Design Method for Subsonic and Transonic Cascade with Prescribed Mach Number Distribution

Transcription

1 THE AMERICAN SOCIETY OF MECHANICAL ENGINEERS 345 E. 47 St., New York, N.Y GT-18 The Society shall not be responsible for statements or opinions advanced in papers or in discussion at meetings of the Society or of its Divisions or Sections, or printed in its publications. m Discussion is printed only if the paper is published in an ASME Journal. Papers are available ]^C from ASME for fifteen months after the meeting. Printed in USA. Copyright 1991 by ASME Design Method for Subsonic and Transonic Cascade with Prescribed Mach Number Distribution. LEONARD and R. A. VAN DEN BRAEMBUSSCHE von Karman Institute Chaussee de Waterloo Rhode-Saint-Genese Belgium ABSTRACT An iterative procedure for blade design, using a Time Marching procedure to solve the unsteady Euler equations in the bladeto-blade plane is presented. A flow solver, which performs the analysis of the flow field for a given geometry, is transformed into a design method. This is done by replacing the classical slip condition (no normal velocity component) by other boundary conditions, in such a way that the required pressure or Mach number distribution may be imposed directly on the blade. The unknowns are calculated on the blade wall using the so-called compatibility relations. Since the blade shape is not compatible with the required pressure distribution, a non-zero velocity component normal to the blade wall evolves from the new flow calculation. The blade geometry is then modified by resetting the wall parallel to the new flow field, using a transpiration technique, and the procedure is repeated until the calculated pressure distribution has converged to the required one. Examples for both subsonic and transonic flows are presented and show a rapid convergence to the geometry required for the desired Mach number distribution. An important advantage of the present method is the possibility to use the same code for the design and the analysis of a blade. SYMBOLS a speed of sound area of the cell i, j A, B matrices of the system d, D dissipative terms F, G components of the flux K characteristic matrix l left eigenvector M isentropic Mach number n normal vector p total pressure p static pressure s; side of the cell i, j S characteristic surface To total temperature U vector of unknowns u, v velocity components V velocity greek symbols flow angle (with resp. axial) A eigenvalue v propagation direction p density v cascade solidity subscripts n normal component t tangential component 1 cascade inlet 2 cascade outlet superscripts a equation number new value at the time n + 1 old value at the time n req required value * intermediate value INTRODUCTION The design of more efficient blades is an important part of compressor or turbine development. In many cases this is still done in an empirical way by successive modifications of the blade geometry and verifications by wind tunnel testing or flow calculations. Such a process, however, can be very time consuming, depending on the designers experience, and may result in high design costs and delays. In the past, blade design methods have been developed, in which the problem is solved in an analytical way. These meth- Presented at the International Gas Turbine and Aeroengine Congress and Exposition Orlando, FL June 3-6, 1991 This paper has been accepted for publication in the Transactions of the ASME Discussion of it will be accepted at ASME Headquarters until September 3, 1991 Downloaded From: on 3/5/216 Terms of Use:

2 ods require severe simplifications and therefore restrictive assumptions in order to allow an analytical solution. The inverse problem for incompressible potential flow has been solved by conformal mapping (Lighthill, 1945, Schwering, 197, Papailiou, 1968) and/or by singularity methods (Murugesan and Railly, 1969, Ubaldi, 1984). Linearization of the compressible potential flow equations is possible by changing the variables and by calculating in the hodograph plane (Bauer et al., 1972, Sanz, 1988). The same equations may be solved in the potential and streamfunction plane (Stanitz, 1953, Schmidt, 198). These methods are restricted to blade design and are not suited for off-design analysis. They eliminate the need of a first guess of the geometry but can complicate the design of the stagnation point region and they are also restricted to shock-free flows. Non potential flow fields require to solve the Euler equations and a first guess of the geometry is necessary to define the calculation domain. This initial geometry may be modified during the flow calculation, until the imposed pressure or velocity distribution is reached (Meauze, 1982, Giles and Drela, 1987, Zannetti et al., 198,1984). Each blade modification is calculated in such a way that the slip condition at the wall is verified. The modification of the geometry can be performed by a purely mathematical algorithm minimizing an error function that expresses the difference between the desired pressure distribution and the one calculated with a direct solver. These methods have the disadvantage to be very expensive in terms of CPU time, because many flow analyses are usually required, but they provide a blade shape even if the required performances do not allow an exact physical solution (Vanderplaats, 1979, Hicks, 1981). Methods using a physical model to relate the geometrical changes to the error function are more efficient. The modification algorithm developed by Van den Braembussche et al. (1989) is based on the singularity methods and has been succesfully combined with both potential and Euler direct solvers. However problems encountered using vortices for transonic blade design have led to develope the present method where the blade modification is calculated by solving the Euler equations. For a system of n first order partial differential equations describing a convection process, the information may propagate like waves, in certain directions v, under some conditions. If the system is written in matrix form A-= k=1,..,m (1) the first condition is that the determinant Akvk must vanish. a +A Ux From this relation, n directions of propagation can be calculated, if arbitrary values of m 1 components of v are chosen. The second condition to have an hyperbolic problem is that the n solutions must be real. In that case, n wave front surfaces may be defined, which separate the points already influenced by the propagating disturbance from the points not yet reached by the information. These surfaces are called characteristic surfaces and transport certain properties of the flow. If one space variable, say x, is singled out and the corresponding Jacobian matrix A- is the unit matrix, this variable is called a time-like variable. The system (1) is written k ( ) = 2 and the characteristic condition becomes an eigenvalue problem where the characteristic directions are obtained as the eigenvalues of the matrix A k vkk=1,..,m-1 (3) If the n eigenvalues are real there are n characteristic surfaces which transport information in the (v1,.., v,,,_ 1 ) direction. Equation (2) is similar to the quasi-linear form of the unsteady two-dimensional Euler equations 8t +A + B 8U (4) y where U, A and B may be written, in function of the primitive variables p, V and p PERMEABLE WALL BOUNDARY CONDITIONS The analysis of a cascade geometry by a classical Euler solver provides a velocity field tangent to the blade wall, as imposed by the slip condition. This velocity field does not necessarily satisfy the required pressure distribution. Imposing simultaneously the pressure and the slip condition on a wall leads to an ill-posed problem from a mathematical point of view since the problem is over-determined. It is possible to impose the required pressure distribution if one allows a velocity component normal to the blade wall, which therefore has to be considered as permeable. This normal velocity component will be used for the geometry modification after convergence of the Time Marching calculation. Characteristic Surfaces and Wave-like Solutions The boundary conditions and the calculation of the unknowns at the blade wall have to be adapted to allow for a normal component of the velocity, since the information propagating along the streamlines does not propagate any more along the blade wall. A detailed discussion of the problem is given by Leonard (199). p u p v p u U 1/p v Uv A u B v 1/p p pat u paz v The matrix introduced in (3) may be written Kn =An.+Bn, VI The eigenvalues of Kn are: Vn, Vn, Vn+a and Vi a, defining the different speeds of propagation in an arbitrary direction K. The four eigenvalues are real and the system is therefore hyperbolic. Boundary Conditions The preceding considerations provide the necessary initial and boundary conditions to be imposed with a given system of differential equations, in order to have a well-posed problem. A solution of the system of n first-order partial differential equations can be written as a superposition of wave-like solutions corresponding to the n eigenvalues of the matrix K. For an hyperbolic problem, all the eigenvalues are real and no amplified mode is generated (an amplified mode is physically not acceptable). Therefore n boundary conditions have to be provided to Downloaded From: on 3/5/216 Terms of Use:

3 determine completely the solution. These boundary conditions have to be distributed along the boundaries at all values of t, according to the way the corresponding waves propagate across the boundary. If the information propagated by one wave front is impinging a boundary point, coming from the inside of the calculation domain (positive eigenvalue if Yc is the outgoing normal vector), the value of the corresponding unknown must be calculated from this information and not from a boundary condition. On the other hand, if the information comes from the outside of the calculation domain (negative eigenvalue), the value of the unknown at this boundary point must be imposed by a boundary condition. These considerations have to be applied in the different boundary problems of a blade-to-blade calculation. For the inlet_boundary (and if a subsonic axial velocity is considered), only Vn + a is positive and three boundary conditions must be imposed, usually the total conditions (p and To) and the inlet flow angle. If the slip condition is imposed on the blade wall (Vn = ) only the eigenvalue Vn a is negative and therefore only one boundary condition must be imposed, i.e. the velocity direction at that point. On the other hand, if the static pressure p is imposed on the blade wall, a velocity component normal to the blade can appear and, depending upon its sign, 2 or additional conditions must be imposed. Three problems have to be solved: what is the sign of the normal velocity component? what boundary conditions to apply in addition to the static pressure? how to calculate the value of the unknowns that are not imposed by the boundary conditions? These problems can be solved by introducing the compatibility relations. Compatibility Relations An alternative definition of characteristic surfaces and hyperbolicity can be obtained from the fact that wave front surfaces carry certain properties and that a complete description of the physical system is obtained when all these properties are known. This implies that the original system of equations, if hyperbolic, can be reformulated as differential relations written along the wave fronts (characteristic surfaces) only. The original system of equations can be transformed, through a linear combination, into an equivalent system of equations containing only derivatives along the characteristic surfaces S. For any equation of the system (2) we have: or l Lat +Ak r9x, (6) law l` at + Y A' ax ] = (7).-i j=1 where the i are Si arbitrary coefficients. The coefficients 1; are chosen such that equation (6) contains only derivatives in a two-dimensional subspace of the (x, y, t) space, namely the characteristic surface S. The condition for S to be normal to v - vj, + ny l z + provides the equation ljvl + j 11 [Kn-Jl; = dj (8) which expresses that l is an eigenvector of the matrix Kn (5), with v1 = A the corresponding eigenvalue. If the unsteady two-dimensional Euler equations are considered, the discretized form of the compatibility relations (6) is, for the equation a: new old (1o)Old p p + ^-oldl +... Ot /f n old (l )o ld P P + sold) = (9) 4 At 4 ( row The terms fold denote numerical approximations of the divergence terms; they can be estimated from the Euler equations. For the density we have: p+_ old p +F = Ot The compatibility relations become: (1 )old ( p ew p*) + (4)old (7 eu, V) (1) + (l4)old( pnew _ P ) = (11) where the superscript * means that this variable is estimated from the discretization of the Euler equations; then a new value of the unknown is computed using the compatibility relation, if the corresponding eigenvalue is positive. The four compatibility relations and the corresponding eigenvalues are: eigenvalue compatibility relation V n a2(pnew _ p p ew p*) = er ) + ( VnVtnew V t* = V,, a pa(vn ew V n)+(phew P')= Vn + apa(vn ew Vn) + (P new p er ) = where the values of the density p and the speed of sound a are taken at the previous time level. The first problem evoked in the previous section can be solved by the fact that the eigenvalue V + a is always positive, if the assumption is made of a subsonic normal component. This means that the corresponding compatibility relation may be used to calculate the value and the sign of Vn, since the information propagating with this wave comes from the inside of the calculation domain. The pressure pn ' in the relation is obviously the required pressure. If the normal velocity is positive, one boundary condition (the required static pressure) must be imposed, since only one eigenvalue Vn a is negative. If the normal velocity is negative, two additional boundary conditions must be imposed, since V, is negative as well. It has been found that the best solution is to impose the total pressure and total temperature at that point. The compatibility relation corresponding to V. a is never used since this eigenvalue is always negative. From the four variables p"', p',vn "" and j/7, new values of the primitive or of the conservative variables are calculated. THE EULER SOLVER A code solving the Euler equations has been developed in order to serve as a basis of the different steps of the development of Downloaded From: on 3/5/216 Terms of Use:

4 the inverse method. The code is based on a Time Marching procedure, a finite volume approach, and on a scheme investigated by Jameson (1981). Non-intersecting finite volumes are used together with a cellvertex approach for the space discretization. C-grids are used for a better leading edge description (fig. 1). n n Fig. 2 Blade modification Fig. 1 C-grid discretization for *a compressor blade This results in: or in discretized form: d(pvvon) = pv,,ds (15) For the np["ii_i time discretization, both second order and fourth order + As py i +2pjnIi-i = (16) viscosities are used. In order to preserve the conservative form of the scheme, the artificial dissipative terms are introduced by adding dissipative fluxes. For the cell i, j the equilibrium is written: The ingoing velocity f/"i i _ i is taken as the mean value of the tangential velocity along the normal direction at the point i 1, namely li-i u ) ij + y Fi,i si,i = Di,i + Vt"' Ii_1 ( 12 ) ^i^= 1 = 2 ( 17) k=1 ( The dissipative operator is defined by: D=,i = di 2 di-' 2 i + di,i+- 2 i (13) where the dissipative flux di+ L i is a blend of first and third order dissipative terms, adapted to the local pressure gradient by means of suitable sensors. A fourth order Runge-Kutta scheme is used for the timestepping, requiring a minimum of computer storage: U s = U td Ul = U id Ot RHS /4 U 2 = U ld At RHS 1 /3 (14) U3 = U Ot RHS 2 /2 U 4 = U id Ot RHS 3 U ew = U 4 This four-stage scheme is second order accurate for nonlinear problems and allows a maximum CFL number of 2^. MODIFICATION OF THE GEOMETRY The geometry modification algorithm calculates the position of the new streamlines, starting from the initial blade wall which is not a streamline any more. It is based on a transpiration model and makes use of the velocity component normal to the initial blade. The mass balance is applied in the cell defined by the points (i)otd (i 1) 1d (i)" and (i 1)" as shown in fig. 2. The outgoing velocity I/ j is calculated in the same way. Expression (16) allows the calculation of the shift Oni if Ani_ 1 is known. The modification starts at the stagnation point, where the value On is set to zero, and is performed separately for the pressure side and the suction side. The new suction and pressure sides are defined as streamlines of the flow satisfying the Euler equations, and therefore can not cross. This guarantees a blade with positive thickness if the numerical integration procedure and the normal velocity calculation is sufficiently accurate. The control of the trailing edge thickness is a very important aspect of any inverse method. Solving the (inviscid) Euler equations, the resulting blade thickness should be larger than the physical one to include also the suction and pressure side boundary layers. This is related to the problem of the existence of a solution. The required velocity distribution must be compatible with the free stream conditions upstream and downstream of the cascade and must result in a realistic blade. No exact formulation of the existency conditions is available for compressible flows. Some criteria have been proposed by Leonard (199), who also indicated a relation between the trailing edge thickness and the velocity level on pressure and suction sides near the trailing edge. If a sufficient thickness can not be achieved using this relation, the suction and pressure side contours can be rotated around the leading edge. In that case, the new blade shape will not provide the required velocity distribution, but will be the best possible approximation of it. 4 Downloaded From: on 3/5/216 Terms of Use:

5 RESULTS 1. In order to decrease the risk of searching a non-existent solution, we started the inverse calculations from existing blades, and we modified the corresponding Mach number distributions, obtained with the direct code. The accuracy of the method has been demonstrated by solving a problem for which the solution is known. The Mach number distribution of an existing geometry (the NACA-65 (12A )1 blade) has been imposed as the required Mach number distribution, starting from the NACA-65 (12A 5 )1 blade as initial geometry. The flow conditions are: =.7, p = 3 bar, T = 34 K, /3 1 = 45 deg, p 2 = bar. The cascade is defined by a stagger angle of 31 deg and a solidity of Z.6 U td S/C _O p,.8.7 /rte.6 LI U OpOp OpOpOOOO OOOO o Oo o Oo Fig. 3c Calculated (+) and required () Mach number distributions after 6 modifications The figure 3a shows the original and the required Mach number distributions. Because of the big difference between both distributions, an under-relaxation factor of.5 has been introduced into the first geometry modification, results of which are shown in fig. 3b. The results after 6 modifications (fig. 3c) show a good agreement between the calculated and the prescribed Mach number distributions. The tiny difference between the final geometry and the (12A 2 Isb)1 blade guarantees the accuracy of the method..1 S/C Fig. 3a Initial (+) and required () Mach number distributions 1..9 o.8.7 o o Z.6 c8.5 O S/C ^ Fig. 3b Calculated (+) and required () Mach number distributions after 1 modification Fig. 4a Original distribution of iso-density lines 5 Downloaded From: on 3/5/216 Terms of Use:

6 Z.9 V , a.i Z r-^ Fig. 4b I 1. 1 Initial (+) and required () Mach number distributions Fig. 4d Calculated (+) and required () Mach number distributions after 3 modifications r 1.4 E 1.Z J" a.a V 7.7 a Fig. 4c is 11 Calculated (+) and required () Mach number distributions after 2 modifications ty lines The second example concerns the design of a transonic compressor blade. The NACA-65 (12A 2Is6)1 is used as starting geometry. The flow conditions are: : M l =.8, p = 3 bar, T = 34 K, /31 = 45 deg, P2 = bar. The cascade geometry is defined by a stagger angle of 31 deg and a solidity of 1. The initial distribution of iso-density lines is shown in figure 4a. The imposed shock-free transonic Mach number distribution is compared to the initial Mach number distribution in figure 4b. Figures 4c and 4d show the convergence of the calculated Mach number distribution to the required one. The new distribution of iso-density lines is shown in fig. 4e. The original and final geometries are compared in fig. 4f. A thinner leading edge can be observed, as well as a less cambered blade shape. metries 6 Downloaded From: on 3/5/216 Terms of Use:

7 as z = a.'k a.2 ono ooo 1^ ' 1 1 '.^4 Fig. 5a Initial (+) and required () Mach number distributions ed a a Fig. 5c a 1.1 Calculated (+) and required () Mach number distributions after 2 modifications 8 /-r Fig. 5d Original anc Fig. 5b Calculated (+) and required () Mach number distributions after 1 modification The third example demonstrates the procedure in the design of a turbine blade. The starting geometry is that of the workshop VKI-LS 82-5 (Arts, 1982). The flow conditions are: p = 1 bar, T O = 278 K, i3 = deg, M 2 = 1.. The cascade geometry is defined by a stagger angle of -6 deg and a solidity of 5. A shock-free transonic Mach number distribution has been imposed on the suction side (fig. 5a). The calculated mach number distribution converges as close as possible to the required one (fig. 5b and 5c) but not completely, probably because there is no physical geometry corresponding to the required Mach number distribution. The original and final geometries are compared in fig. 5d. The main differenece between the two blades is the curvature of the suction side. Off-design performances of the new blade are calculated with the same code and are presented in fig. 5e for an exit Mach number of.9 and in fig. 5f for an exit Mach number of 1.1. CONCLUSIONS An analysis code was succesfully transformed into a design code by changing the boundary conditions and by means of a geometry modification algorithm. The required Mach number distribution is imposed, in terms of static pressure, directly on a permeable blade wall, during the flow calculation. 7 Downloaded From: on 3/5/216 Terms of Use:

8 z U z U V Cd S/C.1 S/C ig. 5e Off design (M 2 =.9) Fig. 5f Off design (M 2 = 1.1) Subsonic and transonic designs have been performed and the method shows a rapid convergence to the blade geometry corresponding to the required pressure distribution. Typical calculation time is twice the time for an analysis. The method is very accurate even for thick leading edges as occuring in turbines. It directly provides the blade shape and does not require post-processing to round off the leading edge. In case there is no blade shape that exactly satisfies the required pressure distribution, the Time Marching procedure will not converge to the required precision since this would imply crossing streamlines. However this method has the advantage to converge as close as possible to the required pressure distribution. The method can handle shock-free design; however since it allows shocks to occur, the shock-free flow is not a result of any limitation of the method, as it can be for irrotational isentropic codes. Another advantage of the present method is that the same code can be used for both design and off-design calculations. REFERENCES ARTS, T., 1982, "Numerical methods for flow in turbomachine bladings", Workshop on 2D and 3D flow calculations in turbine bladings, VKI LS BAUER, F., GARABEDIAN, P. and KORN, D., 1972, "Supercritical Wing Sections", Vols. I,II,III, Springer-Verlag, New York. GILES, M.B. and DRELA, M., 1987, "Two-Dimensional Transonic Aerodynamic Design Method", AIAA Journal, Vol. 25, No. 9, pp HICKS, R.M., 1981, "Transonic Wing Design Using Potential Flow Codes - Successes and Failures", SAE Paper JAMESON, A., SCHMIDT, W. and TURKEL, E., 1981, "Numerical Solutions of the Euler Equations by Finite Volume Methods Using Runge Kutta Time Stepping Schemes", AIAA Paper LEONARD,., 199, "Subsonic and Transonic Cascade Design", AGARD-R-78. LIGHTHILL, J.M., 1945, "A New Method of Two-Dimensional Aerodynamic Design", ARC R&M MEAUZE, G., 1982, "An Inverse Time Marching Method for the Definition of Cascade Geometry", Journal of Engineering for Power (ASME), Vol. 14, pp MURUGESAN, K. and RAILLY, J.W., 1969, "Pure Design Method for Airfoils in Cascade", Journal Mechanical Engineering Science, Vol. 11, No. 5, pp PAPAILIOU, K., 1968, "Blade Optimization Based on Boundary Layer Concepts", VKI Course Note 6. SANZ, J.M., 1988, "Automated Design of Controlled Diffusion Blades", ASME paper 88-GT-139. SCHMIDT, E., 198, "Computation of Supercritical Compressor and Turbine Cascades with a Design Method for Transonic Flows", Journal of Engineering and Power (ASME), Vol. 12, pp SCHWERING, W., 197, "Design of Cascades for Incompressible Plane Potential Flows with Prescribed Velocity Distribution", ASME paper 7-GT-57. STANITZ, J.D., 1953, "Design of Two-Dimensional Channels with Prescribed Velocity Distributions along the Channel Walls", NACA rep. No UBALDI, M., 1984, "Un metodo di progretto per profili in schiera basato sulla teoria delle equizioni integrali, AIMETA, VII Congr. Nazionale, Trieste. VAN DEN BRAEMBUSSCHE, R., LEONARD,. and NEK- MOUCHE, L., 1989, "Subsonic and Transonic Blade Design By Means of Analysis Codes", AGARD-CP-463. VANDERPLAATS, G.N., 1979, "Approximation Concepts for Numerical Airfoils Optimization", NASA TP-137. ZANNETTI, L., 198, "Time-Dependent Methodto Solve the Inverse Problem for Internal Flows", AIAA Journal, Vol. 18, No. 7, pp ZANNETTI, L. and PANDOLFI, M., 1984, "Inverse Design Technique for Cascades", NASA CR Downloaded From: on 3/5/216 Terms of Use: