Investigation of Efficient CFD Methods for the Prediction of Blade ...

Proceedings of ASME Turbo Expo 2013: Turbine Technical Conference and Exposition GT2013 June 3-7, 2013, San Antonio, Texas, USA

GT2013-95005

INVESTIGATION OF EFFICIENT CFD METHODS FOR THE PREDICTION OF BLADE DAMPING

Robin Elder, Ian Woods PCA Engineers Limited Lincoln, UK

Sunil Patil ANSYS Inc Lebanon, NH, USA

ABSTRACT Accurate and efficient prediction of blade damping is one essential element in the engineering of durable and reliable compressors and turbines. Over the years, a variety of empirical and linearized methods have been developed and used, and have served well. Recently, the development of efficient unsteady CFD methods combined with an expansion in available and affordable computing power has enabled CFD analysis of blade damping. This paper looks at the prediction of aerodynamic blade damping using some recently developed CFD methods. Unsteady CFD methods are used to predict the fluid flow in a transonic fan rotor, with tip Mach number of about 1.4. Deformation of the blade is determined from a mechanical pre-stressed modal analysis. In this investigation, blade motion for the first bending moments is prescribed in the CFD code, for a range of nodal diameters. After periodic unsteady solutions are obtained, damping coefficients are calculated based on the predicted blade forces and the specified blade motion. Traditional unsteady CFD methods require the simulation of many blades in a given row, depending on the nodal diameter. For instance, for a nodal diameter of four, a wheel with 22 blades would require simulation of eleven blades. Computational methods have been developed which now enable simulation of only a few (1 or 2) blades per row yet yield the full sector solution, thus providing considerable savings in computing time and machine resources. The properties of the available methods vary, but one method, the Fourier Transformation method, has the property that it is frequency preserving, and hence suitable for the present task.

William Holmes, Robin Steed, Brad Hutchinson ANSYS Inc Waterloo, ON, Canada

Fourier Transformation predictions, for a variety of nodal diameters, are compared with full sector predictions. Positive damping was predicted for this range of nodal diameters at design speed near peak efficiency operating condition indicating a stable system. The Fourier Transformation predictions for blade aerodynamic damping match very closely the reference full sector solutions. The Fourier transformation methods also provide solutions 3.5 times faster than average periodic reference cases. NOMENCLATURE A FT IBPA N N360 ND PAM S SST T U V W k p t x, y, z u,v,w ρ

1

Blade Surface area Fourier Transformation Inter-Blade Phase Angle Number of blades simulated Number of blades in 360 Nodal Diameter Phase Angle Multiplier Source term Shear Stress Transport turbulence model Vibration cycle time period, or transformation matrix Fluid velocity vector Volume of the control volume Velocity of the control volume boundary Turbulent kinetic energy Blade surface pressure Time Cartesian coordinates Modal displacement Density

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σ Г Ф ψ ω ώ

Damping coefficient Diffusion coefficient Conservative variable Phase angle Mode natural frequency Specific dissipation rate of turbulent kinetic energy

1 INTRODUCTION The ‘flutter’ of blades within compressors and turbines has been a serious cause of machine failure which has been difficult to predict and expensive to correct. Furthermore, these issues can arise late in an engine design program and also in early service indicating their sensitivity to environmental conditions and wear. There is clearly a need to avoid the possibility of related failures but until recently this has been beyond the design capability. Historically, empirical design criteria have been used based on parameters involving blade natural frequencies and flow transit times but these methods fail to take into account details which clearly influence matters. The phenomenon involves a large scale harmonic movement of blades within a structure which interacts with the flow causing a sustained cyclic aerodynamic and mechanical distortion. The process occurs at a blade natural frequency and involves blade vibrations resulting from the changing pressure field around an aerofoil as the blade oscillates. For the process to occur it is necessary that over one cycle there is an input of energy from the gas stream to the blade of a sufficient magnitude to overcome any damping. Clearly flutter is dependent on both the aerodynamic and structural characteristics of the blade (and its mounting) and until recent advances in unsteady computational fluid dynamics (CFD) and coupled Finite Element (FE), assessment of the phenomena was difficult to undertake. These advances in coupled CFD/FE methods together with the availability of greatly increased affordable computational power have provided significant progress but primarily within specialized academic and industrial units and using dedicated codes developed specifically for this purpose [1, 2]. The objective of this paper is to demonstrate how similar studies have been achieved using codes which are commercial and have been developed for much more general use. In the example provided here, features of the ANSYS Mechanical and CFD software have been combined to achieve this objective with modest specific development. Progress in using this approach was reported by Mathias, Woods and Elder [3]; but the methods employed there involved the simulation of the unsteady flows around many blades in a given row (depending on the nodal diameter). For instance, for a nodal diameter of four, a rotor with 22 blades required the simulation of eleven blades. More recently, however, computational methods have been developed which permit flutter (and other) phenomena to be captured with only

the simulation of one or two blades per row (yet yielding the full sector solution). This provides considerable savings in computing time and machine resources. The properties of the available methods vary, but the method exploited here involves a Fourier Transformation which preserves frequencies occurring across the periodic surfaces. Details of the method were previously provided in [4-6]. It is worth mentioning that even though the 3D fully non-linear unsteady viscous methods used to be the state-of-the-art for aeroelastic predictions [7], linearized unsteady methods have been found adequate for range of applications [8-10]. This paper presents the aerodynamic damping calculations performed using an efficient transient blade row method. The testcase used here is NASA Rotor 67 [11] as the geometry is in public domain and available for others to use in comparative studies. A range of nodal diameters is investigated with different maximum displacement amplitudes at the design speed and a near peak efficiency operating condition. Accuracy of predictions and savings in computational effort (offered by the Fourier transformation method) are compared with periodic reference solutions. 2 COMPUTATIONAL METHODOLOGIES 2.1 CFD solver ANSYS CFX [12] is a three dimensional finite volume method based time-implicit Navier-Stokes equation solver. A control volume is constructed around each vertex of the mesh (structured, unstructured, or hybrid) elements and fluxes are computed at the integration points on the faces of these control volumes [13]. The discretized equations are then solved using a bounded high resolution advection scheme similar to Barth and Jesperson [14]. Pressure velocity coupling is carried out using the 4th order pressure smoothing Rhie and Chow algorithm [15]. Discretized equations are then solved using the coupled algebraic multi-grid method developed by Hutchinson and Raithby [16], and Raw [17]. Numerical effort of this method scales linearly with the number of grid nodes in the computational domain. Steady state calculations are performed using a pseudo time marching approach until user defined convergence is reached. For unsteady calculations, an iterative procedure updates the non-linear coefficients within each time step while the outer loop advances the solution in time. The Reynolds stresses are computed using the two-equation SST turbulence model developed by Menter [18] with Y+ insensitive near wall treatment [19]. 2.2 Fourier Transformation method The Fourier Transformation (FT) method is based on the phase shifted periodic boundary condition proposed by Erdos [20]. The basic principle is that the pitch-wise boundaries are periodic to one another at different instances in time. The method was modified to avoid storing the signal for a full time period on all pitch-wise boundaries (includes rotor-stator interface for multi-component calculations) using temporal

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Fourier series decomposition (Equation 1), similar to the method proposed by He [21], and double time and azimuthal Fourier series decomposition at the rotor-stator interface [22]. M

f (t ) 



m  M

Am e j (mt )

(1)



). The FT method uses a novel 2 double passage approach (Figure 1), where the Fourier coefficients are collected at the interface between the two adjacent passages, capturing a high quality time signal far from the phase shifted periodic boundaries where the signal is imposed [4, 23]. B1 Blade passage 1 B2 Blade passage 2 B3 Figure 1: Double passage method [12]

Blade Displacement

T ∆T T - Vibration cycle time period ∆T -Time shift

(3)

The signal f B3 (t ) on B3 side in Figure 1 is equal to signal on B2 side, phase shifted by - T .

f B3 (t )  f B2 (t  T )  f B2 (t  T )

This provides significant data compression over the method proposed by Erdos [20]. Only Fourier coefficients ( Am ) need to be stored to reconstruct the solution at an arbitrary time. In the current method, the signal is decomposed into harmonics of fundamental frequency (

f B1 (t )  f B2 (t  T )  f B2 (t  T )

(4)

Where, f B2 (t ) is the reconstructed signal at B2 using the accumulated Fourier coefficients on the same boundary [12]. 2.3 Mesh motion In ANSYS CFX the mesh motion is specified on the selected boundaries. During each timestep, the mesh displacement equations are solved to the user specified convergence criteria and the resulting displacements are applied to update the mesh coordinates. This is followed by solving general transport equations. The integral conservation equations must be modified when the control volumes deform in time. These modifications follow from the application of the Leibnitz Rule [12] d   dV     W j dn j  ( t ) dtv t s V (5) where Wj is the velocity of the control volume boundary. The differential conservation equations are then integrated over a given control volume, however, at this juncture of mesh deformation, the Leibnitz Rule is applied, and the integral conservation equation (for a given conservative variable Ф) become of the form d dV    U j  W j  dn j   .dn j   S dV dt V(t ) S S V (6) The transient term accounts for the rate of change of volume in the deforming control volume, and the advection term accounts for the net advective transport across the control volume's moving boundaries. Erroneous sources of conserved quantities will result if the Geometric Conservation Law (GCL) [12], d dV   W j dn j dtv (t ) S (7)

Time

Blade at higher theta position Blade at lower theta position Figure 2 Time shift between vibrating blades [12] For flutter analysis, The time shift between adjacent vibrating blades is given as

T 

ND



(2)

The signal f B1 (t ) on B1 side in Figure 1 is equal to signal on B2 side, phase shifted by + T .

is not satisfied by the discretized transient and advection terms. The GCL simply states that for each control volume, the rate of change of volume must exactly balance the net volume swept due to the motion of its boundaries. The GCL is satisfied by using the same volume recipes for both the control volume and swept volume calculations, rather than by approximating the swept volumes using the mesh velocities. Unlike other equation classes, the convergence level (i.e., controls and criteria)

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applied to mesh displacement equations is unaffected by changes made to the basic settings for all other equations. 2.4 Mode shape prescription for transient CFD analysis Blade mode shapes and natural frequencies are obtained through a pre-stressed modal analysis in ANSYS Mechanical. The mode shapes normalization is carried out by the mass matrix. The next step is to prescribe the movement of each blade in the CFD computational domain for the desired inter-blade phase angle (IBPA) as defined by Equation 8.

2 ( ND)  N360

(8)

ND is the nodal diameter or phase angle multiplier. Blade movement is accomplished through CFX expression language which computed the coordinates for each blade as a function of time. For a positive rotation about the zaxis as shown in Figure 3, if (x,y,z), and (u,v,w) are the blade coordinates and modal displacements for the 1st blade (shown in green), the co-ordinates and modal displacements for the nth blade (x,y,z)n, and (u,v,w)n are given by

2 N360 The blade position, for the nth blade for a given inter-blade phase angle (IBPA) is given by x   x  u        (11) y     y   v  Sin t  (n  1)  z   z   w  nt  n  n

where b is the blade number and  

for a positive IBPA or forward moving standing wave. or x   x  u        y     y   v  Sin t  (n  1)  z       nt  z n  wn

(12)

for a negative IBPA or backward moving standing wave. In the current investigation, nodal diameters in the range of -8 to 8 have been investigated for the first bending mode and at two different maximum displacement amplitudes 1.5mm and 3mm (corresponding to approximately 1.5% and 3% of the chord length respectively). 3 Testcase Description Rotor 67 case NASA Rotor 67 is chosen as a challenging and practical representative example for which detailed geometry and flow information is available.

Figure 3 First Blade highlighted in green, next blade is second blade, etc.

x  x       y   T n  y  z  z   n  1

(9)

u  u      v   T n v   w  w  n  1 The transformation matrix is Cos(b  1) T   n  Sin(b  1)  0

Figure 4 NASA Rotor 67 geometry

 Sin(b  1) Cos(b  1) 0

0 0  1 

(10)

Figure 4 shows the blade geometry used in the calculations presented. All the calculations were performed at design speed near peak efficiency operating conditions as summarized in Table 1. The design speed of 16043 rpm resulted in an inlet tip Mach number of approximately 1.4. The Reynolds number based on the mid-span chord length is about 1 million.

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Table 1 Rotor 67 details Number of blades Rotational speed Massflow Reynolds number Total pressure ratio Mid-span chord length Tip clearance Average span length Inlet tip Mach number

22 16043 rpm 33.25 kg/s 1.0E6 1.68 94 mm 0.61 mm 132 mm 1.4

4.2 CFD mesh Figures 6 shows the mesh used for the CFD calculations. The “Automated Topology and Meshing” (ATM) method available in ANSYS TurboGrid Release 14.5 was used to create a hexahedral mesh in a single rotor blade passage with tip clearance of about 0.5% of span. The single passage was then replicated in ANSYS CFX to two blade passages for Fourier Transformation analysis and eleven passages for the reference solution of even nodal diameter cases.

4 RESULTS Decoupled fluid-structure interaction analysis performed on Rotor 67 geometry is presented in this section. First, a pre-stressed modal analysis was performed from which mode shapes (displacements on blade surface) were extracted. These mode shapes were used to specify the mesh motion on the blade surface for the desired inter-blade phase angle (IBPA) in the CFD analysis. Aerodynamic damping was computed based on solutions from the transient CFD simulation methods described in the previous section. 4.1 Modal Analysis A pre-stressed modal analysis was carried out using ANSYS Mechanical on a single rotor blade using fixed support at the hub and cyclic symmetry displacement boundary conditions, yielding natural frequencies and mode shapes (eigenvectors). The computed first mode shape is shown in Figure 5. Contours represent the total displacement. As it can be seen in Figure 5, the tip region near leading edge shows the maximum displacement typical of the bending mode. Only the first mode was used in this study. The mode shape, consisting of nodal co-ordinates and normalized modal displacements, was extracted into an ASCII file which was then imported into ANSYS CFX to map it onto the CFD mesh as discussed in previous section.

(a)

(b) Figure 6 Mesh (Medium) (a) on blade and hub surfaces (b) at a mid-span location Three different meshes with approximately 250,000, 800,000, and 1,600,000 elements per blade passage were used to study the effect of mesh refinement on the damping calculations resulting in average blade surface Y+ of 48, 12, and 1 respectively. Figure 5 First mode shape (Natural frequency = 534 Hz)

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4.3 Steady state results Steady state calculations were performed on all three meshes at design speed near the peak efficiency operating conditions presented in Table 1. Predictions are presented in Table 2. All three meshes predict approximately the same total pressure ratio and temperature ratio and efficiency. Percentage change in overall performance prediction going from the coarse mesh to the fine mesh is less than 1 percent. Predicted pressure ratio and the efficiency are in very close agreement with measurement data by Strazisar et al. [11] Table 2 Steady state results

walls using the mode shape and natural frequency obtained from modal analysis. Figure 7 shows the total blade surface displacement at a particular time instance for nodal diameter of 4. 44 time steps per blade vibration cycle were chosen and all calculations (both FT and reference) were run for 20 blade vibration cycles. The effect of the mesh resolution on the prediction of blade aerodynamic damping was studied for a representative nodal diameter of 4. It was observed that the change in the damping coefficient, as calculated by equation 13, is about 1.5% going from coarse to the fine mesh. Hence all further calculations were carried out using the coarse mesh. t0 T

Mesh Coarse Medium Fine

Total pressure ratio 1.680 1.678 1.684

Total temperature ratio 1.174 1.173 1.174

Isentropic efficiency 92.2 92.8 92.6

Measured efficiency [11] 93 93 93

4.4 Aerodynamic damping Unsteady calculations were performed using the Fourier Transformation method with two blade passages and the reference periodic method which requires simulation of half the wheel for even nodal diameter cases and the full wheel for odd (and zero) nodal diameter cases. The k-ώ SST turbulence model [18] with Y+ insensitive near wall treatment [19] was used with assumption that the flow is fully turbulent over the entire blade. The “Total Energy” equation with Viscous Work options were used in the model. The inlet total pressure was set at 1.0 atmosphere and a radially varying static pressure profile was extracted from converged steady calculations and specified at the exit boundary. More details of these inlet and exit boundary conditions can be found in ANSYS R14.5 manual [12].



  pv.ndAdt

t0

A

(13)

2 m. 2 Amax

In equation 13 p is the fluid pressure, v is velocity of the blade due to imposed vibrational displacements, A is the blade surface area, n is the surface normal unit vector, ω is the natural frequency of the given mode, Amax is the maximum displacement amplitude, m=1kg, and T is the vibration cycle time period ( 2 /  ). The numerator represents the work per vibration cycle, and the terms in denominator are used to nondimensionalize the work and are representative of blade kinetic energy. Positive values of damping coefficient indicate a stable system while the negative values indicate that the system will experience destabilizing blade vibrations. Aerodynamic damping calculations were performed for the 1st mode with a range of nodal diameters (or IBPA) and maximum displacement amplitudes. Nodal diameters in the range of -8 to 8 with an increment of 2 were investigated for two different maximum displacement amplitudes 1.5mm and 3mm corresponding to approximately 1.5% and 3% of the chord length respectively. 0.12 0.1

σ

0.08 0.06 0.04 0.02

1.5% Chord 3% Chord

0 Figure 7 Total blade surface displacement (in mm) at arbitrary time (ND = 4, IBPA=65.4 degrees)

-200

-100

0 IBPA

100

200

Figure 8 Aerodynamic damping vs IBPA

These transient simulations were set up in the rotating frame of reference with mesh motion specified on the blade

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Figure 8 represents change in the damping coefficient with the variation in inter-blade phase angle (IBPA). Positive damping coefficients for the range of IBPA covered were observed, indicating a stable system. This observation is consistent with the literature [24] on the same geometry and similar operating conditions. It can be noted from Figure 6 that minimum damping occurs at nodal diameter of -2 (IBPA = 32.7 degrees). The value of damping coefficient remains nearly the same for a given nodal diameter for both maximum displacement amplitudes in the IBPA range investigated, indicating the displacements in the linear range, which further supports the validity of decoupled flutter analysis.

Figure 9 represents the distribution of the time averaged wall power density (the numerator in equation 11) for a representative nodal diameter of 8 (IBPA=130.8 degrees) for both the FT and reference solutions. It is clear from Figure 9 that the region of the blade for which the value of wall work is positive is significantly larger than the negative wall work region, which gives a resultant stabilizing effect. The reference calculations were performed using a half-wheel (11 passages) domain for even nodal diameters, with standard periodicity on the outermost pitchwise boundaries. The reference method has been validated [25] against experimental data [26] and subsequently the FT results were compared with the reference solution. This comparison is shown in Figure 10 and it is clear from these results that the aerodynamic damping predictions from FT calculations match very well with the reference solution. Figure 9 also compares the distribution of time averaged wall power density from FT solution (left) and reference solution (right) for nodal a diameter of 8. It is clear from Figure 9 that the prediction of the local wall power (and work) distribution on the blade surface from a FT calculation is in very close agreement with the reference solution. 0.12 0.1 0.08

(a) 0.06 0.04

FT

0.02

Reference

0 -150

-50

50

150

IBPA

Figure 10 Comparison of damping coefficient predictions with FT and reference solution

(b) Figure 9 Distribution of time averaged wall power density (W/m2) on blade surface for ND = 8 (IBPA=130.8 degrees) (a) Suction surface (b) Pressure surface (FT results are on the left hand side while, reference solution results are on right hand side)

Figure 11 compares the behavior of the area integrated wall power density on the blade surface as a function of solution time for a FT calculation and a reference solution for a representative nodal diameter of 4. During the startup period (first 3-5 periods) of the run, there are differences in values of integrated wall power density between the reference and FT solutions, but after about seven periods the FT solution overlays the reference solution.

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Wall Power Density (W)

800 600 400 200 0 -200 -400 -600 -800 0

100

200

300

400

500

Timesteps Reference

FT

Figure 11 Variation of area integrated wall power density with time (ND =4, IBPA=65.4 degrees) To compare the computational time requirement for both the FT and reference method, a sample calculation for a representative nodal diameter of 8 was performed on a Dell R815 system equipped with 8 processors (AMD Opteron 6178 at 2.3 GHz) and SUSE Linux Enterprise Server 11 SP1 operating system. The FT method takes 17 hours of wall clock time, while the reference method takes 60 hours. Hence, the relative computational time for the reference case calculation was 3.5 times that of the FT calculation. Also, FT calculations required only two blade passages for all nodal diameters while the reference solutions required 11 blade passages (half wheel) for ‘even’ nodal diameters and 22 blade passages (full wheel) ‘odd’ nodal diameters. Hence, for odd nodal diameter cases, the FT method will be approximately 7 times faster than full wheel reference solution for this geometry. It is worth mentioning that practical turbomachinery geometries can have very high blade counts and sometimes that number can be a prime number. In such practical cases, the savings achieved through FT calculation over traditional reference calculation can be even more significant.

match very closely the reference periodic solutions for the cases examined. The Fourier transformation methods also provide solutions 3.5 times faster than periodic reference cases requiring half wheel simulation (even nodal diameter cases). For calculations of odd nodal diameters, FT calculations will be approximately 7 times faster than reference periodic solution for the Rotor 67 case investigated. For practical turbomachinery problems with a higher blade count (or prime number of blades), the savings offered by the Fourier Transformation method becomes increasingly significant. Calculations are being performed across the compressor map at different design speeds and near stall and choke operating conditions. These predictions will be part of the future publications. ACKNOWLEDGMENTS Authors will like to acknowledge Dr. Juan Carlos Morales (ANSYS) for many valuable discussions on flutter theory and setup for the unsteady calculations. Support from Dr. Karl Kuehlert (ANSYS) in making computational resources available for this work is greatly appreciated. REFERENCES 1.

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5 CONCLUSIONS A procedure to perform flutter analysis for practical turbomachinery problems is presented with the use of ANSYS Mechanical and ANSYS CFX. NASA Rotor 67 was considered as a test case. The efficient Fourier Transformation method was used for time accurate calculation of aerodynamic damping. A range of nodal diameters was investigated with different maximum displacement amplitudes. Positive damping was predicted for this range of nodal diameters at design speed near peak efficiency operating condition indicating a stable system. Damping coefficient values were observed to be almost independent of displacement amplitudes for all nodal diameters indicating a linear range for these amplitudes. The Fourier Transformation predictions for blade aerodynamic damping

4.

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