(a) Outline the procedure for drawing the flow net for two-dimensional steady flow of an ideal fluid between the boundaries indicated in Fig. 7-6.

(b) If the uniform velocities at section 2 equal 9.0 m/s and the values of $\Delta n_{2}$ are each 0.03 m, determine flow q and the uniform velocities at section 1, where the values of $\Delta n_{1}$ are 0.09 m.

Question

(a) Outline the procedure for drawing the flow net for two-dimensional steady flow of an ideal fluid between the boundaries indicated in Fig. 7-6.

(b) If the uniform velocities at section 2 equal 9.0 m/s and the values of [latex]\Delta n_{2}[/latex] are each 0.03 m, determine flow q and the uniform velocities at section 1, where the values of [latex]\Delta n_{1}[/latex] are 0.09 m.

Accepted Answer

(a) The procedure of preparing the flow net for this case can be applied to more complex cases. For an ideal fluid we proceed as follows:

(1) Divide the flow at a cross section between parallel boundaries into a number of strips of equal width Δn (using a unit thickness perpendicular to the paper). Each strip represents a streamtube bounded by streamlines or by a streamline and a boundary. The flow between the boundaries is equally divided among the tubes, and [latex]\Delta q \cong v(\Delta n) \cong ext { constant, where } \Delta n[/latex] is measured normal to the local velocity. Since [latex]\Delta q \cong v_{1} \Delta n_{1} \cong v_{2} \Delta n_{2}, v_{1} / v_{2} \cong \Delta n_{2} / \Delta n_{1} \cong \Delta S_{2} / \Delta S_{1}[/latex]. The smaller the values of [latex]\Delta n ext { and } \Delta S[/latex], the closer the approximation is to an exact relationship. A sufficient number of streamlines should be chosen that the accuracy will be acceptable without unnecessary refinement and detail.

(2) To estimate the direction of the streamlines, normal or equipotential lines are drawn. These lines are spaced such that [latex]\Delta S=\Delta n[/latex]. The equipotential lines must be perpendicular to the streamlines at each intersection, and also to the boundaries because boundaries are streamlines.
The diagram will resemble a group of (approximate) squares throughout the flow net.

(3) At or near changes in boundary shape it is impossible to draw good squares. Changes in initial sketching will be necessary, and a useful check is to draw diagonal lines through all the "squares." These diagonal lines, drawn in both directions, should also form approximate squares.

(4) The boundaries themselves usually represent actual streamlines. If this is not so, the flow net will not represent the true flow pattern. For example, where the flow "separates" from the boundary, the boundary itself in this region cannot be used as a streamline. In general, where divergent flows occur, separation zones may develop.

Mathematical solutions for irrotational flows are based upon the definition of the stream function, this definition includes the continuity principle and the properties of a streamline. Flow rate [latex]\psi[/latex] of a streamline is a constant (since no flow can cross the streamline), and if [latex]\psi[/latex] can be established as a function of x and y, the streamline can be plotted. Similarly, the equipotential lines can be defined by [latex]\phi(x, y)=a[/latex] constant. From these expressions we may deduce that

[latex]u=\partial \psi / \partial y \quad ext { and } \quad v=-\partial \psi / \partial x[/latex] for the streamlines.

and [latex]u=-\partial \phi / \partial x \quad ext { and } \quad v=-\partial \phi / \partial y[/latex]. for the equipotential lines

These equations must satisfy the Laplace equation, i.e.,

[latex]\frac{\partial^{2} \psi}{\partial x^{2}}+\frac{\partial^{2} \psi}{\partial y^{2}}=0 \quad ext { or } \quad \frac{\partial^{2} \phi}{\partial x^{2}}+\frac{\partial^{2} \phi}{\partial y^{2}}=0[/latex].

and the continuity equation

[latex]\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}=0[/latex].

In general, equipotential functions are evaluated and plotted. Then the orthogonal streamlines are sketched to indicate the flow.

These exact solutions are presented in texts on advanced fluid mechanics, hydrodynamics, and complex variables.

[latex] ext { (b) Flow per unit width }=q=\Sigma \Delta q=q_{a}+q_{b}+q_{c}+q_{d}+q_{e}=5 v_{2} A_{n_{2}} [/latex].

[latex] ext { For } 1 ext { unit width, } A_{n_{2}}=(1)\left(\Delta n_{2} ight) ext { and } q=(5)(9.0)(1 imes 0.03)=1.35 m ^{3} / s ext { per unit width }[/latex].

[latex] ext { Then for } \Delta n_{1}=0.09 m , 5 v_{1}(0.09 imes 1)=1.35 ext { or } v_{1}=3.00 m / s[/latex].

[latex]v_{1} ext { may also be found from } v_{1} / v_{2} \cong \Delta n_{2} / \Delta n_{1}: v_{1} / 9.0 \cong 0.03 / 0.09, v_{1}=3.00 m / s ext {. }[/latex]

Question 7.13: (a) Outline the procedure for drawing the flow net for two-d...

This Question has been Answered.

Already have an account?

Login

Related Answered Questions