Transactions on Modelling and Simulation vol 30, © 2001 WIT Press, www.witpress.com, ISSN 1743-355X
A new hybrid scheme for turbulent flow
calculations
Y. Noguchi', M.A. Humayunl, T. Shiratori2
I
Scl?ool of Aeronautical Engineering
University of Salford, Greater Manchester, U K
' ~ e p a r t m e nof
t Aerospace Engineering
Tokyo Metropolitan Institute of Technology, Japan
Abstract
A new hybrid scheme to calculate turbulent flows is presented here. The
scheme was first reported during the early stages of its development. In this
report, more detailed test results are presented. T h e scheme uses three
turbulence models of which each of them is switched on at appropriate flow
regions rather than using unnecessarily single complex model for the whole
flow region. The tests on two-dimensional transonic flows over an aerofoil
are carried out. The scheme showed n o deterioration o f accuracy in the
results and 30 to 40% decrease in computing time when compared with using
k-E model alone. This clearly demonstrated the significant potential for the
scheme.
1 Introduction
Most of the flows for engineering applications are turbulent. There are many
types of turbulence models for Computational Fluid Dynamics (CFD). Some
of them are simple but many of them are highly complex. Complex models
require more computing time and memory. It is also quite common that one
particular model performs better for the specific flow conditions such as
separated flows and boundary layers. F o r example k-E model is known to
perform well in wake regions but does not perform particularly well to predict
shock locations. Also a simple model is good enough for regions with
favourable pressure gradient.
Transactions on Modelling and Simulation vol 30, © 2001 WIT Press, www.witpress.com, ISSN 1743-355X
If a particular model can be chosen a t a specific area and another for a
different flow regions. the computing time and memory size may be reduced
with minimum reduction in the accuracy of computation. In addition to the
initial study [l]. a hybrid scheme using three turbulence models is presented to
calculate flows around an aerofoil.
2 Test cases
The flows around single element R A E 2822 aerofoil section at high subsonic
and transonic speeds are used for the tests. The grid is 60 by 241 points as
shown in figure 1. The flow without a shock is called Case1 and the case with
a shock is called Case2. The angle of attack is increased from 3.2 degrees for
Case2 to 4.2 degrees for Case3. There are no experimental data available for
Case3. This case is to observe the behaviour of the hybrid code. The flow
conditions are as follows.
Case 1
Case 2
Case 3
Mach No.
0.676
0.750
0.750
Angle of attack
2.4
3.2
4.2
Reynolds No.
5.7~10~
6.2 X 106
6 . 2 106
~
Transition (xic)
0.1 1
0.03
0.03
-
Table 1 Test cases
3 Methods
3.1 Numerical scheme
The two-dimensional compressible thin-layer approximation of the Reynolds
averaged Navier-Stokes equations in conservation law form in general
curvilinear coordinates are
where U is the vector of conservative variables. E & F are flux vectors, S is
the viscous & heat conduction vector, t is the time, Re is Reynolds number
and 5 & q are the curvilinear coordinate directions. The equations are solved
by the Warming-Beam [ l ]explicit time-marching scheme. T o d a m p high
frequency oscillations, fourth order numerical dissipation terms are added.
The far field boundary is sufficiently far away from the aerofoil to allow the
free stream conditions to be used. The non-slip conditions are used on the
aerofoil surface.
Transactions on Modelling and Simulation vol 30, © 2001 WIT Press, www.witpress.com, ISSN 1743-355X
3.2 Turbulence models
The algebraic turbulence model due to Baldwin and Lomax [3] is one of the
most widely used models for applied compressible flow computations because
of its sin~plicity(BLM). The model has been applied t o a variety of flows
including separated flows. However the method incorporates no "history
effects" and may not be adequate for regions of large pressure gradients.
Johnson and King [4] [5] developed a turbulence model which added some
"history effects7' to the simple algebraic eddy viscosity model (JKM). An
ordinary differential equation is introduced to relate the maximum Reynolds
shear stress (defined as - u'v' but true Reynolds shear stress is
development in the streamwise direction to account for convection and
diffusion effects. The model used here is a slightly modified version [ 6 ] .In this
version, the function F(y), which is used in BLM. is used to avoid defining the
boundary layer thickness.
K-& model (KEM) is one of the standard models in C F D . The version by
Wilcox [7] is used for this study. Menter [8] found that K E M predicts the
shock position downstream of the actual shock location. J K M on the other
hand predicts shock location much more accurately than BLM or KEM
3.3 Manual Hybrid Scheme (MHS)
The combined code uses all three turbulence models. The borders between the
models are defined manually for testing purposes. BLM is used for the
regions with favourable and moderate adverse pressure gradients. This is
because the model is the simplest and therefore is the least demanding in
conlputing. The model still performs well in those areas.
J K M is used at the regions with strong adverse pressure gradients and at
the shock-induced separated areas. The model requires 20'%1 more
computation time than BLM. However it has been proved that the model
performs well with shocks [4] [5].
The wake region is calculated with KEM since it does not require the
definition of the length scale. KEM requires 100 more computing time than
BLM. BLM and J K M require the definition of the length scale. This
requirement can be difficult to achieve accurately if the wake pattern is
complex. Accurate prediction of wake patters may play an important role
such as multi stage cascade flows in turbines and compressors as well as
helicopter rotors.
For Case1 the boundary between BLM and J K M is at x/c=0.7, x/c=0.5 for
Case2 and Case3. The settings up of the boundary and initial conditions
between the different models are the key element of this next stage of the
development. Figure 2 shows the boundaries between the turbulence models
for the current tests.
Transactions on Modelling and Simulation vol 30, © 2001 WIT Press, www.witpress.com, ISSN 1743-355X
58
Conzputational Mrthods and E.yer.inle,ttal Measures
Fig. 2 Boundaries of turbulence models
3.4 Automatic Hybrid Scheme (AHS)
Automatic hybrid scheme switches the boundaries between the turbulence
models automatically with set criteria. At this stage of the development work,
the boundaries between BLM and J K M are moved. T h e switching between
BLM a n d J K M is set with the pressure gradient. When the pressure gradient
is positive, BLM is used. J K M takes over when pressure gradient becomes
negative. K E M is used for the wake region. Further development work is
necessary to change turbulence models in any order.
4 Results and discussion
Figures 3 shows the pressure coefficient of Casel. All three turbulence models
agree well with the experimental results [g]. Figure 4 shows the pressure
distribution for Case2. There are some discrepancies in the results. BLM
predicted the shock position farthest away from the experimental results as
expected from earlier studies. The shock predicted by K E M is slightly
upstream of BLM. As expected J K M predicted the shock position closest to
the experiments, once again as expected from the previous studies [ l ] [6].
Figure 5 shows the pressure coefficient of Case2 a n d Case3 with BLM.
Experimental data for Case3 is not available. The shock wave of Case 3 is
slightly downstream of Case2 as expected for the increase in the angle of
attack.
Figure 6 shows the momentum thickness development for Casel. The
results of BLM and K E M are more or less identical t o each other. The results
predicted by J K M are much closer t o the experimental results. T h e
Transactions on Modelling and Simulation vol 30, © 2001 WIT Press, www.witpress.com, ISSN 1743-355X
Computational Methods and E.rperimenra1 Measures
59
momentum thickness development for Case2 is shown in figure 7 . The results
of BLM and KEM are again identical to each other. The results predicted by
J K M are also much closer to the experimental results.
The comparisons between the momentum thickness development of Case2
with BLM, JKM, KEM and M H S are seen in figure 8. As is in figure 7 BLM
and K E M overlap each other and are difficult to distinguish. The results of
J K M and M H S are also almost identical. M H S uses BLM up to x/c=0.5. In
this region BLM and J K M produce almost identical results because the
effectiveness of the ordinary differential equation in J K M must be very small
in this favourable pressure gradient region. The downstream half of M H S
uses J K M . It is therefore expected that the results of M H S are near identical
to J K M . Using KEM in the wake region does not seem to affect the flow over
the aerofoil section.
Figure 9 shows the momentum thickness of M H S and AHS. The results
are more or less identical. Since AHS is designed to switch turbulence models
near M H S boundaries, this figure confirmed t h a t A H S is working
satisfactorily.
Figure 10 shows the momentum thickness development of Case2 a n d
Case3. This is to test the capability of AHS without adverse effects in the
results. When the angle of attack is increased the location of the boundary
between BLM and J K M moves. However there is n o noticeable abnormality
in the results. This confirms that the scheme is working satisfactorily.
The computing times are compared in table 2. When compared with BLM,
J K M requires 23 to 27% longer computing time. KEM requires 108 to 110%
longer than BLM. The increase of 46 to 50'Y" in computing time is recorded
for M H S and AHS. This is approximately 201)/;,more than J K M but 60 t o
70% less than KEM.
Model
BLM
I
1
Case1
1.00
I
1
Case2
1 S3 (1.00)
2.29 11 SO)
AHS
p
-
P
Table 2 Computing time comparison
This new hybrid scheme showed a great potential of significantly reducing the
computing time without a compromise in accuracy of the results.
5 Concluding remarks
A new hybrid scheme t o calculate turbulent flows has been made. T h e
purpose of this initial study is t o establish the new scheme t o work. This
Transactions on Modelling and Simulation vol 30, © 2001 WIT Press, www.witpress.com, ISSN 1743-355X
60
Conzpumtioual Methods and Esperitnerltal M ~ a s u l . ~ s
hybrid scheme using multiple turbulence models works with considerable
savings in computing time but little sacrifice in accuracy. The principle of this
new scheme obviously works. Potentially any type and number of turbulence
models can be included in the scheme. However these tests have not shown
full potential capability of the scheme. Further development work to exploit
potential capability is necessary. The authors believe that the scheme should
be most effective in unsteady flows.
6 References
[l] Noguchi. Y. Humayun, M.A. and Shiratori, T . , Hybrid approach for
turbulent flow calculations, Computational Methods and Experimental
Measurements IX. WIT Press, pp. 205-212, 1998
[2] Warming. R.F. and Beam. R . M . , Upwind second order difference scheme
and applications in aerodynamics flow, A I A A Journal, Vol. 14, No. 9, pp.
1241-1249, 1976.
[3] Baldwin. B.S. a n d Lomax. H., "Thin Layer Approximation and
Algebraic Model for Separated Turbulent Flows". A I A A Paper 78-0257,
1978.
[4] Johnson, D.A. and King, L.S., "A Mathematically Simple Turbulence
Closure Model for Attached and Separated Turbulent Boundary Layers",
A I A A Jour~zal,Vol. 23. p. 1684. 1985.
[5] Johnson, D.A., "Transonic Separated Flow Predictions with an EddyViscosity/Reynolds-Stress Closure Model", A I A A Jourrzal, Vol. 25, p.
252, 1987.
[6] Noguchi, Y . and Shiratori, T., Behaviour of the Johnson-King turbulence
model in axisymmetric supersonic flows. A I A A Jourrzul, Vol. 32. pp. 13951398, 1994.
171 Wilcox. D.C.. Turbule~zcemodeling for CFD, DCW Industries Inc., 1993.
[8] Menter, F . R . , Assessment of two-equation turbulence models for
transonic flows. A I A A Paper 94-2343. 1991.
[9] Cook, P.H.. McDonald, M.A. and Firmin, M.C.P.. Aerofoil RAE2822 pressure distributions, boundary layer and wake measurements.
Experimental data base for computer program assessment, AGARD-AR138. 1979.
Transactions on Modelling and Simulation vol 30, © 2001 WIT Press, www.witpress.com, ISSN 1743-355X
'igure 1 Part of grid
l
1
Figure 3 Pressure coefficient for Case 1
BLM
m JKM
r KEM
X
L
Figure 4 Pressure coefficient for Case2
EXP
Transactions on Modelling and Simulation vol 30, © 2001 WIT Press, www.witpress.com, ISSN 1743-355X
62
Conlputatiotlal Methods a t d E.vperinlel~talMeasures
Figure 5 Pressure coefficient for Case3
Figure 6 Momentum thickness for Case 1
0.0
0.5
xlc
L
Figure 7 Momentum thickness for Case2
1.0
Transactions on Modelling and Simulation vol 30, © 2001 WIT Press, www.witpress.com, ISSN 1743-355X
Conlputarior~alMethods ar~dE.\-perirner~talM ~ a s u r e s
l
xlc
l
l
Figure 8 Momentum thickness comparison with MHS
xlc
Figure 9 Momentum thickness comparison between MHS & AHS
Figure 10 Momentum thickness comparison between Case2 & 3
63