3.2 Validation of the codes
3.2.1 Blob vortex solution on the boundary
As mentioned in section 2.3.1.3, the modeling of the boundary by blob vortices is only an approximation of the boundary conditions. Therefore one must assess the ac- curacy of the boundary blob vortices used by the discrete vortex method of both the code from Spalart and the Sarpkaya-like code (or C code as it is sometimes written at least until section 3.2.4). As the body moves in translation and in rotation, two invis- cid steady cases will be investigated: uniform translation (section 3.2.1.1) and uniform rotation (section 3.2.1.2). Comparison will be made with the exact potential solution. The leakage on the body surface will also be assessed as well as the smoothness of the solution.
The test uses 1200 boundary vortices with a uniform distribution (i.e. each panel is the same size). Indeed, a distribution where the vortices are concentrated at the edge of the ellipse produces worse results than the uniform distribution considering the velocity field smoothness near the boundary. In the former case, the vortices strength is much too strong at the edge of the ellipse and produces a solution equivalent to a vortex dipole as a model of the ellipse. The error (in velocity direction and magni- tude) and leakage occur mainly at the edges of the ellipse. It originates mainly from the straining of the streamlines at the edges of ellipse, as well as the slight distance of the vortices from the wall (as illustrated in figure 3.5) which is related to some of the method features.
3.2.1.1 Stationary ellipse in steady crossflow
Two different cases were chosen: a symmetric flow with an incidence α = 90o, and
an asymmetrical flow with α = 45o. For each incidence, different plot were used: the
streamlines (figure 3.1 and figure 3.2 for comparison with the potential solution), the error compared to the exact potential solution (figure 3.3), and the leakage on the boundary (figures 3.4).
The potential solution for uniform flow past an ellipse is given by Milne-Thompson [40] through the complex potential:
w=−C ζU∞+ U ∗ ∞ ζ , (3.1)
whereU∞ is a complex number corresponding in the complex plane to −→U∞ and ζ is the
image ofz through the Joukowski transformation
z =C ζ+ λ ζ , (3.2) with C= (a+b)/2,λ= (a2−b2)/(4C2) and z =x+iy.
To compare the solution generated by the boundary vortices to the potential solu-
tion, the velocity field was calculated at points on a uniform 200×200 cartesian grid.
The velocity generated from boundary vortices is named−→Vb = (ub, vb) and the potential flow solution velocity −→Vp = (up, vp). The error was then quantified at a point (x, y) as
ε(x, y) =|−→Vb −−→Vp|/ −→ U∞ .
The streamline plots (figure 3.1) show that on a large scale one obtain a solution broadly similar to the inviscid steady solution (figure 3.2). However a closer examination of the streamlines in figure 3.5 shows the effects of the slight offset of the boundary vortices (section 2.3.3); indeed close to the body the streamlines cross the boundary between the control points with a wavy pattern.
From the velocity error plot in figure 3.3, one can see that the error is largest near the tips of the ellipse where there are high curvature regions and a decreasing error away from the ellipse. One of the sources of error lies in the spacing of the vortices, which is large regarding to the curvature. It is especially pronounced at the tips of the ellipse where much of the vorticity is concentrated. For instance for the steady flow at
α = 90o, 100 boundary vortices at one tip represent about 27 % of the total strength of all vortices. Therefore, in the vicinity of these vortices, the error in the vortices strength solution is magnified. Nevertheless, the overlap ensures that the velocity re- mains smooth and prevents this effect from dominating the solution.
−0.6 −0.4 −0.2 0 0.2 0.4 0.6 −0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5 x* y * (a)α= 45o −0.5 0 0.5 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 x* y * (b)α= 90o
Figure 3.1: Streamline of a flow past a 20:1 ellipse for the blob vortex method
Additionally, there are some effects inherent to the boundary resolution method. As stated earlier, the boundary vortices are slightly off the wall, and though the stream- lines are continuous, they are wavy. Then, as here the points considered are between the wall and the boundary vortices and to a lesser extent up to a distance of about 0.12δ0 (δ0 is the distance between the control points) outside the boundary defined by the boundary vortices, large discrepancies in direction and magnitude appear between the two solutions if one look closely at the velocity field near an edge, as in figure 3.6.
Looking at figure 3.3, the error is concentrated at both tips of the ellipse. The er- ror field (if assimilated as an added velocity to the exact solution) looks similar to two point vortices placed at the edges of the ellipse. The vorticity of one of those point vortices at one edge would be of sign opposite to the vorticity of the point vortex at the other edge. It could thus be possible to roughly model the error field as two virtual
point vortices of opposite vorticity at both edges of the ellipse. −0.6 −0.4 −0.2 0 0.2 0.4 0.6 −0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5 x* y * (a)α= 45o −0.5 0 0.5 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 x* y * (b)α= 90o
Figure 3.2: Streamline of a flow past a 20:1 ellipse for the potential flow solution
Observing the velocity field off the boundary defined by the boundary vortices in figure 3.6, one can see that the velocity is in the same direction but with a larger magnitude than the corresponding potential solution velocity. It can be deduced that the vorticity of these virtual vortices behaves like an added vorticity to the potential solution. This indicates an overestimation of the strength of the blob vortices placed at the ellipse tips.
The normal velocity in figure 3.4 is characterized as (−→Vb.−→n)/
−→
U∞
with −→n the nor- mal unit vector. Again due to the distance of the vortices from the wall, the velocity is not taken directly on the wall surface, but at the boundary defined by the vortices. With the current parameters (20:1 ellipse, 1200 boundary blob vortices) the distance
between the control points and the wall is very small, approximately δN = 8.37.10−4
orδN 'b/30'a/600, where b is half the maximum thickness of the ellipse and a half
the length of the ellipse. Note also that figure 3.5 shows that the velocity field remains continuous in this transition region. It then indicates that the two surfaces (physical and the one defined by the boundary vortices) may be considered equivalent.
If one examine the “leakage” (nonzero local normal velocity) in figure 3.4, one can see some rapid fluctuation along the boundary. These fluctuations in normal velocity mag- nitude observed in figure 3.4 originate from the wavy pattern of the streamlines near
the wall (figure 3.5). Additionally, considering the velocity around θ/π = 1 in figure
3.4(b), one can see a smooth evolution of the local leakage. The leakage is in fact not fluctuating but smooth along the boundary. The fluctuations are caused by the wavy pattern, as well as the scale chosen for the abscissa. Indeed, with the boundary model used, more boundary vortices are used at the edges of the ellipse. Not unexpectedly and following the previous observations on the velocity error plot, this leakage is more important in the region of high curvature, i.e. at the tips of the ellipse (figures 3.3 and 3.4). The maximum leakage is then equal to about O(10−3.5)×
−→
U∞ b.
The error can be reduced by putting more vortices in the region of interest. How- ever as a rule of thumb the ratio between the smallest and largest distance separating vortices should not be greater than two. Otherwise, the matrices resulting from equa- tion 2.30 become ill-conditioned and the solution is unstable. Thus the main parameter one can adjust to improve the velocity accuracy near the wall is preferably the number of boundary vortices.
Interestingly in the case of an asymmetric body, such as with the plate with 45o in-
cidence, the global leakage is a bit more important than at 90o. If one integrate the
normal velocity around the surface, the total leakage at 45o appears to be three times
that of 90o. However, the value of this integral remains very small, of order O(10−10) orO(10−8)×
−→
U∞
b which is close to the machine precision. Therefore, the zero total
leakage boundary condition remains correctly verified in both cases with the blob vortex boundary model.
The blob vortex method has shown to provide a reasonable inviscid non-lifting solution for the range of positions of which the ellipse will be allowed to rotate. Nevertheless, there is still a residual error in velocity near the wall, which for the current geometry is largest near the tips of the ellipse. Despite this error due in part to a wavy local velocity pattern, the local leakage remains at an acceptable level. This cannot be changed unless one use another kind of boundary elements such as a constant vortex sheet.
(a)α= 45o
(b)α= 90o
Figure 3.3: Log10(ε) distribution in the grid for the 20:1 ellipse and uniform translation
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 −11 −10 −9 −8 −7 −6 −5 −4 −3 θ / π
log10 of normal velocity ( log10(|V
b .n / U ∞ |) ) (a)α= 45o 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 −11 −10 −9 −8 −7 −6 −5 −4 −3 θ / π
log10 of normal velocity ( log10(|V
b
.n / U
∞
|) )
(b)α= 90o
Figure 3.4: Logarithm of the normal velocity at the boundary for
0 2 4 6 8 x 10−3 0.986 0.988 0.99 0.992 0.994 0.996 0.998 1 1.002 1.004
(a) Streamlines close to the tip of the ellipse
0 2 4 6 8 x 10−3 0.986 0.988 0.99 0.992 0.994 0.996 0.998 1 1.002 1.004
(b) Velocity vectors close to the tip of the ellipse
Figure 3.5: Velocity and streamline between the boundary vortices and the wall for 90o incidence
−0.0144 −0.0096 −0.0048 0 0.0048 0.0096 0.0144 0.4785 0.4833 0.488 0.4928 0.4976 0.5024 0.5072 0.512 x* y *
Figure 3.6: Comparison of the velocity induced by the blob vortices (red vec- tor) and the velocity from the exact potential solution (blue line). The wall is marked by a black line, and the boundary vortices are marked by green crosses.
3.2.1.2 Stationary ellipse steady flow field rotation
In this subsection, the case study will be the steady plate rotation with no crossflow. As for the steady crossflow case, a comparison will be made between the vortex blob and potential solution (figures 3.7, 3.8 and 3.9), and also a normal velocity plot on the surface so as to assess the leakage (figure 3.10). The frame used for the plot is fixed to the plate, that is to say the flow is rotating around the plate at a constant angular velocity of Ω. −0.6 −0.4 −0.2 0 0.2 0.4 0.6 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 x* y *
Figure 3.7: Streamline of a flow past a 20:1 ellipse for the blob vortex method
Again the potential solution is provided by Milne and Thompson ([40]). Using the convention from section 3.2.1.1, and the transformation defined by equation 3.2, one obtain for the complex velocity:
dw
dz =
2B
Cζ(λ−ζ2), (3.3)
withB defined asB = (i/4)Ω(a2−b2), while Cand λare defined by the transformation in equation 3.2.
From the streamline plots in figure 3.7, the solution is again broadly similar to the inviscid steady solution (see figure 3.8). There still remains some leakage through the boundary as is shown in the plot of normal velocity in figure 3.10. Again, it is more important in the region of high curvature. Note however that over the whole bound- ary the leakage is ten times less pronounced than in the steady inviscid crossflow case. Some irregularities are visible on the streamlines near the body wall but this a problem related to the streamline extraction and does not involve the simulation characteristics.
−0.6 −0.4 −0.2 0 0.2 0.4 0.6 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 x* y *
Figure 3.8: Streamline of a flow past a 20:1 ellipse for the potential flow solution
Like in section 3.2.1.1, to compare solution, a uniform 200×200 cartesian grid was
used. The velocity magnitude error ε(x, y) is defined in the case of the steady rotation asε(x, y) =|−→Vb−−→Vp|/(Ωa). The normal velocity in figure 3.10 is computed in the same manner as in section 3.2.1.1.
As for the non rotating case, errors appear concentrated in the high curvature region of the ellipse if one compare figure 3.9 and the steady crossflow case in figure 3.3. The similar error pattern simply illustrates that the same source of error applies, e.g. the spacing of the vortices which is large regarding the curvature and the wavy nature of the streamlines near the wall.
Figure 3.9: Log10(ε) distribution in the grid for the 20 : 1 ellipse and uniform rotation
The solution for the steadily rotating flow has proven to give a good broad approx- imation for the velocity field, similarly to the steady translating crossflow. Thus, one may be confident that the rotating ellipse can effectively be smoothly implemented us- ing blob vortices as boundary elements. Thanks to the additivity property from the linear system (equation 2.30), the combination of the two kinds of flow does not pose any problem. The existing error at the tips of the ellipse is a result of the boundary vortex distribution off the wall surface and can be reduced for example by using a higher number of boundary vortices, although one must be careful not to have a too high computer cost.
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 −11 −10 −9 −8 −7 −6 −5 −4 −3 θ / π
log10 of normal velocity ( log10(|V
b
.n /
Ω
|) )
(a) Logarithm of the normal velocity magnitude
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 −5 −4 −3 −2 −1 0 1 2 3 4 5x 10 −4 θ / π Normal velocity ( V b .n / Ω )
(b) Normal velocity magnitude