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Chapter 7

Q. 7.6

Unflanged Slot in Streaming Flow at Non-Zero Angle
A plain (unflanged) slot of width 2w (w = 0.020 m) withdraws air that streams past it as seen in Figure 7.15. In the far-field the air velocity (U_0) is uniform with a speed equal to 1.5 m/min. Air approaches the inlet from right to left at an angle (α) of 150° from the positive x-axis. Air enters the slot with a volumetric flow rate (Q/L) of 3.6 m²/min and face velocity (U_{face}) of 90. m/min.
To do:
(a) Predict and plot several streamlines of the air entering the inlet.
(b) Predict the air speed along these streamlines.
(c) Predict the components of the air velocity, U_x and U_y, at some arbitrary point.

7.15

Step-by-Step

Verified Solution

(a) The stream function for the composite flow is given by Eq. (7-115). The stream function for a leftward streaming flow is given in Table 7.2, and the stream function for the plain inlet is expressed implicitly by Eqs. (7-90) and (7-91). The dividing streamlines are given by ψ_{dividing} = ± 1.8. The streamlines between the dividing streamlines are given by Eq. (7-115). The algorithm consists of selecting values of the stream function ψ, specifying a value of \phi_{slot}, and then using a trial and error method to calculate the values of y, x, and ψ_{slot} that satisfy Eqs. (7-90) and (7-91). Once the guessed values have been computed, the process is repeated for the next value of \phi_{slot}. The process is repeated until a selected region upstream of the inlet has been covered. Five streamlines, ψL/Q = -0.499, -0.25, 0, 0.25, and 0.40, are shown in Figure E7.6a, as generated by Mathcad; the file is available on the book’s web site. The authors were not able to get Mathcad to converge for the values of-0.50 or 0.50, which are the dividing streamlines. The reader should note that Mathcad’s Solve Block feature is capable of solving three (or more) simultaneous algebraic equations.

Table 7.2 Potential functions for various inlets.
flow description sketch potential functions
oblique streaming
flow
(x,y) coordinate
system
\phi =-U_0 x \cos\alpha -U_0 y \sin\alpha
ψ =-U_0 y \cos\alpha + U_0 x\sin\alpha
point sink in a plane
(r,θ) coordinate
system
\phi =-\frac{Q}{2\pi} \frac{1}{r}
ψ=-\frac{Q}{2\pi} \cos \theta
line sink in a plane
(r,θ) coordinate
system
\phi =\frac{Q}{L\pi } \ln r
ψ=\frac{Q}{L\pi} \theta
flanged rectangular
inlet
(x,y) coordinate
system
\frac{x}{w} =\cosh(\frac{\phi }{K} )\cos (\frac{ψ}{K} )
\frac{y}{w} =\sinh(\frac{\phi }{k} )\sin (\frac{ψ}{k} )
k=\frac{Q}{\pi L}
unflanged rectangular
inlet
(x,y) coordinate
system
 

\frac{x}{w}=\frac{1}{\pi }\left[2\frac{\phi }{k}+exp(2\frac{\phi }{k} ) \cos (2\frac{ψ}{k} )\right]

\frac{y}{w} =\frac{1}{\pi } \left[2\frac{ψ}{k}+exp(2\frac{\phi }{k} )\sin (2\frac{ψ}{k} ) \right] k=\frac{Q}{\pi L}

Flanged circular inlet
(r,z) coordinate system
 

\phi =\frac{Q}{2\pi w} \arcsin (\frac{2w}{a_1+a_2} )
ψ=\frac{Q}{4\pi w} \sqrt{4w^2-(a_1-a_2)^2}   a_1=\sqrt{z^2+(w+r)^2}
a_2=\sqrt{z^2+(w-r)^2}

ψ=ψ_{\text{streaming flow}}+ψ_{\text{unflanged solt}}= ψ_{sf}+ψ_{solt} = -U_0 y\cos α+U_0 x\sin α+ψ_{solt}  (7-115)

\frac{x}{w} =\frac{1}{\pi} [2\frac{\phi }{K}+exp(2\frac{\phi}{K} )\cos(2\frac{ψ}{K} ) ]  (7-90)

\frac{y}{w} =\frac{1}{\pi} [2\frac{ψ}{K}+exp(2\frac{\phi}{K} )\sin(2\frac{ψ}{K} ) ]  (7-91)

(b) The coordinates (x,y) of points along a particular streamline are contained in Mathcad’s Find matrix obtained in Part (a). A new matrix is created from Part (a) from which values of \phi_{slot} and ψ_{slot} exist at the coordinates (x,y) along streamline ψ. The velocity components produced by the inlet, U_{x, slot} and U_{y, slot}, are computed along the streamline using Eqs. (7-93) and (7-94). The actual velocity components (U_x and U_y) that exist at point (x,y) are found by superposition,

U_x=U_{x,slot} +U\cos α            U_y=U_{y,slot}+U\sin α

Graphs of the velocity components U_x and U_y along the same streamlines as in Figure E7.6a are shown in Figures E7.6b and E7.6c.

U_x=-\frac{K \pi}{2w} \frac{1+exp(2\phi /K)\cos(2ψ/K)}{[exp(2\phi /K)\sin(2ψ/K)]^2+[1+exp(2\phi /K)\cos(2ψ/K)]^2}    (7-93)

U_y=-\frac{K \pi}{2w} \frac{exp(2\phi /K)\cos(2ψ/K)}{[exp(2\phi /K)\sin(2ψ/K)]^2+[1+exp(2\phi /K)\cos(2ψ/K)]^2}    (7-94)

(c) The concept of superposition is applied in which the velocity components (U_x and U_y) for the unflanged inlet alone are computed from Eqs. (7-93) and (7-94), and then vectorially added to the velocity components of the streaming flow (U_0) . To begin the calculation, one needs to compute the value of the potential functions \phi_{slot} and ψ_{slot} at an arbitrary point (x,y), and then substitute these values into Eqs. (7-93) and (7-94). The Solve Block feature of Mathcad can be used to obtain the solution of Eqs. (7-90) and (7-91). At the point (x = 0.23 m, y = 0.211 m), U_x = -2.7 m/s and U_y = -0.60 m/s.

Discussion: The lowest streamline in Figure E7.6a is very close to the lower dividing streamline. The point x = 0.23 m, y = 0.211 m was chosen so that the predicted velocity components U_x and U_y can be compared to the values shown in Figures E7.6b and E7.6c. At these coordinates, the normalized stream function (ψL/Q) is equal to 0.25; the values of U_x and U_y in Figures E7.6b and E7.6c correspond favorably with the values shown in Part (c).

7.6a
7.6b
7.6c