Semi-Unstructured Meshes for Axial Turbomachine Blades

Semi-Unstructured Meshes for Axial Turbomachine Blades

L. Sbardella, A. I. Sayma and M. Imregun



Imperial College of Science, Technology and Medicine Mechanical Engineering Department Exhibition Road, London SW7 2BX, UK

Number of gures : 14 Number of tables : 0 Number of pages : 19

 Author to whom correspondence should be sent

1

Abstract

This paper describes the development and application of a novel mesh generator for the ow analysis of turbomachinery blades. The proposed method uses a combination of structured and unstructured meshes, the former in the radial direction and the latter in the axial and tangential directions and exploits the fact that blade-like structures are not strongly three dimensional since the radial variation is usually small. The proposed semi-unstructured mesh formulation was found to have a number of advantages over its structural counterparts. There is a signi cant improvement in the smoothness of the grid-spacing and in capturing particular aspects of the blade passage geometry. It was also found that the leading- and trailing-edge regions could be discretized without generating super uous points in the far eld and that further re nements of the mesh to capture wake and shock eects were relatively easy to implement. The capability of the method is demonstrated in the case of a transonic fan blade for which the steady-state ow is predicted using both structured and semi-unstructured meshes. A further example is given for a nozzle guide vane in a turbine stage. 1

Introduction

It is well known that the computational mesh must be selected carefully in order to achieve an accurate determination of complex ow elds that are typical of axial- ow turbomachines. The minimization of skewness and the optimization of smoothness generally result in faster convergence and less mesh-dependence of the solution, therefore reducing the computational cost both in terms of in-core memory and CPU time. Consequently, the grid generation procedure should be considered to be an integral part of the numerical method. When performing a numerical simulation of turbulent, viscous turbomachinery ows, the following features are of primary importance: 1) accurate leading- and trailing-edge ow descriptions, 2) wake resolution, 3) proper gridding in the throat area where most of the shock is expected to occur, and 4) imposition of periodicity. Historically, mesh generation techniques for turbomachinery blades use structured hexahedral representations, the most commonly used ones being H-type, C-type, and O-type. These meshes are obtained either by using an algebraic approach or by solving a system of elliptic partial dierential equations [1, 2, 3]. H-type meshes have been by far the most common choice in turbomachinery applications as they are very easy to generate, the imposition of periodicity is straightforward and the mesh density before, inside, and after the blade passage can be controlled easily. However, the leading- and trailing-edge descriptions are poor and a large amount of super uous points is generated in the region between the in ow and the leading-edge. O-type 2

grids are not very eective in capturing the wake and their quality outside the passage is relatively poor. Such shortcomings can smear the outgoing shock of a transonic turbine blade or the bow shock away from the leading edge of a transonic compressor. On the other hand, C-type meshes can capture the wake structure if they are carefully generated but their quality in the region between the in ow and the leading-edge is not suitable to resolve bow shocks accurately. A dierent approach is to use unstructured triangular meshes, especially for 2D turbomachinery calculations. Such grids oer good exibility and most of the ow features can be captured with good accuracy via mesh re nement. 3D unstructured meshes are widely used in external ow applications but they have rarely been applied to turbomachinery cases. One shortcoming would be the requirement for a large number of radial direction grid points so that the leading and trailing edges can be resolved adequately. Moreover, such grids are perhaps best suited to inviscid ow computations. The aim of this paper is to present a novel mesh generation approach for the steady and unsteady ow analyses of turbomachinery blades by using a combination of structured and unstructured meshes, the former in the radial direction and the latter in the axial and tangential directions. The basic idea relies on the fact that blade-like structures are not strongly three dimensional since the radial variation is usually small. It is therefore possible to start with a structured, body- tted two-dimensional O-grid around a given aerofoil section to resolve the boundary layer. This core mesh is then extended in an unstructured fashion up to the far- eld boundaries, the triangulation being performed using an advancing front technique [4, 5]. Once this twodimensional grid is generated, it is projected to the remaining radial sections via quasi-conformal mapping techniques. When all such radial sections are formed, a three-dimensional prismatic grid is obtained by simply connecting the corresponding points of dierent layers. In this way, hexahedral elements are generated in the viscous region and while triangular prisms cover the rest of the solution domain. 2

Generation of Prismatic Grids

This section will describe the overall methodology rst without undue details, while the next section will focus on the main contribution of the paper: the quasi-conformal mapping procedure which links the 2D radial section meshes together. The mesh generation is based on ve main stages:  Mapping procedure to project all radial levels of the blade into 2D planes, using local coordinates.  Generation of a 2D hybrid-mesh at a given radial section.  Generation of a coarse body- tted structured mesh for all radial sections. 3

 Inverse mapping of the unstructured mesh into a structured one at the

same radial level.  Direct mapping to obtain the nal unstructured mesh at all radial levels. The blade geometry is usually de ned at a number of radial sections. In the general case, these radial sections will lie on 3D surfaces i (x; r; ), where i indicates the section index. Using parametric coordinates u and v, a typical surface can be de ned as:

Geometric modelling of the blade

8 >< x = x(u; v) v) >: r == r((u; u; v) 

(1)

The starting point of the present method is the projection of these radial sections into parametric two-dimensional planes, using the local coordinate system u and v. In this way the mesh-generation procedure will deal with plane sections only, the geometric dimension being reduced from three to two. Once all radial sections are mapped into 2D planes, a hybrid quadrilateral-triangular mesh is generated in a dierent 2D plane which corresponds to a certain radial level (usually the middle one). The quadrilateral part of the mesh takes the form of a body- tted O-grid which is generated using a system of elliptic partial-dierential equations. The orthogonality of this structured mesh is very important for an accurate resolution of the turbulent-boundary layer which originates from high Reynolds number ows. The remaining part of the domain is discretized using an unstructured mesh generator which uses an advancing front algorithm [4, 5]. A distinctive feature of this method is that triangles and points are generated simultaneously. Such an approach enables the generation of elements with variable sizes and stretching, and hence it is dierent from Delaunay-type mesh generators [5, 6]. Unstructured mesh generation

An important part of the mesh-generation strategy is the mapping procedure to project the unstructured mesh of a given radial level to all radial blade-de nition levels. A necessary condition for a mapping function is that it must associate a given point of the rst plane with one, and only one, point of the second plane. Moreover a mapping function should also guarantee that a given angle in one plane is mapped into a similar-valued angle in the target plane (quasi-conformal mapping). This last property is essential in order to minimize the skewness of the mesh, especially for highly-twisted fans. As will be illustrated in section 3, the quasi-conformal mapping procedure will make use of structured meshes.

Quasi-conformal mapping.

4

Once the unstructured mesh has been mapped to all radial blade surfaces, a prismatic mesh is obtained by simply connecting the corresponding points at successive levels. Moreover, in order to enhance the quality of the threedimensional mesh, a smoothing procedure is performed. This operation alters the positions of the interior nodes without changing the topology of the mesh. For aeroelasticity calculations, such as fan blade utter and turbine forced response, it is essential to be able to move the aerodynamic mesh according to the structural motion of the blade. Such a requirement can be met by the so-called \spring analogy". The element sides are considered to be springs with stiness values proportional to their length. The nodes are moved until the spring system is in equilibrium, the position of which is found by Jacobi iterations. 3

Quasi-Conformal Mapping

The starting point for quasi-conformal mapping is the generation of coarse, structured quadrilateral grids for all radial sections. Following the approach of Steger et al. [2], these meshes are obtained by solving a system of elliptic partial-dierential equations. An essential requirement for such structured meshes is that they must be generated in exactly the same manner, i.e. with the same number of points and quadrilaterals. Once this is achieved, the mapping procedure is implemented as follows ( g 1):  Geometry searching. Each point J of the unstructured 2D mesh must be located on quadrilateral E , of the structured mesh.  Inverse mapping. The Cartesian coordinates ~x = (x ; y ) of point J , associated with the quadrilateral E are given by: J

~xJ

=

4 X

I =1

J

J

~xI NI (J ; J )

(2)

where ( ;  ) represent the local-coordinates, ~x represents the Cartesian coordinates of nodes I = 1; : : : ; 4 of the quadrilateral E and N is the standard nite-element bilinear shape-function which takes the form [7] 8 >> (1 ) (1 ) < (1 + ) (1 ) 1 (3) N = +  ) (1 + ) 4> >: (1 (1  ) (1 + ) A Newton-Raphson method is used in order to obtain the values of  and  .  Direct mapping. Once all points J of the unstructured mesh are associated with quadrilateral E , and the local-coordinates ( ;  ) are determined, coordinates ~x of all the point on the remaining radial sections can be obtained directly using equation (2). J

J

I

I

I

J

J

J

J

5

J

The mapping method becomes fully conformal if a given element of the 2D hybrid mesh lies within a single quadrilateral of the structured grid. In addition, the angles of this quadrilateral must remain the same for all radial sections [7]. If the above two conditions are not satis ed, the property of conformity is not guaranteed and for this reason the procedure has been labelled quasi-conformal here. Finally, it is worth noting that the algorithm is not CPU-intensive, since both the geometry searching and the inverse-mapping are performed once only. 4

Examples

A semi-unstructured mesh, generated with the proposed method, will now be used to calculate the steady ow eld of a transonic fan blade at 70% of design speed. The diÆculty of obtaining satisfactory part-speed ow solutions is well known in the turbomachinery CFD community and this is why such a case has been attempted here. The Reynolds averaged compressible Navier-Stokes equations, together with the Baldwin and Barth turbulence model [8], are cast in terms of absolute velocity but solved in a relative non-Newtonian reference frame, rotating along with the blade, about the x-axis, with angular velocity

. The ow solver is an implicit, upwind-dierencing algorithm in which the inviscid uxes are obtained on the faces of each control volume using the ux dierence-splitting of Roe [10]. Second-order accuracy is obtained using the gradient informations of the unknown variables at each control volume. In order to guarantee monotonicity of the scheme and not to deteriorate convergence to steady-state, the modi ed van Leer pressure-based switch of [11] is used. The viscous terms are evaluated with a nite-volume formulation which is equivalent to a Galerkin-type approximation and which results in a centraldierence-type formulation of the viscous losses. The solution at each time step is updated using a linearized backward-Euler, time-dierencing scheme. The linear system of equations is solved approximately with a sub-iterative Jacobi procedure. The 2D unstructured mesh at the middle section of the fan, generated with the advancing front technique, is shown in gure 2 together with the structured mesh used for the mapping procedure. The boundary-layer region has been discretized with twelve points in the direction normal to the blade, a mesh density which is suitable for turbulent ow simulations using a wall function. The region behind the blade has been re ned in order to capture the wake as accurately as possible, a feature which is important to be able to predict the passage shock position correctly. This 2D mesh is then mapped to all radial levels using the quasi-conformal mapping procedure of section 3. Two of these sections, namely hub and tip, are shown in gures 3 and 4 together with the corresponding structured meshes that were generated for comparison purposes. The overall semi-unstructured mesh is shown in gure 6

5 with zoom view of the hub section. The steady-state ow results obtained from the semi-unstructured mesh are compared to those computed by the same code, this time using a standard structured H-type mesh ( gure 6). The structured mesh contains the same number of points on the blade surface and at the out ow boundary. However, it has about 20% more points than its semi-unstructured counterpart because of the super uous points at the in ow section. These points are needed for re ning the structured grid around the blade so that the boundary layer can be resolved. Furthermore, it can be seen that the mesh re nement in the wake region is much better for the unstructured mesh. Since the purpose here is to compare the quality of the solutions rather than to validate the solution itself against some other code or measured data, emphasis will be placed on the general features of the two solutions that are being compared. Figures 7 and 8 show the results at 80% blade span. The unstructured mesh solution is seen to be smoother with a better de ned shock. Figure 9 shows the mass ow contours at the out ow boundary of the computational domain. The general features of both solutions are similar but, as can be seen from the mass ow convergence plot of gure 10, the convergence of the mass ow rate is much poorer for the structured mesh. Additionally, the solution converge signi cantly faster in the case of the semi-unstructured mesh, a feature which can be seen from the time history of the residuals plotted in gure 11. The blip in the time history is due to a change in the arti cial dissipation coeÆcient in an attempt to improve the convergence rate for both meshes. The semi-unstructured mesh generator is equally applicable to turbine blades and the discretisation of a typical nozzle guide vane (NGV) will be illustrated here. The hybrid mesh which was split to tetrahedra is plotted at hub and tip section in gure 12. A zoom view of the leading and trailing edges is shown in gure 13. Finally, the mesh of the complete blade, as viewed from the pressure and suction surfaces, is given in gure 14. 5

Concluding remarks

1. A method to generate semi-unstructured prismatic meshes for turbomachinery blades has been presented. The unstructured mesh in the axial and tangential directions oers more exibility than standard structured O-type and C-type meshes, both in terms of skewness minimization and smoothness optimization. 2. Using the same solver, the solution obtained on the semi-unstructured mesh seems to be superior to that obtained on the corresponding structured mesh in terms of convergence rate and smoothness. Additionally, less points are required by the semi-unstructured mesh to provide same level of grid resolution that is required to resolve the boundary layer and 7

the wake behind the trailing edge. Acknowledgments

The authors would like to thank Rolls-Royce plc for both sponsoring the work and allowing its publication. References

[1] Thompson, J. F., Thames, F. C., and Mastin, C. W. (1974) `Automatic Numerical Generation of Body-Fitted Curvilinear Coordinates System for Field Containing any Number of Arbitrary Two-Dimensional Bodies', Journal of Computational Physics, 15, 299-319. [2] Steger, J.L. and Sorenson, R.L. (1979) `Automatic Mesh-Point Clustering Near a Boundary in Grid Generation with Elliptic Partial Dierential Equations', Journal of Computational Physics, 33, 405-410. [3] Brackbill, J. U. and Saltzman, J. S. (1982) `Adaptative Zoning for Singular Problems in two Dimensions', Journal of Computational Physics, 46, 342-368. [4] Morgan, K., Peraire, J. and Peiro, J. (1992) `Unstructured Grid Methods for Compressible Flows', In AGARD Report 787, Special Course on Unstructured Grid Methods for Advection Dominated Flows, 5.1 - 5.39. [5] Mavriplis, D. J. (1995) `Unstructured Mesh Generation and Adaptivity', ICASE Report No. 95-26. [6] Baker, T. J. (1990) `Unstructured Mesh Generation by a Generalized Delaunay Algorithm', In Applications of Mesh Generation to Complex 3-D Con gurations, AGARD Conference Proceedings No. 464, 20.1 20.10. [7] Zienkiewicz, O.C. and Morgan, K. (1983) `Finite Elements and Approximation', John & Sons Ltd. [8] Baldwin, B. S. and Barth, T. J. (1991) `A One-Equation Turbulence Transport Model for High Reynolds Number Wall-Bounded Flows', AIAA Paper 91-0610. [9] Kallinderis, Y. and Ward, S. (1992) `Prismatic Grid Generation with an EÆcient Algebraic Method for Aircraft Con gurations', AIAA Paper 92-2721. [10] Roe, P. (1981) `Approximate Riemann Solvers, Parameter Vectors, and Dierence Schemes', Journal of Computational Physics, 43, 357-372. [11] Swanson, R. C. and Turkel, E. (1993) `On Central-Dierence and Upwind Schemes', Journal of Computational Physics, 101, 292-306. 8

List of Figures

1 2 3 4 5 6 7 8 9 10 11 12 13 14

Mapping procedure . . . . . . . . . . . . . . . . . . . . . . . . . Mapping mesh and resulting unstructured mesh at middle section Mapping mesh and resulting unstructured mesh at hub section Mapping mesh and resulting unstructured mesh at tip section . Semi-unstructured mesh for a fan blade . . . . . . . . . . . . . Structured (top) and unstructured meshes at 80% span . . . . Mach number contours at 80% span . . . . . . . . . . . . . . . Pressure contours at 80% span . . . . . . . . . . . . . . . . . . Mass- ow contours at the out ow boundary: structured (left) and semi-unstructured (right) . . . . . . . . . . . . . . . . . . . Convergence of Mass- ow . . . . . . . . . . . . . . . . . . . . . Residual . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nozzle guide vane mesh . . . . . . . . . . . . . . . . . . . . . . Zoom of the mesh at hub section . . . . . . . . . . . . . . . . . Mesh on the blade surface . . . . . . . . . . . . . . . . . . . . .

9

10 11 11 12 13 14 15 15 16 16 17 18 19 19

Quadrilateral E in all other radial levels

J Quadrilateral E in the radial plane where the 2D unstructured mesh has been generated

Inverse mapping J

Direct mapping η

Ι=4

J

−1

Ι=1

−1

Figure 1: Mapping procedure

10

Ι=3

1

1

Ι=2

ξ

Figure 2: Mapping mesh and resulting unstructured mesh at middle section

Figure 3: Mapping mesh and resulting unstructured mesh at hub section

11

Figure 4: Mapping mesh and resulting unstructured mesh at tip section 12

(a) Suction surface view

(b) View at hub section

Figure 5: Semi-unstructured mesh for a fan blade

13

Figure 6: Structured (top) and unstructured meshes at 80% span 14

(a) Structured mesh

(b) Semi-unstructured mesh

Figure 7: Mach number contours at 80% span

(a) Structured mesh

(b) Semi-unstructured mesh

Figure 8: Pressure contours at 80% span 15

Figure 9: Mass- ow contours at the out ow boundary: structured (left) and semiunstructured (right) 1.0 unstructured structured 0.875

Normalised mass flow

0.750

0.625

0.500

0.375

0.250

0.125

0 0

500

1000

1500

2000 time steps

2500

3000

Figure 10: Convergence of Mass- ow 16

3500

4000

1 unstructured structured

Residual

0.1

0.01

0.001

0.0001 0

500

1000

1500 time steps

Figure 11: Residual

17

2000

2500

3000

(a) Tip section

(b) Hub section

Figure 12: Nozzle guide vane mesh 18

(a) Leading edge

(b) Trailing edge

Figure 13: Zoom of the mesh at hub section

(a) Pressure side

(b) Suction side

Figure 14: Mesh on the blade surface 19