In fluid dynamics, Beltrami flows are flows in which the vorticity vector
and the velocity vector
are parallel to each other. In other words, Beltrami flow is a flow where
Lamb vector In fluid dynamics, Lamb vector is the cross product of vorticity vector and velocity vector of the flow field, named after the physicist Horace Lamb.Truesdell, C. (1954). The kinematics of vorticity (Vol. 954). Bloomington: Indiana University Press. ...
is zero. It is named after the Italian mathematician
Eugenio Beltrami due to his derivation of the
Beltrami vector field, while initial developments in fluid dynamics were done by the Russian scientist
Ippolit S. Gromeka in 1881.
Description
Since the vorticity vector
and the velocity vector
are parallel to each other, we can write
:
where
is some scalar function. One immediate consequence of Beltrami flow is that it can never be a planar or axisymmetric flow because in those flows, vorticity is always perpendicular to the velocity field. The other important consequence will be realized by looking at the incompressible
vorticity equation
:
where
is an external body forces such as gravitational field, electric field etc., and
is the kinematic viscosity. Since
and
are parallel, the non-linear terms in the above equation are identically zero
. Thus Beltrami flows satisfies the linear equation
:
When
, the components of vorticity satisfies a simple
heat equation
In mathematics and physics, the heat equation is a certain partial differential equation. Solutions of the heat equation are sometimes known as caloric functions. The theory of the heat equation was first developed by Joseph Fourier in 1822 for t ...
.
Trkalian flow
Viktor Trkal considered the Beltrami flows without any external forces in 1919 for the scalar function
, i.e.,
:
Introduce the following separation of variables
:
then the equation satisfied by
becomes
:
The
Chandrasekhar–Kendall functions satisfy this equation.
Berker's solution
Ratip Berker obtained the solution in Cartesian coordinates for
in 1963,
:
Generalized Beltrami flow
The generalized Beltrami flow satisfies the condition
:
which is less restrictive than the Beltrami condition
. 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
and since it is also planar we have
. Introduce the stream function
:
Integration of
gives
. So, complete solution is possible if it satisfies all the following three equations
:
A special case is considered when the flow field has uniform vorticity
. Wang (1991) gave the generalized solution as
:
assuming a linear function for
. Substituting this into the vorticity equation and introducing the
separation of variables
In mathematics, separation of variables (also known as the Fourier method) is any of several methods for solving ordinary and partial differential equations, in which algebra allows one to rewrite an equation so that each of two variables occurs ...
with the separating constant
results in
:
The solution obtained for different choices of
can be interpreted differently, for example,
represents a flow downstream a uniform grid,
represents a flow created by a stretching plate,
represents a flow into a corner,
represents an
Asymptotic suction profile etc.
Unsteady planar flows
Here,
:
.
Taylor's decaying vortices
G. I. Taylor
Sir Geoffrey Ingram Taylor OM FRS FRSE (7 March 1886 – 27 June 1975) was a British physicist and mathematician, and a major figure in fluid dynamics and wave theory. His biographer and one-time student, George Batchelor, described him as ...
gave the solution for a special case where
, where
is a constant in 1923. He showed that the separation
satisfies the equation and also
:
Taylor also considered an example, a decaying system of eddies rotating alternatively in opposite directions and arranged in a rectangular array
:
which satisfies the above equation with
, where
is the length of the square formed by an eddy. Therefore, this system of eddies decays as
:
O. Walsh generalized Taylor's eddy solution in 1992.
Steady axisymmetric flows
Here we have
. Integration of
gives
and the three equations are
:
The first equation is the
Hicks equation In fluid dynamics, Hicks equation, sometimes also referred as Bragg–Hawthorne equation or Squire–Long equation, is a partial differential equation that describes the distribution of stream function for axisymmetric inviscid fluid, named after Wi ...
. Marris and Aswani (1977) showed that the only possible solution is
and the remaining equations reduce to
:
A simple set of solutions to the above equation is
:
represents a flow due to two opposing rotational stream on a parabolic surface,
represents rotational flow on a plane wall,
represents a flow ellipsoidal vortex (special case – Hill's spherical vortex),
represents a type of toroidal vortex etc.
The homogeneous solution for
as shown by Berker
:
where
are the
Bessel function of the first kind and
Bessel function of the second kind
Bessel functions, first defined by the mathematician Daniel Bernoulli and then generalized by Friedrich Bessel, are canonical solutions of Bessel's differential equation
x^2 \frac + x \frac + \left(x^2 - \alpha^2 \right)y = 0
for an arbitrary ...
respectively. A special case of the above solution is
Poiseuille flow for cylindrical geometry with transpiration velocities on the walls.
Chia-Shun Yih
Chia-Shun Yih (; July 25, 1918 – April 25, 1997) was the Stephen P. Timoshenko Distinguished University Professor Emeritus at the University of Michigan. He made many significant contributions to fluid mechanics. Yih was also a seal artist.
B ...
found a solution in 1958 for
Poiseuille flow into a sink when
.
[Yih, C. S. (1959). Two solutions for inviscid rotational flow with corner eddies. Journal of Fluid Mechanics, 5(1), 36-40.]
See also
*
Gromeka–Arnold–Beltrami–Childress (GABC) flow
References
{{Reflist, 30em
Fluid dynamics