Two dimensional turbulent jet -- part 2

Posted in math on Sunday, August 10 2014

Continuing on with the analysis of turbulent flow in 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)

I left off last time having arrived at the differential equation:

$$ \frac{1}{2} \left( F^{\prime} \right)^2 + \frac{1}{2} \left( F^{\prime \prime} \right) \cdot \left( F^{\prime \prime} \right) = \lambda \left( F^{\prime \prime \prime} \right) $$

Before I advance trying to solve this I need to figure out some boundary conditions.

Boundary Conditions

The velocity reaches a maximum at the centerline, ξ=0, from the definition of the stream function this means: $$ \frac{\partial}{\partial x} \bar{v}_z = - \frac{\partial^2}{\partial x^2} \psi = \sqrt{\frac{J z}{\rho W}} \cdot \left( \frac{\partial \xi}{\partial x} \right)^2 \cdot \frac{\partial^2}{\partial \xi^2} F(\xi) = 0 \mbox{, at }\xi=0$$

Or, in other words, $$ F^{\prime \prime} (0) = 0$$

The velocity decays down to the bulk fluid velocity, zero, at infinite distance: $$ \bar{v}_z = - \frac{\partial}{\partial x} \psi = \sqrt{\frac{J z}{\rho W}} \cdot \frac{\partial \xi}{\partial x} \cdot \frac{\partial}{\partial \xi} F(\xi) \to 0 \mbox{, as }\xi \to \infty $$

In other words, $$ F^{\prime} (\infty) = 0$$

And it follows that the last boundary condition is $$ F(0) = 0$$

Integration

To get rid of the constant λ and otherwise make the PDE easier to deal with, I've introduced a convenience variable η $$ \eta = \frac{\xi}{4 \lambda} $$

With that substitution I can easily integrate once with respect to η.

page 1

I can use two of the boundary conditions to show that the constant of integration must be zero.

The result can be integrated once more with respect to η

page 2

Here the constant of integration was set to C2 in a kind of revisionist history. The first run through with a regular constant leaves a result with square-roots all over, hence going back and setting the constant of integration to be C2 to make everything nice and pretty.

Solving for the Integration Constant

I can solve for C by plugging it all into my expression for conservation of momentum in the jet.

page 3

The Time-Averaged Velocity Profile

This then leads to a final expression for the velocity

$$ \bar{v}_z = \left(\frac{9}{64 \lambda} \right)^{\frac{1}{3}} \sqrt{\frac{J z}{\rho W}} sech^{2} \left( \left( \frac{3}{64 \lambda^2} \right)^{\frac{1}{3}} \frac{x}{z} \right) $$

Given an empirical value $ \lambda = 0.0102$ $$ \bar{v}_z = 2.3978 \sqrt{\frac{J z}{\rho W }} sech^{2} \left( 7.6662 \frac{x}{z} \right) $$

This figure shows the time-averaged velocity in the z-direction, $ \bar{v}_z$, at various levels of z.

velocity profiles

This figure shows the time averaged velocity vector field. Notice weirdness around z=0.

velocity field

The Mass Flowrate

The mass flow-rate from the jet at any given z can be found by integrating over the cross-sectional area

mass flowrate

Which when plotted looks like plot of mass flowrate

The mass flow-rate in the jet increases as it expands into the fluid because the jet is dragging along fluid that had been quiescent.