# Cross-Correlation of POD Spatial Modes for the Separation of Stochastic Turbulence and Coherent Structures

^{*}

## Abstract

**:**

## 1. Introduction

#### Proper Orthogonal Decomposition

## 2. Methodology

#### 2.1. Experimental Setup

#### 2.2. Flow Characteristics

#### 2.3. Identification Methodology

_{3}is also plotted, exhibiting similar frequency energy distribution across the frequency range, outside of the 2 Hz peak.

**r**and time τ may be defined as:

_{i}and u

_{j}are the velocity components for correlation. It is common to use a simplified form of Equation (6) for correlations at the same point in time, spatial velocity correlation (SVC, τ = 0) or correlations performed at the same point in space, autocorrelation function (ACF,

**r**= 0). For the application presented, it is the cross-correlation of POD spatial modes that is to be considered, thus no true time association exists. However, for the purposes of the comparison, it is convenient to consider the two desired POD modes for comparison as two time-steps at the same spatial location, leading to the following equation:

_{A}= 10 will be compared using Equation (7) to modes S

_{B}= 5:15. This is to account for any minor re-ordering of POD modes which can occur when two or more consecutive POD modes have similar energy and therefore may be slightly re-ordered during the calculation of eigenvalues. This leads to a set of (11) correlation values for each mode considered, as represented in Figure 11. It should be noted that the absolute correlation should be considered as it is possible for two POD analyses of nominally the same velocity fields to produce an equal but opposite velocity spatial mode, with the corresponding reversed temporal co-efficient.

_{A}and S

_{B}. The two datasets may then be recombined, taking note of the randomization carried out earlier in the presented methodology. This is particularly important if subsequent frequency analysis of coherent structures is to be carried out.

## 3. Analysis of Separated Velocity Fields

^{1}L

_{11}= 22 mm and

^{2}L

_{22}= 50 mm in the surrounding region (approximately 1–2 D), compared to a field average length scales of 15 mm and 42 mm respectively. Further, there is a clear link between these structures, and the first two POD modes presented in Figure 8a,c (or Figure 8b,d), underlining the dominance of these modes as expected from the significant energy contained within, as demonstrated in Figure 7. It should be noted that having a masked region within the flow has an impact on the calculated length scales, particularly where these are particularly large. This is most evident in Figure 16b; here, it is not possible to calculate the length scale immediately below the masked region as it is expected to be larger than the distance between the masked region and the edge of the domain. This strip of data should not be regarded any further in analysis. However, in the case where the integral length scale is much smaller, for example in the turbulent field, Figure 16d, there is no evidence of this artifact.

_{0}= 1, the mean upstream velocity. Performing Reynolds decomposition effectively removes this bias, as demonstrated in Figure 17b. While the peak is now centered close to (0,0), the spread is much greater in the cross-flow direction, v, compared to streamwise due to the shape of the large (coherent) structures that can be seen to dominate, for example, in Figure 15b. This is further evidenced when considering the distribution associated with the decomposed coherent velocity data in Figure 17c which has a similar distribution, showing the domination of these motions in the Reynolds decomposed flow fields.

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Ölçmen, S.; Ashford, M.; Schinestsky, P.; Drabo, M. Comparative Analysis of Velocity Decomposition Methods for Internal Combustion Engines. Open J. Fluid Dyn.
**2012**, 2, 70–90. [Google Scholar] [CrossRef] [Green Version] - Stone, R. Introduction to Internal Combustion Engines, 4th ed.; Palgrave Mscmillan: Basingstoke, UK, 2012; ISBN1 9780230576636 (hbk.). ISBN2 023057663X. [Google Scholar]
- Butcher, D.S.A. Influence of Asymmetric Valve Timing Strategy on In-Cylinder Flow of the Internal Combustion Engine; Loughborough University: Loughborough, UK, 2016. [Google Scholar]
- Adrian, R.J.; Christensen, K.T.; Liu, Z.-C. Analysis and interpretation of instantaneous turbulent velocity fields. Exp. Fluids
**2000**, 29, 275–290. [Google Scholar] [CrossRef] - Lumley, J.L. The Structure of Inhomogeneous Turbulent Flows. In Atmospheric Turbulence and Wave Propagation; Publishing House Nauka: Moscow, Russia, 1967; pp. 166–178. [Google Scholar]
- Chatterjee, A. An introduction to the proper orthogonal decomposition. Curr. Sci.
**2000**, 78, 808–817. [Google Scholar] [CrossRef] - Holmes, P.; Lumley, J.L.; Berkooz, G. Turbulence, Coherent Structures, Dynamical Systems and Symmetry; Cambridge University Press: Cambridge, UK, 1996. [Google Scholar]
- Sirovich, L. Turbulence and Dynamics of Coherent Structues. Part I: Coherent Structures. Q. Appl. Math.
**1987**, 45, 561–571. [Google Scholar] [CrossRef] - Brindise, M.C.; Vlachos, P.P. Proper orthogonal decomposition truncation method for data denoising and order reduction. Exp. Fluids
**2017**, 58, 28. [Google Scholar] [CrossRef] - Butcher, D.; Spencer, A.; Chen, R. Influence of asymmetric valve strategy on large-scale and turbulent in-cylinder flows. Int. J. Engine Res.
**2018**, 19, 631–642. [Google Scholar] [CrossRef] - Epps, B.P.; Techet, A.H. An error threshold criterion for singular value decomposition modes extracted from PIV data. Exp. Fluids
**2010**, 48, 355–367. [Google Scholar] [CrossRef] - Raiola, M.; Discetti, S.; Ianiro, A. On PIV random error minimization with optimal POD-based low-order reconstruction. Exp. Fluids
**2015**, 56. [Google Scholar] [CrossRef] - Doosttalab, A.; Dharmarathne, S.; Bocanegra Evans, H.; Hamed, A.M.; Gorumlu, S.; Aksak, B.; Chamorro, L.P.; Tutkun, M.; Castillo, L. Flow modulation by a mushroom-like coating around the separation region of a wind-turbine airfoil section. J. Renew. Sustain. Energy
**2018**, 10. [Google Scholar] [CrossRef] - Norberg, C.; Sunden, B. Turbulence and reynolds number effects on the flow and fluid forces on a single cylinder in cross flow. J. Fluids Struct.
**1987**, 1, 337–357. [Google Scholar] [CrossRef] - Park, J.; Choi, H. Numerical solutions of flow past a circular cylinder at reynolds numbers up to 160. KSME Int. J.
**1998**, 12, 1200–1205. [Google Scholar] [CrossRef] - Behrouzi, P.; McGuirk, J. Experimental data for CFD validation of impinging jets in cross-flow with application to ASTOVL flow problems. In Proceedings of the AGARD Conference Proceedings 534, Computational and Experimental Assessment of Jets in Cross-Flow, Winchester, UK, 19–22 April 1993. [Google Scholar]
- Perrin, R.; Cid, E.; Cazin, S.; Sevrain, A.; Braza, M.; Moradei, F.; Harran, G. Phase-averaged measurements of the turbulence properties in the near wake of a circular cylinder at high Reynolds number by 2C-PIV and 3C-PIV. Exp. Fluids
**2007**, 42, 93–109. [Google Scholar] [CrossRef] - Perrin, R.; Braza, M.; Cid, E.; Cazin, S.; Barthet, A.; Sevrain, A.; Mockett, C.; Thiele, F. Obtaining phase averaged turbulence properties in the near wake of a circular cylinder at high Reynolds number using POD. Exp. Fluids
**2007**, 43, 341–355. [Google Scholar] [CrossRef]

**Figure 1.**Schematic of LU horizontal water flow tunnel. Indicative measurement plane represented with horizontal green line, downstream of test piece.

**Figure 2.**Experimentally obtained flow fields. (

**a**) Instantaneous vector field, (

**b**) Ensemble average stream lines. Colour represents velocity magnitude, normalized by free-stream velocity, U

_{0}.

**Figure 3.**Velocity profiles (y-direction) at selected points downstream. (

**a**) x-component, (

**b**) y-component.

**Figure 6.**Demonstration of velocity convergence with number of samples. (

**a**) PDF of error in u velocity, ${\epsilon}_{u}$ for 1536 snapshots; (

**b**) Trend of standard deviation of error, $\sigma $ with increasing samples calculated from PDFs; (

**c**) Field average velocity RMS convergence.

**Figure 7.**Comparison of energy fraction for each proper-orthogonal-decomposition (POD) mode of Set A and B. Curtailed at mode 25 for clarity.

**Figure 8.**POD spatial modes: (

**a**) S

_{A}= 1, (

**b**) S

_{B}= 1, (

**c**) S

_{A}= 2, (

**d**) S

_{B}= 2, (

**e**) S

_{A}= 3, (

**f**) S

_{B}= 3. Color scale represents magnitude and is normalized to a common factor across all subfigures.

**Figure 9.**POD temporal coefficient analysis for early modes. (

**a**) First, a

_{1}and second, a

_{2}temporal coefficients; (

**b**) Spectral energy content for the first three sets of temporal modes.

**Figure 10.**POD spatial modes for higher order: (

**a**) S

_{A}= 30, (

**b**) S

_{B}= 30. Color scale represents magnitude and is normalized to a common factor.

**Figure 11.**Absolute cross-correlation of POD spatial modes from set A and B. Curtailed at 31 for clarity.

**Figure 12.**Maximum absolute correlation for each POD spatial mode (curtailed at mode 50). Smoothed data is used to produce trend. A cut-off at 60% correlation is indicated.

**Figure 13.**Example of a decomposed instantaneous vector field. (

**a**) Coherent part; (

**b**) incoherent part (vectors shown white for clarity); (

**c**) raw velocity field. Colour scale represents velocity magnitude normalized with free-stream velocity, U

_{0}.

**Figure 15.**Comparison of fluctuating flow fields for an example flow field. (

**a**) Original flow field; (

**b**) All fluctuations (Reynolds decomposition); (

**c**) Coherent part; (

**d**) incoherent part. Colour scale represents velocity magnitude (common scale).

**Figure 16.**Integral length scales associated with (

**a**,

**b**) coherent fields,

^{1}L

_{11}and

^{2}L

_{22}respectively; (

**c**,

**d**) turbulent fields,

^{1}L

_{11}and

^{2}L

_{22}respectively. Note colour scale changes between subfigures. Parentheses values in (

**c**) and (

**d**) are mm.

**Figure 17.**Probability density functions (pdf) of velocity components. (

**a**) Original data; (

**b**) all fluctuations—Reynolds decomposed; (

**c**) coherent part; (

**d**) turbulent part.

**Figure 18.**Spectral energy density for turbulent fluctuations, u’. Five-thirds law plotted for reference.

**Figure 19.**Spectral energy density for coherent fluctuation velocities, u*. Five-thirds law plotted for reference.

**Figure 20.**Integral time scale distributions associated with turbulent parts. (

**a**) uu; (

**b**) vv; (

**c**) Auto-correlation function, uu about two example points, locations indicated in (

**a**) and (

**b**).

**Figure 21.**Spatial distribution of frequency content associated with integral time scales in turbulent field ($1/\tau )$, scaled appropriately to the wake region only.

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**MDPI and ACS Style**

Butcher, D.; Spencer, A.
Cross-Correlation of POD Spatial Modes for the Separation of Stochastic Turbulence and Coherent Structures. *Fluids* **2019**, *4*, 134.
https://doi.org/10.3390/fluids4030134

**AMA Style**

Butcher D, Spencer A.
Cross-Correlation of POD Spatial Modes for the Separation of Stochastic Turbulence and Coherent Structures. *Fluids*. 2019; 4(3):134.
https://doi.org/10.3390/fluids4030134

**Chicago/Turabian Style**

Butcher, Daniel, and Adrian Spencer.
2019. "Cross-Correlation of POD Spatial Modes for the Separation of Stochastic Turbulence and Coherent Structures" *Fluids* 4, no. 3: 134.
https://doi.org/10.3390/fluids4030134