Orthogonal H-type and C-type grid

generation for 2-d twin deck bridge

Xiaobing Liu, Chaoqun Liu, Zhengqing Chen

Mathematic Department, University of Texas at Arlington ， USAWind engineering research center ， Hunan University, China

7TH AIMS CONFERENCE

Main Content

Background

Introduction to grid generation method

Result

Conclusion

BackgroundIn recent years, as traffic amount increasing, decks of bridges are becoming wider and wider. A new kind of bridge, twin deck bridges were built all over the world because of good visual effect and traffic condition.

Fred Hartman bridge (Texas, USA)

Tacoma bridge (Washington, USA)

Qing dao Bay Bridge, Shan dong Province, China

Ping sheng Bridge, Guang dong Province, China

Since two decks of twin deck bridge are very closed to each other, in strong wind, the flow around decks will make wind load, vortex shedding and flutter stability of decks different from those of common single deck. We call this aerodynamic interference effect of twin deck bridge.

The aerodynamic interference effect of twin deck bridge is influenced by many factors, such as geometric shape of deck cross section 、 distance of two decks 、 wind attack angle and so on.

It is more feasible to choose numerical simulation

It will cost lots of money and energy to do wind tunnel test

Basic idea of numerical simulation ：

Discretisise the governing equations of fluid ( partial differential equations) into basic linear equations on numerical grid.

Solve the linear equations to get general variables of fluid, such as velocity, pressure and etc around bridge decks.

Numerical grid is the basis of numerical computing. The quality of numerical grid, such as smoothness and orthogonality plays a crucial role in affecting numerical computing result.

In the following sections, orthogonal H type and C type grids were generated for 2-d Qing dao Bay bridge and 2-d simplified twin deck bridge model using numerical method.

Introduction to grid generation method

The basic idea of grid generation method we used was mentioned firstly by S.P.Spekreijse in his paper “ Elliptic Grid Generation Based on Laplace Equations and Algebraic Transformation”. Here we will give a introduction to this method through following simple geometry.

algebraic transformation

elliptic

transformationcomputational

domain

parameter domain

physical domain

The grid generation mainly consists of following three steps.

The first step aims to generate a simple but not smooth algebraic grid in physical domain based on transfinite interpolation.

According to the grids on the boundary of the physical domain and grids in the computational domain, grid distribution in the interior of physical domain can be obtained based on following equations.

3 4 1 2

4 3 4 3

( , ) (1 ) ( ) ( ) (1 ) ( ) ( )

[ (1) (1 ) (1) (1 ) (0) (1 )(1 ) (0)]E E E E

E E E E

x x x x x

x x x x

3 4 1 2

4 3 4 3

( , ) (1 ) ( ) ( ) (1 ) ( ) ( )

[ (1) (1 ) (1) (1 ) (0) (1 )(1 ) (0)]E E E E

E E E E

y y y y y

y y y y

physical domain

The second step is to make the algebraic grid in physical domain to be smooth and stretched.

elliptic transformation from parameter domain to physical domain

~ ~ ~

2 0ss st ttx x x ~ ~ ~

2 0ss st tty y y

where,

~2 2t tx y ,

~

s t s tx x y y

,

~2 2s sx y

algebraic transformation from computational domain to parameter domain

1( )(1 ) 2( )t E s E s

3( )(1 ) 4( )s E t E t

where 1( )E 2( )E 3( )E and 4( )E

are normalized arc-lengths along four boundary edges.

Compositing these two transformations, we get

2x x x px qx 2y y y py qy

where 2 2x y

,

x x y y

,

2 2x y

( 2 ) ( 2 )s t t t t s s sp

s t t s

( 2 ) ( 2 )s t t t t s s sq

s t t s

By solving above equations using finite difference method, we can get smooth and stretched grid in physical domain .

parameter domain

physical domain

The third step is to make grid in physical domain to be orthogonal on boundaries.

In order to get orthogonal grid on boundary E3, we can first solve two Laplace equations and with following boundary condition to get new s and t.

(0, ) 0s

(1, ) 1s ( ,0) 0t

( ,1) 1t

0

s

n

0xx yys s 0xx yyt t

Where n is the outward normal direction at boundary

E3 and E4

0t

n

Where n is the outward normal direction at boundary

E1 and E2

After that, we re-compute and based on boundary function of above s and t using following Hermite interpolation.

s t

3 0 4 1( ) ( ) ( ) ( )E Es s H t s H t

1 0 2 1( ) ( ) ( ) ( )E Et t H s t H s

where

3( )= ( ,0)Es s 4 ( )= ( ,1)Es s

1( )= (0, )Et t

2 ( )= (1, )Et t

20 ( )=(1 2 )(1 )H s s s

21( )=(3 2 )H s s s

20 ( )=(1 2 )(1 )H t t t

21( )=(3 2 )H t t t

At last we go back to the second step to get new x and y.

parameter domain

physical domain

result

complete grid detail of grid on deck surface

H-type grid for Qing Dao Bay bridge

complete grid detail of grid on deck surface

C-type grid for Qing Dao Bay bridge

H-type grid for twin deck bridge model

complete grid detail of grid on deck surface

C-type grid for twin deck bridge model

complete grid detail of grid on deck surface

Conclusion

There exists aerodynamic interference effect between two decks of twin deck bridge. Both C-type and H-type grids have been generated for a 2-d twin deck bridge and 2-d simplified twin deck bridge model using numerical method. The grids are smooth and orthogonal on deck surface and computational boundary. This provides an effective guarantee for studying on aerodynamic interference effect of twin deck bridge using numerical method.

Thank you !