# Buoyancy-opposed wall jet

## Introduction

The topic chosen for study as the Application Challenge (AC) is that of 2-dimensional buoyancy-opposed plane wall jet penetrating into a slowly moving, counter-current uniform flow. Buoyancy-influenced flows are commonly encountered in nuclear power plants, and these can present particular challenges both to modellers and to experimentalists. One example is found within the side-core region of the steel pressure vessel in magnox gas-cooled reactor systems. In this case, a downward-flowing wall jet of hot gas meets a slow upward flow of cool gas. The downward extent of the penetration of hot gas has a significant influence on the temperature distribution in the side-core region of the reactor pressure vessel.

Experimental study of this flow has been performed at the University of Manchester by Jackson et al (2000) using a water rig. Particle Image Velocimetry (PIV) and Laser Doppler Anemometry (LDA) systems were used to study the mean flow and turbulence fields. Laser light sheet flow visualisation and PIV were used to obtain pictures of the instantaneous flow structure. Detailed measurements of local mean velocity, turbulence and temperature were then made using an LDA system incorporating a fibre optic probe and a traversable rake of thermocouples.

Following the preliminary computations of Dr Kidger, reported in Issue 1, a more extensive set of computations has been undertaken at UMIST, under the direction of Prof. Launder, Dr Iacovides and Dr Craft. The work was carried out by Mr Gerasimov, who continued to use the two-dimensional finite-volume TEAM code, developed by Huang and Leschziner (1983), with a Cartesian geometry for the flow field with a staggered grid storage arrangement. For this particular flow study, convective transport in equations for mean quantities was approximated via the QUICK scheme of Leonard (1979), but with the first order upwind scheme of Patankar (1980) in the turbulence equations. The SIMPLE pressure coupling algorithm of Patankar and Spalding (1972) was employed. As in the preliminary study, four models of turbulence have been considered. Two of these are variants of an eddy-viscosity scheme where the turbulent stresses <uiuj> are approximated by means of an isotropic apparent viscosity mt . The other two use second-moment-closure, where the turbulent stresses are evaluated by means of their own transport equations:

• Model 1: Eddy-Viscosity scheme with wall functions (KEWF)

• Model 2: Low-Reynolds-number Eddy-Viscosity scheme (LRKE)

• Model 3: Basic Second Moment Closure (Basic2mc)

• Model 4: Two-Component Limit Second Moment Closure (TCL2mc)

The modelling of near-wall turbulence within the computationally efficient High-Reynolds-number models (models 1, 3 and 4) received particular attention, because of the important role of the near-wall flow in the overall flow development and of the impracticality (and perhaps impossibility) of adopting a low-Reynolds-number approach for handling the sub-layer. In addition to the conventional wall-function strategy, a novel approach has been developed and introduced. As in the conventional approach, the near-wall control volume is large enough for the near-wall node to lie outside the viscosity-dominated sub-layer. In contrast to the conventional wall-function, the log-law, and the other associated assumptions, are no longer used to provide the wall shear stress to the integral form of the momentum equation and the modifications to the turbulent kinetic energy equation over the near-wall control volume. These are instead obtained from the analytical solution of the near-wall form of the momentum equation, using a prescribed variation of the turbulent viscosity. This novel wall-function strategy has been extended to take into account effects of buoyancy, changes in molecular viscosity due to temperature, streamwise velocity change and high Prandtl number.

During the final stages of this test flow, LES predictions, carried out under the direction of Professor Laurence, have also become available, Addad et al (2003). The objective of these simulations is to supplement the experimental data by providing more extensive data against which to assess the effectiveness of the RANS computations.

The relative simplicity in the particular flow geometry coupled with the complexity of the flow phenomena will, hopefully, enable useful evaluations to be made of turbulence models and computational formulations that are used in the study of buoyancy influenced thermal convection.

## Relevance to Industrial Sector

Buoyancy-influenced flows are commonly encountered in nuclear reactor systems. One example is a buoyancy-opposed plane wall jet flow which is found within the side-core region of the steel pressure vessel in magnox gas-cooled reactor systems.

The reactor pressure-vessel temperatures form part of the input to the structural integrity assessment of the vessel, upon which the safety case for continued operation of the plant is based. Modelling is often used in nuclear plant applications such as this to supplement sparse plant measurements, and quantification of the uncertainties associated with the calculated thermofluid loading of structures is generally needed.

## Design or Assessment Parameters

The preliminary computational experiments performed have raised several numerical and modelling issues. The two main problems encountered so far are the choice (and calibration) of the most appropriate turbulence model as well as an issue of sensitivity of convergence relative to the boundary conditions. For that reason it is flow topology parameters rather than heat transfer parameters that are chosen to assess the quality of the numerical simulations at this initial stage, namely:

• The jet spreading rate (distance from the wall where the mean velocity becomes half the local maximum velocity)

• The jet penetration depth

## Flow Domain Geometry

Figure 1 shows the general arrangement of the test section used in the study of Jackson et al (2000). The test section, which was a vertical passage of breadth 1.2m, width 0.3m and height 2.3m, had transparent walls to enable laser optical measurements of the flow to be made. As shown in Figure 1, a plane jet of warm water issuing downwards from a 18mm gap between a thick glass plate and one wall of the test section encountered a slowly ascending stream of cooler water. The width of the splitter separating the jet from the ascending stream is 20mm. Spreading of the jet and deceleration of the flow occurred until it eventually turned upwards on joining the counter-current stream. The combined flow was withdrawn from the top of the section. The aspect ratio of the jet was 67:1 and that of the test section 4:1. Consequently, the flow pattern was approximately two-dimensional.

Figure 1. Experimental rig

## Flow Physics and Fluid Dynamics Data

Experiments (Jackson et al, 2000) were conducted under both isothermal and non-isothermal conditions. The flow field was generally rather unsteady, particularly under conditions of small buoyancy influence. The temperature varied with time in a highly intermittent manner in the vicinity of the regions where the warm fluid from the jet encountered cooler fluid from the counter-current stream. Buoyancy had the effect of stabilising the flow to some extent and reducing the extent of the mixing region. Jet penetration and lateral spread were both systematically reduced as a result of increase in buoyancy influence.

When the buoyancy influences were strong, a very concentrated mixing layer was formed at the interface between the two flow streams. The turbulence field was modified accordingly and turbulence intensity peaked in this layer.

For a buoyancy-influenced flow of the kind considered here two dimensionless groups appear in the governing equations when they are represented in non-dimensional form. These are the Reynolds number and Grashof number, which are defined respectively as:

${\displaystyle Re={\frac {{\bar {V}}_{j}^{D}}{\nu _{j}}}}$

and

${\displaystyle Gr={\frac {gD^{3}(\rho _{c}-\rho _{j})/\rho _{j}}{\nu _{j}^{2}}}}$

where subscript j refers to jet values and c refers to counter-current values.

For buoyant jet flows the Richardson number Ri (= Gr / Ge2), which can be thought of as being the ratio of buoyancy force to inertia force, is sometimes used in place of the Grashof number. It can be expressed as:

${\displaystyle Ri={\frac {gD(\rho _{c}-\rho _{j})/\rho _{j}}{{\bar {V}}_{j}^{2}}}}$

Another parameter needed to characterise the flow under consideration here is the ratio of background velocity to jet velocity ${\displaystyle V_{chan}/V_{jet}}$. Jackson et al. initially fixed this ratio at a value of about 0.077 which therefore guaranteed that, for most of the cases considered, the effect of buoyancy dominated that of the counter-current flow. Subsequently measurements were also produced for a ${\displaystyle V_{chan}/V_{jet}}$ of 0.15.