Cross-correlation measurements

The purpose of the measurements is to extract a spatial cross-correlation tensor $R_{\alpha \beta }(y,y^{\prime },z,z^{\prime },t,t^{\prime })=\left\langle u_\alpha (y,z,t)u_\beta (y^{\prime },z^{\prime },t^{\prime })\right\rangle$ in the self-similar region of the jet. Here and below Greek letters denote a fluctuation velocity component u₁, u₂or u₃, $\left\langle \cdot \right\rangle$ means the ensemble averaging. For each block Fourier transformation of the velocity vector is performed, $% \hat u_\alpha (y,z,f)=\int u_\alpha (y,z,t)e^{-2\pi ift}dt$. After averaging among all blocks of data a spectral correlation matrix

$$S_{\alpha \beta }(y,y^{\prime },\Delta z,f)=\left\langle \hat... ...{*}(y,z,f)\hat u_\beta (y^{\prime },z+\Delta z,f)\right\rangle$$ (1)

is obtained. The asteric denotes a complex conjugate. $S_{\alpha \beta }$ is the Fourier transform of $R_{\alpha \beta }$ in time. X-wire probes are capable of measuring simultaneously either (u,v) or (u,w) components of velocity, therefore $\alpha$ and $\beta$ are 1&2 or 1&3 respectively. The next step is to perform a spatial Fourier transformation in the homogeneous z-direction to get

$$\Phi _{\alpha \beta }(y,y^{\prime };f,k_z)=\int S_{\alpha \beta }(y,y^{\prime },\Delta z,f)e^{-2\pi ik_z\Delta z}d(\Delta z)$$ (2)

, where k_z is a spanwise wavenumber. This spectral correlation tensor will be a kernel in the integral equation to find POD modes for different fand k_z,

$$\int \Phi _{\alpha \beta }(y,y^{\prime };f,k_z)\varphi _\beta... ...{\prime }=\lambda ^{(n)}(f,k_z)\varphi _\alpha ^{(n)}(y;f,k_z)$$ (3)

Thus, the problem of finding POD modes is reduced to a number of integral equations with f and k_z as parameters. In practice the correlation tensor is known in finite equally spaced experimental points $\left\{ y_i\right\}$, i=1,..,m, where m is the number of the probes at one rake, (m=8). So all the integrals (scalar products in general) should be replaced with an appropriate finite summation. The simplest approach is to consider a scalar product in discrete space $(f,g)=\sum\limits_if_ig_i^{*}$. It corresponds to a rectangular finite approximation of the integral. The equation (3) will become a matrix equation for ${\bf\varphi =}% \left\{ \varphi (y_i)\right\} _{i=1}^m$ and $\lambda$ being an eigenvector and eigenvalue of the $\left[ m\times m\right]$ matrix ${\bf\Phi }_{\alpha \beta }=\left\{ \Phi _{\alpha \beta }(y_i,y_j^{\prime };f,k_z)\right\}$,

 

$$\sum_{j=1}^m\Phi _{\alpha \beta }(y_i,y_j^{\prime };f,k_z)\varphi _\beta ^{(n)}(y_j^{\prime };f,k_z)\Delta y^{\prime } \lambda ^{(n)}(f,k_z)\varphi _\alpha ^{(n)}(y_i;f,k_z)~,$$ = (4)

$$or\qquad {\bf\Phi }_{\alpha \beta }{\bf\varphi }_\beta \frac \lambda {\Delta y}{\bf\varphi }_\alpha$$ = (5)

Here $\Delta y$ is the spacing between probes in the rake. Other higher-order integral finite approximation can be used to increase an accuracy of numerical calculations.

The matrix (5) was solved using IMSL library for UNIX system. Thus, a finite set of m orthogonal spatial modes at the discrete spatial points $\varphi _\alpha ^{(n)}(y_i;f,k_z)$ with the corresponding eigenvalues $\lambda ^{(n)}(f,k_z)$ will be obtained. Because of the spatial aliasing, only the first m/2 eigenmodes are defined unambiguously.