Here is what I have learned: given u,v,w values the angles from each axis to the vector are
acos(u/Um) acos(v/Um) acos (w/Um)
where Um is the vector magnitude. The first two angles are angles directly from the axes to the vector and thus neither is equivalent to atan(v/u), which is the angle from the x axis to the projection of the 3D vector onto the x-y plane. The vertical angle acos(w/Um) is the angle from the z axis to the vector. A value of this angle corresponding to 90 degrees would lie in the x-y plane (i.e. w = 0). Thus, the angle
90 – acos(w/Um)
gives the deviation of the vertical vector from the x-y plane (i.e. the up or down orientation of the vector). Positive values of this angle are upward directed flow and negative values are downward directed flow. This angle is NOT equivalent to atan(v/w) (although it will be for v =0), which I believe is what the quiver function in Matlab gives you for plotting vectors in a 2D plane if you use v and w as the velocity variables.
For example, let’s suppose values of u,v, and w are 50, 10 and 2 respectively. atan(v/w) is +11.3 degrees, whereas 90-acos(w/Um) is only +2.24 degrees. This would make a big difference in the vertical orientation of the 2D v-w vectors as plotted in a cross section.
Hi Frank
Here is what I have learned: given u,v,w values the angles from each axis to the vector are
where Um is the vector magnitude. The first two angles are angles directly from the axes to the vector and thus neither is equivalent to atan(v/u), which is the angle from the x axis to the projection of the 3D vector onto the x-y plane. The vertical angle acos(w/Um) is the angle from the z axis to the vector. A value of this angle corresponding to 90 degrees would lie in the x-y plane (i.e. w = 0). Thus, the angle 90 – acos(w/Um)
gives the deviation of the vertical vector from the x-y plane (i.e. the up or down orientation of the vector). Positive values of this angle are upward directed flow and negative values are downward directed flow. This angle is NOT equivalent to atan(v/w) (although it will be for v =0), which I believe is what the quiver function in Matlab gives you for plotting vectors in a 2D plane if you use v and w as the velocity variables.
For example, let’s suppose values of u,v, and w are 50, 10 and 2 respectively. atan(v/w) is +11.3 degrees, whereas 90-acos(w/Um) is only +2.24 degrees. This would make a big difference in the vertical orientation of the 2D v-w vectors as plotted in a cross section.
Bruce
Professor Bruce L. Rhoads