Two dimensional turbulent jet -- part 1

Posted in math on Thursday, August 07 2014

Moving into turbulent flow now, I'm going to look at a jet issuing out of a small slit running along the x-y axis with length (in the y-direction) W and basically no width. The flow issues forth in the positive z direction into a semi-infinite reservoir of stationary fluid (same fluid as what is coming out of the jet). (Transport Phenomena 5C.1)

This problem is rather long and I am not super keen on typing in piles of latex, so I'm going to break it into two posts and just talk my way through the math with reference to my chicken scratch.

velocity distribution

To start with I assume the flow can be dealt with as two-dimensional flow in the x-z plane. Above is a sketch of the averaged velocity in the z-direction, which spreads out as it moves into the bulk fluid. A critical length is the width, b, which I suppose to be proportional to z.

I would like to say that assuming the width to be proportional to z comes from some deep physical insight, but it really doesn't. In the case of laminar flow the width of the jet is proportional to $ z^{2/3}$ and in turbulent flow the with is proportional to $ z $, empirically.

As a sneak preview, what follows is a series of curves showing a normalized velocity profile at various heights along the z-axis. Each curve is normalized so they all have the same height, just so they fit nicely on the same plot, and show how the velocity spreads out.

expanding jet

Conservation of Momentum

I can write an expression for the conservation of momentum in the jet, introducing a constant J along the way and a convenience variable $ \xi = \frac{x}{z} $. I assume the velocity is proportional to the centerline velocity, $ \bar{v}_{z,max}$ and some function of ξ which is dimensionless.

The choice of ξ comes from some dimensional reasoning: in this case the length scale of interest is the width of the jet, b. So the dimensionless distance should be x/b, but we know b is proportional to z hence $ \xi = \frac{x}{z} $

page 1

Eddy viscosity

I can also introduce the free turbulence eddy viscosity, and using what I figured for the velocity profile, get an expression in terms of a single unknown constant λ (where all those constants of proportionality have been rolled up in λ).

page 2

Equation of Motion

Now that I have some ammunition, it is time to attack a differential equation. I start with the Navier-Stokes equation in two dimensions, reduce it by assuming away some components, and introduce a stream function to reduce it even further.

page 3

Using what I know about the average velocity profile, I can come up with a guess for what the stream function should look like.

page 4

Plugging that form of the stream function into the simplified Navier-Stokes equation lets me reduce the PDE even further.

page 5

I'll pick this up next time solving the PDE