10.1. RANS and energy equations
10.1.1. RANS
Incompressible Navier—Stokes equations.
\[\begin{split}\begin{cases}
\partial_t \mathbf{u} + ( \mathbf{u} \cdot \nabla ) \mathbf{u} - \nu \Delta \mathbf{u} + \nabla P = \mathbf{0} \\
\nabla \cdot \mathbf{u} = 0 \ ,
\end{cases}\end{split}\]
with \(P = \frac{P}{\overline{\rho}}\) with little abuse of notation, and no volume force. For incompressible flows, the following relation holds
\[\nabla \cdot \left( \mathbf{u} \otimes \mathbf{u} \right) = ( \mathbf{u} \cdot \nabla ) \mathbf{u} \ .\]
Using Cartesian coordinates
\[\left\{ \nabla \cdot \left( \mathbf{u} \otimes \mathbf{u} \right) \right}_i = ( u_j u_i )_{/j} = \underbrace{u_{j/j}}_{=0} u_i + u_j u_{i/j} = \left\{ ( \mathbf{u} \cdot \nabla ) \mathbf{u} \right\}_i \ ,\]
as \(u_{j/j} = \nabla \cdot \mathbf{u} = 0\), for the incompressiblity constraint.
Incompressible Reynolds-averaged Navier—Stokes equations, RANS. Applying average operator to NS equations
\[\overline{\text{NS}} \ ,\]
after having defined the relation between the realization of a field, its average and fluctuation,
\[f = \overline{f} + f' \ ,\]
so that applying average \(\overline{f'} = 0\), RANS equations follows
\[\begin{split}\begin{cases}
\partial_t \overline{\mathbf{u}} + ( \overline{\mathbf{u}} \cdot \nabla ) \overline{\mathbf{u}} + \nabla \cdot \overline{\mathbf{u}' \otimes \mathbf{u}'} - \nu \Delta \overline{\mathbf{u}} + \nabla \overline{P} = \mathbf{0} \\
\nabla \cdot \overline{\mathbf{u}} = 0 \ ,
\end{cases}\end{split}\]
if average and differential operators swaps for linear terms. An extra-contribution arises from the non-linear term, as
\[\begin{split}\begin{aligned}
\overline{ a b }
& = \overline{ (\overline{a}+a') (\overline{b}+b')} = \\
& = \overline{ \overline{a} \overline{b} } + \underbrace{ \overline{ \overline{a} b' } }_{\overline{a} \overline{b'} = 0} + \underbrace{\overline{ a' \overline{b} }}_{\overline{a'} \overline{b}=0} + \overline{ a' b'} = \\
& = \overline{a} \overline{b} + \overline{ a' b'} \ .
\end{aligned}\end{split}\]
This can be interpreted as extra stress contribution (as it has the expression of the divergence of a 2-nd order symmetric tensor) felt by the average fields \(\overline{\mathbf{u}}, \overline{P}\), due to fluctuations \(\mathbf{u}'\).
Equation of the fluctuation. Taking the difference between NS equations and RANS equations, the equation for the fluctuation \(\mathbf{u}' = \mathbf{u} - \overline{\mathbf{u}}\) seamlessly follows
\[\begin{split}\begin{cases}
\partial_t \mathbf{u}' + \nabla \cdot (\mathbf{u} \otimes \mathbf{u}) - \nabla \cdot (\overline{\mathbf{u}} \otimes \overline{\mathbf{u}}) - \nabla \cdot \overline{ (\mathbf{u}' \otimes \mathbf{u}')} - \nu \Delta \mathbf{u}' + \nabla P' = \mathbf{0} \\
\nabla \cdot \mathbf{u}' = 0 \ ,
\end{cases}\end{split}\]
and
\[\begin{split}\begin{cases}
\partial_t \mathbf{u}' + \nabla \cdot (\overline{\mathbf{u}} \otimes \mathbf{u}') + \nabla \cdot (\mathbf{u}' \otimes \overline{\mathbf{u}}) + \nabla \cdot (\mathbf{u}' \otimes \mathbf{u}' ) - \nabla \cdot \overline{ (\mathbf{u}' \otimes \mathbf{u}')} - \nu \Delta \mathbf{u}' + \nabla P' = \mathbf{0} \\
\nabla \cdot \mathbf{u}' = 0 \ ,
\end{cases}\end{split}\]
10.1.2. Energy equations
Here, kinetic energy equation, average kinetic energy, kinetic energy of the average field, kinetic energy of the fluctuation and its average,…
Kinetic energy per unit-mass is
\[K := \frac{\mathbf{u} \cdot \mathbf{u}}{2} \ ,\]
and it can be recast using the average and the fluctuation fields
\[\begin{split}\begin{aligned}
K
& = \frac{\mathbf{u} \cdot \mathbf{u}}{2} = \\
& = \frac{( \overline{\mathbf{u}} + \mathbf{u}') \cdot (\overline{\mathbf{u}} + \mathbf{u}')}{2} = \\
& = \frac{\overline{\mathbf{u}} \cdot \overline{\mathbf{u}}}{2} + \frac{\mathbf{u}' \cdot \mathbf{u}'}{2} + \overline{\mathbf{u}} \cdot \mathbf{u}'
\end{aligned}\end{split}\]
being the first tern the kinetic energy of the average flow, the second term the kinetic energy of the fluctuations, and the last term the cross-effects. Taking average, to get the average kinetic energy \(\overline{K}\) the last term disappears, and a relation between the average kinetic energy, the energy of the average flow and the turbulent energy (defined as the average of the kinetic energy of the fluctuations) arises
\[\overline{K} = K_{\overline{\mathbf{u}}} + k \ .\]
Kinetic energy, \(K\). Taking the dot-product of the NS equations with the velocity field, \(\mathbf{u} \cdot \text{NS}\), and exploiting product rule for derivatives,
\[\partial_t \frac{\mathbf{u} \cdot \mathbf{u}}{2} + ( \mathbf{u} \cdot \nabla ) \frac{\mathbf{u} \cdot \mathbf{u}}{2} - \mathbf{u} \cdot \nabla \cdot \mathbb{S} + \mathbf{u} \cdot \nabla P = 0\]
As shown in the box below, the last two contributions can be rearranged to get
\[\frac{D K}{Dt} = \nabla \cdot \left( \nu \nabla K - P \mathbf{u} \right) - \nu | \nabla \mathbf{u} |^2 \ .\]
Assuming uniform viscosity \(\nu\)
\[\begin{split}\begin{aligned}
\mathbf{u} \cdot \nabla \cdot \mathbb{S}
& = \nu \mathbf{u} \cdot \nabla \cdot \left( \nabla \mathbf{u} + \nabla^T \mathbf{u} \right) = \\
& = \nu \mathbf{u} \cdot \Delta \mathbf{u} = \\
& = \nu \nabla \cdot \left( \left( \nabla \mathbf{u} \right) \cdot \mathbf{u} \right) - \nu \nabla \mathbf{u} : \nabla \mathbf{u} = \\
& = \nabla \cdot \left( \nu \nabla \frac{\mathbf{u} \cdot \mathbf{u}}{2} \right) - \nu |\nabla \mathbf{u}|^2 \ ,
\end{aligned}\end{split}\]
as \(\nabla \cdot \nabla^T \mathbf{u} = \partial_i \partial_j u_i = \partial_j u_{i/i} = 0\).
Pressure term can be recast as a divergence contribution as
\[\mathbf{u} \cdot \nabla P = \nabla \cdot ( P \mathbf{u} ) - P \underbrace{ \nabla \cdot \mathbf{u} }_{=0} = \nabla \cdot ( P \mathbf{u} ) \ .\]
Average kinetic energy, \(\overline{K}\). Taking the average of the kinetic energy equation,
\[\partial_t \frac{\mathbf{u} \cdot \mathbf{u}}{2} + ( \mathbf{u} \cdot \nabla ) \frac{\mathbf{u} \cdot \mathbf{u}}{2} - \mathbf{u} \cdot \nabla \cdot \mathbb{S} + \mathbf{u} \cdot \nabla P = 0\]
\[\overline{\partial_t \frac{\mathbf{u} \cdot \mathbf{u}}{2}} = \partial_t \overline{K}\]
\[\overline{\mathbf{u} \cdot \nabla K} = \overline{u}\]
…
todo
Kinetic energy of the average field, \(K_{\overline{\mathbf{u}}}\). Taking the scalar product \(\overline{\mathbf{u}} \cdot \text{RANS}\)
\[\begin{aligned}
\dfrac{\overline{D} K_{\overline{\mathbf{u}}}}{\overline{D}t}
= \nabla \cdot \left( \nu \nabla K_{\overline{\mathbf{u}}} - \overline{\mathbf{u}' \otimes \mathbf{u}'} \cdot \overline{\mathbf{u}} - \overline{P} \overline{\mathbf{u}} \right)
+ \nabla \overline{\mathbf{u}} : \overline{\mathbf{u}' \otimes \mathbf{u}'} - \nu \nabla \overline{\mathbf{u}} : \nabla \overline{\mathbf{u}}
\end{aligned}\]
Source contributions:
turbulent energy «production» (as it will be more clear later), \(\nabla \overline{\mathbf{u}} : \overline{\mathbf{u}' \otimes \mathbf{u}'}\). This contribution has no defined sign. It appears with the opposite side in the equation of the turbulent energy: it could be interpreted as a term that (mainly?) removes energy from the average flow to produce turbulent kinetic energy
viscous dissipation from the average flow (always non positive), \(- \nu \nabla \overline{\mathbf{u}} : \nabla \overline{\mathbf{u}} \le 0\). Is this relevant for large Reynolds number?
Flux contributions: …
\[\dfrac{\overline{D} K_{\overline{\mathbf{u}}}}{\overline{D} t} = \nabla \cdot \boldsymbol\Phi_{K_{\overline{\mathbf{u}}}} - \mathscr{P} - \mathscr{D}_{\overline{\mathbf{u}}}\]
\[\begin{split}\begin{cases}
\partial_t \overline{\mathbf{u}} + ( \overline{\mathbf{u}} \cdot \nabla ) \overline{\mathbf{u}} + \nabla \cdot \overline{\mathbf{u}' \otimes \mathbf{u}'} - \nu \Delta \overline{\mathbf{u}} + \nabla \overline{P} = \mathbf{0} \\
\nabla \cdot \overline{\mathbf{u}} = 0 \ ,
\end{cases}\end{split}\]
\[\partial_t K_{\overline{\mathbf{u}}}\]
\[\overline{\mathbf{u}} \cdot (\overline{\mathbf{u}} \cdot \nabla ) \overline{\mathbf{u}} = \overline{\mathbf{u}} \cdot \nabla K_{\overline{\mathbf{u}}}\]
\[\overline{\mathbf{u}} \cdot \nabla \cdot \overline{\mathbf{u}' \otimes \mathbf{u}'} = U_j ( \overline{ u'_i u'_j } )_{/i} = \nabla \cdot \left( \overline{\mathbf{u}' \otimes \mathbf{u}'} \cdot \overline{\mathbf{u}} \right) - \nabla \overline{\mathbf{u}} : \overline{\mathbf{u}' \otimes \mathbf{u}'} \]
…
\[\begin{split}\begin{aligned}
0 & = \dfrac{\overline{D} K_{\overline{\mathbf{u}}}}{\overline{D}t} + \\
& + \nabla \cdot \left( \overline{\mathbf{u}' \otimes \mathbf{u}'} \cdot \overline{\mathbf{u}} \right) - \nabla \overline{\mathbf{u}} : \overline{\mathbf{u}' \otimes \mathbf{u}'} + \\
& - \nabla \cdot \left( \nu \nabla K_{\overline{\mathbf{u}}} \right) + \nu \nabla \overline{\mathbf{u}} : \nabla \overline{\mathbf{u}} + \\
& + \nabla \cdot \left( \overline{P} \overline{\mathbf{u}} \right)
\end{aligned}\end{split}\]
or collecting divergence (or flux, in an integral formulation) terms and volume source terms
\[\begin{aligned}
\dfrac{\overline{D} K_{\overline{\mathbf{u}}}}{\overline{D}t}
= \nabla \cdot \left( \nu \nabla K_{\overline{\mathbf{u}}} - \overline{\mathbf{u}' \otimes \mathbf{u}'} \cdot \overline{\mathbf{u}} - \overline{P} \overline{\mathbf{u}} \right)
+ \nabla \overline{\mathbf{u}} : \overline{\mathbf{u}' \otimes \mathbf{u}'} - \nu \nabla \overline{\mathbf{u}} : \nabla \overline{\mathbf{u}}
\end{aligned}\]
Kinetic energy of the fluctuation, \(k'\). Taking the scalar product \(\mathbf{u}' \cdot \text{Fluctuation equations}\)
\[\frac{D k'}{Dt} =
- \nabla \overline{\mathbf{u}} : \mathbf{u}' \otimes \mathbf{u}' - \overline{\mathbf{u}'\otimes \mathbf{u}'} : \nabla \mathbf{u}' - \nu \nabla \mathbf{u}' : \nabla \mathbf{u}'
+ \nabla \cdot \left( \overline{\mathbf{u}' \otimes \mathbf{u}'} \cdot \mathbf{u}' + \nu \nabla k' - P' \mathbf{u}' \right)
\]
\[\begin{split}\begin{cases}
\partial_t \mathbf{u}' + \nabla \cdot (\overline{\mathbf{u}} \otimes \mathbf{u}') + \nabla \cdot (\mathbf{u}' \otimes \overline{\mathbf{u}}) + \nabla \cdot (\mathbf{u}' \otimes \mathbf{u}' ) - \nabla \cdot \overline{ (\mathbf{u}' \otimes \mathbf{u}')} - \nu \Delta \mathbf{u}' + \nabla P' = \mathbf{0} \\
\nabla \cdot \mathbf{u}' = 0
\end{cases}\end{split}\]
\[\mathbf{u}' \cdot \partial_t \mathbf{u}' = \partial_t \frac{\mathbf{u}' \cdot \mathbf{u}'}{2}\]
\[\mathbf{u}' \cdot \nabla \cdot \left( \overline{\mathbf{u}} \otimes \mathbf{u}' \right) = u'_i ( U_j u'_i )_{/j} = U_j \{( u'_i u'_i ) / 2\}_{/j} \]
\[\mathbf{u}' \cdot \nabla \cdot \left( \mathbf{u}' \otimes \overline{\mathbf{u}} \right) = u'_i ( u'_j U_i )_{/j} = u'_i u'_j U_{i/j}\]
\[\mathbf{u}' \cdot \nabla \cdot \left( \mathbf{u}' \otimes \mathbf{u}' \right) = u'_i ( u'_j u'_i )_{/j} = u'_j \left\{ \frac{u'_i u'_i}{ 2 }\right\}_{/j} = \left\{ u'_j \frac{u'_i u'_i}{2} \right\}_{/j}\]
\[\mathbf{u}' \cdot \nabla \cdot\overline{ \left( \mathbf{u}' \otimes \mathbf{u}' \right) } = u'_i \overline{ ( u'_j u'_i )}_{/j} = (u'_i \overline{u'_j u'_i})_{/j} - u'_{i/j} \overline{u'_j u'_i} \]
\[\begin{split}\begin{aligned}
u'_i u'_{i/jj}
& = ( u'_i u'_{i/j} )_{/j} - u'_{i/j} u'_{i/j} && = \left(\frac{u'_i u'_i}{2}\right)_{/jj} - u'_{i/j} u'_{i/j} = \\
\end{aligned}\end{split}\]
and putting everything together
\[\partial_t k' + \mathbf{u} \cdot \nabla k' + \nabla \overline{\mathbf{u}} : \mathbf{u}' \otimes \mathbf{u}' - \nabla \cdot \left( \overline{\mathbf{u}' \otimes \mathbf{u}'} \cdot \mathbf{u}' \right) + \overline{\mathbf{u}'\otimes \mathbf{u}'} : \nabla \mathbf{u}' - \Delta ( \nu k' ) + \nu \nabla \mathbf{u}' : \nabla \mathbf{u}' + \nabla \cdot \left( P' \mathbf{u}' \right) = 0\]
Average kinetic energy of the fluctuation, \(k\). Taking the average of the equation for the kinetic energy of the fluctuation…
\[
\dfrac{\overline{D} k}{\overline{D} t} = \nabla \cdot \left( \nu \nabla k - \overline{\mathbf{u}' k'} - \overline{ P' \mathbf{u}'} \right) - \nabla \overline{\mathbf{u}} : \overline{\mathbf{u}' \otimes \mathbf{u}'} - \nu \overline{\nabla \mathbf{u}' : \nabla \mathbf{u}'}
\]
Source contributions:
turbulent energy «production», \(\mathscr{P} := - \nabla \overline{\mathbf{u}} : \overline{\mathbf{u}' \otimes \mathbf{u}'}\). This contribution has no defined sign. It appears with the opposite side in the equation of the kinetic eenrgy of the average field: thus, it could be interpreted as a term that (mainly?) removes energy from the average flow to produce turbulent kinetic energy
viscous dissipation (always non positive), \(\mathscr{D} := - \nu \overline{\nabla \mathbf{u}' : \nabla \mathbf{u}'} \le 0\)
Flux contributions: …
\[\dfrac{\overline{D} k}{\overline{D} t} = \nabla \cdot \boldsymbol\Phi_k + \mathscr{P} - \mathscr{D}\]
\[\partial_t k + \overline{\mathbf{u}} \cdot \nabla k + \nabla \cdot \overline{( \mathbf{u}' k' ) } + \nabla \overline{\mathbf{u}} : \overline{\mathbf{u}' \otimes \mathbf{u}'} - \Delta ( \nu k ) + \nu \overline{ \nabla \mathbf{u}' : \nabla \mathbf{u}' } + \nabla \cdot \overline{ \left( P' \mathbf{u}' \right)} = 0\]
as
\[\overline{\mathbf{u} \cdot \nabla k'} = \overline{ ( \overline{\mathbf{u}} + \mathbf{u}' ) \cdot \nabla k'} = \overline{\mathbf{u}} \cdot \nabla k + \overline{ \mathbf{u}' \cdot \nabla k'} = \overline{\mathbf{u}} \cdot \nabla k + \nabla \cdot \overline{ ( \mathbf{u}' k' ) }\]
Rearranging terms collecting divergence contributions,
\[
\dfrac{\overline{D} k}{\overline{D} t} = \nabla \cdot \left( \nu \nabla k - \overline{\mathbf{u}' k'} - \overline{ P' \mathbf{u}'} \right) - \nabla \overline{\mathbf{u}} : \overline{\mathbf{u}' \otimes \mathbf{u}'} - \nu \overline{\nabla \mathbf{u}' : \nabla \mathbf{u}'}
\]
10.1.3. Dimensional analysis and the assumption of turbulence in equilibrium
A dimensional analysis of the equations of the kinetic energy of the average flow and turbulent energy helps in stating the concept of turbulence in equilibrium and its consequences.
Kinetic energy of the average field.
\[\begin{aligned}
\frac{U^2}{T} \dfrac{\overline{D} K_{\overline{\mathbf{u}}}}{\overline{D}t}
= \frac{1}{L} \nabla \cdot \left( \frac{\nu U^2}{L} \nabla K_{\overline{\mathbf{u}}} - \mathscr{P} L \, \overline{\mathbf{u}' \otimes \mathbf{u}'} \cdot \overline{\mathbf{u}} - P U \overline{P} \overline{\mathbf{u}} \right)
+ \mathscr{P} \nabla \overline{\mathbf{u}} : \overline{\mathbf{u}' \otimes \mathbf{u}'} - \frac{\nu U^2}{L^2} \nabla \overline{\mathbf{u}} : \nabla \overline{\mathbf{u}}
\end{aligned}\]
Without an independent time scale, \(T = \frac{L}{U}\), and in incompressible flows \(P = U^2\), and dividing for \(\frac{U^3}{L}\)
\[\begin{aligned}
\dfrac{\overline{D} K_{\overline{\mathbf{u}}}}{\overline{D}t}
= - \nabla \cdot \left( \overline{P} \overline{\mathbf{u}} \right) + \frac{\nu}{U L} \left\{ \nabla \cdot \left( \nabla K_{\overline{\mathbf{u}}} \right) - \nabla \overline{\mathbf{u}} : \nabla \overline{\mathbf{u}} \right\} - \frac{\mathscr{P} L}{U^3} \left\{ \nabla \cdot \left( \overline{\mathbf{u}' \otimes \mathbf{u}'} \cdot \overline{\mathbf{u}} \right) + \nabla \overline{\mathbf{u}} : \overline{\mathbf{u}' \otimes \mathbf{u}'} \right\}
\end{aligned}\]
Turbulence is a phenomenon that occurs at high Reynolds numbers, i.e. with \(\frac{\nu}{U L} = \frac{1}{Re} \rightarrow 0\). If the other contributions have the same order of magnitude, it follows that \(\mathscr{P} = \frac{U^3}{L}\).
Turbulent energy.
\[
\dfrac{u^2}{T} \dfrac{\overline{D} k}{\overline{D} t} = \frac{1}{L} \nabla \cdot \left( \frac{\nu u^2}{L} \nabla k - u^3 \overline{\mathbf{u}' k'} - p u \overline{ P' \mathbf{u}'} \right) - \mathscr{P} \nabla \overline{\mathbf{u}} : \overline{\mathbf{u}' \otimes \mathbf{u}'} - \mathscr{D} \overline{\nabla \mathbf{u}' : \nabla \mathbf{u}'} \ ,
\]
with \(\mathscr{D} = \frac{\nu U^2}{L^2}\).
Turbulence in equilibrium means \(\mathscr{P} = \mathscr{D}\) and thus
\[\mathscr{P} = \mathscr{D} = \frac{U^3}{L} \ .\]
