The Governing Equations

1 / 1

The Governing Equations - PowerPoint PPT Presentation

time step update. Finite difference form of x momentum equation is arranged to the following form: Coefficients of {F1X k , F2X k , GX k } are obtained form calculation. Finite difference form of y momentum equation is arranged to the following form:

I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.

PowerPoint Slideshow about ' The Governing Equations' - ghalib

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.

- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript

time step update

Finite difference form of x momentum equation is arranged to the following form:

Coefficients of {F1Xk, F2Xk, GXk} are obtained form calculation.

Finite difference form of y momentum equation is arranged to the following form:

Coefficients of {F1Yk, F2Yk, GYk} are obtained from calculation.

1

2

high tide

high tide

(1) and (2) are substituted into the depth averaged continuity equation resulting in system of linear equations with unknown variable of n+1, which is solved by using SOR (Successive Over Relaxation) method.

minimum depth

mwl

mwl

3

low tide

low tide

moving boundary

Using water level n+1 calculated from (3), equations (1) and (2) calculate ukn+1and vkn+1respectively.Then, vertical velocity wkn+1is calculated from the mass conservation equation.

fix boundary

4

b) without wetting-drying process

a) with wetting-drying process

0.25 m

5 m

wLSX,i,j

k=LSX

uLSX,i,j

uLSX,i+1,j

CLSX,i,j

20 km

i,j+1

wLSX+1,i,j

SHXk,i,j = depth of flux faces at i-direction

k=LSX +1

CLSX+1,i,j

uLSX+1,i,j

uLSX+1,i+1,j

vi,j+1

i-1,j

i,j, Ci,j

i+1,j

wLEX-1,i,j

ui,j

ui+1,j

k=LEX -1

CLEX+1,i,j

uLEX-1,i,j

uLEX-1,i+1,j

wLEX,i,j

vi,j

uLEX,i+1,j

k=LEX

CLEX,i,j

Lokasi Studi

uLEX,i,j

1.0 m

i,j-1

6 m

18 km

DEVELOPMENT OF A HYDRODYNAMIC NUMERICAL MODEL

WITH WETTING-DRYING CAPABILITY

Purwanto Bekti Santoso(1), Nastain(2), Wahyu Widiyanto(3)

(1)Department of Civil Engineering, Universitas Jenderal Soedirman, Purwokerto, Indonesia, e-mail: [email protected]

(2)Department of Civil Engineering, Universitas Jenderal Soedirman, Purwokerto, Indonesia, e-mail: [email protected]

(3)Department of Civil Engineering, Universitas Jenderal Soedirman, Purwokerto, Indonesia, e-mail: [email protected]

Introduction

Change of flow pattern due to human intervention cause water quality degradation. The change can create stagnation regions that have limitation on oxygen supply. Representation of physical condition, i.e. the flow pattern, is necessary before any ecological model can be used. In many three-dimensional flow calculation, wetting-drying process hasn’t been included in the calculation.

Calculation of flow with wetting-drying scheme is necessary as part of conservation plan at tidally influenced area, coastal wetland for example. Coastal wetland forms a unique environment that supports nearby waters. The numerical hydrodynamic model with wetting drying scheme will be suitably applied to hydraulic studies in Segara-Anakan because of its shallow bathymetric condition due to sedimentation.

The Numerical Model

This model basically follows the procedure introduced by Sato et al. (1993) with addition of baroclinic term to take into account the effect of density gradient. Similar procedure is also found in Casulli and Cheng (1992). The difference between the two comes from different approaches that are used for solution of water level coupled tridiagonal linear system resulted from semi-implicit dicretization.

Steps of research

Step 1.

Implementation of the wetting-drying scheme on three-dimensional hydrodynamic model.

Coding of Casulli dan Cheng (1992) wetting drying scheme into Fortran 77 code considering the structure of available three dimensional hydrodynamic code.

Step2.

Testing the resulted code by comparison with analytical results

Grid spacing 250 m, Time step 10 s, Tidal period 12 hours, Simulation time 4 tidal period, Friction factor 0.001

a) Linear case 1

b) Linear case 2

a) Horizontal Grid

Fig. 5. Linear cases to be compared with analytical solution

• Step3.
• Testing of hypothetical case.
• Impact on stability due to implementation of wetting drying scheme
• Sensitivity of friction factorto wetting-drying process
• Sensitivity of grid size to wetting-drying process
• There are two cases.
• Linear cases (1 and 2)

Grid spacing 250 m, time step 10 s, tidal period 12 hours, simulation time 4 tidal period, friction coefficient 0.001

Fig. 1. Land-ocean interface of hydrodynamics model a) with and b) without wetting-drying process

b) Vertical Grid

The Governing Equations

The hydrodynamic model adopted here is the one based on the hydrostatic pressure approximation and the boussinesq approximation, and fixed layer divisions in vertical discretization. A set of governing equations is required to compute four unknowns, which are three velocity components and the water level.

Fig. 2. Variables arrangement of a) horizontal grid b) vertical grid

Step4.

Description of Segara Anakan

Bathymetric map, Tidal data, Water level, Current, Wind, River discharge(Citanduy and Cibeureum)

Horizontal Momentum Equations (calculation of horizontal flow velocity)

Fig. 3. Step of calculation

Treatment of the wetting-drying process

Depth averaged continuity equation with kinematics boundary conditions ( calculation of water level)

Fig. 7. Segara anakan

x-direction :

Step 5.

Application of model to real condition (Segara Anakan).

Water level

Simulation will be conductedto calibrate and verify the model by comparison with field data of Segara Anakan

Continuity equation (calculation of vertical flow velocity)

• Verification and calibration will be carried by using data of:
• Water level (2 locations)
• Current (2 locations)

Targets

A hydrodynamics numerical model with wetting-drying capability

Recommendation on friction coefficient and grid size

List of symbols

(u,v,w): flow velocity in direction of (x,y,z)

: free surface elevation

g: gravitation

f: coriolis parameter

0 : constant reference density

’(x,y,z,t): local variation to reference density

h and v : horizontal and vertical eddy viscosity

H: still water level depth

h: water depth

C: concentrations of transported substances/temperature

LSX: vertical index of top layer

LEX: vertical index of bottom layer

Bottom

Fig. 4. Definition of water level in wetting-drying formulation

Literature cited

Casulli, V., and Cheng, R. T., 1992. Semi-Implicit Finite Difference Methods for Three-Dimensional Shallow Water Flow, Int. Jour. for Numerical Methods in Fluids, 15, 629-648.

Sato, K., Matsuoka, M., and Kazumitsu, K., 1993. Efficient calculation method of 3-D tidal current. Proceedings of Coastal Engineering, JSCE, Vol.40, 221-225.(in Japanese)

If H zero friction factor is assumed to be infinite, then the momentum equations become: ui+1,jn+1= 0vi,j+1n+1= 0

If H zero, dry conditionIf H positive, wet condition