Covariant basis¶
Description¶
Velocity-gradient tensor¶
The gradient of a velocity vector is
(∑j→Ej∂∂Xj)⊗(∑i→EiUi)=∑ij(→Ej⊗∂→Ei∂Xj)Ui+∑ij(→Ej⊗→Ei)∂Ui∂Xj.
By using the changes of the basis vectors:
∂→Ei∂Xj=−∑kδijHj2HkHk∂Hi∂Xk→Ek−∑kδijHi2HkHk∂Hj∂Xk→Ek+1Hi∂Hi∂Xj→Ei+1Hj∂Hj∂Xi→Ej,
I find that the vector gradient is given by the sum of the following elements.
∑ij(→Ej⊗→Ei)1HjHj1Hi∂Hi∂XjUi.
∑j(→Ej⊗→Ej)1HjHj1Hj∑k∂Hj∂XkUk.
−∑ijk(→Ej⊗→Ek)δijHj2HkHk∂Hi∂XkUi−∑ijk(→Ej⊗→Ek)δijHi2HkHk∂Hj∂XkUi=−∑jk(→Ej⊗→Ek)Hj2HkHk∂Hj∂XkUj−∑ik(→Ei⊗→Ek)Hi2HkHk∂Hi∂XkUi=−∑ij(→Ej⊗→Ei)1HjHjHjHiHi∂Hj∂XiUj=−∑ij(→Ej⊗→Ei)1HiHi1Hj∂Hj∂XiUj.
∑ij(→Ej⊗→Ei)1HjHj∂Ui∂Xj.
Note that the negative of the third element is the transpose of the first one.
In summary, the velocity gradient is given by the sum of
(→E1→E2→E3)(1H1H1∂U1∂X11H1H1∂U2∂X11H1H1∂U3∂X11H2H2∂U1∂X21H2H2∂U2∂X21H2H2∂U3∂X21H3H3∂U1∂X31H3H3∂U2∂X31H3H3∂U3∂X3)(→E1→E2→E3),
(→E1→E2→E3)(1H1H1∑k1H1∂H1∂XkUk0001H2H2∑k1H2∂H2∂XkUk0001H3H3∑k1H3∂H3∂XkUk)(→E1→E2→E3),
(→E1→E2→E3)(01H1H11H2∂H2∂X1U2−1H2H21H1∂H1∂X2U11H1H11H3∂H3∂X1U3−1H3H31H1∂H1∂X3U11H2H21H1∂H1∂X2U1−1H1H11H2∂H2∂X1U201H2H21H3∂H3∂X2U3−1H3H31H2∂H2∂X3U21H3H31H1∂H1∂X3U1−1H1H11H3∂H3∂X1U31H3H31H2∂H2∂X3U2−1H2H21H3∂H3∂X2U30)(→E1→E2→E3).
Strain-rate tensor¶
The strain-rate tensor is defined as the symmetric part of it; namely the sum of
(→E1→E2→E3)(1H1H1∂U1∂X1121H1H1∂U2∂X1+121H2H2∂U1∂X2121H1H1∂U3∂X1+121H3H3∂U1∂X3sym.1H2H2∂U2∂X2121H2H2∂U3∂X2+121H3H3∂U2∂X3sym.sym.1H3H3∂U3∂X3)(→E1→E2→E3),
(→E1→E2→E3)(1H1H1∑k1H1∂H1∂XkUk00sym.1H2H2∑k1H2∂H2∂XkUk0sym.sym.1H3H3∑k1H3∂H3∂XkUk)(→E1→E2→E3).
Example¶
Cylindrical coordinates
The velocity-gradient tensor is the sum of
(→E1→E2→E3)(∂U1∂X1∂U2∂X1∂U3∂X11X1X1∂U1∂X21X1X1∂U2∂X21X1X1∂U3∂X2∂U1∂X3∂U2∂X3∂U3∂X3)(→E1→E2→E3),
(→E1→E2→E3)(00001X1X1U1X10000)(→E1→E2→E3),
(→E1→E2→E3)(0U2X10−U2X100000)(→E1→E2→E3).
The strain-rate tensor is
(→E1→E2→E3)(∂U1∂X112∂U2∂X1+121X1X1∂U1∂X212∂U3∂X1+12∂U1∂X3sym.1X1X1∂U2∂X2+1X1X1U1X1121X1X1∂U3∂X2+12∂U2∂X3sym.sym.∂U3∂X3)(→E1→E2→E3).
Rectilinear coordinates
The velocity-gradient tensor is the sum of
(→E1→E2→E3)(1H1H1∂U1∂X11H1H1∂U2∂X11H1H1∂U3∂X11H2H2∂U1∂X21H2H2∂U2∂X21H2H2∂U3∂X21H3H3∂U1∂X31H3H3∂U2∂X31H3H3∂U3∂X3)(→E1→E2→E3),
(→E1→E2→E3)(1H1H1∑k1H1∂H1∂XkUk0001H2H2∑k1H2∂H2∂XkUk0001H3H3∑k1H3∂H3∂XkUk)(→E1→E2→E3).
The strain-rate tensor is
(→E1→E2→E3)(1H1H1∂U1∂X1+1H1H1∑k1H1∂H1∂XkUk121H1H1∂U2∂X1+121H1H1∂U2∂X1121H1H1∂U3∂X1+121H3H3∂U1∂X3sym.1H2H2∂U2∂X2+1H2H2∑k1H2∂H2∂XkUk121H2H2∂U3∂X2+121H3H3∂U2∂X3sym.sym.1H3H3∂U3∂X3+1H3H3∑k1H3∂H3∂XkUk)(→E1→E2→E3).
Application
The velocity-gradient tensor is the sum of
(→E1→E2→E3)(1H1H1∂U1∂X11H1H1∂U2∂X11H1H1∂U3∂X11H2H2∂U1∂X21H2H2∂U2∂X21H2H2∂U3∂X21H3H3∂U1∂X31H3H3∂U2∂X31H3H3∂U3∂X3)(→E1→E2→E3),
(→E1→E2→E3)(00001H2H21H2∂H2∂X1U10000)(→E1→E2→E3),
(→E1→E2→E3)(01H1H11H2∂H2∂X1U20−1H1H11H2∂H2∂X1U200000)(→E1→E2→E3).
The strain-rate tensor is
(→E1→E2→E3)(1H1H1∂U1∂X1+1H1H11H1∂H1∂X1U1121H1H1∂U2∂X1+121H1H1∂U2∂X1121H1H1∂U3∂X1+121H3H3∂U1∂X3sym.1H2H2∂U2∂X2+1H2H21H2∂H2∂X1U1121H2H2∂U3∂X2+121H3H3∂U2∂X3sym.sym.1H3H3∂U3∂X3)(→E1→E2→E3).