# UFR 3-07 Best Practice Advice

# Natural and mixed convection boundary layers on vertical heated walls (B)

Underlying Flow Regime 3-07 © copyright ERCOFTAC 2004

# Best Practice Advice

## Best Practice Advice for the UFR

### Key Physics

For buoyancy aided flows, the wall heating gives the near-wall fluid a buoyant up-thrust. This causes a local velocity maximum and a rapid decrease of shear stress with distance from the wall. This leads to a thickening of the viscous sub-layer and a reduction of Nusselt number below that found in fully developed forced convection pipe flow. However, at higher heat loadings, heat transfer can actually increase. This happens if the buoyancy force is appreciable enough for “inverse transition” to occur, when a velocity maximum occurs close to the wall and, as a result of which, the sign of the shear stress reverses in the central region of the pipe and its absolute value increases. The wall layer then exerts an upward force on the core, which leads to an increase in turbulent kinetic energy production and consequent heat transfer enhancement.

For buoyancy opposed flows, buoyancy forces near the heated wall act in the direction opposing the main flow and increase turbulence production. Consequently the heat transfer is always enhanced for this case.

### Generality of Advice Given

The BPA is based on computations from two independent sources, both University based. The test case has not been the subject of a CFD comparison exercise.

### Numerical Issues

*Discretisation method*

Use a higher order scheme (second order or above).

*Grid Resolution*

For low-Reynolds number model simulations right up to the wall, it is necessary to resolve the flow and temperature variation near the wall, and for this, the value of y^{+}, for the near wall node, should be no larger than 2.

For simulations using analytic wall functions, the near wall node should have a y^{+} of less than 150 (because simulations have not been undertaken for larger values than this).

### Computational Domain and Boundary Conditions

The flow should be calculated upstream of the start of the heated section to minimise entry effects at the start of the heated section. At the model inlet, apply an isothermal fully developed flow. The outlet should be positioned downstream of the heated section. For the solution of the thermal energy equation, apply a uniform heat flux to the heated section. Upstream of the test section, the pipe should be adiabatic.

### Physical Modelling

*Turbulence Modelling*

Use the low-Reynolds number k-ε turbulence model of Launder and Sharma (1974), including property variation with temperature. It is also likely that higher order turbulence models are acceptable, but only in conjunction with an analytic wall function. However this has not been checked.

*Near Wall Model*

The choice of wall model is the most important consideration to obtain accurate predictions of flow in a heated vertical tube. The low-Reynolds number model equations can be solved up to the wall. However, this is an expensive approach because it requires resolution of the flow and temperature down to the wall. The alternative is to use an analytical wall function that captures the important physical phenomena, in conjunction with the solution of the low-Reynolds number model equations outside the wall layer. The analytic wall function incorporates the effects of;

(a) convection parallel and normal to the wall,

(b) fluid property variation across the wall layer, including a parabolic variation of molecular viscosity across the viscous sub-layer,

(c) the inclusion of any pressure gradient and buoyancy force,

(d) viscous sub-layer thickening and thinning,

(e) a Prandtl number which may not be close to unity (e.g. water at room conditions) and its significant variation with temperature.

### Application Uncertainties

The alternative approaches suggested above will give good results for most flows. However, if there is weak buoyancy influence and the properties vary significantly near the wall then the results would under-predict the experimental values by ~20%.

The standard k-ε model, using conventional wall functions, may give reasonable predictions only at fairly low heat loadings. However, this practice will under-estimate the enhancement or reduction in heat transfer compared with the equivalent forced flow. In extreme situations, especially if the Prandtl number is different from unity and varies significantly with temperature, conventional wall functions produce absolutely unrealistic and misleading results.

### Recommendations for Further Work

Application of second order turbulence models (e.g. Reynolds stress models), in conjunction with the analytic wall function.

### Acknowledgements

The experimental work described here was undertaken in the Engineering Department at Manchester University, by a team led by Prof. Derek Jackson. Some of the numerical work presented was undertaken in the Mechanical Engineering Department at UMIST, by a team led by Prof. Brian Launder. Alexey Gerasimov and Jian Kang Li have both reviewed this document. Dr. Jeremy Noyce (Magnox Electric) formally reviewed the document for the QNET CFD project.

### Author contact details

Dr Michael Rabbitt, British Energy, E-mail: mike.rabbitt@british-energy.com.

© copyright ERCOFTAC 2004

Contributors: Mike Rabbitt - British Energy