The law of the wall. Indications from DNS, and opinion

Abstract

The law of the wall and its companion the logarithmic law are at the center of our knowledge of wall turbulence, and after almost a century still the subject of intense speculation and proposed amendments, based on experimental and simulation results, as well as putative theories. In particular, the consensus over the value of the Karman constant κ has been lost. However, compensation by the additive constant C is such that the two main contenders cross at y+ = 510, and differ by 1 unit in U+ only when y+ = 40,000; thus, the effect is modest in practice, and any errors in skin-friction measurement are magnified. It has further been suggested that κ has different values in different flows, such as boundary layers and pipes, and in different pressure gradients. This seeming amendment amounts to abandoning the concept of a law of the wall, which would be very damaging to our prediction power in wall-bounded flows. At the same time, it is noted again that the law of the wall fails even for the simplest quantities other than the mean velocity. A model problem is described which generates log laws with artificial values of κ, both for the true flow and for RANS CFD, but only as long as the flows in question are unsettled. Results of Direct Numerical Simulations (DNS) with different enough numerics are found to be consistent with uniqueness of the laws, but these results are still narrow-based and unable to indicate the value of κ, even after the very consequent increase in Reynolds number since the 1980s. They have firmly established the “DNS answer” for the law of the wall well beyond 100 wall units, but not up to 500, and there is now agreement that the log law is entered only near 300 at best, and is entered from above contrary to previous expectations. The general message is to acknowledge the weakness of theory and DNS but to reason, without a proof, that the credibility of the twin laws for mean velocity is intact, and in particular that a true and unique value of κexists.