*2.5.2. Results*

As far as the accuracy of the four numerical schemes in the case of inclined inflow (θ = 45°) is concerned the higher-order and non-linear schemes (hybrid and van Leer numerical scheme) present a similar performance with the first-order upwind numerical scheme and a different

velocity distribution at the longitudinal plane of the domain and (a) 0.2 m from the inlet; (b) 0.4 m

bution predicted by the SUPER scheme presents a more abrupt and accurate profile due to the

A physical problem that has been used to evaluate the performance of the numerical accuracy is the water-vapor condensation of humid air in the three dimensional geometry of a real-scale indoor space. A two-phase flow Euler-Euler mathematical model has been developed, wherein the humid air and water droplets are being treated as separate phases. The two phases exchange momentum and energy and, as the temperature drops below the dew point of humid air, mass transfer and phase change of water vapor to liquid takes place. The flow of humid air inside the room is buoyancy driven in the temperature range of 290–303 K. The properties of humid air (enthalpy, relative humidity, concentration of water vapor, saturation vapor pressure) vary with the temperature [22]. The dimensions of the domain are: width (X)

The interior of the domain is filled with humid air of 303 K and no liquid phase. The temperature on the surface of the walls is 303 K and on the surface of the cold bottom is 290 K. The

and specific heat (4190.0 J/kg ⋅ K) at the dew point temperature. The initial pressure inside the

and the water droplets have a constant density (996 kg/m<sup>3</sup>

velocity distri-

)

performance than the flow-oriented scheme (SUPER scheme). The vertical w<sup>1</sup>

successful minimizing of the false-diffusion errors.

× height (Y) × length (Z) = 4.0 × 4.0 × 8.0 m.

*2.5.1. Boundary and initially conditions*

density of humid air is 1.16 kg/m<sup>3</sup>

**2.5. Heat and mass transfer**

from the inlet; and (c) 0.6 m from the inlet

74 Numerical Simulations in Engineering and Science

**Figure 8.** Vertical w<sup>1</sup>

In **Figure 9** the vertical temperature distribution in the middle of the domain at the height (0–4 m) is presented.

The temperature of humid air near the floor is below the dew point (301.2 K) and water phase humidity covers the whole surface. The hot air close to the floor that comes into contact with the cold surface, reduces its temperature and flows down due to the gravity. The remained hot air flows up to the roof.

In **Figure 10** the vertical temperature distribution in the region close to the floor calculated by three different numerical schemes (HYBRID, van LEER, SUPER) [6, 8, 12] is presented.

Temperature profile in the region of major gradient near the floor surface is predicted more abrupt by the SUPER scheme.

In **Figure 11** the vertical absolute humidity ratio (kg H2O/kg of dry air) predicted by the three different numerical schemes (HYBRID, van LEER, SUPER) is presented at time 360 s.

Heat convection is accompanied by mass transfer and phase change of humid air. The larger gradient of temperature profile predicted by the SUPER scheme [10] leads to the formation of larger amount of water phase. Comparing the performance of the discretization schemes a more accurate solution of the condensation procedure is observed when applying the SUPER scheme.

**Figure 9.** Vertical temperature distribution at time 360 s for initial humidity condition 90%.

errors. The flow oriented SUPER scheme overcomes the phenomenon of the numerical diffu-

Study of the Numerical Diffusion in Computational Calculations

http://dx.doi.org/10.5772/intechopen.75660

77

The first author (D. P. Karadimou) gratefully acknowledges the financial support from the State Scholarships Foundation of Greece through the "IKY Fellowships of Excellence for

sion in most of the cases investigated without increasing the computational cost.

\* and Nikos-Christos Markatos1,2 \*Address all correspondence to: dkaradimou@gmail.com and n.markatos@ntua.gr

1 National Technical University of Athens, School of Chemical Engineering, Athens, Greece

[1] Patel MK, Markatos NC, Cross M. Technical note-method of reducing false diffusion errors in convection diffusion problems. Applied Mathematical Modelling. 1985;**9**:

[2] Patel MK, Markatos NC. An evaluation of eight discretization schemes for two-dimensional convection-diffusion equations. International Journal for Numerical Methods in

[3] Darwish M, Moukalled F. A new approach for building bounded skew-upwind schemes. Computer Methods in Applied Mechanics and Engineering. 1996;**129**:221-233

[4] De Vahl Davis G, Mallinson GD. An evaluation of upwind and central difference approximations by a study of recirculating flows. Computers & Fluids. 1976;**4**(1):29-43

[5] Fromm JE. A method for reducing dispersion in convective difference schemes. Journal

[6] Spalding DB. A novel finite-difference formulation for different expressions involving both first and second derivatives. International Journal for Numerical Methods in

[7] Leonard BP. A stable and accurate convective modelling procedure based on quadratic upstream interpolating. Computational Mechanics and Applied Mechanical

**Acknowledgements**

**Author details**

**References**

302-306

Fluids. 1986;**6**:129-154

of Computational Physics. 1968;**3**:176-189

Engineering. 1972;**4**:551-559

Engineering. 1979;**4**:557-559

Despoina P. Karadimou<sup>1</sup>

Postgraduate studies in Greece - SIEMENS" Program.

2 Texas A & M University at Qatar, Education City, Doha, Qatar

**Figure 10.** Vertical temperature distribution at time 360 s for initial humidity condition 90% in the middle of the domain at the height (0–0.2 m) calculated by three different numerical schemes.

**Figure 11.** Vertical absolute humidity ratio (kg H<sup>2</sup> O/kg of dry air) distribution at time 360 s for initial humidity condition 90% applying the van LEER, the HYBRID, the SUPER numerical schemes at height 0–0.2 m.
