Jump to content

Hicks equation

From Wikipedia, the free encyclopedia
(Redirected from Squire–Long 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 William Mitchinson Hicks, who derived it first in 1898.[1][2][3] The equation was also re-derived by Stephen Bragg and William Hawthorne in 1950 and by Robert R. Long in 1953 and by Herbert Squire in 1956.[4][5][6] The Hicks equation without swirl was first introduced by George Gabriel Stokes in 1842.[7][8] The Grad–Shafranov equation appearing in plasma physics also takes the same form as the Hicks equation.

Representing as coordinates in the sense of cylindrical coordinate system with corresponding flow velocity components denoted by , the stream function that defines the meridional motion can be defined as

that satisfies the continuity equation for axisymmetric flows automatically. The Hicks equation is then given by [9]

where

where is the total head, c.f. Bernoulli's Principle. and is the circulation, both of them being conserved along streamlines. Here, is the pressure and is the fluid density. The functions and are known functions, usually prescribed at one of the boundary; see the example below. If there are closed streamlines in the interior of the fluid domain, say, a recirculation region, then the functions and are typically unknown and therefore in those regions, Hicks equation is not useful; Prandtl–Batchelor theorem provides details about the closed streamline regions.

Derivation

[edit]

Consider the axisymmetric flow in cylindrical coordinate system with velocity components and vorticity components . Since in axisymmetric flows, the vorticity components are

.

Continuity equation allows to define a stream function such that

(Note that the vorticity components and are related to in exactly the same way that and are related to ). Therefore the azimuthal component of vorticity becomes


The inviscid momentum equations , where is the Bernoulli constant, is the fluid pressure and is the fluid density, when written for the axisymmetric flow field, becomes

in which the second equation may also be written as , where is the material derivative. This implies that the circulation round a material curve in the form of a circle centered on -axis is constant.

If the fluid motion is steady, the fluid particle moves along a streamline, in other words, it moves on the surface given by constant. It follows then that and , where . Therefore the radial and the azimuthal component of vorticity are

.

The components of and are locally parallel. The above expressions can be substituted into either the radial or axial momentum equations (after removing the time derivative term) to solve for . For instance, substituting the above expression for into the axial momentum equation leads to[9]

But can be expressed in terms of as shown at the beginning of this derivation. When is expressed in terms of , we get

This completes the required derivation.

Example: Fluid with uniform axial velocity and rigid body rotation in far upstream

[edit]

Consider the problem where the fluid in the far stream exhibit uniform axial velocity and rotates with angular velocity . This upstream motion corresponds to

From these, we obtain

indicating that in this case, and are simple linear functions of . The Hicks equation itself becomes

which upon introducing becomes

where .

Yih equation

[edit]

For an incompressible flow , but with variable density, Chia-Shun Yih derived the necessary equation. The velocity field is first transformed using Yih transformation

where is some reference density, with corresponding Stokes streamfunction defined such that

Let us include the gravitational force acting in the negative direction. The Yih equation is then given by[10][11]

where

References

[edit]
  1. ^ Hicks, W. M. (1898). Researches in vortex motion. Part III. On spiral or gyrostatic vortex aggregates. Proceedings of the Royal Society of London, 62(379–387), 332–338. https://royalsocietypublishing.org/doi/pdf/10.1098/rspl.1897.0119
  2. ^ Hicks, W. M. (1899). II. Researches in vortex motion.—Part III. On spiral or gyrostatic vortex aggregates. Philosophical Transactions of the Royal Society of London. Series A, Containing Papers of a Mathematical or Physical Character, (192), 33–99. https://royalsocietypublishing.org/doi/pdf/10.1098/rsta.1899.0002
  3. ^ Smith, S. G. L., & Hattori, Y. (2012). Axisymmetric magnetic vortices with swirl. Communications in Nonlinear Science and Numerical Simulation, 17(5), 2101–2107.
  4. ^ Bragg, S. L. & Hawthorne, W. R. (1950). Some exact solutions of the flow through annular cascade actuator discs. Journal of the Aeronautical Sciences, 17(4), 243–249
  5. ^ Long, R. R. (1953). Steady motion around a symmetrical obstacle moving along the axis of a rotating liquid. Journal of Meteorology, 10(3), 197–203.
  6. ^ Squire, H. B. (1956). Rotating fluids. Surveys in Mechanics. A collection of Surveys of the present position of Research in some branches of Mechanics, written in Commemoration of the 70th Birthday of Geoffrey Ingram Taylor, Eds. G. K. Batchelor and R. M. Davies. 139–169
  7. ^ Stokes, G. (1842). On the steady motion of incompressible fluids Trans. Camb. Phil. Soc. VII, 349.
  8. ^ Lamb, H. (1993). Hydrodynamics. Cambridge university press.
  9. ^ a b Batchelor, G. K. (1967). An introduction to fluid dynamics. Section 7.5. Cambridge university press. section 7.5, p. 543-545
  10. ^ Yih, C. S. (2012). Stratified flows. Elsevier.
  11. ^ Yih, C. S. (1991). On stratified flows in a gravitational field. In Selected Papers By Chia-Shun Yih: (In 2 Volumes) (pp. 13-21).