Due to the ability of Euler methods to treat rotational, nonisentropic flows and also to correctly transport on the rotation embedded in the flow field it is possible to correctly represent the inflow conditions on the blade in the stationary hovering flight of a helicopter, which are significantly influenced by the tip vortices (blade-vortex interaction) of all blades. It is shown that also the very complex starting procedure of a helicopter rotor can be very well described by a simple Euler method that is to say without a wake model. The algorithm based on the procedure is part of category upwind schemes, in which the difference formation orientates to the actual, local flow state that is to say to the typical distrubance expansion direction. Hence, the artificial dissipation required for the numerical stability is included in a natural way adapted to the real flow state over the break-up error of the difference equation and has not to be included from outside. This makes the procedure robust. An implicit solution algorithm is used, where the invertation of the coefficient matrix is carried out by means of a Point-Gauss-Seidel relaxation

Krämer, Ewald

Wagner, Siegfried

Online Publikationen der Universität Stuttgart

Computational FlUid Dyn;lmi(;~ 'IJ:!, Volumc 2 
Ch. Hirsch cl al. (Editors) 
199:! Elsevier S(;lcnec Publishers B.V 
H5l 
AN INNOVATIVE ALGORITHM TO ACCURATELY SOLVE THE EULER EQUATIONS 
FOR ROTARY WING FLOW 
S. WAG:->ER and E. KRA11ER 
Institut fUr Aerodynamik und Gasdynamik 
Gniversitat Stuttgart 
I. INTRODUCTION 
Due to the ability of Euler methods to treat rota· 
tional, nonisentropic flows and also to correctly trans-
port on the rotation embedded in the flow field it is 
possible to correctly represent the inflow conditions 
on the blade in the stationary hovering flight of a he-
licopter, which are significantly influenced by the tip 
vortices (blade-vortex interaction) of all blades. 
In the following it is mainly reported on the develop-
ment of a robust Euler method (ref. 1) for computing 
the transonic flow around a rotor blade by using the 
Wake-Capturing method. One of the central points of 
this work was to prove that the radial change of the 
transport equation at the rotor blade had to particu-
larly be taken into account in contrast to the method 
used in the calculation affixed wings, where the trans-
port velocity q over the examined control surface of 
a discrete control volume is assumed to be constant. 
This resulted in far-reaching consequences, when the 
flows are calculated, and completely new algorithms. 
Furthermore, it is shown that also the very complex 
starting procedure of a helicopter rotor caD be very 
well described by a simple Euler method that is to say 
without a wake-model. 
The algorithm based on the procedure is part of cat-
egory of upwind schemes, in which the difference for-
mation orientates to the actual, local flow state that is 
to say to the typical disturbance expansion direction. 
Hence, the artificial dissipation required for the nu-
merical stability is included in a nat ural way adapted 
to the real flow state over the break-up error of the 
difference equation and has not to be included from 
outside. This makes the procedure robust. An im-
plicit solution algorithm is used, where the invertation 
of the coefficient matrix is carried out by means of a 
Point-Gaufi-Seidel relaxation. 
After a short presentation of the basic equations and 
the numerical solution method the prinicipal results of 
the dissertation (ref. 1) are presented. 
2. DYNAMICAL EQUATION AND 
SOLUTION METHOD 
2.1 Euler equations in the rotating 
cylindrical reference system 
According to the flow field of a rotor, the com· 
pressible Euler equations are formulated in a blade-
fixed cylindrical coordinate system which rotates to-
gether with the rotation velocity w of the rotor. This 
demands the additional examination of the centrifu-
gal force and coriolis force. The Euler equations are 
(,,1.1), 
M + ~ * + * + ~ - K{z) (2.1) 
with ~ - (p,pu,pV,pW,_)'r (2.2) 
as solution factor and the fl.uxes: 
E -[ p;:;p 1 F -[ J~:p lc -[ :: ] 
pwu pvv pw' + P 
phu phv phw 
as well 11.<; the term (2.3) 
where 
p 
•• 1 
(2.4) 
+ pf (2.5) 
rl"present the specific total enl"rgy per unit volume and 
h -7 (2.6) 
the specific total enthalpy per unit mass. 
The vector of thl" right-hand side is composed of 
terms from the differentiation in the cylindrical coor· 
------------ ---------
852 
dinate system as well as the centrifugal and cariolis 
forces influencing the control volume as a result of the 
induced velocities. 
2.2 Separation of the relative contributions 
from the dependant variables 
Kramer (ref. 1) recognized that one of the elemen-
tary demands on a solution algorithm, namely the 
exact conservation of the original uniform flow in an 
undisturbed flow field, is not fulfilled when the Euler 
equations represented above are used. The reason for 
this difficulty is the use of relative terms that is to say 
the representation of the velocity and energy referred 
to the rotating reference system. 
Kramer (ref, 1) solves the problem by eliminating 
the contributions of the velocity of the free flow from 
initial direction from the conservative flow contribu-
tions, and therefore, Kramer significantly improves the 
quality of the solution. 
2.2.1 Determination of flux balance 
through a control volume 
The transport of mass, momentum and energy 
through the surface of a blade-fixed and thus sta-
tionary control volume is carried out by the velocity 
q, which consists of an induced contribution, the so-
called disturbance velocity, and the rotation velocity. 
The equation for a rotor blade rotating with the ve-
locity w at a body-fixed coordinate system is; 
Ii - q + (Col x r) , (2.7) 
where q describes the pure contribution of the distur-
bance velocity. 
For a cylindrical coordinate system the equation is 
The application of the integral form of the Euler equa-
tion requires the determination of the mass, momen-
tum and energy fluxes through the cell surfaces of the 
single finite volumes: 
II •• dS , If <C"n)d' • (2.9) , 
where ¢ stands in for any element of the solution vec-
tor, and n represents the surface normal unit vector, 
In contrast to the fixed wing calculation, where the 
transport velocity (q·n) over the examined surface is 
assumed to be constant, there will be radial variability 
due to the (w· r)-contribution at the rotorblade flOW 
A discrete approximation of the surface integral 
(2)0) 
where the index i is the ith cell surface of an exam· 
ined volume element, is only possible if all participat-
ing terms over the integrating surface are const~nt 
This is not the case because of equation (2.7). ;.e\'· 
ertheless, to maintain a simple formulation according 
to equation (2.10), Kramer (ref. 1) divided the flo\\' 
through cell surface in two triangles (see fig. 1) and 
examined each of the two triangles separately. 
This method allows the mathematically exact calcu-
lation of the unknown contributing integral in a siIflPie 
way. 
II .(w x r)nd' ' ~t [ .J r*)dr L-
5 i-1 r 
-<.J·t fi·(ri,lsi~i + ri,2s~~;) (2.11 ) 
i-1 
with TI,I, r;,~ as radial coordinates of the surface cen· 
ter of the two triangles and S.(,~), S,(,~) as projection 
surfaces of the two triangles in Ilow direction. 
According to Kramer (ref. 1) the approximation of 
equation (2.9) for a cylindrical coordinate systeIll is 
If .(.'n)dS - II .(q..(wx r»ndS , 
, S 
(2.12) 
2.2.2 Transition to absolute flow terms 
However, the division of the transport velocity and 
the separated approximation of both flux contribu· 
tions are not enough, as long as rP is a function of 
r. If the Euler equations are formulated in relath"e 
flow terms, which are referred to a blade-flxed rotat-
ing coordinate system, it is not possible any longer to 
usume that the elements of the solution vector OYer 
the examined cell surface are constant. This can be 
pointed out by considering the specific total energy 
per volume unit 
(2.13) 
and the specific total enthalpie per unit mass 
(2.14) 
Kramer (ref. 1) shows the difficulty by examining the 
circumferential velocity and the energy in the cylin-
drical system. If in equation (2.12) 
pu - p(u+wr) 
e - ~ + I «u+\Jr)2-+v2+w2) 
is substituted for </I, one obtains 
II pu(q'n)dS - II pu(q'n)dS + JI pu(wr)dS(t) 
s s s 
II e(q'n)dS - II e(q'o)dS + II e(wr)ds(tI) 
s s s (2.15) 
respectively. 
If only the last term is examined that is to say the 
contribution which corresponds to a transport of the 
flow terms with the free stream velocity and wbich is 
about orders higher than eac.h ofthe first contributions 
in some distance from the rotor blade, one obtains for 
the ith surface (the index i is not taken into account 
for the sake of simplicity): 
II pu(wr)ds(t) :: 
S w.pu-J r'r(r)dr + ",-p'I r"r(r)dr 
and II e(wr)dS(t) :: w.:.J r,z(r)dr + 
S r 
wl.pu-j r 2 'z(r)dr + ~'P'J r a z(r)dr 
r r (2.16) 
respectively, with the new term e according to equa-
tion (2.17). 
In contrast to the first integral terms according to 
equation (2.12) which can be simply replaced by a 
summation form, this is not possible any longer if r 
appears in the powers r3 and r4, respectively. Here, 
the only possibility to limit the calculation cost and 
yet to obtain a mathematically exact represE'ntation 
of the velocity and energy balance is to use the rel-
ative flow variables instead of the absolute ones, and 
to eliminate all (;...>. r)-contributions from the terms 
of the solution vector. Hence, the integral tE'rms of 
higher order can be dropped in r. 
Kramer (ref. 1) introduces for this purpose besides 
the disturbance velocities within the solution vector a 
new pair of terms which is totally in dependant of the 
contribution of the free fiow, namely 
(2.17) 
(2.18) 
The terms f and h are defined as absolute specific total 
energy per volume unit and as absolute specific total 
enthalpy per unit mass, respectively. Kramer (ref. 1) 
usps equations (2.17) and (2.1S) and shows that, if the 
algorithm is used, this is advantageous compared with 
a formulation with energy and rothalpy according to 
(2.19) 
e '" e - ~ (w r)2 
The new energy equation formed with f is derived from 
the original energy equations by exact mathematical 
transformations. Thus, the physical content of this 
energy conservation law is maintained, and the nu-
merical difficulties due to discretization which occur 
when I' and e, respectively, are used can be avoided. 
Due to this transformation there are now new re-
lations for the Euler equa.tions. Since the vectorial 
writing has not changed, equation (2.1) is maintained 
formally. The new elements of the vectors 4;, E, r, G 
and K. however, are different! 
[:u] 1 [ p~~+p] [:~:] ~; +;: -:: + p;;:p • 
[. i~ 1 _ :;u_[p:;~:] phv r (2.20) pW1+p vw 
phv z vh 
'" 
with 
For the numerical solution the equations of motion are 
transformed in any curvilinear coordinate system ac-
cording to the common methods of the literature (see 
(refs. 2 and 3)). 
3. RESULTS 
In order to validate the developed method measure-
ments carried out by Caradonna. & Tung (ref. 4) for 
a two-bladed model rotor in the hovering flight were 
used. As a first test case a blade-tip Mach number of 
0.815 and a collective angle of attack of sa are chosen. 
The comparison between the calculated pressure 
distribution and the measured data shows that there 
is a very good agreement (fig. 2) at the first three 
radial stations, 
Only at the outer radial station the used method 
calculates a low reduced pressure that is to say a too 
strong blade-tip loss. It seemed reasonable to assume 
that, for example, the influence of the local grid re-
finement, the grid size or the geometric design of the 
blade-tip etc. are responsible for this low pressure, 
but they did not prove to be applicable for this test 
case according to (ref, 1). Otherwise, the results ob-
tained by the Wake-Capturing method are very satis-
fying, 
Figure 3 shows the distribution of the circulation 
density in a reference plane which cuts the rotor disk 
30" behind the blade in vertical direction. The coo-
tractioo of the tip vortex cao be seen very exactly. 
Below the tip vortex of the actual blade the tip vor-
tex of the blade rotating ahead cao be seen fairly dis-
crete. The maximum circulation density in the cen-
ter is fallen down to 15% of the original value due to 
the decrease of the concentration caused by diffusion. 
With increasing vorticity age this tendency continues 
with reduced gradient. On the whole the decrease of 
the circulation density in the center of the vorticity 
over the lowering according to (ref. 1) has more or 
less hyperbolic character. 
A similar but less significant effect was obtained by 
Ri:ittgermann et aI. (ref. 5) with a linear vortex lattice 
method, where the vorticity diffusion was coupled to 
the expansion of the vorticity layer. 
Kra.mer (ref. 1) compares the obtained contraction 
in the stationary final state with available hot-wire 
measurements, which Caradonna & Tung (ref. 4) have 
carried out for a some of the flow cases on the model 
rotor in addition to the pressure measurements. 
In figure 3 the position of the vorticity in the presetlted 
rase is marked by crosses. 
Kramer (ref. 1) has also drawn the envelope of the 
trajectory nfthe blade-tip vortex in the representation 
of the calculated results as it is given by Kocurek's 
linear method (ref. 6) for this test case. Due to the 
high empirical part of the method in (ref. 6) Kra.mer 
considers the position of the vorticity obtained by this 
method for a validation of the presented algorithm. as 
suitable. 
The agreement of the numerically calculated con-
traction of the tip vortex structure with data obtained 
by Caradonna & Tung (ref. 4) and by Kocurek (ref. 
6) is very good. 
Figure 4 shows results obtained by the Wake-
Capturing procedure (ref. 1) of a second test case 
compared with coupled models (refs. 7 and 8) that use 
a Navier-Stokes procedure. The results of Kra.mer's 
method (ref. 1) are substantially better at the in-
board radial stations that is probably caused by the 
fact that the wake models neglect or do not properly 
respect the influence of the inner vortex and of the 
vortex sheet. These missing components of the down-
wash lead to an effective angle of attack that is too 
large. At r/ R ::::: 0.8 some differences especially in 
the recompression can be detected, which the coupled 
Navier-Stokes methods represent by a too flat gradi-
ent. The pressure rise is better reproduced by the 
method of (ref. 1). It should be mentioned, however, 
that the position of the shock is positioned farther to 
the rear because of the inviscid flow calculation. 
For r / R :::: 0.89 corresponding results are mis~ing 
in (refs. 7 and 9) so tha.t a comparison is only pos-
sible with (ref. 8). There, the pressure loss clo~e to 
the leading edge as well as the minimum pressure aTe 
better represented, whereas the pressure distribution 
on the lower side is less accurate than predicted by 
the method of (ref. 1). At the tip, results of both the 
present method and the one of (ref. 7) are not satisfac-
tory. In the Kramer procedure the tip loss is overpre-
dieted. The same is true for the coupled ~avier-Stokes 
methods. 
This comparison shows that the \\'ake-Capturing 
procedure (ref. 1) allows a remarkable increase in ac-
curacy compared with Euler or Kavier-Stokes proce-
dures that are coupled with an additional wake model. 
This is also true for Wake-Capturing methods of 
other authors, e.g. (refs. 10, l1, 12) who also demon-
strate a remarkable improvement compared with cou-
pled methods. 
At a calculation of a flow around a helicopter rotor 
by a Euler method in blade-fixed coordinate system 
the beginning of the numerical iteration can be com-
pared with the real situation of a rotor which is ac-
celerated by a sudden start up to the given rotational 
velocity. 
Figure 5 shows the time dependant development of 
the wake at different times of the iteration process. As 
an example the distribution of circulation is shown in 
a reference plane that is positioned 30 degrees behind 
the blade in plane normal to the rotor disc. The se-
quence of figures shows very clearly the development 
of the two tip vortex systems. The numericallv cal-
culated starting process of the rotor shows the t}'pical 
behaviour that is also obsereved in experiments. 
4. SUMMARY 
The method developed by E. Kramer (ref. 1) to cal-
culate the transonic flow field of a helicopter rotor was 
presented here. This method shows that the examin-
ing of the wake as a part of the solution is a particular 
feature so that the blade-wake-interference, which is 
typical for rotor aerodynamics, is implicitly given that 
is to say without the help of a separate vortex model 
(Wake-Capturing method). 
The basic system of equations was established in a 
blade-fixed, rotating coordinate system. However, the 
radially varying contributions of the free stream veloc-
ity steming from transport velocity and from terms of 
the conservative solution vector were eliminated. The 
results obtained by the present method for steady hov-
ering show a ver}' good agreement with measurements 
not only concerning the pressure distribution but also 
the contraction of the tip vortex trajectory. 
5. LITERATURE 
1. E. Kramer: Theoretische l'ntersuchungen der 
stationaren Rotorblattumstromung mit Hilfe 
eines Euler- Verfahrens, Dissertation, rnh'er-
siHi.t der Bundeswehr )'liinchen, Xeubiberg, 
1991. VDI Verlag, Fortschrittsberichte, Reihe 1: 
Stromungstechnik, Xr. 197, 1991. 
2. D.A. Anderson, J.C. Tannehill, R.H. Fletcher: 
Computational Fluid Dynamics and Heat Trans-
fer, Hemisphere Publishing Corporation, Xew 
York, 1984. 
3. J.F. Thompson, ZX.A. Warsi, G.W. ~fa5tin: Nu-
merical Grid Generation. ~orth-Holland, Xew 
York,1985. 
4. F.X. Caradonna, C. Tung: Experimental and An-
alytical Studies of a ),lodel Rotor Hover, XASA, 
R'i5 
HI 81232, 1981. 
5. A. Rottgermann, R. Behr, Ch. Schott I, S. 
Wagner: Calculation of Blade-\'ortex Interac-
tion of Rotary Wings in Incompressible Flow 
by an l'nsteady Vortex-Lattice- ~fethod Includ-
ing Free Wake Analysis, Paper presented at the 
7. GA),.L\I-Symposium, Kiel, Jan. 1991. Xotes 
on Xumerical Fluid :-'Iechanics, Vol. 33: Xumer-
iral Techniques for Boundary Element Methods, 
(W. Hackbusch (Ed.)). Vieweg 1991, pp. 153-
166. 
6. J.D. Kocurek, L.F. Berkowitz, F.D. Harris: Hover 
Performance ),Iethodology at Bell Helicopter Tex-
tron, 36th Annual Forum of the American Heli-
copter Society, Washington D.C., Paper Xr. 80-3 
(1980). 
1. R.K. Agarwal, J.E. Deese: :;.;'avier-Stokes Ca.lcu-
lations of the Flowfield of a Helicopter Rotor in 
Hover. AIAA Paper 88-0106 (1988). 
8. G.R. Srinivasan, W.J. :-"·fcCroskey: ~avjer-Stokes 
Calculations of Hovering Rotor Flowfields, Jour-
nal of Aircraft, Band 25, r\r. 10, 1988, pp. 865-
874. 
9. R.K. Agarwal, J.E. Deese: Euler(Navier-Stokes 
Calculations of the Flowfield of a Helicopter Ro-
tor in Hover and Forward Flight, 2nd Interna-
tional Conference on Rotorcraft Basic Research, 
College Park, ~laryland, 1988. 
10. c.L. Chen, \Y.J. McCroskey; Numerical Simu-
lation of Helicopter Multi-Bladed Rotor Flow, 
AIAA Paper 88-0046 (1988). 
11. N. Kroll: Berechnung von Str6mungsfeldern um 
Propeller und Rotoren im Schwebeflug durch die 
Losung der Euler-Gleichungen, Dissertation, TU 
Braunschweig, 1989. 
12. G.R. Srinivasan, J.D. Baeder, S. Obayshi, W.J. 
McCroskey: Flowlleld of a Lifting Hovering Rotor 
- A Nader-Stokes Simulation, 16th European 
Rotorcraft Forum. Glasgow, Paper Nr. 1.3.5. 
(1990). 


An innovative algorithm to accurately solve the Euler equations for rotary wing flow

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An innovative algorithm to accurately solve the Euler equations for rotary wing flow

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