Introduction to turbulence/Turbulence kinetic energy
From CFD-Wiki
It is clear from the previous chapter that the straightforward application of ideas that worked well for viscous stresses do not work too well for turbulence Reynolds stresses. Moreover, even the attempt to directly derive equations for the Reynolds stresses using the Navier-Stokes equations as a starting point has left us with far more equations than unknowns. Unfortunately this means that the turbulence problem for engineers is not going to have a simple solution: we simply cannot produce a set of reasonably universal equations. Obviously we are going to have to study the turbulence fluctuations in more detail and learn how they get their energy (usually from the mean flow somehow), and what they ultimately do with it. Our hope is that by understanding more about turbulence itself, we will gain insight into how we might make closure approximations that will work, at least sometimes. Hopefully, we will also gain an understanding of when and why they will not work.
An equation for the fluctuating kinetic energy for constant density flow can be obtained directly from the Reynolds stress equation derived earlier, equation 3.35, by contracting the free indices. The result is:
|
| (4.1) |
![\begin{matrix}
\left[ \frac{\partial}{\partial t} \left\langle u_{i} u_{i} \right\rangle + U_{j} \frac{\partial }{\partial x_{j} } \left\langle u_{i} u_{i} \right\rangle \right] \\
& = & \frac{\partial}{\partial x_{j}} \left\{ -\frac{2}{\rho} \left\langle p u_{i} \right\rangle \delta_{ij} - \left\langle q^{2} u_{j} \right\rangle + 4 \nu \left\langle s_{ij} u_{i} \right\rangle \right\} \\
& & - 2 \left\langle u_{i}u_{j} \right\rangle \frac{\partial U_{i}}{\partial x_{j}} - 4 \nu \left\langle s_{ij} \frac{\partial u_{i}}{\partial x_{j} } \right\rangle \\
\end{matrix}](/W/images/math/c/d/d/cdd713e7c866ab3a41c1db5df222a7b2.png)
where the incompressibility condition ( 
) has been used to eliminate the pressure-strain rate term, and 
.
The last term can be simplified by recalling that the velocity deformation rate tensor, 
, can be decomposed into symmetric and anti-symmetric parts; i.e.,
|
| (4.2) |
where the symmetric part is the strain-rate tensor, 
, and the anti-symmetric part is the rotation-rate tensor 
, defined by:
|
| (4.3) |
![\omega_{ij} = \frac{1}{2} \left[ \frac{\partial u_{i}}{\partial x_{j}} - \frac{\partial u_{j}}{\partial x_{i}} \right]](/W/images/math/b/7/7/b771bd4a8422189e342acf06c33672da.png)
Since the double contraction of a symmetric tensor with an anti-symmetric tensor is identically zero, it follows immediately that:
|
| (4.4) |
Now it is customary to define a new variable k, the average fluctuating kinetic energy per unit mass, by:
|
| (4.5) |
![k \equiv \frac{1}{2} \left\langle u_{i}u_{i} \right\rangle = \frac{1}{2} \left\langle q^{2} \right\rangle = \frac{1}{2} \left[ \left\langle u^{2}_{1} \right\rangle + \left\langle u^{2}_{2} \right\rangle + \left\langle u^{2}_{3} \right\rangle \right]](/W/images/math/5/f/4/5f49df0b6717a9a52c142a281cda814f.png)
By dividing equation 4.1 by 2 and inserting this definition, the equation for the average kinetic energy per unit mass of the fluctuating motion can be re-written as:
|
| (4.6) |
![\begin{matrix}
\left[ \frac{\partial}{\partial t} + U_{j} \frac{\partial}{\partial x_{j}} \right] k & = & \frac{\partial}{\partial x_{j}} \left\{ - \frac{1}{\rho} \left\langle pu_{i} \right\rangle \delta_{ij} - \frac{1}{2} \left\langle q^{2} u_{j} \right\rangle + 2 \nu \left\langle s_{ij}u_{i} \right\rangle \right\} \\
& & - \left\langle u_{i}u_{j} \right\rangle \frac{\partial U_{i}}{\partial x_{j} } - 2 \nu \left\langle s_{ij} s_{ij} \right\rangle \\
\end{matrix}](/W/images/math/d/3/e/d3e5c43dd0769a3aad2fa9455ba955af.png)
The role of each of these terms will be examined in detail later. First note that an alternative form of this equation can be derived by leaving the viscous stress in terms of the strain rate. We can obtain the appropriate form of the equation for the fluctuating momentum from equation 3.21 by substituting the incompressible Newtonian constitutive equation into it to obtain:
|
| (4.7) |
![\left[ \frac{\partial }{\partial t } + U_{j} \frac{\partial }{\partial x_{j} } \right] u_{i} = - \frac{1}{\rho} \frac{\partial p}{\partial x_{i}} + \nu \frac{\partial^{2} u_{i}}{ \partial x^{2}_{j}} - \left[ u_{j} \frac{\partial U_{i}}{\partial x_{j} } \right] - \left\{ u_{j} \frac{\partial u_{i}}{ \partial x_{j}} - \left\langle u_{j} \frac{\partial u_{i}}{\partial x_{j}} \right\rangle \right\}](/W/images/math/e/0/b/e0b4576b17fa45fb1c66a20067c611f4.png)
If we take the scalar product of this with the fluctuating velocity itself and average, it follows (after some rearrangement) that:
|
| (4.8) |
![\begin{matrix}
\left[ \frac{\partial}{\partial t} + U_{j} \frac{\partial}{\partial x_{j}} \right] k & = & \frac{\partial }{ \partial x_{j} } \left\{ - \frac{1}{\rho} \left\langle pu_{i} \right\rangle \delta_{ij} - \frac{1}{2} \left\langle q^{2} u_{j} \right\rangle + \nu \frac{\partial}{\partial x_{j} } k \right\} \\
& & - \left\langle u_{i} u_{j} \right\rangle \frac{\partial U_{i}}{\partial x_{j}} - \nu \left\langle \frac{\partial u_{i}}{\partial x_{j}} \frac{\partial u_{i}}{\partial x_{j}} \right\rangle\\
\end{matrix}](/W/images/math/5/b/7/5b7fd5594272d4c5ca943c732ada6978.png)
Both equations 4.6 and 4.8 play an important role in the study of turbulence. The first form given by equation 4.6 will provide the framework for understanding the dynamics of turbulent motion. The second form, equation 4.8 forms the basis for most of the second-order closure attempts at turbulence modelling; e.g., the socalled k-e models ( usually referred to as the “k-epsilon models”). This because it has fewer unknowns to be modelled, although this comes at the expense of some extra assumptions about the last term. It is only the last term in equation 4.6 that can be identified as the true rate of dissipation of turbulence kinetic energy, unlike the last term in equation 4.8 which is only the dissipation when the flow is homogeneous. We will talk about homogeniety below, but suffice it to say now that it never occurs in nature. Nonetheless, many flows can be assumed to be homogeneous at the scales of turbulence which are important to this term, so-called local homogeniety.
Each term in the equation for the kinetic energy of the turbulence has a distinct role to play in the overall kinetic energy balance. Briefly these are:
- Rate of change of kinetic energy per unit mass due to non-stationarity; i.e., time dependence of the mean:
|
| (4.9) |
- Rate of change of kinetic energy per unit mass due to convection (or advection) by the mean flow through an inhomogenous field :
|
| (4.10) |
- Transport of kinetic energy in an inhomogeneous field due respectively to the pressure fluctuations, the turbulence itself, and the viscous stresses:
