[Paper Review] Quasi-periodically developed flow in channels with arrays of in-line square cylinders

Geert Buckinx, Arthur Vangeffelen|arXiv (Cornell University)|Jun 25, 2023

Fluid Dynamics and Vibration AnalysisEngineering3 citations

TL;DR

This study investigates quasi-periodically developed laminar flow in channels with in-line square cylinders using direct numerical simulations (DNS). It identifies that velocity and pressure modes decay exponentially along the flow direction, governed by an eigenvalue problem generalizing the Orr-Sommerfeld equation, and demonstrates that the quasi-periodic region dominates the flow development zone, explaining scaling laws for the onset of periodically developed flow.

ABSTRACT

In this stub article, we show that laminar quasi-periodically developed flow is characterized by velocity and pressure modes which decay exponentially along the main flow direction. As the amplitudes of these modes exhibit streamwise periodicity, they can be determined on a single transversal row of the array. Their shape and corresponding decay rate are governed by an eigenvalue problem which generalizes the Orr-Sommerfeld equation for quasi-developed Poiseuille flow. By means of full-scale channel flow simulations, we numerically investigate the onset point, extent, eigenvalues and perturbation sizes of quasi-periodically developed flow in channels with equidistant in-line square cylinders, assuming a parabolic inlet velocity profile. In particular, the dependence on the mass flow rate, the aspect ratio and height of the channel is discussed, considering three porosities of the cylinder array (0.75, 0.89 and 0.94), and Reynolds numbers from 20 to 300, based on a reference length of twice the channel height. It is observed that the region of quasi-periodically developed flow covers a large part of the region of flow development in the channel. Therefore, the corresponding eigenvalues explain largely the observed scaling laws for the onset point of periodically developed flow.

Motivation & Objective

• To characterize the features of quasi-periodically developed flow in channels with periodic arrays of in-line square cylinders.
• To determine the onset and extent of quasi-periodic flow upstream of the periodically developed regime.
• To quantify the eigenvalues and perturbation amplitudes of the decaying modes governing the flow development.
• To investigate the dependence of flow development on Reynolds number, channel aspect ratio, and porosity of the cylinder array.
• To provide reference data for the onset of periodically developed flow in simple geometries, filling a gap in the literature.

Proposed method

• Conduct full-scale direct numerical simulations (DNS) of channel flows with equidistant in-line square cylinders using structured meshes of 140–250 million cells.
• Use a parabolic inlet velocity profile and solve the incompressible Navier-Stokes equations with a finite-volume method and iterative solvers.
• Apply a volume-averaging approach to extract macro-scale pressure gradients and define the periodically developed flow regime.
• Formulate an eigenvalue problem generalizing the Orr-Sommerfeld equation to describe the decay and shape of velocity and pressure modes in the quasi-periodic region.
• Reconstruct the flow field from velocity data on a single transversal row of the array using the eigenmodes and scaling factors.
• Perform mesh convergence studies with coarser grids (34M, 88M, 180M cells) to ensure numerical reliability of the DNS results.

Experimental results

Research questions

• RQ1 How do the onset and extent of quasi-periodically developed flow depend on Reynolds number, channel aspect ratio, and porosity in arrays of in-line square cylinders?
• RQ2 What is the role of the eigenvalues and decay rates of velocity and pressure modes in characterizing the quasi-periodic flow region?
• RQ3 To what extent does the quasi-periodic flow region dominate the flow development zone, and how does it influence the onset of periodically developed flow?
• RQ4 How do the perturbation amplitudes of the eigenmodes scale with Reynolds number and geometric parameters?
• RQ5 Can the eigenvalue problem derived from the generalized Orr-Sommerfeld equation accurately describe the decaying modes in the quasi-periodic flow region?

Key findings

• The region of quasi-periodically developed flow covers a large portion of the flow development zone in the channel, indicating its dominant role in the transition to periodic development.
• Velocity and pressure modes decay exponentially along the streamwise direction, with decay rates and shapes determined by an eigenvalue problem that generalizes the Orr-Sommerfeld equation.
• The eigenvalues of the quasi-periodic modes explain a significant part of the observed scaling laws for the onset point of periodically developed flow.
• For Reynolds numbers between 20 and 300, and porosities of 0.75, 0.89, and 0.94, the onset of quasi-periodic flow is consistently observed upstream of the periodically developed region.
• The perturbation amplitudes of the eigenmodes are sensitive to the channel height and width, with larger aspect ratios leading to stronger mode amplitudes and longer development lengths.
• The scaling factor $ C_{m{ ext{U}}} $, which relates the macro-scale velocity to the eigenmode amplitude, is independent of the mass flow rate, confirming that only the periodic component contributes to the bulk flow.