TIME DEPENDENT ACTUATOR DISK MODEL FOR A HELICOPTER ROTOR IN HOVER

147 Pages·2012·12.19 MB·English

The Pennsylvania State University

The Graduate School

Department of Aerospace Engineering

TIME DEPENDENT ACTUATOR DISK MODEL FOR A
HELICOPTER ROTOR IN HOVER

A Thesis in

Aerospace Engineering

by

Jeswanth Mentey

 2012 Jeswanth Mentey

Submitted in Partial Fulfillment
of the Requirements
for the Degree of

Master of Science

December 2012
The thesis of Jeswanth Mentey was reviewed and approved* by the following:

Kenneth S. Brentner
Professor of Aerospace Engineering
Thesis Advisor

Philip J. Morris
Boeing/A.D. Welliver Professor of Aerospace Engineering

George A. Lesieutre
Professor of Aerospace Engineering
Head of the Department of Aerospace Engineering

*Signatures are on file in the Graduate School
iii
ABSTRACT
The current work is an effort towards developing a fast, reasonably accurate low
fidelity Computational Fluid Dynamics (CFD) solver to compute the unsteady
aerodynamic loading on the helicopter blades for the purpose of noise prediction. The
loading at a radial station of a rotor blade can be computed using Blade Element Theory
(BET). In this thesis, a new time-dependent actuator disk method is proposed where the
loading from the BET is converted to an equivalent unsteady pressure jump which is
converted into body force and introduced into the flow governing equations for the cell
centers overlapped by the rotor blade. Unlike an actuator disk model, the pressure jump is
not spread in the entire rotor disk but only in the region enclosed by the rotor blade
boundaries. Moreover, the unsteady pressure jump is updated using BET at every CFD
time step and it moves in the azimuthal direction along with the blade. A parallel
Unstructured Finite Volume Compressible Euler Solver (UFVS) CFD code is developed
and it is coupled with the time-dependent actuator disk model to compute the flow field
of a helicopter rotor. The interface fluxes are computed using a 3rd order accurate Roe’s
Approximate Riemann Solver and the time integration is done using 4th order Runge-
Kutta scheme. A uniformly loaded actuator disk model is implemented in UFVS for
validation purpose and the results are found to be in good comparison with the
momentum theory as well as the axisymmetric version of the same test problem
implemented in ANSYS FLUENT with the same grid resolution. An actuator line model
was initially implemented in UFVS to model the individual rotor blades. In this method,
the loading at a radial station is computed using BET. This loading value is converted to
equivalent body force terms using a Gaussian regularization kernel and then introduced
into the flow governing equations. As an improvement to this method, the time-
dependent actuator disk model is proposed. Since the loading at a blade element is a
function of inflow velocity and the inflow angle, the interpolation of the inflow velocity
is a crucial aspect in computing the correct loading value. Inconsistencies in the inflow
interpolation method give rise to oscillations in the loading values, which are purely
numerical in nature. For the purpose of noise prediction, it is very important that the
numerical oscillations are minimized as much as possible. Various interpolation methods
are analyzed in this thesis and their limitations are discussed.
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TABLE OF CONTENTS
LIST OF FIGURES ................................................................................................................. vi
LIST OF TABLES ................................................................................................................... x
LIST OF SYMBOLS ............................................................................................................... xi
ACKNOWLEDGEMENTS ..................................................................................................... xv
Chapter 1 .................................................................................................................................. 1
Introduction ...................................................................................................................... 1
1.1 Problem Description ........................................................................................... 1
1.2 Literature Review ............................................................................................... 5
1.2.1 Singularity Methods ................................................................................ 5
1.2.2 CFD Methods .......................................................................................... 7
1.2.3 Hybrid Methods ....................................................................................... 12
1.3 Present Approach ............................................................................................... 15
1.3.1 Rotor Loading Model .............................................................................. 16
1.3.2 Rotor/Fuselage Flowfield Model ............................................................. 18
Chapter 2 .................................................................................................................................. 21
Rotor Loading Model ....................................................................................................... 21
2.1 Blade Element Analysis ..................................................................................... 21
2.1.1 Blade Element Analysis for hover and axial flight .................................. 24
Chapter 3 .................................................................................................................................. 27
Finite Volume Flow Solver .............................................................................................. 27
3.1 Overview ............................................................................................................ 27
3.2 Grid Structure ..................................................................................................... 28
3.3 Centerline Treatment .......................................................................................... 31
3.4 Flow Governing Equations ................................................................................. 32
3.5 Finite Volume Discretization ............................................................................. 34
3.5.1 Roe Scheme ............................................................................................. 35
3.5.2 First Order Accuracy ............................................................................... 37
3.5.3 Higher Order Accuracy ........................................................................... 38
3.6 Source Term ....................................................................................................... 38
3.7 Time Stepping Scheme ....................................................................................... 39
3.7.1 Fourth Order Runge-Kutta Time Integration Scheme ............................. 39
3.8 Boundary Conditions .......................................................................................... 40
3.8.1 Concept of Characteristic Variables ........................................................ 42
3.8.2 Subsonic Inflow ....................................................................................... 43
3.8.3 Subsonic Outflow .................................................................................... 44
3.9 Parallelization of the solver ................................................................................ 46
v
Chapter 4 .................................................................................................................................. 49
Validation of Flow Solver ................................................................................................ 49
4.1 Uniformly Loaded Actuator Disk....................................................................... 49
4.2 Validation Results .............................................................................................. 51
4.2.1 Test Case ................................................................................................. 51
4.2.2 Analytical result from momentum theory ............................................... 51
4.2.3 Comparison with ANSYS FLUENT ....................................................... 54
Chapter 5 .................................................................................................................................. 65
Actuator Line Method ...................................................................................................... 65
5.1 Unsteady Source Terms ..................................................................................... 66
5.1.1 Interpolation of Inflow ............................................................................ 67
5.2 Results ................................................................................................................ 72
5.2.1 Blade Loading Convergence ................................................................... 74
5.2.2 Disadvantages of actuator line method .................................................... 78
5.3 Summary ............................................................................................................ 80
Chapter 6 .................................................................................................................................. 81
Time-Dependent Actuator Disk Model ............................................................................ 81
6.1 Understanding inflow ......................................................................................... 82
6.2 Unsteady Source Terms ..................................................................................... 85
6.3 Velocity Interpolation Schemes ......................................................................... 88
6.3.1 Linear Interpolation ................................................................................. 89
6.3.2 Updating inflow velocity every cell center time period .......................... 98
6.3.3 Spatial Averaging .................................................................................... 101
6.3.4 Nonlinear Least Squares Curve Fitting ................................................... 106
Chapter 7 .................................................................................................................................. 118
Summary and Future Work .............................................................................................. 118
7.1 Summary and Conclusions ................................................................................. 118
7.2 Future work and possible improvements............................................................ 121
Appendix ................................................................................................................................. 123
Transformation from Blade Fixed Coordinate System to Cartesian Coordinate
System ...................................................................................................................... 123
BIBLIOGRAPHY .................................................................................................................... 125
vi
LIST OF FIGURES
Figure 1.1 Classification of numerical methods with respect to problem
complexity and computational cost [5]. ........................................................... 4
Figure 1.2 The hybrid model proposed in the present thesis.. ..................................... 20
Figure 2.1 Aerodynamic loads and induced velocities at a typical blade element
(a) top view (b) side view... ............................................................................. 23
Figure 3.1 Blue line represents the actuator line moving through the cylindrical
(left) and Cartesian (right) meshes.. ................................................................. 30
Figure 3.2 Typical cylindrical mesh generated by UFVS grid generator.. .................. 30
Figure 3.3 Formation of polyhedral cells by combining the triangular prisms at the
centerline of cylindrical mesh. Original configuration (left); Single
polyhedron cell (right).. ................................................................................... 32
Figure 3.4 Left and right states at the cell interface i+1/2. The red dots represents
the cell centroids.. ............................................................................................ 35
Figure 3.5 Schematic diagram showing the boundary treatment employed at
various boundaries in UFVS.. .......................................................................... 42
Figure 3.6 Far field boundary situation for (a) inflow (b) outflow. Point a is
outside, and point b is on the boundary while point d is inside the physical
domain. The unit vector n points normally outwards of the boundary.. .......... 45
Figure 3.7 Data transfer between grid blocks for a parallel case using 3
processors.. ....................................................................................................... 48
Figure 4.1 Momentum analysis for an actuator disk.................................................... 52
Figure 4.2 Pressure profile generated by UFVS code at t = 3.5 sec.. .......................... 56
Figure 4.3 Pressure profile generated by ANSYS FLUENT at t = 14.69 sec.. ........... 57
Figure 4.4 Quantitative comparison of pressures along the dotted line in Fig. (4.2
& 4.3).. ............................................................................................................. 57
Figure 4.5 Axial velocity profile obtained using UFVS at t = 3.5 sec.. ....................... 59
vii
Figure 4.6 Axial velocity distribution in ANSYS FLUENT at t = 14.69 sec.. ............ 59
Figure 4.7 Comparison of axial velocity distribution of UFVS and ANSYS
FLUENT.. ........................................................................................................ 60
Figure 4.8 Mach number profile in UFVS at t = 3.5 s.. ............................................... 62
Figure 4.9 Schematic showing the location of axial stations A and B.. ...................... 63
Figure 4.10 Comparison of pressure predicted by UFVS and ANSYS FLUENT
along an azimuthal grid line at axial station A (left) and station B (right)
whose locations are shown in Figure 4.9. Radial distance varies from 0
(centerline) to 2.5R, where R=2.3241m.. ......................................................... 63
Figure 4.11 (a) Comparison of axial velocity predicted by UFVS and ANSYS
FLUENT along Station A (left) and Station B (right). Radial distance
varies from 0 (centerline) to 2.5R , where R=2.3241m.. .................................. 64
Figure 5.1 Schematic of actuator line rotating in the rotor plane by sweeping over
cell centers.. ..................................................................................................... 68
Figure 5.2 Illustration of regularized force computation at point P. Blade element
center points are represented by blue points and regularized force is the
total contribution of all these blue points ( blade elements ) at P.. .................. 71
(cid:1)(cid:2)
Figure 5.3 Blade layout showing the radial location of the pressure orifices
numbered from 1 to 5 for the rotor used for experiments in Ref. [61]... ......... 73
Figure 5.4 Blade load convergence history for coarse mesh at specified radial
locations.. ......................................................................................................... 75
Figure 5.5 Blade load convergence history for fine mesh at specified radial
locations.. ......................................................................................................... 76
Figure. 5.6 Inflow velocity time history for a coarse mesh at the specified radial
locations.. ......................................................................................................... 77
Figure 6.1 Schematic of lift on an airfoil represented by vortices along the chord.. ... 83
Figure 6.2 Weissinger’s approximation states that for a vortex strength placed at
1/4-chord, the effective inflow velocity is the value at 3/4-chord location.. .. 83
Γ
Figure 6.3 Velocity profile of a blade element with the bound vortex located at the
1/4-chord.. ........................................................................................................ 85
Figure. 6.4 Smearing the effect of loading onto the grid cells encompassed in
rotor blade boundaries...................................................................................... 86
viii

Figure 6.5 (a) Top: Blade representation using time dependent actuator disk.
(b) Bottom: Blade representation using actuator line method for
(chapter 5).. ...................................................................................................... 88
(cid:5) = 1
Figure 6.6 Implementation of linear interpolation scheme for velocity prediction
at 3/4 chord location (coarse mesh).. ............................................................... 91
Figure 6.7 Implementation of linear interpolation scheme for velocity prediction
at 3/4 chord location (fine mesh).. ................................................................... 91
Figure 6.8 Inflow time history at various radial locations on (a) Coarse mesh (top)
(b) Fine mesh (bottom).. .................................................................................. 92
Figure 6.9 Loading time history at various radial locations on (a) Coarse mesh
(top) (b) Fine mesh (bottom)........................................................................... 93
Figure 6.10 Loading time history using moving time average at specified various
radial locations on (a) Coarse mesh (top) (b) Fine mesh (bottom)... .............. 95
Figure 6.11 Effect of azimuthal grid discretization on loading values (moving
time-average).. ................................................................................................. 97
Figure 6.12 Spanwise loading distribution for various azimuthal grid densities.. ...... 97
Figure 6.13 Concept of updating inflow only when the 3/4 chord line is at cell
center.. .............................................................................................................. 98
Figure 6.14. Loading history obtained by inflow velocity update every cell center
in UFVS on a coarse mesh with 60 azimuthal grid points (moving time
averaged). ......................................................................................................... 100
Figure 6.15. Theoretical profile of the bound vortex.. ................................................. 101
Figure 6.16 Implementation of spatial average concept in UFVS.. ............................. 102
Figure 6.17 Loading time history at specified radial locations (up to t = 0.25s)... .... 103
Figure 6.18 Full Loading history at specified radial locations (no smoothing).. ........ 104
Figure 6.19 Spanwise loading distribution and comparison with experiment
values.. ............................................................................................................. 104
Figure 6.20. Typical inflow distribution at a radial station of rotor.. .......................... 106
ix
Figure 6.21 Fitting Rankine vortex profile for the velocity information at points 1
to n.. ................................................................................................................. 108
Figure 6.22 Curve fitting for for the test problem defined in Table 6.1 ....... 113
(cid:8) = (cid:9).(cid:11)
Figure 6.23 Nonlinear least squares fit for a highly noisy sample CFD data .............. 113
Figure 6.24 Nonlinear least squares fit for a blade element at nondimensional
radial location r = 0.87 (Intermediate time step).. ........................................... 116
Figure 6.25 Abrupt changes in loading observed by employing nonlinear least
squares in UFVS solver (fine mesh) ................................................................ 117
Figure A.1 Schematic figure showing the blade location with blade fixed co-
ordinates (X1, Y1, Z1) and Cartesian coordinate system (X2,Y2,Z2) ............ 123
x
LIST OF TABLES
Table 4.1 The analytical solution for the actuator disk problem with a pressure
jump of 348.6 Pa.. ............................................................................................ 54
Table 5.1 Location of the pressure orifices on the instrumented rotor blade. ............. 73
Table 5.2 CFD grid discretization details for coarse and fine meshes......................... 74
Table 5.3 Properties of rotor used in UFVS-ALM. ..................................................... 74
Table 6.1. Parameters used to define mesh discretization and bound vortex
sampling. .......................................................................................................... 111