10.3.3 Test Cases
The accompanying CD-ROM contains six test cases described below. The accompanying CDROM contains the complete input and output for each test case.
Test Case 1
Insert AAA
Test Case 2
The unsteady interactive boundary-layer (UIBL) program can be used for steady and
unsteady airfoil flows. This test case demonstrates its application to steady flows at high
Reynolds numbers (see subsection 7.31) with the onset of transition location computed by
Michel’s method, Eq. (7.31), except where the boundary layer separates upstream of this
location, in which case transition is assumed to correspond to the separation point. It can also be
used for low Reynolds number flows provided the onset of transition is computed with the enmethod since in such flows, the transition location is usually inside the separation bubble
(subsection 7.3.2)
Figure 10.2 shows the variation of the lift coefficient with angle of attack for a chord
Reynolds numbers Rc of 3X106 and 6 X106 . The calculated results agree well with experimental
data as well as with those computed (see Fig. 7.25a) with the airfoil code for steady flows
described in [1].
Fig. 10.2 Comparison between calculated (solid lines) and experimental values (symbols) of the
lift coefficient. NACA 0012 airfoil, (a) Rc = 3X106, (b) Rc = 6X106
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Test Case 3
As discussed in Section 4.6, the unsteady panel method uses steady-state conditions to
initiate the time-dependent inviscid flow solutions. This procedure is quite acceptable at low
frequencies but not at higher frequencies. To illustrate this with the inviscid panel method, we
show the variation of the lift coefficient with angle of attack for a NACA 0012 airfoil undergoing
a rotational harmonic motion,
  5  11sin t
Figure 10.3 shows the results for three cycles for   0.001 . As can be seen from Fig.
10.3a, the solutions that originate at   5o (t  0) show a small spike which continues up to
11o. After that the solutions are free from the initial conditions. Figure 10.3b shows that at the
beginning of the first cycle, the value of the lift coefficient is different than its value at the end of
the third cycle. The difference is small but increases with increasing frequency. This is
illustrated in Fig. 10.4 for   0.01 where the spike in the lift coefficient is much more different
than its value at the end of the third cycle. Figures 10.5, 10.6 and 10.7 show the results for
  0.1, 0.25, 0.5 respectively. As can be seen, in order to obtain truly unsteady flow solutions,
it is necessary to run at least two cycles. This is an important point to remember in performing
inviscid-viscous interactions since the spike due to steady-state effects can cause the breakdown
of the boundary-layer solutions.
Fig 10.3 Variation of lift coefficient cl with (a) angle of attack a and (b) cycle,   0.001
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Fig 10.4 Variation of lift coefficient cl with (a) angle of attack a and (b) cycle,   0.01
Fig 10.5 Variation of lift coefficient cl with (a) angle of attack a and (b) cycle,   0.1
Fig 10.6 Variation of lift coefficient cl with (a) angle of attack a and (b) cycle,   0.25
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Fig 10.7 Variation of lift coefficient cl with (a) angle of attack a and (b) cycle,   0.5
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Test Case 4
This case is again for the same airfoil discussed in the previous test case. The boundarylayer calculations employing quasi-steady and unsteady flow assumptions are performed for
reduced frequencies of 0.0001 (Fig. 10.8), 0.01 (Fig. 10.9), 0.1 (Fig. 10.10), 0.25 (Fig. 10.11)
and 0.5 (Fig. 10.12), for a chord Reynolds number Rc of 6  106 with 1  5 ,    12 and
 T S  30 . In all cases, the results indicate not only the strong effect of the viscosity on the
solutions, but also the effect of quasi-steady and unsteady flow assumption on the solutions.
Fig 10.8 Variation of lift coefficient cl with (a) angle of attack a and (b) cycle,   0.0001
Fig 10.9 Variation of lift coefficient cl with (a) angle of attack a and (b) cycle,   0.01
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Fig 10.10 Variation of lift coefficient cl with (a) angle of attack a and (b) cycle,   0.1
Fig 10.11 Variation of lift coefficient cl with (a) angle of attack a and (b) cycle,   0.25
Fig 10.12 Variation of lift coefficient cl with (a) angle of attack a and (b) cycle,   0.5
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Test Case 5
This test case is for the same airfoil of the previous test case subject, this time to a ramp-type
harmonic motion. The boundary-layer calculations employing quasi-steady and unsteady flow
assumptions are performed for constant pitch rates A of 0.0001 (Fig. 10.13a), 0.01 (Fig. 10.13b),
0.05 (Fig. 10.13c), and 0.1 (Fig. 10.13d) at a chord Reynolds number Rc of 6  106 with 1  5 ,
   12 and  T S  30 . The results indicate that at the lowest pitch rate, viscous effects have a
strong effect with significant differences from those computed with the panel method. While at
low to modest angles of attack, there is practically no difference between the lift coefficients
computed with quassi-steady or full unsteady flow assumptions, at higher angles there is a
difference; the quasi-steady flow assumption shows a stronger viscous effect.
As the pitch rate increases the differences between the inviscid and viscous lift coefficients
become less, with quassi-steady flow assumption again indicating a stronger viscous effect.
Figure 10.13. Variation of lift coefficient cl for ramp-type harmonic motion for constant pitch
rates of A (a) 0.0001 (b) 0.01 (c) 0.05 and (d) 0.1
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Test Case 6
This test case is for the same airfoil except that the airfoil is assumed to undergo translational
harmonic motion with   6o , Rc  3  106 , x  0.1 , y  0.3,   10 for two reduced
frequencies   0.025 and 0.25.
Figure 10.14. Variation of (a) lift coefficient cl and (b) velocity translation (ux,uy) for
translational harmonic motion with   0.025
Figure 10.15. Variation of (a) lift coefficient cl and (b) velocity translation (ux,uy) for
translational harmonic motion with   0.25
Figures 10.14 and 10.15 show cl, ux and uy as a function of cycle for both reduced frequencies.
As can be seen, in both cases, the quasi and unsteady flow assumptions agree very well with
each other and the viscous lift coefficients differ from those given by the inviscid flow theory.
We note that the inviscid flow solutions were obtained for two cycles in order to reduce the
steady-state effects and the viscous flow solutions were started at the end of the first cycle.
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Figure 10.16 shows the variation of the lift coefficient as a function of cycle for both reduced
frequencies. As can be seen, with an increase in , the difference between maximum and
minimum lift coefficient increases and the calculations with quasi-steady and unsteady flow
assumption yield essentially the same solutions. We also note that the inviscid flow solutions
were obtained for two cycles in order to reduce the steady-state effects and the viscous flow
solutions were started at the end of the first cycle.
The variation of lift coefficients for two cycles is shown in Fig. 10.5. As can be seen
from Fig. 10.5 with increase in reduced frequency, the results in cycle one still contain the
steady-state effects. However, in the second cycle, these effects become much less. The results
in Fig. 10.5a, on the other hand, do not show the steady-state effects; the solutions for both
cycles are essentially the same.
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