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- Good morning all!
- Now I will present my talk titled ...
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# Overview
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## Recap
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- Essential quantities: \\( \tau,\,u_* \\)
- Log-law of the wall
- Effect of roughness
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# Overview
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## Recap
## As a boundary condition
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.right-column[
- Roughness parameter \\( z_0 \\)
- Wall model
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## Recap
### Wall shear stress \\( \tau \\) and friction velocity \\( u_* \\)
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- Turbulent shear stress can be expressed as:
\\[ \tau_{ij} = -\rho \overline{u_iu_j } \\]
Reynolds stress tensor
- Friction velocity
\\[ u_* = \sqrt{\frac{ |\tau| }{\rho}} \\]
- Viscous length scale
\\[ \delta_\nu = \dfrac{\nu}{\text{friction velocity}} \\]
]
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## Observations from DNS of channel flows
\\[ \delta_v = \text{viscous length scale},\quad \delta = \text{displacement thickness} \\]
\\[ u^+ = \bar{U} / u_* , \quad y^+ = y / \delta_v \\]

]
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In general (without any assumptions), on dimensional grounds,
\\[
\frac{\partial \bar U}{\partial y} = \frac{u_*}{y} \Phi(y^+, y/\delta)
\\]
## Log-law of the wall
von Karman (1930) postulated that for
- high Reynolds number
- \\( y / \delta \ll 1 \\)
- \\(y^+ \gg 1 \\)
**negligible viscous effects**, which implies velocity profile is free from
dependence of \\( \nu \\) or \\( y/\delta_v \\)
\\[
\Phi_1 = \frac{1}{\kappa} \implies
\frac{\partial u^+}{\partial y^+} = \frac{1}{\kappa y^+} \implies
u^+ = \frac{1}{\kappa} \ln y^+ + B
\\]
]
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## Effect of roughness
Roughness influences the "intercept" \\( B \\)
\\[
u^+ = \frac{1}{\kappa} \ln y^+ + B
\\]
### Monin-Obukhov similarity theory
- Monin-Obukhov similarity theory incorporates stratification effects with a
correction \\( \phi_m \\)
In **neutral** conditions, \\( \phi_m \approx 1\\) and we recover
the classical log-law.
\\[
\frac{\partial U}{\partial z} = \frac{u_*}{\kappa z}
\\]
\\[
U(z) = \frac{u_*}{\kappa} \left[ \ln z - \ln z_0 \right]
\\]
]
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### As a stress boundary condition (Moeng 1984)
\\[
\tau = \rho u_*^2 = \rho \left[ \frac{\kappa U}{\ln (z/z_0)}\right]^2
\\]
evaluated at \\(z = \frac{\Delta_z}{2} \\) half grid height
### What if...
- Mesh starts at \\( z_0 \\) where on average \\(\bar{U} \to 0 \\)
- Mesh coordinates are expressed with \\( z_{ref} \\)
]
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---
## Moeng's variant
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]
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#### Implementation in Nek5000 for neutral conditions
\\[
\tau = \rho u_*^2 = \rho \left[ \frac{\kappa U}{\ln (z/z_0)}\right]^2
\\]
which evaluated at \\( z = \frac{1}{2} z_1 \\)
```fortran
! --------Wall normal cordinate: `y`
KAPPA = 0.41
y0 = 0.1 ! << y_max
! --------Calculate Moeng's model parameters
ie = gllel(eg)
u1_2 = (vx(ix,2,iz,ie) + vx(ix,1,iz,ie))/2
w1_2 = (vz(ix,2,iz,ie) + vz(ix,1,iz,ie))/2
absu = sqrt(u1_2**2 + w1_2**2)
y1_2 = (ym1(ix,2,iz,ie) + ym1(ix,1,iz,ie))/2
! --------Calculate Stresses
trx = -KAPPA**2*(u1_2*absu)/(log(y1_2/y0)**2)
try = 0.0
trz = -KAPPA**2*(w1_2*absu)/(log(y1_2/y0)**2)
```
]
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class: center, middle, inverse
# Thank you for your attention!
### Any questions?

## slides will be uploaded: [ashwin.info.tm/talks](https://ashwin.info.tm/talks.html)
---
# References
1. Pope, S.B. Turbulent Flows. Cambridge University Press, 2000.
1. Wyngaard, John C. Turbulence in the Atmosphere. Cambridge: Cambridge University Press, 2010. https://doi.org/10.1017/CBO9780511840524.
1. Monin, A S, and A M Obukhov. “Basic Laws of Turbulent Mixing in the Surface Layer of the Atmosphere,” 1954, 30.
1. Moeng, Chin-Hoh. “A Large-Eddy-Simulation Model for the Study of Planetary Boundary-Layer Turbulence.” Journal of the Atmospheric Sciences 41, no. 13 (July 1, 1984): 2052–62. https://doi.org/10.1175/1520-0469(1984)041<2052:ALESMF>2.0.CO;2.