LATTICE BOLTZMANN SHCEME TO SIMULATE 2-D HYDRAULIC TRANSIENTS IN CONCRETE SPIRAL CASES

Yongguang Cheng¹  Lisheng Suo²

College of Water Conservancy and Hydropower Engineering, Wuhan University,

Wuhan, 430072, China. Tel: +86-27-87647273. E-mail: chengyg@mailsvr.hhu.edu.cn

College of Water Conservancy and Hydropower Engineering, Hohai University,

Nanjing, 210098, China. Tel: +86-25-3713777-51278. E-mail: lssuo@mailsvr.hhu.edu.cn

Abstract: Basic equations are established for the two-dimensional hydraulic transient simulations in which the third dimension of the flow field varies spatially. A lattice Boltzmann scheme specially for solving these equations is developed by multi-scale analysis. After numerical verifications of the equations and scheme with typical flows, the two-dimensional hydraulic transients in a reinforced-concrete spiral case are simulated and analyzed. The reasonable results show that the lattice Boltzmann method has promising potential to simulate multi-dimensional hydraulic transients in practical engineering.

Keywords: two-dimensional hydraulic transient, numerical simulation, lattice boltzmann method, concrete spiral case

1  INTRODUCTION

Since the 1970s, latticework method [2,5], bi-characteristics method [4], nearcharacteristics method [3], characteristics-like method [1,5] were developed for 2-D hydraulic transient simulation. Though some reasonable simulations were reported, these schemes have several draw backs such as not robust [3], complicated [4] and fixed mesh [1,5], which may be aggravated in 3-D simulations and in cases of complex geometry.

The lattice Boltzmann method (LBM) is a newly developed kinetic discrete method for simulating macroscopic continuous partial differential equations, especially Navier-Stokes equations [6-10]. Simplicity in algorithms, easy treatment of complex boundaries, direct calculation of pressure and suitability for parallel computing are LBM's superior features, which are very appealing to large-scale multi-dimensional simulations of fluid flows. But it is unacquainted in hydraulic engineering owing to the promoters of LBM are mainly in mathematics, physics or computer science fields and very few applied simulations are reported previously. In Ref. [9], a LBM scheme for simulating 1-D transient flows was firstly built and proved to be feasible.

In this paper, a special 2-D scheme that takes the spatial variation of the thickness into account is developed to calculate the hydraulic transients in reinforced-concrete spiral cases. The basic equations and LBM scheme are first established. After verifying the equations and scheme with typical flows, the 2-D hydraulic transients in a reinforced-concrete spiral case are simulated and analyzed.

2  BASIC EQUATIONS

The fluid in volume formed by  planar region  and the  directional thickness  (see Fig. 1) is considered with the Lagrange concepts.

The conservation law of mass insures . Using the formula for total derivative of mass volume, one deducts . Considering the optional selection of  and the continuum of the integrand, we have , which can be expressed by tensor form as

                                                              (1)

where is density,  velocity component, , . Eq. (1) is the continuity equation with an additional term that reflects the thickness variation in  direction.

The conservation law of momentum reads that the change rate of fluid momentum in a control volume equals the total force acting on the fluid in the volume. The change rate of fluid momentum in the concerned volume is  where the formula for total derivative of vector volume is applied. The total body force is  and the total surface force is , in which,  is the force acted by the top and bottom walls, which is considered as slip boundaries. Therefore, we have

and

Then, the momentum equations in tensor form are

                                                      (2)

where indices  and stress tensor . Similarly the last term in the left side of Eq. (2) reflects the effect of thickness variation. With uniform thickness,  and the equations reduce to normal Navier-Stokes equations.

3  LATTICE BOLTZMANN SCHEME

The ordinary LBM models [6-10] can not be used directly because Eqs. (1) and (2) include the additional terms. By attaching an extra term, the original lattice Boltzmann equation is modified here to

                                            (3)

where  is the distribution function;  the equilibrium distribution function;  the added extra term;  the local particle stream velocity;  the location of a computational grid node;  time increment;  the BGK relaxation parameter. Detailed information of Eq. (3) may refer to Ref. [7].

3.1  Multi-scale analysis

The corresponding macroscopic equations of mesoscopic Eq. (3) are to be investigated by multi-scale analysis as follows.

Applying the Chapman-Enskog expansion [7,10] to  leads to

                                                            (4)

where  is small parameter of the same order of the mean free path. Considering definitions [6,7]

  ,                                                       (5)

one obtains

                                                      (6)

The time may also be expanded to different scales as

                                                            (7)

Using Taylor expansion to the left side of Eq. (3) gives [10]

                                           (8)

Substituting Eqs. (4) (7) into (8) and letting , the equation to order  yields

                                                             (9)

and the equation to order  yields

                                                 (10)

Inserting Eq. (9) into (10) leads to

                                                (11)

Summing Eqs. (9) and (11) over and making use of Eqs. (5) and (6) give

                                                                  (12)

                                                               (13)

where

,                                                                (14)

Multiplying Eqs. (9) and (11) by then summing over  yield

                                                                 (15)

                                                     (16)

in which

, ,                                            (17)

The above  is similar to that in the original LB models [6-8]. It is

                                                              (18)

From Eqs. (17), (18), (12) and (15), one gets

                                                      (19)

Then, (12) pulsing (15), (13) pulsing (16) and back substituting  give

                                                      (20)

                                       (21)

in which pressure  and kinematic viscosity . Eqs. (20) and (21) are the corresponding macroscopic equations of Eq. (3).

3.2  LBM model

Giving

, ,                                                    (22)

and comparing Eqs. (1) and (2) with Eqs. (20) and (21), it can be noticed that if

                                                           (23)

                                                (24)

                                                                  (25)

the corresponding macroscopic equations of Eqs. (3) can approach Eqs. (1) and (2) with truncation error .

The often used model for 2-D simulation is the D2Q9 model (see Refs. [6,7]). Based on the layout of discrete particle streaming velocities of model D2Q9 and Eqs. (23), (24) and (25), we choose

                                                               (26)

,                                             (27)

,                                           (28)

The variables at should be used in the calculations of Eqs, (26), (27) and (28). Borrowing Eqs. (12) and (15), the partial derivative  can be replaced by , which may be approached by any second-order differencing schemes.

4  NUMERICAL VERIFICATIONS

For the convenience of analysis and comparison, 1-D incompressible flows are simulated with 2-D scheme here. The 1-D governing equations can be reduced from Eqs. (1) and (2) to

                                                           (29)

                                                    (30)

The flow field is of rectangular shape, with the inflow pressure  and outflow velocity  specified, and the two sides in symmetrical. The thickness (or height) changes linearly from inlet  to outlet  with .

 4.1  Steady flow

The analytical solution of steady flow in the above layout is

                                                                (31)

                                                  (32)

Several examples are shown in Tab.1 and the velocity and pressure curves of No.D are plotted in Fig. 2. Very good agreement between the computed inflow velocity 's and the analytical velocities can be seen. The outflow pressure 's slightly deviate from the analytical pressures. The main reason should be the viscosity term  of Eq. (2) is not included in the LBM model, because the curve of Analytical (2) in Fig. 2 (from Eq. (32) but discarding ) is closer to the computed pressure curve than that of Analytical (1) (directly from Eq. (32)). If viscosity of flow is small enough this error will be negligible.

         Table 1  Comparison of  LBM results with analytical solutions for steady 1-D incompressible flows

4.2  Transient flow

Given , , ,  and , use the method of characteristics (MOC) to simulate the linearized equations of Eqs. (29) and (30). Let the outflow velocity  varies linearly from 0 to 0.1 within 100 LBM time steps (about 57.73 MOC time steps) and then keeps 0.1 thereafter.

The pressure history in the outlet is plotted in Fig.3. The LBM result is in very good agreement with that of MOC.

5  SIMULATIONS OF HYDRAULIC TRANSIENTS IN A CONCRETE SPIRAL CASE

A reinforce-concrete spiral case with subtended angle  is sketched in Fig. 4. The rated head and rated discharge of per height . The head at reservoir is assumed constant and the flow condition of guide vane is treated like valve in the calculations. Wave speed . The wicket gate is closed linearly within  

Two cases are simulated. In Case1, the thickness of spiral case is constant. And in Case 2, the thickness varies linearly with angle from 1 at the spiral case inlet section A-B (see Fig.4) to 1/2 at the nose.

The velocity distributions and velocity vectors at special sections in the normal operating condition are shown in Fig. 5. The flow patterns of Case 1 and Case 2 are the same in the duct before the spiral case inlet section. Behind the inlet section, the velocity in Case 2 becomes larger than in Case 1 due to the reduction of thickness in Case 2. Because the walls are treated as non-slip boundaries and boundary function is not applied, the velocity profiles are similar to those of laminar flow. However, this has no apparent influence on pressure results. The pressure history at different sections (see Fig. 4) after load rejection is presented in Fig. 6. And the pressure distributions at  in the case of load rejection are shown in Fig. 7, which are somehow similar to those of 1-D waterhammer. The closer to the nose of the spiral case, the larger the pressure rise. Naturally, the pressure rises in Case 2 are larger than that in Case 1.

6  CONCLUSIONS

The basic equations and LBM scheme are proposed to simulate 2-D hydraulic transients in reinforced-concrete spiral case, in which the third dimension varies spatially. Practical numerical examples are presented. The preliminary study shows that LBM is feasible for simulating 2-D, as well as 3-D hydraulic transients.       

Acknowledgements

The authors are grateful to Prof. J. D. YANG for helpful discussions. This study is supported by the National Natural Science Foundation of China (Grant No. 50009007).

Fig.1  Sketch of the concerned volume of non-uniform thickness

Fig.2  Velocity and pressure distributions for steady 1-D incompressible flow

Fig.3  Pressure history at the outlet for unsteady 1-D incompressible flow

Fig.4  Sketch of the reinforced-concrete spiral case

Fig.5(a)  Velocity distributions for normal operating condition (Case 1)

Fig.5(b)   Velocity distributions for normal operating condition (Case 2)

Fig.6(a)  Pressure history after load rejection (Case 1) 

 Fig.6(b)  Pressure history after load rejection (Case 2)

 Fig.7(a)  Pressure distribution at  after load rejection (Case 1)

      Fig.7(b)  Pressure distribution at  after load rejection (Case 2)