Journal of Mechanical Science and Technology 29 (11) (2015) 4719~4727 www.springerlink.com/content/1738-494x(Print)/1976-3824(Online)

DOI 10.1007/s12206-015-1019-8

Effect of baffle number on mixed convection within a ventilated cavity† Mohamed Amine Belmiloud* and Nord-eddine Sad chemloul Department of Mechanical Engineering, Ibn Khaldoun University, Tiaret, 14000, Algeria (Manuscript Received Januay 20, 2015; Revised June 9, 2015; Accepted July 8, 2015) ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------

Abstract Transverse mixed convection is studied numerically in a ventilated cavity with uniform heat flux on the bottom wall and the remaining walls are adiabatic. Our purpose was to see the influence of the baffle number on the variation of average Nusselt number. An air flow inlet in to the cavity through an opening in the lower part of the left vertical wall and another opening in the opposite wall (i.e., the bottom part BB configuration; the top part BT configuration). The number of baffles ranged between 1 and 4. The two-dimensional mathematical model includes a system of four equations with partial derivatives of continuity, momentum, and energy, solved by the finite volume method. Flow fields are studied by numerical simulations to the air with a Richardson number in the range 1 £ Ri £ 10 , for the Grashof number is fixed at Gr = 104 , aspect ratio of the cavity A = 2 , the height openings w = 0.1 and Prandtl number Pr = 0.71 . The results show that the variation of the average Nusselt number and the average temperature depend on the type of configuration (BB or BT), and of the baffle number. The basic nature of the resulting interaction between the forced external air stream and the buoyancydriven flow by the heat source is explained by the Nusselt number and the patterns of the streamlines and isotherms. Keywords: Adiabatic partition; Mixed convection; Numerical study; Ventilated cavity ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------

1. Introduction Mixed convection heat transfer in ventilated systems continues to be a fertile area of research, due to the interest of the phenomenon in many technological processes, such as the design of solar collectors, thermal design of buildings, air conditioning, and recently the cooling of electronic circuit boards. In ventilated enclosures, the interaction between the external forced stream and the buoyancy driven flow induced by buoyancy forces can lead to complex flow structures. In the literature, numerous analytical, numerical and experimental studies dealing with mixed convection in ventilated geometries have been reported without radiation effect. The effect of the latter can be neglected for configurations with nonemissive or weakly emissive boundaries, which is not the case in general since the contribution of radiation to the overall heat transfer can be significant. Papanicolaou and Jaluria [1] numerically analyzed mixed convection turbulence from a remote heat source in a square cavity with openings on opposite sides; the results show that the turbulence developed around the right side of the vertical wall cavity. In a numerical study of the mixed convection in a square enclosure with a partition divided partially done by Hsu et al. [2], the results show that the values of average Nusselt number increase more *

Corresponding author. Tel.: +213 7 90 09 49 88 E-mail address: [email protected] † Recommended by Associate Editor Seongwon Kang © KSME & Springer 2015

and more ( Re / Ri ) to a given value of ( Ri / Re ) . Raji and Hasnaoui [3, 4] studied laminar mixed convection in a rectangular ventilated cavity where a vertical wall is subjected to a constant heat flux and the other walls are kept adiabatic. The results showed that the average temperature decreases as the Reynolds number ( Re ) increases, and the average temperature increases when the Rayleigh number ( Ra ) increments. A numerical study of the transient convection in a square enclosure with a partition divided partially was done by How and Hsu [5]. Their results show that the transient heat transfer and the flow structure are strongly dependent on the height and location of the baffle and, in general, are higher values of Reynolds number ( Re ) or lower values of Richardson number ( Ri ) , delaying the implementation of solutions to the state of balance. Hsu and Wang [6] studied mixed convection in a rectangular enclosure. The two-dimensional discrete heat sources were placed on a vertical board located on the bottom wall of the enclosure. The results show that when the heat source is integrated on the right side of the board, the Nusselt number is independent of the variation of the position of the heat source and the plate. When the heat sources are placed to the left of the board, the two numbers strongly depend on the variation of the positions of the two elements. Manca et al. [7] studied the effect of positron heated on the mixed convection in a channel with an open cavity (U-shaped) wall; the results of this study show that the maximum temperature decreases as the Reynolds number and Richardson increase. A study of the

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mixed convection in a shallow housing with a series of components of the heat generation was done by Bhoite et al. [8]. The results show that the velocity induced by free convection at any value of the Grashof number increases first until the Reynolds number reaches a moderate value. A numerical simulation of flows in 2D 3D natural and mixed convection in isothermal closed and ventilated cavities was made by Ezzouhri et al. [9]. Saha et al. [10] performed a numerical analysis of mixed convection in a rectangular cavity. The results show that the positioning of inputs and outputs is a significant variation in the average Nusselt number. Bahlaouia et al. [11], studied digital mixed convection combined with radiation surfaces in a rectangular partitioned cavity. The results of the study show that the effect of radiation leads to a better homogenization of the temperature within the cavity by reducing the space of the cold zone to the inlet zone.

L

g e

H

h Air

w

ui ; TC q

(a) L w

g e

H

2. Problem formulation The configuration under study, together with the system of coordinates, is depicted in Fig. 1. It consists of a ventilated rectangular cavity having an aspect ratio A = 2 . The bottom wall is uniformly heated with a constant heat flux and provided with a vertical adiabatic baffle of width e = 0.025 ; the remains are adiabatic wall. The inflow opening located on the left vertical wall and the outflow opening on the opposite wall are arranged as shown in the schematic figures and may vary in location, either top or bottom position. The cavity presented in Fig. 1(a) is subjected to an external flow which enters via the bottom of the insulated vertical wall and leaves via the bottom of the opposite vertical wall. For reasons of brevity, this case will be referred to as the BB configuration. When the horizontal cold jet enters the enclosure from the bottom of its insulated wall and leaves from the top of the other vertical one, Fig. 1(b), this case will be referred as the BT configuration. The number of baffles ranges between 1 and 4. 2.1 Governing equations The fluid properties were evaluated at a mean temperature and the fluid Newtonian Air flow was assumed to be two dimensional, laminar, incompressible and obeying the Boussinesq approximation. Under these assumptions, the dimensionless governing equations, written in terms of vortices and stream function formulation, are as follows: ¶u + ¶v + + =0 ¶x + ¶y +

(1)

u+

¶Ω + ¶Ω + 1 æ ¶2Ω + ¶2Ω + ç + v+ + = + 2 + Re çè ¶x + 2 ¶x ¶y ¶y +

ö ¶T + ÷ + Ri + , ÷ ¶x ø

u+

¶T + ¶T + 1 æ ¶ 2T + ¶ 2T + ö + v+ + = + ç ÷. + Re Pr çè ¶x + 2 ¶y + 2 ÷ø ¶x ¶y

h Air ui ; TC q

(b) Fig. 1. Schematic of the studied configuration: (a) BB; (b) BT.

locity components by the following expressions:

u+ =

¶v + ¶u + ¶Ψ + + ¶Ψ + ; v = + and W + = + - + + ¶x ¶y ¶x ¶y

(4)

where u + and v + are the velocity components in the x + and y + directions, respectively, T + is the dimensionless temperature and P + is the dimensionless pressure. The governing parameters, namely, the Grashof number Gr , Reynolds number Re , Richardson number Ri and Prandtl number Pr are defined in the following forms: u + , v+ =

λ (T - TC ) u,v + + x,y ,x ,y = , T+ = , ui H qH

P+ =

P r2 g b q H 4 , Gr = , 2 r ui lm 2

Re =

Gr r ui H m Cp , Ri = 2 , Pr = . Re m l

(5)

2.2 Boundary condition (2)

The boundary conditions associated with the problem are as follows:

(3)

The stream function and the vortices are related to the ve-

At the inlet: u + = 1 ; v + = 0 ; T + = 0 . At the lower horizontal hot wall: - ¶T + ¶y + = 1 . At the adiabatic wall: - ¶T + ¶n + = 0 .

(6)

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n: being the normal direction to the considered adiabatic wall.

Table 1. Effect of Re = 100 and Ri = 10 on the mean convective Nusselt number and maximal temperature evaluated on the heating wall.

2.3 Heat transfer

Present work

To determine the heat transfer characteristics in the steady state and mixed convection, in this study, the number of total average Nusselt is presented as Nu =

qH . l (T - TC )

(7)

2.235

2.276

+ Tmax

0.206

0.207

Table 2. Grid effect on the obtained results for Gr = 104 , Hb = 1 , BT configuration and Ri = 1 . Grid

The average Nusselt numbers:

Saha et al. [10]

Nuav

Mesh (81X41)

Mesh (101X51)

Mesh (121X61)

Nuav

1.5394

1.5392

1.5285

Tav+

0.5211

0.5212

0.5288

A

Nuav =

1 1 ¶T + - + × dx + . ò A 0 T ¶y +

(8)

3. Method of solution The mass, momentum, and energy equations were solved by a finite difference algorithm, the Semi-implicit method for pressure linked equations (SIMPLE). Details of this method are described by Patankar. [12] The differential equations are discretized over a control volume. The Power law difference scheme (PLDS) was employed for the calculation of scalar variables and the Quadratic upstream-weighted interpolation for convective kinematics (QUICK) scheme for Hayase et al. [13] vector variables. The main procedure of the numerical simulation in this study is as follows. - Initial values such as velocities, pressure, and temperature are assumed. - Solve the energy equation, Eq. (3), for the temperature field. - The boundary equation, Eq. (6), is solved for updating the temperature distribution on the insulated walls. - Eqs. (1) and (2) are solved to obtain the velocity field according to the SIMPLE algorithm [12]. - An entire new iteration is conducted by repeating steps 24. This procedure is repeated until the convergence criterion is satisfied. The numerical model was validated with the reference solution Saha et al. [10] compared to the case of a ventilated cavity. The comparison of the results shows excellent agreement. The differences in terms of average Nusselt number Nuav and + are, maximum values of dimensionless temperature Tmax respectively, 1.82% and 0.39% (Table 1). The results are given in the form of streamline, isotherm contour, and Nusselt numbers of a hot wall. The properties of air are mainly set at the reference temperature TC = 298.15 K . Richardson number ranged from 1 to 10 for keeping the flow regimes under the laminar boundary layer flow, with a uniform grid of (81x41) for both x and y directions; grid was selected as a reasonable compromise between computational effort and required accuracy. The results obtained for one

baffle at ( Ri = 1) , with this grid were comparable to those obtained with grid of (101x51) and (121x61). In fact, the refinement of the grid to (121x61) generates maximum variations lower than 0.71% and 1.46% in terms of Nuav and Tav+ , respectively (Table 2); it is clearly seen that there is little difference between the three results and a grid of (81x41) was used in all subsequent calculations. The convergence criterion is defined as fn - fn -1 max fn

< 10-5 .

(9)

Here, f represents any dependent variable, the subscript n represents the new results from the present iteration, and n - 1 represents that from the prior iteration.

4. Results and discussion The governing physical parameters in the problem considered were Gr , Ri , and geometrical conditions of the heat source, the outflow opening and the baffle. All these values were varied over with ranges to study the effects on the thermal transport and fluid flow phenomena. Some geometrical relationships, shown in Fig. 1, are specified as follows: A = 2 and B = 0.5 . The inflow opening was fixed at w = 0.1. The value of temperature of external imposed flow was maintained constant at TC = 298.15 K . The value of Grashof number was fixed at Gr = 10 4 , retained in this study, Richardson numbers varied between 1 and 10. In the following, effects of Richardson number, relative position of outflow configuration (BB, BT) and baffle number, 1 £ Hb £ 4 on fluid flow and heat transfer characteristics are illustrated. The numerical results were performed for air with a Pr = 0.71 . 4.1 Streamline and isotherm distribution The results of the study were presented for values of the Richardson number, varying in the range (0.1, 16). This range

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+ + (a) Ψ min = 0.0072; Ψ max = 0.1438

+ + (b) Ψ min = 0.0064; Ψ max = 0.1296

+ + (c) Ψ min = 0.0060; Ψ max = 0.1205

+ + (d) Ψ min = 0.0060; Ψ max = 0.1212

Streamlines

(a')

(b')

(c')

(d') Isothermal lines

Fig. 2. Streamlines and isotherms obtained for Ri = 1 , Gr = 10

4

and BB configuration: (a) Hb = 1 ; (b) Hb = 2 ; (c) Hb = 3 ; (d) Hb = 4 .

of Richardson was selected on the calculation by Roy et al. [14], which covers mixed convection; both the natural and forced convection are dominant. For low values of Richardson ( Ri < 0.1) , forced convection is dominant. For Richardson ( Ri > 16 ) , natural convection is dominant. In fact, the boundary conditions imposed at the entrance of the cavity are not compatible with the physical nature of the problem; a closed cell is formed on the right of the baffle and prevents the free flow of imposed flow characterized by open lines. Figs. 2 and 3 show the influence of the baffle number on the velocity contours and isothermal lines for Ri = 1 , Gr = 104 and for the two sections of the outflow opening of the cavity (BB and BT configurations). Velocity contours show the presence of a vortex just at the entrance of the cavity and another vortex just after each baffle. The vortex prevents circulation of the imposed flow characterized by open streamlines. The effect of the vortex decreases with decreasing their size caused by the increase in the num-

ber of baffles. It is observed for BB configuration that the direction of gravity and the position of the fluid outlet opening have the same direction, and this means that the presence or absence of gravity does not affect the velocity contour. For BT configuration, note that gravity has an influence in the case of the one baffle and is weak when ( Hb > 1) . The isothermal lines are clamped at the hot wall in the case of BB configuration; this shows that there is good heat exchange by convection between the wall and the fluid. For BT configuration, the results are in good agreement with those found by Bahlaouia et al. [11] in the case of a single baffle. When the number of baffles is greater than one ( Hb > 1) , the results obtained are similar to those of the BB configuration. However, the heat exchange between the wall and the fluid for the BT configuration is low compared to that of BB configuration. This reduction in the heat exchange in the case of BT configuration can be due to the air transit time within the cav-

M. A. Belmiloud and N. Sad chemloul / Journal of Mechanical Science and Technology 29 (11) (2015) 4719~4727

+ + (a) Ψ min = 0.0068; Ψ max = 0.13667

+ + (b) Ψ min = 0.0064; Ψ max = 0.1276

+ + (c) Ψ min = 0.0061; Ψ max = 0.1224

+ + (d) Ψ min = 0.0059; Ψ max = 0.119

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Streamlines

(a')

(b')

(c')

(d') Isothermal lines

Fig. 3. Streamlines and isotherms obtained for Ri = 1 , Gr = 10

4

and BT configuration: (a) Hb = 1 ; (b) Hb = 2 ; (c) Hb = 3 ; (d) Hb = 4.

ity which depends on the number of baffles. 4.2 Variations of heat transfer parameters

For the BB configuration (Fig. 4(b)) and for a Richardson number such that 2 £ Ri £ 7 the maximum value the average Nusselt number Nuav is obtained for almost an odd number of baffles Hb = 3 .

Fig. 4 shows the variation of the average Nusselt number. Nuav as a function of the Richardson number Ri , for the

various baffle numbers Hb used. Note that all results are obtained at the lower hot wall. We note that the average Nusselt number Nuav decreases when the Richardson number increases Ri ; this shows that the convective heat transfer increases when the flow rate increases. For the BT configuration (Fig. 4(a)), and when the number of baffles is greater than one ( Hb > 1) , the maximum value of the average Nusselt number Nuav is obtained for a number of baffles equal to two ( Hb = 2 ) . The maximum relative difference between the values corresponding to Nusselt Hb = 2 and Hb = 4 is 1.4%, for cons; it is more important between Hb = 3 and Hb = 4 either 5.2%. This shows that the even number of baffles allows a good heat transfer relative to the odd number.

4.3 Average dimensionless temperature + Fig. 5 shows the variation of the average temperature Tav as a function of the Richardson number Ri , for the various baffle numbers Hb are used. For BT configuration (Fig. 5(a)), + we find that the average temperature Tav increases with increase in Richardson Ri; this temperature is maximum for Hb = 3 . The curves of Hb = 2 and Hb = 4 are combined to Ri = 4 , beyond they deviate, the relative maximum value of this difference is 3.18%. By cons, this difference is important between Hb = 3 and Hb = 4 , and its relative maximum is 19.49%. When the Richardson number Ri ³ 5 , the two curves corresponding to Hb = 1 and Hb = 4 are almost coincident, and the relative maximum value is 0.82%.

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M. A. Belmiloud and N. Sad chemloul / Journal of Mechanical Science and Technology 29 (11) (2015) 4719~4727 1.7

1.8 Hb=1 Hb=2 Hb=3 Hb=4

1.6

1.6 Nuav

Nuav

1.5

Hb=1 Hb=2 Hb=3 Hb=4

1.7

1.4

1.5 1.4

1.3

1.3 1.2

1.2 1

2

3

4

5

Ri

6

7

8

9

1

10

2

3

4

5

(a)

Ri

6

7

8

9

10

(b)

Fig. 4. Average Nusselt number Nuav for several of baffle number Hb with the several of Ri : (a) BT configuration and; (b) BB configuration. 0.8

0.7

0.7

0.6

T+ av

T+ av

0.6 Hb = 1 Hb = 2 Hb = 3 Hb = 4

0.5

Hb=1 Hb=2 Hb=3 Hb=4

0.5

0.4

0.4 1

2

3

4

5

Ri

6

7

8

9

10

(a) Fig. 5. Average temperature

+ Tav

0.3

1

2

3

4

5

Ri

6

7

8

9

10

(b)

for several of baffle number Hb with the several of Ri : (a) BT configuration; and (b) BB configuration.

In the case of BB configuration, we note that when the number of upper the number of baffles to a ( Hb > 1) , the three + curves of the variation of the average temperature Tav are + combined. The average temperature Tav increases with decrease in the number of baffles Hb , for a Richardson number Ri in the interval of 1 to 6. When Ri > 6 , the change in average temperature increases with increasing the number of baffles Hb . 4.4 Variation of the dimensionless velocity Figs. 6 and 7 show the influence of the number of baffles on the variation of the dimensionless velocity along the x axis and, respectively, for the BT and BB configurations for different values of the Richardson number Ri . Fig. 6 shows that at the entrance of the cavity, the vertical velocity component decreases as Ri increases; the result is similar to Ref. [10]. At a position x + = 0.6 before the chi-

cane, increases with Ri . Between the baffle and the position x + = 1.6 , Ri decreases, this to the vortex that starts just after the baffle. Beyond this position and to the exit of the cavity, increases with Ri, passes through a maximum and then tends towards zero. This result is similar to that of Ref. [10]. The results obtained in the case of BT configuration remain valid for the case of BB configuration. The remarks made for the cavity at a chicane are valid for the two chicanes cavity (Figs. 6(b) and 7(b)), for both BT and BB configurations. For cavities in three and four baffles, the variation has the same appearance. Two cases: 1st case: between the input and the first chicane, increases when Ri increases, reaches a maximum and then decreases with increasing Ri . Whereas, in the inlet and outlet pair of the baffles as in the case of an odd number of baffles, the result of this first case is valid for a cavity having an odd number of baffles. 2nd case: between the first and second baffle decreases as

M. A. Belmiloud and N. Sad chemloul / Journal of Mechanical Science and Technology 29 (11) (2015) 4719~4727

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0.6

0.2

v+

0.2

v+

Ri =1 Ri =5 R i = 10

0.4

Ri =1 Ri =5 R i = 10

0.4

0.0

0.0

-0.2 -0.2 -0.4 -0.4 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

x+

(a)

(b) 0.4

0.4 Ri =1 Ri =5 R i = 10

Ri =1 Ri =5 R i = 10

0.2

v+

v+

0.2

1.8 2.0

x+

0.0

0.0

-0.2

-0.2

-0.4 0.0 0.2

-0.4 0.4 0.6

0.8 1.0 1.2

1.4 1.6

1.8 2.0

0.0 0.2

0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

x+

x+

(c)

(d)

Fig. 6. Influence of the number of baffle on the variation of the velocity for the BT configurations and for different values of the Richardson number Ri : (a) Hb = 1 ; (b) Hb = 2 ; (c) Hb = 3 ; (d) Hb = 4 . 0.4

Ri =1 Ri =5 R i = 10

0.4

Ri =1 Ri =5 R i = 10 0.2

0.0 v+

v+

0.2

0.0

- 0 .2 -0.2

-0.4 0.0 0.2

0.4 0.6 0.8 1.0

x+

0.0 0.2

1.2 1.4 1.6 1.8 2.0

0.4

0.6 0.8 1.0

(a)

1.8 2.0

(b)

0.4

0.4 Ri =1 Ri =5 R i = 10

Ri =1 Ri =5 R i = 10

0.2

v+

0.2

v+

1.2 1.4 1.6

x+

0.0

0.0

-0.2 -0.4

- 0 .4 0.0

0.2

0.4 0.6

0.8 1.0 1.2

x+

(c)

1.4 1.6 1.8

2.0

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

x+

(d)

Fig. 7. Influence of the number of baffle on the variation of the velocity for the BB configurations and for different values of the Richardson number Ri : (a) Hb = 1 ; (b) Hb = 2 ; (c) Hb = 3 ; (d) Hb = 4 .

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Ri is increased, not by a minimum and then increase as Ri increases. Considering that the entry is a pair chicane and the output as an odd chicane, the result of this second case remains valid for a cavity having an even number of baffles. At the input and the output v + is zero.

5. Conclusions We analyzed the effect of the baffle number on the mixed convection in a ventilated cavity. Two sections of the outlet of the cavity were considered. In view of the findings, we can draw the following conclusions: The two provisions of the outlet of the cavity (BB and BT configurations) have a significant influence on the number of average Nusselt whose maximum value is obtained for the BB configuration due to streamlines that are clamped to the hot wall. Gravity has no influence on heat transfer and whatever the number of baffles used. The influence of gravity is observed only for the BT configuration and particularly in the case of a single baffle. For a higher number of baffles, a ( Hb > 1) in the case of BT configuration, the maximum value of the average Nusselt number is obtained for a number of baffles equal to two ( Hb = 2 ) . The difference between the values of the Nusselt number Nuav obtained for Hb = 2 and Hb = 4 is very small, by this difference is significant between Hb = 2 and Hb = 3 . This shows that compared to an odd number of baffles, an even number allows good heat transfer. For The BB configuration and a Richardson number as 2 £ Ri £ 7 , the maximum value is obtained for almost an odd number of baffle Hb = 3 .

Nomenclature-----------------------------------------------------------------------A B e g Gr h H Hb L Nu Pr P P+ q Ri T+ TC ui u + , v+ w x+ , y +

: Aspect ratio of the cavity A = L H : Relative height of baffle B = h H : Width of baffle : Acceleration due to gravity : Grashof number : Baffle’s height : Height of the cavity : Baffle number : Length of the cavity : Nusselt number : Prandtl number : Pressure : Dimensionless pressure : Heat flux : Richardson number : Dimensionless fluid temperature : Reference temperature : Velocity of the imposed flow : Dimensionless horizontal and vertical velocities : Height of the openings : Dimensionless coordinates

Greek symbols a b l m W+ Y+

: Thermal diffusivity of fluid : Thermal expansion coefficient of fluid : Thermal conductivity of fluid : Dynamic viscosity : Dimensionless vortices : Dimensionless stream function

Subscripts av

: Average value

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Mohamed Amine Belmiloud received his M.S. in Mechanical Engineering (energy option) at the University Ibn Khaldoun, Tiaret, Algeria and he is currently a Ph.D. candidate there. His research interests include heat transfer and the mass, computational fluid dynamics (CFD).