**2. Numerical problems**

### **2.1. Inclined flow of a scalar quantity**

The physical problem presented in this section describes the transport of a scalar quantity (C1) in a two-dimensional geometry. The flow of C1 is inclined to the grid lines (θ = 45°) and the natural diffusion is assumed equal to zero. This physical problem is considered as a benchmark for comparing the performance of various numerical schemes.

The boundary conditions applied to the grid are the following:

Air enters the domain from the west side with stable mass flow rate (*ρ* ⋅ *u*) kg/m<sup>2</sup> s, uniform values of the two components of velocity (*u*<sup>1</sup> <sup>=</sup> *<sup>v</sup>*<sup>1</sup> <sup>=</sup> <sup>1</sup> <sup>m</sup>/s) and uniform value of the convected quantity (C1) (1 ⋅ kg/m<sup>3</sup> ).

Air enters the domain also from the south side with stable mass flow rate (*ρ* ⋅ *u*) kg/m<sup>2</sup> s, uniform values of the two components of velocity (*u*<sup>1</sup> <sup>=</sup> *<sup>v</sup>*<sup>1</sup> <sup>=</sup> <sup>1</sup> <sup>m</sup>/s) and uniform value of the convected quantity (C1) (1 ⋅ kg/m<sup>3</sup> ).

At the two outlets (north and east side) of the domain the air is supposed to exhaust at an environment of fixed uniform pressure.

The dimensions of the domain are 1 × 1 m. The numerical results presented are based on the independent grid of 33 × 33 cells. For the discretization of the convected term in the transport equation of the scalar quantity C1 the numerical schemes: (a) UPWIND, (b) van LEER, (c) SUCCA, (d) SUPER are applied [6, 8, 11, 12].

In **Figure 1** the velocity vector distribution predicted by the SUPER numerical scheme is presented.

In **Figure 2** the vertical (concentration) distribution of the scalar quantity C1 in the middle of the domain is presented.

The vertical distribution of the scalar quantity C1 that is transferred by air with inclined direction is predicted more abrupt by the numerical schemes (SUCCA, SUPER) that take into account the

**Figure 1.** Air velocity vector distribution predicted by the SUPER scheme.

The dependent variables solved for are:

68 Numerical Simulations in Engineering and Science

**b.** Volume fractions of each phase, *r*

**e.** Temperature (K), enthalpy (J/kg)

<sup>∂</sup>(*ri <sup>ρ</sup><sup>i</sup>*

(CFD) code PHOENICS [16].

**2. Numerical problems**

quantity (C1) (1 ⋅ kg/m<sup>3</sup>

vected quantity (C1) (1 ⋅ kg/m<sup>3</sup>

**2.1. Inclined flow of a scalar quantity**

).

environment of fixed uniform pressure.

SUCCA, (d) SUPER are applied [6, 8, 11, 12].

= phase volume fraction, (m<sup>3</sup>

(m2 /s<sup>2</sup> )

where *<sup>r</sup> <sup>i</sup>*

(m/s); and *<sup>S</sup> <sup>φ</sup>*,*<sup>i</sup>*

**a.** Pressure that is shared by the two phases, *P*(Pa)

**c.** Three components of velocity for each phase, *ui*

*i* (m3 /m<sup>3</sup> )

The phase volume fraction equation is obtained from the continuity equation:

<sup>∂</sup>*<sup>t</sup>* <sup>+</sup> *div*(*ri <sup>ρ</sup><sup>i</sup> <sup>u</sup>*

The discretized by the finite volume method form of all the conservation equations is solved by the SIMPLEST and IPSA algorithms embodied in the Computational Fluid Dynamics

The physical problem presented in this section describes the transport of a scalar quantity (C1) in a two-dimensional geometry. The flow of C1 is inclined to the grid lines (θ = 45°) and the natural diffusion is assumed equal to zero. This physical problem is considered as a

Air enters the domain from the west side with stable mass flow rate (*ρ* ⋅ *u*) kg/m<sup>2</sup> s, uniform values of the two components of velocity (*u*<sup>1</sup> <sup>=</sup> *<sup>v</sup>*<sup>1</sup> <sup>=</sup> <sup>1</sup> <sup>m</sup>/s) and uniform value of the convected

Air enters the domain also from the south side with stable mass flow rate (*ρ* ⋅ *u*) kg/m<sup>2</sup> s, uniform values of the two components of velocity (*u*<sup>1</sup> <sup>=</sup> *<sup>v</sup>*<sup>1</sup> <sup>=</sup> <sup>1</sup> <sup>m</sup>/s) and uniform value of the con-

At the two outlets (north and east side) of the domain the air is supposed to exhaust at an

The dimensions of the domain are 1 × 1 m. The numerical results presented are based on the independent grid of 33 × 33 cells. For the discretization of the convected term in the transport equation of the scalar quantity C1 the numerical schemes: (a) UPWIND, (b) van LEER, (c)

) \_\_\_\_\_

/m<sup>3</sup> ); *<sup>ρ</sup> <sup>i</sup>*

benchmark for comparing the performance of various numerical schemes.

The boundary conditions applied to the grid are the following:

).

= net rate of mass entering phase i from phase j, (kg/m<sup>3</sup>

, *vi* , *wi* (m/s)

→

= phase density, (kg/m<sup>3</sup>

*<sup>i</sup>*) = *S<sup>φ</sup>*,*<sup>i</sup>* (3)

= phase velocity vector,

s), if there is phase change.

); *u* <sup>→</sup> *i*

**d.** Turbulence kinetic energy and dissipation rate of turbulence for the first phase, *k* and *ε*

**Figure 2.** Vertical distribution of the scalar quantity C1 in the middle of the domain applying the numerical schemes: (a) UPWIND, (b) van LEER, (c) SUCCA, (d) SUPER.

flow direction than the conventional numerical scheme (UPWIND) and the non-linear numerical scheme (van LEER). It is concluded that the numerical diffusion errors are minimized when the discretization schemes (SUCCA, SUPER) are applied to predict the transport of the scalar quantity C1 when the air-flow direction appears the major inclination angle 45° to the grid lines. The performance of the van LEER numerical scheme is also satisfactory in predicting the scalar concentration distribution. The conventional UPWIND numerical scheme that does not take into account the phenomenon of false-diffusion presents the poorest accuracy.

In **Figure 3** the numerical prediction of the temperature distribution along the heat exchanger is validated against experimental data. The average% relative error of the numerical predic-

Study of the Numerical Diffusion in Computational Calculations

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

71

In **Figure 4** the numerical results of the enthalpy distribution at the lateral plane in the middle of the heat exchanger applying three different numerical schemes (HYBRID, van LEER,

Comparing the performance of the numerical schemes in predicting the transfer of the scalar quantity H1 it is concluded that the numerical prediction is equally satisfactory for the three

**Figure 4.** Enthalpy distribution at the lateral plane of the heat exchanger applying: (a) the HYBRID, (b) the van Leer, (c)

tion does not exceed the value of 20%.

SUPER) [6, 8, 12] is presented.

discretization schemes.

the SUPER numerical scheme.
