Related Experiment Video
Updated: Jun 28, 2026
12:43
Parallel-plate Flow Chamber and Continuous Flow Circuit to Evaluate Endothelial Progenitor Cells under Laminar Flow Shear Stress
Published on: January 17, 2012
The critical layer in pipe flow at high Reynolds number
1Mathematics Department, University of Michigan, Ann Arbor, MI 48109, USA. divakar@umich.edu
Summary
We computed traveling wave solutions for pipe flow, revealing that stability increases with higher Reynolds numbers (Re). These solutions exhibit streaks and rolls, with a critical layer forming at large Re.
Area of Science:
- Fluid dynamics
- Computational physics
- Nonlinear dynamics
Background:
- Pipe flow is a fundamental problem in fluid dynamics.
- Understanding flow stability is crucial for various engineering applications.
- Lower branch solutions in shear flows often exhibit complex structures like streaks and rolls.
Purpose of the Study:
- To compute traveling wave solutions for pipe flow up to high Reynolds numbers (Re).
- To analyze the stability of these solutions and their behavior as Re approaches infinity.
- To investigate the universality of observed phenomena in lower branch solutions of shear flows.
Main Methods:
- Computation of traveling wave solutions using GMRES-hookstep and Arnoldi iterations.
- Analysis of solution spectra to determine linear stability.
- Asymptotic analysis of solutions in the limit of large Reynolds numbers.
Main Results:
- Traveling wave solutions with prominent streaks and rolls were computed up to Re=75000.
- For large Re, solutions develop a critical layer away from the wall.
- Despite linear instability, solutions become more stable as Re increases, with unstable eigenvalues approaching zero at specific rates (Re^-0.41 and Re^-0.87).
Conclusions:
- The study reveals a counterintuitive increase in stability for pipe flow solutions at high Reynolds numbers.
- The formation of critical layers and asymptotic behavior may be universal to lower branch solutions in shear flows.
- The computational methods employed provide insights into the dynamics of pipe flow.
Related Concept Videos
01:22General Characteristics of Pipe Flow I
Pipe flow refers to the movement of fluids within fully enclosed conduits, typically cylindrical in shape, such as water pipes or hydraulic hoses. These conduits are designed to withstand high-pressure gradients that drive fluid movement, contrasting with open-channel flows, where gravity is the primary driving force. Rectangular conduits, like air conditioning and heating ducts, generally operate at lower pressures and are less suited for high-pressure applications.
The classification of fluid...
The classification of fluid...
01:10Poiseuille's Law and Reynolds Number
Any fluid in a horizontal tube can flow due to pressure differences—fluid flows from high to low pressure. The flow rate (Q) is the ratio of pressure difference and resistance through a horizontal tube. The greater the pressure difference, the higher the flow rate. The flow resistance is expressed as:
01:27Laminar Flow
Laminar flow represents a smooth, orderly fluid motion where particles move along parallel paths, resulting in minimal mixing between layers. Streamlined particle paths characterize this flow regime and occur under conditions where viscous forces dominate over inertial forces. The distinction between laminar, transitional, and turbulent flow is primarily determined by the Reynolds number, a dimensionless quantity calculated as:
01:24Laminar Flow: Problem Solving
Laminar flow occurs when a fluid moves smoothly in parallel layers with minimal mixing and turbulence. In fluid mechanics, ensuring laminar flow within a pipe is essential for precise control of flow characteristics, especially in engineering applications. The key factor in determining whether flow remains laminar is the Reynolds number, a dimensionless quantity that depends on the fluid's velocity, density, viscosity, and the pipe's diameter. A Reynolds number of 2100 or lower indicates...
01:24General Characteristics of Pipe Flow II
When fluid enters a pipe, it first passes through the entrance region, where the velocity profile adjusts due to viscous effects. In this region, a boundary layer forms along the pipe walls and grows until it fully occupies the pipe's cross-section. Once the boundary layer merges, the flow becomes fully developed, with a steady velocity profile that remains consistent along the pipe's length.
The distance to reach a fully developed flow is called the entrance length and depends on the flow...
The distance to reach a fully developed flow is called the entrance length and depends on the flow...
01:09The Buckingham Pi Theorem
The Buckingham Pi theorem provides a structured method to simplify fluid dynamics problems by reducing complex systems of variables to dimensionless terms.