6.2 Representation of Blades Geometry

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Key Symbols
Indexes and Other Signs

6.2.1 File Formats for Blades Storage

Sources of geometric information related to turbines blades are quite varied. These can be drawings in the paper or electronic form, the results of measurement of coordinates of the dots multitude, using mechanical or laser devices, coordinates of cross sections by flat or conical surfaces.

When the surface of the blade is represented by sets of dots its conical (cylindrical for axial machines) cross sections, it is assumed that the program, taking this information, will build the splines on cross sections and then will stretch the spline on surface. In this sense, this description is procedural.

In particular, the BladeGen preprocessor (Ansys CFX) offers two format of procedural form of the blades storing – RTZT and CURVE. Because the information in the CURVE file is not enough for permanent storage of data of the blade wheel, we have developed its extension – CUR format. It additionally includes the number of blades in the crown, the number of the cross sections
and the number of the profile sectors in the cross sections, the number of dots in the sectors, etc.

Figure 6.1 Fragment of the blade wheel stored in the CUR format
Figure 6.1 Fragment of the blade wheel stored in the CUR format (4 sectors on the profile).


The number of sectors on the contour of the cross section can be 1, 2 or 4. In the first case, the surface of the blade is formed by one spline. Accordingly, in the second and third cases, the blades is formed by two (suction and pressure sides) or four (suction side, leading edge, pressure side and trailing edge) spline segments.

The order and type of splines (for example, interpolation or approximation)are not stored in a file, because these options are implementation-dependent. They must be specified in the reading procedure. A fragment of the blade wheel, described using the CUR format with four sectors in each of the five initial cross sections of the blade, is shown at Fig. 6.1. The cross sections, shown at
Fig. 6.1, came out as a result of spline-approximation of original dots (dots on the sites have different colors).

6.2.2 Stacking

The process of drawing up the blades from known cross sections (flat or cylindrical) is called stacking. To do this, specific point of each cross section, which coincides with the stacking line of this cross section, must be selected. Often, for convenience, the centers of the edges or the centers of gravity of the cross sections are chosen as stacking points (Fig. 6.2). In general, this selection can significantly change the shape of the blades.

Any deviation of the stacking line from radial location will be named lean. With a simple lean stacking line remains straight and is characterized by a single parameter –the angle of incline. When we have a complex lean it can take

figure 6.2
Figure 6.2 The simplest methods of the blade forming: a – radial center of gravity;
b – radial center of inlet edge; c – radial center of outlet edge any form.
There is distinguishing among axial and tangential lean.

For easy use, it makes sense to limit parameterization of stacking line by Bezier curves.

6.2.3 Forming the Lateral Surfaces of the Blades

The surface of the blade is described by the parametric functions-interpolation or approximation B-splines based on two parameters: u-along the contour of each cross section and v-along the direction of stacking. Interpolation spline passes exactly through all dots of cross-section of the blade, and approximation spline – in accordance with supporting polygon, which is build using the dots of cross section or by least squares method [36]. All dots of the surface can be found, when u and v parameters taking the values from 0 to 1. In some cases, it is required to allow extrapolation towards the staking and then the v parameter may become a bit less than 0 or greater than 1.

The blade can be described either by one surface or by several. In our implementation (as already noted) 2 or 4 surfaces, that can be useful for some applications, in particular, when constructing grids, are allowed. Since there is no joining on surface boundary, the junction’s error could be managed only changing the spline order along u and v directions. It is usually within the range 2…5.

Figure 6.3
Figure 6.3 The blade surface formation, using one (a), two (b) and four (c) surfaces.
6.2.4 Three-Dimensional the Turbine Blade Parametric Model

One of the key elements of the 3D aerodynamic optimization of the turbine cascades algorithm is the turbine blade model parameterization, which consists in the possibility of changing blade shape (curvature) by variation of the limited number of numeric parameters that describes the stacking line.

The Bezier curve of 3-d (method I) and 4-th order (method 2) it seems convenient to use as the binding line. The second method allows creating a stacking line with practically straight-line the middle segment. (Fig. 6.4, 6.5). The number of independent variables in both cases can be reduced to two ( Yh and Ys ).

Parametric model of turbine blades must provide an opportunity to check the mass flow through the cascade. This requires incorporation in the model parameter, which allows controlling the mass flow during optimization. Most eventually the last gives the possibility to ensure equality of the mass flow between initial and optimized cascades with the same flow parameters before
and after cascades. The changing of stagger angle of the blade, relative to the original, can be taken as such parameter.

Figure 6.4 Bezier curves of 3-rd and 4-th order.

In addition to complex lean, the simple lean was implemented in methodical aim, which consists in turning of the turbine blades profiles relatively of the axis of rotation of the turbine on the specified angle. In general, developed parametric model of turbine blade allows producing its curvature in the circumferential (tangential) direction as well as in axial direction, simultaneously or separately.

Figure 6.5 Shapes of the blades with complex lean.
6.2.5 The Grids Construction

As it is known, the results of the CFD calculations may depend on the type of calculation grids. One of the tasks, needing to be addressed, is to build up a three-dimensional parametric calculation grids, satisfying the form of parameterized blades.

Fast and reliable building of the parametric calculation grids is an integral part of the optimization studies as it implies the calculation of a large number of variants of the geometry of the turbine cascade. Since the developed algorithm of optimization should not be tied up with solitary CFD-solver, a specialized grids builder has been developed.

We will describe in details the work with H-grids, which represent a convenient compromise between complexity of the grids creation and quality of the obtained solutions when flows computation in turbomachine cascades occur. H-grid topologically is equivalent to the cube. Therefore, a data structure is simple enough for description. It intentionally is made redundant to accelerate frequently meeting operations. This, of course, slightly reduces the maximum size of the grid when a limited amount of RAM available to the computer, but it is not critical to the solving problem.

The structured calculation grid for channel between the blades is obtained because of deformation in the direction of each of the coordinate axes of a rectangular parallelepiped (in space) or rectangle (flat case).

Inter-blade channel is formed by concave and convex sides of the two adjacent blades (or profiles in the planar case). For selecting the high pressure and low pressure sides of the blade, the blade is made up by two splines, connecting in the points of minimum and maximum x-coordinate sections. Parametric lines v = const of these splines give the calculated coordinates of D
sections of the grid in the radial direction. Next inter-blade channels are supplemented by input and output sections of the specified length, representing segments of the rings (for circumferential blade cascade) or parallelepiped (for flat cascade).The resulting area in each section is split up to cells of grids in the direction of x-coordinates dimension L. On the inlet and outlet sections is
usually taken by L/4 cells. Other cells are located on the profile and coordinates of the nodes are calculated by interpolation spline via dots of the splines of the high and low pressure sides.

Finally, channels are split on H sections along the directions x = const, that completes the structured grid formation. In the process of grid building a primitives numbering (nodes, edges, verges, and cells), topological ties formation and geometrical data calculation are made. All information is entered into a data structure.

Figure 6.6 The spatial H-grid of inter-blade channel 32*8*16 with thickening
in three directions.

As such, the calculated grids are not yet suitable for conducting reliable calculations of viscous flows in the blades cascades. They should be improved in order to fit the peculiarities of the flow near the walls of the channel.

Thickening structured grid is performed independently for each of the coordinate directions. Law of deformation of the grid can be different and should reflect the physical characteristics of the flow in the area of thickening. For example, near the wall polynomial law for changing the grid can be used, which corresponds to the rate of changing the velocity in the boundary layer. In
the area of input and output edges the deformation may be exponential in nature that is less aggressive. In either case, a number of parameters controlling the thickening as for the rate of deformation as well as the ratio of the sizes of areas of the channel subject to or not to distortion should be entered.

In general three-dimensional case, the grid, suitable for calculations of viscous flows, is presented at Fig. 6.6.

6.2.6 File Format for Grids Storage

The diversity of formats creates some difficulty in reading these files by different CFD-applications. CGNS-standard for CFD calculations data storage is positioned as a “common, portable and extensible”. Software implementation of the standard is an open, cross-platform and well documented that, in principle, precludes differences of various applications.

Data in CGNS format are stored in binary form and access to it is implemented through a set of functions for reading, writing, and modifying of the contents of the files which can be called from application in different programming languages. In general case CGNS file can contain data which is associated with viscous compressible fluid flow, but suitable for solutions of the Euler equations and potential flows.

The standard includes the following data types: structured, unstructured hybrid grids; data of the CFD calculations; information on the sub-grids docking or overlapping; boundary conditions; descriptions of equations of state, turbulence models etc.; nonstationary solutions, including deformation of calculation grids in time; dimension of variables; variables reference points;
history of calculations; user’s and other data.

For the purpose of specific tasks solution there is no need to implement in full all the functionality, supported by the CGNS (this is not currently doing even such advanced products like CFX). It is enough, for example, organize saving of the structured grids and setting of the boundary conditions, satisfying the terms of the calculation task. This significantly speeds up the preparation of data for CFD calculations. Analysis of output information, perhaps, you might need to implement by means of post-processors of used packages, since not all of them conserve the results of calculations in CGNS format.

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