An LU implicit multigrid algorithm is developed to calculate 3-D compressible viscous flows. This scheme solves the full 3-D Reynolds-Averaged Navier-Stokes equation with a two-equation kappa-epsilon model of turbulence. The flow equations are integrated by an efficient, diagonally inverted, LU implicit multigrid scheme while the kappa-epsilon equations are solved, uncoupled from the flow equations, by a block LU implicit algorithm. The flow equations are solved within the framework of the multigrid method using a four-grid level W-cycle, while the kappa-epsilon equations are iterated only on the finest grid. This treatment of the Reynolds-Averaged Navier-Stokes equations proves to be an efficient method for calculating 3-D compressible viscous flows

Yokota, Jeffrey W.

English

NASA Technical Reports Server

NASA Contractor Report 182209
AIAA-88-0467
A Diagonally Inverted LU Implicit Multigrid
Scheme for the 3-D Navier-Stokes
Equations and a Two Equation
Model of Turbulence
Jeffrey W. Yokota
Sverdrup Technology, Inc.
NASA Lewis Research Center Group
Cleveland, Ohio
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October 1988
Prepared for
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Under Contract NAS3-25266
National Aeronautics and
Space Administration
https://ntrs.nasa.gov/search.jsp?R=19890001492 2020-03-20T06:05:14+00:00Z
A Diagonally Inverted LU Implicit Multigrid Scheme
for the 3-D Navier-Stokes Equations and
a Two Equation Model of Turbulence
Jeffrey W. Yokota
Sverdrup Technology, Inc.
NASA Lewis Research Center Group
Cleveland. Ohio 44135
Abstract 2. Analysis
An LU implicit multigrid algorithm is developed
to calculate three-dimensional compressible viscous flows.
This scheme solves the full three-dimensional Reynolds-
Averaged Navier-Stokes equation with a two equation k - e
model of turbulence. The flow equations are integrated
by an efficient, diagonally inverted, LU implicit multigrid
scheme while the k- • equations are solved, uncoupled
from the flow equations, by a block LU implicit algorithm.
The flow equations are solved within the framework of the
multigrid method using a four grid level W-cycle, while the
k - s equations are iterated only on the finest grid. This
treatment of the Reynolds-Averaged Navier-Stokes equa-
tions proves to be an efficient method for calculating three-
dimensional compressible viscous flows.
1. Introduction
As the availability of larger and more powerful comput-
ers increases, so will the attention being directed towards
computing complex three-dimensional compressible viscous
flows. At present the most sucessful approach has been to
solve the Thin-Layer Navier-Stokes equations with an alge-
braic turbulence model. Most notable has been the work of
Pulliam and Steger 1 and their highly sucessful ARC codes.
Attempts to produce efficient numerical schemes for
the calculation of compressible viscous flows have led to
the Diagonalized ADI scheme of Chaussee and Pulliarn 2
and more recently to the multigrid Runge-Kutta scheme of
Jayaram and Jameson. 3 As emphasis increasingly focuses
upon calculating complex three-dimensional flows, the need
to numerically solve the Reynolds-Averaged Navier-Stokes
equations efficiently is correspondingly apparent.
The present work deals with the development of an ef-
ficient LU implicit multigrid scheme for the numericaI- solu-
tion of the Reynolds-Averaged Navier-Stokes equations and
the two equation k - • turbulence model. The Reynolds-
Averaged Navier-Stokes equations, which for mathematical
closure require the modelling of the Reynolds stress tensor,
are solved by the Diagonally Inverted LU Implicit Multi-
grid scheme developed by Yokota, Caughey, and Chima. 4
A Boussinesq eddy-viscosity formulation is used to model
the Reynolds stress term, where the turbulent viscosities
are calculated from a standard high Reynolds number k -
model.
The three-dimensional Reynolds-Averaged Navier-
Stokes equations, with the Boussinesq eddy-viscosity for-
mulation, are written in divergence form and then trans-
formed from the Cartesian coordinate system (x,y, z) to
the generalized system {_, r/, f). The resulting equations
can be written:
off, oF ad a#
+
of. ad_ a#. (x)
a-T + + a-T
with
i pD
pDu
= pool
PDw I
pDE]
i pDU )
pDUu + PD_
pDUv + PD(y
pDUw + PD_,
DU(E + P)
[ pDV )
[ pDVu + PDtl.
o= I pDV: + PDrly ,0 =| pDV ,.v+ PDrb,
\ DV(E + P)
pDW I
pDWu + PDc_
pDWv + PDcy
pDWw + PDc,
DW(E + P)
/°,/
/°,/
=
/.,!
L,/( o /D(_r_x + D(yr_u + D(,r_,D(xr_ u + D(uryy + D(,ru,D(_rx, + D(yr_, + D(,r,,
__7.__" c aT c 2 00--_n+¢3u f ., + v f .s -l- w f ., -t- .y_x l e( t-i(+ aa--_)
{g_* )
[ g"=
d_=|a., =
g""
\ gv5
where
pDVk-F.k(C4 ak c ok ok )Gk* = _- , , ac a,x
#,lc4y_ + csb_ +c6_1
and
= (pDWk- , a_ Ok#ktCT_ + CS_
_rk" \ pDWe , a. o,
P,[c7 b-_ + cs_- 6
ak )
+ c9_)
Ot
+ c9_)
si = D(O - pe)
D(tlO - t2pe)e
B2 ---_
k
#t
#k = I_t + --
o" k
#t
#, = l_t + --
O"c
for which the production rate of the turbulence kinetic en-
ergy is defined as
-5 pk + 7z
+ 2"'Tz- V
a,, a,, av av a,,, aw_ _+., _+_z+_+_+_+ av ]
The turbulence modelling constants axe chosen to be the
standard Launder and Spalding 9 values of
C,, : 0.09 tl = 1.44 t2 - 1.92
Ok = 1.0 a, = 1.3
and were never altered during the course of this work.
(Isotropy is assumed since C_, is a scalar constant rather
than a non-constant vector quantity.) The transformed
Navier-Stokes and k - e equations are discretized by a finite
volume formulation that approximates the spatial differ-
ences as a net flux across the faces of each mesh cell. Global
conservation and the admission of possible uniform flow so-
lutions are insured by evaluating both the inviscid and vis-
cous flux vectors on the faces of the boundary-conforming
mesh cells. This procedure requires that the flow and k-
variables, viscous stresses, and flux-embedded geometric
quantities be defined on the faces of the mesh cells during
the flux evaluations, although it is the cell-averaged flow
and k - _ variables that are calculated during the time and
spatial marching. The viscous stresses and geometric quan-
tities are evaluated directly on the cell faces, while the flow
and k - • variables are averaged over values found in adja-
cent cells. The unsteady equations can be discretized into
an implicit approximation that, when written in a linearised
delta form, produces a numerical scheme whose steady state
solutions are independent of the time step size used in the
time marching. The computational effort required to con-
struct this implicit approximation is kept at a minimum
by treating only the inviscid fluxes. The delta form, pro-
duced by linearizing the changes in the inviscid flux vectors
through a Taylor series expansion about a time level n, can
be written for the flow equations as follows:
[I + #,At(6,A + 6,B + 6,C)] AW_,.k-" =
g "
-At($¢(._- _,,) + 8,,(0 - _,,} + $_(/7 - "))ok (4)
where
and At is the time step size; 0 < #i <_ 1 is a parameter
governing the degree of implicitness; 6 and $ are cell- and
face-centered central differences; I is the identi_y matrix;
and A, B, and C axe the inviscid flux Jacobian matrices
relative to the vectors J_, G, and /7. The implicit form of
the k - e equations can be written as follows:
_n+ .,",(*,Ak. +  .Bk, +  ,ck. - =
-_t(8¢Pk. + 8,dk. + 8,#k. - "
- Sk,)_Sk (5)
and
-- Wkl r
where Ak_, Bk,, and Ck_ are the inviscid flux Jacobian
matrices relative to the inviscid terms found in the vectors
Gk,, and /Jk_; and E_:, is the Jacobian matrix relative
to the source vector Sk_-
3. Artificial Dissipation
The finite volume formulation reduces to a central dif-
ference approximation on a uniform grid and therefore re-
quires the addition of explicit artificial dissipation terms
2) Upper Sweep
: A_,._ (11)[i+ _,A_(sta, + s.+B2+ 6,+e_)]A_,_k -"
A similar explicit sweep, but in directions opposite to those
taken in the lower sweep, is needed to solve this system of
equations. The resulting flow field corrections are then used
to update the flow field.
The LU factorization requires the solution of two block
triangular operators each of which, through back substitu-
tion, can be reduced to simple 5x5 matrix systems at every
mesh cell. These reduced systems can be written:
1) Lower Sweep
I+_iAt(At + Bl + Cl AYiyk =
-- -- _ rt
-At(_(_ g_)+_,(d G)+_,(g-g_)+ ),,_
+u, at(a,A¢,"_l,,,_ + BiAi,?;_l,k + C1A¢,?,.__1) 02)
2) Upper Sweep
I- #iAt(A2 + B2 + C2 AWijk =
_rt _n
AYi_k - I.qAt(AzAWi+ i,j,k
+B2AP_,_i+l,k + C2AI4/,%k+l) (13)
A similar analysis for the k - • equations can be used
to write the following 2x2 matrix equations:
1) Lower Sweep
[ - ]AYksliikI+ F.iAt(Ak,, + Bk., + Ck,, Ek,,} _ '_ =
rt
-At($¢h. + _.dk. + 6¢/ik, -- Ek. + Tk.),i k
n n
+mAt(Ak,, AYk, I,-l,S,k + Bk,,Aek,l,,j-,,k
+Ck,, A k,I,.yk-_)
2) Upper Sweep
_ +_ 1AW_.l,,._" =[I .,At(Ak., + B,., Ck., Ek.,) l
AYk,{i./k- p_At(A_:,3AWk,[_+I.Y,k
(15)
5, Diagonally Inverted LU Factorization
Th_ LU factorisation of the k - * equations produces
simple 2x2 matrix systems which can be inverted alge-
braically, while the matrix systems associated with the
flow equations axe diagonally inverted using the similarity
transformation _ that produces the following diagonal ma-
trix:
Q-l(a + B + C)Q =
which has elements
A11 =A:_=Aas= U+V+W
_,,= u + v + w - _x/_?+ g + _
_ = u + v + w + _ + _ + q
Aii = 0 when i _ 3"
where
q2 = u2 + 112-4-w 2
l_ = _+r/_+C_
la = G+_/-'+C,
and c is the local speed of sound. This similarity transfor-
mation, for a local time step defined as:
eft
At=
(IAI + IBI + lCl)
where Gft is the Courant number used in the time marching,
can be used to transform the lower and upper sweeps into
scalar equations with the following vector components (rn =
1, ..., 5)
1} Lower Sweep
(Q aY,_).. =
-AtQ-'((S_(f-f_)+_.(d-G) )+$¢(/] -/Y.) + T)O._
-,,,(a,a¢,- ,.,.k+., TM +claY,,_.__,))
- AY,,_-L_ m
((1+_)1+ _!A),_
{16)
2) Upper Sweep
(Q aw,,._).,=
T. Steady State Calculations
The calctllation of steady state solutions is made more
efficient by using local time-stepping and the multigrid
method.
Local time-stepping is used to optimize the time step
throughout the flow field. A locally varying time step size,
based on a constant Courant number, is used to create a
warped time integration that can accelerate the calculation
to a steady state without affecting the steady state solution.
These time steps axe defined identically for both the flow
and k - e equations, although the value of the Courant
numbers used in their evaluation may vary.
The multigrid method is incorporated into the Diago-
nally Inverted LU scheme to accelerate the removal of low
frequency errors from the flow solution and thus increase
the efficiency of the time-marching procedure. Following
the work of Jameson, 16 the flow solver is used to smooth
out high frequency errors resolvable on any current grid
level (h), while the multigrid method is used to eliminate
low frequency errors through a sequence of flow calculations
on coarser grids (2h, 4h, 8h,...) . The multigrid sequencing
used is the four-level W- cycle shown below
h
÷h
Coarse grid boundary conditions are identical to those used
on the fine grid with the exception of the inflow/outflow
conditions which axe updated only on the fine grids. The
Reynolds-Averaged Navier-Stokes equations are solved on
the finest grid while only the Euler fluxes are evaluated on
the coaxse grids. Coarse grid residuals are kept smooth by
adding only a constant coefficient second difference artifi-
cial dissipation term {the nonlinear blended terms axe used
only on the fine grid}. This treatment attempts to limit
the amount of high frequency errors reintroduced into the
flow field by the upward interpolation of the coarse grid
corrections. The flow solver, in this case the Diagonally In-
verted LU scheme, is invoked only once on each grid level
and only before transfering the flow field to the next coarser
grid. The multigrid cycle defined above requires approxi-
mately 1.32 work units of computational effort, where work
units are normalized by a single Navier-Stokes calculation
on the finest grid. The k - e equations axe solved only on
the finest grid and axe not accelerated with the rmlltigrid
method. Preliminary attempts at multigridding the k -
equations proved to be less efficient than simply solving
these equations only on the finest grid.
8. Results
Numerical results axe presented to illustrate the Di-
agonally Inverted LU scheme's ability to calculate three-
dimensional compressible viscous flows and the convergence
acceleration produced by the multigrid method.
Turbomachinery calculations were performed on H-
type grids consisting of 96x24x24 mesh cells in the through-
flow, blade-to-blade, and radial directions, respectively.
These grids were generated using a modified version of the
GRAPE code 17 originally developed by Sorenson. Is All cal-
culations were performed on a CRAY X-MP, where a calcu-
lation consisting of 211 work units required approximately
1.5 million words of memory and 40 minutes of CPU time.
The test case used to evaluate the LU scheme is the
Annular Cascade designed and extensively tested at NASA
Lewis. 19'2° The computational grid shown in Figure 1 is
based on the full annular ring of 36 core turbine stator
vanes. The geometry is a 38.10 mm high untwisted blade
of constant profile with an axial chord of 38.23 mm. The
stator has a tip diameter of 508 mm and a 0.85 hub-to-tip
radius ratio. Mesh cells found immediately adjacent to solid
walls axe centered at distances 0.002 of an axial chord away,
which correspond to a value of Y+ _ 60 for the following
flow calculations.
Experimental test conditions of ambient axial inflow
and a 0.65 hub-static to inlet-total pressure ratio produce
a flow field with mean radius inlet and exit critical velocity
ratios of 0.231 and 0.778 respectively. To match the up-
stream flow conditions (an inflow Mach number of 0.211),
the nonrotating (ROT --- 0} calculations were run with a
0.665 hub-static to inlet-total pressure ratio (PR = 0.665).
Figure 2 shows the convergence history of the Annu-
lax Cascade calculation (with nmltigrid} where a drop of
3.5 orders of magnitude in the flow field error was pro-
duced within 200 work units. The residual was reduced
from an initial RES1 = 0.169x101 to a final value of
RES2 = 0.572x10 -3 (0.9628 average rate of convergence).
A Courant number of Cn = 6 was used in the solution of the
flow equations while a Cn --- 4 was used in the ]c - e equa-
tions (no attempt to optimize these numbers was made}.
The resulting flow field is fully subsonic and is com-
pared with experimental data at three spanwise positions.
Figures 3, 4, and 5 compare the calculated blade surface
static pressure distributions {normalized by inlet total pres-
sure} at 13.3, 50, and 86.7 percent span with the exper-
imental data produced by Goldman and Seasholtz. 2° The
[17] Smith, W.A., "Modified Grape Code," Sibley School [19]
of Mechanical and Aerospace Engineering FDA Report
87-10, CorneH University, June 1987.
[18] Sorenson, R., "A Computer Progra_n to Generate Two
Dimensional Grids About Airfoils and Other Shapes by
the use of the Poisson's Equation," NASA TM 81198, [20]
1980.
Goldman, L.J., and McLallin, K.L., "Cold-Air Annu-
lax Cascaxle Investigation of Aerodynamic Performance
of Core-Engine-Cooled Turbine Vanes. I: 8olid-Vane
Performance and Facility Description," NASA TM X-
3224, 1975.
Goldman, L.J., and Seasholta, R.G., "Laser Anemome-
ter Measurements in a Annular Cascaxle of Core Tur-
bine Vanes and Comparison with Theory," NASA TP-
2018, 1982.
I .05
Fig. 1. Computational grid for the Annular Cascade.
Cd
-3
_q.
___l L I I
-5 0 200 _-i]O 600 800
_ORK
1 O0
P.R. 0.o650 ANGY 90.00
POT. 0.000 ANGZ 0.00
RESI O. !6_E Ol #ES2 0.551E-03
NORK 2I ! .8_ C.N. 6.0 QATE 0.9_28
Fig. 2. Convergence history for the multigrid calculation.
0.95
z0.85
0._
\
U)
c:L 0.75
0.65
o.s_0
4.
+
÷
÷g*
÷
*
@
I I J
0
÷
÷
÷
÷
"0
÷ ÷ * @ ÷ ÷,
q.
t 1 I I
0.5
÷
\ .
÷
4-
÷
• I l ]
1°0
x/c
• CALCULATION
° EXPERIMENTAL OATA_
Fig. 3. Staticpressureblade distributionat 13.3 percent
span.



A diagonally inverted LU implicit multigrid scheme for the 3-D Navier-Stokes equations and a two equation model of turbulence

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