A Model for Eddy Conductivity and Turbulent Prandtl Number

1973 ◽  
Vol 95 (2) ◽  
pp. 227-234 ◽  
Author(s):  
T. Cebeci
Keyword(s):  
Heat Transfer ◽  
Mass Transfer ◽  
Prandtl Number ◽  
Turbulent Flows ◽  
Pressure Gradients ◽  
Turbulent Prandtl Number ◽  
Temperature Distributions ◽  
Type Flow ◽  
Prandtl Numbers ◽  
Experimental Findings

This paper presents a model for eddy conductivity and turbulent Prandtl number based on the considerations of a Stokes-type flow. The expressions obtained by the model provide continuous velocity and temperature distributions for turbulent flows and are applicable to flows with pressure gradients, mass transfer, and heat transfer. Close to the wall the turbulent Prandtl number appears to be strongly affected by the molecular Prandtl number; away from the wall it is constant, that is, it is independent of the molecular Prandtl number. Calculated results agree well with experiments, including those with fluids having both low and high Prandtl numbers. In addition the results confirm recent experimental findings, in that the mass transfer has no effect on turbulent Prandtl number.

Asymptotic expansions for laminar forced-convection heat and mass transfer Part 2. Boundary-layer flows

1966 ◽  
Vol 24 (2) ◽  
pp. 339-366 ◽  
Author(s):  
J. D. Goddard ◽  
Andreas Acrivos
Keyword(s):  
Heat Transfer ◽  
Boundary Layer ◽  
Mass Transfer ◽  
Prandtl Number ◽  
Order Term ◽  
Higher Order ◽  
Convection Heat ◽  
Boundary Layer Flows ◽  
Prandtl Numbers ◽  
Higher Order Term

This is the second of two articles by the authors dealing with asymptotic expansions for forced-convection heat or mass transfer to laminar flows. It is shown here how the method of the first paper (Acrivos & Goddard 1965), which was used to derive a higher-order term in the large Péclet number expansion for heat or mass transfer to small Reynolds number flows, can yield equally well higher-order terms in both the large and the small Prandtl number expansions for heat transfer to laminar boundary-layer flows. By means of this method an exact expression for the first-order correction to Lighthill's (1950) asymptotic formula for heat transfer at large Prandtl numbers, as well as an additional higher-order term for the small Prandtl number expansion of Morgan, Pipkin & Warner (1958), are derived. The results thus obtained are applicable to systems with non-isothermal surfaces and arbitrary planar or axisymmetric flow geometries. For the latter geometries a derivation is given of a higher-order term in the Péclet number expansion which arises from the curvature of the thermal layer for small Prandtl numbers. Finally, some applications of the results to ‘similarity’ flows are also presented.

Analytical Study of the Convection Heat Transfer From an Isothermal Wedge Surface to Fluids

2012 ◽  
Vol 134 (6) ◽  
Author(s):  
M. Bachiri ◽  
A. Bouabdallah
Keyword(s):  
Heat Transfer ◽  
Prandtl Number ◽  
Ad Hoc ◽  
Convection Heat Transfer ◽  
Analytical Approximation ◽  
Numerical Results ◽  
Convection Heat ◽  
Governing Equations ◽  
Wedge Surface ◽  
Prandtl Numbers

In this work, we attempt to establish a general analytical approximation of the convection heat transfer from an isothermal wedge surface to fluids for all Prandtl numbers. The flow has been assumed to be laminar and steady state. The governing equations have been written in dimensionless form using a similarity method. A simple ad hoc technique is used to solve analytically the governing equations by proposing a general formula of the velocity profile. This formula verifies the boundary conditions and the equilibrium of the governing equations in the whole spatial region and permits us to obtain analytically the temperature profiles for all Prandtl numbers and for various configurations of the wedge surface. A comparison with the numerical results is given for all spatial regions and in wide Prandtl number values. A new Nusselt number expression is obtained for various configurations of the wedge surface and compared with the numerical results in wide Prandtl number values.

Numerical Investigation of Gas Turbine Combustor Liner Film Cooling Slots

2018 ◽  
Author(s):  
Firat Kiyici ◽  
Ahmet Topal ◽  
Ender Hepkaya ◽  
Sinan Inanli
Keyword(s):  
Heat Transfer ◽  
Prandtl Number ◽  
Surface Temperature ◽  
Gas Turbine ◽  
Film Cooling ◽  
Turbulent Prandtl Number ◽  
Surface Cooling ◽  
Gas Turbine Combustor ◽  
Cooling Effectiveness ◽  
Combustor Liner

A numerical study, based on experimental work of Inanli et al. [1] is conducted to understand the heat transfer characteristics of film cooled test plates that represent the gas turbine combustor liner cooling system. Film cooling tests are conducted by six different slot geometries and they are scaled-up model of real combustor liner. Three different blowing ratios are applied to six different geometries and surface cooling effectiveness is determined for each test condition by measuring the surface temperature distribution. Effects of geometrical and flow parameters on cooling effectiveness are investigated. In this study, Conjugate Heat Transfer (CHT) simulations are performed with different turbulence models. Effect of the turbulent Prandtl Number is also investigated in terms of heat transfer distribution along the measurement surface. For this purpose, turbulent Prandtl number is calculated with a correlation as a function of local surface temperature gradient and its effect also compared with the constant turbulent Prandtl numbers. Good agreement is obtained with two-layered k–ϵ with modified Turbulent Prandtl number.

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. 

UNIVERSAL PRANDTL NUMBER IN TWO-DIMENSIONAL KRAICHNAN-BATCHELOR TURBULENCE

2008 ◽  
Vol 22 (20) ◽  
pp. 3421-3431
Author(s):  
MALAY K. NANDY
Keyword(s):  
Prandtl Number ◽  
Mode Coupling ◽  
Perturbation Expansion ◽  
Dynamic Scaling ◽  
Two Dimensional ◽  
Turbulent Prandtl Number ◽  
Energy Regime ◽  
Self Consistent ◽  
Prandtl Numbers

We evaluate the universal turbulent Prandtl numbers in the energy and enstrophy régimes of the Kraichnan-Batchelor spectra of two-dimensional turbulence using a self-consistent mode-coupling formulation coming from a renormalized perturbation expansion coupled with dynamic scaling ideas. The turbulent Prandtl number is found to be exactly unity in the (logarithmic) enstrophy régime, where the theory is infrared marginal. In the energy régime, the theory being finite, we extract singularities coming from both ultraviolet and infrared ends by means of Laurent expansions about these poles. This yields the turbulent Prandtl number σ ≈ 0.9 in the energy régime.

Validation and Development of DNS Database for Low Prandtl Numbers in Rod Bundle

2019 ◽  
Author(s):  
Jonathan K. Lai ◽  
Elia Merzari ◽  
Yassin A. Hassan ◽  
Aleksandr Obabko
Keyword(s):  
Heat Transfer ◽  
Thermal Diffusivity ◽  
Prandtl Number ◽  
Liquid Metals ◽  
Heat Transfer Characteristics ◽  
Reynolds Analogy ◽  
Transfer Characteristics ◽  
Rod Bundle ◽  
High Thermal Diffusivity ◽  
Prandtl Numbers

Abstract Difficulty in capturing heat transfer characteristics for liquid metals is commonplace because of their low molecular Prandtl number (Pr). Since these fluids have very high thermal diffusivity, the Reynolds analogy is not valid and creates modeling difficulties when assuming a turbulent Prandtl number (Prt) of near unity. Baseline problems have used direct numerical simulations (DNS) for the channel flow and backward facing step to aid in developing a correlation for Prt. More complex physics need to be considered, however, since correlation accuracy is limited. A tight lattice square rod bundle has been chosen for DNS benchmarking because of its presence of flow oscillations and coherent structures even with a relatively simple geometry. Calculations of the Kolmogorov length and time scales have been made to ensure that the spatial-temporal discretization is sufficient for DNS. In order to validate the results, Hooper and Wood’s 1984 experiment has been modeled with a pitch-to-diameter (P/D) ratio of 1.107. The present work aims at validating first- and second-order statistics for the velocity field, and then analyzing the heat transfer behavior at different molecular Pr. The effects of low Pr flow are presented to demonstrate how the normalized mean and fluctuating heat transfer characteristics vary with different thermal diffusivity. Progress and future work toward creating a full DNS database for liquid metals are discussed.

scholarly journals Prandtl Number Effects on Mixed Convection Between Rotating Coaxial Disks

1996 ◽  
Vol 2 (3) ◽  
pp. 161-166 ◽  
Author(s):  
Chyi-Yeou Soong
Keyword(s):  
Heat Transfer ◽  
Prandtl Number ◽  
Mixed Convection ◽  
Boussinesq Approximation ◽  
Density Variation ◽  
Influential Factor ◽  
Rotating Disks ◽  
Temperature Relation ◽  
Flow And Heat Transfer ◽  
Prandtl Numbers

Prandtl number characterizes the competition of viscous and thermal diffusion effects and, therefore, is an influential factor in thermal-fluid flows. In the present study, the Prandtl number effects on non-isothermal flow and heat transfer between two infinite coaxial disks are studied by using a similarity model for rotation-induced mixed convection. To account for the buoyancy effects, density variation in Coriolis and centrifugal force terms are considered by invoking Boussinesq approximation and a linear density-temperature relation. Co-rotating disks(Ω2=Ω1)and rotor-stator system(Ω1≠Ω2=0)are considered to investigate the free and mixed convection flows, respectively. For Reynolds number, Re, up to 1000 and the buoyancy parameter, B=βΔT, of the range of|B|≤0.05, the flow and heat transfer characteristics with Prandtl numbers of 100, 7, 0.7, 0.1, and 0.01 are examined. The results reveal that the Prandtl number shows significant impact on the fluid flow and heat transfer performance. In the typical cases of mixed convection in a rotor-stator system with|B|=0.05, the effects in buoyancy-opposed flowsB=0.05are more pronounced than that in buoyancy-assisted ones.

Heat transfer in a turbulent channel flow with square bars or circular rods on one wall

2015 ◽  
Vol 776 ◽  
pp. 512-530 ◽  
Author(s):  
S. Leonardi ◽  
P. Orlandi ◽  
L. Djenidi ◽  
R. A. Antonia
Keyword(s):  
Heat Transfer ◽  
Heat Flux ◽  
Prandtl Number ◽  
Channel Flow ◽  
Turbulent Channel Flow ◽  
Turbulent Heat ◽  
Turbulent Prandtl Number ◽  
Smooth Wall ◽  
Roughness Elements ◽  
Smooth Channel

Direct numerical simulations (DNS) are carried out to study the passive heat transport in a turbulent channel flow with either square bars or circular rods on one wall. Several values of the pitch (${\it\lambda}$) to height ($k$) ratio and two Reynolds numbers are considered. The roughness increases the heat transfer by inducing ejections at the leading edge of the roughness elements. The amounts of heat transfer and mixing depend on the separation between the roughness elements, an increase in heat transfer accompanying an increase in drag. The ratio of non-dimensional heat flux to the non-dimensional wall shear stress is higher for circular rods than square bars irrespectively of the pitch to height ratio. The turbulent heat flux varies within the cavities and is larger near the roughness elements. Both momentum and thermal eddy diffusivities increase relative to the smooth wall. For square cavities (${\it\lambda}/k=2$) the turbulent Prandtl number is smaller than for a smooth channel near the wall. As ${\it\lambda}/k$ increases, the turbulent Prandtl number increases up to a maximum of 2.5 at the crests plane of the square bars (${\it\lambda}/k=7.5$). With increasing distance from the wall, the differences with respect to the smooth wall vanish and at three roughness heights above the crests plane, the turbulent Prandtl number is essentially the same for smooth and rough walls.

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