and At must be smMl. B_ >> B_,. A_ >> A_,. (139) so that c_ B_. This is only a constant. By applying. (140) to (135), we obtain the following tridiagonal system:.
NASA
Contractor
Report
4540
Multigrid Methods for Flow Transition in Three-Dimensional Boundary
Layers
Surface
Chaoqun and
Roughness
Liu,
Steve
Zhining
Research
Denver,
Colorado
Prepared
for
under
Research Contract
Center NASI-19312
National Aeronautics Space Administration
and
Office of Management Scientific and Technical Information Program 1993
Liu,
McCormick
Ecodynamics
Langley
With
Associates,
Inc.
TABLE
1
INTRODUCTION
2
GOVERNING
3
4
5
OF
CONTENTS
.....................................................
EQUATIONS Form
IN
PHYSICAL
2.1
Original
2.2
Nondimensional
Forms
for Planar
2.3
Nondimensional
Forms
for Flat-Plate-Type
2.4
Initial
and
Boundary
Conditions
2.5
Initial
and
Boundary
Condition
4
..........................
4 Channel-Type
Vector
of the
3.2
General
Transformation
3.3
General
Curvilinear
Flow
EQUATIONS
Equations
5 6
Flow
..........
7
.............
8
.......................
8
.....................
8
................................
METHODS
Discretization
on a Uniform
4.2
Discretization
on a Stretched
4.3
Outflow
4.4
Distributive
4.5
Multigrid
and
.......................
10
TWO-DIMENSIONAL
Staggered
Grid
Staggered
Treatment
Relaxation
9
System
FOR
4.1
Methods
Type
for Flat-Plate-Type
Navier-Stokes
...............
..................
for Planar-Channel
Coordinate
Boundary
Flow
Flow
OF GOVERNING
3.1
NUMERICAL
SPACE
.....................................
TRANSFORMATION Form
1
Grid
PROBLEMS
12
• • •
.....................
12
....................
19
............................. Line-Distributive
23 Relaxation
..............
26
...................................
30
4.5.1
Full-Coarsening
.................................
30
4.5.2
Semi-Coarsening
.................................
33
NUMERICAL
METHODS
FOR
THREE-DIMENSIONAL
PROBLEM
5.1
Discretization
of the
Navier-Stokes
Equations
on a Uniform
5.2
Discretization
of the
Navier-Stokes
Equations
on a Nonuniform
5.2.1
Analytical
Mapping
5.2.2
Governing
Equations
Staggered
Grid
Staggered
35
.... . . . Grid
38
............................... under
Special
35
38 Mapping
.................
40
\
Pl_elOiNiil
PAGE
BLANK
NrJi
PtLMEL_
5.2.3
6
7
5.3
Line-Distributive
5.4
Multigrid
Relaxation
Methods
.................................
49
5.4.2
Semi-Coarsening
.................................
51
OF
TWO
DIMENSIONAL
SIMULATION
Channel
6.2
Two-Dimensional
Spatial
Flat
6.2.1
Parallel
Base
6.2.2
Non-Parallel
Flow
Temporal ?.1.1
Linear
?.1.2
Secondary
Spatial
7.2.2
Secondary
Stage
Instability
Stage
9.2
Stage
Parallel
8.1.2
Non-Parallel
OF
Base
.............................
59
CHANNEL
SIMULATION
.............
70 70
.............................
?0 ?5
Base
82
.............................
82
PLATE
83
SIMULATION
................
85
................................
Flow
Instability
85
............................... Flow
85
.............................
85
..................................
ROUGHNESS
Two-Dimensional
57
..............................
FLAT
8.1.1
EFFECT
57
..............................
Instability
Instability
Secondary
.........................
....................................
THREE-DIMENSIONAL Linear
Flow
Instability
Approach Linear
55
..................................
Instability
7.2.1
55
..........................
Plate
PLANAR
Approach
........................
...............................
Base
THREE-DIMENSIONAL
9.1
48
Full-Coarsening
Spatial
8.2
45
...................................
Two-Dimensional
8.1
41
.............................
6.1
?.2
9
..................................
5.4.1
RESULTS
?.1
8
Discretization
85
ELEMENTS
Roughness
FOR
Element
TRANSITION
for Channel
Flow
..............
90
..............
90
9.1.1
Isolated
Roughness
...............................
90
9.1.2
Multiple
Roughness
Elements
90
Two-Dimensional
Isolated
9.2.1
Grid
Generation
9.2.2
Small-Scale
Roughness
......................... Element
.................................
Roughness
.............................
iv
for Flat
Plate
Boundary
Layer
Flow
97 97 99
9.2.3
Medium-Scale
9.2.4
Large-Scale
9.3
Two-Dimensional
9.4
Three-Dimensional Flow
9.5
............................
Roughness
.............................
Distributed
Multiple
Isolated
Roughness
Roughness Element
109 117 ................. for Flat
Plate
123 Boundary
Layer
...........................................
9.4.1
Base
9.4.2
Disturbance
Flow
Three-Dimensional Flow
Roughness
126
....................................
126
................................... Distributed
Roughness
127 Element
...........................................
9.5.1
Base
9.5.1
Disturbance
10 CONCLUDING
Flow
11 ACKNOWLEDGEMENTS REFERENCES
Boundary
Layer 131
.................................... ...................................
REMARKS
for Flat Plate
................................. .................................
......................................
132 132
139 139 140
SUMMARY In this research project, an efficient multilevel adaptive method has been successfully applied to perform direct numerical simulation (DNS) of flow transition in 3-D channels and 3-D boundary layers with 2-D and 3-D isolated and distributed roughness elements resolved by a curvilinear coordinate system. The approach consists of a fourth-order finite difference technique on stretched and staggered grids, a fuLly-implicit time marching scheme, a semi-coarsening multigrid method based on line distributive relaxation, and an improved outflow boundarycondition treatment, which needs only a very short buffer domain to damp all order-one wave reflections. This approach makes the multigrid DNS code very accurate and efficient. It allows not only spatial DNS for the 3-D channel and flat plate at low computational cost, but also spatial DNS for transition in the 3-D boundary layer with 3-D single and multiple roughness elements. These calculations would have extremely high computational costs with conventional methods. Numerical results show good agreement with linear stability theory, secondary instability theory, and a number of laboratory to transition is also analyzed.
1
experiments.
The contribution
of isolated
and distributed
roughness
Introduction
The main driving force for the study of flow transition is the understanding, prediction, and control of transition to turbulence. The transition process from laminar to turbulent flow in a shear layer is a basic scientific problem in modern fluid mechanics and has been the subject of study for over a century, but it remains a challenging and important area for research (Reed & Saric, 1989). The control of flow transition to turbulent state is a very significant topic. The drag of an aircraft at cruise flight conditions is about 60% friction drag for presentday transport aircraft with turbulent boundary layers on their wetted surface. The principal drag reduction opportunities lie in stabilizing the laminar boundary layer as much as possible (Reshotko, 1988). However, the development of flow transition control is mainly based on our knowledge of flow transition. Therefore, developing an understanding of shear flow at high Reynolds numbers has been a central problem in the theory of fluid motion (Bayly,
Orszag,
Classical
& Herbert,
linear
stability
1988). theory,
which was established
independently
by Orr (1907a,
1907b)
and Sommerfeld
(1908), is based on a parallel base flow assumption on the linearized perturbation form of the Navier-Stokes equations, referred to as the Orr-Sommerfeld equation. Tollmien (1931) and Schlichting (1932) were able to solve the Orr-Sommerfeld equation about 20 years later. Linear stability theory validation was first confirmed by Schubauer and Skramstad (1948), who used a vibrating ribbon to impress a disturbance into the boundary layer and hot wires to take measurements. Linear stability theory is now widely accepted as a tool to predict the initial growth of disturbances, more commonly by solving the Orr-Sommerfeld equation When
the amplitude
of a T-S wave
referred to as Tollmien-Schlichting (Joslin et ai., 1992). reaches
a certain
finite
value
(T-S)
( about
wave,
which
1% of the
can be obtained
base flow
maximum
velocity in root-mean-square (rms) values ), nonlinear effects are felt and the instability waves no longer obey the linear theory. At this stage, the three-dimensional disturbances are energized and streamwise vortices form and intensify. Later, instantaneous streamwise velocity profiles become inflectional, resulting in high shear layers, which is accompanied by the appearance of A-like structures. Eventually the high shear layers develop kinks, leading leading
to the formation to turbulence.
Nonlinear dimensional
instability, structures.
Sargent (1962), now dimensional structure
of hairpin
eddies
followed
by the breakdown
of the flow.
commonly referred to as secondary instability, is associated An aligned three-dimensional structure was first observed
Finally,
turbulent
spots
form,
with the formation of threeby Klebanoff, Tidstrom, g_
referred to as fundamental or K-type after Klebanoff. The staggered patterns of threecharacterized by subharmonic signais were found by Kachanov, Ko,.lov, _r Levchenko
(1978). These experiments showed that the subharmonic staggered patterns) was excited in the boundary layer.
of the fundamental At smaLl amplitudes,
wave (a necessary it produced the
feature resonant
of the wave
2
Liu,Liu,McCornuck,
Final Report
interactions predicted by Craik (1971). This type of instability is referred to as C-type after Cralk. At large amplitudes, Craik's mechanism becomes less important and instability is characterized by the dominance of offresonance modes. This type of instability is referred to as H-type after Herbert (1983a). Benney and Lin (1960) imposed a two-dimensional T-S wave and a pair of oblique three-dimensional wave at the inflow boundary to study the interactions between them. Later, Herbert (1983a, 1983b) derived a theory for an experimentally observed three-dimensional parametric instability. Remarkable agreement between prediction from this new theory and experimental results is obtained, and the secondary instability theory is now generally accepted as a tool to predict later development of transition in boundary layers. The major engineering tool for transition prediction is the e N- method. Smith and Gamberoni (1956) correlated transition data with the amplitude ratio of the most amplified eigenmodes between the onset of instability and the transition location and found that transition occurred when the amplitude ratio reached a value of e 9 ._ 8100. However, the method is semi-empirical and thus requires some foreknowledge of the flow undergoing transition. Although great progress has been achieved in this century, the mechanism of transition remains a mystery. Flow transition is a very complicated, three-dimensional, time-dependent, multistage physical phenomena, and is affected by many factors, including disturbance environment, pressure gradient, surface roughness, threedimensionallty, curvature, Mach number, and many other aspects that influence boundary layer development. For example, the manner of the effect of surface roughness on flow transition is still largely unknown. This is one of the major interests of the current work. The appearance of modern digital computers led to the development of numerical methods applied to studying instability and transition from laminar to turbulent flow. The most successful approach is by direct numerical simulation (DNS). Due to insufficient computer resources, the majority of the first investigations were temporal. Among them are those by Orssag & Kells (1980), Wray & Hussaini (1984), Biringen (1984), Kleiser & Laurien (1985), Zang & Hussaini (1985), and Zang, Krist, Erlebacher, & Hussaini (1987). Temporal simulations follow the time evolution of a single wavelength of the disturbance as it convects with the phase speed of the wave. Available mesh resolution can then be focused on resolving a single wavelength. This enabled simulation into the later stages of transition and revealed good agreement with the experiments. However, the assumption of periodicity along the streamwise direction does not snow direct comparison of temporal numerical simulations with spatially evolving experimental results. Moreover, streamwise growth of the base flow (e.g., in boundary layers) is not snowed in the temporal approach. Hence, boundary layer simulations are restricted to the parallel flow assumption as well. This approach basically lacks a physically realistic representation. On the other hand, the spatial DNS approach provides the needed quantitative information about transition. But with spatial DNS, obstacles have existed that have prevented fully carrying out such a study. Among these are the realistic specification of inflow and outflow boundary conditions and the high demands on computational resources. Even with today's supercomputers, current resources are insufficient to snow conventional methods to perform full simulation of transition to turbulence in a boundary layer in a spatial setting. A number of authors have made some progress in spatial DNS, including Fasel (1976), Fasel & Bestek (1980), Fasel & Konselmann (1990), Spalart (1989), Danabasoglu, Biringen, & Streett (1991), and Joslin, Streett, & Chang (1992). The results provided by spatial DNS show qualitative agreement with linear stability, secondary instability theory, and some available experiments. Kleiser & Zang (1991) gave a more complete list of DNS works in their review paper. It is worth mentioning that significant achievements in spatial DNS have been made by Liu, Liu, & McCormick (1991, 1992, 1993). Supported by the NASA Langley Research Center since 1990, they have developed a very efficient and accurate multigrid approach for DNS. The flow transition process, including linear evolving secondary instability and breakdown in 3-D channels and 3-D flat plates, was fully simulated, showing good agreement with linear stability theory, secondary instability theory, and experiments by Nishioka, Asai, & Iida (1981) and Saric & Reed (1987). Since the approach is computationsny very efficient, it enabled simulation of more complicated transition processes such as with 3-D distributed roughness elements in 3-D boundary layers.
Liu,Liu,McCorn_ck,
Final
Instability
of boundary
to describe, ducted
model,
& Tidstrom
recovery
zone.
angular
two-dimensional ribbon.
range,
showing
mixing
(shear)
has
been
that
the
and
the
layer
in the flow.
other
hand,
concerning
the
roughness
boundary
channels Liu
the 3-D,
flow
s
new
•
•
stretched
implicit
that
of explicit
successfully
and
able
to work
are
and
staggered
for both
general
linear less
general
2-D
only
computational (1977)
numerical an
boundary
in 3-D
hours
for each
solver,
which
can
roughness
in a
to breakdown
has
and
case),
now
successfully
shape
Ap-
elements
6 CRAY
solver
sim-
isolated,
layers.
roughness
instability
This
found
& Schefenacker
2-D
than
of the channel
They
investigated
in
multigrid
solve
simulated
distribution
in
than
of
geometry accepted
convection
is very
efficient
at
each
time
at each schemes
out-flow
boundary
reduces
the
magnitude
and
with
are generally
that
based
channels
that
relaxation
for
way,
for more
directions
step
multiple
from
coordinates.
The
(1993)
waves
a fast
instability in a 3-D
recently.
Bestek,
to a single
it takes
elements
three
time
efficient
transition
(usually
in general
in all
multigrid
the
limited
by a
instability
to
have
higher
accuracy
that
and
diffusion)
that
has much
better
stability
than
that
has
high
accuracy
and
can
reduce
artificial
viscosity
error
approach
linear
grids
(implicit
Semi-coarsening
Since
of instability are
a rect-
of Tollrnien-Schlichting
limited.
al.
point
used
frequencies
a two-dimensional
et
in
methods
distributive
spatially
very
transition
(1990)
by the
most
by Fasel,
investigated
developed
roughness
includes:
•
and
3-D
work
Line
New
way
equations
that
discretization
phase
to explicit
over
and
•
•
the flow
study
time-marching
Fourth-order and
simulating
(1993)
flow
grids
Fully
•
development
efficient
2-D
are
conducted
Danabasoglu
roughness
in a very
in this
coordinates
Multilevel
step.
spatial
roughness
the
elements
is no evidence
itself
of the
in the inviscid
is governed
Edwards
roughness
at several
roughness
there
(1977)
layer
zone
but
were
on surface
have
and
cause
& Kozlov
frequencies
distributed
that
con-
Klebanoff
influence
and
been
layers.
used
non-staggered
Fasel
layers
multiple
boundary
approach
General
the
incompressible
past
and
et al.
on
has
of surface
(1993).
successfully
their
time-dependent,
channels The
After
boundary
continued
transitional 3-D
3-D
studies
with
have
elements.
showed
Dovgal
boundary
recirculation
Dybbs,
flow
studies
the
downstream,
a backward-facing
element
numerical
layers.
and
& Liu
with
Reshotko,
of a separated
& Streett
layer
all existing
numerical
and
of instability
that
and
destabilising
direction
(1990)
excited
in the case
as it moves
stability
Biringen,
two-dimensional
2-D
existing
and growth
realistic
by Tadjfar, rapidly
spatial
of a boundary
parently,
A more
studied
waves
element, the
the
applications
studies
roughness
element via
in the streamwise
of disturbances
edge.
roughness
of its practical
experimental
or distributed
transition
& Shcherbakov
revealed
development
at its
amplify
Kozlov,
because
Several
isolated
early
topic
design.
as an isolated causes
roughness
is amplified
Danabasoglu,
ulation
isolated
wire
rapidly
Dovgal,
is an important
airplane
dimensional
instead
investigation
experimentally
On the studies
that
trip
but
Boiko,
These
oscillation
instability
three-
and
disturbances
upstream.
roughness
two-
a cylindrical
These
more
surface
for efficient
disturbances,
to move
vibrating
with transition
with
used
introduce
3
layers control
layers
(1972)
not
the
and
in boundary
does
Report
new
2-D fiat
of buffer
also and plates.
on
the line
step
developed
that
this
secondary
instability
new
numerical
simulation
roughness
elements,
for
Navier-Stokes
nonlinear
equations
algebraic
relaxation
systems
to make
arising
fully
implicit
reflections
from
in
the
schemes
implicit
comparable
cost
efficiently
and 3-D
scale
eliminates
to as short
which
in
large
distributive
in CPU
domain
disturbance,
approach
growth but
treatment
size
of inflow
solving
smoother
greatly
as less enhances
work
has
in 3-D regimes with
high
channels have general
all wave
than the
one
T-S
efficiency
accuracy, and been shape
wavelength of spatial
stability,
flat
plates
reached: and
been
outflow
boundary
small
and
large
DNS
and
efficiency,
successfully
successful general
the
for both
not
simulation
distribution
only
simulated
has in an
of transition in
2-D
and
3-D
4
Liu,Liu,McCormick,
2
Governing
2.1
Original
Equations
in Physical
Final Report
Space
Form
Low speed flows are governed by the incompressible Navier-Stokes equations, in which momentum and mass conservation are represented by three momentum equations and s continuity equation, respectively. Considering only Cartesian coordinates, z-momentum equation:
the governing
equations
Du* PDt,---;-: v-momentum
continuity
OP" - 0z---;--t- p,A*u*
as follows:
4- f=,
(1)
+ f,.
(2)
+ f,,
(3)
equation: De*
_
OP*
Dw" P Dr*
-
8P* az*
PDt" z-momentum
can be expressed
O_.+ _,A.
v*
equation:
+ _._"
equation:
Dp au' a,,' a=') D,_--; + p('_="+ _ + a="
(4)
0,
where, u*, v*, are dimensional streamwise, normal, and spanwise velocity components, respectively, P* is the dimensional pressure, p represents the density of the fluid, f=, fy, and fz are body forces in the three directions, p is the dynamic viscosity coefficient, A is the three-dimensional Laplacian, and D denotes the material derivative. In Cartesian coordinates, we have 02
It is assumed
that
no density
A"
-
02 02 + -+ --, az* = Oy* = Oz*=
D Dr'
-
O +u" O,_'
variation
exists
_z'
+ v* _ O
in the investigation
(5)
O + w* 0z---:" domain,
(6)
thus,
D__pp= 0, D_" and the continuity
equation
becomes Ou*
Or*
oz-; + _ Furthermore, written as
assuming
the effect of the body
Ow"
+ Oz: =o
force f = (f=, fy, .f,) is negligible,
Ou* , Ou* v* Ou* w* Ou* 10P* aQ_;_+ ,., __;.=,+ IOy. + __' + -_-,o Oz'
_ :
(s) the momentum
02u * 02u * 02u * "(a--_+ _ + a-_-)
equations
can be
= 0,
(9)
Or* u* Or* Or* w* Or* 10P* 02v * 82u * 82v * _ + "" -+ --) = o, --at" + Oy* + _ p Oy ° - _'(IOz*' + IBy*" + _.,
(lo)
Ow* . Ow* . Ow* . cgw* 10P* at-i;- + u Oz--="+ v _ + w _ + P Oz* Here,
(7)
_P is the kinematic
viscosity
coefficient.
O=w * 02w * O2w * ="(O--_ -;_- + "---_Oy + a-'_ -) = 0.
(11)
Liu,Liu,McCorrruck,
2.2
Final
Report
5
Nondimensionalized Equations
the
flow,
half
height
(8)-(11)
can
for instance, h as
Define
the
Forms be
Planar
nondimensionalized
reference
the
for
velocity
reference
length
nondimensional
variable
by dividing
and
reference
the
center
and
Channel-Type
line
velocity
1_*
reference
constants
channel-type
U0 of the
steady
flow
appropriate
flow,
we use
the
reference
as
the
to
channel velocity.
_=_'
y*
Z*
Y=_-' P*
P
1/)*
v0
• *
Reynolds
some
planar
V*
v
_=X'
the
by
For
as:
_=_,
and
them
length.
Flow
"=_-'
t*Uo g-
pU_'
h
(12)
'
number
U0h Re=
--,
(13)
V
we finally
obtain
the
following
nondimensionalized
au --
Ou
governing
Ou + --
Ou
Ov av av av ---+ at + u-o-_ + ray W-_z aw
aw
aw
aw
a-/-+__-_+"-_-y+_-a-;
equations:
+
OP
1(0%
+
oe Oy
1 Re
(Oar + cgz 2
1
aaw
ap
+ az
Re(a='
+
+
0%
+
0%)
= 0,
a2v _y2
+
c92v az 2
)=o,
IO2w
ay2
au
+
O2w ) :
0,2
av
can
be
derived
to from
use
the
conservative
Ouu
Our
O-Y +_+ Ov
major
equations
flow
into
variables
steady and
Ow
Ouw
0--_-+
_
purpose plays base prime
of a more flow the
this
+-_z
Ovv
momentum
Ovw
_
equation
Oww
is to
=
in the
variables,
The
1
ay
(16)
0.
(17)
numerical
simulation,
which
_
. c92v
+
02v
1
. 02w
the
behavior
:0'
(18)
= 0,
(19)
02v.
02w
02w.
+ oz-WV) = O.
of disturbances,
equations
perturbation.
_-¥z2)
+ _-fi)
Re(=_-o.+ _
perturbation
unsteady
Oau.
R--ee(_2z 2 + _
OP
+ a_
02u
Re(O-_z 2 +
Oz
investigate
role.
1 . 02u
OP +
+ _
a fluctuating
fluctuating
+
+ _
+ _ work
OP
Ovw
+ _
important and
Ouw
_-y Our
0--_-+ _
The
of the
0,
(14)-(17): Ou
the
form
(15)
Ow
O_ + Oy + It is recommended
(14)
are
Letting
so the
obtained the
by
subscript
(20) perturbation decomposing 0 denote
the
form
total
base
flow
we have
_.(., y, z,t) = _o(*,y, z) + ,,'(., y,.,t), v(., y,_, t) = vo(*,y,_) + ¢(*, y, _,t), _(_, y, _,t) = _0(_, y, _) + w'(., y,z, t), P(z,y,z,t) = Po(z,y,z) + P'(z,y,z,t). Substituting have
(21)
into
(17)-(20),
and
noting
of
the
that
the
base
flow
itself
also
satisfies
(21)
the
Navier-Stokes
equations,
we
6
Liu,Liu,McCornuck,
z-momentum
Final Report
equation: a(,_¢
a,,,--+ a(2,,.o,,'+ ,,',,') + Ot a(uow'
+ u'wo + u%')
__
8y aP'
az y-momentum
+ u'_0 + u'¢)
8z 1
+ a,,
a2u '
a=u _
Re(_Z_-=' + _
8_u ' + _-,_ ) = o,
(22)
equation:
__a'/+a(,,,',,o+ ,,o,/+ ,,',/) + a(2,,o,/+ ,,',/) _at
z-momentum
Oz
O(vow'
+ v'wo + v'w')
aw'
a(_,o:'+ _,'_o+,,',,,')
OP'
am a(2w'wo + w'w')
02_ '
=
o,
(23)
aP'
+
a(,,',.,,o+ _o_'+ v'_,') Oy 1 .a:w'
a:w I
=
o.
(24)
4-
a:w'
we will omit all primes henceforth, so that all variables without any sub/superscript part of the flow in the perturbation equations. In planar channel flow, for example, and u0 : u0(y), the above momentum equations can be simplified to au --at
a(2uou
+
+ uu)
a=
#w a_
+
O(uow + uw) oz
Nondimensionalized are several
methods
auv
auw
+-ff-;+--_-,+
av a(uov + uv) a= --at. +
There
02_ '
equation:
For simplicity, the fluctuating and wo vanish
2.3
1 (O2v '
equation: --+ or,
continuity
Oy
a_v
avw
OP
a= aP
+-F;+--_-,__ +_.ay
O'vw Oww +-_-+-=--+ oy
Forms
oz
for
aP az
1 .a2u 1 .a2v
a_v
_(_-J=,+Y_+ 1 . 02w 02w _(-_-+-_-+ 11ce-
o_-
Flat-Plate-Type
to nondimensionalize
a2u
R_(_-_=_+_ +
the Navier-Stokes
oy-
will denote since both v0
a_u Oz _ )
=
o.
(26)
a2_ az 2 )
=
o.
(27)
) =
o.
(28)
8_w az 2
Flow equations
for flat-plate-type
flow.
Since
the choice of reference length for this kind of flow seems to be somewhat arbitrary, we can for instance choose the distance from the leading edge to the inflow boundary, L0, or the thickness of the boundary layer at inflow, 50, or the thickness of the displacement at inflow, 6_, or something else, as the reference length. In this paper, we choose 6_ as the reference length, and the free stream velocity Uoo as the reference velocity. Under this setting, the Reynolds number is defined as Re_-
U_,6_
(29)
/J
The general perturbation
form of the governing flow, if the fiat plate
equations is the same as (14)-(17) if we substitute Re by Re_. For the is used, the base flow is two-dimensional, and w0 vanished from equations
Liu,Liu,McCorrnick,Final Report
7
(22)-(24).Thus, the resultinggoverning equations are 1
OP Om
O-t +
+
Oy
Ov O(uv + uov + uvo) O-t+ am Ow --+ Ot
O(uw
+
+
a(vv + 2,,or)
+ uow) Oz
+
a(vw
+
ay
+
Oz + vow) Oz
O(vw + vow) Oy
_
8z
+
.O2u
aP
1
Oy
_C_ 1
Oww OP + _ + Oz
a2u
02u
(_-_-_.= + _ .02v
+ _-i-_) = o,
02v
02v.
+ _
.02w
+ _Z_) = o,
02w
+ -g-_ ) = O, Ov
Initial
and
For a planar
Boundary
channel,
Conditions
Poiseuille
flow is considered V(y)
The initial
conditions
for
for the perturbation
Planar-Channel
2,
flow for temporal
P =
-- O.
(33)
Flow
as the base flow:
= 1-(y-l)
(_, v, w) =
Type
(32)
Ow
0-_ + ayy + _
2.4
(31)
O2w
Re; C-g-__ + _ Ou
(30)
yE
[0,2].
(34)
simulation
are
._dReal{(¢., ¢., o)_e'("')} +,,_+Rear{(¢., ¢., ¢. )3d+¢(_" +_')} +'3d-Re_t{(¢., ¢., ¢.)_._ e'(:'-_')}, 0,
(3s)
where eu, ¢, and ¢,_ are eigenfunctions for u, v, and w obtained from Orr-Sommerfeld equation, e2_, e3d+ and end- are amplitudes of the 2-D and 3-D perturbation waves, al and as are streamwise wave numbers for 2-D and 3-D perturbation, and/_ is the spanwise wavenumber. For spatial simulation, the initial conditions are
On solid
walls,
No boundary
no-slip
condition
For temporal directions:
conditions
(36)
(_,v, w)l..,, = 0.
(3z)
are assumed:
for pressure
simulation,
(u, v, w, P) -- 0.
is needed
periodic
since
boundary
we use a staggered
conditions
are
grid for the numerical
imposed
in both
the
simulation.
streamwise
and
spanwise
,7(. + L., y, _) = ¢(., u,z), P(_ + L., y, _) = e(_, y,.), P(z, For spatial wave pair.
simulations, the inflow Because of the specialty
_,(o,y,_,t) V"
A_
(--_-, y,_,t)
w
Az
(--_-, u, _,t)
y, z + L, ) = P(z,
y, z).
(38)
boundary conditions are obtained from the 2-D T-S wave of staggered grid, the condition can be written as:
-:
_2dReal{qbu2d e-i'_t}
+
e3d- Real{¢_3_-ei(-I_z-"t)},
=
e2dReal{_b,2_e'(-_-_--_t)}
+
ezd_Real{¢,3d_ei(-*_-a'-_t)},
=
e3d+Real{¢_sd+ei(-°_+#'-_')}
"_- _3d4-Real{qbu3d+
and
a 3-D oblique
ei(_z-'_t)}
+ e3d+Real{_,3d+e
i(-_-+#'-_t)}
+ esd_Real{¢,3d_ei(-_-#'-_t)}.
(39)
8
Liu,Liu,McCorrnick,
Final Report
The outflow boundary conditionsare much more complicated,and willbe discussedin detailin latersections. Periodicityconditionsare alsoused in the spanwise directionfor3-D temporal and spatialsimulationsbecause the flow in this directionis not confined,and the laboratory experiments also support the spanwise periodic structures.
2.5
Initial and
Boundary
Conditions
for Flat-Plate-Type
Flow
For flows over a flatplate,the Blasius similarity solutionis applied as the base flow. For rough plates,the Blasiussolutionwillbe used as the initial guess of the base flow. Detailswillbe given in laterchapters. Only spatialsimulation isconducted forsmooth or rough platesinthisstudy. Thus, similarto the lastsection, Benny-Lin (1960) type disturbancesare imposed st the inflowboundary:
_(o, y, z, t)
=
_2_Re_t{C,,,2_e-")
+
ead+Real{¢ua_+e
+
e3d_Real{q_.ad_ei(-/J*-_O),
=
_Real{¢,_d#
+
ead+Real(¢,3d+e
+
ead_Recd_vad_ei(-a_-_z-_t)),
=
ead+Real_wad+e
i(_'-'_')}
Az
_(-T'
Az _n(-_-,y,
y'z't)
z _" , )
-a_-_')} i(-a-_+_z-_t)
}
i(-a_+/jz-_t))
+ _3__ ReaZ{__ #-° _-_'-_0L
(40)
where o_ is the real frequency of the disturbance,_ is a realnumber that representsthe spanwise wavcnumber, and c_-- aR + ial isthe streamwisc complex wavenumbcr obtained from linearstability theory. A no slip boundary condition is applied at the solid wall. According to the linear stabilitytheory, the disturbancesvanish at infinity, so, we obtain the boundary conditionsat farfieldgiven by
u(z, y -, oo, z, t) = O, v(z, !l _ oo, z, t) = O, w(z, !/---' co, z, t) = O. Also, no pressure The
3 3.1
outflow
condition boundary
Transformation Vector
is needed condition
at the inflow, will be discussed
of Governing
Form
(41)
solid wall, and far field since
a staggered
grid
is used.
later.
Equations
of the Navier-Stokes
Equations
The dimensionless3-D incompressibleNavier-Stokesequationsin Cartesian coordinatesare given in (17)-(20). Itisconvenient to rewritethem to a general vector form. First,we introducethe followingvectors: 1
1)
u2+P
,
P=
U
v 2 +P
Ill
UYO
,
O=
W_
(42) w_+P
Liu,Liu,McCormick,
Then
equations
Final
(17)-(20)
Report
9
can be rewritten
in a simple
a-q-+ _ + _
form:
(43)
+ o, - Re_0,
where 02
a2
_= _-j, + _ is the Laplacian 3.2
operator
General
in the physical
02
+ o_--i
(z, y, z) space.
Transformation
In order to treat arbitrarily shaped _, r], (, which are related to Cartesian
boundaries, coordinates
it is often convenient by a transformation
to employ general curvilineax of the general form
coordinates
= _(_,., O, y = y(_,., ¢), z = z(_,., O, or the corresponding
inverse
(44)
transformation
= _(-., y,_), . = .(_, y,_), = _(_,y,_). Using
the chain
rule, the partial
derivatives
become
a 8 --
(45)
a
a
a
8
a a
=
I
8
a
a
a
a--;= "*"_ + " _ + ¢"_"
(46)
If the inverse anMyticM transformation (45) is easy to obtain, the metric (,, _7,, _ffi, _y, _7_, G, _,, _7,, (, can be obtained directly. Otherwise, the following procedure is recommended. Rewriting the differential expressions in matrix form, we have _Z
=
r]z
(. and,
_7_ (y
r]s (,
dy dz
,
(47)
similarly,
[d.][.,._,][,] dy dz
Defining
the Jacobian
=
of the transformation
y_ z¢
y,_ z, 1
y( z¢
dr] d_
,
(48)
J as
r]y _z
(,
(,
(49)
10
Liu,Liu,McCorn_ck,
Final
Report
from (47) and (48) we obtain
[. and,
[.....]_1[
yqz¢ - y_ z_
_7= _7y
th
_
G
_
=
y_
Y_
Y(
z_
z,1
z(
-(zqz¢
-(_z_ - _ =_)
= J
"1
zny ¢ - z(yn
|
z_z¢ - z(z_
-(z_z_
y_z,_ -- yetaz_
- zCz_) - z_z_)
-(ztzt¢ zTyq - - zeus) z_y¢ J ,
thus,
Q =
-J(z,_z¢
- z_z_),
n= =
-J(y_z_
- y_z_),
rty =
J(z_z¢
- z_z_),
m =
-J(z_yc
- z_yt),
_y =
-J(z_z,1
- z,_zt), (50)
After
the proper
computational
3.3
General Based
transformation,
a domain
with general
geometry
can be transferred
to a cubic domain
in the
(_, 7/, ¢) space.
Curvilinear
on the above
Coordinates
transformation,
System
we now define
the new vectors
in the computational
space:
0 Vl
_
--
=
_($_=
j'
+ $'_ + GG),
(51) (43) then becomes
--+ #t where/_
is the physical
Laplacian
-_-+ T_j + -_=
operator
_ : (_=
transferred
computational
(_, _7,i) space:
0 +¢=_)(_=_ 0 0 +,_=_ + ¢= ) + _=N
o o +(_,_+_,_+_, Though discretized
to the
(52)
T_,v,,
_ )(_,_+_,_+_, o o
_ ).
(53)
(52) looks very simple, it is not convenient to discreti_e. To obtain equations that are more easily for numerical simulation, we define the new variables U, V, and W in the computational (_, _/, _)
space: U
: _(,4= + _, + _oG), J
(54)
Liu,Liu,McCormick,
Final Report
11 1
v = y(un.
(55)
+ v% + wn,),
1
(56)
w = y(_¢, + _¢, + wG),
Our objective and consider both variable triples u, v, w and U, V, W as unknowns in the computational space. is to develop relatively simple expressions for the governing equations in general coordinates that are easy to discretize. Using the above transformation, we rewrite the vectors bY1, El, F1 and G1 in the following form: U
1
uU +
IA E1
=
vU +-_L wu + P--_1 •
111
W
V uV+
,G'I
uW
=
The governing
equations
then
aVu -_--)+ --_-
a,, _.our a-_ -{- d(--_at
(57)
ww
+ P-_] "
become
au + S( _OUu --0t
--OW
+ P_]
vW + -_
vV + -_ w V + _1"
+ OWu.
4- or,, -_+ a --
+ ((= ___ + n= _o
,,) + (_,
. O U 'w O V 'w a W'w +./(--_+--_+--_)+(_,O-_+n,_-_+¢,
.
4- ny-_
a
+ (, ff_._)P
T (y
a
_._
1A
--
_--_e
o-7+
a[u(W
s(a[_,(v
a_
+
a[u(y
+ Wo) a a a( + uoW] )4-(_._-4-n,_4-_,
av j(a[v(U a--7+ a[v(W
+ Uo) + _u]
O,
(59)
)P-
.__e O1h l .t
=
0,
(60)
-
O,
(6,)
+ a( Wo) + voW] ) + (_,_-_a
P.
aW a_
Similarly,
+ Vo) + _,oV]
a.
___)P-
+ Vo) + yoU] a[v(v a_ +
(58)
:
a--( + -O--,in+
au
0,
--_eAlV
aV
u, v, w, U, V, W and
=
)P-
aU
The system (54)-(56) and (58)-(61) has 7 unknowns, bation form of the governing equation as follows:
u
1
we can obtain
the pertur-
_-
_1 A Iu
=0,
(62)
= O,
(63)
Al_O
=
O,
(64)
aw a(
-
0,
(65)
+ Vo) + voV] an + _-_e 1 h iv
a 4- _, 0-_ )P+ ny _--_
a,o j(a[_(v+ Uo)+ ,_oU} a[_(v+ Vo)+ ,_oV] a--_-+
a(
a[w(W + Wo) + woW] a
o,F$>O, F$>O, a_,>O, G_.>o, we obtain
laal> Ib=l> o,
I=_1> Ib_l÷ I_l,
[a_.y_zl>
Ic,=.¢_x[ > O,
,
30
Liu,Liu,McCorn_ck,
and
thus
we can
the
above
update
system
U j and
can
VJ
be solved
by LU
decomposition
without
pivoting
very
rapidly.
Final
With
Report
6j computed,
by
j =
2,3,-..,n
i -- I,
(144)
and
v_ _- v_ + 6j_1- 6j, j = Finally,
the
pressure
corrections
")'i are
determined
'Y2
=
-
1.
(145)
-
1,
(146)
by
2
A_
3,4,...,nj
2
+ A c C=_C
W20'),
B_(6j -6j_,) :
_Y
"Yi-
1 -)-
Dj c
,
j : and
P is updated
3,4,...,nj
via
P$,-- P$ +.yj, j = 2, 3,...,.j The
programming •
process
Freezing
P and
computational •
Freezing
P and
For
each
and
Pi,y
i :
different
convergence
4.5.1
perform
U,
2, 3,-
using
perform
•., hi-
the
as follows:
line
(or
point)
Gauss-Seidel
relaxation
on (111)
to
obtain
new
U in the
whole
line
(or
point)
Gauss-Seidel
relaxation
on
to obtain
new
V in the
whole
ej.,
(111)
-(112),
known
levels.
process
and
for all j =
6j,
and are
2, 3,.-.,
"ri in the
unchanged
n 1 - 1 at once,
form and
depicted the
change
in Figure
corresponding
U/.,y,
Ui,.+l,3'
15 so that continuity
the
, V/,j.,
V/,j.+I
,
corresponding
equations,
(113),
are
for its For
efficiency
our
specific
in attenuating
all the
applications,
several
errors
with
different
different
wave
techniques
are
numbers developed
by relaxation to
make
the
faster.
Full-Coarsening For
simplicity
approximation form
of
discussion,
scheme(FAS)
we to
consider
accommodate
only
the
be described
loosely
two-grid
nonlinearities.
LhW may
(112)
Methods
is well grid
1 in turn,
corrections
equations,
Multigrid Multigrid
on
described
domain.
momentum satisfied.
4.5
V,
be
(147)
domain.
computational •
can
- 1.
case
h = fh
relax
on
LhW
ii)
solve
LahW
_' : 2h =
fh, LahI_aW
t' + i_'(f
h -
iii) replace W h ,---W h + I)_,(W 2_'- I_'W").
depicted FAS
in algorithm
Figure for
16.
We
use
an equation
a full of the
(148)
as follows:
i)
as
A two-level
Lt'W_'),
Liu,Liu,MeCormiek,
The notation
Final
Report
we have introduced
the approximation)
and
_h
I -_
C)
-
"_
[
O
--
includes
the difference
(for the residual),
operators
L _ and L 2s, the restriction
and the interpolation
operator
_.-.-A-.-.-...-...-A,...._ ..I...&I..M....A......, I I I I I I I
---o
÷
1
÷
O
_
o-I
I
--+_
O
O--_
I •
I
O-
_
I
••
I-
-- -"0---4-"0-
0
I-
_O--_
t
l
O--'_
O--
----_
0
-- ÷
0
---_
0
-- -_
_ /
I. 1-
_ I
• • r
I
I-
I
•w
I
I-
I
-.-
0
-+.-
0
_
Uh
Figure
l
vh
_
16. Two-level
__..
p_h
staggered
_t2h
grid
••
O--
,...!., .......,VM, © ph
--
I
-,- 0
-,-0--4-"0-
I_ h (for
I
---_ 0
I
operators
I_ h.
,.....&../..M/A,.-,-> I I I
•'
--
31
-'_
I 0
--
t "_
0
•• --
"_
--
I 0
--
,,..,I,,,,
t/..4./..;
i
l
V2h
(full coarsening).
We first perform distributive relaxation for (111)-(113), then calculate the residuals for z, y momentum and continuity equations (R_, R_, R_) on the fine grid. Relaxation on the coarse grid requires restriction of the approximations and residuals from the fine grid to the coarse grid to provide initial guesses and right-hand sides, respectively. For restriction of approximations, we use the following stencils:
[1] 5
I_(_) : [ I _1, 1
i_a(p)
:
i
(149)
4
For restriction of residuals, we use the following developed by Liu (1989) (see Figure 17):
°
full-weighting
stencils,
ii r_(_)
:
After
all these
restrictions
are done,
we perform
coarse
grid
come
from
the so-called
area law
,
i i 4
which
4
(150) "
relaxation.
32
Liu,Liu,McCormick,
Figure The next step
is to interpolate
17. Restriction
the corrections u_
,-
Final Report
for P_.
from the coarse
grid to the fine grid:
u h + l_h(u)(u 2_ - I_h(U)Uh),
,./" ,-.- ,," + x_,,.(,.,)(,,'' - .,,g"(.v).,,"), ph Here we use hilinear
Figure 18 depicts the above stencils.
the
interpolation,
relations
_
given by the following
between
2_ - I_h(P)PP').
ph_t_ I_h(P)(P
stencils:
I_(_)
: [_3
x_(Ap)
:
Au 2_' and
1
i]'
_
_
16
16
(152) "
Au h, Av 2h and
•
0
(151)
•
Av h, and
Ap2h
and
Aph
corresponding
to
AP 2h
Aunh © ©
"Au h
Av 2h
© Aph
©
©
©
© Figure
18. Bilinear
interpolation
for u, v and
P.
The FAS V-cycle based on the ingredients described here performed well in all of the cases we have considered for 2-D channel problems on this study. Typical convergence factors per V(2,2) cycle investigated were about 0.3 for the fully-implicit
scheme.
Liu,Liu,McCormieJt,
Final Report
TI
33
TI
TI
TI
TI
TI
rl
TI
t
. ph ___.U h I Figure 4.5.2
19. Two-level
vh
© p2h
staggered
grid
....._
(normal
U_h I
direction
vZh
semi-coarsening).
Semi-Coarsenlng
The full coarsening approach works well when the grid ratio Az/Ay is not too large. For a fiat plate, because of the use of highly stretched grids, in some regions, this ratio may increase an order of magnitude or more. Under such circumstances, efficiency of the full-coarsening method described in the above subsection degenerates. To avoid this inefficiency, we use a semi-coarsening multigrid algorithm based on the modified point distributive relaxation described in the previous section. To accommodate the high degree of anisotropy caused by the dense concentration of grid points near the solid wall, we use a seml-coarsening scheme that involves grids that are coarsened in the vertical direction only. See Figure 19, which depicts the location of the velocity and pressure nodes for grid h and 2h. Note the advantage here in treating the vertical boundaries by semi-coarsening since they coincide on all levels. The
relaxation
interpolation
process
are changed
is the same
as described
in the last
subsection,
but the
stencils
of restriction
and
to follows:
/_h(V)
:
[°1 1 0
,
(153) This means that U 2h and p2h depend only on their corresponding lower and upper nearest neighbors on the finer level grid, and V 2n is just V _ evaluated at the same point (see Figure 19). For restriction of residuals, we use the following fuLl-weighting stencils for semi-coarsening grids:
(154)
34
Liu,Liu,McCormick,
Here,
Ru,
Rv
and
RM
are the
residuals
for the
z-momentum,
y-momentum
and
continuity
Final
equations,
Report
respec-
tively. We
use
bilinear
interpolation
for
the
variables,
given
by
Ii] .Ha[i]'
I°l i1 1
and
_
,
0
[i] "°d[i] Let
A be
used
to
denote
corrections,
(155)
so that
av_ = i_(p 2_- ipp _) denotes
the
use.
example,
For
fine-grid the
correction
to pressure,
AU h = are
determined
for
example.
Figure
and
AV
20 illustrates
these
interpolation
processes
we
problem.
An
corrections I_hAU
ah
_ = I_hAV
2h
as follows:
Av_
-
_
3Arch
1AV2h
AU_
=
"41AU2hN +
43-AUah' S ,
AV_
=
1Ag2h
1AV2h
u+
4
_+2
s,
s,
AV_ = _v_'.
.
o
__
AV_ h
• _,,_
_
,',u_ AU_
o Figure
By average
using
semi-coarsening
convergence
factor
•
h
20.
AV_h
Bilinear
multigrid, investigated
It
AV;
interpolation
the
convergence
under
proper
time
0
Ap_
0
AP_
•
Ap_'
for
corrections
to
rate
becomes
step
is about
U, V and
P.
much
better
the
0.2
for each
V(2,
flat
plate
2) cycle.
Liu,Liu,McCormiek,
5
Numerical
5.1 The
of the
perturbation
form
Navier-Stokes
of the
Equations
three-dimensional
parts of the flow fields in a 3-D planar Ou Ouou Ouu Our -at +-_-z +_-z +-_-y VOv
Ovuo
Navier-Stokes channel
vovu
vow
vOwu
vOwuo
vO-_+ -_--_+ _
vOwv
a Uniform
equations
that
1 .O2v
Ovw
Staggered govern
1 .a2w
vOww
O2v
=0,
02v)
OP
a2w
a2w.
aP
av
parts
Ow
of the flow field, and uo is the steady
(156) (157)
(158)
+ -_-v + 0z
the fluctuating
of the
OP
a--;+ _ + _ P denote
behavior
_(-g_, + o_---_ + az, + -_y = o,
Ou
Here, u, v, w, and
the
Grid
as
1 .02u 02u 02u Re(-_z2 +_-2y 2 + Oz= )+
you° Ouw --_-y + Oz
+
on
can be written
o-/+ -_-_ + -_-_+ -_-y+ az vOw
Problem
for Three-Dimensional
Methods
Diseretization
fluctuating
35
Final Report
=0. undisturbed
(159)
Poiseuille
flOW.
y
2.0
uo(y)
= 1 - (y-
1) 2
Z
Figure
21. 3-D planar
channel
flow.
For temporal channel problems, periodic boundary conditions are assumed in both the _:- and z- directions, while no-slip boundary conditions are imposed on the solid walls. Let Lffi and L_ be the nondimensional wavelengths of one T-S wave in the z- and z-direction, respectively. The nondimensional height of the channel is set to 2. Referring to Figure 21, the boundary conditions may then be written as u(z,0,
z, 0
=
v(z,0,
z,0=w(z,0,
z,t)=0,
,,(_, 2,,_, _)
=
,,0,, 2, ,_,,_)= ,,,0,, 2, _, _) = o,
,,(_,y,z,t)
=
_,(_ + Lffi,v,_,O,
36
Liu,Liu,McCormJek,
_(.,_,_,_) = _(-,_,_,,) = _(,,_,,..,,,) = _(.,y,_,,) = _(,,_,_,t) : PCz,_/,z,t) =
Final Report
_(.+ %,y,_,t), PC.+L.,_,_,,), ,_(,,_,,.,+z,,,,_), _(.,_,_+L.,,), _(-,_,_+L.,_), P(z,71,z+L.,Z).
(160)
A uniform staggered grid is used for the discretisation (Figure 22). Values of P are defined at its cell centers, u at the centers of the cell surfaces parallel to the (11,z)-plane, v at the centers of the cell surfaces parallel to the (z, z)-plane, and w at the centers of the cell surfaces parallel to the (z, ll)-plane. In the
same
differences
way as in Section
in space
4, second-order
are used to discretise
backward
system
Euler di/Ferences
in time
and
fourth-order
central
(156)-(159).
Y
///// 1
U
/
/
/
///// /////
X
1
/
/
/
/
/
/
,/
///V/ ///I//
Figure Using
methods
analogous
AEEuEE AFFUFF
+
Cg_wgE CFF_IJFF
of section
+ AE_*E + Awuw "4-AEuB
AFUF
BEEVEE BFFVFF
to that
+
BFVF
+ CEWE q- CFVJF
q- BaYS + Cwww _- Cs_13B
DUEEUEE DVsvs
grid structure.
4, we can write
the resulting
+ Awwuww
ABBUBB
+ BEVE + Bwvw q-
22. Staggered
-- Acuc
+ Bwwvww + BBBvBB
+ ANNUNN -t- Dww
+ Cwwwww
Pww
+ BNNVNN -- BCvc
-- CcloC
+ DUsum
+ DUwuw
- DVcvc
+ DWFFWFF
_-
in the following
+ Dw Pw + DEPE
+ ESPS
+ CNWN
FBBPBB
+ DWFwF
form:
+ DWBwB
:
S_,
(161)
=
S,,
(162)
=
S., (163)
=
Sin, (164)
+
-- EcPc
+ CSSWSS+
+ FFPF
+ DVNNVNN
-DcPc
+ Bssvss
+ ENPN
+ CSWS
+ FBPB
- DUcuc
general
+ Asus + Assuss+
+ BNVN + Bsvs
+ EssPss
+ CNNWNN
q- CBBIDBB
+ ANUN
scheme
--FcPc
q- DVNVN+ - DWcwc
Liu,Liu,McCorraJck,
Final Report
37
where the subscript C refers to the central point of the control volume, E to its nenxest neighbor to the east, W to the west, N to the north, S to the south, B to the back, F to the front, EE is nearest point to E from the east, WW the nearest point to W from the west, and similarly for NN, SS, BB and FF. As an illustration of the notation we use, the relevant symbols for the discrete z-momentum equation are depicted in Figure 23. Though it is straightforward to list all of the coefficients and source terms restrict ourselves to one set of the coefficients for equation (161) as an example. for the interior
points
of equation
(161) with respect
AEB
_
to uc
1 12ReAz
1 2 + 1-_--_z(uEE
4 AE Aw Aww ANN AN AS ASS AFF
-
3R a '
--
4 3ReAz2 1
AB ABB AC
+ uo_),
2 + 3--_z(uw 1
+_w), [
12Az _uww 1
+
--
12ReAz 1
2
--
12ReAy 4
2 + =w'r--"v'_'_'12_y 2
=
"_ReAy 2 4
--
3ReA_I 2 + 3A-----_v,, 1 1
=
- 12ReAy2
= --
below:
2
_OWW
),
3Ay v'_' 2
1 12ReAz
12All v's' 1 2 + 1--_-_vwyf,
4 AF
now are listed
in the above discrete system, we The coefficients and source term
2
3ReAz2 3Az wl' 4 2 3ReAz 2 + "_A--_wb, 1 1
--
12ReAz
2
--
3 5(1 2At + 2Re
12Az
Wbb,
1 Az 2 + ---_ Ay
1 + -_z a)'
1 DE
--
24Az'
=
Dc-
27 DW
24Az
'
1 DWW
-
24Az' _,--1
s_
_
-4u_
+ uc 2At
2ycvc.
(165)
Here, superscripts n and n - 1 are used to indicate values at previous time steps (superscript n + 1, indicating current time, is dropped for convenience). !/c is the vertical coordinate of the node corresponding to uc. Lower case subscripts denote the approximate values of the v and w at points where the values of u with associated capital case subscript are located (see Figure 23), and should be obtained by at least 4th-order interpolation (see Section
4).
38
Liu,Liu,McCormJck,
u_tt Ay
Final
Report
w1! --
--_ UN
Az
N
--_ UF
tt_
F
w!
--
.-._
UN
--
-_UCo
-_
Up
_e uWW
uW 0
UB 0
PWW
Pw
UBB
_WW -"
--
0
Pc
-PWW
Um
--
Y
I
uB _UG
0
0
Pw
UBB
--
Pc to b
-.-._ V, S
Az
uB
"i
wbb
Az
_'SS
z
Z
UW --_
y
(_)
uBB
z
(b)
Figure 23. Neighbor points for z- momentum equation: (a) cross-section in (z, y)-plane, z-axis orthogonal to view, (b) cross-section in (z, z)-plane, y-axis orthogonai to view. For spatial simulation, the discretization should be treated very carefully to obtain
5.2
Discretization
procedure is the same, but the inflow and outflow boundary reasonable results. This will be discussed in Section 5.4.
of the Navier-Stokes
Equations
on a Nonuniform
Staggered
conditions
Grid
A nonuniform grid must be employed for the problem with general geometry, or even for the boundary layer problems on a flat plate. Since transition is very sensitive to numerical error, the grid transformation, or more precisely, the Jacobian coefficients used in the governing equations, must be very accurate. It is believed that at least sixth-order accurate Jacobian coefficients should be used to maintain fourth-order accuracy where a numerical mapping is used. To avoid this complexity, we instead use the analytical mapping that maintains full accuracy. Since no orthogonality of the grid is required for the governing equations (see Section 3), we are able to use a simpler grid.
5.2.1 First,
Analytical we consider
Mapping a very special
but relatively
simple
mapping
z=_,
: y({, ,7,¢), (166) or its inverse
transformation
,7: ,7(=,y,z), (167)
Liu,Liu,McCormick,
Under
this
Final
special
Report
mapping,
39
according
to
Section
3, we obtain J = T}_, _.
=(,
=1,
G = G = o. The
remaining
task
flat-plate-type
now
problem,
expressed
is to
find
a proper
analytical
mapping
so only
one
solid
is located
in the
wall
r/ =
t/(z,
y,z).
computational
Assume
domain,
the
we
are
shape
considering
of which
a
can
be
as Y0 =
First,
(168)
a rectangulation
rectangulation
of the
mapping
can
domain be
should
be
this
map,
to transfer space.
the transferred
the
In our
stretched study,
domain
grid
this
(169)
obtained.
Letting
1/ be the
associated
intermediate
variable,
the
written y, =
Under
f(z,z).
in the
mapping
(y - f(z, y,,,,, -
becomes
rectangular.
intermediate
is chosen
z))y,,tu f(z, z)
(170)
Next,
(z, y', z)
space
an analytically to a uniform
grid
stretched
mapping
in the
computational
is employed (_, _7,()
as 77 _
(a
+ Ym,**)Y'
(171)
a+y' The
resulting
mapping
from
the
physical
to the
uniform
y,.aaz(a _7 = and
its
inverse
mapping
and
Y,,,_z
is the
a is the (173)
maximum
parameter
expresses
the
(z, y, z) space
r/aCymaz
are
now
the
expressed
r/z
- f(z,
z))
z)Ca
+ ymatf(z,
y,,ta,(a height
used
that
+ Ymffiffi)(Y -- f(z,
+ y) - fCz,
grid
is then
just
z))
+ Y,,taz)'
(172)
is
y=
where
y,,_=Ca
computational
for
complete
of the
computational
adjusting map,
matches
the
the
which
solid
+ Y,,_a,
density
transfers
wall.
The
z)(q
domain of grids the
useful
+ Yma®
-- 77)
-- _7)
near
uniform derivatives
' in both the grid
the lower
in the
that
are
(173)
physical solid
and
computational
(_, 17, () required
space
to
the general
in the governing
.f.
_
=
o/ az'
grid
in
equations
below:
y,,,azf.a(a + Yr,,az)(Y - Y,,,..) [Vm..(o + V)-/(_,*)(_ + V._--)]2'
(174)
[W-.(° + V)- f(_, z)(_ + V,_..)P'
(175)
y,,_..f,e(e + y,_.f)(y - Y,,_a.) [w._(_ + v) -/(_,*)(_ + w.f)P'
(176)
y,ac,zaCo'+ym¢,z)Cy--yr_..)fzf[y'_'®(a+y) --f(z'z)(cr+Ym'f)]+2f_ffi(a+Y"") [v=..(_ + v) - 1(_, _)(_ + v,_.z)]_ -2y_._:a(cr + Ymaz)(Y,_,,z-/(z, z)) [y,._,_.(a+ y) - f(z, z)(_ + y=ff)]a '
(177)
yr,,azaCcr + Yr,,az)CY -- Ymaz)fz*[ymaz(a+ y) -- f(z,z)(a + Y,n.=)]+ 2fza(o"+ Y,naz) [y,.,.ffi(a+ y) - f(z, z)(a + y,,_..)]a , where
space,
wall.
A,
--
_ _, Oz
fz
=
o/ Oz'
and
f,,
=
_-__" az
(178) (179)
40
Liu,Liu,McCorrrdc_,
5.2.2
Governing
Equations
under
Special
Final Report
Mapping
Under the above mapping in (171), the governing equations derived in Section 3 can be simplified. First, the variable transformation between the u, v, and w originally defined in the physical (z, y, z) space and the U, V and
H/"originally
defined
in the computational
(_, 77,() space
u :
is now represented
as
[f_Ty,
w = WUy , : V-
(180)
U_ffi - W_,,
or
U = u/_y, W = _/_y, V : Most importantly,
A1 :
the transferred
Laplacian
operator
02 02 82 _--_ + (_Tffi; + 77_ + _7_)_--_-2+ _-_
As we described
in Section
(u_= + v_y -I- w17s)/17y. in the computational
0_ + 2_7®a--_
3, in order to make
(181)
82 + 2_, 8--_
the numerical
space
is relatively
-I- (17ffiz-I- _,,
procedure
+ _z, ) -,
relatively
simple,
simple:
.
(182)
we consider
both
(u, v, w) and (U, V, 14T) as unknowns, and combine three momentum equations and one continuity equation gether with (180) or (181) to generate a closed system. The resulting momentum and continuity equations now be written as follows: • Original
equation
(for base flow) Ou
OuU
8uV
8uW.
0
8
1
0-7+ n"( --#C + T_ + -_-C ) + (N + _" _ )P - _ z_lu = o, 0v + Ir/$,C .8vU 8vV 0"-_" --a--_--I- Tu
Ow + "(--_. OwU + _OwV + --_--) OwW. O-T
-F _
) -F r/, 8P ar /
+ (7, _
8 + _)e OU
1 .ReAl v--
1 - _a,w OV
0--_--I- _ where u, v, w, P, U, V, W" are all variables. • Perturbation
tocan
equation
(183)-(186)
together
0,
= O,
(lS3) (184) (185)
OW -I- -_-
----O,
with (180) contains
7 equations
(186) for 7 unknowns.
(for disturbance)
ou + Uo)+ uoU] a[u(v + Vo)+ ,,ov] -_ + '7"(a[u(u 0(. + 0,7 +
a[u(w + Wo)+ uoW] 0 o 1 0( ) + (_ + ,7._)P - _A,u
0,,
a[,,(U + Uo)+ ,,oU] + 77,(
0(
-_ a[,,(w + Wo) + 0(
= O.
(18_)
a[,,(V + Vo)+ ,,oV] +
v°W])
0,I aP
+ r/_ Or/
1
Re _lv
=
o,
(188)
Liu,Liu,McCormick,
Final
Report
41
+
+ a(
Uo)+
+ vo)+ oV] av
+
woW].) + (_,
+
)POU a_
where u,v, w, P, U, V, W are all fluctuating represent the base flow. 5.2.3
parts
_ _Alw 8V
= 0,
(189)
- 0.
(190)
8W
+ _
+
of the corresponding
a(
variables,
and
uo, vo, wo, Uo,Vo,
W0
Discretlsation
In the computational (_, '7, ¢) space, the grids are uniform. Suppose u, v, w and U, V, W are defined in terms of a staggered grid in the computational space (see Figure 24). Here, the values of P are associated with its cell centers, u and U with centers of the cell surfaces parallel to the (_, ¢) plane, v and V with centers of the cell surfaces parallel to the (_, ¢) plane, and w and W with centers of the cell surfaces parallel Note that though Figures 22 and 24 looks similar, they are defined in a different space.
to the (_, _) plane.
///// /
/
/
/
/
/
/
/
/
,/
/
/
,/
///// /
///// /
///// /
/////
Figure
24. Staggered
grid structure
in the computational
(_, ,7, () space.
Second-order backward Euler differences are used in the time direction, and fourth-order central are used in space. Details of such an approach can be found in section 4. We can write the discretised equations symbolically as follows: AEEUEE AFFUFF
-_ AFUF
+ AEUE
_- ABUB
B_Ev_E BFF1)FF
-_- BF_F
Cs_wE_ CFF_FF
+ CF_F
-I-ABBUBB
+ BE_E +
+ Awuw
BBVB
+ Awwuww --
+ Bw_w -_- BBB_BB
+ CE,ZE + CW_W + CB_V_ + C_V_
Acuc
+ ANNUNN
+ Dww
Pww
+ Bww_ww --
BCvC
+ CWW_WW --CCWC
+ +
+ Dw Pw
BNNI)NN
EssPss
+ CNN_NN + FB_P_
+ ANUN
+
+ DEPE
BN_N
+ EsPs
+ Asus
+ ENPN
+ Assuss+
-DcPc
_- Bays
differences governing
+
: Su,
(191)
BSSI)SS_-
- EcPc
= S,,
(192)
+ CN_N + CS_S + Css_ss+ + F_P_
+ FFPF
- FcPc
= S.,
(193)
42
Liu,Liu,McCormick,
DUEEUEE DVsVs
+ DUEUs
- DVcVc
+
+ DUwUw
DWFFWFF
+
- DUcUc
DWFWF
+
+ DVNNVNN DWBWB
--
Final Report
+ DVNVN+
DWcWc
= S..,
uc = UycUc, wc = UycWc,
_c : vc - u, ouc - _,oWc.
(194) (195) (196)
(197)
As an illustration of the notation we use, relevant symbols for the discrete _-momentum equation are depicted in Figure 25. Though both the original and perturbation forms are used, we only give the formulas for the latter here since formulas for the original equations can be obtained from the perturbation form simply by setting the values of base flow to zero. The coefficients and source term for the interior points of the discrete _-momentum equation
(191)
associated
AEE
=
As
-
Aw
--
Aww
ANN AN AS ASS AFF AF AB ASB AC
with uc
arc given as follows:
1 I?yc I,, -- 12ReA_. 2 + 1--2-_-__uss 4
2r/_c (Us
hT"
4 3ReA_2
2r/_c (Uw 4- -_-'_,
--
1 12ReA_2
--
ac V_c 12ReAl7 2 + I--_
=
4ac 3ReA_
-
4ac 3ReAl72
+
2u0s______ EE), %ss
+ 2uos
+ 2uow ), _TyW
2r}_c ..... 1-_ t_ww
2uow____ww ), _?_WW
+
Iv Vo_) _''_'_ +
2_/_c(V,,+ Vo,,)+ 27c 3A_ 3ReA--------_'
2
+ 2_/Yc[v "_-_t , + Vo,)
_c 12ReA_f
:
1 12ReA(
-
3ReA(2
-
4 2_yc {Wb Wob), 3ReA_ 2 + '3_-(" + _7_b
-
_I_c iv 12-_'""
27c 3ReA17'
--
-
7c 12ReAl7'
7c + Vo.,)+ 12ReAr/'
'r/yc [,x, wof! ), 2 + 1--_-_ _"1! + U_!!
4
2_?yc
wo! ),
3_((W!+
1 12ReA(2 3 5 2At + 5-_
%,!
wOb___!b
¢I_C i,x, 1-_,r_bb
1 (-_-_
"4- Uybb )'
ac + -_¢
I + _-_)'
1 DE
24A_' 27
Dw
=
Dc---
24A_' 1
Dww
s=
: :
- 24A_' --4u_
-P,,,= + u c,_-z + _/=c 2At --_$oNN
_7_,c(
Vn.
"I-8UONV
+ 8P,= - 8P, + Po, 12A_? .
--
12A_ 7
8t/.osV
a -[-UossV.,
+
Liu,Liu,McCorrnJck,
Final Report
43
___ -- I_oF
F Wl
l
+
811.OF
_rl
--
gl/,oB_V
b -_- BOB
B Wbb
) --
12A( 1
-uO.
6R, ('7=c
_ + 8_0_ - Su0 + u,t.
+ _=c -u(..
A,7
+ 8uo. - 8u 0 + u O, ).
(198)
A,7
Here, superscripts n and n - 1 are used to indicate values at previous time steps, and superscript n + 1, which indicates the current time step, is dropped for convenience. Lower case subscripts denote the approximate values of the u and _v at points where the associated vMues of u with capital subscript are located (Figure 25). Other symbols
used in the above
formulas
are as follows: 2
2
r/f= +rbu 0u
7
=
u(
-
a("
other
than
+ r/==,
0u All function interpolations
values
that
are
required
in the computational c
+
space.
Vc
:
(9(V
P.
:
(9(Ps +Psw)
VN
+
VN
at
For example W
+
Vw
) _
(199) the canonical (see Figure
(Vsw
w
+
locations
are
obtained
by fourth-order
26),
VNNW
W
+
VSB
+
VNNE))/32,
(200)
- (PsE + Psww))/16.
(201)
Wy! --_ UF P
w1 -_ up
We
uB
uW 0
--
Pww
Pww
-_
0
uBB
--
Pw W. "4" u B
W&b -'_ UB
Figure
25. Neighbor
The coefficients continuity equation
points
for _-momentum
equation
(U are at the same points
as u and are
for the 17- and _- momentum equations are defined in an analogous is developed simply by applying central differences to each term.
For the points adjacent to solid wall or the far field boundary, second order, and maintain fourth order in both the _- and _in Section 4, and is thus omitted.
we change directions.
B
not shown
way, and
here).
the discrete
finite differences in the q-direction to The method is the same as described
44
Liu,Liu,McCorn_ck,
I
Final
Report
I ½
_NNWW
VN NE
T l ½
VSWW
VsE
T
t
o
o t .o i o
Fsww
Figure
26.
Psw
Neighbor
points
for
Fs
fourth-order
FsB
approximation
P_,4
P/-2_4
O
for
Vc and
_+
1,4
0
Pa.
0
Pi.,_ P/-
2,3
P/j3
0
o
Pi-
2,2
Pi+
0
Figure
We depart the
solid
by (201),
from
wall.
Section
P is fit
27.
Extrapolation
4 in how
Specifically,
of eP _,
we treat
the
order
polynomial
at grid
gradient
_o_/,.,_(°_ , is estimated,
by a second
0
'_.,½
using role
0
point
adjacent
of pressure
in the
extrapolation.
near
the
solid
1,2
Letting
to solid
r/-direction Pi.,i ,_
wall.
at the P,.,},
point
and
P,,,,}
adjacent be
to given
wall:
(202)
P(,7) = a'72 + b,7+ c, where
=
b c This
yields
the
extrapolated
(-P,
_
(15P,
is used
.-)la,7
It, j 2
, +3P,
(203)
,)/8
value
oP
method
,-P,
lip 3
, -10P,
-3P,.,_
(_-_)'"'_ : A similar
,_+3P, 2
to treat
the
point
adjacent
+ 4P,.,_ - P_.,{
2,",,7 to the
far-field
(204) boundary.
Liu,Liu,McCormlek,
5.3
Fina2 Report
Line-Distributive
45
Relaxation
The line-distributive
relaxation
scheme
used here is basically
the same as that
described
in Section
4. Though
the specific governing equations are different, the general form is just (191)-(197). By using the predicted corrections distributed in Figure 28, for the Dirichlet boundary value problem we can describe the three-dimensional line-distributive relaxation as follows: •
Freezing P, U, V, W, v, and w, perform point domain to obtain a new u.
Gauss-Seidel
relaxation
on (191) over the entire computational
Freezing P, U, V, W, u, and w, perform domain to obtain a new v.
point
Gauss-Seidel
relaxation
on (192) over the entire computational
Freezing P, U, V, 141, u, and v, perform domain to obtain a new w.
point Gauss-Seidel
relaxation
on (193) over the entire computational
• Use the inverse
transformation
(195)-(197)
to obtain
new U, V, 147.
• For all j = 2, 3,..., nj - 1 at once: change Uok , Ui+l i_, _jk, WOk, and Wi_ h+l to satisfy the associated continuity equations, then update Pok so that the new U, V, W and P as well as the associated transferred u, v, w satisfy the three momentum equations. Since
all of the u,
v,
w have
been
previously
relaxed,
and
the U,
V,
W
are
updated,
we may
equations (191)-(193) hold exactly. Let e, _, a and 3' represent the corrections for U, V, W and The correction equations for cube ijk corresponding to (191)-(193) are written as -'tS
_Ty
+Ac
U_
)ej--_c
n_(6; - 6;+,) - _R_ik(6j._l (.-djk
_b+a
t. F r/y
+t.
_ijk
c
_i_x
rb
7j
6i ) _ -o v O'L7, )_rj--l,
,_ijk
6
7j
assume
P, respectively.
=
O,
(205)
:
0,
(20e)
=
0,
(207)
(DU_' + DU_'),j + (OW__ + DWS*)_,+ DV_/_(6,- 6j+_)- DV_(6j_I - _,) = S_', j = 2,3,..-,ni-
that
(208)
1.
This system has 4(nj - 2) equations and 4(nj - 2) variables. Unfortunately, coupling between the correction variables makes the problem somewhat complicated. To develop a simpler approximate system, define
_e_
__
with fixed i and k. Then
_zj
c
(B_ _ +
E6 _
_N
(As
FciJk (_ ...... +
From (198), we find that diik 3 for high Bijk ' 3 c _ _ and C_ _ "_ _E_" These yields
Re
_
vj+1 - _c
uj_,
(209)
+ "_c _7_ _o_
B_)t_ - B'g%+, - _i_h, _o o,-1
and
small
At,
which
is much
(21o) larger
than
aijk "'s • Similarly,
A'_
(211)
_n
(212)
_C/'/Y
F_ _c_
A07_
_7_
46
Liu,Liu,McCorn_ck,
Here, the superscript points.
_,j,k,
vi,j,k
and wi,j,k
denote
the coordinate
values
of the corresponding
Final Report
discrete
u, v, w
Vinik = 0
I [f i6k
--
Piek + "re
_6 --_
Ui+l
6k
-_
(6
0
Ke, + 6s - 6e
t Uisk
--
PiSk + 75
_5
Ui+1 Sk + _s --
--e.-
T Ui4k
--
Pi4k + "r4
_4 --_
Ui+; 4k + e4
0
Vi4k + 6a - 64
Ui3k
--
Pia_ + 7a
E3 --_
Ui+x al, + _a __
0
...._.
V_a_+ 82 - 8a
T Ui2k
--
_2
P_2k + ")'a
Ui+l
21, + e2
0
_ak
Figure
28. Distribution
With the above approximations, _:j and w,j written in terms of only the unknowns 6j:
= 0
of corrections
can be treated
in the (_, 7) plane.
as known
parameters,
so equation
(208)
can be
[(DV__+ DVo,_k),_.,.+ (Dv__ + Dv__) + (Dw__ + Dw$_*)_.,]6_- Dv_J%_I- _,,,jk, ..,, o,+, = s':. .,_ (21s) Let
_, = (DU__+ Du_'_)_._ + (Dv;_+ '_ Dv__) + (DW__ + Dw_)_,,, ci
(_14)
_
_ nvO'k
(215)
=
-DVc _*,
(216) j = 2,3,...,ni-1.
Liu,Liu,McCormick,
Then we obtain
Final Report
the tridiagonal a2
b2
C3
a3 "..
47
system k
62
63
b3 •.
(217)
Cnj - 2
Thus,
6i,
j : 2, 3,..-,
an j- a
bnj-2
cnj-1
omj_1
nj - 1 can be determined ej
very efficiently. :
The
updated
Wij
other
k 1 k
velocity
corrections
are given
by
- 1.
on all cells in the i, k y-line Ui+ljk
stain j-
w=j6j,
j = 2,3,...,n$ The u, v, and w are then
s_-2
6ni-2 • 6,_i- I .
as follows:
.-
ui+ljk+ej,
U _ik
*-
U ijk-ej,
k+l
+-
W_j k+l_4_aj, (218) j = 2,3,...,n_.
V _k
_
-
V °'k+6_'_1-6J',
(219)
j = 3,4,...,nj Finally,
the pressure
corrections
-yj are determined
1,
- 1.
as follows:
"Y2
P is then updated
For large
time
step,
via
the above
method
for pressure Ac >>
is not valid. Poisson type
A_,
correction Bc
>>
may
BN,
For such cases, we use the above method to obtain problem for P with Neumann boundary conditions:
O_P a_---_+ _
(,7,_0,'_)
Cc
cause >>
some trouble
because
the assumption
CF
an initial
guess
for P, then
solve
= S,,
OP -= O. Here, Sp is the source term for the pressure good approximation to P, thus the increases
an induced
equation• Only a few relaxations cost is very limited.
(223) are needed
to obtain
a relatively
48
5.4
Liu,Liu,MeCorndck,
Multigrld
Final
Report
Methods
Basically, the multigrid strategy for three-dimensional problems is the same as that for two-dimensional problems, although there are a few technical differences. To simplify the discussion, we limit ourselves to two grids. Note that the full approximation scheme (FAS) must again be used in order to accommodate nonlinearities of the Navier-Stokes equations. For a generic equation of the form LhWh a two level FAS process
may be described
loosely
i)
relax
on LhW h = fh
ii)
solve
L2_W 2h = L_hI_W
iii)
replace
W h _
= $h
as foUows:
h + i_(f
h - LhWh),
W _ + I_h(W 2_ - I_hW_').
Here we have introduced the difference operators L _ (fourth order for fine grid) and L 2_' (second order for coarse grid), the restriction operators I_ _ (for the approximation) and i_ _ (for the residual), and the interpolation operator I_. The coarse grid levels use the lower-order difference operator, which is less expensive and adequate for correcting smooth fine grid errors. Two coarsening strategies are used here: full coarsening, which is suitable for grid ratios Az/A_I near one and semi-coarsening, where the grid is coarsened only in directions of strong influence of the grid operators.
l
---P-
(_)
!
l
I ®
--
®
:J
l
:J
I
--
®
-_
®
--
-.-
m..._
®
t
t _
®
:J
I
---.._
®
®
÷
-.-
/
®
m
m._.
m--4_
--
m-4_
®
t
--
®
m
4 I
--
®
®
m
m
w.-..
--_
®
--
a
t
.-_
®
--
®
I ®
®
t
I -i_
__.
t
t ®
®
.', 5,
m-4_
_
--
t
--
--
¢.
t
t
t
t _)
m
t
__.__
---4s-
t
i
m...
®
':? --
I ._
®
--
t
x
I Figure 29. Projection of two-level staggered grid on the (z, _/)-plane (z-axis is orthogonal to view, and the actual u, v, w and P points in the computational domain are located in different (z, _/)-planes).
Liu,Liu,McCormick,
5.4.1
Final
Report
49
Full-Coarsening Figure
Under
29
depicts
the
full-coarsening,
u, v, and
w,
projection
the
we use
the
of
restriction
a two-level
and
full-coarsening
interpolation
means,
for
example,
that
u 2h takes
2h
one
1
the
pressure
P, the
stencil
quarter
the
restricted
structure
in
as follows.
For
the
(z, y)
restriction
plane. I_ h of
p_h
will
P_+I il
momentum
from
each
come
2i -- 2,
from
+
jl
1
1
1
1
all eight
i,j,+l
k, +
j,}, -'t- ph_+1
:2i-2,
:
2j -- 2,
1
1
1
1 ph
]2_,
1
1
1
we use
2
2
16
2
2
1
1
1
_1
in the
same
(y, z)-plane:
2k - 2.
=
(225)
i,i,k,+l
contained
+
30. the
1
1
1
1
in a cell
t,y,+l
kx+l
of which
p2h
is located
at
is based
on
+
_+1 k, + P_+I j,k,+l + P_+, j,+l k,+l), k 1:
2k-2.
continuity
1
1
it located
'
points
j1:2j-2,
Figure
1
u h around _h
equations
For restriction of residuals, the so-called volume law :
(224)
I_ h is
_(Pi,j,k,
1
grid
defined
'
l_h
1
which means the center:
1] 1
1
uh
il = For
are
stencil
4111 which
staggered
operators
Full
weighting
following
equation
restriction.
full-weighting
stencils
for the
discrete
momentum
for
discrete
continuity
the
(see
Figure
equations,
equation.
30),
which
and
(226)
50
Liu,Liu,McCorndck,
Taking
the residual,
rl,
of the z-momentum
:hijk : 1(
r_ h
rl
is-1
h is-1
jlhz
"_ rw
+2r_ +rh
where,
_1
:
Tfilinear
is-{-1
2i - 2, jl = 2j - 2, interpolation
,,j,k,
j_ik, +
]C1 :
equation
2k
+'_
hz
-{- 2r_ ',js@l
h 7". iljr
--
js÷l
as an example,
I ys÷l
h, + 2r_ ,,y,k,+1
h "_- _'-
ks
we use
i.k,÷l +'._ i1-1
i!-1
Final Report
j1÷1
ks÷l
-{- 2r_ ,,j,+1/,,+1 h
11-)-1
jsks-#-I
"_- r.
i,÷1
yi-}-1
(227)
hi÷1)'
2.
is used for the transfers
I_
applied
to the corrections
A W _, _ W _, _ l_hW
h.
From Figure 29, taking u as an example, we first interpolate u on the (y, z)-plane, then on the (z, y)-plane. the (9, z)-plane where both u h and u 2h are located (Figure 31), the stencil is of the bilinear form
[9 3] [, 11 16
_
16
Once simply
the _u h have been by linear
determined
interpolation
[3 9] [1
16
for /Xu,_, J'"
16
in the
(y, z)-plane,
in the (z, y)-plsne.
16
16
_
_
16
16
all other
points
o A"L
(228)
for _ku),.
can be determined
For v and w, the procedures
For
from
these
values
are analogous.
o _"._,
O
Z
Figure
Z
31. Bilinear
interpolation
for _u
in the (_, z)-plane.
Interpolation, I_h, of the pressure corrections is quite complicated. The stencils matrix form. Referring to Figure 32, the stencil for the specific _ph point is
where
the first block
front plane.
of the matrix
denotes
27
9
_4
_4
e4
e4
the weights
9
3
_4
_4
64
64
obtained
can be expressed
,
in block
(229)
from the rear
plane,
and
the second
from
the
Liu,Liu,McCormick, Final Report
51
AP=h
_,"
...r"i
w
.."" [
i
ah
_.....
!
._'.•
_
e":........ t---_........o"l i
front plane
I
I
I ._,........ i..........-..::
i .. "__L !-...-"....
"//
rear plane
a / I
Figure
5.4.2
32. Trilinear
interpolation
for pressure
points.
Semi-Coarsening
A fuU-coarsening strategy is generally ineffective for problems that favor special coordinate directions (e.g., anisotropie problems). To overcome this limitation, we consider now a special combination of seml-coarsening and line-distributive relaxation. The basic idea is, to use line-distributive relaxation in one direction (say the y-direction) projected
and
coarsening
in the (z, y)-pleme
Full weighting can be expressed
restriction as follows:
only
in the other two directions
and
(z, z)-plane
(z-
is given in Figure
i_h(R_)
A two-level
staggered
grid
33.
is still used here for transferring
:
and z-directions).
the residual
_
_
$
4
_
from fine to coarse
grid.
The stencils
,
: i (230)
These stencilscan be explained geometricallyas shown in Figures 34-35.
52
Liu,Liu,McCorn_ck,
t
1 ---4_
t
_h
m._.
.ph--
t
1
t
m--I_
t
Final Report
•
t
m
t n.-_.
t
t
t (a) _2h
1
I
I
I I t
I I /
I I t
•
--÷
Q
,
--_
_2h
•
(b) _2h
t m-i_
•
t
I
l _h
I
m
L 1•
I
t I
÷
Lh
t
1/,2h
•
t
t •
__
_
I
Lk IF
) I
1 I
t
t
_
1 I
4I
t
•
t
-- ---_-
:
1 I
LL IF
m.4_
t
--_
t
t
•
Lh 5F --÷
4
l • t
i |
z
(c)
Figure 33. Two-level staggered grid structure for z - z direction semi-coarsening: (a) fine grid projection on (z, !/) plane, (b) coarse grid projection on (z, _¢) plane, (c) fine and coarse grid projection on (z, z) plane.
Liu,Liu,McCorrruck,
Figure
Final
34.
the
restriction
53
Full-weighting
Figure
For
Report
35.
restriction
Full-weighting
of variables,
for
(a) z-momentum
restriction
bilinear
equation,
for y-momentum
interpolation
is used.
and
Its stencils
(b)
z-momentum
continuity
equation.
equations.
are
_,._../_-'W: °,.-_o'o/_
0
is then
employed.
z zo.
From
(224),
we find that
118
Thus,
Liu,Liu,Mc
the mixed
scheme
for the base flow is modified
hybrid: central Both separation about 20 times the
if _ difference:
Cormick,
Final Report
to:
> 0,
if o°-_,_ 0.
bubbles in front and rear of the roughness element are then produced. The second bubble is height of the roughness element. Figure 115 gives the contour plots of the streamfunctions of
the computed base flow. The comparison of the streamwise profile of our numerical results with the experimental results obtained by Dovgal & Koslov (1990) is depicted in Figure 116. Basically, they agree each other well for a large portion of the domain. The major difference appears upstream of the hump. The experiments show a larger separation zone ahead of the hump, but the computation gives a smaller separation sone, due apparently to the artificial viscosity introduced by the up-winding strategy and the surface shape difference between Case I and the experiment. The streamwise shape factor for our results and those of Dovgal & Koslov (1990) are also compared in Figure 117, which shows qualitatively good agreement. Note that all the geometric parameters appearing in the figures in this section have been interpreted according to those used in the experiments. A periodic perturbation with w = 0.0968 (69Ha) and e =0.003 is then imposed at the inflow. In this case, one T-S period is divided to 400 time steps. Comparison of the computed amplification factors along the streamwise direction with those obtained by Dovgal & Koslov (1990) for the hump is shown in Figure 118. Similar to the experiment, we obtained an exponential growth region downstream of the roughness element, which is mainly due to the effect of the rear separation bubble. Unfortunately, because the shape of roughness elements is different, the front separation bubble we obtained is smaller than that obtained by experiment for flow over a square hump. The disturbance almost has behavior similar to the linear instability stage before it reaches the rear separation bubble, and so, we do not fully obtain the first high rate growth upstream of the hump. The comparison of the amplitude of disturbance at select cross-sections is given in Figure 119, showing good qualitative agreement. Case
II:
Similar to the last subsection, we then changed the roughness square roughness element. The shape function now is chosen to be f(z) Similar
to Case I, the mixed
scheme
:
1.12/cosh_[O.OOS(z
is used to discretise
shape
so that
it a better
approximates
-- 44.0)'].
the base flow.
the
(245) Both front and rear separation
bubbles
are
larger than for Case I (see Figure 120). The shape-factor obtained is in quite good agreement with experimental results (Figure 121). The streamwise profiles also show better agreement with experiment (Figure 122). The amplification factors of the disturbance (Figure 123) and the normalized perturbation streamwise velocity profile (Figure 124) are also compared with those obtained by experiment, and both show better agreement than for case I. The perturbation streamfunction contours are given in Figure 125. Note that some additional interactions between the experimental measurement instrument and the base disturbance level of the wind tunnel may generate some 3-D disturbance, which is not possible for this 2-D simulation to describe.
Liu,Liu,McCormi_,
Final
Report
119
2.940 2.160
• 2.940
1.500
1 600
_-.-----_
o 96o-----
u._ou -- 0.540 __J_-_ __.,___..__
__
0.24.0 ------____
o.z4o _
J/J'__
- 0.060
Figure
115.
Streamfunction
0.540
contours
of the
base
o
flow
for
.060
large-scale
roughness
case
I. Flow
direction
is from
left
to right. x(mm)
-50
-15
-.5
10
40
70
100
130
:I t)oJjJ,tJJ
4
"
'_
•"
2
.'
o"
._
'
o'°"
.e."
.
o"
"
.o
._"
"
o"
°
_" .o" @
numerical
Figure
116.
mental
results
Comparison obtained
of the
numerical
by Dovgal
results
o
streamwise
& Kozlov
o
velocity
(1990,
over
experiment
profile
a square
numerical
results
experiment
of
Dovgal
•
,
0
..../ /
for
o"/""
".._/. ._
0
u/U.
of
Dovgal
large-scale
....
,
-100
Figure obtained
117.
Comparison
by Dovgal
of the
& Koslov
-50
numerical (1990,
....
over
streamwise a square
?
.
..
,
0
disturbance hump).
.
et
1.0 al
roughness
case
I and
experi-
hump).
et
al
,
....
o 2
"
50
,
0 ....
100
of shape
factor
150
H
and
the
experimental
results
120
Liu,Liu,McCormick,
exponential
i I I
growth
I
5 4
......
o
-iO0
-50
Final
Report
,
0
50
100
(_) enu"
I
expo-
_ o'°_
nentiel'
._"
I _rowth l .-'"
4
I
-50
0
50
lO0
tU
(b) Figure
118.
roughness
Streamwise
element,
distribution
(b)
x(mm)
_4 v
results
-15
-50
6
of the
experimental
5
amplification obtained 10
factors: by Dovgal
40
(a) numerical 8z Kozlov
70
results
(1990,
over
for large-scale a square
hump).
3O
100
)l.
0 0
=_/u'.,.
Z.O
(a) _C. =-50_'u
-15
-5
10
40
70
-
100
130
-.,.,-.,.,
(b)
Figure large-scale
119.
roughness
Normalized case
I, (b)
amplitude experimental
of streamwise results
perturbation obtained
velocity:
by Dovgal
(a)
& Kozlov
numerical (1990,
results over
for
a square
hump).
Liu,Liu,McCormici,
Final
Report
121
2.940
---.-__
• 2.940
-2.160 • 1.500 • 0.960 0.540 • 0.240 - 0.060 • N_Ofl4
Figure
120.
Contour
plots
of the
base
flow
for large-scale
roughness
case
II.
ai numerical o
o
results
experiment
of Dovgal
et al
6
4
2
-I00 Figure
121.
results
obtained
Comparison by
of numerical
Dovgal
i
,
i
-50
0
50
iO0
shape
& Kozlov
x(mm)
,
factor
(1990)
-50
distribution
for a square
-15
for large-scaie
150 roughness
case
II and
hump.
-5
10
40
70
iO0
130
6
_4
•"
o"
,•
,•
"
.
.
2 01 0 numerical
Figure
122.
Re_
663.12,
=
Comparison t¢ =
1.12.
of streamwise
velocity
results
profile
o
o
at selected
experiment
locations
of
_/_.
Dovgal
with
eL
1.0 al
experimental
results.
experimental
122
Liu,Liu,McCormick,
Final Report
I
5
J I
exponential
4 °
43 -=
=-z
-100
-50
0
50
I00
(a) -al.
'-';_;h_'
4
I I
.7""-"
I o.D"
16 r°wth
o,_1" I
J d
./%_" |
-50
0
50
loo
(b) Figure 123. Comparison (b) experimental results -50
x(mm)
of amplification factors: (a) laxge-scale roughness case II, obtained by Dovgal & Koslov (1990) over a square hump.
-15
-5
10
40
70
tO0
t30
8 6 A
_4 2
o I,:/='._
1.o
(a) --5Omm -15
-5
10
40
70
100
130
(b) Figure 124. Comparison for laxge-scale roughness
of the normalized streamwise disturbance velocity profiles: (a) numerical case II, (b) experimental results of Dovgal & Kozlov (1990) over a square
results hump.
Liu,Lia,McCorndck,
Final Report
Figure
9.3
123
125. Contours plots of instantaneous Re_ : 663.12, w : 0.0968, e : 0.003.
Two-Dimensional
Distributed
Multiple
perturbation streamfunctions at t : Flow direction is from left to right.
9T,
Roughness
Since the major contribution to the disturbance amplification comes from the separation bubbles, it is expected that the multiple roughness will induce larger amplification factors through the whole domain. To test this, five roughness
elements f(z)
The computational
are now chosen,
--
0.5/coah2[O.OO5(z
+
0.7/cosh2[O.OO5(zdomain
described
by
- 44) 4] ÷ 0.7/coah2[O.OO5(z
- "/9)4] -k 0.9/coah2[O.OOS(z
149) 4] + 0.5/cosh2[O.OOS(z
- 114) 4]
- 184)4].
(246)
is z E [0,242],
1/E [0, 50],
(247)
the grid stretch parameter _r : 4.5, and the grid sise is 362 x 50 (9 T-S wavelengths physical domain T-S wavelength buffer). The contour plots of the base flow are depicted in Figure 126, and the associated parameter H and both the displacement and momentum thickness are shown in Figure 12"/.
and 1 shape
For the perturbation flow, the parameters are the same as for the large roughness cases, but the amplitude of disturbance is reduced to _=0.0005. As expected, the amplification factors of the disturbance reach a higher value. Since the roughness elements are distributed quite randomly, the fundamental after passing a few roughness elements. Figure 128 displays the amplification factors 129 gives the instantaneous streamfunction contours at _ = 10T.
T-S wave is almost invisible for both u and v, and Figure
124
Liu,Liu,McCormick,
Figure 126. Streamfunction contours of the base flow for multiple the domain is shown, flow direction is from left to right).
distributed
roughness
Final Report
elements
case (part
of
6 5 4 3 2 -100
0
I00
200
3OO
200
300
2.0 1.5
1.0 0.5
J
.............
.,...*-
.....
.....
.....
,'_
...
0.0 -100
Figure 127. Nondimensional base flow with the presence
0
I00
shape parameter H, displacement of multiple roughness elements.
thickness
6'
and momentum
thickness
O of the
Liu,Liu,McCormiek,
Fined Report
125
.'...
-100
0
100
200
300
I00
200
300
(a)
6
.j
_4 p,.
2
0 -I00
0
(b) Figure
128.
Streamwise
(a) amplification
distribution
factors
of amplification
for u, (b) amplification
factors;
factors
for v.
F / I
I
/ I
Figure
129. Contour plots of the instantaneous perturbation streamfunctions at t = 10T for the distributed multiple roughness elements (part of the domain is shown, flow direction is from left to right).
126
Liu,Liu,McCormic_,
9.4
Three-Dimensional Flow
Isolated
Roughness
Element
In this subsection, the effect of a three-dimensional isolated The parameters for this study are listed in Tables 6 and 7. The computational
domain
of the surfaces
Reynolds
element
Boundary
with different
scales
Layer
is studied.
y _ [0, _,,,],
of the roughness f(z,
The roughness
roughness
Plate
is set to _ [0, mL.],
and the shape
for Flat
Final Report
is defined
by
z) = g/coah2[bl(m
- zo) 4 + bzz4)].
number Ug/_
Re,` = _,
(248)
V
where
U, is the undisturbed
size small medium large
velocity
at height
g, are also listed
Re_
Re_
_
_R
663.12 663.12 663.12
31.83 123.34 418.14
0.0968 0.0968 0.0968
0.25970 0.25970 0.25970
Table 6. Flow parameters
size small medium large
_2d 0.003 0.003 0.003
length (3+I)T-S (3q-1)T-S (3+1)T-S
T/At 360 360 400
Table 7. Computational 9.4.1
Base
in Table
6.
al -2.6244 --2.6244 --2.6244
ZO
x 10 -3 x 10 -a x 10 -3
for 3-D roughness
grids 98 x 34 x 34 98 x 34 x 34 114 x 34 x 34 parameters
_r 4.0 4.0 4.0
0.25 0.25 0.25
44.0 44.0
0.30 0.60
44.0
1.12
flow.
y_.ffi bl 50 0.005 50 0.005 50 0.005
for 3-D roughness
b2 0.0025 0.0025 0.0025
flow.
Flow
The base flow is calculated by the hybrid scheme, which yields better stability and faster convergence than central differences, though less accurate for large mesh Reynolds number. Three cases are under consideration. 1. Small-Scale Roughness In this case, we set g : 0.3 and the roughness Reynolds number Re,_ = 31.83. Other parameters can be found in Tables 6 and 7. Though the base flow is still streamwise dominant, it is really three-dimensional, i.e., the spanwise velocity component, uJ, is no longer zero. Figure 130 depicts the velocity vector plots of the base flow field on the j : 2 and j = 3 grid surfaces. No obvious separation is observed at either the front or rear of the roughness element, smooth and the roughness height is really small.
because
the roughness
surface
function
that
we used
is quite
2. Medium-Scale Roughness We then change the roughness height to _ = 0.6, and the corresponding roughness Reynolds number to Re,, = 123.3 < 300. A separation zone is found behind the roughness, which is much smaller than what we observed in the 2-D roughness case with the same roughness Reynolds number. The vector plots of the base flow on j = 2 grid surface is show in Figure 131, demonstrating that a very small separation zone exists in front of the roughness and another separation zone appears at the rear of the roughness. The front separation can form a horseshoe-shaped filament, which wraps around the front of the roughness element, as it travels downstream and is further increased.
Liu,Liu,McCorndck,
3.
Large-Scale
Final Report
Roughness
127
As the height
of the roughness
_ is increased
to 1.12 (the same as that
used for
2-D case), the roughness Reynolds number increases to Re_ : 418.14 > 300. Two separation bubbles are clearly found in front and at the rear of the roughness element. The front separation forms a horseshoe-shaped vortex filament (see Figure 132 for the contour plots of streamwise vorticity on selected (y, z)-planes), and may cause a secondary instability when it travels downstream. Besides, a counter-rotating pair of vertical spirals is found close to the rear wall and near the top of the roughness element. Figure 133 gives the vector plots of the velocity field on the j = 3 grid surface. The magnitude of vorticity contours on the y = 0.245 plane are depicted in Figure 134, which clearly exhibits the existence of the pair of vertical spirals. This base flow pattern obtained by our computation qualitatively supports the experimental results obtained by Mochisuki (196In,b). However, Mochisuki used a sphere as the isolated roughness element, and assumed elements of nearly equal height, span, and length, which is very different from ours. In our examples, the length and span are several times larger than the height. Therefore, it is difficult to obtain any quantitative comparison between our computational results and Mochisuki's experiment; yet the qualitative agreement between computation and experiment is nonetheless encouraging.
9.4.2
Disturbance
After obtaining the base flows numerically, we then impose a 2-D periodic small disturbance (T-S wave) at the inflow boundary and study the behavior of the developing perturbation. Again, three cases described in Tables 6 and 7 in the last subsection are investigated. 1. Small-Scale Roughness Figure 135 depicts the magnitude of disturbance vorticity contours on the j : 2 and j : 3 grid surfaces, which clearly shows that the disturbance has become three-dimensional, even though no 3-D disturbance is imposed at inflow. This is different from the case of 2-D roughness elements. Figure 136 gives the amplification curve for the velocity magnitude of perturbation and its u- and wcomponents. Since the flow has become three-dimensional, we use the amplification factor of disturbance velocity magnitude given by
as the measure refers to inflow flow meets decreases.
of amplification. values. A similar
the roughness
Here, u _, v' and w' are perturbation velocity components, and the subscript 0 method can be used for the 2-D problem. From Figure 136 we find that when the
element,
the flow perturbation
gains
2. Medium-Scale Roughness Evidently, In this case, the small-scale roughness case, and the disturbance in the spanwise contour plots of the magnitude of disturbance vorticity on the 2-D T-S wave is apparently stimulated by streamwise vorticity
more
energy,
but past
the roughness
sone
energy
disturbance is enlarged much faster than for the direction grows very rapidly. Figure 137 gives the j : 2 and j : 3 grid surfaces. The fundamental carried by the travelling horseshoe vortex.
3. Large-Scale Roughness The original version of the code has difficulty with disturbances in the case of large-scale roughness elements. Because of the dramatic amplification of the disturbance, central differences induce the flow to break up in front of the roughness element. Although the separation area is much smaller than that for 2-D roughness, the flow more easily breaks down for 3-D roughness. It seems that the formation and development of the horseshoe vortex strongly undermines flow stability. In this computation, though the major disturbance has not passed the roughness region, we still observe some 3-D disturbance behind the roughness element. Thus, it seems that the behavior of the disturbance is changed not only in space, but also in time. Some disturbance
with greater
travelling
speed is generated.
128
Liu,Liu,MeCormick,
Figure
130.
Vector 3-D
Figure
131.
Vector
plots
of the
base
roughness
case.
Re_
plots
3-D
of the
roughness
the case.
flow
on j :
= 663.12,
base
flow
Re_
=
2 grid Re_
surface
on j = 2 grid
663.12,
Re,_
for the
= 31.83,
=
small-scale
m = 0.3.
surface 123.3,
Final
for
the
medium-scale
_; = 0.6.
:ii!
Figure
132.
(V,z)-planes
Contour for
plots the
of the
large-scale
streamwise 3-D
vorticity
roughness
case.
(base Re_
flow) =
on
663.12,
i :
45, 50, 55, 60, 65, 70, 75, 80
Re,_ :
418.14,
_; :
1.12.
Report
Liu,Liu,McCormick,
FJns_
Figure
Report
133.
Vector 3-D
Figure
134. for
Figure bottom
135.
Contour the
plots
for the
plots
roughness
plots
large-scale
Contour
to top)
129
of the 3-D
of the small-scale
of the
base
case.
Re_
spanwise
roughness
disturbance 3-D
flow :
on j =
663.12,
vorticity case.
Re_
vorticity
roughness
3 grid
Re,_ :
(base :
flow)
663.12,
magnitude case
surface
for
418.14_
at t --- 5T.
_ :
on/e Re_
on
j = Re_
: :
the
large-scale
1.12.
12,
17,
418.14,
2 and -- 663.12,
22 _ :
j =
(z, y)-planes 1.12.
3 grid _ --- 0.3,
surface
(from
_ -- 0.003.
130
Liu,Liu,McCormick,
Final
Report
m
0.8
3-D
isolated
roughness
0.6 0.4 0.2 0.0 -0.2
,
J
I
I
0
,
i
i
20
,'I
.....
I''i
i
i
40
I
,
,
,
I
00
80
(a)
t.O :
_
3-D
isolated
roughness
A
0.8 0.6 0.4 O.Z 0.0 -0.2
I
0
,
,
20
,I
.....
i''..
,
,
40
I
,
,
,
I
60
80
(b)
3
I
0
20
40
80
80
(_)
Figure -scale
136.
(a) Comparison
isolated
roughness
between spanwise
2-D
smooth velocity
of amplification case.
fiat
plate
magnitude
(b)
factors
Comparison
and
3-D
w for the
of the
small-scale 3-D
small
of u between disturbance isolated roughness
the
2-D
smooth
velocity
roughness case.
Re_
fiat
magnitude
plate
and
amplification
case.
(c)
Amplification
=
663.12,
_ =
0.3,
3-D
small factors
factors e = 0.003.
of
Liu,Liu,McCormick,
Final Report
131
\
Figure 137. Contour plots of the perturbation surfaces, for the medium-scale 3-D roughness
9.5
Three-Dimensional Layer
Distributed
vorticity on Ca) j = 2 and (b) j = 3 grid case. Re_ = 663.12, _ = 0.6, e = 0.003.
Roughness
Element
for
Flat
Plate
Boundary
Flow
The multiple roughness case is as a collection of single three-dimensional roughness elements randomly in the domain, with arbitrary height and spacing. According to Morkovin and Corke (1984), roughness will cause • complex,
predominantly
• long induction
distances
streamwise
vorticity
distributed distributed
patterns;
from the process-initiating
roughness
to the onset
of turbulence.
General multiple roughness simulation is not yet possible. In this section, a simplified model with seven roughness elements in the computational domain is used (see Figure 138). By assuming the spanwise wavenumber _/- OR, the computational domain is then restricted to z E [0, 121], and the shape f(z,Z)
of the surfaces
with seven
z • [-12.1,
y E [0, 50], roughness
elements
is defined
12.1],
by
_/cosh2[b(z
- 29.8) 4 ÷ bz4)] q- g/cosh2[b(z
- 54.0) 4 q- bz4)] q-
_/cosh2[b(z
- 78.2) 4 -I- bz4)] q- _/cosh2[b(z
- 41.9) 4 ÷ b(z - 12.1)4)1 q-
_/cosh2[b(z
- 41.9) 4 ÷ b(z _- 12.1)4)]
_fcosh2[b(z
- 66.1) 4 q- b(z _- 12.1)4)].
A 122 x 34 x 34 grid is used, domain.
which includes
+ _/cosh2[b(z
a 4.3 wavelengths
physical
- 66.1) 4 ÷ b(z -
domain
and
12.1)4)]
÷
a 0.7 wavelength
buffer
132
Liu,Liu,McCormick,
9.5.1
1.
Base
Final Report
Flow
Two cases
with different
Small-Scale
Distributed
sizes of roughness
are considered.
Roughness
First, let b = 0.005 and _ : 0.35 (Re,, ,._ 43.5), which is considered to be small-scale roughness. Figure 139(a) depicts the vector plots of velocity field on the j = 2 and j = 3 grid surfaces). Yet the base flow is no doubt threedimensional with a w-component. In addition, the so-called horseshoe vortices are generated downstream of the computational domain, which is different from the isolated roughness flow. Figure 139(b) gives the streamwise vorticity contours on the selected (y, z)-planes, clearly shows the development of horseshoe vortices. 2.
Large-Scale
Distributed
Roughness
We then increase the height of roughness to _ = 1.12, yielding a corresponding roughness Reynolds number is of Re_ .._ 418, which is considered to be large-scale roughness. Similar to what we described in the case of a single 3-D roughness element, two separation zones in front and at the rear of each roughness element are found. The rear are stronger than the front vortices. The rear vortex energy comes from the downwash of the flow from the high velocity fluid at the top of the roughness element (high momentum region), while the front horseshoe vortex energy comes from stagnation of the fluid in front of the roughness element (low momentum region). Figure 140(a) depicts the vector plots of velocity field on the j = 2 and j = 3 grid surfaces, and Figure 140(b) gives the streamwise vortex contours on the selected (y, z)-planes, clearly showing the strong development of horseshoe vortices. Because of the separation at the rear of the roughness elements, the profiles of the base flow are distorted from the Blasius solution. Figure 141 depicts the u-velocity profiles between three roughness elements (in the central (z, y)-plane of the computational domain), which shows qualitatively agreement with experimental measurements by Tadjfar et al.(1993).
9.5.2
Disturbance
We then impose a 2-D periodic perturbation (T-S wave) on the base flow. Unfortunately, the central difference scheme shows numerical instability for the large-scale roughness case with early breakup. On the other hand, the code works well for the small-scale roughness case. With e2a = 0.005, and one T-S period divided into 360 time steps, we obtained the expected results. Figures 142 and 143 depict the amplification curve of the maximum perturbation velocity and its u-component, which is several times larger than what was obtained for the 2-D T-S wave past the smooth fiat plate around the roughness region. According to the experiments of Leventhal & Reshotko (1981), for flow over multiple roughness elements, the maximum amplification rates are about three times as large as those for T-S instability over a smooth plate. Our computation qualitatively supports their experiments. Figure 144 gives the contours of spanwise vorticity on the j = 2 and j = 3 grid surfaces, which shows that the T-S waves are apparently weakened when the disturbance passes the distributed roughness zone, while the h-wave gradually develops downstream, and thus the flow stability worsens. This is also qualitatively in agreement with the experiments of Tadjfar et al. (1993) in which the oscillation of disturbance amplified rapidly as it moved downstream, although no evidence of T-S waves in the boundary layer was observed. To investigate the effect of the amplitude of the imposed disturbance, we then increased the amplitude of disturbance e2d = 0.015. The H-type like pattern now can he seen clearly (see Figure 145). When the intensity of the disturbance is increased further (_2a = 0.05), the hairpin vortex (cf. Morkovin, 1990) clearly appeared (see Figure 146), and the laminar flow then broke down.
Liu,Liu,McCormick,
Final
Report
Figure
Figure grid
133
138.
Distribution
139(a). surfaces
Vector for
the
of the
plots
3-D
distributed
of velocity
small-scale
3-D
fields distributed
roughness
on j =
2 and
roughness
surface.
j = case.
3
134
Liu,Liu,McCormlch,
Figure
139(b). Contour plots of stresmwise vorticity on i = 30, 42, 54, 66, 78, 90 (y, z)-planes for the small-scale 3-D distributed roughness case.
Figure 140(a). Vector plots of velocity fields on j = 2 and j = 3 grid surfaces for the large-scale 3-D distributed roughness case.
Final Report
Liu,Liu,McCorn_ck,
Final
Figure
Report
140(b).
135
Contour
(y, z)-plsnes
plots
for
of streamwise
the large-scale
vorticity
3-D
on i :
distributed
30, 42, 54, 66, 78, 90
roughness
case.
2.0
1.5
"_ 1.0 .:-:.:.:.:.:.:.:.:.:.:. ...................
_!iiiiiiiiiiiiii!iii!iiiii 0.5
,, iiiiiiiiiiii
0.0
(a) zldm
I0
¢mlmoc
Xlj
-
19].26e,_.
AE t
-
]15
Y
Y
• t
t
t
t:\1
I
II
]l
Y t
V
'
50
t
• t
Y
t
1
•
t
i
26
_l
(b) Figure 3-D
141.
Comparison
distributed
of u-velocity
roughness
case,
profiles (b)
for the
experiment
base
of Tadjfar
flow.
(a)
Numerical
et al (1993,
over
results the
for the
sphere
large-scale
roughness).
136
Liu,Liu,
McCormick,
Final
Report
3-D distributed roughness smooth flat plste
1.0 0.8 0.8
0.0_ -0._
I
0
Figure
142.
roughness
.-"'1"'-
20
,
Amplification obtained
on
factor a
82
40
of
× 34
the
× 18
O.O
3-D
0.0
smooth
t ....
maximum (4.4
T-S
distributed flat
;"
,
,
i
O0
:"
.
'
80
perturbation
velocity
wavelengths
._ ....
physical
for
100
the
domain
small-scale
+
0.6
3-D
wavelength
distributed buffer)
grid.
roughneBB plate
0.4 '_
0.2
0.0 -0.2
i 0
Figure
143.
Amplification
obtained
on
a
.,.---]--.
20
factor 82
× 34
× 18
of
,
,
,
40
the
(4.4
perturbation T-S
wavelengths
..----:-.
.
,
60
u-velocity physical
for
.-.';..
80
the
domain
small-scale ÷
0.6
,
i 1 O0
3-D wavelength
distributed buffer)
roughness grid.
Liu,Liu,McCormi_,
Final
Report
137
(
Figure grid
144.
Contours
surfaces e2d :
(from
0.005,
of spanwise bottom
_; :
0.35,
perturbation
to top) Re,_ :
for
43.5,
vorticity
the
small-scale
flow
direction
on j :
3,
distributed is from
left
5 and
roughness. to right.
>:
Figure t :
145.
Contour
5T for the :
0.35,
plots
small-scale
Re,_ :
43.5,
of relative distributed flow
direction
helicity
of the
roughness, is from
total e2a :
left
to
flow 0.015,
right.
at
7
138
Liu,Liu,McCormlck,
Figure t =
146.
Contour
5T for the =
0.35,
plots
small-scale
Re,_ =
43.5,
of relative distributed flow
direction
helieity
of the
roughness, is from
total
flow
e2d = 0.05, left
to right.
at
Final
Report
Liu,Liu,McCormJck,
10
Final
Concluding
Report
139
Remarks
A very accurate and efficient multigrid code has been developed in Phase II of the project. This new code supports direct numerical simulation for 3-D smooth channels, flat plates, and flows with boundary layers incorporating 2-D and 3-D single and multiple roughness elements. • The fourth-order rate and stable
and fully-implicit time-marching for transition problems.
• The semi-coarsening fully-implicit scheme
scheme
method based on line-distributive comparable to explicit schemes
on a staggered
and stretched
relaxation is very efficient, at each time step.
grid
making
is highly
the
cost
accu-
of the
• The new version of the governing equations in general coordinates, which considers all u, v, w, P, U, V, W as unknowns, together with a high-order mapping successfully performs DNS for transitional flow with roughness elements, and is extendible to problems with more general geometry. • The
new
spatial
treatment
simulation
of the
outflow
boundary
dramatically
reduces
computational
cost,
making
realistic
much more feasible.
• The surface roughness contributes to early transitions in different ways for 2-D or 3-D single or multiple roughness elements. The current DNS computation shows quantitative agreement with experiments by Dovgal and Kozlov (1990) for a single 2-D roughness elements and qualitatively agreement with experiments by Mochizuki (1961a,b) for a single 3-D roughness element and by Tadjfar et al. (1993) for multiple 3-D roughness elements.
11
Acknowledgements
The authors thank NASA Langley Research Center for the financial support of this work. The authors are pleased to acknowledge helpful discussions with Drs. Tom Zang, Craig Streett, and Ron 2oslin at NASA Langley Research Center, Dr. Helen Reed at Arizona State University, Dr. Achi Brandt at Weizmann Institute of Science, Rehovot, Israel, and Dr. Pat Roache at Ecodynamics Research Associates, Inc.
140
Liu,Liu,McCormick,
Final Report
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[22] Klebanoff, P. S. & Tidstrom, K. D., 1972 Induces Boundary Layer Transition', Phys. [23] Kleiser, L., & Laurien, No.85-0566.
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[26] Kovasznay, L. S. G., Komoda, H., Vasudeva, Heat Transl. and Fluid Mech. Inst., 1.
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[27] Leventhal, L. & Reshotko, Layer Due to Distributed versity.
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DOCUMENTATION
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