## PHY454H1S Continuum Mechanics. Lecture 16: Boundary Layers. Taught by Prof. K. Das.

Posted by peeterjoot on March 14, 2012

# Disclaimer.

Peeter’s lecture notes from class. May not be entirely coherent.

# Boundary Layers.

Suppose we have an obstacle to fluid flow, as in figure (\ref{fig:continuumL16:continuumL16Fig1})

\begin{figure}[htp]

\centering

\includegraphics[totalheight=0.2\textheight]{continuumL16Fig1}

\caption{Flow lines around circular obstacle}

\end{figure}

we have a couple conditions on fluid flow.

\begin{enumerate}

\item No fluid can cross the solid boundary

\item Due to viscosity the tangential velocity of the fluid should match the velocity of the solid boundary.

\end{enumerate}

In study of this type of flow, we can consider the flow separated into two portions, one is a flow that is largely viscous, and the other that is largely inertial. This is depicted in figure (\ref{fig:continuumL16:continuumL16Fig2})

\begin{figure}[htp]

\centering

\def\svgwidth{0.5\columnwidth}

\caption{Viscous and inviscous regions in boundary layer flow.}

\end{figure}

We call the study of these two regions boundary layer flow.

# Unsteady rectilinear flow.

Given

Our continuity equation (incompressibility assumption) is

Our non-linear term of Navier-Stokes is then killed

so that Navier Stokes takes the form

\begin{subequations}

\end{subequations}

Taking the partial of 3.4a, we have

Since we also have zero partials in the and directions from 3.4b, and 3.4c, we must then have

So, after integrating we find for the pressure

where

In general is a function of , but constant in space. Given this, we have for our Navier-Stokes equation

## Impulsively started flow.

Reading: section 2 from [1]

Let’s consider a flow driven by a moving boundary. We have two ways that we can look at such a flow, the first of which is with the fluid fixed and the boundary moving and the second is with the fluid moving and the boundary fixed. This are depicted respectively in figure (\ref{fig:continuumL16:continuumL16Fig3a}) and figure (\ref{fig:continuumL16:continuumL16Fig3b})

\begin{figure}[htp]

\centering

\includegraphics[totalheight=0.2\textheight]{continuumL16Fig3a}

\caption{Lagrangian view.}

\end{figure}

\begin{figure}[htp]

\centering

\includegraphics[totalheight=0.2\textheight]{continuumL16Fig3b}

\caption{Eulerian view.}

\end{figure}

These two possible viewpoints can be called the Eulerian and the Lagrangian views where

\begin{itemize}

\item Lagrangian: the observer is moving with the fluid.

\item Eulerian: the observer is fixed in space, watching the fluid.

\end{itemize}

With a flow of the form

NS equation

Our boundary value constraints are

latex t < 0$} \\ U & \quad \mbox{for }\end{array}\right.\end{aligned} \hspace{\stretch{1}}(3.12)$

and as .

If we make a transformation to dimensionless arguments

so that

Then we require of the parameters

so that we have a characteristic length scale of the form

We can find an approximate solution

We can introduce a similarity variable (often hard to find), of the form

FIXME: try attacking this more systematically using Fourier or Laplace transforms (probably a Laplace transform, because of our initial conditions).

Let’s use our similarity variable and see what happens. With

we find

where

We then find

and

Putting these all together we have

or

With , we have

With solution

or

Integrating once more, and writing the integral in terms of the error function

We find

From our boundary value condition at the origin we have

so that

Since , we must have . For our boundary value constraint far from the impulse, we have

but since , we must have . Our solution is then found to be

where (again)

Explicitly, this is

## boundary layer thickness.

If we look at the thickness of the boundary layer for different viscosities, sampled at different times we may end up with curves as in figure (\ref{fig:continuumL16:continuumL16Fig4a})

\begin{figure}[htp]

\centering

\def\svgwidth{0.3\columnwidth}

\caption{Plots of separation thickness for different viscosities.}

\end{figure}

However, it turns out that for any fluids, regardless of the viscosities, the thickness of the boundary layers generally vary as a linear function of so if is plotted against that as in figure (\ref{fig:continuumL16:continuumL16Fig4b}) we see a linear relationship.

\begin{figure}[htp]

\centering

\def\svgwidth{0.3\columnwidth}

\caption{Linear relation between separation thickness and }

\end{figure}

# Appendix. Error function properties.

Let’s verify the value of . In the square that is

# References

[1] D.J. Acheson. *Elementary fluid dynamics*. Oxford University Press, USA, 1990.

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