Elementary flow

In the larger context of the Navier-Stokes equations (and especially in the context of potential theory), elementary flows are a collection of basic flows from which it is possible to construct more complex flows with different techniques. In this article the term flows is used interchangeably to the term solutions due to historical reasons.

The techniques involved to create more complex solutions can be for example by superposition, by techniques such as topology or considering them as local solutions on a certain neighborhood, subdomain or boundary layer and to be patched together. Elementary flows can be considered the basic building blocks (Fundamental solutions, local solutions and solitons) of the different types of equations derived from the Navier-Stokes. Some of the flows reflect specific cases constraints such as incompressible or irrotational flows, or both, as in the case of potential flow, and some of the flows are often limited in the case of 2 dimensions. ^[1]

Due to the relation from fluid dynamics to all field theory it is important to understand how all these flows are relevant not only to aerodynamics but to all field theory in general. To put it in perspective boundary layers can be interpreted as topological defects on generic manifolds, and considering fluid dynamics analogies and limit cases in electromagnetism, quantum mechanics and general relativity one can see how all these solutions are at the core of recent developments in theoretical physics such as the ads/cft duality, the SYK model, the physics of nematic liquids, strongly correlated systems and even to quark gluon plasmas.

Two-dimensional uniform flow

Potential flow streamlines for an ideal uniform flow

Given a uniform velocity of a fluid at any position in space:

${\displaystyle \mathbf {V_{0}} =v_{0}\cos(\theta _{0})\mathbf {e} _{x}+v_{0}\sin(\theta _{0})\mathbf {e} _{y}}$

This flow is incompressible because the velocity is constant, the first derivatives of the velocity components are zero, and the total divergence is zero: ${\displaystyle \nabla \cdot \mathbf {v} =0}$

Given the circulation is always zero the flow is also irrotational, we can derive this from the Kelvin's circulation theorem and from the explicit computation of the vorticity:

${\displaystyle \omega _{z}={\frac {\partial v_{x}}{\partial y}}-{\frac {\partial v_{y}}{\partial x}}=0}$

Being incompressible and two-dimensional, this flow is constructed from a stream function:

${\displaystyle v_{x}={\frac {\partial \psi }{\partial y}}}$

${\displaystyle v_{y}=-{\frac {\partial \psi }{\partial x}}}$

from which

${\displaystyle \psi =-v_{0}\sin(\theta _{0})x+v_{0}\cos(\theta _{0})y}$

and in cylindrical coordinates:

${\displaystyle v_{r}=-{\frac {1}{r}}{\frac {\partial \psi }{\partial \theta }}}$

${\displaystyle v_{\theta }={\frac {\partial \psi }{\partial r}}}$

from which

${\displaystyle \psi =-v_{0}r\sin(\theta -\theta _{0})}$

As usual the stream function is defined up to a constant value which here we take as zero. We can also confirm that the flow is irrotational from:

${\displaystyle \nabla ^{2}\psi =0}$

Being irrotational, the potential function is instead:

${\displaystyle v_{x}=-{\frac {\partial \phi }{\partial x}}}$

${\displaystyle v_{y}=-{\frac {\partial \phi }{\partial y}}}$

and therefore

${\displaystyle \phi =-v_{0}\cos(\theta _{0})x-v_{0}\sin(\theta _{0})y}$

and in cylindrical coordinates

${\displaystyle v_{r}={\frac {\partial \phi }{\partial r}}}$

${\displaystyle v_{\theta }={\frac {1}{r}}{\frac {\partial \phi }{\partial \theta }}}$

${\displaystyle \phi =-v_{0}r\cos(\theta -\theta _{0})}$

Two-dimensional line source

Potential flow streamlines for an ideal line source

The case of a vertical line emitting at a fixed rate a constant quantity of fluid Q per unit length is a line source. The problem has a cylindrical symmetry and can be treated in two dimension on the orthogonal plane.

Line sources and line sinks (below) are important elementary flows because they play the role of monopole(s) for incompressible fluids (which can also be considered examples of solenoidal fields i.e. divergence free fields). Generic flow patterns can be also de-composed in terms of multipole expansions, in the same manner as for electric and magnetic fields where the monopole is essentially the first non-trivial (e.g. constant) term of the expansion.

This flow pattern is also both irrotational and incompressible.

This is characterized by a cylindrical symmetry:

${\displaystyle \mathbf {v} =v_{r}(r)\mathbf {e} _{r}}$

Where the total outgoing flux is constant

${\displaystyle \int _{S}\mathbf {v} \cdot d\mathbf {S} =\int _{0}^{2\pi }(v_{r}(r)\,\mathbf {e} _{r})\cdot (\mathbf {e} _{r}\,r\,d\theta )=\!2\pi \,r\,v_{r}(r)=Q}$

Therefore,

${\displaystyle v_{r}={\frac {Q}{2\pi r}}}$

This is derived from a stream function

${\displaystyle \psi (r,\theta )=-{\frac {Q}{2\pi }}\theta }$

or from a potential function

${\displaystyle \phi (r,\theta )=-{\frac {Q}{2\pi }}\ln r}$

Two-dimensional line sink

The case of a vertical line absorbing at a fixed rate a constant quantity of fluid Q per unit length is a line sink. Everything is the same as the case of a line source a part from the negative sign.

${\displaystyle v_{r}=-{\frac {Q}{2\pi r}}}$

This is derived from a stream function

${\displaystyle \psi (r,\theta )={\frac {Q}{2\pi }}\theta }$

or from a potential function

${\displaystyle \phi (r,\theta )={\frac {Q}{2\pi }}\ln r}$

Given that the two results are the same a part from a minus sign we can treat transparently both line sources and line sinks with the same stream and potential functions permitting Q to assume both positive and negative values and absorbing the minus sign into the definition of Q.

Two-dimensional doublet or dipole line source

Potential flow streamlines for an ideal doublet, or dipole, line

If we consider a line source and a line sink at a distance d we can reuse the results above and the stream function will be

${\displaystyle \psi (\mathbf {r} )=\psi _{Q}(\mathbf {r} -\mathbf {d} /2)-\psi _{Q}(\mathbf {r} +\mathbf {d} /2)\ \simeq \mathbf {d} \cdot \nabla \psi _{Q}(\mathbf {r} )}$

The last approximation is to the first order in d.

Given

${\displaystyle \mathbf {d} =d[\cos(\theta _{0})\mathbf {e} _{x}+\sin(\theta _{0})\mathbf {e} _{y}]=d[\cos(\theta -\theta _{0})\mathbf {e} _{r}+\sin(\theta -\theta _{0})\mathbf {e} _{\theta }]}$

It remains

${\displaystyle \psi (r,\theta )=-{\frac {Qd}{2\pi }}{\frac {\sin(\theta -\theta _{0})}{r}}}$

The velocity is then

${\displaystyle v_{r}(r,\theta )={\frac {Qd}{2\pi }}{\frac {\cos(\theta -\theta _{0})}{r^{2}}}}$

${\displaystyle v_{\theta }(r,\theta )={\frac {Qd}{2\pi }}{\frac {\sin(\theta -\theta _{0})}{r^{2}}}}$

And the potential instead

${\displaystyle \phi (r,\theta )={\frac {Qd}{2\pi }}{\frac {\cos(\theta -\theta _{0})}{r}}}$

Two-dimensional vortex line

Potential flow streamlines for an ideal vortex line

This is the case of a vortex filament rotating at constant speed, there is a cylindrical symmetry and the problem can be solved in the orthogonal plane.

Dual to the case above of line sources, vortex lines play the role of monopoles for irrotational flows.

Also in this case the flow is also both irrotational and incompressible and therefore a case of potential flow.

This is characterized by a cylindrical symmetry:

${\displaystyle \mathbf {v} =v_{\theta }(r)\,\mathbf {e} _{\theta }}$

Where the total circulation is constant for every closed line around the central vortex

${\displaystyle \oint \mathbf {v} \cdot d\mathbf {s} =\int _{0}^{2\pi }(v_{\theta }(r)\,\mathbf {e} _{\theta })\cdot (\mathbf {e} _{\theta }\,r\,d\theta )=\!2\pi \,r\,v_{\theta }(r)=\Gamma }$

and is zero for any line not including the vortex.

Therefore,

${\displaystyle v_{\theta }={\frac {\Gamma }{2\pi r}}}$

This is derived from a stream function

${\displaystyle \psi (r,\theta )={\frac {\Gamma }{2\pi }}\ln r}$

or from a potential function

${\displaystyle \phi (r,\theta )=-{\frac {\Gamma }{2\pi }}\theta }$

Which is dual to the previous case of a line source

Generic two-dimensional potential flow

Given an incompressible two-dimensional flow which is also irrotational we have:

${\displaystyle \nabla ^{2}\psi =0}$

Which is in cylindrical coordinates ^[2]

${\displaystyle {\frac {1}{r}}{\frac {\partial }{\partial r}}\left(r{\frac {\partial \psi }{\partial r}}\right)+{\frac {1}{r^{2}}}{\frac {\partial ^{2}\psi }{\partial \theta ^{2}}}=0}$

We look for a solution with separated variables:

${\displaystyle \psi (r,\theta )=R(r)\Theta (\theta )}$

which gives

${\displaystyle {\frac {r}{R(r)}}{\frac {d}{dr}}\left(r{\frac {dR(r)}{dr}}\right)=-{\frac {1}{\Theta (\theta )}}{\frac {d^{2}\Theta (\theta )}{d\theta ^{2}}}}$

Given the left part depends only on r and the right parts depends only on ${\displaystyle \theta }$, the two parts must be equal to a constant independent from r and ${\displaystyle \theta }$. The constant shall be positive^{[ clarification needed ]}. Therefore,

${\displaystyle r{\frac {d}{dr}}\left(r{\frac {d}{dr}}R(r)\right)=m^{2}R(r)}$

${\displaystyle {\frac {d^{2}\Theta (\theta )}{d\theta ^{2}}}=-m^{2}\Theta (\theta )}$

The solution to the second equation is a linear combination of ${\displaystyle e^{im\theta }}$ and ${\displaystyle e^{-im\theta }}$ In order to have a single-valued velocity (and also a single-valued stream function) m shall be a positive integer.

therefore the most generic solution is given by

${\displaystyle \psi =\alpha _{0}+\beta _{0}\ln r+\sum _{m>0}{\left(\alpha _{m}r^{m}+\beta _{m}r^{-m}\right)\sin {[m(\theta -\theta _{m})]}}}$

The potential is instead given by

${\displaystyle \phi =\alpha _{0}-\beta _{0}\theta +\sum _{m\mathop {>} 0}{(\alpha _{m}r^{m}-\beta _{m}r^{-m})\cos {[m(\theta -\theta _{m})]}}}$

References

• Fitzpatrick, Richard (2017), Theoretical fluid dynamics, IOP science, ISBN   978-0-7503-1554-8
• Faber, T.E. (1995), Fluid Dynamics for Physicists, Cambridge university press, ISBN   9780511806735

Specific

1. Oliver, David (2013-03-14). The Shaggy Steed of Physics: Mathematical Beauty in the Physical World. Springer Science & Business Media. ISBN   978-1-4757-4347-0.
2. Laplace operator

Further reading

• Batchelor, G.K. (1973), An introduction to fluid dynamics, Cambridge University Press, ISBN   978-0-521-09817-5
• Chanson, H. (2009), Applied Hydrodynamics: An Introduction to Ideal and Real Fluid Flows, CRC Press, Taylor & Francis Group, Leiden, The Netherlands, 478 pages, ISBN   978-0-415-49271-3
• Lamb, H. (1994) [1932], Hydrodynamics (6th ed.), Cambridge University Press, ISBN   978-0-521-45868-9
• Milne-Thomson, L.M. (1996) [1968], Theoretical hydrodynamics (5th ed.), Dover, ISBN   978-0-486-68970-8

External links

• Richard Fitzpatrick University of Texas, Austin (2017). "Fluid Mechanics". University of Texas, Austin . Retrieved 2018-02-07.

• (c) Aerospace, Mechanical & Mechatronic Engg. 2005 University of Sydney (2005). "Elements of Potential Flow". University of Sydney . Retrieved 2019-04-19.