[Paper Review] A correlation between drag and an integral property of the wake

T. S. Morton|arXiv (Cornell University)|May 24, 2006

Fluid Dynamics and Vibration Analysis67 references3 citations

TL;DR

This paper introduces a new integral quantity—defined as the ratio of kinetic energy to vorticity in the fluid boundary—that correlates strongly with drag in two-dimensional flows across Reynolds numbers from 9,000 to 144,000. The quantity is proportional to angular momentum in the wake bubble for laminar flow and enables consistent drag correlation across diverse geometries, including cylinders, wedges, and flat plates, suggesting drag scales with flow speed and mass flow rate in the fluid boundary including the wake.

ABSTRACT

An integral quantity is presented that relates the wake of a body in nominally two-dimensional flow to its drag, for Reynolds numbers ranging from 9,000 to 144,000. It is defined as the ratio of the kinetic energy to the vorticity in the fluid boundary and, for the special case of laminar flow, is proportional to the angular momentum in the wake bubble. The new quantity is useful for correlating drag data for circular and rectangular cylinders, wedges, v-gutters, and normal flat plates with and without splitter plates. The correlation indicates that the drag force is proportional to the flow speed and the mass flow rate stored in the boundary of the fluid, where the fluid boundary is defined so as to include the wake bubble. Order-of-magnitude arguments indicate that, absent any quantization of vortex size, this mass flow rate, and hence the drag force, can become unbounded as the vortices contained in the wake becomes finer.

Motivation & Objective

• To identify a universal integral property of the wake that correlates with drag across diverse bluff body geometries.
• To bridge the gap between wake structure and drag force using a dimensionless, flow- and geometry-invariant quantity.
• To provide a predictive framework for drag that extends beyond empirical correlations, especially for complex or non-circular bodies.
• To explore the physical basis of drag in terms of momentum and vorticity transport in the wake region.
• To assess the implications of vortex scale fineness on drag limits, suggesting potential unbounded growth if vortex quantization is absent.

Proposed method

• Define a new integral quantity as the ratio of kinetic energy to vorticity magnitude in the fluid boundary, including the wake bubble.
• Apply the quantity to numerically simulated or experimentally measured flows around circular and rectangular cylinders, wedges, v-gutters, and flat plates with and without splitter plates.
• Use Reynolds numbers ranging from 9,000 to 144,000 to test the robustness of the correlation across flow regimes.
• Demonstrate proportionality between drag force and the product of flow speed and the mass flow rate stored in the fluid boundary.
• Derive order-of-magnitude arguments to explore the theoretical limit of drag as vortex structures in the wake become finer.
• Establish that for laminar flow, the quantity reduces to a form proportional to angular momentum in the wake bubble.

Experimental results

Research questions

• RQ1 Can a single integral property of the wake reliably correlate with drag across diverse bluff body geometries?
• RQ2 How does the proposed kinetic energy-to-vorticity ratio relate to known momentum and angular momentum balances in the wake?
• RQ3 To what extent does the correlation hold across varying Reynolds numbers and body shapes?
• RQ4 What physical insight does this quantity provide into the mechanisms of drag generation in separated flows?
• RQ5 Under what conditions might drag become unbounded, and what does this imply about vortex structure and scale?

Key findings

• The proposed integral quantity—kinetic energy divided by vorticity magnitude in the fluid boundary—shows strong correlation with drag across all tested geometries.
• For laminar flows, the quantity is proportional to the angular momentum in the wake bubble, linking it to rotational momentum in the separated flow.
• Drag force is found to scale linearly with both flow speed and the mass flow rate stored in the fluid boundary, which includes the wake region.
• The correlation holds across Reynolds numbers from 9,000 to 144,000, indicating broad applicability to turbulent and transitional flows.
• Order-of-magnitude analysis suggests that without quantization of vortex size, the mass flow rate and hence drag could become unbounded as vortices in the wake become finer.
• The method successfully correlates drag data for circular and rectangular cylinders, wedges, v-gutters, and normal flat plates with and without splitter plates, demonstrating its generality.