Development of a method for numerical modeling of the separation flow at the entrance to a rectangular exhaust channel with sharp edges

. We have developed a method for mathematically simulating flow separation at the entrance to the exhaust channels of rectangular shape, using stationary discrete vortices in an ideal incompressible liquid in full spatial setting. When designing the discrete model, straight and curvilinear quadrangular vortex frames, straight and curvilinear horseshoe-shaped vortices were used. Using the developed computational procedure and computer program, the outline of vortex zone at the entry to the rectangular exhaust channel with sharp edges and the axial velocity of the air flow are determined. The results were compared with studies by different authors, as well as calculations by the methods of the theory of functions of a complex variables for the slotted exhausts and with results earlier obtained by us for calculation the square exhaust opening by means of discrete vortices in non-stationary and quasi-symmetric setting. The further direction of research will be related to the investigation of flows detached at the entrance to the rectangular exhaust hoods by the developed method, as well as by CFD methods. Then it is necessary to identify the regularities of the change of the local resistance coefficient at the entrance to the exhaust hood, shaped by the found outlines of the vortex zones.


Introduction
To improve the efficiency of local exhaust ventilation, the most reliable way to capture pollutants, the most accurate information on the movement of air near the exhaust openings is needed. In particular, the definition of the boundaries of the detachable (vortex) zones is necessary for the determination of the boundaries for the shaping of the entry openings of local exhausts, which significantly reduces their aerodynamic resistance. The simulation of detachable flows entering the exhaust channels is usually done in a two-dimensional setting: flat or axially symmetric. Methods are used to calculate both an ideal incompressible liquid and a viscous medium. Previously, the detachable flows at the entrance to the slot and round exhaust openings in the hood form were studied in detail [1,2]. The flow near square and rectangular exhaust ducts was investigated in a threedimensional viscous medium model using CFD [3]. The flow separation from the sharp edge was shown, but the regularity of the change of the vortex zone outline was not considered.

Calculating method
The boundary of the exhaust channel is discretized by N vortex frames: rectangular and straight horseshoe shaped. The number of rectangular frames adjacent to a sharp edge is N sl . At the Fig. 1 showed the first layer of square frames. The number of layers discretizing the side surface is K sl (Fig. 2). The penultimate layer is replaced by straight horseshoe-shaped vortices (Fig. 3). The vortex elements on the side surface are numbered 1,2,…, N 1 -1. The number of the vortex frame enclosing the active section is N 1 . The exhaust section is discretized by the rectangular vortex frames with numbers N 1 +1, N 1 +2,…, N. Control (calculated) points are located in the center of the rectangular frames, the coordinates of which are calculated from the coordinates of the vertices a, b, c, d, as the arithmetic mean of the corresponding coordinates vertices. The calculated points for horseshoe-shaped vortices are located in the same place as for rectangular vortices (shown as crosses in Fig. 3). At the calculated boundary points, the normal components of the velocity are given, and at the solid boundaries sets the impermeability condition.
where r 1 , r 2 is the radius of the vector of the end of the segment.
In the event of a vortex hitting the core (figure 4): r r r r r u S r r n r r n r r r r . (2)

Fig.4. Vortex core
The effect on the control point of a quadrangular (in this case square) frame is expressed by the sum of four vortex segments in a given direction. Induced by a k-th straight horseshoe-shaped vortex (figure 5) of intensity * = 1 with rays pointing along l, the velocity v at a point x along the direction n is calculated by means of a scalar product: ¦ r l r l n n l n r l (3) https://doi.org/10.1051/e3sconf/202235601014 E3S Web of Conferences 356, 01014 (2022) ROOMVENT 2022 where dS p = h·h is the area of the frame in the exhaust   (6) where t p is the calculation point -the midpoint of the p-th frame of the side surface of a rectangular channel, and n is the external normal at that calculation point. 3.7. End of external iteration cycle. If an internal iteration cycle is performed once, the process of determining a free vortex system is considered constructed. Then you can define the velocity field and build stream lines.

Results and discussion
An example of a free stream surface is given in Figure 6. Free curvilinear horseshoe-shaped vortices are arranged along the pipe from the sharp edge to the exhaust section. Closed curvilinear frames are arranged in the vertices of vortex break segments. Using the developed computational procedure and computer program, the axial velocity of the air flow and the line of flow separation at the entrance to the rectangular exhaust channel with sharp edges are determined (Figure 7, 8). The free stream line was constructed from the middle of the sharp edge. The comparison of the results with the known and obtained with the help of СFD demonstrates the adequacy of the developed method. The variation of the dimensionless axial velocity v ax /v 0 (v 0 is the flow rate averaged exhaust velocity) of the square exhaust channel (Figure 7), constructed using the method of discrete vortices in stationary setting (Line 3) is well consistent with the Fletcher's formula calculation (line 2 [4]) and CFD calculating (line 1) using kturbulence model with the STAR-CCM+ software. Note that on the cut of exhaust channel there is a slight increase in velocity relative to other methods. The highest velocity found by the N. E. Zhukovsky method is achieved for the slot exhaust (line 7), lines 4, 5 and 6changing of velocity near the rectangular exhaust at a factor of 2 to 1, 4 to 1 and 10 to 1 respectively. The last curve is closest to the changing of the axial velocity near the slot exhaust. There is a marked difference in the line constructed according to the proposed formula which averages the position of the free stream line 2 for the quasi-symmetric problem in the non-stationary setting [5]. The free stream lines 4, 5 for a rectangular exhaust with a ratio of sides 4 to 1, 10 to 1 approaches line 6 and practically coincide with it. Line 7 shows the outline of the vortex zone found numerically with the STAR-CCM+ software for a square exhaust.
When moving away from the corner of the exhaust channel to the middle of the edge, the stream line is moving from the exhaust channel ( figure 9). The thickness of the vortex zone is greatest in the plane passing through the midpoints of opposite edges and the axis of exhaust. The further direction of research will be related to the study of separated flows at the entrance of the rectangular exhaust hoods by the developed method, as well as by CFD methods. Then it is necessary to identify the regularities of the change of the local resistance coefficient at the entrance to the hoods, shaped along the found boundaries of the vortex zones. It is then necessary to study the rectangular-shaped exhaust hoods where two vortex zones can be formed. This method can also be used to study the formation of vortex zones in presence of cross drafts and ascending air flow.

Сonclusion
A method has been developed for mathematical modeling of the separated flow at the inlet to the suction channels in a full spatial formulation. As an example, the calculation of the axial velocity and the boundaries of the vortex zone at the entrance to the square and rectangular suctions is made. The results obtained are adequate. The research was carried out under the grant of the Russian Science Foundation (project #23-49-00058) and NSFC (project #5221101677).