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Incorporation of actuator disk model into Navier-Stokes CFD codeNES

S.Peigin*,B.Epstein

,S.Seror AND G.Hoffman

Israel Aerospace Industries, Lod, 70100, Israel

P. Dhandapani**

ADE, Bengalore, India

This paper describes incorporation of actuator disk model in the form of blade element

theory into Navier-Stokes CFD code NES. This code can by used for flow analysis over

aircraft with propellers in tractor/pusher configurations. Sections 1 and 2 describe the

technical aspects of the tool, covering the theoretical basis of the method, and details of

the code logic, respectively, while in Section 3 results of validation for several practical

propeller configurations are presented.

Introduction

This work aims at the enhancement of the code NES by implementing a mathematical

model to simulate in quasi-steady mode the presence of a propeller. The work consists in

developing the same capability existing in the RANS code FUN3D of NASA Langley

Research Center based on the following references [1-6].

The method for predicting propeller steady aerodynamics involves the use of the Navier-

Stokes equations. Although not yet operational, the computer program NES based on the the

Navier-Stokes equations is nearly complete. This approach promises new insight into

propeller flow fields especially in the areas of blade boundary layers and blade viscous wakes

as well as improved accuracy for blade leading edge and tip vortex development.

It is common to divide the running propellers influence on the aircraft characteristicsinto two parts: the direct and the indirect effects. The direct effects include thrust that acts

along the shaft axis. This force can produce also moments such as pitching moment if it has a

vertical displacement from the center of gravity. Another direct effect is the force acting

normal to the propeller plane. This force is produced when the local flow has an inclination to

the propeller disk. It can affect the pitching and yawing moments. The third direct effect is the

rolling moment due to the torque applied to the propeller shaft. Indirect effects are the

changes in the configuration forces and moments excluding the direct contribution. They arise

from changes in local dynamic pressure on aircraft parts and from changes to local angle of

attack including changes in downwash and side wash angles. The analysis tool under

development will be the NES CFD code. This code solves the RANS equations on arbitrary

configurations, using a multiblock multigrid system. The propellers' effect is modeled in aspecial module that uses the following approach:

Actuator Disc + Blade Element Theory provide jump conditions for steady-statecalculation at the propeller.

Induced velocity at the disc via a quasi-steady Prantdl-Glauert correction term todecrease the load at the tip.

*Senior Research Scientist, Technical Fellow, IAI CFD Department, speigin@iai.co.ilCFD Consultant for IAI CFD Department,Senior Aerospace Research Engineer, Technical Fellow, Head CFD Department, sseror@iai.co.il

Engineer, Aerodynamic Department, Head of UAV Aerodynamics, ghoffman@iai.co.il**Scientist M.Tech, Aeronautical Development Establishment, Defence Research & Development OrganisationMinistry of Defence, Government of India

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Note that reverse flow areas if existing will be naturally predicted by the RANS equations

even through the propeller. This CFD method gives both the direct and the indirect effects of

the propellers.

A.Blade Element Theory

The Blade Element Theory (BET) is an analysis method that may be applied to a rotor,

propeller, fan, and even a lightly loaded compressor. BET is the foundation for almost all

analyses of helicopter aerodynamics because it deals with the detailed flow and loading of the

blade. The theory gives basic insights into the rotor performance as well as other

characteristics. William Froude originally conceived of BET in the 1870's.

Stefan Drzewiecki however, was the first to rigorously examine and apply the BET. He

performed his work between 1892 and 1920. BET is very similar to the Strip Theory for fixed

wing aerodynamics. The blade is assumed to be composed of numerous, miniscule strips with

width 'dr' that are connected from tip to tip.

The lift and drag are estimated at the strip using the 2-D airfoil characteristics of the section.

Also, the local flow characteristics are accounted for in terms of climb speed, inflow velocity,

and angular velocity. The section lift and drag may be calculated and integrated over the blade

span.

The BET is a very useful tool for the engineer to perform a fairly detailed local analysis of the

rotor in a short amount of time. .

In contrast to the BET, the Momentum Theory is a global analysis which gives useful resultsbut can not be used as a stand-alone tool to design the rotor. It was originally intended to

provide an analytical means for evaluating ship propellers (Rankine 1865 & Froude 1885).

Later Betz (1920) extended Rankine and Froude's work to include the rotation of the

slipstream. Momentum Theory is also well known as Disk Actuator Theory. Momentum

Theory assumes that the flow is inviscid and steady, also the rotor is thought of as an actuator

disk with an infinite number of blades, each with an infinite aspect ratio. The useful results

from momentum theory that are applied to BET are listed below.

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The downwash is twice as fast as the inflow The ideal power is a simple function of the thrust If the down wash is uniform, the ideal power is minimized The inflow is a simple function of the thrust

When the two theories are combined, it is possible to evaluate a field of induced velocity

around the rotor or propeller, and therefore correct the inflow conditions assumed in the basic

blade element theory. The induced velocities aren't known until the blade loads are computed.

With the loading available one can re-compute the field of induced velocities. This is an

iterative method; generally the quantity that is iterated for is the thrust coefficient. The

combined Blade Element Momentum Theory is a fairly accurate analytical tool (for lightly

loaded rotors or propellers) that can be used by the engineer early in the design of a rotor.

B.Actuator Disk Model

The basic idea of the actuator disk model in connection with propeller aerodynamics is to

replace the real propeller with a permeable disk of equivalent area where the forces from the

blades are distributed on the circular disk. In fact, in actuator disk approach, the propeller is

represented as an infinitely thin disk with given distribution of external forces on the disk

surface. The distributed forces on the actuator disk depend on the local velocities through the

disk and in general the entire flowfield around the rotor disk. Specifically, in the current codethe forces distribution along the disk surface is based on a blade element theory (see the

general sketch which is given below.

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Let us consider 2-bladed propeller which is shown in side view. The rotation speed is , and

the axis vector is , normal to the disc plane, in the downstream direction. A section at radialstation r, of the blade closest to the viewer, is shown. The reference chord of the section is

oriented at an angle (r) with respect to the disc plane. The section has a velocity in the

(circumferential) direction, of . The inflow velocity at this section is

n

r 1q

.

Relative to a stationary blade section, the inflow velocity at the section is 1 rq

.

Neglecting the radial component of this velocity, and considering the local speed of sound

from the flowfield, the 2D angle of attack and Mach at the section is known.

Assuming that data are available for various sections along propeller blade, as a function of

and Mach, then the 2D lift, drag and pitch acting on the section (in fact the above mentioned

external forces) may be calculated.In reality the actuator disk is a limiting case in which the number of blades goes to infinity.

In contrast real propellers have finite number of blades which produce a system of distinct tip

vorticity structures in the wake. Thus, a different vortex wake is produced by a rotor with

infinite number of blades as compared with one with a finite number of blades. Prandtl

derived a formula for the tip-correction, quantified the factor F, in order to compensate the

finite number of blades. In order to include tip-correction effects with generalized actuator

disk model, the external aerodynamic forces are corrected using the Prandtl tip-correction

factor F.

sin2

)(expcos.

2 1

r

rRBF

In the literature, actuator disk implementations fall into two general categories: a boundary

condition approach and a source term approach. The primary difference between the two

approaches is how the propeller is treated in the fluid control volume. In the boundary

condition approach, the control volume is wrapped around the actuator disk in such a way that

the actuator disk lies outside of the control volume. Conversely, in the source approach the

actuator disk is contained inside the control volume. These differences are illustrated in Fig. 1

using a one-dimensional duct example.

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Figure 1. Comparison of the boundary condition and source based actuatordisk methods.

A typical internal cell without an actuator disk at face 23 will have the same flux on the left

and right sides of the face. For the boundary condition method and source method an actuator

disk will be present between cells 2 and 3. The boundary condition method will update the

flow variables in the same fashion as the typical internal cell. However, the flux at the

actuator disk face will no longer be the same on the left and right sides of the face. Instead,the fluxes are related by some proportionality condition, which in this case is the imposed

actuator disk boundary condition. In the source term approach the fluxes are computed just as

they were for a typical internal cell, but the update of the flow variables is different. To make

the presence of the actuator disk known to the flow solver, an extra source term is added to

the equation.

The computational implementation of each of these approaches differs significantly. The

source implementation was found to be more robust when solving the RANS equations in the

framework of multiblock / multiface CFD code NES. The source term is obtained once the

force acting on the source elemental area is known. The force is obtained using a pre-

specified distribution like a blade element method, which computes the forces based on the

local flow condition. The sources are then added to the governing equations.

C.Addition to the code NES

This section of the report describes additional modules needed to incorporate actuatordisk model in the form of blade element theory into the existing multiblock / multiface

Navier-Stokes CFD code NES

1.Identification of disk location

Several actuator disks are allowed in the same configuration input. It is assumed that

Each disk represents a planar circle

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Each disk is located on the face (or a number of faces) of the blocks comprising the

configuration mesh

Each face may contain only one disk or part of the disk

The number of actuator disks is an input parameter.

To determine the disk location(s), the following data are inputted for each actuator disk

Cartesian coordinates (X,Y,Z) of the center

Radius of the disk

Three components of the unit vector normal to the disk surface. The direction of the

unit vector coincides with the direction of thrust.

Based on these data, disk location(s) are automatically identified by the code. As a result

of this identification, each merged block face is flagged in the following way:

If the face contains even a part of a disk it is flagged by the positive number of this

disk

In the opposite case the flag is set to zero

The identification of the disks and the corresponding face flags is performed in the stage

of preprocessing of the code NES where all the geometrical issues are treated.

2. Computation of source terms

Mathematically source terms are calculated dynamically (that is in each numerical

iteration). They depend on the data base provided by the blade element theory (which

includes the specific propeller(s) characteristics) and, on the other side, on the current

flowfield.

The above data base is inputted together with other solver input data such as flow

conditions and numerical parameters. A commonly used arrangement of such data is the

Rotorcraft format. This contains a header noting the number of radial data stations, and the

chords and the twist angles of the sections relative to the disc plane, for each radial station.

Then follows a matrix of lift, drag and pitching moment data for each station, as a function

of the angle of attack and Mach value.

Source terms are computed for each "merged" block face (that is a face common to two

blocks). If the disk actuator flag is set to zero, the corresponding source terms are also set

to zero. Otherwise the source terms are calculated by interpolating the basic data baseaccording to the local radius of the point, local Mach value, local velocity components,

local angle of attack in relation to the 2D blade profile under consideration and its chord

value.

D.Validation test cases

3.Limitations of the method:

a. Steady state assumptionIn accordance with the actuator disc/blade element approach, this method ignores the true

unsteady nature of flow at the propeller. Parts of the configuration near the propeller

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experience only a time averaged effect, and are unaware of the cyclic variations in the

flow field generated by the rotating blades. In this respect this method is similar to the vast

majority of practical propeller codes currently used in industry, which can calculate a

range of propeller operating conditions in reasonable time and with modest computer

resources. By contrast, current state-of-the-art unsteady propeller codes may run for weeks

on powerful multiprocessor computers, to calculate a single operating point.

b. 2D blade characteristics

According to blade element theory, the aerodynamic forces acting on a section, at some

radial station along the blade, can be estimated based on 2D data for this section, viz. lift

and drag as a function of local Mach and angle of attack, as experienced by a stationary

profile immersed in steady 2D flow. Even after subtracting the circumferential component

rfrom the actual velocity of the propeller blade section, and introducing the tip effect, it

is clear that the 2D section data is only an approximation to the true aerodynamic force

exerted on the blade element. The method of this report does not consider, for example,

the effect of Coriolis forces acting on the rotating flow. This report does not deal with thevarious techniques for adjusting the straightforward 2D profile data, in order to improve

the accuracy of the calculated propeller characteristics.In the validation cases, the

propeller databases were prepared using a 2D code capable of providing reliable 2D

aerodynamic coefficients for a typical airfoil. The 2D code was applied to several blade

sections between the hub and the tip. For each of these sections, the appropriate Mach and

Reynolds numbers were found (taking into account the circumferential speed r at the

section), and the 2D code was run for the widest range of angles of attack possible. It was

then assumed that the coefficient varied linearly from the the value at the minimum

calculated angle of attack to zero at -180 degrees, and linearly from the the value at the

maximum calculated angle of attack to zero at +180 degrees. This technique should suffice

for normal operating conditions, where each blade section should enjoy optimal, stall-freeflow. Obviously this technique is inappropriate if the blade section experiences 2D angles

of attack outside of the range calculated by the 2D code.

4. Validation test cases:

Two cases are presented. The first is from a well known series of experiments at NASA

Langley, documented in NACA Reports 640, 642, dealing with 10 ft. diameter tractor

propellers on typical nacelles. The second is an older (1922) test case described in the

A.R.C reports 829 and 830, which contains comprehensive data from both tractor and

pusher propellers, operating on nacelles of varying degrees of bluntness.

a. NACA Report 642

The Navier-Stokes code computation results, compared to the NACA Report 642 test case

and MGAERO data [7], are shown in Figures 2 for the propeller thrust coefficient (CT),

Figure 3 for the propeller power coefficient (CP) and in Figure 4 for the propeller

efficiency (ETA). Here

,42DN

ForceCT

,53DN

PowerCP

P

T

C

JC ,

ND

VJ

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were - free-stream velocity, D diameter of the propeller, N angular propeller

velocity (revs/sec), J advanced ratio coefficient. Additionally, the corresponding flow

visualization over this propeller for pitch angle 20 degree are presented in Figures 5-7.

V

Acceptable agreement between the Navier-Stokes code results and the test results can be

observed at most of the operational advance ratio (J) range.

The discrepancy with experiment at low advance ratio values is due to stall that may occurin this regime, leading to inappropriate 2D database values for high angles of attack. For

larger advance ratios, the inaccuracy is compatible with an error of about 1 degree in the

pitch angle of the propeller.

Figure 2: NACA Report 642 and MGAERO results vs NES code Thrust coefficient

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Figure 3: NACA Report 642 and MGAERO results vs NES code Power coefficient

Figure 4: NACA Report 642 and MGAERO results vs NES code Efficiency

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Figure 5: Flow visualization over NACA 642 propeller for J=0.66, pitch=20 deg.

Figure 6: Flow visualization over NACA 642 propeller for J=0.66, pitch=20 deg.

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Figure 7: Flow visualization over NACA 642 propeller for J=0.66, pitch=20 deg.

b. A.R.C. Reports 829, 830

Figures 8, 9 & 10 presents the comparison of the NES code computation results, compared

to the ARC 829 Report test results and MGAERO data, for CT, CP & ETA respectively.

Figure 8: Thrust coefficient for a 4-bladed tractor propeller on a minimum body.

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Figure 9: Power coefficient for a 4-bladed tractor propeller on a minimum body.

Figure 10: Efficiency coefficient for a 4-bladed tractor propeller on a minimum body

The ARC 830 Report presents the effect of different bodies P, Q and R, on the

performance of a pusher propeller configuration, as shown in Figure 11. The ratios of the

maximum body diameter to the propeller diameter are 0.40, 0.60 and 0.75, respectively

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Figure 11: Comparison of pusher body configurations P, Q and R.

Figures 12, 13 & 14 presents the comparison of the NES code computation results, compared

to the ARC 830 Report test results and MGAERO data, for CT, CP & ETA respectively.

As it can be observed from these figures, Navier-Stokes computations provide a better

comparison with experiment than the corresponding Euler results. This is due to a

significantly more adequate simulation of the separation zone after the blunt body.

Figure 12: ARC Report 830 and MGAERO results vs NES code Thrust coefficient

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Figure 13: ARC Report 830 and MGAERO results vs NES code Power coefficient

Figure 14: ARC Report 830 and MGAERO results vs NES code Shaft Efficiency

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I. Conclusion

The results obtained by the Navier-Stokes code NES with incorporated actuator disk model

in form of blade-element theory demonstrate accuracy comparable to standard stand-alone

propeller codes, with the additional benefit of the inclusion of the interference effects between

the propeller and the configuration geometry.

Acknowledgments

This work was done with the financial support of MAFAT IMOD and ADE in the

framework of a commercial agreement between Israel Aerospace Industries and ADE.

References

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13, 2005, AIAA 2005-0468

3. Clark, D. R. and Maskew, B. Study For Prediction Of Rotor/Wake/Fuselage InterferencePart I: Technical Report, NASA CR-177340, 1985

4. Robert Mikkelsen, Actuator Disc Methods Applied to Wind Turbines, PhD Thesis

Department of Mechanical Engineering, Technical University of Denmark, June, 2003

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Speed Propeller-Nacelle Aerodynamic Performance Prediction, NASA Technical Report

R79-912949-19, June 1979.

7. Theodore Rubin, Improvements of the propeller computation module of IAIs version of

MGAERO Eulerian CFD code, IAI Technical Report 100852