Computed quantities#
The following tables list the different quantities computed by the post-processing scripts implemented in VaMPy.
In table Table 1 we present the hemodynamic indices computed by the script/command compute_hemodynamic_indices.py/vampy-hemo.
Quantity |
Abbreviation/Symbol |
Definition |
Unit |
|---|---|---|---|
Wall shear stress |
WSS, \(\tau\) |
\(\displaystyle \mu \frac{\partial u}{\partial n}\) |
[Pa] |
Time averaged wall shear stress |
TAWSS |
\(\displaystyle \frac{1}{T}\int_0^T \left| \tau \right| \, d t\) |
[Pa] |
Temporal wall shear stress gradient |
TWSSG |
\(\displaystyle \frac{1}{T}\int_0^T \left| \frac{\partial \tau}{\partial t} \right| \,d t\) |
[Pa/s] |
Oscillatory shear index |
OSI |
\(\displaystyle \frac{1}{2}\left(1- \frac{\left| \int_0^T \tau \,d t \right|}{\int_0^T \left| \tau \right|\,d t} \right)\) |
[ - ] |
Relative residence time |
RRT |
\(\displaystyle \frac{1}{(1-2\cdot \text{OSI})\cdot \text{TAWSS}}\) |
[1/Pa] |
Endothelial cell activation potential |
ECAP |
\(\displaystyle \frac{\text{OSI}}{\text{TAWSS}}\) |
[1/Pa] |
In table Table 2 we present the fluid mechanical metrics and simulation quantities that are computed by the script/command compute_flow_and_simulation_metrics.py/vampy-metrics.
Quantity |
Abbreviation/Symbol |
Definition |
Unit |
|---|---|---|---|
Velocity |
\(u\) |
\(\displaystyle u(x,y,z,t) = (u_x, u_y, u_z)\) |
[m/s] |
Mean velocity |
\(\bar{u}, u_{\text{mean}}\) |
\(\displaystyle \frac{1}{T} \int_0^T u \,dt\) |
[m/s] |
Turbulent velocity |
\(u'\) |
\(\displaystyle u - \bar{u}\) |
[m/s] |
Kinematic viscosity |
\(\nu\) |
\(\displaystyle \frac{\mu}{\rho}\) |
[m\(^2\)/s] |
Time interval |
\(T\) |
\(\textit{User defined}\) |
[s] |
Time step |
\(\Delta t\) |
\(\displaystyle \frac{T }{ N}\) |
[s] |
Characteristic edge length |
\(\Delta x, h\) |
\(\displaystyle \texttt{CellDiameter(mesh)}\) |
[m] |
Courant–Friedrichs–Lewy condition |
CFL |
\(\displaystyle |u|\frac{\Delta t}{\Delta x}\) |
[ - ] |
Rate of strain |
\(S_{ij}\) |
\(\displaystyle \frac{1}{2} \left(\frac{\partial u_i}{\partial x_j} + \frac{\partial u_j}{\partial x_i} \right) \) |
[1/s] |
Turbulent rate of strain |
\(s_{ij}\) |
\(\displaystyle \frac{1}{2} \left(\frac{\partial u'_i}{\partial x_j} + \frac{\partial u'_j}{\partial x_i} \right) \) |
[1/s] |
Absolute rate of strain |
Strain |
\(\displaystyle \sqrt{\langle S_{ij}, S_{ij}} \rangle \) |
[1/s] |
Dissipation |
\(\mathcal{E} \) |
\(\displaystyle 2\nu \langle S_{ij}, S_{ij} \rangle \) |
[m\(^2\)/s\(^3\)] |
Turbulent dissipation |
\(\varepsilon \) |
\(\displaystyle 2\nu \langle s_{ij}, s_{ij} \rangle \) |
[m\(^2\)/s\(^3\)] |
Kinetic energy |
KE, \(E_k\) |
\(\displaystyle \frac{1}{2} \left( u_x^2 + u_y^2 + u_z^2 \right)\) |
[m\(^2\)/s\(^2\)] |
Turbulent kinetic energy |
TKE, \(k\) |
\(\displaystyle \frac{1}{2} \left( (u'_x)^2 + (u'_y)^2 + (u'_z)^2 \right)\) |
[m\(^2\)/s\(^2\)] |
Friction velocity |
\(u^{\star},u_\tau\) |
\(\displaystyle \sqrt{\nu S_{ij}}\) |
[m/s] |
Generalized length scale |
\(\ell^+\) |
\(\displaystyle \frac{u^\star \Delta x}{\nu}\) |
[ - ] |
Generalized time scale |
\(t^+\) |
\(\displaystyle \frac{(u^\star)^2 \Delta t}{\nu}\) |
[ - ] |
Kolmogorov length scale |
\(\eta\) |
\(\displaystyle \left( \frac{\nu^3}{\varepsilon} \right)^{\frac{1}{4}} \) |
[m] |
Kolmogorov time scale |
\(\tau_\eta\) |
\(\displaystyle \left( \frac{\nu}{\varepsilon} \right)^{\frac{1}{2}} \) |
[s] |
Kolmogorov velocity scale |
\(u_\eta\) |
\(\displaystyle \left( \varepsilon \nu \right)^{\frac{1}{4}} \) |
[m/s] |