A Hyperbolic Tangent Damping Function for the Turbulence Model: Mathematical Formulation and Analytical Properties

Abstract

The standard \(k-\varepsilon\) turbulence model as we know, requires damping functions to correct the over - prediction of eddy viscosity in low turbulent Reynolds number regions. The existing damping functions, such as the Launder - Sharma exponential formulation are mathematically complex, containing multiple empirical constants, and lack rigorous and analytical characterization. This paper therefore proposes and mathematically analyzes a novel damping function \(f_\mu=\tanh \left(\frac{R_{e t}}{A}\right)\), where \(R_{e t}=\frac{k^2}{\nu \varepsilon}\) is the turbulent Reynolds number and \(A>0\) is a single calibration constant. In this paper, the hyperbolic tangent function is shown to possess all essential mathematical properties for a damping function: boundedness ( \(0 \leq f_\mu<1\) ), smoothness (infinitely differentiable), monotonicity (strictly increasing in \(R_{e t}\) ), and correct asymptotic behavior \(\left(f_\mu \rightarrow 0\right.\) as \(R_{e t} \rightarrow 0 ; f_\mu \rightarrow 1\) as \(\left.R_{e t} \rightarrow \infty\right)\). Closed form expressions are derived for all derivatives, enabling analytical sensitivity. The function is compared mathematically with existing damping functions, revealing that the tanh formulation achieves comparable or even superior mathematical properties with single parameter simplicity. This research work has established the rigorous mathematical foundation for the proposed novel damping function, positioning it as a better alternative to existing low Reynolds number corrections.