Similarity solution

In fluid dynamics, a similarity solution is a form of solution in which at least one co-ordinate lacks a distinguished origin; more physically, it describes a flow which 'looks the same' either at all times, or at all length scales. These include, for example, the Blasius boundary layer or the Sedov-Taylor shell.[1]

Concept

A powerful tool in physics is the concept of dimensional analysis and scaling laws; by looking at the physical effects present in a system we may estimate their size and hence which, for example, might be neglected. If we have catalogued these effects we will occasionally find that the system has not fixed a natural lengthscale (timescale), but that the solution depends on space (time). It is then necessary to construct a lengthscale (timescale) using time (space) and the other dimensional quantities present - such as the viscosity \nu. These constructs are not 'guessed' but are derived immediately from the scaling of the governing equations.

Example - The impulsively started plate

Consider a semi-infinite domain bounded by a rigid wall and filled with viscous fluid.[2] At time t=0 the wall is made to move with constant speed U in a fixed direction (for definiteness, say the x direction and consider only the x-y plane). We can see that there is no distinguished length scale given in the problem, and we have the boundary conditions of no slip

u = U on y = 0

and that the plate have no effect on the fluid at infinity

u \rightarrow 0 as  y \rightarrow \infty .

Now, if we examine the Navier-Stokes equations

\rho \left( \dfrac{\partial \vec{u}}{\partial t} %2B \vec{u} . \nabla \vec{u} \right) =- \nabla p %2B \mu \nabla^{2} \vec{u}

we can observe that this flow will be rectilinear, with gradients in the y direction and flow in the x direction, and that the pressure term will have no tangential component so that \dfrac{\partial p}{\partial y} = 0. The y component of the Navier-Stokes equations then becomes

\dfrac{\partial \vec{u}}{\partial t}  = \nu \partial^{2}_{y} \vec{u}

and we may apply scaling arguments to show that

 \frac{U}{t} \sim \nu \frac{U}{y^{2}}

which gives us the scaling of the y co-ordinate as

y  \sim (\nu t)^{1/2}.

This allows us to pose an self-similar ansatz such that, with f and \eta dimensionless,

u = U f \left( \eta \equiv \dfrac{y}{(\nu t)^{1/2}} \right)

We have now extracted all of the relevant physics and need only solve the equations; for many cases this will need to be done numerically. This equation is

- \eta f'/2 = f''

with solution satisfying the boundary conditions that

f = 1 - erf (\eta / 2) or u = U \left(1 - erf \left(- y / (4 \nu t)^{1/2} \right)\right)

which is a self-similar solution of the first kind.

References

  1. ^ Pringle and King, 2007, Astrophysical Flows, p54
  2. ^ Batchelor (2006 edition), An Introduction to Fluid Dynamics, p189