# The Lagrangian Picture of Fluid Motion and its Implication for Flow Structures

• J. T. Stuart
Conference paper

## Abstract

In modern studies in fluid dynamics it is quite common to describe the velocity and pressure fields in the Eulerian way, with these quantities being measured and defined at a given point in space. Having found this Eulerian velocity field, u i(x j,t), where i,j range over 1,2,3 and u i is associated with the coordinate x i, we can then study the equations
$$\frac{dx_1}{u_1} = \frac{dx_2}{u_2} = \frac{dx_3}{u_3} = dt$$
, in order to obtain the particle paths and properties associated with them. Such knowledge can be important for the understanding of flows visualized experimentally by dye or smoke.

An alternative approach is that of the Lagrangian description, in which the individual particles are marked and followed in a time-dependent way. A time derivative on a given marked particle gives its velocity, and this gives a connection with the Eulerian description mentioned above.

The partial differential equations for the Eulerian and Lagrangian schemes look superficially different, but are connected by the ordinary differential equations quoted above. However, there are some phenomena of relevance and importance in connection with turbulence and with transition to turbulence, in which an approach from the Lagrangian point of view gives rise to simpler and less intuitive nonlinear mathematics, and leads to illuminating insights. Thus a Lagrangian approach to such problems will be described in this paper, but with reference only to an inviscid fluid satisfying a simple pressure (p)-density (ρ) law of the form p = f(ρ) or more simply still for ρ = constant.

The particular class of problems to be addressed is that connected with longitudinal vorticity filaments in boundary layers or other shear flows. Of relevance here is the notion of the near collision of particles at spanwise locations where there is a convergence of the flow, and in the neighborhood of which eruptions of the flow field can take place. In an inviscid process it is shown that there is a distinct possibility of a singularity forming in a finite time, depending on the initial conditions given for the problem.

The Lagrangian picture of flows of this type will be described from a mathematical point of view but with a physical interpretation.

## Keywords

Pressure Field Fluid Motion Lagrangian Description Longitudinal Vortex Flow Convergence
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## References

1. 1.
Stuart, J.T., The Production of Intense Shear Layers by Vortex Stretching and convection, AGARD Report 514, (1965).Google Scholar
2. 2.
Stuart, J.T., Instability of Laminar Flows, Non-Linear Growth of Fluctuations and Transition to Turbulence in Turbulence and Chaotic Phenomena in Fluids (IUTAM Symposium Kyoto 1983), ed. T. Tatsumi, 17–26, North-Holland (1984).Google Scholar
3. 3.
Stuart, J.T., Nonlinear Euler Partial Differential Equations: Singularities in Their Solution, Proc. Symp. Honor C.C. Lin; eds. Benney, D.J., Shu, F.H., Chi Yuan; World Scientific Press, Singapore (1987).Google Scholar
4. 4.
Childress, S., Ierley, G.R., Spiegel, E.A., and Young, W.R., Blow up of unsteady two-dimensional Euler and Navier-Stokes solutions having stagnation point form, J. Fluid Mech. 203, 1–22 (1989).
5. 5.
Hoskins, B.J. and Bretherton, F.P., Atmospheric Frontogenesis Models: Mathematical Formulation and Solution, J. Atmos. Sci., 29, 11–37 (1972).Google Scholar
6. 6.
Stern, M.E. and Paldor, N., Large-amplitude Long Waves in a Shear flow, Phys. Fluids, 26, 906–919 (1983).Google Scholar
7. 7.
Calogero, F., A Solvable Nonlinear Wave Equation, Stud. Appl. Math., 70, 189–199 (1984).Google Scholar
8. 8.
Russell J.M. and Landahl, M.T., The Evolution of a Flat Eddy near a Wall in an Inviscid Shear Flow, Phys. Fluids, 27, 557–570 (1984).Google Scholar
9. 9.
Van Dommelen, L.L. and Shen, S.F., The Genesis of Separation, in Num. Phys. Aspects Aero. Flows. (Proc. Symp. 1981), ed. T. Cebeci, 293–311, Springer, New York (1982).Google Scholar
10. 10.
Van Dommelen, L.L. and Cowley, S.J., On the Lagrangian Description of Unsteady Boundary-Layer Separation, Part I: General theory, J. Fluid Mech., 210, 593–626 (1990).Google Scholar
11. 11.
Lamb, H., Hydrodynamics, 6th Edition, Cambridge University Press, 1932.Google Scholar
12. 12.
John, F., Two-Dimensional Potential Flows with a Free Boundary, Comm. Pure Applied Math. 6, 497–503, (1953).