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OCR for page 815
The Flow Past a Wit Body Junction
An Experimental Evaluation of Turbulence Models
W. Devenport, R. Simpson
(Virginia Polytechnic Institute and State University, USA)
ABSTRACT
Detailed three-component LDV measurements
have been made in the flow of a turbulent
boundary layer past an idealized wing-body
junction. These measurements, which show great
variety and three-dimensionality in the vortex-
dominated turbulence structure of this flow, are
here used to evaluate a number of turbulence
models. Many of these models require or imply a
relationship between the angles of the turbulence
shear-stress and mean-velocity vectors. In the
present flow these angles are not only different
but do not follow any simple relationship. To
predict the shear-stress angle, accurate modeling
of the full shear-stress transport equations is
clearly needed. In particular, new models based
on measurements are needed for the pressure-
strain term. The ability of six turbulence models
to predict the magnitude of the shear-stress
vector from the mean-velocity and/or turbulence
kinetic energy is examined. Among the best are
the Cebeci-Smith and algebraic-stress models.
Other models, particularly the Johnson-King, are
not well suited to this vertical flow.
INTRODUCT ION
This paper follows several (Davenport and
Simpson, 1986, 1987, 1988a, 1988b, 1988c, 1990a)
in which we have presented detailed velocity
measurements made in the flow of a turbulent
boundary layer around the nose of a wing-body
junction. The purposes of this paper are; (i) to
briefly review these measurements, (ii) to
present new measurements made around the rest of
the junction, and (iii) to use the whole data set
to evaluate the usefulness and generality of a
variety of turbulence models and modelling
parameters.
For a review of other experimental work on
wing-body junction flows see Devenport and
Simpson (1990a).
EQU I PMENT
Only abbreviated descriptions are given
here; for complete details see Devenport and
Simpson (1990b).
The Wing and Wind Tunnel
The wing (figure 1) is cylindrical, has a
maximum thickness (T) of 71.7mm, a chord of 305mm
and a span of 229mm. In cross section its shape
(figure 2) consists of a 3:2 elliptical nose
(major axis aligned with the chord) and a NACA
0020 tail joined at the maximum thickness. Trips
are attached to both sides of the wing to ensure
steady and fixed transition.
The wing is mounted at zero sweep and
incidence at the center of the flat 0.91-m wide
test wall of the Virginia Tech Boundary Layer
Tunnel, forming the junction. In the absence of
the wing this tunnel produces a flow of zero
streamwise pressure gradient, consisting of a
closely uniform (to within 1%) low turbulence
(0.2%) free stream and an equilibrium two-
dimensional turbulent boundary layer (see Ahn
(1986)) on the test wall. With the wing in place,
inserts attached to the wind tunnel side walls
are used to minimize blockage-induced pressure
gradients.
TV optics:
U. v system
B
~~~.~!
~ ~ .,` .. x,u
LDV optics: i/,/\\
0, W systemic
B
Figure 1. Perspective view of
junction.
Laser Doppler Velocimeter (LDV)
AA unshifted
BB -IS MHz shirts
CC +21.5 I\1Hz shirtea
the wing-body
A 3-component LDV was used to measure
detailed profiles of mean-velocity and turbulence
quantities in 6 planes surrounding the wing.
These planes (numbered 1,3,4,5,8, and 10 for
organizational reasons) are illustrated in figure
2.
The LDV has three sets of sending optics,
two of which are shown schematically in figure 1.
W.J. Devenport and R.L. Simpson, ACE Dept., VPI&SU, 215 Randolph Hall, Blacksburg
VA 24061
815
OCR for page 816
- 2
-1 .5
-.5
~ O
.5
1
1.5
2
Each set produces an arrangement of beams
sensitive to a different pair of velocity
components and their associated Reynolds shear
stress. Only one set is used at a time. The flow
is seeded using dioctal phthalate smoke (typical
particle diameter 1 micron). Light scattered from
the measurement volume is focussed onto the
pinhole of a single photomultiplier tube. Data
are obtained from the photomultiplier signal
using either fast sweep rate sampling spectrum
analysis (see Simpson and Barr, 1975) or a DANTEC
55N10 Burst Spectrum Analyzer. Velocity
statistics are obtained by time (not particle)
averaging and thus should be free of bias.
Measurements presented here have been
corrected, where necessary, for velocity gradient
broadening and finite transit time broadening
using the techniques described by Durst et al.
(1981). Uncertainty estimates for 95% confidence
limits are listed in table 1.
.
Uncertainty
+.03 Uref
~1
Plane 8
Plane 5
Plane 3
_
l l
-
-2 -1 0
Plane 10
o
O ~' ~ '''t 5tO
// ,°.~;~ :,~L~
1 ,. 4 ~
- 1~.
~`~ .
\ ~ ,
C'/'j' 1~`'~"'' - ~: {! ~ //
. - , , , , ~ ~ ~- _
5:o 5~°
~1 ° o
O
~ O
'AN
o;
/of ~ ~ ~\
1,
1
,,, 1
, , , , 1 , 1 1 1 1
2 3 4
X / T
Figure 2. Contours of mean-surface pressure coefficient Cp on the wall
surrounding the wing, 0 - locations of LDV profiles, line of separation,
~ "line of low shear, ~~. locus of peak turbulence kinetic energy in the
vortex.
Quantity
Mean velocity
Turbulence kinetic
energy, k
Turbulence _ear-
stress -uv
Turbulence _ear-
stress -vw
Turbulence shear-
stress magnitude
Table 1 Typical uncertainties
measurements. 95% confidence limits.
1 1 1
COORDINATE SYSTEMS, TEST CONDITIONS
+.00033 Uref2
+.00027 Uref2
Most results and discussion will use the
lab fixed coordinate system X,Y,Z and U,V,W
centered at the intersection of the wing leading
edge and wall (figure 1). X is measured
downstream from the leading edge, Y normal to the
wall and Z completes a right-handed system. In
presenting LDV measurements the additional
coordinate S will be used. S is measured along
any of the LDV measurement planes from the wing
surface or flow centerline, as shown in figure 2.
In discussing turbulence models and parameters
other coordinate systems will be used
distinguished by subscripts. Subscripts 'f', 's'
and 'g' refer to coordinates fixed in the local
free-stream direction, the local mean-flow
direction and the local direction of the mean-
velocity gradient vector, respectively. In
calculating these directions V component
velocities will be ignored. In all coordinate
systems, upper case and lower case symbols will
be used to denote the mean and fluctuating
components of velocity respectively.
Distances will in general be non
+ 00026 U. ' dimensionlized on the maximum thickness of the
- ref wing (T), velocities on the undisturbed approach
free-stream velocity, Uref. Under nominal test
conditions the momentum th~ckness Reynolds number
of the approach boundary layer, measured in the
plane of symmetry 2.15T upstream of the wing
leading edge, was 6700, corresponding to a total
boundary layer thickne~s ~ of 36mm (.50T) and Urf
of 27 m/s. e
+.00033 Uref2
in LDV
816
OCR for page 817
EXPERIMENTAL RESULTS
Figure 2 shows contours of mean surface
pressure coefficient C (based on undisturbed
free-stream conditions) Pand principle features of
an oil-flow visualization performed on the wall
surrounding the wing. Figures 3 and 4 show m~an-
velocity vectors and contours of turbulence
kinetic energy k/Uref2 measured in planes 1
through 10. The mean-velocity vectors represent
components normal to the centerline of the
horseshoe vortex defined as the locus of peak
turbulence kinetic energy (see figure 2). Other
projections of the mean-velocity field (e.g.
normal to the wing, parallel to the measurement
planes) do not clearly show the secondary-flow
velocities associated with the vortex. Note that
the measurements presented here in planes 1, 3
and 4 have previously been published by Devenport
and Simpson (1987, 1988a, 1988b and 1990a).
This flow is dominated by the pressure
field produced by the wing and the velocity field
generated by the horseshoe vortex that is wrapped
around the junction between the wing and wall. In
the plane of symmetry upstream of the wing (plane
1) the oncoming boundary layer experiences an
adverse pressure gradient that causes it to
separate 0.47T upstream of the leading edge
(figure 2). The separation region formed
(figure 3(a)) is dominated by the recirculation
associated with the horseshoe vortex. This
roughly elliptical structure, centered at X/T = -
.2, Y/T = .05, generates an intense backflow by
reversing fluid impinging on the leading edge of
the wing. The backflow reaches a maximum mean
velocity of -0.48Uref and then decelerates, giving
the appearance of reseparation between X/T = -.25
and -.3. Reseparation, however, does not occur as
a thin region of weak reversed flow is sustained
adjacent to the wall. This region is then all
that remains of the backflow upstream to the
separation point. The near reseparation of the
backflow produces a distinct line in the surface
oil-flow visualization known as the line of low
shear (figure 2).
In the vicinity of the horseshoe vortex the
turbulence stresses (and thus the turbulence
kinetic energy) become very large reaching values
an order of magnitude greater than in the
approach boundary layer (figure 4(a)). These
large stresses are associated with bimodal
(double-peaked) histograms of velocity
fluctuations like those shown in figure 5, and
are produced by intense low-frequency bistable
unsteadiness in the structure of the vortex. This
unsteadiness is a result of the turbulent/non-
turbulent intermittence of fluid entrained into
the corner between the wing and wall (Davenport
and Simpson (199Oa)).
Moving out of the plane of symmetry, fluid
experiences a strong favorable pressure gradient
(figure 2) that accelerates''~as it moves around
the nose. Close to the wing in planes 3, 4 and 5
this acceleration, acting in concert with the
rotational motion of the vortex (which here is
bringing low-turbulence high-momentum fluid from
the free-stream down close to the wall), locally
relaminarizes the boundary layer (Davenport and
Simpson (1988b and c)). Turbulence shear stresses
in this region are therefore much smaller than
elsewhere. Turbulence kinetic energy (figures
4(b), (c) and (d)) is also reduced. Although the
intensity of turbulent fluctuations in the
vicinity of the vortex falls in the favorable
pressure gradient the peak values of turbulence
kinetic energy remain many times those in the
surrounding boundary layer because of the bimodal
unsteadiness. Bimodal histograms are seen in the
vicinity of the vortex in planes 3, 4 and 5
(figures 4(b), 4(c) and 4(d)). Despite the
favorable pressure gradient the vortex clearly
grows in this region moving away from the wall
and the wing (figures 3 and 4). (Projected onto
i _
Y/ T _
0.0024
_ ~
('~'°~1 1
0.0136 ~
._ 11; ~ 1 )! .~
0.0031 11 0.0159 ~ 0.0789
-0.0038 Ill 0.01946 - 0.0963
0.00~- 0.02~ ~ 7,
0.0292 0.1434
. ~ ~ J ~
0.0357 0.1 756
,fL
0.0432 0.'147 - 1
0.()527
0.0644
0.0963
[~
0.1175 -
/ \
1
n 1414
~ 1756
0 3 -3 0 3 -3 0 3
_,
1,u2.'
Figure 5. Histograms of U-component velocity
fluctuations measured at X/T = -.2 in plane 1.
the wall the centerline of the vortex fairly
closely follows the line of low shear, see figure
2.) In addition mean secondary flow velocities
fall by a factor of about 2 between planes 1 and
5 (figure 3).
Downstream of the maximum thickness this
flow is subjected to an adverse pressure gradient
(figure 2) that appears to cause rapid growth in
the vortex and a dissipation of the bimodal
unsteadiness. (Note the change in scales between
different parts of figures 3 and 4.) Bimodal
histograms were not observed in planes 8 and 10
and peak turbulence kinetic energies are much
lower here than upstream. Secondary-flow
velocities, which are also reduced in the adverse
pressure gradient, become much more difficult to
817
OCR for page 818
.Z5
. 15
.05
C
35
4
L
, L_
-
. 3 ~1-
c ~: ~
~-
.2 L
, ~
~ t
.cs
~_
0 ~ I
.4 ~
_
.35 _
.3 _ .,
. 25 _
-;_ ~
.15 _
.1 _ _
ns
.6
(b) Plane 3
\
_ ~ ~ \ \
_ _ ~ ~ \ \ ~ ~
,-~a,,,~',j,~,,T,
5 .4 .3 .2 .1 0
S / T Uret
(c) Plane 4
Figure 3. Mean secondary flow field generated by
the vortex.
.5 .4 .3 .2 .1 0
S / T
Ure ~2
(a) Plane 1
. ~s
~.
.05 _
~..
v :! ~ I .... I
-.6 - .5 - .4 - .3 - .2
X / T
(a) Plane 1
~_ :~
4 0
(b) Plane 3
.4
.35
.3
.2S
.2
. 15
. ~
.05
o
o
~.~
.6 .5 .4 .3 .2 _
S / T
(c) Plane 4
1 n
Figure2 4. Contours of turbulence kinetic energy
k/Uref in the vicinity of the vortex. Dotted
lines enclose the regions in which bimodal
histograms are observed
818
OCR for page 819
: ~
Is
t
.4 ~
r~ -~ -_
. B _
7 _
.6 _
.8
,4
.2
-
(f) Plane 10
Figure 3. Mean secondary flow field generated by
the vortex.
819
.
.s _
r . _ _
, ~ . _
~ ~9
(d) Plane 5
(e) Plane 8
_ -_^ -_ - ~
~ \ ~ ~ N..
w..~...L,..:
_ .3 .2 .1
S / T ~
o
.2
o
.8 .7 .6 .5 .4
S / T
(d) Plane 5
.3 .2 . ~O
i:
l.2 1 .B .6 .4 .c
S / T
(e) Plane 8
0
.8
.6
. 4
.2
o
(f) Plane 10
.6 .4 .2 0
Figure 4. Contours of turbulence kinetic energy
k/Uref2 in the vicinity of the vortex. Dotted
lines enclose the regions in which bimodal
histograms are observed
OCR for page 820
distinguish from the rest of the mean-velocity
field (figures 3(e) and (f)). Despite these
changes the region between the vortex and the
wing remains one of low turbulence shear-stresses
because of the free-stream fluid entrained here
by the vortex.
Figures 3 and 4 represent only a fraction
of the mean-velocity and turbulence information
we have collected. All mean-velocity and Reynolds
stress components, some triple products and
histograms of fluctuations in all three
components have been measured at over 1400 points
in this flow. The quantity of experimental data
and the variety of turbulence structure in this
flow make it, in our opinion, ideal for testing
the generality and therefore usefulness of
turbulence models.
EVALUATION OF TURBULENCE MODELS
General Remarks
Before evaluating the validity of
turbulence models it is appropriate to discuss
the relationship between the turbulent shear
stress and velocity gradient directions since
many models use or imply such a relationship. The
shear-stress and velocity-gradient vectors are
defined as having components -uv, -vw and OU/3Y,
aW/BY in the X and Z directions respectively.
Their directions are given by the angles,
a~tan~~( w) and ag=tan~~(aWu/3Y) (1)
Most often the shear-stress and velocity gradient
angles are assumed to be the same, i.e.
- vw ~W/dY - uv - vw
or
-uv au/ay au/aY aw/aY
A" shown this implies that the streamwise and
cross-flow eddy viscosities are the same.
Although this is ideal for converting turbulence
models designed for two-dimensional flows to
three dimensions, it is not supported by the
present or past experiments (see Johnston (1970),
van den Berg and Elsenaar (1972), Fernholz (1981)
and others). Figure 6 shows a plot of spanwise
vs. streamwise eddy viscosity for all points
inside the boundary layer in planes 3 through 10.
Points outside the line of separation, where the
direct effects of the horseshoe vortex and its
bimodal unsteadiness are much smaller, are
plotted with different symbols to those inside.
In neither region does there appear to be any
significant correlation between these two
parameters.
A possible improvement has been suggested
by Rotta (1977) who derives an alternative
relationship between the eddy viscosities using
the transport equations for the shear stresses
approximated for thin shear layers,
D(-UV) _v2 U_p/ (aUy+ax)+,~3 (~;71+UV2) (3)
Dt -v aY- p ( aY+ aZ ) + By ( p +wv ) (4)
The terms from left to right represent
convection, production, pressure strain and
diffusion. By substituting the Poisson equation
for the fluctuating pressure p' it can be shown
that the pressure strain is composed of two
terms, the first ~1 is associated with the
interaction of the mean strain and fluctuating
velocities and the second ~2 with the interaction
of the fluctuating velocities alone. .2 is usually
004
.,~
~n
o
~n
a,
3
o
1
CO
o- 002
c~
002
o
-.004
. -o
- o
o c
o
O o OO~b
,` .
~. ~ ;
o
o
c:
n
1 , , , 1 , , ,.W , , , ~ , , . . .
- .004 - .002 0 .002 .004
Streamwise eddy v i scos i ty
Figure 6. Spanwise vs. streamwise eddy viscosity
in local mean flow coordinates for all
me ~urement points inside the boundary layer
(Vu /U ~ 5%). Squares represent points inside the
line of separation.
approximated by the shear stress itself
multiplied by a factor related to the turbulence
kinetic energy (see Rodi (1984)), that factor
being the same in both equations. Rotta
approximated .1 using the Poisson equation for the
pressure fluctuation p' and by assuming local
symmetry in the turbulence structure. Neglecting
convection and diffusion, which can be shown to
be higher order terms for thin shear layers, and
dividing equation (2) by equation (1), he then
obtained the expression,
tan(a ~-a) - T.tan(ag-a)
- V~Ws - u,svg ( 5 )
or / - T. /
i.e. the cross-flow eddy viscosity is an
empirical constant T times the spanwise eddy
viscosity, in local flow coordinates.
Unfortunately, as can be seen from figure 6 this
equation is no more valid than equation (2) in
the present flow. This result is confirmed by
figure 7 in which values of T deduced from these
measurements are plotted together as a histogram.
This shows a large spread with T varying over a
range of at least +2.
We have tested a number of other
hypothetical relationships between the shear-
stress and velocity gradient angles also without
success. These have included a relationship
between at ~ a and the local cross-flow velocity
Wf/Ue, one bet9ween the spanwise and streamwise
eddy viscosities, and one between at and a9 based
on van den Berg's (1982) hypothesis.
There are two principle reasons for the
failure of the above concepts. The first becomes
apparent if we transform the problem to
coordinates based on the direction of the local
mean velocity gradient vector (subscript 'g'). In
this system the cross-flow shear stress exactly
represents the lag or lead of the angle of the
shear stress vector over that of the velocity
gradient vector, i.e.
_g ~ tan(a~-ag) (6)
-u`~v
820
OCR for page 821
qn
so
20
0
o
-4 -3 -2
~R~T!
_ ~
=:t':~
0 1 2 3 4
l
Figure 7. Histogram of values of Rotta's T
parameter compiled from all--measurement points
inside the boundary layer (Ju2/U > 5%).
Also in this system, however, the transport
equation for the~ross-flow stress (equation (4))
looses the term v dW/8Y since by definition dW/BY
is zero. The cross-flow stress and the lag of the
shear-stress vector are therefore determined
entirely by the unknown pressure strain and the
neglected convection ~ d diffusion terms which
in the absence of v2aW/aY are likely to be
important. The second reason for the failure is
the way in which the pressure strain terms are
usually modelled. It is simple to show that
(without the boundary-layer approximation) the
pressure strain and pressure diffusion can be
combined into a single term with the form,
. . .
~ WAX +u Spy (7)
in the uv transport equation, and
-v ,a~Z +w pappy (8)
in the vw transport equation. These terms
obviously cannot be modelled by substituting for
B~'/6X, Bp'/3Y and Bp'/8Z from the Navier-Stokes
equations or an approximation to them (as done by
van den Berg (1982)) since this will lead to an where
identity or an expression of the error in the
approximation. By the same argument, substituting
for p' using the Poisson equation (which is just
the Navier Stokes rearranged) or an approximation
(as is done in effect by Rotta (1977)) must also
eventually lead to an identity or an expression
of the error in the approximation. Neither of and
these approaches are therefore valid.
The pressure strain terms can in general
only be modelled by substituting different
moments of Navier Stokes equations and/or by
using information derived from experiments.
In our opinion only the latter approach is likely
to prove successful since higher moments of the
Navier-Stokes will only introduce more unknowns.
Careful experiments in which the pressure strain
terms are (presumably) measured by difference are
therefore needed.
Turbulence models
In this section the assumptions of several
prescribed eddy viscosity models, the k-e model
an algebraic-stress model and Bradshaw's (1971)
model are tested. For each model, predictions of
the magnitude of the shear-ntress vector from the
measured mean-velocity or turbulence kinetic
energy distributions are compared with
measurements. For models that use the eddy-
viscosity at the shear-stress magnitude is assumed
to be given by
1pl ~ (~j2+~2)~ Vt[( aaU)2+( ~aW)2] 2 (9)
i.e. the cross-flow and streamwise eddy
viscosities are assumed equal. Although the
results of the previous section show that this is
not the case, there appears to be no better
alternative. Since skin-friction data are not yet
available for the present flow the wall
treatments employed by most of the models are not
tested and are ignored in the following
discussion. Comparisons with experimental data do
not include points in the near-wall region
Y/T < .02 (y+ less than about 120).
The authors concede that several of the
models examined here were never intended for use
in flows as complex as this one. However, they,
or models like them, are often used in complex
flows. It is therefore important that their
limitations be known.
In the first and simplest turbulence model
considered here the eddy viscosity is prescribed
as a function of Y entirely in terms of mixing
length '1',
1 - BY Y/6 < A/~
1 ~ 18 Y/8 2 A/x
where the eddy viscosity is given by
vie - 12[( BU)2+( aW)2] ~
(10)
( 11)
A and K are empirical constants and ~ is the
boundary-layer thickness. From two-dimensional
test calculations Patankar and Spalding (1970)
suggest A = .09 and K ( the van Karman
constant) - .435.
The Cebeci-Smith and Johnston-King
turbulence models, described for three-
dimensional flows by Abid (1988), are variations
on this basic form. The Cebeci-Smith model is
described by the relations
V ~ - V t (1 - exp(-v~i/v`O)) (12)
V ~ 12~( ~U)2~( ~W)2] 2 (13)
1 - lCY, 1C - . 4
a
V t - O . 0168ykll(Q.-p)dyl (14)
o
Yk ~ [1+5.5(~)6] 1 (15)
Q jU2+w2 Qua - Qly-a (16)
the principal difference with the basic model
being the explicit prescription of the eddy
viscosity in the outer region in terms of the
Klebanoff intermittence function Yk and the use of
a smoothing function between the inner and outer
regions. The Johnson-King model uses the same
smoothing function but def ines
821
OCR for page 822
V ~ ~ 1 ( _ ) 2
1 - Icy, tic - . 4
and
where
v t - 0.ol680ykli(0e-Q)dyI (18)
o
p
m ~( ~;2 +~;j2 ) 2 I m ( 19 )
the maximum turbulence shear stress in the
profile. The Johnson-King model was originally
designed for two-dimensional adverse pressure
gradient and separated boundary layers in which
the maximum shear stress appears to be an
appropriate scaling parameter. In the turbulence
model Am is determined from a differential
transport equation. The parameter ~ is chosen so
that the equation
am as V [ ( )2+ ( ey) 2] (20)
is satisfied at the location of maximum shear
stress in each profile. This requires an
iterative procedure.
The above three models were used to
calculate the turbulence shear-stress magnitude
from the measured mean-velocity field. In the
case of the Johnson-King model the maximum shear
stress and its location were also provided from
the experimental data.
The k-e model is one of the most widely
used in calculating two-and three-dimensional
turbulent flows. Coupled with the wall treatment
of Chen and Patel (1988) it has been used by Deng
(1990) to calculate the flow past a wing-body
junction very similar to that studied in the
present experiments. The k-e model defines the
eddy viscosity in terms of the turbulence kinetic
energy k and the dissipation c,
k2
V t ~ Cp - (21)
k and ~ are determined from approximate transport
equations (see Rodi (1984) or Abid and Schmitt
(1984)),
Dk _ - Uv aU _ ~ aw ~
+ any E ( v + o t ) sky ]
a [( ~' aft (23)
(22)
The empirical constants are usually given the
values,
C~-O.O9, ok-1, Ce1-1 57, 24
CC2-2 O. a~ 1.3 ( )
The ~ equation could not be tested using the
present measurements. The k equation was tested
by substituting the eddy-viscosity and the
velocity gradients for the Reynolds stresses and
substituting equation (21) for the dissipation.
This gives,
(17) Dt ~ v`[t~y)2~(~y)2]_ ~(25)
+ aids+ V t) Ok]
Using the measured distributions of k and the
mean-velocity gradients this equation was solved
iteratively for vt. Initial and boundary values
for at required for this calculation were
determined from the measurements. Convection of k
normal to the LDV measurement planes was ignored
in this calculation since it could not be deduced
from the measurements. This term was almost
certainly negligible at most points.
The algebraic stress model uses transport
equations for k and ~ similar to those above.
However, instead of relating the turbulent
stresses to k and ~ through an eddy viscosity,
algebraic equations for the individual stresses
are used. These are derived from the full
(differential) stress transport equations by
assuming, among other things, that the convection
and diffusion of the individual stresses is
proportional to that of k. The algebraic stress
equations, written in standard tensor notation in
their full form (Rodi (1984), Abid and Schmitt
(1984)), are
En and. Cl+P/~-l
where Pj. is the production of ujuj, P is the
production of k, djj is 1 of i=j and zero
otherwise, and ~iw is a component of the
pressure-strain correlation that accounts for
wall proximity effects. ~iw is an algebraic
function of only k, a, ~ and the stresses
themselves (see Abid and Schmitt (1984)). The
following values for the empirical constants,
suggested by Abid and Schmitt (1984) and Launder
(1982), were used
y-0.55, C1-2.2, C/-0.5, ~-0.3 (27)
C'1 and C' appearing in the equations for Id
Equation (26), together with the definition of of,
give seven algebraic equations for the six
Reynolds stresses and ~ in terms of k and the
mean-velocity field. If the latter are provided
from experimental data then the stresses and ~
can be deduced. This requires an iterative
Newton-Raphson procedure since the equations are
non-linear. Combining uv and vw then gives the
magnitude of the shear stress. In performing this
calculation production terms associated with
gradients of V and W normal to the measurement
planes were ignored since they could not be
obtained from the measurements. These terms were
almost certainly negligible at most points. Note
that equation (26) does not involve boundary
layer approximations. Without these the algebraic
stress model can, at least in theory, predict a
lag or lead in the angle of the turbulence shear-
stress vector.
Bradshaw' s ( 1971 ) turbulence model for
three-dimensional boundary layers uses
approximate dif ferential transport equations for
uv and vw. These are derived by analogy with the
transport equation for k assuming a simple
constant of proportionality between k and the
shear-stress magnitude,
a tu~+vw2) 2 (28)
822
OCR for page 823
By analogy with two-dimensional flows Bradshaw
suggests a value of 0.15 for al. Bradshaw's model
was tested simply by multiplying measured values
of k by 0.15 to obtain estimates of the shear
stress magnitude.
Results
~ turbulence model
calculations are presented in figures 8 and 9.
Figures 8(a) through 8(i) show measured and
computed contours of shear-stress magnitude in
plane 8 located towards the trailing edge of the
wing (figure 2). Because of space limitations
detailed comparisons in other planes are not
presented. However, figures 9(b) through 9(i)
show, for each turbulence model, histograms of
the ratio of computed to measured shear-stress
magnitude compiled from data in all planes. For
reference figure 9(a) shows a probable histogram
of experimental error in the measured values of
the shear-stress deduced from uncertainty
estimates. Note that figures 9(a) through 9(g) do
not contain data from close to th~-wall (Y/T <
.02) or from in the free stream (Vu2/U ~ 5%).
Of the prescribed eddy-viscosity models
(figures 8 and 9 (b) through (e)) the Cebeci
Smith appears to be the best. Although there are
some obvious qualitative differences in the
shapes of the measured and computed shear-stress
contours in plane 8 (compare figures 8(a) and
(d)) these do not represent large quantitative
differences at most points. In other planes there
are some large differences, however, as indicated
by the histogram in figure 9(d). According to
this histogram the r.m.s. error in predictions
with the Cebeci-Smith model is about 70% while
the mean error is only +3%.
Clearly the worst of these three models is
the Johnson-King which produces an unrealistic
shear-stress field in the vortex (figure 8(e)).
This model fails because the eddy viscosity
distribution it prescribes depends not just on
the peak shear-stress magnitude but also,
implicitly, on the distance from the wall at
which it occurs. Moving across plane 8, or any
other plane through the vortex, the peak jumps
from the near-wall region to the center of the
vortex producing a sudden and unrealistic change
in the prescribed eddy-viscosity profile. There
are also problems with the smoothing function
used in this model, equation (12). This function
requires at ~ vj at the maximum shear-stress
location, a condition not always met near the
center of the vortex. Note that the histogram of
calculated to measured shear-stress magnitude for
the Johnson-King model (figure 9(e)) is
misleading since it, and the mean and r.m.s.
errors stated on it, do not include many points
where the computed shear-stress magnitude
exceeded 4 times that measured. Also, the peak
near 1 in this histogram does not necessarily
represent any accuracy in the model since the
maximum shear-stress magnitude and its location
were provided to the model from the experimental
data. The model is therefore bound to produce
accurate estimates of the shear-stress magnitude
at and near this point.
Although the shear-stress magnitude
distributions produced by the basic mixing length
model of equations (10) and (11) appear
qualitatively realistic (figure 8(b)), the
histogram in this case (figure 9(b)) shows large
quantitative discrepancies (mean and r.m.s.
errors +57% and 97% respectively). A detailed
comparison with the measurements shows that most
of the larger discrepancies occur in the near-
wall region where the mixing length is prescribed
a" a linear function of Y with slope K = 0. 435
(equation (10)). This suggests that a different
value of K might improve the predictions. Figures
8(c) and 9(c) show that some improvement is
achieved by optimizing this constant to 0.3. The
Results for the
r.m.s. error with ~ = 0.3 is still 86% however.
Unlike the above models the k-e, algebraic
stress and Bradshaw's (1971) model were given the
measured distribution of turbulence kinetic
energy from which to calculate the shear-stress
magnitude. It is surprising then that these
models appear to perform little better (see
figures 8 and 9 (f) through (i)).
The k-e model does not accurately reproduce
the features of the shear-stress field in plane 8
(figure 8(f)), the vortex being much flatter and
the point of maximum shear-stress magnitude
occurring much closer to the wing than in the
measurements. As indicated by the histogram
(figure 9(f)) the k-e model also does poorly in
the other planes, the mean and r.m.s. errors in
its predictions being +58% and 83% respectively.
Bradshaw's model produces slightly better
qualitative agreement in plane 8 but over
estimates the shear-stress magnitude at most
locations (figure 8(h)). This model seems unable
to account for the low shear stress levels in the
region adjacent to the wing where the vortex is
bringing low-turbulence fluid down close to the
wall. As shown by the histogram (figure 9(h)) the
shear-strens magnitude is also over-predicted in
other planes, the mean and r.m.s. errors being
+63% and 72% respectively. These errors can be
reduced somewhat by optimizing the value of a
(see figures 8(i) and 9(i)). A value of 0.12
seems to best suit the present data set reducing
the r.m.s. error to 68%. Much of this remaining
error results from over-estimation of the shear
stress magnitude in the low turbulence region
between the vortex and wing, which still
persists .
The algebraic stress model, perhaps the
best of these three, produces the most realistic
shear-stress contours in plane 8 (figure 8(g))
especially away from the wall. Overall
comparisons with shear-stress measurements in
other planes (figure 9(g)) give mean and r.m.s.
errors of +12% and 75%. Theoretically, at least,
the algebraic stress model is capable of
predicting leads and lags in the angle of the
shear-stress vector relative to the mean-velocity
gradient vector. As shown in figure 10, lag angle
predictions appear much smaller than, and largely
uncorrelated with, measured angles.
1
5
&
u
In
O
1
4~
-
- 5
- 1
n
~ 0 0 ~
lo lo
lo
lo
-1 - 5
1 o
lo
O 0~
lo
,,,, [ ]
i
n
lo
tan(at ~ as) measured
0
lo
To $oo°
] or ~ ~ o o
o to
lo
lo
lo
5 1
Figure 10. Comparison of angles between the
turbulence shear-stress and mean-velocity
gradient vectors measured and computed using the
algebraic stress model.
823
OCR for page 824
3
7
6
5
4
3
2
a .
.7
. B _
.6
1.2 1 .B .6 .4 .2
S / T
(c) Basic mixing-length model ~ = .3
Figure 8. Contours of measured and computed
turbulence shear-stress magnitude V(~+~)/Uref2
in plane 8.
7000
6000
5000
4000
\ 1 2000
_ ~
1000
,,7
~_-~ . ~
t P
in
-
Irk
V)
to
D
z
_ ~ ~64~ \
~ ~ l
2 - ~ ,.
=~:
V V
t . 2 ~. ~ . 6 . 4 2 ~a
(b) Basic mixing-length model ~ = .435
_. _ _ .. . . . ..
Il!,e~l - into
Standard dev~ati0~4 = 0 0q7
0
1~ t. ........
lo .s 1 15 2 2.5 3
., ~-
3~ ~
(a) Measurements (a) Probable histogram of error in shear-stress
magnitude measurements.
~-
Mean = I .56
Standard deviation = 0q7
(b) Basic mixing-length model K = .435
Mean = I . 21
Jard deviation = ~ 86
2
]
4
I_
3 3.5 4
(c) Basic mixing-length model K = . 3
Figure 9. Histograms of the ratio of computed to
measured turbulence shear stress magnitude
compiled from co baritone at points inside the
boundary layer (Vu /U > 5%) but outside the near-
wall region (Y/T > .02).
824
OCR for page 825
-
. B _
.6 _
.5 _
S .
.6
.5
.2
. ~
(d) Cebeci Smith model
~ D
.e
(e) Johnson-King model
B _
.7
.6 _
.5
.4
1.2 1 .8 5 / T .2
(f) k-e model
. 1
v
Figure 8. Contours of measured and cO2mputed2
turbulence shear-stress magnitude ](uv +vw )/Uref
in plane 8.
80
70
60
50
4(
2(
(d) Cebeci Smith model
60
I 50
i
1
40
cat
us
TO 30
a,
A 20
10
n
~ 1 o ~
_
60
50
in
-
Q
~ 40
U7
to
~ 30
a,
D
go
20
10
a
Mean - 1.03
Standard deviation = (570
l
11 Hi r
2.5 3 3.5 4
Meall = 1.12
Standard deviation = C)-7z
rT ~ r..~
~ 1.5 2 2.5 3 3.5 4
(e) Johnson-King model
l
Mean = ~ .58
Standard deviation = 0-83
1
l
1 ill
0 .5
(f) k-e model
Figure 9. Histograms of the ratio of computed to
measured turbulence shear stress magnitude
compiled from comparisons at points inside the
boundary layer (VuZ/U > 5%) but outside the near-
wall region (Y/T > .02).
825
OCR for page 826
. B
.7
6
.5
. 4
. 3
.2
. ~
B
.6 _
.5 _
~ ' _
50
7 t
4 _
U 1
1.2
Me~n = 1.12 1
n(larn nP ~'~ t I nn = n 7'i
\
~ s -~
G:
~_=4 'w~ ~ ~ ~ ~,~,3
1 .6 5 / T .2 ~o 5 1 1.5 2 2.5 3 3.5 4
(g) Algebraic stress model
v,
q)
a,
-
. 4
(h) Bradshaw's (1971) model, a1 = .15.
B
. 7
.6:
.5 _
. 4 _
> _
.3
. ~
o
V
0 ~.5
60
50
40
~0
20
10
O
v,
-
u)
~ ~ ~0
~ z
1.2 1 .8 .6 ._
S / T
90
~0
70
60
50
40
30
20
10
o
(i) Bradshaw's (1971) model, a1 = .11. (i)
Figure 8. Contours of measured and cO2mputed2
turbulence shear-stress magnitude V(uv +vw )/Uref
in plane 8.
(g) Algebraic stress model
Me~n = I .62
StandaPt deviation = 07
~h~
3 3.5 4
(h) Bradshaw's (1971) model, a1 = .15.
.
Mean = I .29 i
Standard deviation = 068
.5 1
.~.~
2 2.5 3 3.5 4
Bradshaw's (1971) model, a1 = .11..
Figure 9. Histograms of the ratio of computed to
measured turbulence shear stress magnitude
compiled from co ~arisons at points inside the
boundary layer (Vu /U > 5%) but outside the near-
wall region (Y/T > .02).
826
OCR for page 827
In summary, it appears that more complex
turbulence models do not necessarily do better
than simpler ones. It could be argued that, out
of the 6 models tested here, the Cebeci Smith is
the best. Despite the fact that the Cebeci Smith
uses a prescribed eddy viscosity profile intended
for much simpler flows than the present, its
predictions of shear-stress magnitude from the
mean-velocity field alone are on the whole better
than those of the more complex models. This
implies, contrary to conventional thinking, that
the more complex models are no more general.
CONCLUSIONS
New velocity measurements, made in the flow
past a wing body junction, have been presented.
Combined with earlier results these show the
formation and development of the horseshoe vortex
and its three-dimensional turbulence structure
around the entire wing.
A number of turbulence models have been
examined using these data. Many of these models
require or imply a relationship between the
angles of the turbulence shear-stress and mean-
velocity vectors. In the present flow these
angles are not only different but do not follow
any simple relationship. To predict the shear-
stress angle, accurate modeling of the full
shear-stress transport equations is clearly
needed. In particular, new models based on
measurements are needed for the pressure-strain
term.
The assumptions of several prescribed eddy
viscosity models, the k-e model, an algebraic-
stress model and Bradshaw's (1971) model have
been tested. For each model, predictions of the
magnitude of the shear-stress vector from the
measured mean-velocity or turbulence kinetic
energy distributions were compared with
measurements. The Cebeci-Smith eddy-viscosity
model is among the best. The algebraic stress and
Bradshaw's model also do well but appear to gain
little from their relative complexity. Other
models, particularly the Johnson-King, are not
well suited to this vertical flow.
All the experimental data presented and
referred to here is available in tables and on
magnetic disc from the authors.
ACKNOWLEDGEMENTS
The authors would like to thank Dr. S.
Olamen and Mr. A. Obst for their help in taking
some of the above measurements. This work was
sponsored by NAVSEA through NSWC contract N00014-
87-K-0421.
REFERENCES
Abid R. 1988, "Extension of the Johnson-King
turbulence model to the 3-D flows", AIAA paper
88-0223, 26th Aerospace Sciences Meeting, Reno,
Nevada, January 11-14.
Abid R and Schmitt R. 1984, "Critical examination
of turbulence models for a separated three
dimensional turbulent boundary layer", Rech.
Aerosp., No 6.
Ahn S. 1986, "Unsteady Features of Turbulent
Boundary Layers", M.S. Thesis, Dept. of Aerospace
and Ocean Engineering, VPI&SU.
Bradshaw PP, 1971, "Calculation of three-
dimensional turbulent boundary layers", Journal
of Fluid Mechanics, vol.46, p.417-445.
Chen H C and Patel V C, 1988, "Near-wall
turbulence models for complex flows including
separation", AIAA Journal, vol. 26, p.641-648.
Deng G. 1989, "Resolution des equations Navier
Stokes tridimensionelles. Application au calcul
d'un raccord plaque plane-alla", PhD thesis,
Universite de Nantes, France.
Devenport W J and Simpson R L, 1990a, t'Time-
dependent and time-averaged turbulence structure
near the nose of an wing-body junction", Journal
of Fluid Mechanics, vol. 210, pp 23-55.
Devenport W J and Simpson R L, 1990b, "An
experimental investigation of the flow past an
idealized wing-body junction: preliminary data
report", AOE Dept., VPI&SU.
Devenport W J and Simpson R L, 1988a, "The
turbulence structure near an appendage-body
junction", 17th Symposium on Naval Hydrodynamics,
The Hague, The Netherlands.
Devenport W J and Simpson R L, 1988b, "LDV
measurements in the flow past a wing-body
junction", 4th International Symposium on
Applications of Laser Anemometry to Fluid
Mechanics, Lisbon, Portugal.
Devenport W J and Simpson R L, 1988c, "Time-
dependent structure in wing-body junction flows",
Turbulent Shear Flows 6, Springer Verlag.
Devenport W J and Simpson R L, 1987, "Turbulence
structure near the nose of a wing-body junction",
AIAA paper 87-1310, AIAA 19th Fluid Dynamics,
Plasma Dynamics and Lasers Conference, Honolulu,
Hawaii.
Devenport W J and Simpson R L, 1986, "Some time-
dependent features of turbulent appendage-body
juncture flows", 16th Symposium on Naval
Hydrodynamics, Berkeley, California.
Durst F, Melling and Whitelaw J H, 1981,
Princinles and Practice of Laser Do~pler
Anemometry, NY: Academic Press.
Fernholz H H and Vagt J D, 1981, "Turbulence
measurements in an adverse pressure gradient
three dimensional turbulent boundary layer along
a circular cylinder", Journal of Fluid Mechanics,
vol. 111, p233.
Johnston J P, 1970, "Measurements in a three-
dimensional turbulent boundary layer induced by a
swept forward-facing step", Journal of Fluid
Mechanics, vol. 42, p823.
Launder B E, 1982, "A generalized algebraic
stress transport hypothesis", AIAA Journal, vol.
20, p. 436-437.
Patankar S V and Spalding D B, 1970, Heat and
Mass Transfer in Boundary Layers, Second edition,
Intertext, London.
Rodi W, 1984, Turbulence Models and Their
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Netherlands.
Rotta J C, 1977, "A family of turbulence models
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van den Berg B, 1982, "Some notes on three-
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827
OCR for page 828
van den Berg B and Elsenaar A, 1972,
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DISCUSSION
Fred Stern
The University of Iowa, USA
Is there a reason that the Baldwin-Lomax turbulence model was not
chosen for evaluation? As I'm sure you are aware, this model is the
workhorse of the aerospace industry and also used extensively by the
Navy laboratories and others. Also, was it possible through the
comparisons to reach any conclusion with regard to the quasi-steady
assumption, which is made in most current turbulence models?
AUTHORS' REPLY
The Baldwin-Lomax model is identical to the Cebeci-Smith model in
the inner region of an attached two~imensional turbulent boundary
layer. The form of the outer region model is also identical to the
Cebeci-Smith model, with the differing length scales. Stock and
Haase, AIAA Journal, Vol. 27, pp. 5-14, 1989, show that the
Cebeci-Smith model performs better than the Baldwin-Lomax model
for the two-dimensional flows tested.
828
Representative terms from entire chapter:
turbulence models
