intake were studied. Results showed that correct combinations of the mentioned parameters can suppress the intake unstart problem and in addition can prevent the reduction of the intake efficiency.

Numerical simulations of turbulent ionized gas flows in the circumsolar protoplanetary disk

An axisymmetric protoplanetary disk model that takes into account the interaction of turbulent gas flows with the magnetic field is considered. A closed system of equations of homogeneous compressible magnetohydrodynamics in the regime of developed turbulence in the gravitational and magnetic fields of a star has been constructed. Apart from the traditional probability-theoretical averaging of the MHD equations, the weighted Favre averaging is used. The approach by A.V. Kolesnichenko and M.Ya. Marov to modeling the turbulent transport coefficients in a weakly ionized disk has been implemented. It allows the inverse effects of the generated magnetic field on a turbulent gas flow and the dissipation of turbulence through kinematic and magnetic viscosities to be taken into account. A parallel code for numerically solving the system of averaged MHD equations has been developed. The averaged gas density and velocity distributions as well as the configuration of the disk’s intrinsic magnetic field at a distance of 1 AU from the star have been obtained through numerical simulations. The assumption that the vertical (parallel to the disk’s rotation axis) magnetic induction component changes much more profoundly in height than in radius and, hence, gives grounds to take into account its gradient in the model of the turbulent kinematic viscosity coefficient has been confirmed.

Small-time solvability of primitive equations for the ocean with spatially-varying vertical mixing

The small-time existence of a strong solution to the free surface problem of primitive equations for the ocean with variable turbulent viscosity terms is shown in this paper. In this model, the turbulent viscosity coefficients, which include the Richardson number depending on unknown variables, are explicitly formulated. In addition, following the formulation of practical models, the kinematic condition is assumed on the free ocean surface. As in preceding works, we consider the problem in the three-dimensional strip-like region, and assume the f -approximation. Under some conditions on the initial and boundary data and the topography of the bottom of the ocean, we construct a strong local-in-time solution in Sobolev-Slobodetski˘ i spaces. The boundedness of the temperature and salinity is also shown in the present paper.

Modeling the Atmospheric Boundary Layer Wind Response to Mesoscale Sea Surface Temperature Perturbations

Abstract The wind speed response to mesoscale SST variability is investigated over the Agulhas Return Current region of the Southern Ocean using the Weather Research and Forecasting (WRF) Model and the U.S. Navy Coupled Ocean–Atmosphere Mesoscale Prediction System (COAMPS) atmospheric model. The SST-induced wind response is assessed from eight simulations with different subgrid-scale vertical mixing parameterizations, validated using Quick Scatterometer (QuikSCAT) winds and satellite-based sea surface temperature (SST) observations on 0.25° grids. The satellite data produce a coupling coefficient of sU = 0.42 m s−1 °C−1 for wind to mesoscale SST perturbations. The eight model configurations produce coupling coefficients varying from 0.31 to 0.56 m s−1 °C−1. Most closely matching QuikSCAT are a WRF simulation with the Grenier–Bretherton–McCaa (GBM) boundary layer mixing scheme (sU = 0.40 m s−1 °C−1), and a COAMPS simulation with a form of Mellor–Yamada parameterization (sU = 0.38 m s−1 °C−1). Model rankings based on coupling coefficients for wind stress, or for curl and divergence of vector winds and wind stress, are similar to that based on sU. In all simulations, the atmospheric potential temperature response to local SST variations decreases gradually with height throughout the boundary layer (0–1.5 km). In contrast, the wind speed response to local SST perturbations decreases rapidly with height to near zero at 150–300 m. The simulated wind speed coupling coefficient is found to correlate well with the height-averaged turbulent eddy viscosity coefficient. The details of the vertical structure of the eddy viscosity depend on both the absolute magnitude of local SST perturbations, and the orientation of the surface wind to the SST gradient.

Numerical analysis of two ensemble eddy viscosity numerical regularizations of fluid motion

This report analyzes an efficient ensemble regularization algorithm for under‐resolved and convection dominated flows (including ones at higher Reynolds numbers). Computing an ensemble simultaneously allows each realization to access ensemble data. This allows use of means and fluctuations in regularizations used for each realization. The combined approach of ensemble time stepping and ensemble regularizations allows direct calculation of the turbulent viscosity coefficient and gives an unconditionally stable algorithm. It also suggests reconsidering an old but not as well‐developed definition of the mixing length. This mixing length vanishes at solid walls without van Driest damping, increases stability, and improves flow predictions in our preliminary tests. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 630–651, 2015

A stability condition for turbulence model: From EMMS model to EMMS-based turbulence model

The closure problem of turbulence is still a challenging issue in turbulence modeling. In this work, a stability condition is used to close turbulence. Specifically, we regard single-phase flow as a mixture of turbulent and non-turbulent fluids, separating the structure of turbulence. Subsequently, according to the picture of the turbulent eddy cascade, the energy contained in turbulent flow is decomposed into different parts and then quantified. A turbulence stability condition, similar to the principle of the energy-minimization multi-scale (EMMS) model for gas–solid systems, is formulated to close the dynamic constraint equations of turbulence, allowing the inhomogeneous structural parameters of turbulence to be optimized. We name this model as the “EMMS-based turbulence model”, and use it to construct the corresponding turbulent viscosity coefficient. To validate the EMMS-based turbulence model, it is used to simulate two classical benchmark problems, lid-driven cavity flow and turbulent flow with forced convection in an empty room. The numerical results show that the EMMS-based turbulence model improves the accuracy of turbulence modeling due to it considers the principle of compromise in competition between viscosity and inertia.

Models of turbulent viscosity and mixing in channels and packed flow-through mixers

The use of packed flow-through mixers for intense mixing of liquid media and of liquids with a finely dispersed phase is considered. Ways to determine the mixing and turbulent exchange coefficients for evaluating the mixing efficiency in channels and in packed flow-through mixers are presented. The results are consistent with the experimental data on turbulent viscosity in pipes. An expression is obtained for calculating the turbulent viscosity coefficient in a packing bed. Comparative characteristics of mixers with Raschig rings and Inzhekhim packing are presented. The calculated values of the Peclet number for various packings in relation to the Reynolds number are given.

Calculation of the flow about a sphere and the drag of the sphere under laminar and strongly turbulent conditions

To investigate the influence of a strongly turbulent incoming flow on the hydrodynamic drag of a body and occurrence of the early crisis of drag, a numerical experiment is conducted in which a free gas flow about a sphere is simulated for two cases, namely, for a laminar flow and for a strongly turbulent flow. Turbulence is simulated by assuming a high kinematic coefficient of turbulent viscosity. Calculation data lead us to conclude that the early crisis of drag at Reynolds numbers near 100, which shows up as a considerable (four-to sevenfold) decrease in the hydrodynamic force and the drag coefficient of the body, can be explained by the strong turbulence of the incoming flow.

Characteristics of Fiber Suspension Flow in a Turbulent Boundary Layer

The equations of averaged momentum, turbulence kinetic energy, turbulence dissipation rate with the additional term of the fibers, and the equation of probability distribution function for mean fiber orientation are derived and solved numerically for fiber suspension flowing in a turbulent boundary layer. The mathematical model and numerical code are verified by comparing the numerical results with the experimental ones in a turbulent channel flow. The effects of Reynolds number, fiber concentration and fiber aspect-ratio on the mean velocity profile, turbulent kinetic energy, Reynolds stress, turbulent dissipation rate and eddy viscosity coefficient are analyzed. The results show that the velocity profiles become full, and the turbulent kinetic energy, Reynolds stress and eddy viscosity coefficient increase, while turbulent dissipation rate decreases, as the Reynolds number, fiber concentration and fiber aspect-ratio increase. The effect of the fiber aspect-ratio on the turbulent properties is larger than that of the Reynolds number, but smaller than that of the fiber concentration in the range of parameters considered in this paper.

A Modified k-ε Model for Computation of Flows with Large Streamline Curvature

This paper presents an improved method for inclusion of system rotation and streamline curvature effects into existing two-equation eddy-viscosity turbulence models. A new formulation for calculation of the turbulence viscosity coefficient, which is implemented into the traditional k-ε model with a two-layer near-wall treatment in a commercial computational fluid dynamics solver, is proposed. In contrast to precious model, the modified rotation rate which appears in the formulation for turbulent viscosity coefficient is herein expressed exactly and universally. Thus, it provides an effective alternative for turbulence modeling.

Mean velocity profiles of two-dimensional fully developed turbulent flows

In the work presented in this Note, an Indirect Turbulence Model (ITM) is proposed to derive the mean velocity profiles in wall-bounded flows in hydraulically smooth channels having a very wide rectangular cross section. The analytical expression of the mean velocity distribution is given. The connection between the velocity distribution parameters and Reynoldsʼ number is indicated. The thickness of the viscous sublayer is evaluated. The skin friction coefficient is computed, and the analytical expression of the turbulent viscosity coefficient is provided.The validity of the proposed model is verified with reference to the velocity distributions, available in the literature, obtained with Direct Numerical Simulation (DNS) of Navier–Stokesʼ equations.

Solvability of primitive equations for the ocean with vertical mixing

Small time existence of a strong solution to the free surface problem of primitive equations for the ocean with the variable turbulent viscosity terms is shown in this paper. The turbulent viscosity coefficients, which include the Richardson number depending on the unknown variables, are formulated explicitly. We consider the problem in the 3-dimensional stripe-like region, and construct the strong local-in-time solution in Sobolev-Slobodetskiĭ spaces. The details of the proofs will be provided in another full paper.

Numerical simulation and parametric study of a supersonic intake

A computational fluid dynamics code was developed to compute the flow inside and around a supersonic external compression axisymmetric intake. The code solves the Reynolds-averaged Navier–Stokes equations using an explicit finite volume method in a structured grid and uses the Baldwin–Lomax algebraic model to compute the turbulent viscosity coefficient. Experiments were performed to validate the predicted results and good agreements are achieved. In the next part of the research, a parametric study was undertaken using the designed base case at a constant Mach number of 2 and at 0° angle of attack. The effects of various important parameters such as free stream Mach number, spike deflection angle, and back pressure ratio on the total pressure recovery, mass flow ratio, flow distortion, and drag coefficient of the intake were then numerically investigated. The results showed that when the spike deflection angle of the intake was changed from 28° (designed base case) to 30°, the intake drag coefficient was reduced up to 9%. In addition, the intake performance degraded for very low values of the back pressure ratio.

Investigation of a new flux scheme for the numerical simulation of the supersonic intake flow

A numerical code for supersonic intake design with a proper simulation of the normal and/or oblique shocks, boundary layer development, interaction of the shock and the boundary layer, as well as prediction of the flow separation is of great help to the designers. In this research, a numerical code is developed to solve the inner and outer flow fields of the intake and validated with various experimental tests. The intake is an axisymmetric external compression one. Roe scheme and new schemes, AUSM+-up (for all speed) and Advection Upstream Splitting Method with Pressure-Based Weight function (AUSMPW), are used to compute the convective fluxes. The original version of the AUSMPW scheme has serious problems concerning the stability and accuracy. In this investigation, several modifications are implemented to the original version of the AUSMPW scheme for the first time which resulted in a better stability and accuracy. Turbulent viscosity coefficient is computed using the Baldwin–Lomax algebraic model. The numerical results are compared well with the experimental data. It is further found that the Roe scheme has the best accuracy; however, the AUSM+-up (for all speed) scheme is numerically very efficient.

Turbulent viscosity variability in self-preserving far wake with zero net momentum

The profile of the self-preserving far wake with zero net momentum depends on the effective turbulent viscosity coefficient. The current model is based on the assumption of uniform viscosity in the wake cross section. It predicts the self-similar shape of the wake where the width W depends on the distance z from the body as W∝z(1/5) for the axisymmetric case (or z(1/4) for the plane case). The observed wake width, however, demonstrates the dependence W∝z(α) (where α≤1/5). We generalize the model of a self-preserving far wake for the case of the turbulent viscosity coefficient depending on the radius. Additional integrals of motion allow a new family of self-similar profiles with α≤1/5.