N87-22204
CALCULATING ROTORDYNAMIC COEFFICIENTS OF SEALS
BY FINITE-DIFFERENCE TECHNIQUES
F.J. Dietzen and R. Nordmann
University of Kalserslautern
Kalserslautern, Federal Republic of Germany
For modelling the turbulent flow in a seal the Navier-Stokes equations in con-
nection with a turbulence model (k-_-model) are solved by a finite-difference
method. A motion of the shaft round the centered position is assumed. After cal-
culating the corresponding flow field and the pressure distribution, the rotor-
dynamic coefficients of the seal can be determined. These coefficients are com-
pared with results obtained by using the bulk flow theory of Childs [i] and with
experimental results.
INTRODUCTION
It is well known that the fluid forces in seals, which are described by equa-
tion (I)
have a strong influence on the dynamic behaviour of rotating turbo-machinery.
While there exist some good theories for calculating the coefficients of
straight seals [I], no satisfactory model is known to describe the effects of
grooved seals. Reference [2] presents a survey and comparison of results of
existing theories. The authors' opinion is that the existing methods are not
at all satisfactory. The main weakness of these theories is the fact, that they
are using so called 'bulk-flow-theories' which connect the wall shear stress with
the mean flow-velocity relative to this wall. Howeve_in the region of a groove
there occur stresses in the fluid which cannot be neglected. Calculating the flow
by using the Navier-Stokes equations in connection with a turbulence model elimi-
nates this disadvantage. Therefore, a finite difference model is presented which
allows the calculation of the coefficients by using these equations.
77 PR CF.DmiGPAGE NOT
Nomenclature:
Fz , Fy
K, k
D, d
M, m
U, V_ W
P
k
s
We' PI' _t
P
t
x, r, 8
q
o k , oS , K
Cp, C1, C2
So
C
0
6
r o
r o
e -
Co
k
Forces on the shaft in z and y direction
direct and cross-coupling stiffness in eq. (1, 24)
direct and cross-coupling damping in eq. (1,24)
direct and cross-coupling inertia in eq. (1, 24)
axial, radial and circumferential velocity
pressure
turbulence energy
energy dissipation
effective, laminar and turbulent viscosity
density
time
axial, radial and circumferential coordinate
radial coordinate after transformation
Constants od the k-s-model
Constants of the k-s-model
general variable standing for u, v, w, p, k or s
general source term
seal clearance by centric shaft position
seal clearance by eccentric shaft position
radius of the precession motion of the shaft
perturbation parameter
rotational frequency of the shaft
precession frequency of the shaft
entrance lost-coefficient
Length of the seal
78
r i
r a
Subscripts
0
1
R
S
radius of the rotor (shaft)
radius of the stator
zeroth order variables
first order variables
rotor
stator
MATHEMATI CAL MODEL
To describe turbulent flow by the Navier-Stokes equations the velocities and the
pressure are separated into mean and fluctuating quantities.
u = u + u' v = v + v'
w = w + w' p = p + p'
Time-averaging of the Navier-Stokes equations leads to terms of the following
form: _-r_-r, _-r_-r, _-r_T.
TO substitute these terms one can use the Boussinesq's eddy-viscosity concept.
For example:
Pt 3u 3v
= - 7 (_ + Tx ) (2)
Pt is the turbulent viscosity, which is not a fluid property but depends strongly
on the state of flow. Summing up the laminar and turbulent viscosity to an effec-
tive viscosity
_e = _I + _t ' (3)
one obtains the following time-averaged Navier-Stokes equations for turbulent
flow. (In the following the overbars are omitted.)
I. axial momentum:
3u 3 3 , 3u, 13 13 3u 13 , wu" la ,1 3u,
p_ + _-_(puu) - _t_eT_; + _(rpvu) - _(rl_e_- _) + _-_tp ; - _-ot_Pe_; =
_@__p a , au, la , 3v, 13 , Bw,
3x + _t_ea-x; + r_e-@x; + raOtPe_-£;
(4)
79
2. radial momentum:
3v
P_ + __x(pUv) 3 , 3v, 13 "r " 13 r 3v, 13 13 ,1 3v,
- _-_,pe_-_) + -_-_ pvv) - r_-{(pe_--_-) + _--6(pwv) - _-6_--_e_-_; :
__p 13 @v 3 , 3u, 13 3 w 2 3w 2 _0 2
3r + r_-r(rPe_ ) + 3-x_Pe_-r ) + r3-c)(rPe_-r(-r)) - -r2"e_ - r2Pe v + r w (5)
3. tangential momentum
3w 3 _ , 3w, 13 13 3w 13 13 ,i 3w,
p_ + _-x(pUw) - _tpe_-_) + _(rpvw) - _-_(rlJe_ ) + r_(pww) - r_-6trlJe_-_)=
_lBp 13 _v 3 ,1 3u, 1 3v
r30 + r_(_e_ ) + _£r_e_-O _ + rZ_e_
4. continuity equation
w 3 13 ,2 v" 13 ,1 3w, Pvw
rz_(r]_ e) + r_-_r]_e ) + r_-E)_Pe_-_).-
(6)
13 'r " 13
x(PU)+ pv; + T (pw): o
(7)
To describe Pt we use the k-_ turbulence model [3, 4]. This model determines Pt
as a function of the kinetic energy k of the turbulent motion and the energy
dissipation _. It is relative simple and often used to calculate the turbulent
flow in seals [12, 13, 14, 15]. Stoff [12], for example, compares his flow meas-
urements in a labyrinth seal with calculations on base of the k-e model. He
observes that both agree well.
k2 (8)
Pt = c p_--
The equations for k and c can be derived in exact form from the Navier-Stokes
equations
5. turbulence energy k
. 3 _Pe3k_ . 13 13 , Pe3k
3_k 3 _puk) .... _ -
_St " 3-x' 3X_OkSX_ • _(rpvk_ _-_£r_kk_-_)
G - pe
+ l_(pwk) 13 _IPeSk_
- =
(.9)
80
6. energy dissipation
ae a _(laeae la la , Ueae,
p_ + _(pue) - ax'a a-x) + _(rpve) -_-r_r_--a--6)
la la (l_eae_ =
+ Fge(pwe) - Fa-o'F6-g6'
ez (10)
C2p_-
CI[G -
G = lae{ 2(/av_ 2 tau/2 /law + v 2) av au_2 flav
'_' + W_' + 'rae -_) + (gi + ag' + 'Fa_ +
aw w = aw lau_2}
ar r) + (a-x + raC)'
C = 0.09 CI = 1.44 C2 = 1.92
K
< = 0.4187 o k = 1. o = C_(Cl - Cz)
(II)
To model the flow in the case of a shaft moving on an eccentric orbit, a coordi-
nate-transformation [5, 6] is made. (Fig. 1)
r e-r
n = r a - _ CO (12)
8(O,t) is the seal clearance, varying with angle @ and time t. By this trans-
formation the eccentric moving shaft is reduced to a shaft rotating in the
centre of the seal.
We must note that the following relations of the transformation must be used.
+ (an)O(_-_)r
aq_ aq_ aq_ an
(Y[)r = (_)n + (a-n)t(a-%-lr (131
(_x) r a@
:
81