Equation Used in Aircraft Design Found to Be Flawed
By Lynn Yarris, LCYarris@LBL.gov April 25, 1996

BERKELEY -- Planes do not always stay aloft. Engines have been known to fail. The search for what went wrong has not always yielded answers. An explanation that may solve some of these mysteries has been brought forward by two Berkeley mathematicians.

The "Law of the Wall," an equation that for the past 60 years has served as a basis for the design of aircraft, engines, wind tunnels, compressors, even the space shuttle and many other devices subjected to the stresses of turbulence can produce substantially erroneous predictions.

This is the assertion of Alexandre Chorin and Grigory Barenblatt. Chorin holds a joint appointment with the Ernest Orlando Lawrence Berkeley National Laboratory (Berkeley Lab) and the University of California at Berkeley (UCB). Barenblatt is a professor of fluid mechanics at Cambridge University, who is visiting at UCB's Miller Institute for Basic Research.

Since 1938, the Law of the Wall has been the standard equation presented in aerodynamics and engineering textbooks for calculating the forces exerted on a solid object or "wall" by turbulence. This equation has been so thoroughly accepted as correct that when data from wind tunnel experiments did not agree with the Law's predictions, researchers sought ways to correct their experimental methods, such as polishing the walls of their wind tunnels.

However, according to the calculations of Chorin and Barenblatt, under the high-speed turbulent conditions that commonly occur in many industrial situations as well in aerodynamics, the Law of the Wall breaks down. In some cases, the forces exerted by turbulence can be as much as 65-percent greater than what the Law of the Wall predicts.

"The Law of the Wall was viewed as one of the few certainties in the difficult field of turbulence and now it has been dethroned," Chorin says. "Generations of engineers who learned the law will have to abandon it. Many textbooks will have to be revised."

Test results from the latest and most accurate wind tunnel experiments support the theoretical work of Chorin and Barenblatt. Chorin says a key to their theoretical breakthrough was to ask the right question for experimental data to answer so that the Law of the Wall could be checked.

"For years, people have looked at turbulent flow without understanding what they saw," he says. "The theory we developed led to predictions of a beautiful and complex structure in turbulence near walls that could be seen in the experimental data."

In mathematical terms, Chorin and Barnblatt say that the correct descriptions of the effects of turbulence on solid objects requires a family of "scaling laws" rather than a single "Law of the Wall." They offer such a family and explain the failure of the Law of the Wall in an upcoming Proceedings of the National Academy of Sciences.

Berkeley Lab is a U.S. Department of Energy national laboratory located in Berkeley, California. It conducts unclassified scientific research and is managed by the University of California.