Bonus: example of the self-similar turbulent jet

Bonus: example of the self-similar turbulent jet#

Weakly non parallel flow scaling laws#

Fig. 21 shows examples of weakly non parallel flows.

../_images/weakly_non_parallel_flows.jpg — Fig. 21 Different weakly non parallel flows. From top to bottom: jet, mixing layer, wake and flat boundary layer.#

Streamwise component of the Navier-Stokes equation:

\[{\partial}_x {\langle u \rangle}^2 + {\partial}_x {\langle u'^2 \rangle} + {\partial}_y {\langle v \rangle}{\langle u \rangle} + {\partial}_y {\langle v'u' \rangle} = - {\partial}_x p + \nu {\boldsymbol{\nabla}}^2 {\langle u \rangle}\]

Neglect terms: weakly non parallel flow and large \(Re\):

\[{\partial}_x {\langle u \rangle}^2 + {\partial}_y {\langle v \rangle}{\langle u \rangle} + {\partial}_y {\langle v'u' \rangle} = 0\]

Flux of Conserved quantities in the jet#

Cylindrical coordinate

Flux of mass (here volume)

\[\mathcal{M} = 2\pi \int_0^\infty {\langle u_z \rangle} r dr.\]

Flux of momentum

\[\mathcal{P} = 2\pi \int_0^\infty {\langle {u_z}^2 \rangle} r dr.\]

Flux of kinetic energy

\[\mathcal{E} = 2\pi \int_0^\infty {\langle u_z {{\boldsymbol{u}}}^2/2 \rangle} r dr.\]

The flux of momentum is constant along the streamwise direction (\(z\)).

Self-similarity, decay law, prediction#

Hypothesis (observed in experimental data):

\[\begin{split}\begin{aligned} {\langle u_z \rangle} &= U(z) F(\xi), \\ {\langle u_r \rangle} &= U(z) G(\xi),\end{aligned}\end{split}\]

where \(\xi = r/b(z)\) and \(b(z)\) is a characteristic width of the jet.

This hypothesis is related to a symmetry of the Euler equation.

Consequence of the self-similarity + divergence free flow:#

\[{\boldsymbol{\nabla}}\cdot {\langle {\boldsymbol{u}}\rangle} \Rightarrow \frac{1}{r} {\partial}_r (r {\langle u_r \rangle}) + {\partial}_z {\langle u_z \rangle} = 0.\]

Change of variables \((z, r) \rightarrow (\tilde z=z, \xi=r/b(z))\)

\[\frac{1}{\xi} d_\xi (\xi G) + \frac{bU'}{U} F(\xi) - b' \xi d_\xi F = 0.\]

Theorem \(\Rightarrow\) \(b'\) and \(z d\log U / dz\) are 2 constants. This implies that \(b(z) = \alpha z\) and \(U \propto z^{-1}\).

Consequences on fluxes of conserved quantities#

Momentum#

With self-similarity, we show that

\[\mathcal{P} = \lambda_2 b^2 U^2,\]

where \(\lambda_2\) is a constant.

Since \(\mathcal{P}\) is conserved, we find that \(U \propto 1/b \propto 1/x\).

Mass: entrainment of ambient fluid#

\[\mathcal{M} = \lambda_1 b^2 U \propto x.\]

The mass flux increases.

Kinetic energy: dissipation#

The flux of kinetic energy

\[\mathcal{E} = \lambda_3 b^2 U^3 \propto 1/x\]

decreases because of dissipation of energy due to viscosity.