TURBINE FORCED RESPONSE PREDICTION USING AN INTEGRATED NONLINEAR ANALYSIS A.I. Sayma, M. Vahdati & M. Imregun Abstract The forced response due to flow

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1 TURBINE FORCED RESPONSE PREDICTION USING AN INTEGRATED NONLINEAR ANALYSIS A.I. Sayma, M. Vahdati & M. Imregun 1 Imperial College of Science Technology and Medicine Mechanical Engineering Department Exhibition Road, London SW7 2BX, UK a.sayma@ic.ac.uk, m.vahdati@ic.ac.uk, & m.imregun@ic.ac.uk Number of pages : 45 Number of figures : 23 Number of tables : 1 Running title: Turbine forced response prediction 1 Author to whom correspondence should be addressed Turbine forced response prediction 1

2 TURBINE FORCED RESPONSE PREDICTION USING AN INTEGRATED NONLINEAR ANALYSIS A.I. Sayma, M. Vahdati & M. Imregun Abstract The forced response due to flow defects caused by the upstream blade rows is predicted for two turbines: intermediate pressure (IP) and low pressure (LP). The prediction method is based on an advanced numerical tool where the compressible viscous flow field is modelled by solving Favre-averaged Navier-Stokes equations with Baldwin and Barth turbulence model. The flow solution is coupled to a modal model of the structure and information is exchanged every time step between the fluid and the structural domains. The hybrid unstructured mesh is moved at each time step to follow the structural motion using a spring analogy. For the IP turbine, the method was used to rank two different designs of nozzle guide vanes. For the LP turbine, special emphasis was placed on predicting vibration amplitudes due to high and low engine order excitations. Predictions and measurements were found to be in good agreement for both turbines. Due to insufficient experimental data, it was difficult to assess the accuracy of the low engine order computations, although it was shown that the model was capable of undertaking such a task. Key words: Aeroengine vibrations, turbine forced response, aeroelasticity, turbomachinery CFD, unsteady time-accurate viscous flows. Turbine forced response prediction 2

3 Nomenclature A + o turbulence model constant = 26 A + 2 turbulence model constant = 10 B i C boundary integral of the flux structual damping matrix C μ turbulence model constant = 0.09 C ffl1 turbulence model constant = 1.2 C ffl2 turbulence model constant = 2.0 D IJs D 1 ;D 2 artificial dissipation along the side IJ s turbulence model parameters ~F invisicd flux vector F IJs numerical inviscid flux along side IJ s ~G viscous flux vector G IJs K M P Pr R (U) R e numerical viscous flux along side IJ s stiffness matrix mass matrix turbulence production term Prandtl number vector of fluxes Reynolds number ~R T turbulent Reynolds number S T U U I e f 2 h source term in the Navier-Stokes equations temperature vector of conservative vatiables solution vector at node I interal energy wall damping function enthalpy Turbine forced response prediction 3

4 ~n unit normal vector n s number of nodes connected toanode p; p(t) unsteady pressure ~q vector of the principal or modal co-ordinates ~u velocity vector in the absolute frame of reference ~v velocity vector in the relative frame of reference y + non-dimensional wall distance ~x mode shape vector x i spatial coordinate in direction i control volume boundary fi pseudo time step t physical time step Ω control volume [Φ] mode shape matrix Ω I fl ffi ij area of the control volume constant specific heat ratio Kronecker delta i vector of modal damping ~ IJs weight of the side IJ s generic primitive variable» turbulence model constant = 0.41 μ t μ l ν ρ ff ij ff ffl eddy viscosity molecular viscosity kinematic viscosity density viscous stress tensor turbulence model parameter! natural frequency Turbine forced response prediction 4

5 1 Introduction The forced response of bladed disks is a very common vibration problem during the development phase of new gas turbines. A primary mechanism of blade failure is high-cycle fatigue (HCF) caused by vibrations at levels exceeding material fatigue endurance limits. Preventing turbomachinery blade failures is a necessary goal for engine manufacturers and, to this end, it is essential to prevent excessive vibration in turbomachinery blading due to forced response under aerodynamic excitation. From the outset, it is appropriate to distinguish between two types of forced response. The first type, or classical forced response, is due to the excitation forces generated by the rotation of the bladed system past a pressure field, the strength of which varies periodically with angular position around the turbine. Suchflowvariations are mainly caused by the stator blades which act as upstream obstructions, and the rotor blades experience their wakes as time-varying forces with a frequency, or periodicity, based on the rotational speed. The spatial distribution of the forcing function will primarily be determined by the number of upstream stator blades and by its aliases with respect to the rotor blades. A Fourier transform of this forcing function will reveal the harmonics that will excite the assembly modes. Typically, such harmonics will excite high nodal diameter modes as their actual order is related to the blade numbers in the rotor/stator row of interest. Although it is difficult to predict the corresponding rotor blade vibration levels accurately, turbomachinery designers rely on Campbell diagrams (or their variants) which indicate the likelihood of encountering forced response resonances of the first type within the operating range. In principle, it is then possible to design the rotor wheels away from the primary resonances, subject to being able to predict the assembly's dynamic behaviour to a required degree of Turbine forced response prediction 5

6 accuracy. The second type of forced response, far less studied in the open literature, is much more difficult to deal with as the controlling parameters and the exact excitation mechanisms are poorly understood. However, the unsteady aerodynamic forcing function is known to be composed of low-order harmonics as it is responsible for exciting low-order nodal diameter assembly modes, hence the term low engine order (LEO) forced response. A detailed overview of turbomachinery aeroelasticity methods, including forced response prediction techniques, is discussed by Marshall & Imregun (1996). Most existing models are based on calculating the unsteady forces from the upstream and downstream distortions, independently of the blade's motion. The contribution of the blade vibration to the unsteady forces is usually calculated separately. The (positive) aerodynamic damping from the blade displacement is added to the (negative) contribution from the unsteady forcing to obtain the overall blade response. The advantage of performing forced response calculations in a coupled manner is that both the forcing and vibration effects are allowed to interact with each other, in an integrated fashion. There is experimental evidence that the unsteadiness due to blade motion can be significant (Manwaring et al. 1996). It is now appropriate to outline the sources and nature of various unsteady phenomena that are present in rotor-stator interactions. In the main, there are two sources of unsteadiness due to the relative motion of the stator and rotor rows. 1.1 Wake/rotor interaction The stator wakes, which can be assumed to be approximately steady in the stator frame of reference, are unsteady in the rotor frame of reference since the rotor is moving through them. Although the generation of the stator wakes is a viscous phenomenon, their subsequent interaction with the rotor blades is primarily an inviscid process. Therefore, it may be possible to perform an unsteady inviscid Turbine forced response prediction 6

7 computation for the rotor domain, with the wakes being specified as unsteady inflow conditions obtained from a previous viscous calculation. However, the preferred approach is to perform a single viscous calculation for both domains. 1.2 Potential stator/rotor interaction The unsteadiness can be explained by considering the three components of the pressure in the region between the stator and rotor bladerows: (i) uniform and steady, (ii) non-uniform but steady in the rotor frame, and (iii) non-uniform but steady in the stator frame. With rotation, both the stator and rotor blades experience unsteady forces that are due to the non-uniform pressure components. Such aninteraction is purely inviscid and can be modelled via the Euler equations. However, there is a need to consider the rotor and stator domains together when the spacing between the stator and rotor rows is small since the definition of unsteady rotor inlet boundary conditions will be very difficult in such cases. 1.3 Computational methods In recent years, the rapid development of numerical methods for the solution of the flow equations and the availability of powerful computers led to various prediction systems which are based on structured grids (Dawes 1988, Arnone 1995). However, the requirement to include complex geometric features such as tip gaps, cooling holes, snubbered fan blades, non-symmetric intake ducts, struts, etc can only be met via unstructured grids. Due to grid generation difficulties and flow solver limitations, unstructured grids generally use tetrahedral elements, even for turbomachinery applications (Vahdati & Imregun 1996). Although such grids are relatively easy to generate and efficient for capturing the inviscid features of the flow, the situation becomes more complicated in boundary layers, where large aspect ratio cells are required for computational efficiency. Such considerations led to the development ofhybrid grid models where hex- Turbine forced response prediction 7

8 ahedral or prismatic cells can be used in the boundary layers and tetrahedral and prismatic cells can be used to fill the domain away from the walls. For turbomachinery blades, Sbardella et al. (1997) presented a method to generate hybrid semi-structured grids where the boundary layers are filled with hexahedral cells and the rest of the domain is filled with triangular prisms. The same discretisation strategy will also be used here. 2 The flow model The unsteady, compressible, Favre-averaged Navier-Stokes equations, written in an arbitrary Eulerian Lagrangian (ALE) conservative form for a control volume Ω with boundary, take the form: Z I d U dω+ ~F 1 ~G dt Re ~n d = Z Ω S dω (1) ~n represents the outward unit vector of the control volume boundary. The viscous term ~ G on the left-hand side of equation (1) has been scaled by the reference Reynolds number for non-dimensionalization purposes. vector of conservative variables U is given by: 2 3 ρ U = 6 ρ~u ρe The solution (2) The inviscid flux vector ~ F has the following components: ~F = U~v pffi ij u j p (3) where ffi ij represents the Kronecker delta function and ~v is the velocity in the relative frame of reference. The pressure p and the total enthalpy h are related to Turbine forced response prediction 8

9 density ρ, absolute velocity ~u and internal energy e by two perfect gas equations: p =(fl 1) ρ "e j~uj2 2 # ; h = e + p ρ where fl is the constant specific heat ratio. The viscous flux vector ~ G has the following components 2 ~G = 6 4 u k ff ik + fl fl 1 0 ff ij μl Pr l + μ t i The viscous stress tensor ff ij is expressed using the eddy viscosity concept which assumes that, in analogy with viscous stresses in laminar flows, the turbulent stresses are proportional to the mean velocity gradients: ff ij =(μ l + μ t ) j + ffi ij ~r i μ l represents the molecular viscosity given by the Sutherland's formula, μ t denotes the turbulent eddy viscosity, which must be determined by a suitable turbulence model. The value of is given by the Stokes relation = 2 μ while the laminar 3 Prandtl number, Pr l, is taken as 0.7 for air. The turbulent Prandtl number, Pr t, is taken as 0.9. The eddy viscosity μ t is calculated using the one-equation turbulence model of Baldwin & Barth (1991) as follows: The equation is then solved for the variable ν ~ (4) (5) (6) ν t = μ t ρ = C μν ~ RT D 1 D 2 (7) ν ~ R T j ν ~ R T =(Cffl2 f 2 C ffl1 ) + (ν l + ν t =ff ffl ν ~ RT where the production term P is given by: P = ν k 1 ff k! 2 j k q ν R ~ T P ν RT k (8) Turbine forced response prediction 9 (9)

10 The wall damping function is given by: f 2 = C ffl1 C ffl2 + ψ 1 C ffl1 1 C " ffl2 q D 1 D 2 + y+»y + + D 1D 2! p D1 D 2 ψ D2 A + o e y+ =A + o + D 1 e y+ =A + 2 A + 2!# (10) where the parameters D 1 and D 2 are defined as: D 1 =1 e y+ =A + o and D 2 =1 e y+ =A + 2 (11) The model closure coefficients are defined as: C ffl1 =1:2; C ffl2 =2:0; C μ =0:09; A + o =26; A + = q =(C ffl2 C ffl1 ) C μ =» 2 ;» =0:41 (12) ff ffl 3 Numerical methodology The three dimensional spatial domain is discretised using unstructured grids which, in principle, can contain cells with any number of boundary faces. The solution vector is stored at the vertices of the cells. The present work uses semi-structured grids for their computational efficiency, although the solver is written for general hybrid unstructured grids. To achieve further computational efficiency, the mesh is represented using an edge base data structure. In this approach, the grid is presented to the solver as a set of node pairs connected by edges. The edge weights representing the inter-cell boundaries are computed in a separate pre-processor stage. Consequently, the solver has a unified data structure for which the nature of the hybrid mesh is concealed from the main calculations loops. The flow model will now be explained in more detail. For clarity, the numerical discretization of the flow equations will be illustrated on a 2D mesh. However, the resulting formulation is equally applicable to 3D cells. Using an edge-based scheme, the typical 2D mesh of Fig. 1 can be discretised by connecting the Turbine forced response prediction 10

11 median dual of the cells surrounding an internal node. For internal node I, the semi-discrete form can be written as: d (Ω I U I ) Xn s 1 + dt 2 j~ IJ s j (F IJs G IJs )+B i = Ω I S I (13) s=1 where Ω I is the area of the control volume (shaded area in Fig. 1), U I is the solution vector at node I, n s is the number of sides connected to node I, F IJs and G IJs are the numerical inviscid and viscous fluxes along side IJ s, and B i is the boundary integral. The side weight ~ IJs is given by the summation of the two dual median lengths around the side times their normals. For example, the weight of the side connecting nodes I and J 1 is given by: ~ IJ1 = ~ J1 I = ~ AB + ~ BC (14) 3.1 Inviscid fluxes The inviscid fluxes in (13) are expressed using a central difference formulation with a suitable artificial dissipation which is required to stabilize the scheme. Thus, inviscid fluxes can be written as: ~FI + F ~ Js D IJs (15) where D IJs F IJs = ~ IJs j~ IJs j is the artificial dissipation along the side IJ s. The artificial dissipation is based on an upwind scheme developed by Swanson & Turkel (1992), and Jorgenson & Turkel (1993). The scheme consists of a mixture of second and fourth order artificial viscosity. The fourth order terms ensure the stability of the scheme in smooth flow regions The second order terms are required to damp numerical oscillations in the vicinity of discontinuities where the scheme reverts to first order using a pressure based sensor. 3.2 Viscous fluxes The viscous fluxes are treated within the same edge-based data structure framework, provided the gradients of the primitive variables are known at the mesh Turbine forced response prediction 11

12 nodes. Using the edge weights of (14), these gradients can be calculated from the formula: " x j I = Xn s s=1 1 2 IJs ( I + Js )+B i (16) where represents a generic primitive variable and B i is the boundary integral arising from the contributions of the boundary faces at the domain boundaries. This formulation results in a viscous flux scheme which uses information from two layers of points surrounding the point under consideration. The choice of this scheme is purely for computational efficiency and storage economy. The use of a finite element approach will require the storage of nine additional quantities per side. In any case, numerical experiments show that there is negligible difference between the two approaches. 3.3 Implicit temporal discretization Equation (13) can be expressed in the form: as: d (Ω I U I ) = R (U) (17) dt A second order implicit backward time integration of (17) can be expressed 3(ΩU) n +1 4(ΩU) n I I +(ΩU)n 1 I 2 t = R (U n +1 ) (18) where n denotes the time level. The implicit non-linear system of equations given by (18) needs to be solved every time step. An iterative equation is constructed from (18) by simply adding a pseudo-time derivative term U fi side. Indicating with U m U fi + 3(ΩU)n +1 4(ΩU) n I I +(ΩU)n 1 I 2 t the m th approximation to U n +1, equation (19) can be rewritten as 1 Ω n +1 I fi U I + 2 t to the left-hand = R (U n +1 ) (19) 3Ωn +1 I U m I 4(ΩU)n I +(ΩU)n 1 I 2 t = R m +1 (20) Turbine forced response prediction 12

13 where U I = U m +1 I U m I, fi represents the pseudo-time step. Linearizing the right-hand side of (20) around U m I, the pseudo-time integration, which advances the solution from t n to t n +1, becomes: ψ Ω n +1 I Ω n +1 I fi t E n I J m! U I = R m 3 2 Ω n +1 I t Um I E n I (21) involves the portion of the physical time derivative at previous time steps and is invariant during the iteration process. E n I = 4(ΩU)n I (ΩU)n 1 I 2 t (22) The left-hand side of (21) contains a portion of the physical-time derivative in order to reduce the pseudo time-step in regions of the flow where the ratio fi pseudo/physical time step,, becomes large ( Melson et.al 1993). Equation t (21) is iteratively solved until the term U I is driven to a specified small tolerance. Within this iteration level, a Jacobi sub-iteration procedure is performed to solve the linearized system of equations described by (21). Time accuracy is guaranteed by the outer iteration level where the time step is fixed throughout the solution domain, while the inner iteration procedure can be performed using traditional acceleration techniques such as local time stepping and residual smoothing. 3.4 Boundary conditions Whereas the far-field boundaries for isolated airfoils can be taken many chords away, the boundaries are typically less than one chord away for most turbomachinery applications. This situation may lead to computational inaccuracies if the boundary conditions are not suitably formulated. Various techniques have been developed to minimize the reflection of the outgoing waves (Engquist & Majda 1977, Higdon 1986, Ferm 1995, Giles 1990) and an overview is given by Givoli (1991). Here two different set of treatments, one for steady state computation and the other for unsteady computation, are used. The steady-state boundary Turbine forced response prediction 13

14 treatment is based on the characteristics of the Euler equations. In particular, the steady-state boundary conditions are obtained from the linear, harmonic unsteady non-reflecting boundary conditions, as a limiting case of zero-frequency unsteadiness. The resulting non-reflecting boundary conditions are exact for linear solutions at the far-field boundary. The treatment of such boundary conditions for 2D turbomachinery applications can be found in Giles (1988). An extension to 3D is reported by Giles (1990). Two further types of boundary conditions are needed for turbomachinery calculations: solid wall and periodicity. On solid walls, the pressure is extrapolated from the interior points and the slip (inviscid) or no-slip (viscous) conditions are used to compute the other quantities. An extra boundary condition for the heat flux is employed for viscous flow calculations. Using an edge-based data structure, the periodicity is handled in a straightforward way as long as the points in the two periodic boundaries are located at same axial and radial coordinates. 3.5 Treatment of the sliding boundaries From a computational point of view, one major difficulty in simulating rotorstator flows arises because of the relative motion between the rotor and stator blades. One of the most straightforward solutions to this problem is to use two grids that move relative to each other. Typically, one would use a stationary grid to discretize the stator blades and a moving grid (stationary with respect to the rotor) to represent the rotor blades. For a given rotation between the stator and the rotor bladerows, the flow solution at the interface must be determined. In the procedure used here, the solution is updated at the interface in a conservative manner by linearly interpolating the variables in the stator computational domain to obtain rotor fluxes, and in the rotor computational domain to obtain the stator fluxes. The fluxes are computed by using a characteristic technique which allows the correct propagation of the information and the interface is up- Turbine forced response prediction 14

15 dated at the end of each time step. In other words, flow data are exchanged between the two grids via specially-formulated boundary conditions at the interface. Numerical experience indicates that the current implementation, based on the formulation of Rai (1986) is numerically stable, spatially and temporally accurate, and conservative so the flow discontinuities can move from one grid to another without causing distortions. 4 Structural model 4.1 Coupling of the fluid and structural domains As long as the structural behaviour can be assumed to be linear, the structural equations of motion can be uncoupled by a co-ordinate transformation via the modeshape matrix. The reduced modal equations can be solved by timemarching, with the modal matrix providing the link between the principal coordinates in the equation of motion (EOM) and the physical co-ordinates of the structure. The modeshape vectors can then be interpolated onto the aerodynamic grid points at the start of the calculation. Although no further interpolation is required, the mesh is still moved at each time step to accommodate the aeroelastic motion, a feature that will be discussed later. The aeroelastic EOM can be written as: [M] n ~ ẍ o +[C] n ~_x o +[K] f~xg = fp (t) ~ng (23) where M; C; K are the mass, structural damping and stiffness matrices, ~x is the mode shape vector, p (t) is the pressure and ~n is the normal unit vector of the blade's surface. The free vibration problem can be solved to yield natural frequencies,! i, and the mass-normalised modeshape matrix [Φ]. The centrifugal stiffening effects are taken into account by adding the appropriate terms to the stiffness matrix. Equation (23) is usually solved for a typical sector, which also Turbine forced response prediction 15

16 includes the disk for turbine cases. The modeshapes of the full assembly are then obtained by expanding the cyclic symmetry mode shapes. Many such typical sector analyses are performed to cover all assembly modes of interest. The required co-ordinate transformation is: f~xg =[Φ] f~qg (24) where ~q is the vector of the principal or modal co-ordinates. Using equations (23) and (24) and pre-multiplying by [Φ] T : [Φ] T [M][Φ] n ~ qo +[Φ] T [C][Φ] n ~_qo +[Φ] T [K][Φ] f~qg =[Φ] T fp (t) ~ng (25) The structural equations of motion can be reduced further by removing both the co-ordinates and modes that are of no interest in the flutter calculations. Assuming proportional damping (with no loss of generality) and using the orthogonality properties of the system matrices with respect to the mode shape matrix, one obtains: n o n o i ~ q ~_q N + [diag (2 i! i )] N N N + h diag! 2 i N N f~qg N = [Φ] T m N fp (t)g m = h ~Π(t) i N where! i and i are the natural frequency and modal damping for mode i, N is the number of structural coordinates and m is the number of modes. The RHS vector of modal forces is formed as follows: ( na ) X ~Π(t) =[Φ] T fp (t) ~ng = p i ffi i;r ~n i (27) i=1 r=1;m (26) where n a is the number of aerodynamic nodes on the blade's surface. In this form, the equations of motion can be solved by any standard numerical integration scheme and the self-starting, second-order accurate and unconditionally stable Newmark discretization (Newmark 1959) was used in the currentwork. Turbine forced response prediction 16

17 4.2 Mesh movement When undertaking a forced response aeroelasticity analysis, it is desirable to move the fluid mesh according to the instantaneous position of body under consideration so that the blade vibration can be included in the calculations. This requirement is met by using an algorithm which considers the mesh as a network of springs whose extension/compression is prescribed by the mode shape at the blade surface and becomes zero at the farfield. At each node, the spring stiffnesses are allocated values that are inversely proportional to the length of the shared edge lengths. The CFD algorithm takes full account of the unsteady fluxes which arise due to cell volume changes at each time step. 5 Case studies The above methodology was used to develop the code AU3D which deals with both steady and unsteady flow predictions as well as aeroelasticity analyses consisting of flutter and forced response calculations. The present work compares AU3D predictions to measured data for two cases which are presented in the following subsections 5.1 Case 1: IP turbine The forced response for an intermediate pressure (IP) turbine stage with 26 NGVs and 130 rotor blades will be studied first. The aim was to compare the effects of two different types of NGVs, hereafter referred to as Configuration R and Configuration S, on the vibration levels of the rotor blade. The simulations were performed using a sector of 1 NGV and 5 rotor blades. Two part-speed conditions, associated with 26 nodal diameter assembly modes, were investigated. The first condition corresponds to the first torsion (1T) mode while the second one corresponds to the second bending (2F) mode. Initially, it was decided to Turbine forced response prediction 17

18 perform separate computations for the stator and rotor domains. Since the wake generation is a viscous process, the NGV domain was modelled via a viscous flow representation while the rotor domain was treated inviscidly as the interaction of the wake with the rotor blade row is, essentially, aninviscid phenomenon. Steady-state solutions were obtained for single NGV and rotor blade passages using the measured boundary conditions. The inviscid rotor domain mesh containined about 22,000 points. Steady-state pressure contours at mid section are shown in Fig. 2. The viscous meshes of both types of NGVs contained about 100,000 points and these are shown in Figs. 3 and 4. The steady state pressure contours for both NGVs for the 1T resonance case are shown in Figs. 5 and 6 respectively. Similarly, solutions were also obtained for the 2F case. A view of the assembly mesh used in the unsteady computations is shown in Fig. 7. The relative orientation of the rotor blades and the two sets of NGVs is important in understanding the resulting interactions. The two blade rows are very close at the hub section, a feature that indicates that coupled rotor-stator computations may be necessary. The structural mode shapes for the 1T and 2F modes are shown in Fig. 8. These mode shapes where interpolated onto the aerodynamic mesh and used throughout the unsteady computations. The expanded single-stage steady-state solutions were used as initial conditions for starting the unsteady flow computations. The model allows the rotor blades to vibrate and the modal and actual displacement time histories can thus be predicted for the 1T and 2F resonances Study of the 1T resonance The unsteady coupled viscous simulations were performed for about 1.5 rotations of the rotor. The circumferential variation of the pressure distribution at the mid-passage line at the outflow surface is shown in Fig. 9. It can bee seen that the unsteady pressure at mid section for Configuration S is about 1.4 times Turbine forced response prediction 18

19 that of Configuration R. Moreover, there is a shock crossing the middle section in Configuration S which cannot be seen in Configuration R. The effect of this shock is very clearly visible from the instantaneous pressure contours of Fig. 10. In Configuration S, the shock interacts with the rotor blades, and thus produces a higher level of unsteady forcing than that in Configuration R. It is interesting to note that the shock in Configuration S is predicted to be weaker by the steady-state computation for which circumferentially averaged outflow boundary conditions were used. This finding confirms the need to perform multi-row computations. The modal displacements are compared in Fig. 11 for both sets of NGVs. From these figures and the magnitude of the actual displacements of the rotor blade, two observations can be made. The first one is that Configuration S NGVs produce an unsteady force, the magnitude of which is about 1.4 times that produced by Configuration R NGVs. The measured force ratio is about 1.5, confirming the validity of the analysis. Quantitatively, the stress levels were under-predicted by about 15%, an agreement which is considered to be satisfactory. The second observation is that there is a noticeable amount of aerodynamic damping for both configurations. Since the calculations were performed assuming zero structural damping, the amount ofaerodynamic damping was obtained directly from the time histories. Using such an approach, an equivalent damping value of 5% was estimated. Assuming 1% inherent material damping, the amount of aerodynamic damping is found to be 4%, which is, again be consistent with the measured value. A second set of calculations was performed for an inviscid NGV domain. These showed very little difference from the previous set of viscous computations. This finding indicates that the forcing in this case is mostly due to potential effects, a feature that is consistent with the fact that the stress pressure ratios were similar to the unsteady pressure ratios of Fig. 9. Turbine forced response prediction 19

20 5.1.2 Study of the 2F resonance The circumferential variation of the pressure distribution along the mid-passage line at the outflow surface is shown in Fig. 12. It is seen that the unsteady pressure distribution is only about 6% higher for Configuration S than Configuration R. There is no shock at this particular speed and, consequently, the contribution of the potential unsteady effects is expected to be lower. This expectation was confirmed by undertaking an inviscid analysis of the stator domain as before. The modal forces of the viscous analysis were found to be 30% higher than those of the inviscid analysis. Similarly, the modal displacements for the two sets of NGVs are compared in Fig. 13. The resulting stress ratio is about 1.5. Again, this value compares very well with the measured ratio of 1.4. However, quantitatively, the predictions were about 50% lower than the measurements for both configurations. explanation is the possibility of the forcing being dominated by the viscous effects. Such a situation would require a much finer mesh to predict the magnitude of the unsteady forcing correctly. 5.2 Case 2: LP turbine An LP turbine stage with 24 stator and 96 rotor blades will be studied next. The structural model is obtained via a standard finite element formulation and it is incorporated into the aeroelasticity calculations in the form of a modal representation. The main unsteadiness is due to the 24th engine-order (EO) excitation which affects the rotor blade's third bending mode (3F). A further unsteadiness is known to result in a 3rd EO excitation which drives the first bending mode (1F) into resonance. As mentioned earlier, the excitation mechanisms for the 3F/24EO and the 1F/3EO modes are fundamentally different in nature. The first resonance is due to the wakes from the stator blades while the second one is generated by some general unsteadiness of the flow field. These mechanisms will now be investigated in more detail. Turbine forced response prediction 20

21 5.2.1 Study of the 3F/24EO resonance The steady flow analysis was conducted for a single passage for both the stator and rotor blades (Figs. 14 & 15). The resulting pressure contours are shown in Fig. 16. The unsteady flow analysis was conducted in a multi-row fashion using a sector of 1 stator and 4 rotor blades. The simulation was carried out for about 0.02 seconds, a time interval that corresponds to about 4.8 revolutions of the rotor. The mesh, shown in Fig. 17, consisted of about 210,000 points. About 100,000 grid points were placed in the stator domain which was modelled in a viscous fashion. The steady-state results were expanded so that they could be used as initial conditions for the unsteady flow analysis. The instantaneous Mach number and pressure contours are plotted in Fig. 18. These plots show an interaction of the shock on the suction side of the stator blade and the wake of the next blade. There is a further interaction of the weak stator blade trailing edge shock with the strong suction side shock of the previous stator blade. One can also observe an increase in the stator boundary layer which is induced by the shock. The strong influence of the rotor domain flow field on the stator domain flow field indicates a close coupling of the flow through the two sets of bladerows. The position of the rotor blades relative to the stator blades dictates the strength of the stator blade trailing edge shock, as well as the position of the wake, a feature that can be seen by observing the flow field for several rotor. The pressure variation along the mid-passage line at the outflow surface is shown in Fig. 19. The predicted modal displacement time histories showed that the effects of the fluid damping were small. Therefore, the resonant amplitude is likely to be controlled by the amount of inherent mechanical damping. The actual displacements were computed for Q-factors of 36, 50 and 100. The corresponding maximum displacements are summarised in Table 1 which also includes the corresponding measured value. It is clearly seen that a Q-factor of about 80 is likely Turbine forced response prediction 21

22 Q factor 2 Normalised Displacement Measured Displacement Table 1: Variation of the maximum response with the Q-factor to provide a very good agreement with the measured value. Given the uncertainties in estimating the mechanical damping and measurement errors, the level of agreement was considered to be satisfactory Study of the 1F/3EO resonance The low-engine order excitation arises from a loss of symmetry in the flow and here it is proposed to investigate the effects of stator blade throat width variations. Although such variations are known to be random, it was decided to use a certain pattern so that it would be easier to establish a relationship between this imposed non-uniformity and the ensuing unsteady forcing. Three stator blade passages, placed at 120 degrees, were assumed to be 3% wider than the rest and a wholeannulus, two-row mesh was created for the investigation of the low engine order excitation. The mesh, shown in Fig. 20, consisted of about 400,000 points, of which 100,000 were in the stator domain. To reduce the computational cost, the blade flexibility was not included in the calculations, a feature that also neglects fluid damping. The unsteady flow calculations were performed for about seconds, which corresponds to about 4.3 revolutions of the rotor. The pressure contours at the outflow of the stator domain are shown in Fig. 21. Selecting a constant radial distance halfway between the hub and the tip, the pressure distribution of Fig. 21 can be plotted along a circumferential line, the x axis representing the time instants seen by the rotor domain (Fig. 22a). This 2 Q-factor = 1/(hysteretic damping) Turbine forced response prediction 22

23 result can be compared to that plotted in Fig. 22b which corresponds to an assembly with equally-spaced stator blades. A Fourier transform of the two pressure distributions is shown in Fig. 23. As expected, non-uniform blade spacing produces additional lower-order harmonics which can produce low engine order excitation. 6 Conclusions A general non-linear coupled aeroelasticity analysis method was presented for the forced response analysis of turbine blades. The method was applied to two turbine rotors. In the first case, it was used to rank two different NGV designs for an IP turbine. Although good agreement was achieved against measured data, the forcing levels were under-predicted at low speeds for which the viscous effects may be important. It is anticipated that finer meshes are needed to capture such viscous effects and hence to improve the accuracy of the predictions. In the second case, forced response predictions were performed for an LP turbine stage to evaluate the vibration amplitudes under both high and low engine order excitation. For high engine order excitation, the predictions were satisfactory, provided a good estimate of the mechanical damping was available. For low engine order excitation, unavailability of measured data did not allow an accurate assessment of the predictions. However, it was shown that the numerical model was able to deal with low engine order excitation. 7 Acknowledgements The authors would like to thank Rolls-Royce plc for both sponsoring the work and allowing the publication of this paper. Turbine forced response prediction 23

24 References Arnone, A. (1995). Multigrid Methods for Turbomachinery Navier-Stokes Calculations. In W. Habashi (Ed.), Solution Techniques for Large-Scale CFD Problems. Baldwin, B.S. and Barth, T.J. (1991). A one-equation turbulence transport model for high reynolds number wall-bounded flows. In AIAA 29 th Aerospace Sciences Meeting. AIAA Dawes, W. N. (1988). Development of a Three-dimensional Navier-Stokes Solver for Application to all Types of Turbomachinery. ASME paper 88- GT-70. Engquist, B. and Majda, A. (1977). Absorbing Boundary Conditions for The Numerical Simulation of Waves. Mathematics of Computation 31, Ferm, L. (1995). Non-Reflecting Boundary Conditions for the Steady Euler Equations. Journal of Computational Physics 122, Giles, M. B. (1988). Non-Reflecting Boundary Conditions for the Euler Equations. Technical Report TR-88-1, Computational Fluid Dynamics Lab, MIT. Giles, M. B. (1990). Non-Reflecting Boundary Conditions for the Euler Equations Calculations. AIAA Journal 28, Givoli, D. (1991). Non-reflecting Boundary Conditions. Journal of Computational Physics 94, Higdon, R. L. (1986). Absorbing Boundary Conditions for Difference Approximations to the Multi-Dimensional Wave Equation. Mathematics of Computation 47, Jorgenson, P. C. and Turkel, E. (1993). Central Difference TVD Schemes for Time Dependent and Steady State Problems. Journal of Computational Turbine forced response prediction 24

25 Physics 107, Manwaring, S. R. Rabe, D. C., Lorence, C. B., and Wadia, A. R. (1996). Inlet distortion generated forced response of a low aspect ratio transonic fan. ASME paper 96-GT-376. Marshall, J. G. and Imregun, M. (1996). A review of aeroelasticity methods with emphasis on turbomachinery applications. Journal of Fluids and Structures 10, Melson, N. D., Sanetrik, M. D. and Atkins, H. L. (1993). Time-accurate Navier- Stokes Calculations with Multigrid Acceleration. In Proceedings of the Sixth Copper Mountain Conference on Multigrid Methods. Newmark, N. M. (1959). A method of computation for structural dynamics. ASCE Journal of Engineering Mechanics Division 8, Rai, M. M. (1986). implicit conservative zonal boundary scheme for Euler equations calculations. Computers and Fluids 14 (3), Sbardella, L., Sayma, A. I. and Imregun, M. (1997). Semi-Unstructured Mesh Generator for Flow Calculations in Axial Turbomachinery Blading. In T. Fransson (Ed.), 18th International Symposium on Unsteady Aerodynamics and Aeroelasticity of Turbomachines. Kluwer Academic Publishers. Swanson, R. C. and Turkel, E. (1992). On Central-Difference and Upwind Schemes. Journal of Computational Physics 101, Vahdati, M. and Imregun, M. (1996). Non-linear Aeroelasticity analysis of a fan blade using unstructured dynamic meshes. IMechE Journal of Mechanical Engineering Science 210, Turbine forced response prediction 25

26 List of Figures 1 Typical 2D mixed-cell mesh Steady-state pressure contours for the rotor passage at mid-section 28 3 Viscous mesh for the Configuration R NGV Viscous mesh for the Configuration S NGV Steady-state solution at mid-section - Configuration R NGV Steady-state solution at mid-section for Configuration S NGV View of the assembly mesh for Configuration R NGV Structural modes used in the analysis Pressure distribution along the mid-passage line Pressure contours at a section close to hub Comparison of modal displacements for the 1T case Pressure distribution along the mid-passage line - 2F Comparison of results for the 2F case Single passage mesh for the stator blade Single passage mesh for the rotor blade Steady-state pressure contours for the stator and rotor blades Multi-row sector mesh for aeroelasticity calculations Instantaneous Mach number and pressure contours Unsteady pressure distribution along a circumferential line Whole assembly multi-row mesh for low engine order excitation Unsteady pressure contours at the stator outflow Unsteady pressure distribution along a circumferential line Fourier transform of the pressure distribution in Figs. 22a & 22b. 45 Turbine forced response prediction 26

27 J4 J3 J2 I A J5 B J1 C η IJ1 J6 J7 Figure 1: Typical 2D mixed-cell mesh Turbine forced response prediction 27

28 Figure 2: Steady-state pressure contours for the rotor passage at mid-section Figure 3: Viscous mesh for the Configuration R NGV Turbine forced response prediction 28

29 Figure 4: Viscous mesh for the Configuration S NGV Figure 5: Steady-state solution at mid-section - Configuration R NGV Turbine forced response prediction 29

30 Figure 6: Steady-state solution at mid-section for Configuration S NGV Turbine forced response prediction 30

31 Figure 7: View of the assembly mesh for Configuration R NGV Turbine forced response prediction 31

32 (a) 1T mode (b) 2F mode Figure 8: Structural modes used in the analysis Configuration R Configuration S 3.55 Pressure Coefficient Tangential coordinate Figure 9: Pressure distribution along the mid-passage line Turbine forced response prediction 32

33 (a) Configuration R NGV (b) Configuration S NGV Figure 10: Pressure contours at a section close to hub Turbine forced response prediction 33

34 1.5 x Modal displacement (m/sqrt(kg))) Configuration S Configuration R Time (seconds) Figure 11: Comparison of modal displacements for the 1T case 1.0 Configuration R Configuration S Normalised Pressure Coefficient Tangential coordinate Figure 12: Pressure distribution along the mid-passage line - 2F Turbine forced response prediction 34

35 2 x 10 5 Modal displacement (m/sqrt(kg))) Configuration S Configuration R Time (seconds) Figure 13: Comparison of results for the 2F case Turbine forced response prediction 35

36 Figure 14: Single passage mesh for the stator blade Turbine forced response prediction 36

37 Figure 15: Single passage mesh for the rotor blade Turbine forced response prediction 37

38 (a) Stator (b) Rotor Figure 16: Steady-state pressure contours for the stator and rotor blades Turbine forced response prediction 38

39 Figure 17: Multi-row sector mesh for aeroelasticity calculations Turbine forced response prediction 39

40 (a) Mach number (b) Pressure Figure 18: Instantaneous Mach number and pressure contours Turbine forced response prediction 40

41 Pressure theta (rad) Figure 19: Unsteady pressure distribution along a circumferential line Turbine forced response prediction 41

42 Figure 20: Whole assembly multi-row mesh for low engine order excitation Turbine forced response prediction 42

43 Figure 21: Unsteady pressure contours at the stator outflow Turbine forced response prediction 43

44 Pressure Time (s) x 10 3 (a) Non-uniform spacing Pressure Time (s) x 10 3 (b) Uniform spacing Figure 22: Unsteady pressure distribution along a circumferential line Turbine forced response prediction 44

45 db pairs of passages, 3% larger Equal passages Frequency Hz Figure 23: Fourier transform of the pressure distribution in Figs. 22a & 22b Turbine forced response prediction 45