A Near-Wall Eddy Conductivity Model for Fluids With Different Prandtl Numbers

1994 ◽  
Vol 116 (4) ◽  
pp. 844-854 ◽  
Author(s):  
R. M. C. So ◽  
T. P. Sommer
Keyword(s):  
Kinetic Energy ◽  
Prandtl Number ◽  
Turbulent Kinetic Energy ◽  
Dissipation Rate ◽  
Reynolds Numbers ◽  
Turbulence Statistics ◽  
Temperature Variance ◽  
Prandtl Numbers ◽  
Good Agreement ◽  
Near Wall

Near-wall turbulence models for the velocity and temperature fields based on the transport equations for the Reynolds stresses, the dissipation rate of turbulent kinetic energy, and the temperature variance and its dissipation rate are formulated for flows with widely different Prandtl numbers. Conventional high-Reynolds-number models are used to close these equations and modifications are proposed to render them asymptotically correct near a wall compared to the behavior of the corresponding exact equations. Thus formulated, two additional constants are introduced into the definition of the eddy conductivity. These constants are found to be parametric in the Prandtl number. The near-wall models are used to calculate flows with different wall thermal boundary conditions covering a wide range of Reynolds numbers and Prandtl numbers. The calculated Nusselt number variations with Prandtl number are in good agreement with established formulae at two different Reynolds numbers. Furthermore, the mean profiles, turbulence statistics, heat flux, temperature variance, and the dissipation rates of turbulent kinetic energy and temperature variance are compared with measurements and direct numerical simulation data. These comparisons show that correct near-wall asymptotic behavior is recovered for the calculated turbulence statistics and the calculations are in good agreement with measurements over the range of Prandtl numbers investigated.

scholarly journals A Warm Rain Microphysics Parameterization that Includes the Effect of Turbulence

2008 ◽  
Vol 65 (6) ◽  
pp. 1795-1816 ◽  
Author(s):  
Charmaine N. Franklin
Keyword(s):  
Kinetic Energy ◽  
Size Distribution ◽  
Turbulent Kinetic Energy ◽  
Dissipation Rate ◽  
Drop Size ◽  
Reynolds Numbers ◽  
Liquid Water Content ◽  
Drop Size Distribution ◽  
Collision Kernel ◽  
Warm Rain

Abstract A warm rain parameterization has been developed by solving the stochastic collection equation with the use of turbulent collision kernels. The resulting parameterizations for the processes of autoconversion, accretion, and self-collection are functions of the turbulent intensity of the flow and are applicable to turbulent cloud conditions ranging in dissipation rates of turbulent kinetic energy from 100 to 1500 cm2 s−3. Turbulence has a significant effect on the acceleration of the drop size distribution and can reduce the time to the formation of raindrops. When the stochastic collection equation is solved with the gravitational collision kernel for an initial distribution with a liquid water content of 1 g m−3 and 240 drops cm−3 with a mean volume radius of 10 μm, the amount of mass that is transferred to drop sizes greater than 40 μm in radius after 20 min is 0.9% of the total mass. When the stochastic collection equation is solved with a turbulent collision kernel for collector drops in the range of 10–30 μm with a dissipation rate of turbulent kinetic energy equal to 100 cm2 s−3, this percentage increases to 21.4. Increasing the dissipation rate of turbulent kinetic energy to 500, 1000, and 1500 cm2 s−3 further increases the percentage of mass transferred to radii greater than 40 μm after 20 min to 41%, 52%, and 58%, respectively, showing a substantial acceleration of the drop size distribution when a turbulent collision kernel that includes both turbulent and gravitational forcing replaces the purely gravitational kernel. The warm rain microphysics parameterization has been developed from direct numerical simulation (DNS) results that are characterized by Reynolds numbers that are orders of magnitude smaller than those of atmospheric turbulence. The uncertainty involved with the extrapolation of the results to high Reynolds numbers, the use of gravitational collision efficiencies, and the range of the droplets for which the effect of turbulence has been included should all be considered when interpreting results based on these new microphysics parameterizations.

Assessment of the Turbulent Prandtl Number Effect on Simulating Heat Transfer Characteristics of Supercritical Water Flow in Bare Tubes

2017 ◽  
Author(s):  
Amjad Farah ◽  
Glenn Harvel ◽  
Igor Pioro
Keyword(s):  
Heat Transfer ◽  
Fluid Dynamics ◽  
Kinetic Energy ◽  
Prandtl Number ◽  
Turbulent Kinetic Energy ◽  
Supercritical Water ◽  
Dissipation Rate ◽  
Turbulence Models ◽  
Turbulent Prandtl Number ◽  
Wide Range

Computational Fluid Dynamics (CFD) is a numerical approach to modelling fluids in multidimensional space using the Navier-Stokes equations and databases of fluid properties to arrive at a full simulation of a fluid dynamics and heat transfer system. The turbulence models employed in CFD are a set of equations that determine the turbulence transport terms in the mean flow equations. They are based on hypotheses about the process of turbulence, and as such require empirical input in the form of constants or functions, in order to achieve closure. By introducing a set of empirical constants to a model, that model then becomes valid for certain flow conditions, or for a range of flows. Of those constants, the turbulent Prandtl number appears in multiple equations; energy, momentum, turbulent kinetic energy, turbulent kinetic energy dissipation rate, etc. and the value it takes in each equation is different and chosen empirically to fit a wide range of flows in the subcritical region. The studies that attempt to find the effect of varying the turbulent Pr number on simulation results, often only mention one number; presumably the one that appears in the energy equation (although it is never explicitly explained). The rest of the constants are treated as universally acceptable for generalized flow and not tested for their effect on flow parameters. A numerical study on heat transfer to supercritical water flowing in a vertical tube is carried out using the ANSYS FLUENT code and employing the Realizable k-ε (RKE) and the SST k-ε turbulence models. The 3-D mesh consists of a 1/8 slice (45° radially) of a bare tube. The study explored the effects of turbulent Pr numbers, and their variations, in order to understand their significance, and to build on previous knowledge to modify the turbulence models and achieve higher accuracy in simulating experimental conditions. The numerical results of 3D flow and thermal distributions under normal and deteriorated heat transfer conditions are compared to experimental results. The distributions of temperature and turbulence levels are used to understand the underlying phenomena of the heat transfer deterioration in supercritical water flows. Reducing the energy turbulent Pr number produced the most accurate prediction of the deterioration in heat transfer, by altering the production term due to buoyancy, which appears in the equations for turbulent kinetic energy as well as its dissipation rate. The buoyancy forces in upward flows act to reduce the turbulent shear stress, resulting in localized increase in wall temperatures.

Large-scale modes of turbulent channel flow: transport and structure

2001 ◽  
Vol 448 ◽  
pp. 53-80 ◽  
Author(s):  
Z. LIU ◽  
R. J. ADRIAN ◽  
T. J. HANRATTY
Keyword(s):  
Shear Stress ◽  
Kinetic Energy ◽  
Shear Layer ◽  
Turbulent Kinetic Energy ◽  
Large Scale ◽  
Reynolds Shear Stress ◽  
Reynolds Numbers ◽  
Upstream Region ◽  
Two Dimensional ◽  
Normal Plane

Turbulent flow in a rectangular channel is investigated to determine the scale and pattern of the eddies that contribute most to the total turbulent kinetic energy and the Reynolds shear stress. Instantaneous, two-dimensional particle image velocimeter measurements in the streamwise-wall-normal plane at Reynolds numbers Reh = 5378 and 29 935 are used to form two-point spatial correlation functions, from which the proper orthogonal modes are determined. Large-scale motions – having length scales of the order of the channel width and represented by a small set of low-order eigenmodes – contain a large fraction of the kinetic energy of the streamwise velocity component and a small fraction of the kinetic energy of the wall-normal velocities. Surprisingly, the set of large-scale modes that contains half of the total turbulent kinetic energy in the channel, also contains two-thirds to three-quarters of the total Reynolds shear stress in the outer region. Thus, it is the large-scale motions, rather than the main turbulent motions, that dominate turbulent transport in all parts of the channel except the buffer layer. Samples of the large-scale structures associated with the dominant eigenfunctions are found by projecting individual realizations onto the dominant modes. In the streamwise wall-normal plane their patterns often consist of an inclined region of second quadrant vectors separated from an upstream region of fourth quadrant vectors by a stagnation point/shear layer. The inclined Q4/shear layer/Q2 region of the largest motions extends beyond the centreline of the channel and lies under a region of fluid that rotates about the spanwise direction. This pattern is very similar to the signature of a hairpin vortex. Reynolds number similarity of the large structures is demonstrated, approximately, by comparing the two-dimensional correlation coefficients and the eigenvalues of the different modes at the two Reynolds numbers.

scholarly journals MODELING HEAT EXCHANGE DEPENDING ON THE PRANDTL NUMBER FOR VARIOUS GEOMETRIC AND REGIME PARAMETERS

2020 ◽  
Vol 46 (4) ◽  
pp. 91-101
Author(s):  
I. E. Lobanov
Keyword(s):  
Heat Transfer ◽  
Shear Stress ◽  
Prandtl Number ◽  
Energy Equation ◽  
Transport Model ◽  
Reynolds Numbers ◽  
Structured Grids ◽  
Shear Stress Transport Model ◽  
Small Reynolds Numbers ◽  
Prandtl Numbers

Objectives. The aim is to study the dependency of the distribution of integral heat transfer during turbulent convective heat transfer in a pipe with a sequence of periodic protrusions of semicircular geometry on the Prandtl number using the calculation method based on a numerical solution of the system of Reynolds equations closed using the Menter’s shear stress transport model and the energy equation on different-sized intersecting structured grids.Method. A calculation was carried out on the basis of a theoretical method based on the solution of the Reynolds equations by factored finite-volume method closed with the help of the Menter shear stress transport model, as well as the energy equation on different-scaled intersecting structured grids (fast composite mesh method (FCOM)).Results. The calculations performed in the work showed that with an increase in the Prandtl number at small Reynolds numbers, there is an initial noticeable increase in the relative heat transfer. With additional increase in the Prandtl number, the relative heat transfer changes less: for small steps, it increases; for median steps it is almost stabilised, while for large steps it declines insignificantly. At large Reynolds numbers, the relative heat transfer decreases with an increase in the Prandtl number followed by its further stabilisation.Conclusion. The study analyses the calculated dependencies of the relative heat transfer on the Pr Prandtl number for various values of the relative h/D height of the turbulator, the relative t/D pitch between the turbulators and for various values of the Re Reynolds number. Qualitative and quantitative changes in calculated parameters are described all other things being equal. The analytical substantiation of the obtained calculation laws is that the height of the turbuliser is less for small Reynolds numbers, while for large Reynolds numbers, it is less than the height of the wall layer. Consequently, only the core of the flow is turbulised, which results in an increase in hydroresistance and a decrease in heat transfer. In the work on the basis of limited calculation material, a tangible decrease in the level of heat transfer intensification for small Prandtl numbers is theoretically confirmed. The obtained results of intensified heat transfer in the region of low Prandtl numbers substantiate the promising development of research in this direction. The theoretical data obtained in the work have determined the laws of relative heat transfer across a wide range of Prandtl numbers, including in those areas where experimental material does not currently exist. 

Evaluation of Turbulent Kinetic Energy Dissipation Rate in a Free Jet Flow from PIV Measurements

2012 ◽  
Vol 7 (1) ◽  
pp. 53-69
Author(s):  
Vladimir Dulin ◽  
Yuriy Kozorezov ◽  
Dmitriy Markovich
Keyword(s):  
Energy Dissipation ◽  
Kinetic Energy ◽  
Turbulent Kinetic Energy ◽  
Dissipation Rate ◽  
Energy Dissipation Rate ◽  
Turbulent Velocity ◽  
Small Scale ◽  
Velocity Fluctuations ◽  
Piv Measurements ◽  
Free Jet

The present paper reports PIV (Particle Image Velocimetry) measurements of turbulent velocity fluctuations statistics in development region of an axisymmetric free jet (Re = 28 000). To minimize measurement uncertainty, adaptive calibration, image processing and data post-processing algorithms were utilized. On the basis of theoretical analysis and direct measurements, the paper discusses effect of PIV spatial resolution on measured statistical characteristics of turbulent fluctuations. Underestimation of the second-order moments of velocity derivatives and of the turbulent kinetic energy dissipation rate due to a finite size of PIV interrogation area and finite thickness of laser sheet was analyzed from model spectra of turbulent velocity fluctuations. The results are in a good agreement with the measured experimental data. The paper also describes performance of possible ways to account for unresolved small-scale velocity fluctuations in PIV measurements of the dissipation rate. In particular, a turbulent viscosity model can be efficiently used to account for the unresolved pulsations in a free turbulent flow

scholarly journals Resolved and subgrid dynamics of Rayleigh–Bénard convection

2019 ◽  
Vol 867 ◽  
pp. 906-933 ◽  
Author(s):  
Riccardo Togni ◽  
Andrea Cimarelli ◽  
Elisabetta De Angelis
Keyword(s):  
Kinetic Energy ◽  
Turbulent Kinetic Energy ◽  
Theoretical Framework ◽  
Temperature Fields ◽  
Temperature Variance ◽  
Eddy Simulation ◽  
Large Eddy ◽  
Subgrid Scales ◽  
Rayleigh Benard Convection ◽  
Rayleigh Benard

In this work we present and demonstrate the reliability of a theoretical framework for the study of thermally driven turbulence. It consists of scale-by-scale budget equations for the second-order velocity and temperature structure functions and their limiting cases, represented by the turbulent kinetic energy and temperature variance budgets. This framework represents an extension of the classical Kolmogorov and Yaglom equations to inhomogeneous and anisotropic flows, and allows for a novel assessment of the turbulent processes occurring at different scales and locations in the fluid domain. Two relevant characteristic scales, $\ell _{c}^{u}$ for the velocity field and $\ell _{c}^{\unicode[STIX]{x1D703}}$ for the temperature field, are identified. These variables separate the space of scales into a quasi-homogeneous range, characterized by turbulent kinetic energy and temperature variance cascades towards dissipation, and an inhomogeneity-dominated range, where the production and the transport in physical space are important. This theoretical framework is then extended to the context of large-eddy simulation to quantify the effect of a low-pass filtering operation on both resolved and subgrid dynamics of turbulent Rayleigh–Bénard convection. It consists of single-point and scale-by-scale budget equations for the filtered velocity and temperature fields. To evaluate the effect of the filter length $\ell _{F}$ on the resolved and subgrid dynamics, the velocity and temperature fields obtained from a direct numerical simulation are split into filtered and residual components using a spectral cutoff filter. It is found that when $\ell _{F}$ is smaller than the minimum values of the cross-over scales given by $\ell _{c,min}^{\unicode[STIX]{x1D703}\ast }=\ell _{c,min}^{\unicode[STIX]{x1D703}}Nu/H=0.8$, the resolved processes correspond to the exact ones, except for a depletion of viscous and thermal dissipations, and the only role of the subgrid scales is to drain turbulent kinetic energy and temperature variance to dissipate them. On the other hand, the resolved dynamics is much poorer in the near-wall region and the effects of the subgrid scales are more complex for filter lengths of the order of $\ell _{F}\approx 3\ell _{c,min}^{\unicode[STIX]{x1D703}}$ or larger. This study suggests that classic eddy-viscosity/diffusivity models employed in large-eddy simulation may suffer from some limitations for large filter lengths, and that alternative closures should be considered to account for the inhomogeneous processes at subgrid level. Moreover, the theoretical framework based on the filtered Kolmogorov and Yaglom equations may represent a valuable tool for future assessments of the subgrid-scale models.

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