One-dimensional turbulent mass transfer at air-water interfaces: details of discontinuities of derivatives using the RSW method

Abstract

The one-dimensional turbulent mass transfer was quantified using the nonlinear unclosed statistical governing equations derived from the traditional statistical methods. Further, this study considers the a priori simplifications of the bimodal Random Square Waves (RSW) approximation. This model enables the formulation of parametric equations for the variables of the statistical governing equations, but intrinsic aspects of this method still need to be clarified. In this sense, this study considers details of a version of the RSW equation for turbulent mass transfer which uses a constant reduction function. The mentioned details are related to discontinuities of higher order derivatives of the nondimensional concentration profile in the vicinity of a singular point. Numerical integrations were conducted at the air-water interface in both directions: from the liquid to the gas phase, and from the gas to the liquid phase. The results suggest that the continuity and smoothness of the calculated concentration profile does not depend on the discontinuities of the higher order derivatives, but the behaviour of the higher order derivatives depends strongly on the values of the nondimensional parameters existing in the formulation. Adequate boundary conditions were defined in the liquid phase, involving a new condition for the second derivative of the concentration profile. A fourth order Runge-Kutta method was used. This paper presents the proposed methodology; the comparison between results of calculated and measured concentration profiles,

Turbulence aspects of mass transfer in the thin interfacial region of the concentration boundary layer in gas–liquid systems

“…Data of Ref [6],. presented as a grey cloud, were Experimental data of α varying along z*, and the two constant values used here.compared with the calculated profiles, and the agreement with the solution obtained for θ 1 =2 is remarkable ([16,17]). 3 STATISTICAL FUNCTIONS FOR EQUATIONS (5A) AND (5B)3.1 Central Moments: the Concentration RMS Function n, α, β, and ω 2 allow obtaining any statistical function related to mass transfer for the boundary layer situation under study.…”

Solutions of scalar mean profiles close to gas–liquid interfaces under turbulent free slip motion

“…Numerical solutions for transient cases are found in Schulz et al [26,27], Lopes Jr. and Schulz [28], Gonçalves and Schulz [29] and Gonçalves [30]. Eqn (2) is an equivalent integrated second order equation obtained for eqn (1), described by Souza et al [31].…”

“…Using a so called constant "reduction function, ", the 1D scalar transport is governed by a non-linear third order differential equation. Transient cases were studied numerically by Lopes Jr. and Schulz [28] and Gonçalves and Schulz [29], discussing discontinuities of the higher derivatives.…”

Turbulent One-dimensional Interfacial Scalar Transport: Statistical Random Square Waves Solution

“…Theoretical improvements (Janzen 2006;Schulz et al 2010;Lopes Jr 2012) gradually empowered the formulation, so that the statistical functions defined in the RSW method could be adequately used in the scalar advection-diffusion equation. Normalized analytical solutions were then obtained for the interfacial mass transport (Gonçalves and Schulz 2013;Gonçalves 2014;Schulz et al 2018), firstly for the stationary regime (steady state) and a constant reduction function (a basic statistical element of the RSW method). In the sequence, for more general situations (transient flows with variable reduction function), the possibility of using Taylor series to obtain adequate concentration profiles and related statistical parameters was introduced (Lavín and Schulz 2019; Lavín 2020).…”