???
- Good morning all!
- Now I will present my talk titled ...
- Thank: SU for the opportunity to come here
---
layout: false
# Overview
.left-column[
## References
## Turbulence fundamentals
]
.right-column[
- Essential quantities: \\( \tau,\,u_* \\)
- Regions inside the boundary layer
- Log-law of the wall
- Effect of roughness
]
---
# Overview
.left-column[
## References
## Turbulence fundamentals
## Monin-Obukhov similarity theory
]
.right-column[
- Derivation using Buckingham Pi theorem
- Physical meaning
- Moeng's boundary condition variant
]
---
# Overview
.left-column[
## References
## Turbulence fundamentals
## Monin-Obukhov similarity theory
## Possible compressible formulation
]
.right-column[
- "Buoyancy term"
- Expression for stability parameter, \\( N \\)
]
---
# Overview
.left-column[
## References
## Turbulence fundamentals
## Monin-Obukhov similarity theory
## Possible compressible formulation
## Appendix: project update
]
.right-column[
- Filtering
- Next steps?
]
---
# 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.
1. Cushman-Roisin, Benoit, and Jean-Marie Beckers. “Stratification.” In International Geophysics, 101:347–64. Elsevier, 2011. https://doi.org/10.1016/B978-0-12-088759-0.00011-0.
1. Svensson, Gunilla, and Albert A. M. Holtslag. “Analysis of Model Results for the Turning of the Wind and Related Momentum Fluxes in the Stable Boundary Layer.” Boundary-Layer Meteorology 132, no. 2 (August 2009): 261–77. https://doi.org/10.1007/s10546-009-9395-1.
1. Maronga, Björn, Christoph Knigge, and Siegfried Raasch. “An Improved Surface Boundary Condition for Large-Eddy Simulations Based on Monin–Obukhov Similarity Theory: Evaluation and Consequences for Grid Convergence in Neutral and Stable Conditions.” Boundary-Layer Meteorology, October 29, 2019. https://doi.org/10.1007/s10546-019-00485-w.
---
class: center, middle, inverse
# Turbulence fundamentals
---
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## Essential quantities
### Wall shear stress \\( \tau \\) and friction velocity \\( u_* \\)
.pull-left[
Shear stress can be driven by:
- molecular diffusion
\\[ \tau_{ij} = \mu \frac{\partial u_i}{\partial x_j} \\]
... Newton's law of viscosity
- turbulent diffusion
\\[ \tau_{ij} = -\rho \overline{u_iu_j } \\]
... Reynolds stress tensor (statistical quantity)
Friction velocity
\\[ u_* = \sqrt{\frac{\tau}{\rho}} \\]
]
.pull-right[
![Courtesy: Wyngaard](https://cdn.jsdelivr.net/gh/ashwinvis/talks@3ebeb930f997b97dfcf87ce65fc5253a74eb9735/./images/most_tau.png)
]
---
## Observations from DNS of channel flows
.pull-left[
N.B: \\( y \\) is the wall normal direction in engineering
\\[ \delta_v = \text{viscous length scale},\quad \delta = \text{displacement thickness} \\]
\\[ u^+ = \bar{U} / u_* , \quad y^+ = y / \delta_v \\]
![Courtesy: Pope, Kim et al.](https://cdn.jsdelivr.net/gh/ashwinvis/talks@3ebeb930f997b97dfcf87ce65fc5253a74eb9735/./images/most_log_law_dns.png)
]
.pull-right[
![Courtesy: Pope](https://cdn.jsdelivr.net/gh/ashwinvis/talks@3ebeb930f997b97dfcf87ce65fc5253a74eb9735/./images/most_bl_regions.png)
]
---
## Mean velocity profiles
In general (without any assumptions), on dimensional grounds,
\\[
\frac{\partial \bar U}{\partial y} = \frac{u_*}{y} \Phi(y/\delta_v, y/\delta)
\\]
For \\( y / \delta \ll 1 \\), *tends asymptotically to*
\\[
\frac{\partial \bar U}{\partial y} = \frac{u_*}{y} \Phi_1(y/\delta_v) \implies
\frac{\partial u^+}{\partial y^+} = \frac{1}{y^+} \Phi_1(y^+)
\\]
## Log-law of the wall
von Karman (1930) postulated that for high \\( Re ,\, 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
\\]
---
## Effect of roughness
In general, the velocity gradient depends on 3 parameters. Including \\( s \\)
the roughness scale:
\\[
\frac{\partial \bar U}{\partial y} = \frac{u_*}{y} \Phi(y/\delta_v, y/\delta, s/\delta_v)
\\]
For the general case of roughtness size \\( s \sim \delta_v \\) we get,
\\[
u^+ = \frac{1}{\kappa} \ln (y / s) + B(s / \delta_v)
\\]
other relations exist for extremes cases of small and large roughness scale
\\( s \\).
## Note
- The log-law is one among many established results in turbulence.
- A similar approach is used in developing Monin-Obukhov similarity theory.
---
class: center, middle, inverse
# Monin-Obukhov similarity theory
---
## Governing parameters and assumptions
Turbulence in the surface layer is determined by:
1. length scale \\( l \sim z \\)
1. velocity scale \\( u \sim u_* \\)
1. surface stress \\( \tau = \rho u_*^2 \\)
1. buoyancy parameter \\(\sim g/\theta_0 \\)
1. surface temperature flux \\( Q_0 \\)
1. ~~surface flux of a conserved scalar \\( C_0 \\)~~
## Buckingham Pi theorem
- \\( m = 5 \\) parameters
- \\( n = 3 \\) dimensions: length, time, temperature, ~~scalar~~
implies the model can be rewritten using \\( m - n = 2 \\) independent
dimensionless quantities.
---
## M-O functions
.pull-left[
Similarity variable taken as \\( z/L \\), where *Monin-Obukhov length*:
\\[ L = -u_*^3 \theta_0 / \kappa g Q_0 \\]
![](https://cdn.jsdelivr.net/gh/ashwinvis/talks@3ebeb930f997b97dfcf87ce65fc5253a74eb9735/./images/most_mo_func1.png)
![](https://cdn.jsdelivr.net/gh/ashwinvis/talks@3ebeb930f997b97dfcf87ce65fc5253a74eb9735/./images/most_mo_func2.png)
\\( L \\) being a function of the turbulence statistic \\( u_* \\) restricts
a closed form solution of the M-O functions.
These remained merely a theory until...
]
.pull-right[
The 1968 Kansas experiment measured the LHS for different z and L and confirmed
these functions are truly similar:
![](https://cdn.jsdelivr.net/gh/ashwinvis/talks@3ebeb930f997b97dfcf87ce65fc5253a74eb9735/./images/most_mo_func.png)
]
---
## M-O functions and log-law
In **neutral** conditions, \\( z/L \to 0 \implies \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]
\\]
Compare the term \\( \ln z_0 \\) to roughness parameter \\(B\\). For **stable**
and **unstable** regimes, curve fitting gives,
![](https://cdn.jsdelivr.net/gh/ashwinvis/talks@3ebeb930f997b97dfcf87ce65fc5253a74eb9735/./images/most_fit1.png)
![](https://cdn.jsdelivr.net/gh/ashwinvis/talks@3ebeb930f997b97dfcf87ce65fc5253a74eb9735/./images/most_fit2.png)
---
## Moeng's variant
.pull-left[
![](https://cdn.jsdelivr.net/gh/ashwinvis/talks@3ebeb930f997b97dfcf87ce65fc5253a74eb9735/./images/most_moeng.png)
]
.pull-right[
#### 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)
```
]
---
.pull-left[
## Physical aspects
### Meaning of \\( z/L \\)
\\[ z / L = \frac{\text{buoyant production rate of turb. kinetic energy}}{\text{shear
production rate of turb. kinetic energy}}
\\]
![](https://cdn.jsdelivr.net/gh/ashwinvis/talks@3ebeb930f997b97dfcf87ce65fc5253a74eb9735/./images/most_ekmann.png)
]
.pull-right[
### Effect of boundary conditions
- Average turning angle of the Ekmann spiral (bottom left)
- Log-layer mismatch (bottom right)
![](https://cdn.jsdelivr.net/gh/ashwinvis/talks@3ebeb930f997b97dfcf87ce65fc5253a74eb9735/./images/most_log_layer_mismatch.png)
]
---
class: center, middle, inverse
# Possible compressible formulation
---
# Buoyancy frequency / stability
.pull-left[
![](https://cdn.jsdelivr.net/gh/ashwinvis/talks@3ebeb930f997b97dfcf87ce65fc5253a74eb9735/./images/most_comp0.png)
![](https://cdn.jsdelivr.net/gh/ashwinvis/talks@3ebeb930f997b97dfcf87ce65fc5253a74eb9735/./images/most_comp0b.png)
![](https://cdn.jsdelivr.net/gh/ashwinvis/talks@3ebeb930f997b97dfcf87ce65fc5253a74eb9735/./images/most_comp1.png)
]
.pull-right[
![](https://cdn.jsdelivr.net/gh/ashwinvis/talks@3ebeb930f997b97dfcf87ce65fc5253a74eb9735/./images/most_comp2.png)
![](https://cdn.jsdelivr.net/gh/ashwinvis/talks@3ebeb930f997b97dfcf87ce65fc5253a74eb9735/./images/most_comp3.png)
]
---
class: center, middle, inverse
# Appendix: project update
---
.pull-left[
# Filtering
- Excessive filtering in last meeting
- Filtering parameters were reduced
- If time permits ... some visuals
]
.pull-right[
# Next steps?
Stay with neutral stratification
- Oscillations
- Sponge layer
- Rayleigh radiative BC
- Improved boundary condition: Robin conditions in literature
- Comparison of statistics
- Generated but unsure of what is plotted.
- Need contact with a Ph.D. student using Nek5000.
]
---
class: center, middle, inverse
# Thank you for your attention!
### Any questions?
![](https://cdn.jsdelivr.net/gh/ashwinvis/talks@3ebeb930f997b97dfcf87ce65fc5253a74eb9735/./images/noun_questions_inv.svg)
## slides will be uploaded: [ashwin.info.tm/talks](https://ashwin.info.tm/talks.html)