Determination of Rotationality in a Two-Dimensional Flow
Consider the following steady, incompressible, two-dimensional velocity field:
\vec{V}=(u, v)=x^{2} \vec{i}+(-2 x y-1) \vec{j} (1)
Is this flow rotational or irrotational? Sketch some streamlines in the first quadrant and discuss.
We are to determine whether a flow with a given velocity field is rotational or irrotational, and we are to draw some streamlines in the first quadrant.
Analysis Since the flow is two-dimensional, Eq. 4–31 is applicable. Thus,
\vec{\zeta}=\left(\frac{\partial v}{\partial x}-\frac{\partial u}{\partial y}\right) \vec{k} (4-31)
Vorticity: \vec{\zeta}=\left(\frac{\partial v}{\partial x}-\frac{\partial u}{\partial y}\right) \vec{k}=(-2 y-0) \vec{k}=-2 y \vec{k} (2)
Since the vorticity is nonzero, this flow is rotational. In Fig. 4–49 we plot several streamlines of the flow in the first quadrant; we see that fluid moves downward and to the right. The translation and deformation of a fluid parcel is also shown: at Δt = 0, the fluid parcel is square, at Δt = 0.25 s, it has moved and deformed, and at Δt = 0.50 s, the parcel has moved farther and is further deformed. In particular, the right-most portion of the fluid parcel moves faster to the right and faster downward compared to the left-most portion, stretching the parcel in the x-direction and squashing it in the vertical direction. It is clear that there is also a net clockwise rotation of the fluid parcel, which agrees with the result of Eq. 2.
Discussion From Eq. 4–29, individual fluid particles rotate at an angular velocity equal to \vec{\omega}=-y \vec{k}, half of the vorticity vector. Since \vec{\omega} is not constant, this flow is not solid-body rotation. Rather, \vec{\omega} is a linear function of y. Further analysis reveals that this flow field is incompressible; the area (and volume) of the shaded regions representing the fluid parcel in Fig. 4–49 remains constant at all three instants in time.
