Beltrami flow

Ratip Berker obtained the solution in Cartesian coordinates for ${\displaystyle \mathbf {g} (\mathbf {x} )}$ in 1963,[4]

${\displaystyle \mathbf {g} =\cos \left({\frac {cx}{\sqrt {2}}}\right)\sin \left({\frac {cy}{\sqrt {2}}}\right)\left[-{\frac {1}{\sqrt {2}}}\mathbf {e_{x}} +{\frac {1}{\sqrt {2}}}\mathbf {e_{y}} +\mathbf {e_{z}} \right].}$

For steady generalized Beltrami flow, we have ${\displaystyle \nabla ^{2}{\boldsymbol {\omega }}=0,\ \nabla \times (\mathbf {v} \times {\boldsymbol {\omega }})=0}$ and since it is also planar we have ${\displaystyle \mathbf {v} =(u,v,0),\ {\boldsymbol {\omega }}=(0,0,\zeta )}$. Introduce the stream function

${\displaystyle u={\frac {\partial \psi }{\partial y}},\quad v=-{\frac {\partial \psi }{\partial x}},\quad \Rightarrow \quad \nabla ^{2}\psi =-\zeta .}$

Integration of ${\displaystyle \nabla \times (\mathbf {v} \times {\boldsymbol {\omega }})=0}$ gives ${\displaystyle \zeta =-f(\psi )}$. So, complete solution is possible if it satisfies all the following three equations

${\displaystyle \nabla ^{2}\psi =-\zeta ,\quad \nabla ^{2}\zeta =0,\quad \zeta =-f(\psi ).}$

A special case is considered when the flow field has uniform vorticity ${\displaystyle f(\psi )=C={\text{constant}}}$. Wang (1991)[6] gave the generalized solution as

${\displaystyle \zeta =\psi +A(x,y),\quad A(x,y)=ax+by}$

assuming a linear function for ${\displaystyle A(x,y)}$. Substituting this into the vorticity equation and introducing the separation of variables ${\displaystyle \psi (x,y)=X(x)Y(y)}$ with the separating constant ${\displaystyle \lambda ^{2}}$ results in

${\displaystyle {\frac {d^{2}X}{dx^{2}}}+{\frac {b}{\nu }}{\frac {dX}{dx}}-\lambda ^{2}X=0,\quad {\frac {d^{2}Y}{dy^{2}}}-{\frac {a}{\nu }}{\frac {dY}{dy}}+\lambda ^{2}Y=0.}$

The solution obtained for different choices of ${\displaystyle a,\ b,\ \lambda }$ can be interpreted differently, for example, ${\displaystyle a=0,\ b=-U,\lambda ^{2}>0}$ represents a flow downstream a uniform grid, ${\displaystyle a=-U,\ b=0,\lambda ^{2}=0}$ represents a flow created by a stretching plate, ${\displaystyle a=-U,\ b=U,\lambda ^{2}=0}$ represents a flow into a corner, ${\displaystyle a=-V,\ b=-U,\lambda ^{2}=0}$ represents an Asymptotic suction profile etc.

Here,

${\displaystyle \nabla ^{2}\psi =-\zeta ,\quad {\frac {\partial \zeta }{\partial t}}=\nabla ^{2}\zeta ,\quad \zeta =-f(\psi ,t)}$.

G. I. Taylor gave the solution for a special case where ${\displaystyle \zeta =K\psi }$, where ${\displaystyle K}$ is a constant in 1923.[7] He showed that the separation ${\displaystyle \psi =e^{-K\nu t}\Psi (x,y)}$ satisfies the equation and also

${\displaystyle \nabla ^{2}\Psi =-K\Psi .}$

Taylor also considered an example, a decaying system of eddies rotating alternatively in opposite directions and arranged in a rectangular array

${\displaystyle \Psi =A\cos {\frac {\pi x}{d}}\cos {\frac {\pi y}{d}}}$

which satisfies the above equation with ${\displaystyle K={\frac {2\pi ^{2}}{d^{2}}}}$, where ${\displaystyle d}$ is the length of the square formed by an eddy. Therefore, this system of eddies decays as

${\displaystyle \psi =A\cos \left({\frac {\pi x}{d}}\right)\cos \left({\frac {\pi y}{d}}\right)e^{-{\frac {2\pi ^{2}}{d^{2}}}\nu t}.}$

Here we have ${\displaystyle \mathbf {v} =\left(u_{r},0,u_{z}\right),\ {\boldsymbol {\omega }}=(0,\zeta ,0)}$. Integration of ${\displaystyle \nabla \times (\mathbf {v} \times {\boldsymbol {\omega }})=0}$ gives ${\displaystyle \zeta =rf(\psi )}$ and the three equations are

${\displaystyle {\frac {\partial }{\partial r}}\left({\frac {1}{r}}{\frac {\partial \psi }{\partial z}}\right)+{\frac {1}{r}}{\frac {\partial ^{2}\psi }{\partial z^{2}}}=-\zeta ,\quad \nabla ^{2}\zeta =0,\quad \zeta =rf(\psi )}$

The first equation is the Hicks equation. Marris and Aswani (1977)[8] showed that the only possible solution is ${\displaystyle f(\psi )=C={\text{constant}}}$ and the remaining equations reduce to

${\displaystyle {\frac {\partial ^{2}\psi }{\partial r^{2}}}-{\frac {1}{r}}{\frac {\partial \psi }{\partial r}}+{\frac {\partial ^{2}\psi }{\partial z^{2}}}+Cr^{2}=0}$

A simple set of solutions to the above equation is

${\displaystyle \psi (r,z)=c_{1}r^{4}+c_{2}r^{2}z^{2}+c_{3}r^{2}+c_{4}r^{2}z+c_{5}\left(r^{6}-12r^{4}z^{2}+8r^{2}z^{4}\right),\quad C=-\left(8c_{1}+2c_{2}\right)}$

${\displaystyle c_{1},c_{4}\neq 0,\ c_{2},c_{3},c_{5}=0}$ represents a flow due to two opposing rotational stream on a parabolic surface, ${\displaystyle c_{2}\neq 0,c_{1},c_{3},c_{4},c_{5}=0}$ represents rotational flow on a plane wall, ${\displaystyle c_{1},c_{2},c_{3}\neq 0,\ c_{4},c_{5}=0}$ represents a flow ellipsoidal vortex (special case – Hill's spherical vortex), ${\displaystyle c_{1},c_{3},c_{5}\neq 0,\ c_{2},c_{4}=0}$ represents a type of toroidal vortex etc.

The homogeneous solution for ${\displaystyle C=0}$ as shown by Berker[9]

${\displaystyle \psi =r\left[A_{k}J_{1}(kr)+B_{k}Y_{1}(kr)\right]\left(C_{k}e^{kz}+D_{k}e^{-kz}\right)}$

where ${\displaystyle J_{1}(kr),Y_{1}(kr)}$ are the Bessel function of the first kind and Bessel function of the second kind respectively. A special case of the above solution is Poiseuille flow for cylindrical geometry with transpiration velocities on the walls. Chia-Shun Yih found a solution in 1958 for Poiseuille flow into a sink when ${\displaystyle C=2,\,c_{1}=-1/4,\,c_{3}=1/2,\,c_{2}=c_{4}=c_{5}=B_{k}=C_{k}=0}$.[10]