For the curvature-correction, the main idea is to find a neighboring vector in the vicinity (
\(r \rightarrow 0,\) see Fig.
3) of the second-order shift vector such that both vectors connect two points on the same circle. Under this condition, the following relation holds between the angles of the vectors
\({\varvec{\delta}}_{\bf 1}\) and
\({\varvec{\delta}}_{\bf 2}\) and the second-order shift vector
$$ \beta_{1}+\phi-{\phi_{n}}=\beta_{2} $$
(2.5)
Where
\(\phi\) and
\(\phi_n\) are the angle of the second-order shift vector and the neighboring vector. The starting point of the neighboring vector must be close to the starting point of the shift vector in order to align the vector
\({\varvec{\delta}_{\bf 1}}\) tangentially to the circle. Due to the finite vector spacing of PIV data points, a local (first-order) series expansion for the neighboring vector
$$ \Updelta{\bf X}^{\left(2\right)}\left(\left(X_{0},Y_{0}\right)+{\varvec{\delta}_{1}}\right)= \Updelta{\bf X}^{\left(2\right)}\left(X_{0},Y_{0}\right)+\frac{\partial\Updelta{\bf X}^{(2)}}{\partial{\varvec{\delta}_{\bf 1}}}\cdot{\varvec{\delta}_{\bf 1}}+\cdots $$
(2.6)
is required in order to interpolate between the discrete data points. The second-order shift vector
\(\Updelta{\bf X}^{\left(2\right)}\left(X_{0},Y_{0}\right)\) and a neighboring shift vector that fulfills the condition in Eq.
2.5 differ by the vector
\({\varvec{\delta}_{\bf 2}}-{\varvec{\delta}_{\bf 1}}\):
$$ \Updelta{\bf X}^{\left(2\right)}\left(X_{0},Y_{0}\right)+{\varvec{\delta}_{\bf 2}}= {\varvec{\delta}_{\bf 1}}+\Updelta{\bf X}^{\left(2\right)}\left(\left(X_{0},Y_{0}\right)+{\varvec{\delta}_{\bf 1}}\right) $$
(2.7)
In order to find a suitable neighboring shift vector, vectors surrounding
\(\left(X_{0},Y_{0}\right)\) are analyzed. For a fixed distance between the second-order vector and the neighboring vector, the angle
α is varied (see Fig.
3). The angles
β
1 and
β
2 depend on
α as follows:
$$ \begin{aligned} \beta_{1}\left(\alpha\right)& =-\varphi \\&\quad +\hbox{arctan }\left(\frac{r\cdot\left(2\cdot\sin\alpha-V_{X}\cos\alpha-V_{Y}\sin\alpha\right)}{r\cdot\left(2\cdot\cos\alpha-U_{X}\cos\alpha-U_{Y}\sin\alpha\right)}\right) \end{aligned}$$
(2.8)
$$ \begin{aligned} \beta_{2}\left(\alpha\right)&=\varphi \\&\quad -\hbox{arctan }\left(\frac{r\cdot\left(2\cdot\sin\alpha+V_{X}\cos\alpha+V_{Y}\sin\alpha\right)}{r\cdot\left(2\cdot\cos\alpha+U_{X}\cos\alpha+U_{Y}\sin\alpha\right)}\right) \end{aligned}$$
(2.9)
where
\(\left(U_{0},V_{0}\right)\) are the components of the second-order shift vector at
\(\left(X_{0},Y_{0}\right),\) and
U
X
,
U
Y
,
V
X
, and
V
Y
are the partial derivatives with respect to
X and
Y, respectively. The angle
\(\varphi\) is the orientation of the second-order shift vector. It is important to note that
β
1 and
β
2 are not given explicitly in Eq.
2.8 and
2.9. However, it is possible to compute
\(\beta_{1}\left(\alpha\right)\) and
\(\beta_{2}\left(\alpha\right)\) for a set of
α (ranging from
\(\varphi-\pi/2\) to
\(\varphi+\pi/2\)) and find the solution for
\(\beta_{1}\left(\alpha\right)=\beta_{2}\left(\alpha\right)\) numerically at fairly low computational cost. Once the angle
β
1 is found, the radius
R of the circle can be computed
$$ R=\frac{\left|\Updelta{\bf X}^{(2)}\left(X_{0},Y_{0}\right)\right|}{2\cdot\sin\frac{\xi}{2}} $$
(2.10)
where the arc’s angle is given by
$$ \frac{\xi}{2}=\beta_{1} $$
(2.11)
The sign of
R defines the direction of the curvature: If
R > 0, the streamline follows clockwise rotation, and for
R < 0, the streamline follows counter-clockwise rotation in a right-handed system. Finally, from the radius
R and the arc’s angle
ξ, the curvature-correction is applied to the second-order shift vector field in three steps as follows:
-
$$ {{\varvec{\delta}}}_{\rm cc}=R\cdot\left(1-\cos\left[\frac{\xi}{2}\right]\right)\cdot \left(\begin{array}{c} -\sin\varphi\\ \cos\varphi \end{array} \right) $$
(2.12)
It should be emphasized that the estimation of the angle
β
1 is based on the gradients of the second-order shift vector field (see Eqs.
2.8 and
2.9). Thus, for reliable curvature-correction, the gradients must be estimated accurately first.