NASA Technical Memorandum 1070_5_1 .............
///_O//
A k-o) Turbulence Model for
Quasi-Three-Dimensional
Turbomachinery Flows
Rodrick V. Chima
Lewis Research Center
Cleveland, Ohio
Prepared for the
34th Aerospace Sciences meeting
sponsored by the American Institute of Aeronautics and Astronautics
Reno, Nevada, January 15-18, 1996
National Aeronauticsand
Space Administration
.y
(NASA-TM-107051) A K-GMEGA
TURBULENCE MODEL FOR
QUASI-THREE-CIMENSIONAL
TUR_CMACHINERY FLOWS (NASA.
_; Research Center) 14 p
Lewis
N96-14786
Unclas
G3/OI 0065453
A k-o3 Turbulence Model for Quasi-Three-Dimensional
Turbomachinery Flows
Rodrick V. Chima*
NASA Lewis Research Center
Cleveland, Ohio 44135
Abstract
A two-equation k-co turbulence model has been devel-
oped and applied to a quasi-three-dimensional viscous
analysis code for blade-to-blade flows in turbomachinery.
The code includes the effects of rotation, radius change,
and variable stream sheet thickness. The flow equations
are given and the explicit Runge-Kutta solution scheme is
described. The k-co model equations are also given and
the upwind implicit approximate-factorization solution
scheme is described. Three cases were ca!cula_ted: transi-
tional flow over a flat plate, a transonic compressor rotor,
and a transonic turbine vane with heat transfer. Results
were compared to theory, experimental data, and to results
using the Baldwin-Lomax turbulence model. The two
models compared reasonably well with the data and sur-
prisingly well with each other. Although the k-co model
behaves well numerically and simulates effects of transi-
tion, freestream turbulence, and wall roughness, it was not
decisively better than the Baldwin-Lomax model for the
cases considered here.
Introduction
A large percentage of computational fluid dynamics
(CFD) analysis codes for turbomachinery use the Bald-
win-Lomax turbulence model [1]. This was evident in the
results of the blind test case for turbomachinery codes
sponsored by ASME/IGTI at the 39th International Gas
Turbine Conference held in The Hague in June of 1994.
The results have not yet been published. Of the 12 partic-
ipants, nine used the Baldwin-Lomax turbulence model,
one used an algebraic mixing length model, and two used
k-e models. One of the objectives of that test case was to
investigate the effects of turbulence models. However,
because of differences in grids, large variations between
the computed solutions, and lack of experimental mea-
surements in the boundary layers, it was not possible to
draw any conclusions regarding the effect of turbulence
models.
*Aerospace Engineer, Associate Fellow AIAA
The Baldwin-Lomax model is popular because it is
easy to implement (at least in 2-D) and works fairly well
for predicting overall turbomachinery performance. How-
ever, the model has both numerical and physical problems.
Numerical problems include awkward implementation in
3-D, difficulty in finding the length scale [2], and slow
convergence if the length scale jumps between grid points.
Physical problems include a crude transition model and
the neglect of freestream turbulence, surface roughness,
and mass injection effects which are often important in
turbines. These effects are sometimes added to the Bald-
win-Lomax model using techniques developed for bound-
ary layer codes [3, 4]. Physical problems also include
poor prediction of separation [5], which is important in
compressors, and underprediction of wake spreading [2].
A few researchers have used other turbulence models
for turbomachinery problems. Choi et. al. have used the q-
co model [6], Hah (who participated in the blind test case)
used a k-e model [7], and Kunz and Lakshminarayana
used an algebraic Reynolds stress k-e model [8]. Unfortu-
nately none of these researchers have used a Baldwin-
Lomax model in the same code for comparison. Ameri
and Arnone have compared the q-co, k-e, and Baldwin-
Lomax models for turbine heat transfer problems [9, 10].
Two papers have compared the k-o_ and Baldwin-
Lomax models for turbomachinery problems. Bassi, et. al.
examined a film-cooled turbine cascade [11], and Liu et.
al. examined a low pressure turbine cascade [12]. Both
papers compared the computed results primarily with
experimental pressure distributions.
In the present work the k-co model developed by Wil-
cox [13] was incorporated in the author's quasi-three-
dimensional (quasi-3-D) turbomachinery analysis code
[14]. The code includes the effects of rotation, radius
change, and stream surface thickness variation, and also
includes the Baldwin-Lomax turbulence model. The k-c0
model was chosen for several reasons. First, the effects of
freestream turbulence, surface roughness, and mass injec-
tion are easily included in the model [13]. Second, transi-
tion can be calculated using the low-Reynolds-number
version of the model [15]. Third, Menter has shown that
the k-co model does well for flows with adverse pressure
gradients [5,16]. Finally, the k-c0 model should behave
well numerically since it avoids the use of the distance to
the wall and complicated damping functions.
1
z
Figure 1. Quasi-three-dimensional stream surface for a
compressor rotor.
This paper describes the quasi-3-D flow equations and
the explicit Runge-Kutta scheme used to solve them. The
paper also gives the k-t0 equations written in a quasi-3-D
form, gives the boundary conditions, and describes the
implicit upwind ADI scheme used to solve the turbulence
model equations. The model was tested on three cases and
compared to the Baldwin-Lomax model and to experimen-
tal data. The cases included a fiat plate boundary layer
with transition, a transonic compressor rotor, and a tran-
sonic turbine vane.
Quasi-3-D Navier-Stokes Equations
The Navier-Stokes equations have been developed in
an (m, 0) coordinate system as shown in figure 1. Here
m is the arc length along the surface,
dm 2 = dz 2+dr 2 (1)
and the 0-coordinate is fixed to the blade row and rotates
with angular velocity O.
The radius r and the thickness h of the stream surface
are assumed to be known functions of m. The equations
have been mapped to a body-fitted coordinate system, sim-
plified using the thin layer approximation, and nondimen-
sionalized by arbitrary reference quantities Po' Co' and
I.to . The Reynolds number Re and Prandtl number Pr are
defined in terms of these reference quantities. The final
equations are given in [14] and are summarized below.
Otq+_E+Orl_F-Re-ls), , = K (2)
where
q = J-l[p, pu, pvr, e] T
K = J-1IO, K2, 0, OlT
S = J-I[o, s2, $3,$41 T
(3)
0. ]
j-1 puU + _mP
E = (pvU+_oP)r I
(e + p) U + _orfIpJ
pV
F = j-I puV+ rim p
(pvV + _oP) r
( e + p) V+ rlorfI l
1 (u z + vZ)]e = p [CvT+
is the total energy per unit volume,
1 v2)lp = (7-1)[e-_(u2+
is the pressure,
hat l dh
r._m= ldr_.[_, and w = -
r rdm" h hdm
are derivatives of the streamtube geometry, and
(4)
(5)
(6)
(7)
(8)
K2 (pv2 _ -1 "_rm (p Re-t633]_ (9)
= +p-l,(e CY22)r+ -
The viscous fluxes are given by
S 2 = rlmC_ll +rloCrl2
S3 = (1]m(_12 + "1"100"22) r (10)
I.t 2
S4= (T-- i)'Pr( TIm+ r12] _rla2 + uS2 + vS3
a2 = Vp/p is the speed of sound squared. Using Stokes"
2
hypothesis, _. = -gll, the shear stress terms are given by
fill = 211OmU+ _,V . _"
211 (OoV + urm) + _.V .
0"22 = r
0"33 --- 21.tu_ + XV. _' (11)
o_2= _tIO,.v-vr+ !oou)
2 .___] + _ov ]_,V. ' = -_12[OmU + U(_ + hm
U and V are relative contravariant velocities
2
u = _mu + _e"
(12)
V = _tau + TloW
where w is the relative tangential velocity, w = v - rO.
The 0-metrics are scaled by 1/r and the Jacobian is
scaled by rh. The metric terms are found using central
differences and
,r,.-1
n L-o m /rJ (13)
J = [rh (m_0rl - mrl0 _) ] -1
The effective viscosity is
= gL + gT (14)
where the molecular (laminar) viscosity I.tL is evaluated
using a power law function of the temperature, and the tur-
bulent viscosity [J'T is evaluated using either the Baldwin-
Lomax model [1] or Wilcox's k-co model [13, 15]. Minor
modifications to the coefficients and blending functions
used in the Baldwin-Lomax model are described in [2].
Boundary Conditions
At the inlet the total pressure, total temperature, and
tangential velocity component are specified and the
upstream-running Riemann invariant based on the axial
velocity is extrapolated from the interior. At the exit, three
of the four conserved variables are exwapolated and the 0-
averaged pressure is specified using the method described
by Giles in ref [17]. Periodic boundaries between the
blades are solved like interior points using a dummy grid
line outside the domain.
Multistage Runge-Kutta Scheme
The flow equations are discretized using finite differ-
ences and solved using an explicit Runge-Kutta scheme.
A spatially-varying time step and implicit residual
smoothing axe used to enhance convergence. Details of
the solution scheme used here are given in (18) and are
described briefly below.
The discrete equations are solved using the explicit
multistage Runge-Kutta scheme developed by Jameson,
Schmidt, and Turkel [19]. A four-stage scheme is used.
For efficiency, physical and artificial dissipation terms are
computed only at the first stage. The Baldwin-Lomax
model is updated every five time steps. The k-(o model is
usually updated every two time steps with twice the At of
the flow solution.
The spatially-varying time step is calculated as the
harmonic mean of inviscid and viscous components in
each grid directional.
Artificial dissipation consisting of blended second and
fourth differences is added to prevent point decoupling
and to enhance stability. Eigenvalue scaling, as introduced
by MartineUi and Jameson [20] but modified by Kunz and
Lakshminarayana [8], is used to weight the artificial dissi-
pation in each direction. The scaling is based on a blend
of the one-dimensional time step limits at each point. The
artificial dissipation is also reduced linearly by grid index
near the wall and wake centerline to minimize the effects
on the boundary layer.
The explicit four-stage Runge-Kutta scheme has a
Courant stability limit of about 2.8. Implicit residual
smoothing introduced by Jameson and Baker in [21] can
be used to increase the time step, and hence the conver-
gence rate, by a factor of two to three. On high aspect
ratio grids the stability limit is dominated by the grid spac-
ing in the finest direction, and it is sufficient to use implicit
smoothing in that direction only. The stability analysis
given in [21] is used to calculate the smoothing parameter
required at each point, then the same Eigenvalue scaling
used for the artificial dissipation is used to reduce or elim-
inate the smoothing parameter in grid directions where it
is not needed. The use of Eigenvalue scaling for both the
artificial dissipation and implicit smoothing greatly
increases the robustness of the numerical scheme.
k-_ Turbulence Model
The k-c0 turbulence model was first postulated by
Kolmogorov in 1942 and later independently by Saffman
in 1970 (see Wilcox's book [13] for references.) It has
been under development by Wilcox for many years and is
described in detail in [13]. The model solves two turbu-
lence transport equations for the turbulent kinetic energy k
and the specific dissipation rate c0. The model has a basic
formulation for fully turbulent flows that satisfies the law
of the wall without knowledge of the distance to the wall
or complicated near-wall damping terms. There is also a
low-Reynolds-number formulation used for modeling
transition [15]. Boundary conditions can be specified to
simulate mass injection or surface roughness.
Most of Wilcox's development of the model used
boundary layer codes, but recently Menter has shown sev-
eral applications to Navier-Stokes codes [16]. Menter
found that the model exhibited strong dependence on
freestream values of co and proposed a somewhat ad hoc
fix. In this work many of Menter's suggestions for numer-
ical implementation of the model have been used, but his
fix for the problem of freestream dependence has not.
The quasi-3-D form of the k-equation has been
derived by writing the m- and 0-momentum equations in
non-conservative form, multiplying each by its fluctuating
3
velocity component, and Favre averaging. The usual tur-
bulence modeling approximations are made, i. e., the
Boussinesq model is used for the Reynolds stress terms,
pressure work, diffusion, and dilatation are all neglected,
and turbulent dissipation is taken to be proportional to
k x to. The production term is written in terms of the vor-
ticity magnitude using Menter's suggestion [16]. Source
terms that arise from the quasi-3-D equations are
neglected. The to-equation is derived from the k-equation
by dimensional considerations. Wilcox's constants are
used without modification. The final form of the model
equations is as follows:
_,q+U_q+ V_rlq-RCl JG = I (p-D) (15)
P P
where
q = [k, o)] T (16)
_tr = (x"pk (17)
to
The molecular plus turbulent diffusion terms G are
written using the thin-layer approximation giving
(18)
Menter's form of the production terms is used [16].
(19)
where
rr a= 8,nv- 8e u + v-- (20)
T
is the vorticity. The destruction terms are given by
p J
The baseline k-to model has five coefficients:
[3 = 3/40, [3" = 9/100, (r = 1/2, (r*= 1/2, ct = 5/9,
and the trivial constant o_° = 1.
The low-Reynolds-number model replaces three of
the constants with the following bilinear functions of the
turbulence Reynolds number Re T:
6*= (9/100)F_, ct= (5/9) (Fa/FIX) , and co*= Fix,
where
5/18 + (ReT/RI3 ,)4
FI3 =
1 + (ReT/R p) 4
a 0 + ReT/Rto
Fct = 1 + ReT/R(o
c(_ + ReT/R k
Fix = I+ReT/R _
Re T = pk
_tL0)
(22)
with % = 1/10, % = 13/3 = 1/40,
and RI3 = 8, R_ = 27/10, R k = 6.
Boundary Conditions
At the inlet the turbulence intensity Tu and turbulent
viscosity IxT are specified. Then k and co are found from
k = 3Tu2U.2
2 an
O) --- C{*pk
l.t7-
(23)
where (x* = 1 for the baseline model or ix* = F_t for the
low-Reynolds-number model. Substituting equation (22)
for ix" into equation (17) for I.tT gives a quadratic for
Re T . The solution is
(24)
and co may be found from
to = __EL_ (25)
l.tLRe T
A turbulent length scale can be defined using (17) as
l.tT ct*P-_k = p,g_,----_-'--,: p,a_l (26)
The effects of varying the inlet values of to or l is dis-
cussed with the results.
On solid walls k = 0, and co is set using Wilcox's
roughness model.
4