A modified defect function for wave boundary layers

In the present paper, a new defect function is proposed to predict the first harmonic motion in turbulent wave boundary layer (WBL) flows, based on the existing and newly acquired experimental evidences over a universal range of the governing flow parameters (A/k_s & Re), taking the advantage of the existing defect function models. The defect function, denoted as D(z), having a complex form, describes the spatio-temporal velocity deficit in the WBL as the only function of z. For the new defect function, two sets of model parameters (p₁ & λ₁, p₂ & λ₂) are introduced to formulate the amplitude and phase defects, namely −ln|D|=(z/λ₁)^p₁ and arg(D)=(z/λ₂)^p₂. Correlation relationships between model parameters (p₁ & λ₁, p₂ & λ₂) and governing flow parameters (A/k_s & Re) were established. Through (i) matching the logarithmic and the defect laws of velocity distribution and (ii) the ‘one particle analysis’ of turbulent diffusion process, we demonstrated that the model parameters λ₁ and λ₂ represent the characteristic length scales, and the model parameters p₁ and p₂ reflect the nonlinearities of amplitude defect and phase defect respectively. The new defect function not only leads to better predictions of the flow in the phase space than the existing models but also extends the applicable parameter ranges of the existing models.