Resolvent Analysis for Turbulent Channel Flow with Riblets

Abstract

This paper extends the resolvent formalism for wall turbulence proposed by McKeon and Sharma [2010] to account for the effect of streamwise-constant riblets. Under the resolvent formulation, the Navier-Stokes equations are interpreted as a forcing-response system: the nonlinear convective term is interpreted as a feedback forcing on the remaining linear terms, which generates a velocity and pressure response. A gain-based decomposition of the linear forcing-response transfer function -the resolvent operator -yields highly amplified velocity and pressure modes, which can be considered key building blocks of the turbulent flow field. Previous work has shown that these high-gain modes provide substantial insight into turbulence statistics, structure, and control of smooth-walled flows. To introduce the effect of riblets within this framework, a linear spatially-varying body force is added to the governing equations. In other words, volume penalization is used to approximate the surface features. Predictions for spanwiseperiodic and streamwise-constant riblets show that specific high-gain modes identified from the modified governing equations reproduce observations made in prior direct numerical simulations with limited computation. The deterioration in performance with increasing riblet size is predicted well and so is the emergence of spanwise rollers resembling Kelvin-Helmholtz vortices. This new modeling framework is also used to pursue limited riblet shape optimization.

Model-based design of riblets for turbulent drag reduction

“…While the former study used a change of coordinates to translate spatially periodic geometry into spatially periodic differential operators, the latter utilized a volume penalization technique to represent the effect of riblets as a feedback term in the dynamics. Moreover, Chavarin & Luhar (2019) showed that the dependence of the resolvent gain on the spacing of riblets closely follows previously reported drag-reducing trends in turbulent flows.…”

“…In this paper we extend the framework developed in Moarref & Jovanovic (2012) to quantify the effect of riblets on turbulent channel flow. Following Chavarin & Luhar (2019), we use a volume penalization technique (Khadra et al 2000) to approximate the effect of the spatially periodic surface on the turbulent flow. This method introduces a static feedback term that captures the shape of riblets via a resistive function in the momentum equation.…”

“…The self-regular model for wall turbulence regeneration proposed by Bandyopadhyay & Hellum (2014) accounts for the spatio-temporal evolution of flow structures over patterned surfaces and matches experimental results for transitional and turbulent flows at low Reynolds numbers. More recently, the receptivity of channel flow over riblets was studied using the H 2 norm of the linearized dynamics (Kasliwal, Duncan & Papachristodoulou 2012) and the resolvent analysis (Chavarin & Luhar 2019). While the former study used a change of coordinates to translate spatially periodic geometry into spatially periodic differential operators, the latter utilized a volume penalization technique to represent the effect of riblets as a feedback term in the dynamics.…”

“…The permeability function K takes on two values: within the fluid, K → ∞ yields back the original mean flow equations (2.5a,b); and within the riblets, K → 0 forces the velocity field to zero. Following Chavarin & Luhar (2019), we account for streamwise-constant, spanwise-periodic corrugation by considering the harmonic resistance…”

“…Prior model-based efforts have shown promise in predicting the energetics of turbulent flows in the presence of riblets. However, apart from Chavarin & Luhar (2019), such studies do not account for the interactions among harmonics of flow fluctuations that are induced by spatially periodic geometry. Furthermore, in the absence of a systematic framework to quantify the influence of background turbulence on the mean velocity, prior studies cannot provide accurate predictions of skin-friction drag in the presence of riblets.…”