Standard $k\!-\!\varepsilon$ Model
For flows that satisfy the Boussinesq approximation,
the buoyancy source term in the transport equations for the turbulent kinetic energy \(k\) and its dissipation rate \(\varepsilon\) can be neglected
\(\Longrightarrow \frac{\partial \varepsilon}{\partial t}
+ U_k \frac{\partial \varepsilon}{\partial x_k}
= -\frac{\partial}{\partial x_j}
\left(
\nu u_j\overline{\frac{\partial u_i}{\partial x_k},\frac{\partial u_i}{\partial x_k}}
-\nu\frac{\partial \varepsilon}{\partial x_k}
\right)
- 2\nu\frac{\partial^2 U_i}{\partial x_j\partial x_k}
\overline{u_j
\frac{\partial u_i}{\partial x_k}}
- 2\nu
\overline{\frac{\partial u_i}{\partial x_j}
\frac{\partial u_i}{\partial x_k}
\frac{\partial u_j}{\partial x_k}}
- 2\nu^{2}
\overline{\frac{\partial^{2} u_i}{\partial x_j\partial x_k}
\frac{\partial^{2} u_i}{\partial x_j\partial x_k}}\)
When analyze the order of magnitude of each term on its right-hand side and discard those of smaller order, the following sets of turbulent scales should be considered
-
The characteristic length scale $L$ and velocity scale $U$ of the turbulent mean motion
-
The characteristic length scale $\ell$ of the energy-containing eddies in the inertial range and the characteristic turbulent-fluctuation velocity scale $u$, with $u\sim k^{1/2}$
-
The smallest eddy length scale \(\eta\) and velocity scale \(v\) associated with the dissipation of turbulent kinetic energy (the Kolmogorov scales).
Dimensional analysis gives
$$
\eta=\left(\frac{\nu^{3}}{\varepsilon}\right)^{1/4},\qquad
v=\left(\nu\,\varepsilon\right)^{1/4}
$$
Velocity fluctuations that appear inside spatial derivative operators
(e.g., \(\frac{\partial u_i}{\partial x_k}, \frac{\partial^2 u_i}{\partial x_j \partial x_k}\)) should be interpreted as
velocity fluctuations generated by dissipation of eddies,
whereas velocity fluctuations outside the space derivative operators should be interpreted as fluctuations generated by energy-containing eddies.
For example,
the order of magnitude of the "small-eddy stretching/production" term may be estimated as
$$
-2\nu
\overline{\frac{\partial u_i}{\partial x_j}
\frac{\partial u_i}{\partial x_k}
\frac{\partial u_j}{\partial x_k}}
\;\sim\;
\varepsilon\,\Big(\frac{\varepsilon}{\nu}\Big)^{1/2}
$$
For correlation quantities involving velocity fluctuations and spatial derivatives of velocity fluctuation, such as
\(
\overline{u_j \frac{\partial u_i}{\partial x_k}}
\),
since the correlation between large-scale energy-containing eddies and
small-scale dissipative eddies decreases as the turbulent Reynolds number \(Re_t\) increases,
its order of magnitude should include a decay factor \(Re_t^{n}\).
The exponent \(n\) of this factor can be determined as
\[
\overline{u_i \frac{\partial u_i}{\partial x_j}} = \frac{\partial k}{\partial x_j}
\Rightarrow
\sqrt{k \left(\frac{\varepsilon}{\nu}\right)} Re_t^{n} \sim \frac{k}{l}
\]
\(\because \varepsilon \sim k^{1.5}/l, Re_t \sim k^{0.5} l / \nu \Rightarrow n = -0.5\)
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