# Natural Velocity Decomposition: A Review

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## Abstract

A review of the natural velocity decomposition (NVD) formulation for the analysis of Navier–Stokes flows developed by the authors is presented. The decomposition consists in expressing the velocity field as the sum of two terms, one of which is given as the gradient of a potential \(\varphi \), whereas the other, \(\mathbf{w}\), is rotational and is governed by its own evolution equation. Hence, contrary to all the related approaches such as the Helmholtz decomposition, the NVD does not require the evaluation of the vorticity. The theoretical formulation is presented in detail for incompressible viscous flows and briefly outlined for compressible viscous flows. Of particular significance is the relationship between the present approach for thin vortical layers (almost-potential flows) and that for zero-thickness layers (quasi-potential flows), when this is combined with the transpiration-velocity correction by Lighthill. The methodology may also be a valuable computational tool. To assess this, two problems are addressed. The first consists in the planar flow around a disk, for low Reynolds numbers (separated laminar flows). The computational scheme used is discussed thoroughly. The corresponding numerical results compare favorably with data available in the open literature. These include the separation angle and the length of the recirculation bubble. Also, the flow-field results are in very good agreement with experimental flow visualizations available in the open literature. The second problem that we addressed is the flow due to a jet. This is tackled by the almost-potential flow approximation of the NVD formulation. Again, the results are in good agreement with experimental data available in the open literature. Theoretical innovations (and limitations) of the methodology are discussed, along with its advantages and drawbacks.

## Keywords

Computational fluid dynamics Viscous flows Potential/vortical-field decomposition## Notes

### Acknowledgements

We wish to express our appreciation to the Hariri Center of Boston University, and its Director, Prof. Azer Bestavros, for hosting P.G. and L.M., as well as the Engineering Faculty of Embry-Riddle Aeronautical University and its Dean, Prof. Anastasios Lyrinzis, for hosting M.C. and L.S., in connection with the activity reported here.

### Compliance with Ethical Standards

### Conflict of interest

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

## Supplementary material

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