# Downward flow in a heated annulus

## Introduction

This AC concerns turbulent downward flow in an annulus with a uniformly heated core and an adiabatic outer casing. An investigation is made on the influence of buoyancy on mixed convection flow, heat transfer and turbulence. The Reynolds number of the flows ranges from 1000 to 6000, and the Grashof number (based on heat flux) ranges from 1.1x108 to 1.4x109. Such flows with strong buoyancy influences are found in a number of nuclear power plant situations, and pose a variety of challenges to experimentalists and CFD modellers.

The experimental rig is housed in the Simon Engineering Laboratory of the School of Engineering, University of Manchester. Temperatures, velocity and turbulence data have been obtained in a water flow, using thermocouples to allow the heat-transfer rates from the inner cylinder to be mapped both axially and circumferentially, and LDA to make detailed measurements of the velocity and turbulence. The experiments are reported in Jackson et al (2000, 2003)

Computational studies have been carried out at UMIST, aimed at developing and testing turbulence models for buoyancy-aided flows, with particular emphasis being placed on the representation of the near-wall region. Three-dimensional calculations with a circumferential grid covering the entire 360o of the cross-section were performed by Guy et al (1999). These, in line with experiments, showed that the flow could be treated in an axisymmetric manner, and subsequent explorations, Gerasimov (2002), have thus used the finite volume solver TEAM (Huang & Leschziner, 1983), which employs a two-dimensional or axisymmetric Cartesian geometry with a staggered grid storage arrangement.

For vertical buoyancy-aided flows it has long been known (Cotton & Jackson, 1987; Ince & Launder, 1989) that the major effects of buoyancy on the flow development can be reasonably captured with the low-Reynolds-number k-ε model of Launder & Sharma (1974). However, when a high-Reynolds-number k-ε scheme is employed with standard wall functions, the effects of buoyancy in the near-wall region are poorly reproduced. New wall functions which have recently been developed at UMIST have therefore been tested. These do not rely on the traditional log-law assumptions and can account for the effects of buoyancy on the flow structure in the near-wall layer.

In this particular challenge the flow direction is reversed compared with the above-cited examples, so that buoyant forces oppose the forced-convection circulation. While the geometry of the flow is simple, the strong buoyancy effects result in complex flow physics. The case thus provides a stringent test of the turbulence models’ performance in predicting buoyancy-influenced convective heat transfer.

## Relevance to Industrial Sector

This is a test case by which the competency of CFD for use in nuclear power stations can be judged. However, it only shows the validity for buoyancy influenced flows with vertical boundary layers. This AC is well understood due to the available experimental data.

## Design or Assessment Parameters

The main parameter of interest is the heat transfer coefficient at the inner wall, and this is particularly influenced by the flow structure of the near-wall, viscosity-affected sublayer. The modelling of this layer is thus crucial, and the computational study has therefore focused on different near-wall modeling treatments, whilst the primary parameter chosen to assess the accuracy and quality of the numerical simulations is the distribution of Nusselt number along the annulus.

## Flow Domain Geometry

Figure 1: Experimental test rig.

Figure 1 shows the arrangement of the experimental test section employed by Jackson et al (2000, 2003). The test section is a vertical passage of annular cross section which has a heated core of outside diameter 76mm and an adiabatic outer casing of internal diameter 140mm. The core is made from stainless steel which is uniformly heated by resistive means over a section of length 3m, and which is preceded by an unheated length of 1.5m and followed by a further unheated section of length 0.5m.

Thermocouples distributed along the length and around the circumference of the inner core allow the surface temperature to be mapped, whilst the outer cylinder is made of perspex, to allow optical access for LDA measurements to be made.

Water from a header tank flows to the top of the flow domain where it passes through a manifold and flow conditioning arrangement. It then flows downwards through the test section. On leaving the test section, the water passes through a further manifold from which it is drawn by a pump, which returns it to the header via an orifice plate flowmeter and a shell and tube cooler.

## Flow Physics and Fluid Dynamics Data

In these buoyancy-opposed mixed convection flows, the buoyancy has the effect of increasing the heat-transfer relative to that which would be found in forced convection at the same Reynolds number. Heat is transferred from the heated core to the downward flowing water, and the velocity magnitude in the near-wall region is thus reduced as the buoyancy becomes stronger. At sufficiently high levels of buoyancy flow reversal can occur adjacent to the wall.

The non-dimensional groups which appear when the governing equations are non-dimensionalized are the Reynolds, Prandtl and Grashof numbers, defined as:

${\displaystyle Re={\frac {U_{b}d_{\text{eff}}}{\nu }}}$

${\displaystyle Pr={\frac {\mu {C}_{p}}{k}}}$

${\displaystyle Gr^{*}={\frac {\beta {g}d_{\text{out}}^{4}q_{w}}{k\nu ^{2}}}}$

where ${\displaystyle U_{b}}$ is the bulk velocity, ${\displaystyle d_{eff}}$ the effective diameter (the difference between the outer and inner diameters, ${\displaystyle d_{out}}$ and ${\displaystyle d_{in}}$, of the annular section) and ${\displaystyle q_{w}}$ is the heat flux.

The Reynolds, Grashof and Prandtl numbers can be combined in the Buoyancy parameter Bo, identified by Jackson & Hall (1979) and Launder (1986), and defined as

${\displaystyle Bo={\frac {8\times 10^{4}Gr^{*}}{Re^{3.425}Pr^{0.8}}}}$

A high buoyancy parameter can thus either be obtained either by operating at a low flow rate (giving a low Reynolds number), or at a high heating rate (giving high Gr*).

The working fluid is water, at room pressure. The thermo-physical properties of water at atmospheric pressure are well known, and are therefore not specified here. Since the temperature variation across the boundary layer is significant, to undertake CFD calculations it may be necessary to specify fluid properties that vary with temperature (see section 3.1).