Chapter 3 Methodology
3.2 Coherent structure identification
where a1, a2 and a3 are the derived coefficients. Once δpb(kx, y, kz) is solved, the
pressure in physical space can be obtained by the inverse Fourier transform. Up to this point, the whole procedure for the implicit fractional step method is finished. Three are three second-order accuracy approximations, i.e., equations (3.13), (3.16) and (3.17), thus it gives an overall second-order temporal accuracy. A minimum second-order central difference is used for the spatial discretisation, there- fore it gives an overall spatial accuracy of second-order as well. The temporal and spatial accuracy tests were given by Kim et al. (2002) and Talha (2012), respectively. To avoid the checking board effect, a staggered mesh is used,i.e., the three velocity components are defined on different surfaces of the mesh cell, and the pres- sure is defined in the cell centre. The body force vector is treated in the same way as the velocity vector. For detailed staggered mesh arrangement, please refer to (Talha, 2012).
To make the code capable for high Reynolds number flow simulation, it is paralleled using the MPI library. Since direct solvers are used for all the matrix inversions, a 2DECMP&FFT library (Li and Laizet, 2010) is implemented to satisfy this requirement, and also to guarantee the scalability of the code. Details about the implementation can be found in Hurst (2013). The paralleled code is highly scalable up to 1024 cores tested.
3.2
Coherent structure identification
Theλ2 criterion proposed by Jeong and Hussain (1995) is used in the present study to identify the vortex structures. λ2 is the second largest eigenvalue of the gradient tensorS2+Ω2, whereS is the strain tensor,i.e.,Sij = 12
∂ui ∂xj + ∂uj ∂xi , andΩis the vorticity tensor,i.e., Ωij = 12
∂ui ∂xj − ∂uj ∂xi
. According to Jeong and Hussain (1995), the vortex cores correspond to regions, where the gradient tensor S2+Ω2 has at least two negative eigenvalues, equivalent toλ2 <0. Figure 7.5(a) shows a snapshot of the quasi-streamwise vortices in the buffer region identified by negative λ2 value at Reτ = 800. Some hairpin structures can be observed, but majority of them are
single legged (Robinson, 1991). Essentially, there are two types of structures: the positive one rotating in clockwise direction (ω′x >0) and the negative one rotating in anti-clockwise direction (ωx′ < 0). However, the high population density and irregular shapes of these near wall structures make the view very difficult. Jeong et al. (1997) proposed an ensemble average method to extract the shapes of the positive and negative structures. This ensemble average method is adopted in the present study with a certain improvement. Detail of the ensemble average method and its validation is given in the following part.
Generally, the ensemble average method includes two procedures: 1) identi- fying the cores of the vortex structures; 2) selecting the identified vortex structures and conditionally average them. To identify the vortex cores, the local minima of theλ2value in a small window with a diameter ofDc+= 20 are detected in everyyz
plane. The two local minima in two adjacentyz planes are connected if they satisfy the following two criteria: a) the two points form a vector which has a streamwise angle −45◦ < θ < 45◦; b) the signs of the streamwise vorticity fluctuation ω′
x are
the same at the two points. A typical identified λ2 structure is shown in figure 3.3. This is a negative λ2 structure, since ω′x < 0 for all the identified local minima.
The quasi-streamwise vortices are within the buffer layer and have a typical length of λ+
x ≈300, which can be read through the energy peak in pre-multiplied density
spectrumkxkzΦvv(figure 3.12(b)), thus only those structures longer thanλ+x = 150
and within the near wall regiony+<60 are selected for the next procedure.
Figure 3.3: A typical identifiedλ2 structure. Iso-surface is λ+2 =−0.01 coloured by wall distance. Black spheres indicate the selected local minima on each yz plane and blue arrows indicate the searching directions.
Once the positive (negative)λ2 structures in the flow field are all identified, a crude averaged positive (negative) structure can be obtained by simply aligning the streamwise centres of the positive (negative) λ2 structures and taking the average. Then the correlation between the crude positive (negative) averaged structure and each individual positive (negative) structure is calculated in a window size of 150× 60×40 wall units by shifting the individual positive (negative) structure for up to 30 wall units in x and z directions, while the y location is kept unchanged. If the maximum correlation is higher than 0.4, then the individual positive (negative) structure is retained; otherwise it is discarded. The remaining positive (negative) individual structures are averaged again by aligning the streamwise centre points of the structures. This procedure can be repeated for more than once to make sure the the remaining individual positive (negative) structures are highly correlated. At the same time, the velocity and pressure fields associated with each individual positive (negative) structure are also averaged to get the ensemble averaged flow fields. The final ensemble averaged positive (negative)λ2 structures are calculated based on the ensemble averaged velocity field.
The ensemble averaged positive and negativeλ2structures are shown in figure 3.4. A total of 10 well separated flow fields in time are used, and the total number of selected positive and negative λ2 structures are 1601 and 1609, respectively. As observed from figure 3.4, the positive and negative structures are highly symmetric about streamwise direction. A rough measure by eyes suggests a ∓5◦ tilting angle for positive and negative λ2 structures; and a 10◦ inclination angle for both. This result is very close to the∓4◦ tilting angles, 9◦ inclination angle reported by Jeong et al. (1997), and the∓6◦ tilting angles, 9◦ inclination angle reported by Jung and Sung (2006).
(a) (b)
Figure 3.4: Ensemble averagedλ2 structures (λ+2 =−0.005) inyz,xz andxy plane views: (a) positive one; and (b) negative one.
The quasi-streamwise vortices are Reynolds stress carrying eddies. The pat- terns for the three velocity fluctuation components, i.e., u′, v′, w′, and the three Reynolds shear stress components, i.e.,−u′v′, −v′w′, −u′w′ in a yz plane cutting through the streamwise centre of the ensemble averaged λ2 structures (x = 0) are shown in figure 3.5. Again, these patterns are symmetric or anti-symmetric between the positive and negativeλ2structures. The Reynolds shear stress components show a very good agreement with the ensemble averaged data shown by Jeong et al. (1997) (figure 13 of their paper). The high- and low-speed streaks can be well observed on both sides of theλ2 structures, with the high-speed streak peaking at y+≈23 and the low-speed streak peaking aty+ ≈11. The high-speed streak is associated with Q4 event (sweep), and the low-speed streak is associated with Q2 event (ejection). Both Q4 and Q2 create positive streamwise Reynolds shear stress−u′v′, as shown by two positive peaks in the left and right sides of the positive and negative λ2 structures in figure 3.5(d). The negative contribution to −u′v′ from Q3 and Q1 events are on the top and bottom sides of the λ2 structures. The different peak locations of the high- and low-speed streaks agree with the quadrant analysis result by Kim et al. (1987), who reported that sweep events dominated in the near wall region, and ejection events dominated in the region further away from the wall, with the crossing point aty+≈12. The high skin-friction region associated with the λ
2 structures is clearly displayed in figure 3.5.