[Paper Review] A Theoretical Closure for Turbulent Flows Near Walls

Trinh Khanh Tuoc|arXiv (Cornell University)|Jan 14, 2010

Fluid Dynamics and Turbulent Flows40 references3 citations

TL;DR

This paper proposes a theoretical closure for turbulent flows near walls by modeling the wall layer as a transient laminar sub-boundary layer governed by the Stokes solution for an impulsively started flat plate, while matching it to a steady-state log-law solution via a damping function. The method derives the logarithmic law's coefficient theoretically and enables accurate, Reynolds-number-independent wall functions for CFD simulations.

ABSTRACT

This paper proposes a simple new closure principle for turbulent shear flows. The turbulent flow field is divided into an outer and an inner region. The inner region is made up of a log-law region and a wall layer. The wall layer is viewed in terms of the well known inrush-sweep-burst sequence observed since 1967. It is modelled as a transient laminar sub-boundary layer, which obeys the Stokes solution for an impulsively started flat plate. The wall layer may also be modelled with a steady state solution by adding a damping function to the log-law. Closure is achieved by matching the unsteady and steady state solutions at the edge of the wall layer. This procedure in effect feeds information about the transient coherent structures back into the time-averaged solution and determines theoretically the numerical coefficient of the logarithmic law of the wall The method gives a new technique for writing accurate wall functions, valid for all Reynolds numbers, in computer fluid dynamics (CFD) programmes. Keywords: Reynolds equations, modelling, closure technique, wall layer, log-law, CFD.

Motivation & Objective

• To develop a theoretically grounded closure for turbulent shear flows near walls that accounts for transient coherent structures.
• To resolve the uncertainty in the logarithmic law's coefficient by deriving it from first principles.
• To enable accurate wall functions in CFD that are valid across all Reynolds numbers.
• To unify transient wall-layer dynamics with time-averaged mean flow via matching unsteady and steady solutions.

Proposed method

• Divides the turbulent boundary layer into an outer region and an inner region comprising a log-law region and a wall layer.
• Models the wall layer as a transient laminar sub-boundary layer using the Stokes solution for an impulsively started flat plate.
• Represents the steady-state wall layer using a modified log-law with an added damping function.
• Achieves closure by matching the unsteady Stokes solution and the steady-state solution at the edge of the wall layer.
• Uses the matching condition to determine the logarithmic law's coefficient theoretically.
• Applies the resulting formulation to derive wall functions suitable for CFD simulations across all Reynolds numbers.

Experimental results

Research questions

• RQ1 How can the logarithmic law's coefficient in wall-bounded turbulent flows be derived theoretically rather than empirically?
• RQ2 What is the role of transient coherent structures like inrush-sweep-burst sequences in determining the mean wall stress?
• RQ3 Can a consistent theoretical closure be established by matching unsteady and steady solutions at the wall layer edge?
• RQ4 How can wall functions in CFD be made universally valid across all Reynolds numbers?

Key findings

• The logarithmic law's coefficient is derived theoretically through matching the unsteady Stokes solution and the steady-state log-law with a damping function.
• The method provides a physically based closure that incorporates transient coherent structures into the time-averaged mean flow.
• The resulting wall functions are valid for all Reynolds numbers, eliminating the need for empirical tuning.
• The approach establishes a theoretical link between the dynamics of wall-attached coherent structures and the mean velocity profile.