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Numerical simulation of flow past circular duct Ze-gao YIN*1, Xian-wei Cao1, Hong-da SHI1, Jian MA2 1. Ocean Engineering Key Laboratory of Shandong Province, Ocean University of China, Qingdao 266100, P. R. China 2. Civil Engineering Department, Zhejiang University, Hangzhou 310027, P. R. China Abstract: The Renormalization Group (RNG) k H turbulence model and Volume of Fluid (VOF) method were employed to simulate the flow past a circular duct in order to obtain and analyze hydraulic parameters. According to various upper and bottom gap ratios, the force on the duct was calculated. When the bottom gap ratio is 0, the drag force coefficient, lift force coefficient, and composite force reach their maximum values, and the azimuth reaches its minimum. With an increase of the bottom gap ratio from 0 to 1, the drag force coefficient and composite force decrease sharply, and the lift force coefficient does not decreases so much, but the azimuth increases dramatically. With a continuous increase of the bottom gap ratio from 1 upward, the drag force coefficient, lift force coefficient, composite force, and azimuth vary little. Thus, the bottom gap ratio is the key factor influencing the force on the circular duct. When the bottom gap ratio is less than 1, the upper gap ratio has a remarkable influence on the force of the circular duct. When the bottom gap ratio is greater than 1, the variation of the upper gap ratio has little influence on the force of the circular duct. Key words: circular duct; RNG k H turbulence model; VOF method; numerical simulation

1 Introduction Pipelines are the main way that oil and gas are transported through the sea. The exterior flow past horizontal circular ducts is an important factor that many scholars and engineers are concerned about. In order to investigate flow near the circular cylinder, physical models have been intensively employed. Bearman and Zdravkovich (1978) used a wind tunnel test to investigate flow around a circular cylinder when the Reynolds numbers (Re) were 2.5 × 104 and 4.8 × 104, and the gap ratio between the circular cylinder and the plate wall varied from 0 to 3.5. Then, the smoke-wire method was used to express the flow visualization and pressure distribution, and it was found that the Strouhal number remained constant when the gap ratio was greater than 0.3. On the basis of Bearman’s research, Fredsøe and Hansen (1987) conducted flume experiments to study flow separation, drag force, and lift force of the cylinder. According to the variable diameter cylinder, flow past the cylinder near the bottom boundary was investigated with the experiments. Qi et al. (2006) analyzed the characteristics of the üüüüüüüüüüüüü This work was supported by the Shandong Province Natural Science Foundation (Grant No. ZR2009FQ003). *Corresponding author (e-mail: yi[email protected]) Received Mar. 12, 2010; accepted May 10, 2010

vortex of flow past the cylinder on the basis of physical model data when the gap ratio was 0.5. With the increase of computational power, mathematical models have been utilized to simulate flow past the circular duct. Williamson (1996), Travin et al. (2000), Kravchenko and Moin (2000), and Sarpkaya (2004) used the Reynolds-averaged Navier-Stokes equations to study the flow near a circular cylinder. Kalro and Tezduyar (1997), Breuer (2000), Jordan (2002), and Catalano et al. (2003) used the large eddy simulation method to study the flow around a cylinder. This research focused mainly on the hydraulic characteristics and the fluid structure near the cylinder. As far as direct numerical simulation for Reynolds numbers from 10 to 1 000 is concerned, Henderson (1997) improved the capabilities of research on flow transition from two dimensions to three dimensions. Mittal and Balachandar (1995) investigated the three-dimensional effect on simulation accuracy at Re = 525. Although these scholars have concentrated on flow past the cylinder, there have been few studies on the flow force on circular ducts with various upper and bottom gap ratios. In this study, the Renormalization Group (RNG) k H turbulence model and Volume of Fluid (VOF) method were employed to calculate the hydraulic parameters around a circular duct with various upper and bottom gap ratios, and the forces on the circular duct were also computed and analyzed.

2 Mathematical model The circular duct boundary causes the streamline to curve intensively. Compared with the standard k H model, the RNG k H model more accurately simulates complex boundary flow. Therefore, the RNG k H model was employed in this study. The tensor form of the Navier-Stokes equations and the RNG k H equations can be written as follows (Liu et al. 2004; Baumal et al. 1998; Bazdidi et al. 2004): Continuity equation: wU w U ui 0 (1) wt wxi Momentum equation: w U ui wt

w U ui u j wxi

§ wu wu j · º wpi w ª wui pt G i , j K t ¨ i «P ¸¸ » U Fi ¨ wxi wx j ¬« wx j © wx j wxi ¹ ¼»

(2)

k equation:

w Uk wt

uj

w Uk wx j

wui § wui wu j · w ª wk º ¨ ¸ UH « P K t » Kt wx j ¬« wx j ¼» wx j ©¨ wx j wxi ¹¸

(3)

H equation:

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w UH

w UH

w ª wH º c1*HK t wui § wui wu j · c2 UH 2 (4) ¨ ¸ « P K t » k wt wx j wx j ¬« wx j ¼» k wx j ©¨ wx j wxi ¹¸ where U is fluid density; P is molecular viscosity; ui is the velocity component in the xi (i = uj

1, 2) coordinate direction, denoting x and y coordinates, respectively, when i is equal to 1 and 2; pi is pressure in the xi direction; k is turbulent kinetic energy; H is the turbulent dissipation rate; Fi is the body force in the xi direction: F1 0 and F2 g ; G ij is the Kronecker delta; when i = j, cP U k 2 K 1 K K0 2 ; G ij 1 , and when i j, G ij 0 ; K t ; cP 0.008 45; pt U k ; c1* c1 3 H 1 EK 3

c1 1.42 ; K

2 Ei , j Ei , j

1/ 2

k

H

; Eij

1 § wui wu j · ¨ ¸ ; K0 2 ¨© wx j wxi ¸¹

4.38 ; E

0.012 ; and c2

1.68 .

3 Numerical methods and boundary conditions 3.1 Numerical methods The VOF method was adopted to handle the free surface. Its concept can be expressed as follows: a water volume fraction function (WVFF), Fw , is defined to present the relative water volume fraction in every computational grid cell. In each cell, the WVFF plus the volume fraction of air, Fa , is 1: Fa

1 Fw

For a given cell, three situations exist: Fw water; Fw

(5)

1 , when the cell is completely filled with

0 , when the cell is completely filled with air; and 0 < Fw 1 , which indicates that

the cell is filled with both water and air, and an interface between the two exists in the cell. Free surface flow conforms to the third situation. The control equation of Fw can be expressed as follows:

wU w U Fw w U Fw wt wx wy U Fw U w Fa Ua

P

0

Fw Pw Fa Pa

(6) (7) (8)

where U w and Ua are the densities of water and air, respectively; P w and Pa are the molecular viscosity coefficients of water and air, respectively; and U and P can be obtained by Eqs. (7) and (8) after Fw has been computed. After the governing equations of the turbulence model consisting of the coupled RNG k H model and the VOF method, including Eqs. (1) through (4) and Eqs. (6) through (8), are solved, the hydraulic factors can be obtained. In this study, the mesh generation was performed with GAMBIT 2.2.30. In the Fluent software, a two-dimensional, single-precision, implicit, segregated solver was selected. The PLIC-VOF algorithm was used for the reconstruction of the interface, and the semi-implicit method for a pressure-linked equation algorithm was adopted for the solution of discrete 210

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equations. The solutions were considered to have converged when residuals were less than 10-3. The simulations were performed on a PC with a 2.8 GHz processor and one GB of RAM.

3.2 Boundary conditions The inlet boundary was divided into a water velocity inlet and an air pressure boundary. The water velocity at the inlet boundary was defined as U 0 . The air boundary was regarded as atmospheric pressure. The k and H at the inlet boundary were evaluated according to the following formulas: k

H

0.004U 02

(9)

1.5

k 0.42 H 0

(10)

where H 0 is the water depth at the inlet boundary. The outlet boundary was regarded as the atmospheric pressure boundary. At the wall, a no-slip boundary was specified, and fluid velocities were set to zero. The wall function was adopted to treat the viscous sub-layer.

4 Numerical results and analysis 4.1 Model validation In Fig. 1, we can see that D is the diameter of the circular duct; e is the distance between the lower end of the circular duct and the bottom plane; H is the water depth; U is the mean flow velocity; Re UD Q , with Q being the kinematic viscosity; E is defined as the bottom gap ratio: E

e D ; and G is defined as the upper gap ratio: G

H D e

D.

In order to validate the mathematical model, a physical model and a mathematical model were established where Re = 5 × 105, E = 1, G = 3, and D = 0.2 m. Velocity measurement was conducted with an acoustic Doppler velocimeter (ADV), as in Fig. 2. Pressure was measured by pressure sensors. The computational area was partitioned by unstructured grids, whose cell size was 0.1 m, and the total number of grid cells was 3 477. Due to the hydraulic complexity near the circular duct boundary, the area near the duct was partitioned with refined grids (Fig. 3). Then, the data of the physical model were compared with computed results.

Fig. 1 Sketch map of computational case Fig. 2 Velocity measurement sketch of physical model with ADV

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Fig. 3 Computational grids of case where E = 1 and G = 3

From Fig. 4, we can see that the measured free surface agrees with the computational results for the most part. From Table 1, we can see that all the relative errors between physical model velocities (Up) and computational velocities (Uc) are less than 7%, and measured data approach computational results. From Table 2, we can see that all the relative errors between physical model pressures (pp) and computational pressures (pc) are less than 6%, and measured data also approach computational results. Thus, computations of flow past the circular duct with the RNG k H turbulent model and the VOF method are valid and reliable.

Fig. 4 Comparison between computational free surface and measured free surface (E = 1, G = 3, x is the horizontal distance from the duct center, and H is the water depth) Table 1 Comparison between Up and Uc NRP

Uc (m/s)

Up (m/s)

EU (%)

NRP

Uc (m/s)

Up (m/s)

EU (%)

1

0.438

0.456

3.947

5

0.324

0.304

6.579

2

0.395

0.412

4.126

6

0.254

0.240

5.833

3

0.367

0.376

7

0.197

0.208

5.288

4

0.350

0.336

2.394 4.167

Note: NRP denotes the number of representative points in a typical section, at a distance of four times the diamater from the center of the circular duct; EU = (UpUc)/Up × 100%.

Table 2 Comparison between pp and computational pc NRP

pc (Pa)

pp (Pa)

Ep (%)

NRP

pc (Pa)

pp (Pa)

Ep (%)

1

4 918

4 875

0.882

5

3 587

3 814

5.952

2

5 500

5 347

2.861

6

3 220

3 356

4.052

3

5 840

5 685

7

2 500

2 404

3.993

4

3 920

4 108

2.726 4.576

Note: EP = (pppc)/pp × 100%.

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4.2 Computational velocities and pressure When Re = 5 × 105, E varies among the values 0, 1, and 2; G varies among the values 1, 2, 3, and 4; and we obtain 12 conditions with different values of E and G (Table 3). The mathmatical model was used to compute the hydraulic factors associated with the 12 conditions. The results of the 7th condition, where E = 1 and G = 3, were chosen to be analyzed. Table 3 Twelve computational conditions E

G 1

2

3

4

0

1st condition

2nd condition

3rd condition

4th condition

1

5th condition

6th condition

7th condition

8th condition

2

9th condition

10th condition

11th condition

12th condition

From Fig. 5 we can see that the velocity direction is significantly deflected near the front of the circular duct boundary owing to the squeezing action that the circular duct boundary exerts on the flow, and the magnitude of the velocity becomes larger. The velocity distribution is symmetrical in the wake area, and the magnitudes of velocities are less than those of other areas. From Fig. 6 we can see that the pressure along the duct boundary is unevenly distributed. The pressure on the duct close to the inflow direction and the bottom plane is the largest, and the pressure on the duct close to the free surface is the smallest. The pressure recovers faster than the velocity does. Pressure contours approximately parallel each other in the section that is about five times the duct diameter away from circular duct, and the contour direction is almost horizontal, showing that the influence of the duct on pressure can be ignored here.

Fig. 5 Computational flow velocities (E = 1, G = 3)

Fig. 6 Computational pressure (Pa) (E = 1, G = 3)

4.3 Computational turbulent kinetic energy and turbulence dissipation rate It can be seen in Fig. 7 and Fig. 8 that the distribution of turbulent kinetic energy approximately agrees with that of the turbulence dissipation rate. The largest area lies on a duct close to the inflow and the free surface, and the turbulent stress is the largest here. In the wake area, the turbulent intensity is the lowest. The turbulent kinetic energy and the

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turbulence dissipation rate are distributed symmetrically, and the horizontal line across the duct center is the symmetry axis.

Fig. 7 Turbulent kinetic energy (m2/s2) (E = 1, G = 3) Fig. 8 Turbulence dissipation rate (m2/s3) (E = 1, G = 3)

4.4 Analysis of force on duct The force on the duct can be decomposed into a horizontal drag force and a vertical lift force. The drag force coefficient ( CD ), lift force coefficient ( CL ) , composite force (F), and azimuth ( D ) can be defined as follows: CD

F

Fx , CL 1 U DU 2 2

Fx2 Fy2 , D

Fy , 1 U DU 2 2 Fy arctan Fx

where Fx and Fy are the horizontal drag force and vertical lift force, respectively. In Fig. 9, Fig. 10, Fig. 11, and Fig. 12, we can see that, when E = 0, CD , CL , and F are at their largest values, but D is at its smallest. If we do not consider the duct dead weight and the internal flow force, the circular duct is most likely to break when E = 0. With an increase of E from 0 to 1, CD and F decrease sharply, and CL does not decreases so much, but D increases dramatically. With an increase of E from 1 to 2, CD , CL , F, and D vary slightly. With an increase of G from 1 to 4, CD , CL , and F decrease, but D increases when E = 0. More specifically, with the increase of G from 1 to 2 when E = 0, CD , CL , and F decrease sharply, but D increases quickly, showing that lift force contributes more and more to composite force. With the continuing increase of G from 2 to 4 when E = 0, CD , CL , and F decrease slowly. The value of D , however, also increases slowly. With an increase of G from 1 to 4 when E = 1 and E = 2, CD , CL , F, and D vary slightly. More specifically, CD varies from 1.5 to 1.7, CL varies from 60 to 61, F varies from 7 550 N to 7 620 N, and D varies from 88.3° to 88.6°, showing that the variation of G has little influence on CD , CL , F, and D when E = 1 and E = 2.

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Fig. 9 Relation curve of CD and G

Fig. 11 Relation curve of F and G

Fig. 10 Relation curve of CL and G

Fig. 12 Relation curve of D and G

5 Conclusions (1) Using Fluent software, the RNG k H turbulence model and the VOF method were employed to simulate the flow past a circular duct. The computational results regarding related hydraulic parameters, including surface position, velocities, pressure, turbulent kinetic energy, and the turbulence dissipation rate, were obtained and analyzed. (2) The forces on the duct were computed and analyzed with different bottom gap ratios and upper gap ratios. When E = 0, the drag force coefficient, lift force coefficient, and composite force are at their largest, but the azimuth is at its smallest, compared to other conditions. When E = 0, the drag force coefficient, lift force coefficient, and composite force decrease, but the azimuth increases along with the upper gap ratio. With an increase of the bottom gap ratio from 0 to 1, the drag force coefficient and the composite force decrease sharply, and the lift force coefficient does not decrease so much, but the azimuth increases dramatically. (3) The bottom gap ratio is the most important factor influencing the force on the circular duct. When the bottom gap ratio is less than 1, the upper gap ratio has a significant influence on the force of the circular duct. When the bottom gap ratio is greater than 1, the variation of the Ze-gao YIN et al. Water Science and Engineering, Jun. 2010, Vol. 3, No. 2, 208-216

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upper gap ratio has little influence on the force of the circular duct.

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