Chapter 3 Methodology
3.3 Proper orthogonal decomposition
Proper orthogonal decomposition (POD), also known as Karhunen-Lo´eve decom- position (KL) provides a set of bases to study multi-scale turbulent flow (Berkooz et al., 1993), in the sense that the energy decays the fastest across all the modes. In this case, the key turbulent dynamics can be captured by a small number of leading order POD modes. The modes in POD are not pre-fixed, but in the periodic direc- tion the POD modes are statistically equivalent to Fourier modes. In the present numerical channel flow, POD is only used for the wall normal direction, while direct Fourier transform is used for streamwise and spanwise directions.
For the one dimensional POD modes in wall normal direction, the three velocity fluctuation components form a vectorφ(y) as below,
φ(y) = [u′(y1), u′(y2), ..., u′(yj), ..., u′(yNy−1), u′(yNy), v′(y1), v′(y2), ..., v′(yj), ..., v′(yNy−1), v′(yNy), w′(y1), w′(y2), ..., w′(yj), ..., w′(yNy−1), w′(yNy)]
T.
(a) (b) (c) (d) (e) (f)
Figure 3.5: Patterns of Reynolds stresses in a yz plane (x = 0) cutting through the streamwise centres of the ensemble averaged positive (top row) and negative (bottom row) λ2 structures: (a) u′; (b) v′; (c) w′; (d) −u′v′; (e) −v′w′ and (f)
−u′w′. λ2 structures in the yz planes are indicated by contour lines λ+
2 =−0.005. Bright (dark) colour is for high (low) value. Negative contour lines are dashed.
x
+ -100 -50 0 50 100z
+ -50 0 50 ∆Cf 3 1 -1 -3 -5 -7 ×10-3x
+ -100 -50 0 50 100z
+ -50 0 50 ∆Cf 3 1 -1 -3 -5 -7 ×10-3 (a) (b)Figure 3.6: Skin-friction associated with: (a) positiveλ2 structure; and (b) negative
λ2 structure. Full wall surface averagedCf has been subtracted.
Thus the POD modes are defined in the sense to give the fastest spectrum decay rate for the turbulent kinetic energy, k = 12||φ(y)||2. A space and time averaged correlation tensor can be defined,R=hφ(y)φ(y′)Ti
x,z for the velocity vector φ(y).
The POD essentially solves the following eigenvalue problem,
Z 1 0
Rψ(n)(y)dy =λ(n)ψ(n)(y), (3.23) whereψ(n)(y) andλ(n) are thenth eigenfunction and eigenvalue. ψ(n)(y) represents
the nth POD mode, and λ(n) represents the energy associated with the nth POD mode, and ||φ(y)||2 = 3Ny X n=1 λ(n). (3.24)
Since the correlation tensor R is Hamiltonian, all the eigenvectors are orthogonal to each other. Moreover, they are normalised, so that,
Z 1 0 ψ(n)(y)ψ(m)(y)dy = 1, if m=n, 0, if m6=n. (3.25)
To be noticed that, this decomposition can be applied in any wall normal region [ymin, ymax]. The smaller (ymax−ymin) is, the faster convergence can be achieved.
The NAG@ library is employed to solve equation (3.23). Since a non-uniform grid is used in they direction (equation (3.30)), following Moin and Moser (1989) and Ball et al. (1991), a coordinate transformation is employed to preserve the symmetry of
R, thus equation (3.23) can be transformed to a symmetric matrix problem,i.e.,
ζ1/2Rijζ1/2 ζ1/2ψ(n)
=λ(n)ζ1/2ψ(n), (3.26) whereζ =ζ1/2ζ1/2 = [∆y1,∆y2, ...,∆yNy,∆y1,∆y2, ...,∆yNy,∆y1,∆y2, ...,∆yNy]T.
The above approach is called direct POD method, which is suitable for low freedom vector φ (Berkooz et al., 1993). For example, in the present one dimen- sional case, the freedom ofφis 3Ny, and the size ofRis (3Ny)2. However, when the
freedom of vector φis large, which is especially true for two dimensional and three dimensional turbulent databases, solving equation (3.23) is computational expen- sive. Fortunately, in two or three dimensional cases, the number of available flow fields (snapshots) Nt is typically lower than the freedom of a single flow field, thus
the snapshot method can be used to save the computational cost (Sirovich, 1987). The idea is to change the kernel in equation (3.23) from a spatial correlation tension
Rto a temporal correlation tensor K, which is defined as below, Kij =
Z 1 0
φ(y, ti)φ(y, tj)dy. (3.27)
Thus, equation (3.23) becomes
Z
tK
where max{m}=Nt, smaller than the eigenvector space in the direct POD method.
Then the final eigenvector can be calculated by projecting all the snapshotsφ(y, t) into the vectorϕ(m)(t),i.e.,
ψ(m)(y) =
Z
t
φ(y, t)ϕ(m)(t)dt. (3.29) For more detailed theory, please refer to Sirovich (1987).
Figure 3.7 shows the first four POD modes for streamwise, wall normal and spanwise velocity fluctuation components, u′, v′ and w′. Following the assumption given by Moin and Moser (1989), w′ is taken to be uncorrelated with u′ and v′ in the current channel flow, thus w′ is set to be zero in φ(y) when calculating the POD modes foru′ and v′; similarly, both u′ andv′ are set to be zeros inφ(y) when calculating the POD modes forw′. Both direct POD and snapshot POD methods are used for this calculation, and they show a very good agreement with each other, and also with the result from Moin and Moser (1989) at a slightly lower Reynolds number,Reτ = 180. The peak locations calculated by Moin and Moser (1989) are
all slightly closer to the wall, which may come from the effect of different wall normal coordinate used. Similar to the Fourier modes, higher POD mode has more local minima and maxima.
Table 3.1 shows the turbulent kinetic energy contributions from the first four POD modes compared with Moin and Moser (1989), Sen et al. (2007). A reasonable agreement is achieved. To capture 90% total kinetic energy, 15 POD modes are needed for the present data, while only 10 POD modes for Moin and Moser (1989), Sen et al. (2007), which suggests that POD is indeed a powerful tool for turbulent dimension reduction analysis.
Table 3.1: Contributions to turbulent kinetic energy kfrom different POD modes. Case λ1/(2k) λ2/(2k) λ3/(2k) λ4/(2k) 90% of TKE
Present study 0.291 0.155 0.085 0.059 15
Moin and Moser (1989) 0.32 0.16 0.08 - 10
Sen et al. (2007) 0.28 0.16 0.085 0.05 10
Even though snapshot POD can significantly reduce the computational cost, the convergence rate is slow for three dimensional flow fields. By taking advantage of the periodic boundary in the streamwise and spanwise directions, the three di- mensional POD can be reduced to a one dimensional POD in Fourier space for each wave pair (m, n), wherem andnare the integer numbers of sinusoidal waves in the
0 50 100 150 200 -1 0 1 2
y
+ u′ v′ w′√
λ
1ψ
(1 ) 0 50 100 150 200 -1 -0.5 0 0.5 1 1.5y
+√
λ
2ψ
(2 ) (a) (b) 0 50 100 150 200 -1.5 -1 -0.5 0 0.5 1y
+√
λ
3ψ
(3 ) 0 50 100 150 200 -1 -0.5 0 0.5 1y
+√
λ
4ψ
(4 ) (c) (d)Figure 3.7: The first four POD modes for velocity fluctuations, u′,v′,w′: (a) first mode√λ1ψ(1); (b) second mode √λ2ψ(2); (c) third mode√λ3ψ(3); and (d) fourth mode√λ4ψ(4). Solid lines are calculation using direct POD method; dashed lines are calculated using snapshot POD method; and symbols are data from Moin and Moser (1989).
streamwise and spanwise directions, i.e., λx = Lmx,λz = Lnz. Then equations (3.22)
to (3.26) can be written in the complex space for vector φb(m, n, y) (see detail in Moin and Moser (1989), Ball et al. (1991)). After solving the eigenvalue problem, a total number ofq (an integer number) POD modes inydirection are found for each (m, n) pair. Thus a single three dimensional POD mode is indicated by a quantum group (m, n, q). Considering the first POD mode in wall normal directionq= 1 for all the wave pairs (m, n), this gives the characteristic eddy in the turbulent field, as defined by Moin and Moser (1989). Figure 3.8 shows the characteristic eddy in a three dimensional view when the phase difference between each mode and its reference phase is zero. It shows a high-speed streak with two low-speed streaks accompanying aside. From ayz plan view, the spanwise spacing between the high- and low-speed streaks isλ+z ≈50 in the near wall region, but the spacing increases
as the wall normal distance becomes larger. The high-speed streak has a very long tail in the near wall region, and a round head in the channel centre. This agrees with that channel turbulence becomes more isotropic as it leaves the near wall region to the channel centre, and again it is reminiscent of Townsend’s double cone eddies (Townsend, 1976). Due to the homogeneity of the flow in the spanwise direction, when a phase difference of π is used between each POD mode with its reference phase, a similar characteristic eddy is expected, but the high- and low-speed streaks swap positions in the spanwise direction. The captured characteristic eddy is similar to those shown by Moin et al. (1989) and Moarref and Jovanovi´c (2012), and it is a perfect low dimensional structure for understanding the effect of flow control, which is going to be discussed in the following chapters.
Figure 3.8: The characteristic eddy identified in CH200 case. Iso-surfaces areu′ = −0.35 (red) and u′ = 0.35 (yellow).