Some details of the PARSOL method

Contents

  1. Introduction
  2. How cut-cell edges are detected
    1. The two-dimensional-projection method
    2. The two-dimensional-section method
  3. An example


a. Introduction

PARSOL is the technique in PHOENICS for improving the accuracy of flow simulations for situations in which a fluid/solid boundary intersects obliquely some of the cells of a cartesian or polar-coordinate grid by:

(1) introducing new variables, for example the temperatures in each part of the intersected cell, and

(2) providing equations from which their values can be deduced.

PARSOL is capable of calculating the fluid-flow phenomena with improved accuracy, whether the flow is laminar or turbulent, and whether heat transfer is present or absent.

It is currently a requirement that the information about the geometry of the solid-fluid interface should be conveyed to EARTH by means of a FACETDAT file, of the kind which can be produced by the Virtual- Reality interface of PHOENICS.

This represents the surface as a continuous sequence of triangular or rectangular facets, the x-y-z coordinates of which are contained in FACETDAT.

b. How cut-cell edges are detected

The two-dimensional-projection method (2DPM)

In this, the first-used method now superseded, the facets of the object were projected on to a plane normal to one of the coordinate directions, as triangles or quadrilaterals.

Cell edges appeared in the projection plane as points; and the existence of an intersection was evidenced by the cell-edge point lying within the projected facet area. This was detected by calculating the areas of triangles having facet-edge projections as base, and the grid-line projection as apex. The determination of whether the grid-line projection lay inside or outside the facet projection depended on the signs of the sums of these areas.

Thereafter, the normal-to-plane location of the intersection could be calculated, and the geometrical properties of the cut-cell computed and stored for use.

The disadvantages were the slowness of the area calculation, the hit-or-miss nature of the decision as to whether points lay inside or outside, and the obscurity of the intersection calculation.

Moreover, as implemented, no advantage was taken of the economies which can be made when the flow-situation to be simulated is two- rather than three-dimensional.

The 2DPM coding was thus found, after intensive study, to have several drawbacks, of which the most serious were:

  1. It could not be relied upon always to detect intersections between facets and cell edges, because of lack of control of 'tolerances', i.e. the differences of distance between what was and what was not an intersection.
  2. It could not directly treat the commonly-occurring two-dimensional flow situations, but had to convert them into pseudo-three-dimensional ones, which was at best uneconomical and at worst contributed to the 'missed-intersection' phenomenon.
  3. Even when intersections were correctly deteced and their positions computed, the excessive amount of computation involved imposed a serious delay on the start-up of the true CFD calculations.

The two-dimensional-section method (2DSM)

It was for these reasons that the alternative 2DSM method was invented.

The 2DSM proceeds as follows:

  1. The object intersection is detected plane by plane. Preliminary selection is used to make sure the detection is applied only to those facets which could possibly intersect with the plane by checking the dimensions of the bounding box of the object in question.

  2. The intersection between a plane and a facet edge is a segment with start and end points. The segment complies with the 'in-on-the-right' convention, which means the right side of the segment is in the object. The facet is slightly extended to avoid missing an intersection when the plane just touches the facet.

  3. Once all the segments within the plane are obtained together with the indications of whether the left side is fluid and the right side is solid, or vice versa, the intersections between grid lines and segments are calculated. A slight extension is also used to avoid missing an intersection, e.g. the line just touches the end of the segment, which could lead to objects missing in a column of cells.

  4. Each grid line coincides with the edges of four columns of cells. The edge may be classified in three ways:
    1. if the edge is wholly within a pair of intersection points, it means the edge is inside the object;
    2. if it is outside the pair of intersection points, the edge is in fluid;
    3. if one of the paired points falls on the edge, it is on its cut edge.

    Then the edge fraction is used to indicate which case edge is in question.

  5. Consequently, the cells are divided into three groups:
    1. a cell with all twelve edges inside fluid is an open cell;
    2. a cell with all twelve edges inside the object is a blocked cell; and
    3. a cell with edges partly in object and partly in fluid is a cut cell, which is split into a fluid sub-cell and a blocked (solid) sub-cell.

  6. Then the necessary number of cut-cell geometry quantities, including centre coordinates, areas and volumes of sub-cells, etc, are calculated and stored in the PHOENICS F-array, and used by the PARSOL solver.

c. An example

The following pictures exemplify the ability of PARSOL to simulate the flow around curved bodies of complicated shape (in this case louvres) while using a cartesian grid.

Figure 1 Pressure contours with velocity vectors for flow between louvres

Figure 2 Velocity contours for flow between louvres

The following pictures show the flow past a spherical body. In this case both PARSOL and Fine-grid embedding are used, PARSOL is used to represent the spherical body, and FGEM to show details of the flow near the body.

Figure 3. Velocity vectors for flow past a spherical body

Figure 4. Pressure contours for flow past a spherical body