application of nasir flow-solver for finite volume...

Application of NASIR1 Flow-Solver for Finite Volume Solution of

Wind Effects on a Set of Cooling Tower

S. R. Sabbagh Yazdi*, B. Haghighi** and M. Torbati***

Email: [email protected]

AbstractComputer simulation of wind flow around a set of three cooling towers in the KAZERUN power station, which are under construction in south part of IRAN is presented in this paper. A Numerical Analyzer for Scientific and Industrial Requirements (NASIR) is developed to investigate the wind effects on structural surfaces of these towers with particular arrangement. This software which solves incompressible Euler equations for steady flow on unstructured finite volumes facilitates considering special treatments required for practical problems, like open louvers on cooling towers footing. In this model, the equation of continuity can be simultaneously solved with the equations of motion in a coupled manner for the steady state problems by application of the pseudo compressibility technique. This technique helps coupling the pressure and the velocity fields during the explicit computation procedure of the incompressible flow problems and therefore speeds up the convergence of the solution. The discrete form of the three-dimensional flow equations are formulated using the Cell Vertex Finite Volume Method for unstructured mesh of tetrahedral cells. Using unstructured meshes provides great flexibility for modeling the flow in geometrically complex domains. The computed results are presented in terms of color coded maps of pressure and velocity fields as well as velocity vectors on boundary surfaces of the solution domain.

Key words: Wind Effect on Cooling Towers, Steady Pseudo Compressible Euler Equations, Tetrahedral Unstructured Mesh, Cells Vertex Finite Volume Method,

1 Numerical Analyzer for Solving Industrial and Scientific Requirements* Academic Staff , Civil Engineering Department, KNT University of Technology, Iran** Head of Structural Design Office of BOLAND-PAYEH Co.*** Research Assistant , Civil Engineering Department, KNT University of Technology, Iran

Proceedings of the 10th WSEAS International Confenrence on APPLIED MATHEMATICS, Dallas, Texas, USA, November 1-3, 2006 34

1- Introduction It is common to use standard formulations and graphical solution advised by design manuals for calculation of the wind pressure on cooling towers. However, the geometrical features of the proposed cooling towers and surrounding structures as well as ambient environments may considerably change he pressure filed. More over, wind flow through particular arrangement of cooling towers set may produce unexpected pressure fields, which may cause disastrous structural loading condition. Therefore, modeling of final design is recommended by most of the codes of practice to evaluate actual pressure loads on the cooling towers. The availability of high performance digital computers and development of efficient numerical models techniques have accelerated the use of Computational Fluid Dynamics. The control over properties and behavior of fluid flow and relative parameters are the advantages offered by CFD which make it suitable for the simulation of the applied problems. Consequently, the computer simulation of complicated flow cases has become one of the challenging areas of the research works. In this respect, many attempts have been made to develop several efficient and accurate numerical methods suitable for the complex solution domain. The assumption of incompressibility is valid for common civil and environmental engineering problems. For the most civil engineering flow problems, the boundary layer is confined to thin regions close to the solid surfaces. Since these regions are negligible comparing to the main domain of interest, the effect of viscous stresses can be omitted in the equations of the motion. The resulted set of equations, which is known as the incompressible form of the Euler equations, provides considerable simplicity in the absence of second order spatial

derivative terms. This simplification of the governing equations provides the ease of the solution procedures, and consequently, saves the computational efforts.In present work, the Cell Vertex Finite Volume Method is used to derive the discrete formulas of the governing equations on triangular unstructured meshes. The problem of growing up numerical errors, which usually disturbs the explicit solution of the formulations are overcome by adding artificial dissipation terms suitable for the unstructured meshes. These extra terms are used to damp out the unwanted errors and stabilize the numerical solution procedure while preserving the accuracy of the solution. In order to increase the computational efficiency, some numerical technique such as Runge-Kutta multi-stage time stepping, upwind flux averaging and face-base algorithm are applied in the frame work of a numerical analyzer for scientific and industrial requirements (NASIR). In this paper, the ability of the NASIR finite volume solver to simulate wind flow effects on a set of three cooling towers in KAZERUN power station (in south part of IRAN) is presented. The results are demonstrated using color coded maps of velocity and pressure fields.

2- Governing EquationsIn the with high Reynolds number flows, the boundary layer is usually limited to a thin layer close to solid walls. Therefore, the effects of viscosity are ignorable in the major part of the flow field. The assumption of inviscid behavior of the fluid flow is acceptable for the regions outside the boundary layer.The wind flow around ground structures subsonic flow problems is normally subsonic (Mach<0.3), and therefore, the air density is constant. In such cases, the fluid flow is considered incompressible and the changes


in temperature field are negligible. Hence, considering the isothermal condition for the flow problem, the equations of continuity and motions represent the mathematical model equations of the incompressible flow, which is known as the incompressible Euler equations.The conservative vector form of the governing equations in Cartesian coordinates can be written as:

0)( =∂

∂+∂

∂+∂

∂+∂∂

zG

yF

xE

tQ

Where

=

wvup

Q

2/ β

,

+

=

uwuv

puu

E oρ/2

,

+=

vwpv

vuv

Foρ/2 ,

+

=

opwwvwuw

G

ρ/2

Q represents the conserved variables while, E, F and G are the components of convective flux vector kGjFiEH ˆˆˆ ++= of Q in x and y directions, respectively. u, v and w the components of velocity, p pressure are four dependent variables by considering oρ as the constant density. The parameter β is introduced using the analogy to the speed of sound in equation of state of compressible flow. Application of this pseudo compressible transient term converts the elliptic system of incompressible flow equations into a set of hyperbolic type equations [1]. Ideally, the value of the pseudo compressibility is to be chosen so that the speed of the introduced waves approaches that of the incompressible flow. This, however, introduces a problem of contaminating the accuracy of the numerical algorithm, as well as affecting the stability property. On the other and, if the pseudo

compressibility parameter is chosen such that these waves travel too slowly, then the variation of the pressure field accompanying these waves is very slow. Therefore, a method of controlling the speed of pressure waves is a key to the success of this approach. The theory for the method of pseudo compressibility technique is presented in the literature [2]. Some algorithms have used constant value of pseudo compressibility parameter and some workers have developed sophisticated algorithms for solving mixed incompressible and compressible problems [3]. However, the value of the parameter may be considered as a function of local velocity using following formula proposed [4]

|)|( 22min

2 UCorMaximum ββ = In order to prevent numerical difficulties in the region of very small velocities (ie, in the vicinity of stagnation pints), the parameter

2minβ is considered in the range of 0.1 to 0.3,

and optimum C is suggested between 1 and 5 [5].The method of the pseudo compressibility can also be used to solve unsteady problems. For this propose, by considering additional transient term. Before advancing in time, the pressure must be iterated until a divergence free velocity field is obtained within a desired accuracy. The approach in solving a time-accurate problem has absorbed considerable attentions [6]. In present paper, the primary interest is in developing a method of obtaining steady-state solutions.

3- Numerical Method The governing equations can be changed to discrete form for the unstructured meshes by the application of Cell Vertex (overlapping) scheme of the Finite Volume Method. This method ends up with the following formulation [7]:


∑=

+ ∆+∆+∆Ω∆−=

facesN

k

nkzyx

i

ni

ni SGSFSEtQQ

1

1 ])()()([.

Where, Qi represents conserved variables at the center of control volume Ωi (Fig.1).

Here, E , F and G are the mean values of fluxes at the control volume boundary faces. Superscripts n and n+1 show nth and the n+1th computational steps. ∆t is the computational step (proportional to the minimum mesh spacing) applied between time stages n and n+1. In present study, a five-stage Runge-Kutta scheme is used for stabilizing the computational process by damping high frequency errors, which this in turn, relaxes CFL condition [8].The explicit solution of above formulation on the equally spaced grids presents the behavior of the central schemes, and hence, the absence of damping nature of physical viscosity near the high gradient regions may give rise to numerical oscillations. In order to damp unwanted numerical oscillations a fourth order (Bi-Harmonic) numerical dissipation term is added to the convective term, ∑ =

∆+∆+∆= sidesN

kknkzyxi SGSFSEQC

1])()()([)( [8].

The numerical dissipation term, )()( 22

1 ijN

k ki QQQD faces ∇−∇= ∑ =λε , is formed by using

the Laplacian operator, ∑ =−=∇ facessN

k iji QQQ1

2 )( . The Laplacian operator at every node i, is computed using the variables Q at two end nodes of all facesN faces (meeting node i).Here, kλ is the minimum of jΛ , the scaling factors of the edges associated with the end nodes j of the edge k. This formulation is adopted using the local maximum value of the spectral radii Jacobian matrix of the governing equations and the size of the mesh spacing as [8]:

kkk

N

kkj SnUnU

e

)ˆ..(ˆ.. 22

1∆++=Λ ∑

=

β

Where,kwjviuU ˆˆˆ ++=

, SkSjSiSn zyx ∆∆+∆+∆= /]ˆ)(ˆ)(ˆ)[(ˆ

and 222 )()()( zyx SSSS ∆+∆+∆=∆ .

Similar to the most numerical formulations, this formulation is somehow mesh-dependent. For obtaining the accurate results, the minimization of the coefficient ε is the key point in the application of the numerical dissipation term on the specific mesh ( 256/3256/1 ≤≤ ε ). The revised final algebraic formula can be written in the following form [8].

)]()([1ii

i

ni

ni QDQCtCFLQQ −

Ω∆−=+

The quantities Q at each node is modified at every time step by adding a residual term of

iiii QDQCtQR Ω−∆= /)]()([)( which is computed using the quantities Q at the nodes of boundary sides of the control volume Ωi

(Fig. 1). Hence, the edges are referred to all over the computation procedure. Therefore, it would be convenient to use the edge-base data structure for definition of unstructured meshes. It has been shown that using the face-base computational algorithm reduces the number of addressing to the memory, and therefore, provides considerable saving in computational CPU time [9].

4- Application of the Model The performance of the developed solver is examined by solving inviscid wind flow around a set of three cooling towers in the KAZERUN power station located in south part of IRAN (Fig. 3). In order to assess the effects of surrounding structures the flow solver is applied to solve the wind induced flow patterns on three meshes which geometrically represent; first single cooling tower without any ambient structure, second three cooling towers without any ambient structures and third three cooling tower with major ambient structures. In order to generate unstructured mesh of tetrahedral around the major objects in the wind flow field, geometry of the far-field flow boundaries as well as surfaces of ground and structures are digitally modeled in the first stage (Fig. 4). The general view of the


second and the third unstructured meshes colored with different types of boundary conditions are presented in Fig. 5. In this work, unit free stream velocity and pressure is imposed at inflow and outflow boundaries of computational meshes (in left and right sides of the site map, respectively). Free slip condition is considered at the solid wall nodes by setting zero normal components of computed velocities at wall nodes. The free stream flow parameters (Outflow pressure and inflow velocity) are set at every computational node as initial conditions. The results on the single cooling tower are plotted in terms of color coded maps of computed velocity and pressure field in Fig. 6. The effect of opening louvers and top of the tower is investigated on this test case (Fig.7). The computed results of the developed model show that there are major differences in computed velocity patterns and pressure field on the wall surface of the cooling tower (Fig. 8).For the case of three cooling towers without any ambient structures the computed color coded maps of velocity and pressure patterns are presented in Fig. 9. As can be seen the flow parameters on three towers are different due to neighboring effects. For the case of three cooling towers with major ambient structures the computed color coded maps of velocity and pressure fields are presented in Fig. 10. As can be seen, due to effects of the upstream structures considerable differences are appeared in the computed flow parameters on three cooling tower surfaces (Fig. 11) in comparison with the previous case.

5- DiscussionThe artificial compressibility technique is used to overcome the numerical problem associated with the coupled solution of the equations of continuity and motion for the incompressible inviscid flow problems. The

results of the cell vertex finite volume solution of chosen bench mark tests on unstructured meshes present encouraging results. Hence, adding a pressure time derivative term to the continuity equation successfully couples the pressure and velocity fields for speeding up the convergence behavior of the explicit solution procedure without any degradation in the accuracy of the results. Such an efficient algorithm for computation of both velocity and pressure fields on certain Cartesian unstructured grid facilitates three-dimensional numerical modeling of the incompressible inviscid flow problems. From the computed results, it can be stated that complicated physical conditions around a geometrically complex object can accurately modeled using the presented algorithm. The computed results of the developed model show that, there are major differences in computed velocity patterns and pressure fields on the wall surface of the cooling tower due to opening of louvers and top of the towers. Neighboring a number of cooling towers will result considerable changes in computed flow parameters. The upstream structures not only the patterns but also the minimum and maximum values of the flow parameters on three towers are different from the case without ambient structures. The computed results prove that the computer simulation can serve as a powerful means for investigation of the complex engineer problems. However, more complexities need development of powerful and capable computational model.

6- AcknowledgementThe finical support of this research work is provided by BOLAND-PAYEH Co.

7- References1- Chorin A. ,“A Numerical Method for Solving Incompressible Viscous Flow Problems”, Journal of Computational Physics, Vol. 2, (1967), 12-26.


2- Chang J.L and Kwak D., "On the Method of Pseudo Compressibility for Numerically Solving Incompressible Flow ", AIAA 84-0252, 22nd

Aerospace Science Meeting and Exhibition (1984) Reno. 3- Turkel E.,., Preconditioning Methods for Solving the Incompressible and Low Speed Compressible Equations", ICASE Report 86-14 , (1986)4- Dreyer. J., “Finite Volume Solution to the Steady Incompressible Euler Equation on Unstructured Triangular Meshes”, M.Sc. Thesis, MAE Dept., Princeton University, (1990).5- Rizzi A. and Eriksson L., "Computation of Inviscid Incompressible Flow with Rotation", Journal of Fluid mechanic, Vol. 153, (1985), 275-312.6- Belov A., Martinelli L. and Jameson A., "A New Implicit Algorithm with Multi-grid for Unsteady Incompressible Flow Calculations", AIAA 95-0049, 33rd Aerospace Science Meeting and Exhibition (1995) Reno.7- SabbaghYazdi R.S., Hadian A., “Accuracy Assessment of Solving Pseudo Compressible Euler Equations for Steady Subsonic Flow on Unstructured Finite Volumes", ANZIAM J. 46 (E) pp. 1222-1238, (2005).8- Jameson A., “A Vertex-Based Multi-Grid Algorithm for Three-Dimensional Compressible Flow Calculations”, ASME Symposium on Numerical Methods for Compressible Flow, Annahim , Dec.1986.11- Jameson A., Baker T.J. and Weatherill N. P. , “Calculation of Inviscid Transonic Flow over a Complete Aircraft”, AIAA Paper 86-0103, Jan. 1986.

8- Figures

Fig. 1: Ωi , Control volume of tetrahedral

Fig. 2: Velocity vector and unit normal at a C.V. face

501

502

503

50 4

505

506

50 7

508

509

560

Z =827.248

SE RVICE WATERELEVATED TANK

PARKING

U=908.929V=722.445

SE RVICE WATERELEVATED TANK

WAT

ER P

OOL

400 m

3

565

Fig. 3: Geometric features (left) and location layout (right) of cooling towers in the KAZERUN power station


Fig. 4 : Geometry modeling of boundary surfaces including cooling tower surfaces and surrounding structures

X

-400

-200

0

200

400

600

Y

0

100

200

300

400

Z

-400

-200

0

200

400

X

Y

Z

NASIR

X

-400

-200

0

200

400

600

Y

0

200

400

Z

-400

-200

0

200

400

X

Y

Z

NASIR

Fig. 5 : Unstructured mesh; without surrounding structures (left) and without surrounding structures (right)

XY Z

Q

1.51.41.31.21.110.90.80.70.60.50.40.30.20.1

3D FLOW SIMULATION

NASIR

WindDir

ection

XY Z

P

2.42.221.81.61.41.210.80.60.40.20-0.2

3D FLOW SIMULATION

NASIR

WindDire

ction

Fig. 6: Color coded map of the computational results without surrounding structures (closed louvers):Velocity vectors (left) and pressure field on ground surface (right) around a cooling


-100 0 100

Y X

Z

Q: 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5

3D FLOW SIMULATION

NASIR

Wind Direction

-100 0 100

Y X

Z

Q: 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5

Wind Direction

NASIR

Fig. 7 : Velocity vectors computed on an unstructured mesh (without surrounding structures):Closed louvers and top (left) and open louvers and top (right)

ZX

YP

2.42.221.81.61.41.210.80.60.40.20-0.2

3D FLOW SIMULATION

NASIR

WindDire

ction

Fig. 8 : Color coded map of pressure field computed on an unstructured mesh (without surrounding structures):Closed louvers and top (left) and open louvers and top (right)


X

Y

Z

Q

1.41.31.21.110.90.80.70.60.50.40.30.20.1

3D FLOW SIMULATION

NASIR

Wind Direction

X

Y

Z

P

21.81.61.41.210.80.60.40.2

3D FLOW SIMULATION

NASIR

Wind Direction

Fig. 9 : Color coded map of the computational results (without surrounding structures); Velocity field ((left) and pressure field (right) on surface mesh

0

200

X

Y

Z

Q

1.31.21.110.90.80.70.60.50.40.30.20.1

3D FLOW SIMULATION

NASIR

0

200

X

Y

Z

P

1.81.71.61.51.41.31.21.110.90.80.70.60.50.40.3

3D FLOW SIMULATION

NASIR

Fig. 10 : Color coded map of the computational results plotted on mesh (with surrounding structures); Velocity field ((left) and pressure field (right) on surface mesh

00

0

00

00

Y

X

Z

Q

1.31.21.110.90.80.70.60.50.40.30.20.1

3D FLOW SIMULATION

NASIR 00

0

00

00

Y

X

Z

P

1.81.71.61.51.41.31.21.110.90.80.70.60.50.40.3

3D FLOW SIMULATION

NASIR

Fig. 10 : Color coded map of the results plotted on ground and structure surfaces (with surrounding structures);


Velocity field ((left) and pressure field (right) on surface mesh


application of nasir flow-solver for finite volume...

Documents