Question 4.1: A Steady Two-Dimensional Velocity Field A steady, incompress...
A Steady Two-Dimensional Velocity Field
A steady, incompressible, two-dimensional velocity field is given by
\vec{V} = (u , \upsilon ) = (0.5 + 0.8x) \vec{i} + (1.5 – 0.8 y)\vec{j} (1)
where the x- and y-coordinates are in meters and the magnitude of velocity is in m/s. A stagnation point is defined as a point in the flow field where the velocity is zero. (a) Determine if there are any stagnation points in this flow field and, if so, where? (b) Sketch velocity vectors at several locations in the domain between x = −2 m to 2 m and y = 0 m to 5 m; qualitatively describe the flow field.
For the given velocity field, the location(s) of stagnation point(s) are to be determined. Several velocity vectors are to be sketched and the velocity field is to be described.
Assumptions 1 The flow is steady and incompressible. 2 The flow is twodimensional, implying no z-component of velocity and no variation of u or 𝜐 with z.
Analysis (a) Since \vec{V} is a vector, all its components must equal zero in order for \vec{V} itself to be zero. Using Eq. 4–4 and setting Eq. 1 equal to zero,
\vec{V} = (u, \upsilon , w) = u(x, y, z, t)\vec{i} + \upsilon (x, y, z, t)\vec{j} + w(x, y, z, t)\vec{k} (4.4)
Stagnation point: u = 0.5 + 0.8x = 0 → x = −0.625 m
𝜐 = 1.5 − 0.8y = 0 → y = 1.875 m
Yes. There is one stagnation point located at x = −0.625 m, y = 1.875 m.
(b) The x- and y-components of velocity are calculated from Eq. 1 for several (x, y) locations in the specified range. For example, at the point (x = 2 m, y = 3 m), u = 2.10 m/s and 𝜐 = −0.900 m/s. The magnitude of velocity (the speed) at that point is 2.28 m/s. At this and at an array of other locations, the velocity vector is constructed from its two components, the results of which are shown in Fig. 4–4. The flow can be described as stagnation point flow in which flow enters from the top and bottom and spreads out to the right and left about a horizontal line of symmetry at y = 1.875 m. The stagnation point of part (a) is indicated by the blue circle in Fig. 4–4.
If we look only at the shaded portion of Fig. 4–4, this flow field models a converging, accelerating flow from the left to the right. Such a flow might be encountered, for example, near the submerged bell mouth inlet of a hydroelectric dam (Fig. 4–5). The useful portion of the given velocity field may be thought of as a first-order approximation of the shaded portion of the physical flow field of Fig. 4–5.
Discussion It can be verified from the material in Chap. 9 that this flow field is physically valid because it satisfies the differential equation for conservation of mass.
