It is obvious that the use of nonstaggered grids can reduce the number of storage allocations, requires fewer interpolations, and makes code development simpler compared to staggered grids. For this arrangement, there is the pressure oscillation problem. To solve the problem, Rhie and Chow [3] proposed a scheme based on the momentum interpolation method for nonstaggered grid arrangement.
This paper presents the application of a finite-volume method for 3D fluid flow and heat transfer using non-orthogonal coordinates and a nonstaggered grid. In the discretization procedure the same control volume for all variables is used. Over all method is applied to predict the laminar flow through a 90 degree banded square duct and the natural convection in the LHRE''s cooling passage.
2. Numerical Method
2.1 Governing equations
If yi(I=1,2,3) and xi(I=1,2,3) reprense the Cartesian
coordinate system and the new coordinate system (Curvature Coordinate system), respectively. The governing
equations based on Cartesian velocity components for heat convection in
3D complex geometries are written in common form:
where
Hear Y represent vj, T K and
e, separately, GYand SY
are the associated "exchange coefficients" and "source terms", gkj
and gkj are the contravariant and covariant metric components.
2.2 Discreted equation
The transport equation (1) was discrete d using a finite-volume method
based on a nonstaggered grid system, where all variables are stored at
their geometrical centers. The form of the discreted equation was written
as follows:
where
here I2 is the integral of the cross-derivation component
arising from nonorthogonality . SCYand
Spare the linearized source terms.
2.3 Pressure correction and pressure equations
If the imperfect velocity components and pressure are denoted by vj*
and p*, and these velocity components and pressure are corrected by vj/
and p/, the estimated values of corrected velocity components
vj/ are obtained from the momentum equation as:
thus
where
The substitution of the velocity correction equations to the continuity
equation represented by the equation (2) with Y=1,
GY=0
and SY=0, and integration of the
written continuity equation over the control volume by using the central
difference scheme, results in an equation for pressure correction:
where
here Bno represents the influence of the pressures at the neighbor
nodes.
In the present scheme, the coupling of the velocity and the pressure
is achieved by using the SIMLER algorithm, in which the pressure field
itself was evaluated by solving a Poisson-link equation. The steps in the
derivation of the pressure equation are similar to those for the pressure
correction equation outlined previously. In the derivation of the P/
equation, the discreted continuity equation was combined with a truncated
form of the momentum equations. Now the full-momentum equations are used
and the starred velocities used earlier are replaced by the pseudo-velocity,
defined as follows:
Therefore, the source term b in the pressure equation is calculated
by using the pseudo-velocities rether then the actual velocities.
2.4 Cell-face velocity interpolation for the nonstaggered arrangement
To eliminate the unrealistic fields of the velocity and pressure, the
cell-face velocity interpolation was used and further extended to the 3D
situation. As an example, the application of the method on w-face is present
here:
On the w-face, the contravariant velocity components in equation (5)
is divided into a mean part and a correction part:
The mean part 
can
be evaluated by following equation as:
Here a is the linear interpolating coefficient,
which takes a value between 0 and 1. Based on the method of Date [4], the
correction part is expressed as:
where
here b takes a value between 0 and 1. If
b=0.5
is chosen, the equation (9) is the formula proposed by Rhie and Chow [3].
3. Laminar flow in a 90 degree square duct
To check the capability of the above mentioned method to treat 3D incompressible
flows, a laminar flow through a 90 degree curved duct of square cross
section was calculated. The predictions are compared with the experimental
and computational results of Humphrey et al[5]. Considering the symmetry
of the flow, only half of the solution domain is solved to reduce the computer
storage requirement and the computational time. A non-uniform 56x12x16
grid was used in the study.
Fig. 1 shows the stream-wise velocity profiles at
the various stream-wise stations. The present computational results agree
well with the experimental data of Humphrey et al[5] at the planes upstream
of the 60 degree plane. The correlation is poor at the 60 degree and 90
degree planes, where strong secondary motion exits. The grid in this region
is too coarse to solve the strong gradients in this region.
[1] P. Tamamidis and D. N. Assanis, Numerical Heat Transfer,
Vol. 24, PP. 57-76 (1993).
[2] K. C. Karki and S. V. Patankar, Numerical Heat Transfer,
Vol. 14, PP 295-307 (1988).
[3] C. M. Rhie and W. L. Chow, AIAA Journal, Vol. 21, PP. 1525-1532
(1983).
[4] A. W. Date, Int. J. Heat Transfer, Vol. 36, PP. 1913-1922,
(1993).
[5] J. A. C. Humphery, A. M. K. P. Tayer and J. H.
Whiletaw, J. Fluid Mech., Vol. 83, PP. 509-527, (1977)