[ieee 2013 international conference on wavelet analysis and pattern recognition (icwapr) - tianjin,...

Proceedings of the 2013 International Conference on Wavelet Analysis and Pattern Recognition, Tianjin, 14-17 July, 2013

WAVELET MULTI-RESOLUTION ANALYSIS ON VORTICAL STRUCTURES OF A DUNE WAKE BASED ON LARGE EDDY SIMULATION

YAN ZHENG, ARIKA RINOSHIKA

Department of Mechanical Systems Engineering, Yamagata University 4-3-16 Jonan, Yonezawa-shi, Yamagata 992-8510, Japan

E-MAIL: [email protected]

Abstract: The three-dimensional orthogonal wavelet multi-resolution

technique was applied to analyze flow structures of various scales behind a dune model. The three-dimensional dune wake flow was evaluated by using large eddy simulation (LES) at a

Reynolds number of 5530. The instantaneous velocity and vorticity were decomposed into the large-, intermediate- and

relatively small-scale components by the wavelet multi-resolution technique. The coherent structure are

visualized by Q-criterion. It is found that the rollers and

horse-shoe structure in the separation bubble are mainly

contributed from large-scale structures, furthermore, some horse-shoe structures can be clearly identified by

intermediate-scale structures, the coherent structures are the combined effect of large-scale and intermediate-scale structures.

The velocity and vorticity of large-scale structure dominate the dune wake flow and the vorticity concentration makes main

contribution, and the intermediate-scale as well as the relatively

small-scale ones tends to become more active as the flow flows downstream.

Keywords: Multi-scale; Wavelet; Large eddy simulation; Dune;

Turbulent structure; Coherent structure,

1. Introduction

The dunes, known as barchans, transverse dunes, longitudinal dunes, network dunes, star dunes and some other types [1] , are considered to be the most beautiful pattern formed by nature. As the simplest and most studied type of dune, the barchans dune is formed when the wind mainly blows from one direction with a restricted sediment supply [2]. Since the first pioneer works by Bagnold [3], the investigation of dune morphology and dynamics has been much studied [4], [5], [6]. Such studies are believed to have potential application in fighting against desertification and other fields. As an example, Rinoshika and Suzuki [7] have developed a dune model in a horizontal pneumatic conveying system, and successfully reduced the sediment of particles

978-1-4799-0417-4/13/$31.00 ©2013 IEEE

and conveying air velocity. In recent years, large eddy simulation (LES) have

become rather common for numerical simulations and provides good agreement with both the time-averaged velocities as well as the turbulent fluctuations measured experimentally [8], [9]. In the field of flow over dune, Yue et al. [10] and Grigoriadis [11] performed LES with 2D dune and concentrate on the coherent structures behind the dune crest. Up to now, little attention has been paid to the analysis of the complex three-dimensional multi-scale turbulent structures in the dune wake from either the numerical simulation or experimental measurement. To give further understanding of flow dynamics over dune, the detailed information on three-dimensional as well as multi-scale turbulent structures of dune wake should be acquired. This is of fundamental significance and has not been previously investigated, thus motivating the present work.

Recently, as one of important multi-scale analysis tool, the orthogonal wavelet transform has been widely used to analyze turbulent structures since Yamada & Ohkitani [12] and Meneveau [13] have first decomposed the one-dimensional experimental data of turbulent flows into different scales for statistical analysis. As an application in numerical simulation, Farge et al. [14] developed a coherent vortex simulation method decomposing the turbulent structures into coherent and incoherent structures based on orthogonal wavelets. In the experimental study, Rinoshika and Zhou [15], have applied the one-dimensional orthogonal wavelet multi-resolution technique to the analysis of the turbulent wakes. This technique is further potentially capable of separating and quantitatively characterizing, not only coherent and incoherent structures in a flow field, but also the turbulent structures of various scales.

The purpose of this study is to reveal the 3D multi-scale turbulent structures of dune wake by using three-dimensional wavelet analysis and to provide both quantitative and qualitative information on the three-dimensional flow structures of various scales.

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2. Details of nnmerical simulation

2.1. Numerical solver

The commercial CFD software, Fluent 6.3, is used to compute unsteady 3D incompressible unsteady flow around the dune model. The LES turbulence model, based on the application of a filtering operation to the 3D unsteady Navier-Stokes equations was adopted in this study, by which the large scale velocity field is directly calculated, and small scales are modeled. The finite volume method (FVM), which conducts numerical simulations with discretizing computational domain into some small volumes, is employed for the discretization of governing equations. The second-order implicit scheme is selected for unsteady formulation, and the SIMPLEC algorithm was applied to solve pressure-velocity coupling. The Central difference schemes and second-order scheme were used for space discretization and pressure discretization. For more details, the interested readers can refer to the Fluent user's guide (Fluent Inc. 2006).

2.2. Computational domain and boundary conditions

A barchans dune model, which has a height of h = 20 mm, was adopted. Computational domain, as shown in Figure 1, the inlet flow locates at 80mm (4h) upstream of dune, and the outlet flow locates at 400mm (20h) downstream of dune. The height and width of computational domain is 100mm (5h), 180mm (9h) respectively. Gambit pre-processing tool was used for computational grid generation in this study. The computational domain contains approximately 1.1 million cells and about 85% cells were generated around the dune model.

The inlet boundary condition was specified using constant velocity Uo=0.285m!s, which is same as the experimental free-stream velocity. The average turbulent intensity at the inlet is about 2%. The symmetric boundary conditions were employed at the upper boundary and two side boundaries. The pressure outlet condition was used in the outlet of the flow domain. No slip wall boundary conditions were applied on the surface of the dune and the bottom boundary.

3 Three-dimensional orthogonal wavelet decomposition

Orthogonal discrete wavelet transform provide a time-frequency representation of the signal, and the wavelet coefficients specified by a particular set of numbers are a minimal number of independent and orthogonal to each other,

Figure 1. Computational domain and boundary conditions

which can be used to decompose data into different scales. Furthermore, the wavelet components of various frequency bandwidths can be uniquely reconstructed by orthogonal inverse wavelet transform. The three-dimensional orthogonal discrete wavelet transform is briefly described in this section.

For a three-dimensional data matrix Mijk with the size of i x:i x2k. Its three-dimensional wavelet transform is conducted by transforming the array sequentially on its first index (for all values of its other indices), then on its second, and last on its third; i.e. making repeated use of the one-dimensional wavelet transform in the i- (horizontal), )(vertical) and k- (longitudinal) direction. The Daubechies wavelet basis matrix with an order 10 is used in this study, the matrix W, wi, W represent the wavelet basis matrix with the size of i xi, :i x:i, 2k x2k respectively. In order to determine the levels of wavelet components clearly, the value of k is assumed to be smaller than i and) in this section.

The first orthogonal wavelet decomposition (see Fig.2) is performed by multiplication of the following matrix:

Xl =(r XWk (pj xWj (p xW XMijJrr (1)

Here the superscript T denotes transpose of matrix, the

transpose process is performed by M ijk � M}ki � M�j . The

permuting matrix P is employed to permute the matrix such that the odd rows of matrix are moved forward as first half elements and the even rows of matrix are moved backward as the last half elements. By the first orthogonal wavelet decomposition, the original data matrix is divided into eight distinct sub-matrices (see Figure 3), all of which have the same size of fl xfl x2k-l. The sub- -matrices which consist of smooth coefficients (S) and difference coefficients (D) are generated by convolution operations of wavelet matrix, they are Si�Sk> Si�Dk> S;DjSk> S;DPk> Di�Sk> DiSPk> D;DjSk and D;DPk> the subscripts of D and S represent the wavelet transform's direction. It is evident that seven sub-matrices contain the difference coefficients (D) and they are called wavelet coefficients at level k-2.

301


MOl ---'-i d-::-i r -""--:-tio-n----->l

S,----->I j direction

0,---->1 j direction

___ --;>'I� JJPtSl O,sj k direction � Op/Jl

O,o} 'L OPf.

k direction O,oP.

Figure 2. Schematic diagram of three-dimeusioual orthogoual wavelet transform

Figure 3. Schematic diagram of decomposed three-dimensional wavelet coefficients

The second orthogonal wavelet decomposition is to apply the wavelet basis matrices W-I, wi-I, JiII"-I and the permuting matrices pi-I, p-I, pk-I, which are the halves of W, wi, Jill" and pi, P, pk, respectively, to the smooth block matrices of Si�Sk along x-, y- and z-direction in the similar way to the transform of first level.

X2 = (r-1 XWk-1(pj-l X Wj-1(p-l XWi-1 Xsisjskyrr (2)

The wavelet coefficients of level k-3, (0, SlSlDkl , , J

D1SlS1 D1SIDI DID1Sl and DIDIDI) with the size of l } k' I J k' l ] k l J k

f2 xj-2 x2k-2 are generated in this operation. This pyramidal procedure is repeated until the last hierarchical level 1 of eight 2(i-k+3j xiJ-k+3j x23 sub-matrices is obtained. The coefficients of three-dimensional wavelet transform X can be expressed as:

Xijk =( Ckx (Cj x (C XMijJYJ (3)

Here the matrix C is constructed based on a cascade algorithm of an orthogonal wavelet basis function:

c =p i-k+3Wi-k+3 . . . p i-IWi-Ip iWi c! =pj-k+3Wj-k+3 . . . pj-IWj-IpjWj C =PW4 . . . p-1W k-lp kW k

(4) (5) (6)

Since the analyzing wavelet matrix satisfies WTW = I, where I is a unit matrix, the discrete wavelet transform is an orthogonal linear operator and invertible. The orthogonal discrete wavelet reconstruction (inverse wavelet transform) can be simply performed by reversing the procedure, starting with the lowest level of the hierarchy, can be written as:

Mijk =(Cy X ((Cj)T X ((ckf X X;kr r (7)

The orthogonal wavelet transform produces coefficients that contain information on the relative local contribution of various frequency bandwidths to the transformed data instead of the frequency components of the original data. In order to obtain the frequency bandwidth components of the transformed data, the orthogonal wavelet coefficient Xijk is decomposed into the sum of all levels:

X Bl B2 Bm Bk-2 ijk = ijk + ijk + . .. ijk + ... + ijk (8)

where B'; consists of a wavelet coefficient matrix of level

m and zero matrix, having size of i xi x2k. For example,

B�;2 of level k-2, is composed of the following eight block

matrices: 0, Si�Dh S/JjSh S/Jl!h Di�Sh DiSPh D/JjSkand D/JPk with the size of 2,-1 x2'- x2k-1 respectively.

Then the above Eq.(6) is substituted into the inverse orthogonal wavelet transform

M;f,=(Cf x(Cff x«C"f X(B�k)TrY +(Cf X (Cf)T x «C")T X(�JrY + ... (C)T Xk)T X(C")T x(B;fofrY +··-(C)T xkf x (C"f x (Ii;,,2frY

(9)

where the first term and last term represent the components of the original data at wavelet level 1 (the lowest frequency band) and level k-2 (the highest frequency band). It is clearly that the sum of all wavelet components reconstructs the original data. This decomposition method is referred to as the three-dimensional wavelet multi-resolution technique.

4 Spectral Characteristics of Wavelet Components

In order to analyze the three-dimensional flow structures of various scales, the velocity, vorticity and pressure data of LES simulation are decomposed by three-dimensional wavelet multi-resolution technique in this study. As shoWfl in FigJ, an analyzing domain, with the volume of 8hx8hx4h, is divided into a mesh of 128x128x64. To determine the scale characteristics of each wavelet component, fast Fourier transform is used to analyze each wavelet component of different levels, the scale at which a pronounced peak occurs is defined as central scale. The wavelet level 1 having central scale of 50 mm represents the large-scale structure, and the

302


wavelet level 2 having central scale of 26 mm represent the intermediate-scale structure, the wavelet level 3 and level 4 were added together and referred as new level 3, having central scale of smaller than 12mm, represents the relatively small-scale.

5 Multi-resolution analysis on vertical structures

In order to evaluate the coherent vortical structures of different scales, the Q-criterion [16] is calculated by using the wavelet components of velocity in different levels for the first time.

Figure 4 shows the instantaneous coherent structures of LES and wavelet components identified by the Q-criterion. Spanwise rollers are observed near dune crest side in the separation bubble due to flow separation at the dune crest (FigAa), rollers in the similar position are identified by large-scale structure (FigAb), besides, some relatively smaller roller structures can also be observed by intermediate-scale structure (FigAc), while rollers of relatively small-scale structure (Fig Ad) is not identifiable. As for LES (FigAa), two typical hairpin vortices and a kolk are generated in the boundary of separation bubble, in the downstream of separation bubble, streamwise vortical structures propagating toward the two side boundary in the developing boundary layer leads to the generation of hairpin vortices in the near outlet region. The large-scale structure, as presented in FigAb, two horse-shoe structures are detected in the boundary of separation bubble which corresponding to the two hairpin vortices mentioned in FigAa, the legs of these structures tends to induce the streamwise structures at the downstream, and the streamwise structures lead to the upward shifting of coherent structures near the outlet, in addition, at the downstream of separation bubble, the coherent structures do not display a hose-shoe like shape with enlarged legs or distorted structure observed. In FigAc, it is found that some reduced hose-shoe like structure exist around the hairpin vortices (FigAa) and large-scale structure (FigAb), a streamwise vortex which is unobserved by large-scale structure is clearly identified. As for relatively small-scale structure (FigAd), no evident structure can be detected, implying that coherent structure is mainly composed of largeand intermediate-scale structures.

Figure 5 shows instantaneous velocity vectors and

vorticity contours OJy of the LES and wavelet components in

sliced (x, z)-plane (see FigAa) at the location of y/h=3. A large spanwise roller, which is originated from the dune crest, is clearly observed by LES (Fig.5a) and large-scale structure (Fig.5b). Besides, relatively small spanwise rollers of LES (Fig.5a) and intermediate-scale structures (Fig.5c) are detected at the downstream of large roller, and the rollers in

the near-wall region are considered to induce the downstream horse-shoe structures.

Figure 4. Instantaneous coherent structures (a) LES, (b) Large-scale, (c) Intermediate-scale, (d) Relatively small-scale

.':!OJ -"" " fill I:!OJ

·u ·oot • .!oJ � -'" f><I .51! ·15 "

cell M.;II·$}

"II!

�l

� :elf

0 1 u �\ u !-

Figure 5. Instantaneous velocity vectors and vorticity contours OJ y at

y/h=3 (a) LES, (b) Large-scale, (c) Intermediate-scale, (d) Relatively small-scale

Figures 6 and 7 present instantaneous velocity vectors

and vorticity contours OJx of the LES and wavelet

303


components in sliced (y, z)-plane (see FigAa) at the location of xlh=7 and xlh=9 respectively. As shown in Fig.6b, two streamwise vortices and two pair of counter-rotating horse-shoe leg are observed by large-scale structure. They correspond well to the large-scale structures of the LES (Fig.6a). Besides, some horse-shoe leg like vortices and small-scale vortices are identifiable by intermediate-scale structure (Fig.6c) and relatively small-scale structure respectively. The two stremwise vortices seem to induce the generation and uplifting of horse shoe structure at the downstream (Fig.7). As presented in figure 7, several one-legged horse shoe vortices are detected by both LES (Fig. 7 a) and large-scale structure (Fig. 7b). However, two streamwise vortices and some other vortical structures are unidentifiable by large-scale structure, which can be clearly identified by intermediate-scale (Fig. 7 c). Besides, the relatively small-scale (Fig.7d), seems to become active at the downstream, this may because of the uplifting of vortical structure which lead to large-scale vortical structure break into small-scale vortices.

Figure 8 shows instantaneous streamlines and vorticity

OJ z contours of the LES and wavelet components in sliced

(x, y)-plane (see FigAa) at the location of zlh=O.5. Two pairs

'�1

h:) �

�l

� rII II I ••

Figure 6. Instantaneous velocity vectors and vorticity contours OJx at

x/h=7 (a) LES, (b) Large-scale, (c) Intermediate-scale, (d) Relatively small-scale

Figure 7. Instantaneous velocity vectors and vorticity contours OJx at

x/h=9 (a) LES, (b) Large-scale, (c) Intermediate-scale, (d) Relatively small-scale

, • , . • � .. �.

a 10

.. �. •

Figure 8. Instantaneous velocity vectors and vorticity contours OJz at

z/h=O.5 (a) LES, (b) Large-scale, (c) Intermediate-scale, (d) Relatively

small-scale

of large-scale vortices are observed in the (x, y)-plane as the quasi-periodical large-scale vortices in Fig.8a and Fig.8b, which corresponds to the horse-shoe like structures presented in FigAa and FigAb. This behavior perhaps implies that the

304


large-scale coherent structures induce the vibration of the dune surface. The intermediate- and small-scale vortices, as shown in Fig.8c and Fig.8d, are unidentifiable in the LES (Fig.8a), and are extracted by the wavelet multi-resolution technique. It is found that intermediate-scale vortices correspond well to the intermediate-scale coherent structure (Fig.4c).

6. Conclusions

The vortical structure at the downstream of dune is simulated by LES. The three-dimensional orthogonal wavelet multi-resolution technique is applied to decompose LES data into various scales. The important results are summarized as follows: 1 Three-dimensional velocity and vorticity of the LES are decomposed into three wavelet components having large-, intermediate- and relatively small-scale. 2 The Q-criterion calculated by LES and wavelet components is used to visualize horse-shoe vortices. Two large-scale horse-shoe structure and spanwise rollers similar to LES are observed in the separation bubble, indicating the horse-shoe structures induced by rollers are mainly contributed from large-scale structures. Some intermediate-scale hose-shoe structures can't be seen by large-scale are clearly identified at the downstream of separation bubble. Streamwise vortices observed by large-scale structures and intermediate-scales are considered as a reason for the height increase of horse-shoe structures. The distribution of horse-shoe structures is the combined effect of large-scale and intermediate-scale structures. 3 In the streamswise and vertical direction, the large-scale structure dominates the dune wake flow and its vorticity concentration makes main contribution, several intermediate-scale vortices, which cannot be observed by LES, are clearly identifiable, besides, some relatively-small structures are also observed. In the spanwise direction, the intermediate-scale as well as the relatively small-scale ones, tend to become more active as the flow flows downstream.

References

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[3] Bagnold R.A "The physics of blown sand and desert dunes", London, 1941.

[4] Lancaster N, Nickling WG, McKenna Neuman C, Wyatt VE, "Sediment flux and airflow on the stoss

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[6] Livingstone I., Wiggs GFS., Weaver CM, "Geomorphology of desert sand dunes: A review of recent progress", Earth-Science Reviews vol. 80, pp. 239-257, 2007.

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[10] Yue, W., Lin, C. L., and Patel, V. C., "Coherent structures in open channel flows over a fixed dune" J. Fluids Eng., vo1.127(5), pp. 858-864, 2005.

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[12] Yamada M. and Ohkitani K., "Identification of Energy Cascade in Turbulence by Orthonormal Wavelet Analysis", Prog Theor Phys, Vol. 86, pp. 799-815, 1991.

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[15] Rinoshika A and Zhou Y., "Orthogonal Wavelet Multi-Resolution Analysis of a Turbulent Cylinder Wake", J. Fluid Mech., Vol. 524, pp. 229-248, 2005.

[16] Hunt, J. C. R., Wray, A A & Moin, P. "Eddies, stream, and convergence zones in turbulent flows", Center for Turbulence Research Report CTR-S88, 1988.

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