Eulerian results

The following are a few Eulerian statistics plotted as profiles across the channel.

All of these plots can be recreated by downloading the the data, the wall_scalesand the matlab plotting script

The data is quite simply structured---it consists of several vectors which represent various quantities from bins at various distances from the wall. The data within each bin is averaged, and the location of that quantity is the center of gravity of all those points, i.e.

\[ y_{center} = \frac{1}{N} \Sum y \]

ybinCenter: Center of gravity of y location for that bin

v_x_mean, v_y_mean, v_z_mean: Simple mean of full velocity components

a_x_mean, a_y_mean, a_z_mean: Simple mean of full acceleration components

Vxx, Vyy, Vzz, Vxy, Vxz, Vyz: Components of the Reynolds stress tensor, defined as:

\[ R_{ij} = ( V_i - \langle V_i \rangle )( V_j - \langle V_j \rangle ) \quad \quad \text{Where} \quad \text{Vxx} = R_{xx} \quad \text{& etc.} \]

I used an interpolated velocity profile when calculating the fluctuating velocity, which is a fitted curve through the measured mean velocity points and is V=0 at the wall. This has some minor effects on the fluctuating velocity near the wall where the shear is very large compared to other methods of calculating the fluctuating velocity.

Axx, Ayy, Azz, Axy, Axz, Ayz: Components of acceleration tensor, defined analogously to Reynolds stress tensor:

\[ AT_{ij} = ( A_i - \langle A_i \rangle )( A_j - \langle A_j \rangle ) \quad \quad \text{Where} \quad \text{Axx} = AT_{xx} \quad \text{ and etc.} \]

AVxx, AVyy, AVzz, AVxy, AVxz, AVyx, AVyz, AVzx, AVzy: Components of $AV_{ij}$ using the full accelerations and velocites, defined as:

\[ AV_{ij} =  A_iV_j  \]

av_mean is simply the trace of $AV_{ij}$

av_var is defined as:

\[  \langle ( A_iV_i - \langle A_iV_i \rangle )^2 \rangle) \]

source:trunk/Figures/Eulerian_profiles/mean_vel_linear.png

source:trunk/Figures/Eulerian_profiles/mean_vel_scaled.png

source:trunk/Figures/Eulerian_profiles/vel_variance.png

source:trunk/Figures/Eulerian_profiles/vel_variance_scaled.png

source:trunk/Figures/Eulerian_profiles/tke.png

source:trunk/Figures/Eulerian_profiles/tke_scaled.png

source:trunk/Figures/Eulerian_profiles/Reynold_stress.png

source:trunk/Figures/Eulerian_profiles/Reynold_stress_scaled.png

source:trunk/Figures/Eulerian_profiles/mean_acc_linear.png

source:trunk/Figures/Eulerian_profiles/mean_acc_scaled.png

source:trunk/Figures/Eulerian_profiles/acc_variance.png

source:trunk/Figures/Eulerian_profiles/acc_variance_scaled.png

source:trunk/Figures/Eulerian_profiles/Acceleration_tensor.png

source:trunk/Figures/Eulerian_profiles/Acceleration_tensor_scaled.png

source:trunk/Figures/Eulerian_profiles/av_mean.png

source:trunk/Figures/Eulerian_profiles/av_mean_scaled.png

source:trunk/Figures/Eulerian_profiles/av_var.png

source:trunk/Figures/Eulerian_profiles/av_var_scaled.png

source:trunk/Figures/Eulerian_profiles/AV_tensor.png

source:trunk/Figures/Eulerian_profiles/AV_tensor_scaled.png

Comparison with DNS

source:trunk/Figures/Eulerian_profiles/Comparison_DNS/vel_mean.png

source:trunk/Figures/Eulerian_profiles/Comparison_DNS/vel_rms2.png

source:trunk/Figures/Eulerian_profiles/Comparison_DNS/acc_mean.png

source:trunk/Figures/Eulerian_profiles/Comparison_DNS/acc_rms2.png