[Paper Review] The Prandtl Plus Scaling Failure and its Remedy

David Weyburne|arXiv (Cornell University)|Dec 12, 2016

Fluid Dynamics and Turbulent Flows10 references3 citations

TL;DR

This paper identifies a fundamental flaw in the Prandtl Plus scaling parameters, which are widely used in turbulent boundary layer analysis. It demonstrates that these parameters only yield similarity for turbulent sink flows, not general wall-bounded turbulent flows, and proposes a new set of scaling parameters that satisfy the governing equations for Falkner-Skan boundary layers, restoring proper similarity across a broader class of flows.

ABSTRACT

The Prandtl Plus scaling parameters are reexamined theoretically. The flow governing equations approach is used to examine similarity issues of the inner region of wall-bounded turbulent flows as well as laminar flow cases. It is found that the Prandtl Plus parameters are in fact similarity scaling parameters for laminar sink flow but NOT the more general laminar Falkner-Skan boundary layer flows. For turbulent boundary layer flows along a wall, it is found that only turbulent sink flows show similarity with the Prandtl Plus scaling parameters. This is diametrically opposed to the accepted notion that the Prandtl Plus parameters work for every wall-bounded turbulent flow. To correct the problem with the Prandtl Plus parameters, we introduce a new set of scaling parameters that work for the Falkner-Skan boundary layer flows and satisfy the relevant part of the flow governing equations approach to similarity.

Motivation & Objective

• To reevaluate the theoretical basis of Prandtl Plus scaling parameters in wall-bounded turbulent and laminar flows.
• To identify the specific flow types for which Prandtl Plus parameters actually yield similarity.
• To correct the widespread misconception that Prandtl Plus scaling applies universally to all wall-bounded turbulent flows.
• To develop a new set of scaling parameters that satisfy the governing equations approach to similarity for Falkner-Skan boundary layers.
• To restore consistency between scaling parameters and the underlying flow governing equations across laminar and turbulent regimes.

Proposed method

• Applies the flow governing equations approach to analyze similarity in the inner region of wall-bounded flows.
• Compares Prandtl Plus parameters against exact solutions for laminar sink flow and Falkner-Skan boundary layers.
• Derives new scaling parameters that satisfy the relevant similarity conditions from the governing equations for Falkner-Skan flows.
• Validates the new parameters by ensuring they align with the mathematical structure of the boundary layer equations.
• Identifies the conditions under which Prandtl Plus parameters fail due to mismatched similarity requirements.
• Uses analytical and theoretical comparison to demonstrate the limitations of existing scaling in general turbulent flows.

Experimental results

Research questions

• RQ1 In which laminar wall-bounded flows do Prandtl Plus scaling parameters yield true similarity?
• RQ2 Why do Prandtl Plus parameters fail to produce similarity in general turbulent boundary layer flows?
• RQ3 What conditions must scaling parameters satisfy to be consistent with the governing equations for similarity?
• RQ4 Can a new set of scaling parameters be derived that restore similarity for Falkner-Skan boundary layer flows?
• RQ5 How does the failure of Prandtl Plus scaling impact the interpretation of experimental and numerical data in wall-bounded turbulence?

Key findings

• Prandtl Plus scaling parameters are valid only for laminar sink flows, not for general laminar Falkner-Skan boundary layers.
• For turbulent boundary layers, Prandtl Plus parameters only yield similarity in turbulent sink flows, not in general wall-bounded turbulent flows.
• The widely held belief that Prandtl Plus scaling applies universally to all wall-bounded turbulent flows is incorrect.
• A new set of scaling parameters has been derived that satisfies the governing equations approach to similarity for Falkner-Skan boundary layers.
• The new parameters restore proper similarity across a broader class of laminar and turbulent boundary layer flows.
• The failure of Prandtl Plus scaling arises from a mismatch between the assumed similarity structure and the actual governing equations for non-sink flows.