Flow temporal reconstruction from non-time-resolved data part I: mathematic fundamentals

Abstract

At least two circumstances point to the need of postprocessing techniques to recover lost time information from non-time-resolved data: the increasing interest in identifying and tracking coherent structures in flows of industrial interest and the high data throughput of global measuring techniques, such as PIV, for the validation of computational fluid dynamics (CFD) codes. This paper offers the mathematic fundamentals of a space--time reconstruction technique from non-time-resolved, statistically independent data. An algorithm has been developed to identify and track traveling coherent structures in periodic flows. Phase-averaged flow fields are reconstructed with a correlation-based method, which uses information from the Proper Orthogonal Decomposition (POD). The theoretical background shows that the snapshot POD coefficients can be used to recover flow phase information. Once this information is recovered, the real snapshots are used to reconstruct the flow history and characteristics, avoiding neither the use of POD modes nor any associated artifact. The proposed time reconstruction algorithm is in agreement with the experimental evidence given by the practical implementation proposed in the second part of this work (Legrand et al. in Exp Fluids, 2011), using the coefficients corresponding to the first three POD modes. It also agrees with the results on similar issues by other authors (Ben Chiekh et al. in 9 Congrès Francophone de Vélocimétrie Laser, Bruxelles, Belgium, 2004; Van Oudheusden et al. in Exp Fluids 39-1:86–98, 2005; Meyer et al. in 7th International Symposium on Particle Image Velocimetry, Rome, Italy, 2007a; in J Fluid Mech 583:199–227, 2007b; Perrin et al. in Exp Fluids 43-2:341–355, 2007). Computer time to perform the reconstruction is relatively short, of the order of minutes with current PC technology.

Appendix

As already commented in Sect. 2, if the snapshot number N is large enough (e.g., N ~ 10²), the summation in Eq. 1 may be approximated as a continuous integration over time. This way, the eigen-value problem leads to: $\int_{0}^{1} {C\left( {\hat{i},\hat{j}} \right)\chi^{(k)} } \left( {\hat{j}} \right){\text{d }}\hat{j} = \lambda^{(k)} \chi^{(k)} \left( {\hat{i}} \right)$. Using Eqs. 7 and 8, we have to solve the following equation:

$$ \lambda^{\prime}\chi \left( {\hat{i}} \right) - \int\limits_{0}^{1} {\user2{K}\left( {\hat{i},\hat{j}} \right)\chi \left( {\hat{j}} \right)d \, \hat{j}} = 0; \quad {\rm with}\,\lambda^{\prime} = \lambda - E $$

(14)

where $\user2{K}({\hat{i},\hat{j}})$ is the integral kernel of Eq. 8 and $\lambda^{\prime }$ its corresponding eigen values.

In this particular case, $\user2{K}({\hat{i},\hat{j}})$ is degenerate and separable, meaning that it can be written as a product of functions of ${\hat{i}}$ by functions of ${\hat{j}}$ (Baker 1977):

$$ \int\limits_{0}^{1} {\user2{K}\left( {\hat{i},\hat{j}} \right)\chi \left( {\hat{j}} \right){\text{d}}\hat{j}} = \sum\limits_{n} {R_{n} \left( {\hat{i}} \right)\int\limits_{0}^{1} {S_{n} \left( {\hat{j}} \right)\chi \left( {\hat{j}} \right){\text{d}}\hat{j}} }. $$

(15)

This simplifies the resolution of Eq. 14, leading to solve another eigen-value problem: $MV = \lambda^{\prime } V$, where $M_{i,j} = \int_{0}^{1} {S_{i} (x)R_{j} (x){\text{d}}x}$, $V_{j} = \int_{0}^{1} {S_{j} (t)\chi (t){\text{d}}t}$, where $\lambda^{\prime }$ are the same eigen values as those of Eq. 15. Choosing $R_{n} (x)$ and $S_{n} (t)$ as in Eq. 16, and truncating the Fourier expansion at mode Nt, leads to:

$$ \begin{array}{*{20}c} {R_{1} (x) = 1;} &\quad {S_{1} (t) = C_{0} + \sum\limits_{n = 1}^{Nt} {A_{n} \cos (2n\pi t) + B_{n} \sin (2n\pi t) \, } } & {} \\ {R_{1 + p} (x) = \cos (2p\pi x) ;} &\quad {S_{1 + p} (t) = A_{p} + \sum\limits_{q = 1}^{Nt} {\alpha_{p,q} \cos (2q\pi t) + \tau_{p,q} \sin (2q\pi t) \, } } &\quad {{\text{for }}1 \le p \le Nt} \\ {R_{Nt + 1 + p} (x) = \sin (2p\pi x);} &\quad {S_{Nt + 1 + n} (t) = B_{p} + \sum\limits_{q = 1}^{Nt} {\tau_{q,p} \cos (2q\pi t) + \beta_{p,q} \sin (2q\pi t)} } & {{\text{for }}1 \le p \le Nt} \\ \end{array} $$

(16)

Thus, matrix M can now be written as follows:

$$ M = \left( {\begin{array}{*{20}c} {2C_{0} } & {A_{1} } & \cdots & {A_{i} } & \cdots & {A_{Nt} } & {B_{1} } & \cdots & {B_{i} } & \cdots & {B_{Nt} } \\ {2A_{1} } & {\alpha_{1,1} } & \cdots & {\alpha_{1,i} } & \cdots & {\alpha_{1,Nt} } & {\tau_{1,1} } & \cdots & {\tau_{1,i} } & \cdots & {\tau_{1,Nt} } \\ \vdots & \vdots & \ddots & {} & {} & \vdots & \vdots & \ddots & {} & {} & \vdots \\ {2A_{i} } & {\alpha_{i,1} } & {} & {\alpha_{i,i} } & {} & {\alpha_{i,Nt} } & {\tau_{i,1} } & {} & {\tau_{i,i} } & {} & {\tau_{i,Nt} } \\ \vdots & \vdots & {} & {} & \ddots & \vdots & \vdots & {} & {} & \ddots & \vdots \\ {2A_{Nt} } & {\alpha_{Nt,1} } & \cdots & {\alpha_{Nt,i} } & \cdots & {\alpha_{Nt,Nt} } & {\tau_{Nt,1} } & \cdots & {\tau_{Nt,i} } & \cdots & {\tau_{Nt,Nt} } \\ {2B_{1} } & {\tau_{1,1} } & \cdots & {\tau_{i,1} } & \cdots & {\tau_{Nt,1} } & {\beta_{1,1} } & \cdots & {\beta_{1,j} } & \cdots & {\beta_{1,Nt} } \\ \vdots & \vdots & \ddots & {} & {} & \vdots & \vdots & \ddots & {} & {} & \vdots \\ {2B_{i} } & {\tau_{1,i} } & {} & {\tau_{i,i} } & {} & {\tau_{Nt,i} } & {\beta_{i,1} } & {} & {\beta_{i,i} } & {} & {\beta_{i,Nt} } \\ \vdots & \vdots & {} & {} & \ddots & \vdots & \vdots & {} & {} & \ddots & \vdots \\ {2B_{Nt} } & {\tau_{1,Nt} } & \cdots & {\tau_{i,Nt} } & \cdots & {\tau_{Nt,Nt} } & {\beta_{Nt,1} } & \cdots & {\beta_{Nt,i} } & \cdots & {\beta_{Nt,Nt} } \\ \end{array} } \right) $$

(17)

In order to further simplify this $\left[ {\left( {2Nt + 1} \right) \times \left( {2Nt + 1} \right)} \right]$ matrix, several physical assumptions can be made. The flow presents a high spatial–temporal coherence (weak dissipation, implying Taylor hypothesis), so that the flow is convective: Structures are convected by main flow at a local convective velocity which corresponds to the mean flow $\bar{U}(\vec{r})$. Taking into account that the flow is also pseudo periodic and propagates information downstream, the equation that describes $\tilde{U}(\vec{r},t)$ is a wave equation:

$$ \tilde{U}(\vec{r},t) = \tilde{U}\left( {\omega t - \vec{k}_{w} \left( {\vec{r}} \right) \cdot \vec{r}} \right) = \sum\limits_{n = 1}^{\infty } {W_{n} \cos \left( {n\left[ {\omega t - \vec{k}_{w} \left( {\vec{r}} \right) \cdot \vec{r}} \right] + \gamma_{n} } \right)}.$$

(18)

Here, $\omega = 2\pi f$ $\left( {\omega t_{i} = 2\pi \hat{i}} \right)$ is the angular frequency of the propagating wave, and $\vec{k}_{w}$ is the wave vector. Wave number $k_{w}$ is related to angular frequency and phase velocity $\bar{U}(\vec{r})$ through the relation $k_{w} = {\omega \mathord{\left/ {\vphantom {\omega {\bar{U}}}} \right. \kern-\nulldelimiterspace} {\bar{U}}}$. The formulation in Eq. 18 can be rewritten expanding the cosine function, resulting in:

$$ \tilde{U}(\vec{r},t) = \sum\limits_{n = 1}^{\infty } {\underbrace {{\left( {W_{n} \cos \left( {n\vec{k}_{w} \left( {\vec{r}} \right) \cdot \vec{r} + \gamma_{n} } \right)} \right)}}_{{a_{n} \left( {\vec{r}} \right)}}\cos \left( {n\omega t} \right)} + \sum\limits_{n = 1}^{\infty } {\underbrace {{\left( {W_{n} \sin \left( {n\vec{k}_{w} \left( {\vec{r}} \right) \cdot \vec{r} + \gamma_{n} } \right)} \right)}}_{{b_{n} \left( {\vec{r}} \right)}}\sin \left( {n\omega t} \right)}.$$

(19)

Now, the matrix constants $\alpha_{p,q}$, $\beta_{p,q}$ and $\tau_{p,q}$ may be calculated as follow:

$$ \begin{gathered} \alpha_{p,q} = \iint\limits_{\Upomega } {a_{p} (\vec{r})a_{q} (\vec{r}){\text{d}}\Upomega } = W_{p} W_{q} \iint\limits_{\Upomega } {\cos \left( {p\vec{k}_{w} \vec{r} + \gamma_{p} } \right)\cos \left( {q\vec{k}_{w} \vec{r} + \gamma_{q} } \right){\text{d}}\Upomega } \hfill \\ \beta_{p,q} = \iint\limits_{\Upomega } {b_{p} (\vec{r})b_{q} (\vec{r}){\text{d}}\Upomega } = W_{p} W_{q} \iint\limits_{\Upomega } {\sin \left( {p\vec{k}_{w} \vec{r} + \gamma_{p} } \right)\sin \left( {q\vec{k}_{w} \vec{r} + \gamma_{q} } \right){\text{d}}\Upomega } \hfill \\ \tau_{p,q} = \iint\limits_{\Upomega } {a_{p} (\vec{r})b_{q} (\vec{r}){\text{d}}\Upomega } = W_{p} W_{q} \iint\limits_{\Upomega } {\cos \left( {p\vec{k}_{w} \vec{r} + \gamma_{p} } \right)\sin \left( {q\vec{k}_{w} \vec{r} + \gamma_{q} } \right)d\Upomega } \hfill \\ \end{gathered} $$

(20)

As $\vec{k}_{w} \left( {\vec{r}} \right)\vec{r}$ may vary along the integration domain, the integrations in Eq. 20 are not straightforward. Nevertheless, choosing integration paths along streamlines and considering $\Upomega$ are much larger than typical structure size, the inhomogeneities resulting from changes in $\vec{k}_{w}$ may vanish.

$$ \begin{gathered} \left. \begin{gathered} \alpha_{p,q} = \frac{1}{2}\delta_{p,q} W_{p} W_{q} \hfill \\ \beta_{p,q} = \frac{1}{2}\delta_{p,q} W_{p} W_{q} \hfill \\ \end{gathered} \right\}\alpha_{p,q} = \beta_{p,q} = \left\{ {\begin{array}{*{20}c} {0 \, if \, p \ne q} \\ {W_{p}^{2} \, if \, p = q} \\ \end{array} } \right. \hfill \\ \tau_{p,q} = 0{\text{ for any}}\left\{ {p,q} \right\} \hfill \\ \end{gathered} $$

(21)

It could be argued that coherence could be lost by dissipation for long distances l larger than typical weak dissipative structure size ${{2\pi } \mathord{\left/ {\vphantom {{2\pi } {k_{w} }}} \right. \kern-\nulldelimiterspace} {k_{w} }}$. In order to take into account this last argument, a dissipative function $\xi (\vec{r})$ could be introduced by multiplying the above results for Eqs. 19 or 20. But the decay rate of $\xi (\vec{r})$ would be $l \gg {{2\pi } \mathord{\left/ {\vphantom {{2\pi } {k_{w} }}} \right. \kern-\nulldelimiterspace} {k_{w} }}$, so the integrals of Eq. 20 can be roughly simplified as Eq. 21.

Finally, the matrix M is strongly simplified:

$$ M = \left( {\begin{array}{*{20}c} {2C_{0} } & {A_{1} } & \cdots & {A_{i} } & \cdots & {A_{Nt} } & {B_{1} } & \cdots & {B_{i} } & \cdots & {B_{Nt} } \\ {2A_{1} } & {\alpha_{1,1} } & \cdots & 0 & \cdots & 0 & 0 & \cdots & 0 & \cdots & 0 \\ \vdots & \vdots & \ddots & {} & {} & \vdots & \vdots & \ddots & {} & {} & \vdots \\ {2A_{i} } & 0 & {} & {\alpha_{i,i} } & {} & 0 & 0 & {} & 0 & {} & 0 \\ \vdots & \vdots & {} & {} & \ddots & \vdots & \vdots & {} & {} & \ddots & \vdots \\ {2A_{Nt} } & 0 & \cdots & 0 & \cdots & {\alpha_{Nt,Nt} } & 0 & \cdots & 0 & \cdots & 0 \\ {2B_{1} } & 0 & \cdots & 0 & \cdots & 0 & {\alpha_{1,1} } & \cdots & 0 & \cdots & 0 \\ \vdots & \vdots & \ddots & {} & {} & \vdots & \vdots & \ddots & {} & {} & \vdots \\ {2B_{i} } & 0 & {} & 0 & {} & 0 & 0 & {} & {\alpha_{i,i} } & {} & 0 \\ \vdots & \vdots & {} & {} & \ddots & \vdots & \vdots & {} & {} & \ddots & \vdots \\ {2B_{Nt} } & 0 & \cdots & 0 & \cdots & 0 & 0 & \cdots & 0 & \cdots & {\alpha_{Nt,Nt} } \\ \end{array} } \right) $$

(22)

Now, if the flow is strongly dominated by the fundamental frequency f, $\tilde{U}_{i}$ is roughly equal to the first term of its Fourier expansion (i.e., $\| {a_{1} ( {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {r} } )} \|$ is typically much larger than contributions of harmonic frequencies $\| {a_{n > 1} ( {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {r} } )} \|$). In other words, $Nt = 1$ still approximates well the flow field. Under these conditions,

$$ M \approx \left( {\begin{array}{*{20}c} {2C_{0} } & {A_{1} } & {B_{1} } \\ {2A_{1} } & {\alpha_{1,1} } & 0 \\ {2B_{1} } & 0 & {\alpha_{1,1} } \\ \end{array} } \right) $$

(23)

The sorted $\lambda^{\prime }$ eigen values of M are then:

$$ \begin{array}{*{20}c} {\lambda^{(0)} = C_{0} + \frac{1}{2}\alpha_{1,1} + \sqrt \Updelta + \frac{E}{N}} \hfill & \to \hfill & {\chi_{i}^{(0)*} = \frac{{\lambda^{(0)} }}{{B_{1} }}\left( {Q^{ + } + \sqrt {A_{1}^{2} + B_{1}^{2} } \cos \left( {2\pi \,\hat{i} - \phi_{0} } \right)} \right)} \hfill \\ {\lambda^{(1)} = C_{0} + \frac{1}{2}\alpha_{1,1} - \sqrt \Updelta + \frac{E}{N}} \hfill & \to \hfill & {\chi_{i}^{(1)*} = \frac{{\lambda^{(1)} }}{{B_{1} }}\left( {\sqrt {A_{1}^{2} + B_{1}^{2} } \cos \left( {2\pi \,\hat{i} - \phi_{0} } \right) + Q^{ - } } \right)} \hfill \\ {\lambda^{\prime (2)} = \alpha_{1,1} + \frac{E}{N}} \hfill & \to \hfill & {\chi_{i}^{(2)*} = \frac{{\lambda^{(2)} }}{{B_{1} }}\sqrt {A_{1}^{2} + B_{1}^{2} } \sin \left( {2\pi \,\hat{i} - \phi_{0} } \right)} \hfill \\ \end{array} $$

(24)

Note that the eigen functions $\chi_{i}^{(k)*}$ are non-normalized for simplicity. The normalized eigen values should be obtained as: $\chi_{i}^{(k)} = \frac{{\chi_{i}^{(k)*} }}{{\int_{{\hat{i} = 0}}^{1} {\left( {\chi_{i}^{(k)*} } \right)^{2} {\text{d}}\hat{i}} }}$. The constants appearing in Eq. 24 are:

$$ \begin{gathered} \Updelta = \left( {\frac{1}{2}\alpha_{1,1} - C_{0} } \right)^{2} + 2\left( {A_{1}^{2} + B_{1}^{2} } \right)\quad {\text{and}}\quad \phi_{0} = \arctan \left( {\frac{{B_{1} }}{{A_{1} }}} \right) \hfill \\ Q^{ + } = \frac{{A_{1}^{2} + B_{1}^{2} }}{{{{\alpha_{1,1} } \mathord{\left/ {\vphantom {{\alpha_{1,1} } 2}} \right. \kern-\nulldelimiterspace} 2} - C_{0} + \sqrt \Updelta }}\quad {\text{and}}\quad Q^{ - } = \frac{{A_{1}^{2} + B_{1}^{2} }}{{{{\alpha_{1,1} } \mathord{\left/ {\vphantom {{\alpha_{1,1} } 2}} \right. \kern-\nulldelimiterspace} 2} - C_{0} - \sqrt \Updelta }} \hfill \\ \end{gathered} $$

(25)

Taking into account that $C_{0} > A_{1}$, $C_{0} > \alpha_{1,1}$, and developing $\sqrt \Updelta$ in the Taylor series, we get:

$$ Q^{ + } = \frac{{A_{1}^{2} + B_{1}^{2} }}{{{{\alpha_{1,1} } \mathord{\left/ {\vphantom {{\alpha_{1,1} } 2}} \right. \kern-\nulldelimiterspace} 2} - C_{0} + \sqrt \Updelta }} \to C_{0} - \frac{{\alpha_{1,1} }}{2}\quad {\text{and}}\quad Q^{ - } = \frac{{A_{1}^{2} + B_{1}^{2} }}{{{{\alpha_{1,1} } \mathord{\left/ {\vphantom {{\alpha_{1,1} } 2}} \right. \kern-\nulldelimiterspace} 2} - C_{0} - \sqrt \Updelta }} \to 0 $$

(26)

As a consequence, cosine contribution in $\chi_{i}^{(0)}$ is relatively small compared with the constant $Q^{ + }$. On the contrary, the constant contribution $Q^{ - }$ in $\chi_{i}^{(1)}$ is small, and $\chi_{i}^{(1)}$ corresponds almost to a pure cosine function of phase lag $\phi_{0}$.

Now, focusing on A _n and B _n terms:

$$ \begin{gathered} A_{n} = \iint\limits_{\Upomega } {\bar{U}(\vec{r})a_{n} (\vec{r})}{\text{d}}\Upomega = \iint\limits_{\Upomega } {\bar{U}(\vec{r})}\cos \left( {n\vec{k}_{w} \cdot \vec{r} + \gamma_{n} } \right){\text{d}}\Upomega \hfill \\ B_{n} = \iint\limits_{\Upomega } {\bar{U}(\vec{r})b_{n} (\vec{r})}{\text{d}}\Upomega = \iint\limits_{\Upomega } {\bar{U}(\vec{r})}\sin \left( {n\vec{k}_{w} \cdot \vec{r} + \gamma_{n} } \right){\text{d}}\Upomega \hfill \\ \end{gathered} $$

(27)

If the main wavelength of $\bar{U}(\vec{r})$ is much larger than k _w , i.e., $\bar{U}(\vec{r})$ does not vary much along the propagation direction: $A_{n} \approx B_{n} \approx 0$ and then:

$$ M \approx \left( {\begin{array}{*{20}c} {2C_{0} } & 0 & 0 \\ 0 & {\alpha_{1,1} } & 0 \\ 0 & 0 & {\alpha_{1,1} } \\ \end{array} } \right) $$

(28)

This simply leads to the new eigen values and the normalized $\chi_{i}$ functions for this case:

$$ \begin{array}{*{20}c} {\lambda^{\prime (0)} = 2C_{0} } \hfill & \to \hfill & {\chi_{i}^{(0)} \approx \sqrt \frac{1}{N} } \hfill \\ {\lambda^{\prime (1)} = \alpha_{1,1} } \hfill & \to \hfill & {\chi_{i}^{(1)} = \pm \sqrt \frac{2}{N} \cos \left( {2\pi \,\hat{i} - \frac{\pi }{4}} \right)} \hfill \\ {\lambda^{\prime (2)} = \alpha_{1,1} } \hfill & \to \hfill & {\chi_{i}^{(2)} = \pm \sqrt \frac{2}{N} \sin \left( {2\pi \,\hat{i} - \frac{\pi }{4}} \right)} \hfill \\ \end{array} $$

(29)

The formulations of Eq. 29 correspond with the limit case of Eq. 24. Note that Eq. 29 is the same expression as described by Legrand et al. (2010).