An alternative two-equation turbulent heat diffusivity closure

The objective of this paper is to present an alternative closure for turbulent heat transport, that has been established on the ground of v²-f model with so-called elliptic relaxation equation and the two-equation model for turbulent thermal field. New formulae on the eddy heat diffusivity is proposed. Additionally the production term of destruction rate of temperature variance transport equation is reformulated. No damping functions are involved in this model. Results of numerical computation have been compared with the experimental data for fully developed thermal field in the pipe and DNS heat transfer prediction for two-dimensional channel flow. The model predicts reasonably well the near-wall distribution of basic turbulence statistics.     

Treatment of turbulent heat fluxes with the elliptic-blending second-moment closure for turbulent natural convection flows

In this paper a comparative study on the treatment of the turbulent heat fluxes with the elliptic-blending second-moment closure for natural convection flows is presented. Three different cases for treating the turbulent heat fluxes are considered. Those are the generalized gradient diffusion hypothesis (GGDH), the algebraic flux model (AFM) and the differential flux model (DFM). These models are implemented in a computer code especially designed for an evaluation of turbulent models. Calculations are performed for turbulent natural convection flows in an 1:5 rectangular cavity (Ra = 4.3 × 10¹⁰) and in a square cavity with conducting top and bottom walls (Ra = 1.58 × 10⁹). The calculated results are compared with the available experimental data. The results show that the GGDH, AFM and DFM models produce sufficiently accurate solutions for the turbulent natural convection in an 1:5 rectangular cavity where the strength of the thermal stratification is weak in a central region of the cavity. However, the GGDH model produces very erroneous solutions for the turbulent natural convection in a square cavity with conducting walls where the Rayleigh number is relatively small and the thermally stratified region is dominant. The AFM and DFM produce very accurate solutions for both cases without invoking any numerical problems.     

Convection heat transfer coefficients for turbulent flow between parallel plates with unequal heat fluxes

This paper contains a semi-theoretical analysis of asymmetric heat transfer in fully developed two dimensional turbulent flow between parallel walls.

An extension of the analogy between the transfer of heat and the transfer of momentum (due to Mizushina), is used to determine the heat transfer coefficients for the respective boundaries. The results of the analysis are presented in the form of working formulae from which the relative magnitudes of the heat transfer coefficients may be determined. The formulae may be applied to practical engineering problems, but care is necessary because the assumptions made in the theory impose restrictions on the Prandtl and Reynolds number ranges for which the predictions are acceptable.

It is immediately obvious from the graphical presentation of the results for a particular fluid, that the heat flux ratio is an important additional parameter, and that cognizance of this should be taken in cases of non-uniform heating.

By its nature, the present analysis is preliminary, and an experimental programme has been devised to study the general problem of asymmetric heat transfer and to test the theory.

While channel flow has been considered, the results are thought to be valid for axisymmetric flow in annuli with small outside diameter/inside diameter ratios.     

A forced convection subcooled boiling model for nonuniform axial heat fluxes

The modeling of convective subcooled boiling of water flowing in round tubes subjected to nonuniform axial heat fluxes is described. The effects of different axial heat flux profiles are modeled using a local hypothesis; i.e. flow and thermal development are assumed to occur very rapidly in the subcooled boiling (SCB) flow regime. A computer code has been developed to predict the pressure drop, heat transfer coefficient and wall temperature for nonuniform axial heat fluxes, starting with a well-validated code for uniform axial heat fluxes. The predictions for some common nonuniform axial heat profiles are compared to the uniform heat flux case.     

Simulation of turbulent natural convection in a porous cylindrical annulus using a macroscopic two-equation model

This work presents numerical computations for laminar and turbulent natural convection within a horizontal cylindrical annulus filled with a fluid saturated porous medium. Computations covered the range 25 < Ra_m < 500 and 3.2 × 10⁻⁴ > Da > 3.2 × 10⁻⁶ and made use of the finite volume method. The inner and outer walls are maintained at constant but different temperatures. The macroscopic k–ε turbulence model with wall function is used to handle turbulent flows in porous media. First, the turbulence model is switched off and the laminar branch of the solution is found when increasing the Rayleigh number, Ra_m. Subsequently, the turbulence model is included and calculations start at high Ra_m, merging to the laminar branch for a reducing Ra_m. This convergence of results as Ra_m decreases can be seen as an estimate of the so-called laminarization phenomenon. Here, a critical Rayleigh number was not identified and results indicated that when the porosity, Prandtl number, conductivity ratio between the fluid and the solid matrix and Ra_m are kept fixed, the lower the Darcy number, the higher is the difference of the average Nusselt number given by the laminar and turbulent models.     

Mathematical model for turbulent flow,heat transfer,and solidification

A steady‐state, two‐dimensional numerical model has been used to describe coupled liquid steel's turbulent flow and heat transfer with solidification for Fe‐C binary alloy in a crystallizer of inverse casting. The solid‐liquid phase change phenomena have been modeled by using continuum formulations and considering the mushy zone as porous media. The turbulence flow in the crystallizer has been accounted for using a modified version of the low‐Reynolds‐number κ?ε turbulence model. The flow pattern in the liquid zone and the temperature distribution in the solid, mushy, and liquid regions have been predicted. The numerical analysis indicates that the residence time of the mother sheet in the crystallizer is one of the key parameters. The effects of some other main parameters on the solidification behavior have also been studied, such as the thickness and the initial temperature of the mother sheet, and the superheat degree of liquid steel. © 2003 Wiley Periodicals, Inc. Heat Trans Asian Res, 32(7): 582–592, 2003; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/htj.10112     

A new comprehensive model for nucleate pool boiling heat transfer of pure liquid at low to high heat fluxes including CHF

Based on the fractal distribution of nucleation sites present on heating surfaces, a new comprehensive model is developed for the nucleate pool boiling of pure liquid at low to high heat fluxes including the critical heat flux (CHF). The proposed model is expressed as a function of total number, minimum and maximum sizes of active nucleation sites, fractal dimension, superheat temperature, and properties of fluids. No additional empirical constant is introduced in the proposed model. This fractal model contains less empirical constants than the conventional models. The model predictions are in good agreement with the available experimental data.     

Effect of magnetic field on near-wall coherent structures and heat transfer in magnetohydrodynamic turbulent channel flow of low Prandtl number fluids

A numerical study is carried out of the magnetic field effects on the coherent structures and the associated heat transfer in a turbulent channel flow with constant temperature at the bottom (cold) and top (hot) walls. Results from direct numerical simulations are conditionally sampled in order to extract the dominant coherent structures in the near-wall region for flows with and without a uniform external magnetic field in the wall-normal direction. The Reynolds number based on the bulk velocity and the wall distance is 5600, while only a representative small Stuart number of 0.01 is explored. Two fluids with Prandtl numbers of 0.01 and 0.71 are studied. It is shown that the conditionally averaged quasi-streamwise vortices are modified by the magnetic field with their size being increased and their strength decreased. The underlying organized fluid motions are damped by the Lorentz force and the turbulent heat transfer related to the action of quasi-streamwise vortices is decreased by the magnetic field. For the higher Prandtl number fluid, a similarity between the coherent temperature and the coherent streamwise velocity fluctuations is observed for both types of flow. This is diminished for the lower Prandtl number fluid, especially in the magnetohydrodynamic flow, inhibiting the intrusion of cold (hot) fluid from the cold (hot) wall towards the central region.     

Analytical solution of forced convective heat transfer in tubes partially filled with metallic foam using the two-equation model

In this study, an analytical solution for fully developed forced convection in a tube partially filled with open-celled metallic foams is presented. In the foam region, the Brinkman flow model is used to describe the fluid transport, and the local thermal non-equilibrium model is adopted to represent the fluid–solid energy exchange. At the foam–fluid interface, interfacial coupling conditions for temperature are proposed and used to derive the analytical solution. Velocity and temperature profiles are derived from this solution, and explicit expressions for the friction factor and the Nusselt number are obtained. A parametric study is conducted to study the influences of various factors on flow resistance and heat transfer performance. The present analytical solution establishes a benchmark for similar work hereafter.     

On a subgrid-scale heat flux model for large eddy simulation of turbulent thermal flow

A non-linear subgrid-scale (SGS) heat flux model is introduced in large eddy simulation for turbulent thermal flows. Unlike the linear isotropic eddy diffusivity model, the proposed model accounts for the SGS heat flux in terms of the large-scale strain-rate tensor and the temperature gradients. This is equivalent to using a tensor diffusivity. The model is to some extent similar to a scale-similarity model subjected to a Taylor expansion for the filtering operation. The formulation leading to the present proposal is discussed. The model is examined in LES for a buoyant flow in an infinite vertical channel with two differentially heated side walls. It is shown that the proposed model reproduces reasonable results as compared with the isotropic SGS diffusivity model and DNS data.     

A filter-independent model identification technique for turbulent combustion modeling

In this paper, we address a method to reduce the number of species equations that must be solved via application of Principal Component Analysis (PCA). This technique provides a robust methodology to reduce the number of species equations by identifying correlations in state-space and defining new variables that are linear combinations of the original variables. We show that applying this technique in the context of Large Eddy Simulation allows for a mapping between the reduced variables and the full set of variables that is insensitive to the size of filter used. This is notable since it provides a model to map state variables to progress variables that is a closed model.As a linear transformation, PCA allows us to derive transport equations for the principal components, which have source terms. These source terms must be parameterized by the reduced set of principal components themselves. We present results from a priori studies to show the strengths and weaknesses of such a modeling approach. Results suggest that the PCA-based model can identify manifolds that exist in state space which are insensitive to filtering, suggesting that the model is directly applicable for use in Large Eddy Simulation. However, the resulting source terms are not parameterized with an accuracy as high as the state variables.     

A POD-Galerkin reduced-order model for isotropic viscoelastic turbulent flow

In this article, a proper orthogonal decomposition (POD) reduced-order model for isotropic turbulent flow of viscoelastic fluid is established for the first time. Particularly, since the present studies about viscoleastic fluid are mainly for revealing the mechanism of turbulence, we try to establish the reduced-order model for momentum equations and constitutive equations, finally get both velocities and deformation rates calculated. Through decomposing the sampling matrices which are obtained by direct numerical simulation (DNS) using finite volume method (FVM), the velocity basis functions and deformation rate basis functions are generated respectively. According to the Galerkin projection method, the equations for velocity spectrum coefficients and deformation rate spectrum coefficients are deducted, which are coupled pluralistic nonlinear equations and solved iteratively by the Newton–Raphson method. To illustrate the performance of the proposed model for the viscoelastic fluid flow, a two-dimensional decaying isotropic turbulence testing case is designed in Example 1. It is found that the established reduced-order model obtains good accuracy when the decaying flow is at its early stage, but the errors get considerable when the flow steps into transition flow. In addition, a three-dimensional forced isotropic viscoelastic turbulence testing case is designed in Example 2. It is indicated that the errors of viscoelastic forced isotropic turbulent flow are acceptable. Finally, the calculation speed of the established reduced-order model is found to be much faster than that of DNS.     

Roughness enhanced mechanism for turbulent convective heat transfer

The mechanism of turbulent convective heat transfer enhancement was experimentally investigated by measuring the heat transfer in two dimensional roughness tubes with different roughness heights at various Reynolds numbers. The results show that there is a maximum Nusselt number ratio (Nu/Nu₀) for a fixed roughness height with increasing Reynolds numbers. For water as working fluid, heat transfer can hardly be increased when the roughness height is lower than the thickness of the viscous sublayer, and both heat transfer and flow friction begin to increase when the roughness height is higher than the viscous sublayer. When the roughness height is more than five times of the viscous sublayer thickness, the flow friction begins to increase sharply but heat transfer is slowly enhanced. So the best heat transfer enhancement for a given pumping power is reached when the roughness height is about three times of the viscous sublayer thickness. The Prandtl number influences to the turbulent heat transfer enhancement by roughness were also analyzed.     

A new heat eddy diffusivity equation for calculation of heat transfer to drag reducing turbulent pipe flows

A new semi-empirical equation of heat eddy diffusivity in terms of friction drag reduction ratio and Weissenburg number is presented. The proposed equation was validated with heat transfer experimental results of Kwack [1] and our recent experimental results, both for aqueous solutions of polyacrylamide (Separan AP-273) for concentrations ranging from 10 to 1000 ppm in turbulent flow through pipes under constant wall heat flux condition. The predictions of heat transfer coefficients with the use of the proposed equation are in good agreement with both sets of independent experimental results. The results of this study indicate that the proposed equation for eddy diffusivity of heat has predictive capability provided experimental measurements of pressure drop and the fluid time scale are available. The fluid time scale for these prediction was estimated using the Powell-Eyring fluid model and apparent viscosity measurements.     

A statistical model for predicting the fluid displaced/added mass and displaced heat capacity effects on transport and heat transfer of arbitrary-density particles in turbulent flows

The objective of the paper is twofold: (i) to present a new statistical model for predicting the transport and heat transfer of arbitrary-density particles suspended in turbulent flows and (ii) to examine the performance of this model in an isotropic velocity flow field without and with a mean temperature gradient as well as in a near-wall turbulent flow. The model presented is based on a kinetic equation for the probability density function (PDF) of velocity and temperature distributions and coves the entire range of the particle-to-fluid density ratio (from heavy particles in a gas to bubbles in a liquid).