Drag Reduction by Polymers in Wall Bounded Turbulence

Abstract

We elucidate the mechanism of drag reduction by polymers in turbulent wall-bounded flows: while momentum is produced at a fixed rate by the forcing, polymer stretching results in the suppression of momentum flux to the wall. On the basis of the equations of fluid mechanics we develop the phenomenology of the "maximum drag reduction asymptote" which is the maximum drag reduction attained by polymers. Based on Newtonian information only we demonstrate the existence of drag reduction, and with one experimental parameter we reach agreement with the experimental measurements.

Identification and Calculation of the Universal Asymptote for Drag Reduction by Polymers in Wall Bounded Turbulence

“…drag reduction. It is easy to argue [4] that the slope of the new log-law is larger that the slope in vonKármán's log-law, and hence drag reduction is obtained. Nevertheless, since neither c V nor the constants a and b in (8) are known apriori, the actual slope of the MDR could not be determined.…”

“…The law (1) is universal, independent of the nature of the Newtonian fluid; it is one of the shortcomings of the theory of wall-bounded turbulence that the von-Kármń constant κ K ≈ 0.436 and the intercept B ≈ 6.13 are only known from experiments and simulations [1,2]. One of the most significant experimental findings [3] concerning turbulent drag reduction by polymers is that in channel and pipe geometries the velocity profile (with polymers added to the Newtonian fluid) is bounded between von-Kármán's log-law and another log-law which describes the maximal possible velocity profile (Maximum Drag Reduction, MDR) [4,5,6,7],This law, which had been discovered experimentally by Virk (and hence the notation κ V ), is also claimed to be universal, independent of the Newtonian fluid and the nature of the polymer additive, including flexible and rigid polymers [8]. The numerical value of the coefficient κ V is presently known only from experiments, κ V −1 ≈ 11.7, giving a phenomenological MDR law in the form [3] V + (y + ) = 11.7 ln y + − 17 .…”

“…The law (1) is universal, independent of the nature of the Newtonian fluid; it is one of the shortcomings of the theory of wall-bounded turbulence that the von-Kármń constant κ K ≈ 0.436 and the intercept B ≈ 6.13 are only known from experiments and simulations [1,2]. One of the most significant experimental findings [3] concerning turbulent drag reduction by polymers is that in channel and pipe geometries the velocity profile (with polymers added to the Newtonian fluid) is bounded between von-Kármán's log-law and another log-law which describes the maximal possible velocity profile (Maximum Drag Reduction, MDR) [4,5,6,7],…”

“…To derive the functional form of the MDR [4] one asserts that the terms containing ν(y) in the balance equations (7) and (8) overwhelm the inertial terms, and then together with (10) one derives S(y) ∼ Const./y which is the log-law for the MDR. Consisten with this law the effective viscosiy turns to be linear in y.…”

Phenomenology of wall-bounded Newtonian turbulence

“…We note that in previous applications [6][7][8][9][10][11][12][13][14] we have employed a model in which only the trace of W (y) and its xy component were kept in a simplified description. For the present purposes we consider all the component of this tensor, paying a price of having more equations to balance, but reaping the benefit of a significantly improved phenomenology.…”