Free Shear Flows

Self-Preserving Wakes

Turbulent Kinetic Energy Budget

Recall the general form of the turbulent kinetic energy

\frac {\overline{\mathrm D}}{\mathrm Dt}\left(\frac 12\overline{u_iu_i}\right)=-\frac {\partial}{\partial x_j}\left[\frac {\overline{u_jp}}{\rho}+\frac 12\overline{u_iu_iu_j}-2\nu\overline{u_i\mathfrak s_{ij}}\right]-\overline{u_iu_j}\overline{S_{ij}}-2\nu\overline{s_{ij}s_{ij}} \tag{1}

where the substantial derivative is defined as

\frac {\overline{\mathrm D}}{\mathrm Dt}=\frac {\partial}{\partial t}+\overline U_j\frac {\partial}{x_j}

Now, make the following assumptions:

  1. Statistically stationary, so the statistics of the fluctuations do not vary across time and \frac {\partial}{\partial t}(\overline{\hspace{5mm}})=0.

  2. The wake is thin, meaning the spanwise thickness, \ell, is much smaller than the streamwise length, L.

  3. Stream-wise mean flow is much larger than the cross-stream flow, \overline U_1\gg\overline U_2

  4. Approximately homogeneous in the x_1 direction and completely homogeneous in the x_3 direction

The simplification of Eq. 1 from order of magnitude analysis becomes

-U_{\infty}\frac {\partial}{\partial x}\left(\frac 12\overline{q^2}\right)-\frac {\partial}{\partial y}\left(\frac {\overline{vp}}{\rho}+\frac 12\overline{vq^2}\right)-\overline{uv}\frac {\partial\overline U}{\partial y}-\varepsilon=0 \tag{2}

Starting from Eq. 1, the substantial derivative is purely composed of the spatial convective derivative terms due to statistical stationarity. When expanded out, then we get

\begin{aligned} \frac {\overline{\mathrm D}}{\mathrm Dt}\left(\frac 12\overline{u_iu_i}\right) & =\overline U_j\frac {\partial}{x_j}\left(\frac 12\overline{u_iu_i}\right) \\ & =\overline U_1\frac {\partial}{\partial x_1}\left(\frac 12\overline{u_1u_1}\right)+\overline U_2\frac {\partial}{\partial x_2}\left(\frac 12\overline{u_2u_2}\right)+\overline U_3\frac {\partial}{\partial x_3}\left(\frac 12\overline{u_3u_3}\right) \end{aligned}

Because of homogeneity in the x_3 direction, \overline U_3\approx0 and \frac {\partial}{\partial x_3}\left(\frac 12\overline{u_3u_3}\right)\approx0. For the convective term in the x_2 direction, we use continuity and order of magnitude analysis on the stream-wise and cross-stream flow directions. Consider a turbulent wake with a point A positioned close to the bluff body, and another point B positioned far away downstream. The velocity at point A is subject to the velocity defect and is \overline U_A\sim U_{\infty}-\overline U_s. At point B, since it is far downstream, \overline U_s is negligible and \overline U_B\sim U_{\infty}. Denoting the length scale in the x_1 and x_2 directions as L and \ell respectively, then

\frac {\partial\overline U}{\partial x_1}\sim\frac {\overline U_B-\overline U_A}{\partial x_1}\sim\frac {U_{\infty}-\left(U_{\infty}-\overline U_s\right)}L\sim\frac {\overline U_s}L

From continuity for constant density flows, then

\frac {\partial\overline U_2}{\partial x_2}=-\frac {\partial\overline U_1}{\partial x_1}\sim\frac {\overline U_s}L

Since the length scale in the x_2 direction is \ell, then

\overline U_2\sim\frac {\overline U_s\ell}L

By assuming a thin turbulent wake, then \ell\ll L and \overline U_2\ll\overline U_1. As a result, the convective term in the x_2 direction, \overline U_2\frac {\partial}{\partial x_2}\left(\frac 12\overline{u_2u_2}\right) becomes negligible when compared to the x_1 direction due to the \overline U_2 term. Using the fact that \overline U_1=U_{\infty}-\overline U_s\approx U_{\infty} far downstream, and letting \overline{q^2}=\overline{u_iu_i}, then the substantial derivative becomes

\frac {\overline{\mathrm{D}}}{\mathrm Dt}\left(\frac 12\overline{u_iu_i}\right)=U_{\infty}\frac {\partial}{\partial x}\left(\frac 12\overline{q^2}\right)

The right-hand side of Eq. 1 can be simplified by first observing that \frac {\partial}{\partial x_1}\sim\frac 1L whereas \frac {\partial}{\partial x_2}\sim\frac 1{\ell}. Hence, \frac {\partial}{\partial x_1}\ll\frac {\partial}{\partial x_2}. As a result, the entire \frac {\partial}{\partial x_1} term may be neglected. Since the flow is homogeneous in the spanwise x_3 direction, we may also neglect the entire \frac {\partial}{\partial x_3} term. What is left from the turbulent transport term is

\frac {\partial}{\partial x_2}\left[\frac {\overline{u_2p}}{\rho}+\frac 12\overline{u_iu_iu_2}-2\nu\overline{u_is_{i2}}\right]

Viscous turbulent transport may be neglected. The fluctuating strain rate, s_{i2}, shares a similar order of magnitude as s_{i2}\sim\mathcal U/\ell