Prandtl–Batchelor theorem

In fluid dynamics, Prandtl–Batchelor theorem states that if in a two-dimensional laminar flow at high Reynolds number closed streamlines occur, then the vorticity in the closed streamline region must be a constant. The theorem is named after Ludwig Prandtl and George Batchelor. Prandtl in his celebrated 1904 paper stated this theorem in arguments,^[1] George Batchelor unaware of this work proved the theorem in 1956.^[2]^[3]

Mathematical proof

At high Reynolds numbers, Euler equations reduce to solving a problem for stream function,

\nabla ^{2}\psi =-\omega (\psi ),\quad \psi =\psi _{o}{\text{ on }}\partial D.

As it stands, the problem is ill-posed since the vorticity distribution $\omega (\psi )$ can have infinite number of possibilities, all of which satisfies the equation and the boundary condition. This is not true if the streamlines are not closed, in which case, every streamline can be traced back to infinity, where $\omega (\psi )$ is known. The problem is only when closed streamlines occur inside the flow at high Reynolds number, where $\omega (\psi )$ is not uniquely defined. The theorem exactly addresses this issue.

The vorticity equation in two-dimension reduces to

\mathbf {u} \cdot \nabla \mathbf {\omega } ={\frac {1}{Re}}\nabla ^{2}\omega .

where even though the Reynolds number is very large, we keep the viscous term for now. Let us integrate this equation over some surface $S$ enclosed by a closed contour $C$ in the region where we have closed streamlines. The convective term gives zero contour since $C$ is taken to be one of those closed streamlines. Then, we have

{\frac {1}{Re}}\int _{C}\nabla \omega \cdot \mathbf {n} \ ds=0

where $\mathbf {n}$ is the unit normal to $C$ with small element $ds$ . This expression is true for finite but large Reynolds number since we did not neglect the viscous term before. In the above expression, $\omega \neq \omega (\psi )$ since this is not the inviscid limit. But for large $Re$ but finite, we can write $\omega =\omega (\psi )+{\rm {small\ corrections}}$ , and this small corrections become smaller and smaller as we increase the Reynolds number. Neglecting these corrections,

{\frac {1}{Re}}\int _{C}\nabla \omega \cdot \mathbf {n} \ ds={\frac {1}{Re}}\int _{C}{\frac {d\omega }{d\psi }}\nabla \psi \cdot \mathbf {n} \ ds=0.

But $d\omega /d\psi$ is constant for any streamlines, and that can be pulled out of the integral,

{\frac {1}{Re}}{\frac {d\omega }{d\psi }}\int _{C}\nabla \psi \cdot \mathbf {n} \ ds=0.

We also known the circulation in these closed streamline is non-zero, i.e.,

\Gamma =-\int _{C}\nabla \psi \cdot \mathbf {n} ds.

Therefore, we have

{\frac {\Gamma }{Re}}{\frac {d\omega }{d\psi }}=0.

The only way this can satisfied for finite $Re$ is if and if only

{\frac {d\omega }{d\psi }}=0,

i.e., vorticity is not changing across these closed streamlines, thus proving the theorem. Of course, the theorem is not valid inside the boundary layer regime.

References

↑ Prandtl, L. (1904). Über Flussigkeitsbewegung bei sehr kleiner Reibung. Verhandl. III, Internat. Math.-Kong., Heidelberg, Teubner, Leipzig, 1904, 484–491.
↑ Batchelor, G. K. (1956). On steady laminar flow with closed streamlines at large Reynolds number. Journal of Fluid Mechanics, 1(2), 177–190.
↑ Davidson, P. A. (2016). Introduction to magnetohydrodynamics (Vol. 55). Cambridge university press.

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[1] Prandtl, L. (1904). Über Flussigkeitsbewegung bei sehr kleiner Reibung. Verhandl. III, Internat. Math.-Kong., Heidelberg, Teubner, Leipzig, 1904, 484–491.

[2] Batchelor, G. K. (1956). On steady laminar flow with closed streamlines at large Reynolds number. Journal of Fluid Mechanics, 1(2), 177–190.

[3] Davidson, P. A. (2016). Introduction to magnetohydrodynamics (Vol. 55). Cambridge university press.