Weighted divergence correction scheme and its fast implementation

Abstract

Forcing the experimental volumetric velocity fields to satisfy mass conversation principles has been proved beneficial for improving the quality of measured data. A number of correction methods including the divergence correction scheme (DCS) have been proposed to remove divergence errors from measurement velocity fields. For tomographic particle image velocimetry (TPIV) data, the measurement uncertainty for the velocity component along the light thickness direction is typically much larger than for the other two components. Such biased measurement errors would weaken the performance of traditional correction methods. The paper proposes a variant for the existing DCS by adding weighting coefficients to the three velocity components, named as the weighting DCS (WDCS). The generalized cross validation (GCV) method is employed to choose the suitable weighting coefficients. A fast algorithm for DCS or WDCS is developed, making the correction process significantly low-cost to implement. WDCS has strong advantages when correcting velocity components with biased noise levels. Numerical tests validate the accuracy and efficiency of the fast algorithm, the effectiveness of GCV method, and the advantages of WDCS. Lastly, DCS and WDCS are employed to process experimental velocity fields from the TPIV measurement of a turbulent boundary layer. This shows that WDCS achieves a better performance than DCS in improving some flow statistics.

Appendix

Deriving the fast algorithm

The optimization problem for WDCS shown as Eq. 3 could be rewritten as a matrix form:

$$\left\{ \begin{aligned} \min (\mathbf{U}_{{\exp }} - & \mathbf{U}_{c} )^{\text T} \varvec{\alpha} ^{2} (\mathbf{U}_{{\exp }} - \mathbf{U}_{c} ) \\ & {\text{s.t.}}\; {\mathbf{AU}_{c} = 0} \\ \end{aligned} \right.$$

(20)

where $\mathbf {U}_{\mathrm {exp}}$ and $\mathbf {U}_c$ are column matrices collecting all the three components of velocities from all the spatial positions, with subscripts of ‘exp’ and ‘c’ denoting the experimental data and corrected field. $\varvec{\alpha}$ is the corresponding weighting matrix and $\mathbf {A}$ is the divergence operator. The specific forms for $\varvec{\alpha }$ and $\mathbf {A}$ are dependent on the arrangement orders of velocity elements in $\mathbf {U}_{\mathrm {exp}}$ and $\mathbf {U}_c$. In this work, the velocity elements are arranged by the following order: the l-th velocity component element in the spatial position (i, j, k) is arranged at $(i+(j-1)\times nx+(k-1)\times nx\times ny+(l-1)\times nx\times ny\times nz)$-th position in the vector matrix $\mathbf {U}_{\mathrm {exp}}$ or $\mathbf {U}_c$. In this way,

$$\begin{aligned} \varvec{{\alpha }}=\begin{bmatrix} \alpha_1\mathbf{I}_{nx}& &\\ &\alpha_2\mathbf{I}_{ny}&\\ & &\alpha_3\mathbf{I}_{nz}\\ \end{bmatrix} \end{aligned}$$

(21)

$$\begin{aligned} \mathbf {A}=[\mathbf {I}_{nz}\otimes \mathbf {I}_{ny}\otimes \mathbf {d}_{nx},\mathbf {I}_{nz}\otimes \mathbf {d}_{ny}\otimes \mathbf {I}_{nx},\mathbf {d}_{nz}\otimes \mathbf {I}_{ny}\otimes \mathbf {I}_{nx}], \end{aligned}$$

(22)

with $\mathbf {I}_{nx}$, $\mathbf {I}_{ny}$, $\mathbf {I}_{nz}$ denoting the identity matrices with the dimensions of nx, ny, nz, respectively. $\otimes$ denotes the Kronecker tensor product between two matrices. $\mathbf {d}_{nx}$, $\mathbf {d}_{ny}$, $\mathbf {d}_{nz}$ are derivative operators with dimensions of nx, ny, nz, respectively, which has been introduced in the article.

By introducing a Lagrange multiplier $2\varvec{{\mu }}$, the constrained minimization problem (Eq. 20) could be transformed as the solution of a set of linear equations (Azijli and Dwight 2015) as:

$$\begin{aligned} {\left\{ \begin{array}{ll} \mathbf {AU}_c=\mathbf {0}\\ \mathbf {A}^\mathrm {T}\varvec{{\mu }}+\varvec{{\alpha }}^2\mathbf {U}_c=\varvec{{\alpha }}^2\mathbf {U}_{\mathrm {exp}} \end{array}\right. }, \end{aligned}$$

(23)

Solving $\mathbf {U}_c$ in the second equation and replacing it in the first equation, a new set of equations are obtained as

$$\begin{aligned} {\left\{ \begin{array}{ll} \mathbf {A\varvec{{\alpha }}^{-2}A}^\mathrm {T}\varvec{{\mu }}=\mathbf {AU}_{\mathrm {exp}}\\ \mathbf {U}_c=\mathbf {U}_{\mathrm {exp}}-\varvec{{\alpha }}^{-2}\mathbf {A}^\mathrm {T}\varvec{{\mu }} \end{array}\right. }. \end{aligned}$$

(24)

The critical procedure to solve these equations is to derive the inverse of $\mathbf {A\varvec{{\alpha }}}^{-2}\mathbf {A}^\mathrm {T}$. The specific form of $\mathbf {A\varvec{{\alpha }}}^{-2}\mathbf {A}^\mathrm {T}$ is

$$\begin{aligned} \mathbf {A\varvec{{\alpha }}}^{-2}\mathbf {A}^\mathrm {T}=\alpha _1^{-2}\mathbf {I}_{nz}\otimes \mathbf {I}_{ny} \otimes \mathbf {d}_{nx}\mathbf {d}_{nx}^\mathrm {T}+\alpha _2^{-2}\mathbf {I}_{nz}\otimes \mathbf {d}_{ny}\mathbf {d}_{ny}^\mathrm {T}\otimes \mathbf {I}_{nx}+ \alpha _3^{-2}\mathbf {d}_{nz}\mathbf {d}_{nz}^\mathrm {T} \otimes \mathbf {I}_{ny}\otimes \mathbf {I}_{nx}. \end{aligned}$$

(25)

Considering $\mathbf {d}_{nx}\mathbf {d}_{nx}^\mathrm {T}$, $\mathbf {d}_{ny}\mathbf {d}_{ny}^\mathrm {T}$, $\mathbf {d}_{nz}\mathbf {d}_{nz}^\mathrm {T}$ are all real symmetric matrices, they could be decomposed as

$$\begin{aligned} \left\{ \begin{array}{l} \mathbf {d}_{nx}\mathbf {d}_{nx}^\mathrm {T}=\mathbf {\Phi }_{nx}\mathbf {\Lambda }_{nx}\mathbf {\Phi }_{nx}^\mathrm {T}\\ \mathbf {d}_{ny}\mathbf {d}_{ny}^\mathrm {T}=\mathbf {\Phi }_{ny}\mathbf {\Lambda }_{ny}\mathbf {\Phi }_{ny}^\mathrm {T}\\ \mathbf {d}_{nz}\mathbf {d}_{nz}^\mathrm {T}=\mathbf {\Phi }_{nz}\mathbf {\Lambda }_{nz}\mathbf {\Phi }_{nz}^\mathrm {T} \end{array}\right. . \end{aligned}$$

(26)

where $\mathbf {\Phi }_{nx}$, $\mathbf {\Phi }_{ny}$, $\mathbf {\Phi }_{nz}$ and $\varvec{{\lambda }}_{nx}$, $\varvec{{\lambda }}_{ny}$, $\varvec{{\lambda }}_{nz}$ are the eigenvalue and eigenvector matrices of $\mathbf {d}_{nx}\mathbf {d}_{nx}^\mathrm {T}$, $\mathbf {d}_{ny}\mathbf {d}_{ny}^\mathrm {T}$, $\mathbf {d}_{nz}\mathbf {d}_{nz}^\mathrm {T}$. $\mathbf {\Phi }_{nx}$, $\mathbf {\Phi }_{ny}$, $\mathbf {\Phi }_{nz}$ are all orthogonal matrices, which would significantly simplify the complex inverse operation.

Employing the decomposition results of $\mathbf {d}_{ny}\mathbf {d}_{ny}^\mathrm {T}$, $\mathbf {d}_{nz}\mathbf {d}_{nz}^\mathrm {T}$, and considering the operation properties for the Kronecker tensor product, $\mathbf {A\varvec{{\alpha }}}^{-2}\mathbf {A}^\mathrm {T}$ could be decomposed as

$$\begin{aligned} \mathbf {A\varvec{{\alpha }}}^{-2}\mathbf {A}^\mathrm {T}=(\mathbf {\Phi }_{nz}\otimes \mathbf {\Phi }_{ny} \otimes \mathbf {\Phi }_{nx})\mathbf {\Gamma } (\mathbf {\Phi }_{nz}\otimes \mathbf {\Phi }_{ny} \otimes \mathbf {\Phi }_{nx})^\mathrm {T}, \end{aligned}$$

(27)

where

$$\begin{aligned} \mathbf {\Gamma }=\alpha _1^{-2}\mathbf {I}_{nz}\otimes \mathbf {I}_{ny}\otimes \mathbf {\Lambda }_{nx}+\alpha _2^{-2}\mathbf {I}_{nz}\otimes \mathbf {\Lambda }_{ny}\otimes \mathbf {I}_{nx}+\alpha _3^{-2}\mathbf {\Lambda }_{nz}\otimes \mathbf {I}_{ny}\otimes \mathbf {I}_{nx}. \end{aligned}$$

(28)

The matrix $\mathbf {\Phi }_{nz}\otimes \mathbf {\Phi }_{ny} \otimes \mathbf {\Phi }_{nx}$ is an orthogonal matrix, which facilitates the inverse operation by a simple transposing. Therefore, the Lagrange multiplier could be derived by an explicit function, as

$$\begin{aligned} \varvec{{\mu }}=(\mathbf {\Phi }_{nz}\otimes \mathbf {\Phi }_{ny} \otimes \mathbf {\Phi }_{nx})\mathbf {\Gamma }^{-1}(\mathbf {\Phi }_{nz}\otimes \mathbf {\Phi }_{ny} \otimes \mathbf {\Phi }_{nx})^\mathrm {T}\mathbf {AU}_{\mathrm {exp}}. \end{aligned}$$

(29)

The final correction velocity $\mathbf {U}_c$ could be determined by the second equation in Eq. 24. For more application about the operation of Kronecker tensor product and similar decomposition work, the reader should refer to Buckley (1994) and Golub et al. (1998).