## Fluid flow through small channels in building envelopes

Jax Kneppers Associates (JKA) specializes in forensic investigation of construction problems, from construction management and scheduling failures to water damage and manufacturing flaws to retrofits on the Golden Gate Bridge, they have investigated a wide variety of practical problems in the building and construction industry.

In one particular problem JKA was interested in whether water that leaked through a hole in a particular siding product could pool in a channel formed by the overlapping of two siding courses and travel significant distances laterally along the wall.

During the investigation a physical model of the system was built and the water migration pattern was observed in this case. However, this gave results for only one configuration. To identify how the water might have migrated under alternative conditions, a mathematical model of open channel flow was developed and investigated via a parametric study in which parameters were varied and the variation in results plotted. Which variables had the biggest effect on the lateral extent of migration? The mathematical model allowed us to determine the answer to these questions and guided the follow-up of the physical testing.

The model involved a one dimensional partial differential equation that incorporated the forces of gravity, the normal force on the bottom of the channel, the leakage of water through gaps in the channel, and the surface tension and surface wetting resistance of the material. Some of the results were counterintuitive, such as predicting that in the presence of leakage, there would be less migration when the slope of the channel was larger (for slopes that are all near zero) due to the more rapid increase in water depth accelerating the leakage.

The mathematical model allowed us to investigate how the different parameters affected the extent of water damage and led to several practical consequences:

1. Coating the back of the product in water repelling material would not only protect the material but increase the surface resistance to wetting which would reduce the migration of water.
2. Spray painting the siding would lead to sealing up the channels and with dramatically reduced drainage would lead to accumulation of water and large lateral spreading.
3. Purposefully providing scallops in the channels could reduce the Bond number $\rho g L^2/\sigma$ allowing for much better drainage. Since this is proportional to the length scale squared, small increases in the size of drainage openings, from say 1mm to 5mm could have dramatic effects (25 times the drainage).
4. As always, both manufacturing and installation need to work together. A product designed for a specific installation method which is installed with what seems like a minor variance in method to the builder (spray painting) can nevertheless fail dramatically.