Canonical flows

source: J. H. Lee et al. (link)

typical canonical flows

• free shear flows
  • round jet
  • mixing layer
  • wake
• wall flows
  • flat plate boundary layer
  • pipe flow

Why do we study canonical flows?

• reduced complexity
• superposition as approximation of
  more complex flows
• guidance for turbulence modeling
• model validation

Reynolds stresses

conservation of mass

$\nabla\cdot \overline{\mathbf{u}} = 0$ and $\nabla\cdot \mathbf{u}^\prime = 0$

in index notation

$\partial_i \overline{u}_i = 0$ and $\partial_i u^\prime_i = 0$

$\rightarrow$ try to spell this out in Cartesian coordinates

conservation of momentum

$$ \nabla\cdot\left(\overline{\mathbf{u}}\overline{\mathbf{u}}\right) = -\nabla \overline{\mathbf{p}} + \nabla\cdot\left[\nu\left(\nabla\overline{\mathbf{u}} + \nabla \overline{\mathbf{u}}^T\right) -\overline{\mathbf{u}^\prime \mathbf{u}^\prime}\right] $$

in index notation

$$ \partial_i(\overline{u}_i\overline{u}_j) = -\partial_j \overline{p} + \partial_i\left[ \nu \left(\partial_iu_j + \partial_j u_i\right) - \overline{u_iu_j}\right] $$

$\rightarrow$ try to spell this out in Cartesian coordinates

Reynolds stress tensor

$$ \begin{pmatrix} \overline{u^\prime_xu^\prime_x} & \overline{u^\prime_xu^\prime_y} & \overline{u^\prime_xu^\prime_z}\\ \overline{u^\prime_yu^\prime_x} & \overline{u^\prime_yu^\prime_y} & \overline{u^\prime_yu^\prime_z}\\ \overline{u^\prime_zu^\prime_x} & \overline{u^\prime_zu^\prime_y} & \overline{u^\prime_zu^\prime_z} \end{pmatrix} $$

mathematical properties

• symmetry: $\overline{u^\prime_iu^\prime_j} = \overline{u^\prime_ju^\prime_i}$
• $\overline{u^{\prime^2}_i}$ always positive
• Cauchy-Schwarz inequality:
  $|\overline{u^\prime_iu^\prime_j}|^2\leq \overline{u^{\prime^2}_i}\ \overline{u^{\prime^2}_j}$ (no summation)

Valid Reynolds stress tensor?

$$ \begin{pmatrix} 0.5 & 0.1 & 0.0\\ 0.1 & 0.3 & 0.1\\ 0.0 & 0.1 & -0.1 \end{pmatrix} $$

Valid Reynolds stress tensor?

$$ \begin{pmatrix} 0.21 & -0.05 & 0.01\\ -0.06 & 0.5 & 0.0\\ 0.01 & 0.0 & 1.0 \end{pmatrix} $$

Valid Reynolds stress tensor?

$$ \begin{pmatrix} 1.0 & 1.2 & -0.2\\ 1.2 & 1.0 & 0.0\\ -0.2 & 0.0 & 1.0 \end{pmatrix} $$

statistically two-dimensional flow

$$ \begin{pmatrix} \overline{u^\prime_xu^\prime_x} & \overline{u^\prime_xu^\prime_y} & \overline{u^\prime_xu^\prime_z}\\ \overline{u^\prime_yu^\prime_x} & \overline{u^\prime_yu^\prime_y} & \overline{u^\prime_yu^\prime_z}\\ \overline{u^\prime_zu^\prime_x} & \overline{u^\prime_zu^\prime_y} & \overline{u^\prime_zu^\prime_z} \end{pmatrix} $$

assumption: $\overline{u}_z = 0$, uncorrelated fluctuations

closure problem

• unknown: $\overline{u}_i$, $\overline{p}$, $\overline{u^\prime_iu^\prime_j}$ (10)
• equations: momentum (3) + pressure (1)

$\rightarrow$ Reynolds stresses need to be modeled by additional transport equations or algebraic relations

Idea: can we model $$\overline{u^\prime_iu^\prime_j} = f(\overline{S}_{ij}, ...)$$?

mean strain rate tensor
$\overline{S}_{ij} = \frac{1}{2}(\partial_i \overline{u}_j + \partial_j \overline{u}_i)$

Reynolds stress transport equations

approach to derive equations (Wilcox p. 41)

$$ \overline{u^\prime_i N(u_j) + u^\prime_j N(u_i)} = 0$$

$$N(u_i) = \partial_t u_j + \partial_i (u_iu_j) + \partial_j p - \partial_i (\nu (\partial_i u_j + \partial_j u_i)) = 0$$

$\rightarrow$ 6 additional transport equations for $\overline{u^\prime_iu^\prime_j}$

22 additional unknowns

summary

• canonical flows are essential for turbulence modeling
• velocity fluctuations up to $\approx 30\%$ of mean flow component
• we can't eliminate physical complexity by mathematical transformations