Mark Cotton

Studies of mixed convection in vertical tubes

Abstract The early study of convective heat transfer considered the branches of forced and free convection independently with only passing reference to their possible interaction. In fact the two are extreme cases of the general condition of “mixed” or “combined” forced and free convection where both mechanisms operate simultaneously. The present contribution aims to provide an up-to-date review of those works concerned with mixed convection heat transfer in vertical tubes. The review is divided into two sections, the first dealing with laminar flow, and the second with turbulent flow; further subdivisions are made according to whether the work is theoretical or experimental. Comparisons between theory and experiment are made where possible, expressions defining the conditions for onset of buoyancy effects are presented and equations for determining heat transfer are given. The paper ends with some general comments and recommendations. The survey is restricted to fluids of moderate Prandtl number; mixed convection in liquid metals can display very different characteristics which will be discussed in a future paper.

A variant of the low-Reynolds-number two-equation turbulence model applied to variable property mixed convection flows

Abstract Previous work has demonstrated that the low-Reynolds-number model of Launder and Sharma (1974) offers significant advantages over other two-equation turbulence models in the computation of highly non-universal buoyancy-influenced (or “mixed convection”) pipe flows. It is known, however, that the Launder and Sharma model does not possess high quantitative accuracy in regard to simpler forced convection flows. A variant of the low-Reynolds-number scheme is developed here by reference to data for constant property forced convection flows. The re-optimized model and the Launder and Sharma formulation are then examined against experimental measurements for mixed convection flows, including cases in which variable property effects are significant.

RANS AND LES INVESTIGATIONS OF VERTICAL FLOWS IN THE FUEL PASSAGES OF GAS-COOLED NUCLEAR REACTORS

Coolant flows in the cores of current gas-cooled nuclear reactors consist of ascending vertical flows in a large number of parallel passages. Under post-trip conditions such heated turbulent flows may be significantly modified from the forced convection condition by the action of buoyancy, and the thermal-hydraulic regime is no longer one of pure forced convection. These modifications are primarily associated with changes to the turbulence structure, and indeed flow laminarization may occur. In the laminarization situation heat transfer rates may be as low as 40% of those in the corresponding forced convection case. The heat transfer performance of such ‘mixed’ convection flows is investigated here using a range of refined ReynoldsAveraged-Navier-Stokes (RANS) turbulence models. While all belong to the broad class of Eddy Viscosity Models (EVMs), the various RANS closures have different physical parameterizations and might therefore be expected to show different responses to externally-imposed conditions. Comparison is made against experimental and Direct Numerical Simulation (DNS) data. In addition, Large Eddy Simulation (LES) results have been generated as part of the study. Three different CFD codes have been employed in the work: ‘CONVERT’, ‘STAR-CD’, and ‘ Code_Saturne ’, which are respectively in-house, commercial, and industrial packages. It is found that the early EVM scheme of Launder and Sharma [1] is in the closest agreement with consistentlynormalized DNS results for the ratio of mixed-to-forced convection Nusselt number ( Nu/Nu 0). However, in relation to DNS and experimental data for forced convection Nusselt number, other models perform better than the LaunderSharma scheme. The present investigation has revealed discrepancies between direct-simulation, experimental, and the current LES studies.

On Modelling Periodic Motion with Turbulence Closures

In periodic flows near walls transport effects may be considerably larger than in a steady turbulent boundary layer. The question explored in this contribution is, therefore, whether providing separate transport equations for each of the Reynolds stresses consequently leads to a better modelling of a periodic flows evolution than an eddy viscosity scheme whose constitutive equation is inherently linked to the generation and dissipation terms being in balance (local equilibrium). Our conclusion is that, while a stress-transport scheme is indeed better equipped to reproduce the phenomena, it does not consistently out-perform the EVM over the range of flows studied. In some cases it is suggested that more attention must be paid to modelling diffusive transport in order to secure the benefits of second-moment closure. To illustrate sensitivity to diffusive transport, two different diffusion models are tested, one of which leads to different effective transport coefficients in each stress component.

Resonant responses in periodic turbulent flows: computations using a k-e eddy viscosity model

Periodically-oscillated pipe flows in which the bulk velocity is varied about a non-zero mean level (Ub = Ub0[1 + γ cos ωt]) are computed using a low-Reynolds-number k–ε turbulence model. Comparison is made with data for periodic pipe flow and it is shown that model is capable of resolving the principal features of the highly non-universal turbulence profiles that occur under conditions of harmonic forcing. There follows an examination of the frequency response of the phase-averaged turbulent kinetic energy, k, which is analysed in terms of its first harmonic variation (k(r, ωt) = k 0 + k 1 cos(ωt + ψ); k 0, k 1,ψ = f(r)). In confirmation of the stress-transport model results of Cotton and Guy [J. Hydraul. Res. 42 (2004) 293], it is found that the modulation of the turbulent kinetic energy, k 1/γ k0 first responds in a quasi-steady manner and then, with increasing frequency, exhibits resonant behaviour, which is itself succeeded by a frozen response at higher frequencies. The resonant condition occurs when the time scale of large-scale turbulence is an order of magnitude less than the period of the imposed oscillation. The paper concludes with a discussion of the parallels that may be drawn between the present results and the experimental study of Mizushina et al. [J. Chem. Engng. Jpn. 6 (1973) 487] in which the effect of external pulsation on the turbulence “bursting” process was investigated

Computations of post-trip reactor core thermal hydraulics using a strain parameter turbulence model

Abstract Coolant flows in the cores of nuclear reactors consist of ascending vertical flows in a large number of parallel passages. Under post-trip conditions such heated turbulent flows may be modified strongly from the forced convection condition by the action of buoyancy, in particular exhibiting impaired levels of heat transfer with respect to corresponding forced convection cases. The heat transfer performance of these ‘mixed convection’ flows is investigated here using two physically distinct eddy viscosity turbulence models: the recent ‘strain parameter’ (or k – e – S ) model of Cotton and Ismael [A strain parameter turbulence model and its application to homogeneous and thin shear flows. Int. J. Heat Fluid Flow 19 (1998) 326] is examined against the benchmark low-Reynolds-number k – e model of Launder and Sharma [Application of the energy-dissipation model of turbulence to the calculation of flow near a spinning disc. Lett. Heat Mass Transfer 1 (1974) 131]. Comparison is made with three sets of heat transfer data for ascending mixed convection flows, and it is demonstrated that both turbulence models are generally successful in resolving the Nusselt number distributions occurring along the lengths of mixed convection flow passages. The mechanisms by which the strain parameter model generates reduced turbulence levels, and hence impaired heat transfer rates, is explored in comparison with a fourth set of experimental data for mixed convection flow profiles.

Bed frictional characteristics in a turbulent flow driven by nonlinear waves

Abstract Waves may have highly nonlinear features that will affect the near-bed flow, particularly in shallow water regions. Here we examine such effects in both the wave analysis and the calculation of the turbulent periodic boundary layer over rough surfaces. Attention is focused upon the peak bed shear stress and the rate of energy dissipation due to bed friction. The boundary layer computation is based upon a k–e turbulence model which is of established reliability in relation to linear wave boundary layers characterized by moderate-to-high ratios of freestream particle amplitude to surface roughness. The computations reported for 30≤a/ks≤1000 apply to natural conditions of relative roughness from the lower end of the range to values in the region of 100 and above. It is found that energy dissipation rates are reduced sharply under nonlinear conditions, a result that is considered in relation to wave power. A number of questions concerning modelling strategy and numerical procedures are addressed in the course of the study.

Analytical and Reynolds stress transport model results for periodically oscillated turbulent flows

Abstract Turbulent pipe flows in which the bulk velocity is oscillated harmonically about a mean level (U b = U bo [1 + γ cos ω t]) are considered from both analytical and turbulence modelling perspectives. The analytical development is concerned with the response of the phase-averaged turbulent kinetic energy (k = k0 + k 1 cos(ω t + ψ)) to imposed unsteadiness. An approximate quasi-steady analysis is used to obtain an asymptotic low frequency limit for the modulation of the turbulent kinetic energy, k 1/γ k 0 → 1.75. The analysis is extended to examine the response of ∂k/∂t, the unsteady rate-of-change term of the k-transport equation. A Reynolds stress transport model (RSTM) of turbulence is compared with data for steady and periodic pipe flows and, in most cases, satisfactory agreement is demonstrated. When the model is run for periodic flow over a wide range of frequency it is found that a “resonant” response in k 1/γ k 0 occurs at mid-frequencies; at higher frequencies the RSTM shows the turbulence to become “frozen” at a mean condition (k 1/γ k 0→0). In the final stage of the study the RSTM is used to explore departures from the quasi-steady variation of ∂k/∂t, in particular identifying the onset of frozen behaviour in the balance of the k-equation at higher frequencies.

Method of solution of the time-dependent fully-developed energy equation.

In the early 1960s Hall, Jackson and Price published a paper in the Journal of Mechanical Engineering Science in which they obtained the fundamental solution of the fully developed energy equation in steady flow. Some attention in the same research group is now turned to the measurement and calculation of transient convective heat transfer and it is shown here that the analysis of Hall et. al. may be extended to such time-dependent flows. The analysis is restricted to forced convection conditions but applies equally to laminar and turbulent flows.

Parameterization of buoyancy effects in generic PWR boron dilution scenarios

A computational investigation is undertaken into the role of buoyancy in a PWR boron dilution transient following a postulated Small Break Loss of Coolant Accident (SB-LOCA). In the scenario envisaged there is flow of de-borated and relatively high temperature water from a single cold leg into the downcomer; flow rates are typical of natural circulation conditions. The study focuses upon the development of boron concentration distributions in the downcomer and adopts a 3D-unsteady formulation of the mean flow equations in combination with the standard high-Reynolds-number k-e turbulence model. It is found that the Richardson number (Ri = Gr/Re2 ) is the most important group parameterizing the course of a concentration transient. At Ri values characterizing a ‘baseline’ scenario the results indicate that there is a stable, circumferentially-uniform, descent through the downcomer of a stratified region of low-borated fluid. Qualitatively the same behaviour is found at higher Richardson number, although at Ri values of approximately one-fifth the baseline level there is evidence of large-scale mixing and a consequent absence of concentration stratification.Copyright