Beltrami flow

Trkalian flow

Viktor Trkal considered the Beltrami flows without any external forces in 1919 for the scalar function ${\displaystyle \alpha (\mathbf{x},t)=c=\text{constant}}$, i.e.,

${\displaystyle \frac{\mathrm{\partial }\mathit{\omega }}{\mathrm{\partial }t}=\nu {\mathrm{\nabla }}^2\mathit{\omega },\mathit{\omega }=c\mathbf{v}.}$

Introduce the following separation of variables

${\displaystyle \mathbf{v}=e^{-c^2\nu t}\mathbf{g}(\mathbf{x}),}$

then the equation satisfied by ${\displaystyle \mathbf{g}(\mathbf{x})}$ becomes

${\displaystyle \mathrm{\nabla }\times \mathbf{g}=c\mathbf{g}.}$

The Chandrasekhar–Kendall functions satisfy this equation.

Generalized Beltrami flow

The generalized Beltrami flow satisfies the condition

${\displaystyle \mathrm{\nabla }\times (\mathbf{v}\times \mathit{\omega })=0}$

which is less restrictive than the Beltrami condition ${\displaystyle \mathbf{v}\times \mathit{\omega }=0}$. Unlike the normal Beltrami flows, the generalized Beltrami flow can be studied for planar and axisymmetric flows.

Steady planar flows

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

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

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

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

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

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

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

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

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.

Unsteady planar flows

Here,

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

Taylor's decaying vortices

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

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

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

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

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

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

O. Walsh generalized Taylor's eddy solution in 1992. Walsh's solution is of the form ${\displaystyle \mathbf{v}=e^{-\nu \lambda t}\mathbf{u}(x,y)}$, where ${\displaystyle {\mathrm{\nabla }}^2\mathbf{u}=-\lambda \mathbf{u}}$ and ${\displaystyle \mathrm{\nabla }\cdot \mathbf{u}=0.}$

Steady axisymmetric flows

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

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

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

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

A simple set of solutions to the above equation is

${\displaystyle \psi (r,z)=c_1{r}^4+c_2{r}^2{z}^2+c_3{r}^2+c_4{r}^{2}z+c_5\left(r^6-12r^4{z}^2+8r^2{z}^4\right),C=-\left(8c_1+2c_2\right)}$

${\displaystyle c_1,c_4\ne 0,c_2,c_3,c_5=0}$ represents a flow due to two opposing rotational stream on a parabolic surface, ${\displaystyle c_2\ne 0,c_1,c_3,c_4,c_5=0}$ represents rotational flow on a plane wall, ${\displaystyle c_1,c_2,c_3\ne 0,c_4,c_5=0}$ represents a flow ellipsoidal vortex (special case – Hill's spherical vortex), ${\displaystyle c_1,c_3,c_5\ne 0,c_2,c_4=0}$ represents a type of toroidal vortex etc.

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

${\displaystyle \psi =r\left[A_k{J}_1(kr)+B_k{Y}_1(kr)\right]\left(C_k{e}^{kz}+D_k{e}^{-kz}\right)}$

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 ${\displaystyle C=2,c_1=-1/4,c_3=1/2,c_2=c_4=c_5=B_k=C_k=0}$.

Beltrami flow in fluid mechanics

Beltrami fields are a classical steady solution to the Euler equation. Beltrami fields play an important role in (ideal) fluid mechanics in equilibrium, as complexity is only expected for these fields.