A Flaw in the Law
Berkeley Lab Mathematicians Dethrone 'Law of the Wall' for More Accurate Turbulence Calculations
January 26, 1998
Turbulence is a primal force that can be seen all around us, from the scudding of clouds to the frothing of the sea, from the wrinkling of flames to the swirling of leaves. It is especially important to the study of aerodynamics, in which correctly calculating the effects of turbulence can determine whether or not an airplane remains aloft. Since 1938, the standard equation presented in aerodynamics and engineering textbooks for calculating the forces exerted on a solid object by turbulence has been the "Law of the Wall."
This equation has been so thoroughly accepted as correct that when data from wind tunnel experiments did not agree with the Law of the Wall's predictions, researchers sought ways to correct their experimental methods.
However, according to calculations by Berkeley Lab's Alexandre Chorin and Grigory Barenblatt, under the high-speed turbulent conditions that commonly occur in aerodynamics as well as many industrial situations, it is the Law of the Wall that breaks down. The forces exerted by turbulence can be very different from what the Law would predict. A more accurate model should lead to better, safer and more economical designs.
"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 said when this discovery was announced. "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 assertions of Chorin and Barenblatt. A key to their theoretical breakthrough was to ask the right question for experimental data to answer. This enabled them to finally check the accuracy of the Law of the Wall.
"For years, people have looked at turbulent flow without understanding what they saw," said Chorin. "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 Barenblatt showed that correct descriptions of the effects of turbulence on solid objects-referred to as "walls" in aerodynamic studies-require a family of "scaling laws" rather than a single "Law of the Wall."
Mathematically, the solutions of the equations of motion in fluid mechanics are chaotic at high speeds, even the most minute perturbations change them. "Because of this," says Chorin, "what is of interest is not specific solutions, which we may never observe, but the ability to make statistical predictions based on collections of solutions. Such collections of solutions are amenable to rigorous analysis by statistical methods."
Chorin and Barenblatt have made progress in this rigorous analysis that has contributed to their debunking of the Law of the Wall. "What was a general mathematical theory has now been validated by the data in a specific problem," says Chorin. "This happy agreement of theory and experimental data constitutes a step toward a fundamental theory of turbulence."
»Download the science highlight (PDF, 2.8MB)
About Computing Sciences at Berkeley Lab
The Lawrence Berkeley National Laboratory (Berkeley Lab) Computing Sciences organization provides the computing and networking resources and expertise critical to advancing the Department of Energy's research missions: developing new energy sources, improving energy efficiency, developing new materials and increasing our understanding of ourselves, our world and our universe.
ESnet, the Energy Sciences Network, provides the high-bandwidth, reliable connections that link scientists at 40 DOE research sites to each other and to experimental facilities and supercomputing centers around the country. The National Energy Research Scientific Computing Center (NERSC) powers the discoveries of 6,000 scientists at national laboratories and universities, including those at Berkeley Lab's Computational Research Division (CRD). CRD conducts research and development in mathematical modeling and simulation, algorithm design, data storage, management and analysis, computer system architecture and high-performance software implementation. NERSC and ESnet are DOE Office of Science User Facilities.
Lawrence Berkeley National Laboratory addresses the world's most urgent scientific challenges by advancing sustainable energy, protecting human health, creating new materials, and revealing the origin and fate of the universe. Founded in 1931, Berkeley Lab's scientific expertise has been recognized with 13 Nobel prizes. The University of California manages Berkeley Lab for the DOE’s Office of Science.
DOE’s Office of Science is the single largest supporter of basic research in the physical sciences in the United States, and is working to address some of the most pressing challenges of our time. For more information, please visit science.energy.gov.