Flow, Turbulence and Combustion 63: 175–191, 1999.
© 2000 Kluwer Academic Publishers. Printed in the Netherlands.
175
Two-Equation Turbulence Modelling of a
Transitional Separation Bubble
R.J.A. HOWARD
Equipe MOST, LEGI, BP 53, INP Grenoble, 38041 Grenoble Cedex 9, France
M. ALAM and N.D. SANDHAM
Aeronautics and Astronautics, University of Southampton, Southampton SO17 1BJ, U.K.
Abstract. Calculations of the Reynolds averaged equations using two different turbulence models
have been compared with direct numerical simulation of a transitional separation bubble. Three
methods of transition modelling were investigated. The first had no transition adjustment, the second
involved fixing the transition point at the location observed in the simulation and the third was a direct
transformation of a method proposed by Wilcox [1] which involved sensitising the eddy viscosity and
transport equations to the local turbulent Reynolds number. The models captured the general features
of the flow but were unable to show the recovery behaviour of the flow behind the bubble. Reasons
for the failure are discussed using a priori analysis of terms in the model equations.
Key words: transition, turbulence modelling, separation.
1. Introduction
Separation occurs on a flat surface when a boundary layer experiences an adverse
pressure gradient strong enough to separate the boundary layer from the surface.
Since separation involves inflexional velocity profiles which are known to be unstable [2, 3] it is particularly susceptible to disturbances and laminar boundary
layers undergo a rapid transition to turbulence. Transition is characterised initially
by the growth of disturbances which may form large structures which then breakdown to turbulence (e.g., [4]). In bubble flows the turbulent flow reattaches and
relaxes downstream towards a standard turbulent boundary layer.
Spalart and Strelets [5, 6] carried out a direct numerical simulation of a transitional separation bubble along with several one and two equation Reynolds averaged model tests while Hadzic and Hanjalic [7] carried out second moment
Reynolds-averaged modelling of the same flow. In these cases the transition point
was not fixed in the modelling and the bubble length predicted was reasonably
accurate. However, the models showed problems in capturing the reverse flow and
recovery behaviour.
Papanicolaou and Rodi [8] computed separated flow transition behind a semi
circular leading edge using a two layer model with a transition sensitising inter-
176
R.J.A. HOWARD ET AL.
Figure 1. A streamwise velocity contour plot with negative an zero contours as dotted lines
and two wall normal velocity profiles showing mean, u and total u = u + u0 velocities in the
laminar and turbulent regions respectively.
mittency model. Their method seemed to compare quite well with the reference
experiments although the relative position of the end of transition and reattachment
points was not well defined in the experiments. The bubble behaviour observed in
the model solutions is however very similar to that observed in the DNS studies.
Lei [9] carried out two-equation modelling of a transitional separation bubble
with inflow turbulence values based on the free stream turbulence intensity and
found that the bubble length was strongly affected by the free stream turbulence.
In this study we used data from the Alam and Sandham [10] simulation. The
discretisation was spectral (Fourier–Chebychev) in space and Runge–Kutta in time
and solved the incompressible Navier–Stokes equations. Since the problem investigated was not periodic in the streamwise direction a fringe was applied in order
to provide the correct inflow conditions. Further details can be found in Alam and
Sandham [10]. In contrast to the Spalart and Strelets [5, 6] work these bubbles are
quite short and the recovery behaviour downstream of reattachment is a significant
part of the flowfield to be modelled. This provides a different test case for validation
of turbulence models.
In order to model this type of flow it is necessary to capture some effects of the
changes the flow goes through. Firstly, the flow is laminar with small disturbances
superimposed and use of linear stability theory (the Orr–Sommerfield equation)
177
TRANSITIONAL SEPARATION BUBBLE
is valid. Finally the flow is fully turbulent and standard turbulence models are
appropriate. Figure 1 shows a contour plot of the main region of interest in the
flow and velocity profiles at two streamwise locations. Lengths are normalised by
the inflow boundary layer displacement thickness and velocities by the free stream
velocity. The velocity profiles show both the instantaneous and mean velocities at
the locations shown by the dotted lines on the contour plot. From this figure it is
possible to see the small disturbances prior to transition and also the strongly mixed
state that is generated in the fully turbulent region. The major stumbling block as
found by many of the workers is the late stage of transition where disturbances are
large but not readily accommodated by turbulence models.
In this paper three methods of carrying out two-equation modelling for the
transitional separation bubble of Alam and Sandham [10] will be investigated. The
first is observation of the model performance with no transition treatment while the
second method involves fixing the transition point using information obtained from
DNS or experiments. The third method is the transition method outlined in Wilcox
[1] which involves sensitising the model to the local turbulent Reynolds number.
All three methods were investigated for the k–g (transformed k–ω) model while
the first two methods were investigated for the Launder and Sharma [11] model.
2. Turbulence Models
Two equation modelling makes the Boussinesq assumption that turbulent flows
have an additional viscosity due to the turbulence of the flow. The Reynolds stress
u0i u0j is a function of this turbulent viscosity νt such that
2
−u0i u0j = 2νt sij − kδij ,
3
where
1 ∂ui
∂uj
sij =
+
2 ∂xj
∂xi
(1)
and νt is a function of turbulence length scale and a turbulence velocity scale. In
two-equation turbulence modelling these scales are modelled using two turbulence
quantities for which transport equations must be solved to close the relation. The
form of these transport equations is generally similar, despite there being many
variants, and we take two examples in common use. An equation for the turbulence
kinetic energy, k, is solved
Dk
νt ∂k
∂
ν+
(2)
= Pk − +
Dt
∂xj
σk ∂xj
with either
D ˜
˜
˜ 2
∂
= C1 Pk − C2 f +
Dt
k
k
∂xj
2 2
νt ∂ ˜
∂ ui
ν+
+ 2ννt
σ ∂xj
∂xj 2
(3)
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R.J.A. HOWARD ET AL.
with C1 = 1.44, C2 = 1.92, σk = 1, σ = 1.3 and f = 1 − 0.3 exp[−Re2t ] or
Dg
νt ∂g
αg
β
∂
ν+
= −Pk
+ ∗ +
Dt
k
2β g ∂xj
σg ∂xj
νt 3 ∂g 2
− ν+
,
(4)
σg g ∂xj
where Pk = {2/3kδij − νt 2sij }[2sij ], α = 5/18, β = 3/40, β ∗ = Cµ , σk = σg = 2.
The first of these relations takes νt = Cµ fµ k 2 /˜ from Launder and Sharma [11]
2
where Cµ = 0.09 and fµ = exp[−3.4/(1 + Ret /50)√
] is a low Reynolds number
damping function and k and = D + ˜ , (D = 2ν(∂ k/∂xj )2 ) are the turbulence
intensity and dissipation respectively. This model is widely used and is often found
to provide the ‘best’ results for a range of different flows. Another common relation
is νt = k/ω of Wilcox [1] where ω represents the ratio of dissipation to turbulence
√
intensity ω = /(Cµk) or νt = cµ kg 2 from Kalitzin et al. [12] where g = k/.
This is a direct transformation of the ω relation with useful properties for near-wall
calculation. The k–ω model of Wilcox must be solved using a near wall function
for ω,
ω=
2ν
,
Cµ y 2
(5)
for grid points below y + = 2.5 where y + = uτ y/ν. This makes the k–ω model
break down to a one-equation model near the wall and presents a problem in separated flow where the use of a local y + variable as a reference becomes unsuitable
since uτ moves through zero and becomes negative. There are several multi-layer
models which use different levels of model complexity for different regions of the
flow. However, in this paper, these models will not be investigated and the k–ω
model solution will not be examined. The k–˜ and k–g models can be solved right
down to the wall without modification to a two-layer approach. The k–g model
does contain a limiter
ymin
gmin = √
,
(6)
νn(n − 1)
where ymin is the height of the grid point nearest to the wall, with n = 3.23, which
is based on the k–ω near wall function but this does not fix the near wall values
explicitly.
The eddy viscosity relation associates the turbulence in a flow with the mean
strain field. While this seems to be appropriate for fully turbulent flow, it is not very
helpful for laminar and transitional flows, as these have strain fields but low turbulence levels. An eddy viscosity model amplifies any turbulence present through
its use of the strain field. This results in over prediction of the turbulence in these
flows. One method of getting around this is to multiply the eddy viscosity by a
‘flag’ that is equal to zero in laminar flow and one in turbulent flow. This is also
TRANSITIONAL SEPARATION BUBBLE
179
known as ‘point’ transition. In order to allow some diffusion between the laminar and turbulent regions, it has been found more appropriate in practice (Gould,
private communication) to attach the flag to the production term, Pk , in the equations. This means that the turbulence quantities can diffuse in laminar regions but
no turbulence is generated. The transition point must be known in order to set the
location for the flags. This information can be obtained from DNS or experiments
where such data is available. The major deficiency is that the model itself contains
no sensitivity to actual mechanisms of flow transition.
When a turbulence model is solved for laminar flow there are several potential
numerical problems. These are all based on the fact that laminar or transitional
flows have
p zero or low levels of turbulence. This means that k ≈ 0, ≈ 0 and also
g[≡ 1/ (Cµ ω)] 6 = 0. As a result of this, any terms which are divided by either k or
, may not be defined. For the k–˜ model the eddy viscosity itself, νt = Cµ fµ k 2 /˜ ,
and the dissipation term in the ˜ equation, −C2 f (˜ 2 /k), have this problem. This
can be overcome by using methods such as ensuring that the free stream turbulence level and dissipation inflow profiles are above certain levels corresponding
to roughly 0.5–1% free stream turbulence (Savill, personal communication) or including fixes to prevent spikes in the data from occurring. However, this is not a
very satisfactory situation for modelling where robustness is often a factor used to
recommend a model. The production term in the k–g model potentially has this
problem but can be rewritten as −Pk (αg/k) ≡ −αg2sij (2/3δij − Cµ g 2 2sij ) so the
problem is eliminated.
Wilcox [1] presents a more general method for modelling transition with a twoequation model. The formulation starts with the observation from the transport
equations that the combination of the relevant production and dissipation terms in
each equation describes whether the turbulence quantity is amplified or reduced
in magnitude. Then, a series of criteria based on the minimum critical Reynolds
number, asymptotic consistency, and the observation that transition will not occur
if the production dissipation combination in the ω equation reaches zero before the
same combination in the k equation, were brought together. Wilcox postulated a
functional dependence on turbulent Reynolds number, Ret , which satisfied these
criteria while returning to fully turbulent values for Ret → ∞. The list of relations
are as follows
νt = α ∗ kg 2 ,
α∗ =
α=
α0∗ + Ret /Rk
,
1 + Ret /Rk
5 α0 + Ret /Rω ∗ −1
(α ) ,
9 1 + Ret /Rω
β ∗ = Cµ
5/18 + (Ret /Rβ )4
,
1 + (Ret /Rβ )4
(7)
(8)
(9)
(10)
180
R.J.A. HOWARD ET AL.
where α0∗ = β/3, α0 = 1/10, Rβ = 8, Rk = 6, Rω = 27/10 with local turbulent
Reynolds number Ret = Cµ kg 2 /ν. If this relation is applied to steady turbulent
boundary layer or channel flow, its presence is similar to that of a low Reynolds
number damping function based on Ret . This is because in these flows Ret 6 = ∞,
and, very near walls, Ret becomes quite small. In this respect the functions must
have the correct behaviour with distance from the wall. However, in transitional
flows, the changes in the turbulence also occur in the streamwise direction so the
same set of functions must have the correct behaviour for streamwise variations
of the turbulence as well. These relations have been inserted appropriately into the
k–g model system.
3. Numerical Method, Grid and Boundary Conditions
The model calculations make use of the unsteady RANS equations. Thus the averaged Navier–Stokes equations are solved using an Adams–Bashforth time advancement scheme with second order central differences for the spatial discretisation.
Solution of the Poisson equation is carried out using fast Fourier transforms with
modified wavenumbers to be consistent with the finite difference scheme. The grid
is staggered with velocities calculated on the face of each cell and the pressure,
turbulence kinetic energy and dissipation at the centres.
An important feature of the modelling approach is that exactly the same fringe
and suction profiles are used as in the DNS. The fringe is just as necessary in the
RANS calculation due to the streamwise periodicity of the code. Each model was
set up as a two-dimensional problem with 128 × 60 grid points in the streamwise and wall normal directions respectively with points clustered near the wall to
resolve the high wall shear.
In order to provide a suitable incoming disturbance, the turbulence kinetic energy and dissipation (or g) profile from the DNS was used as the inflow to the
model problem (along with the laminar boundary-layer profile). The DNS flow
also included a small spanwise turbulence strip which was applied by imposing
oscillating suction and blowing at a thin region upstream of the separation. This
region is modelled by using the DNS turbulence kinetic energy and dissipation (or
g) values at this location and applying these values to the wall at the same location
in the model. The location is x = 10 but the results indicate that the level of
turbulence due to this disturbance is insufficient to have a significant effect on the
models.
4. Modelling
It was found that several models had unsteady solutions. A possible explanation is
that the two-dimensional problem has a vortex shedding instability and the models
are picking this up. One can consider the unsteady Reynolds averaged equations
as transport equations for locally averaged quantities. Thus it can be physically
TRANSITIONAL SEPARATION BUBBLE
181
Figure 2. Turbulent kinetic energy distribution for the DNS and models. L–S denotes the
Launder and Sharma model, flagged means that the turbulence production is switched on at
x = 20 in the domain. The contour lines correspond to k levels of 0.001, 0.005, 0.01, 0.015
and 0.02.
correct for RANS calculations to have oscillations in u and k where the average
of these quantities corresponds to the classical definition. In order to compare the
results and evaluate models for practical applications, averaged model solutions are
shown here.
Lei [9] has also carried out two-equation model calculations of a flat plate transitional flow. In his study the Jones and Launder [13] and Launder and Sharma [11]
models were investigated. These represent essentially the same model with slightly
different constants. The steady state equations were solved and artificial dissipation
was added to the right hand side of the discrete equations to damp numerical oscillations. This type of treatment could potentially absorb the oscillations produced
in the current flow. Lei attempted to obtain a solution for another separation bubble
[14] near the bursting state, but found that the solution broke down. This could
have been caused by the steady state system of equations being unable to model an
unsteady problem.
In Lei’s work, functions based on the free stream turbulence intensity were used
to provide the inflow turbulence levels. This had important implications as it was
found that the model responses were highly sensitive to free stream turbulence
intensity. In the current work we have the benefit of a DNS flowfield from which
such data can be extracted providing a more solid base for comparison.
182
R.J.A. HOWARD ET AL.
Figure 3. Turbulent kinetic energy distribution for the k–g models. Flagged means that the
turbulence production is switched on at x = 20 in the domain and Wilcox indicates that the
Wilcox transition sensitising has been carried out. The contour lines are set to the same limits
as Figure 2.
Figure 2 shows turbulent kinetic energy contours for the DNS and the Launder
and Sharma [11] model with and without point transition (denoted as flagged) and
Figure 3 shows the turbulent kinetic energy distribution for the k–g model with
and without point transition and with the Wilcox transition functions (denoted as
Wilcox). The models with no extra transition treatment generated large amounts
of turbulence right from the inflow and their response to the adverse pressure
gradient was much weaker. Indeed no separation was observed using the Launder
and Sharma [11] model and only very weak separation occurred in the k–g model
(see Figure 4).
Introduction of point transition allowed the models to predict separation bubbles
in qualitative agreement with the DNS bubble. The Wilcox sensitised model produced higher values of turbulence downstream of reattachment which could probably be corrected for in this model problem by tuning of the additional constants for
this model. The original constants were derived for a flat plate attached boundary
layer flow so it is encouraging that they produce the right general behaviour when
introduced to a different flow.
The skin friction profiles in Figure 4 also give indications of the instabilities
in the models. The recovery region downstream of the separation is not accurately
captured by any of the models, although this is strongly affected by how well the
transition itself is modelled.
TRANSITIONAL SEPARATION BUBBLE
183
Figure 4. Skin friction profiles for the DNS and models. L–S denotes the Launder and Sharma
model, flagged means that the turbulence production is switched on at x = 20 in the domain
and Wilcox indicates that the Wilcox transition sensitising has been carried out.
Figure 5 shows the skin friction in the vicinity of the separation itself. On this
plot four streamwise locations are highlighted. These correspond to: (1) x = 7.8,
where the disturbance is small and the flow is laminar; (2) x = 25, near the point
of separation; (3) x = 35.9, within the separated region and (4) x = 50, after
transition at the early stage of recovery. Figures 6 to 9 show the streamwise velocity, turbulence kinetic energy, dissipation and turbulent Reynolds number, Ret , for
the models and the DNS at each of these stations. Although the velocity profiles
seem reasonable it is clear that there are significant problems in modelling the
turbulence quantities. The untreated models overpredict the turbulence kinetic energy and dissipation upstream of transition at stations (1) and (2), and underpredict
the same quantities after transition at station (4). Introduction of point transition
by flagging the production meant that the turbulence kinetic energy was effectively
zero upstream of transition, while the dissipation remained at high levels relative to
the DNS. Downstream of transition however, the models all underpredict the level
of dissipation. The turbulent Reynolds number at the same locations shows that
only the k–g and the flagged k–g models give reasonable results after transition.
The remaining models significantly overpredict the levels of turbulence present.
Spalart and Strelets [6] describe the mechanism for ‘natural’ transition in turbulence models as ‘transition by contact’ by which non-zero model quantities expand
due to the diffusive nature of the turbulence model transport equations. The sensitivity of the turbulence models is in general not tuned for the transition problem
and some models under certain conditions give early transition and no separation
as was observed for the SST model of Spalart and Strelets [6] and the untreated
models in this study.
184
R.J.A. HOWARD ET AL.
Figure 5. Skin friction profiles for the DNS and models in the vicinity of the separation.
L–S denotes the Launder and Sharma model, flagged means that the turbulence production
is switched on at x = 20 in the domain and Wilcox indicates that the Wilcox transition
sensitising has been carried out. The streamwise locations to be examined in more detail are
labelled (1), (2), (3) and (4).
Figure 6. Streamwise velocity profiles at four different streamwise stations corresponding to
x = 7.8 (1), x = 25 (2), x = 35.9 (3), x = 50 (4). Lines and symbols as for Figure 5.
TRANSITIONAL SEPARATION BUBBLE
185
Figure 7. Turbulence kinetic energy profiles at four different streamwise stations corresponding to x = 7.8 (1), x = 25 (2), x = 35.9 (3), x = 50 (4). Lines and symbols as for
Figure 5.
Figure 8. Dissipation profiles at four different streamwise stations corresponding to
x = 7.8 (1), x = 25 (2), x = 35.9 (3), x = 50 (4). Lines and symbols as for Figure 5.
186
R.J.A. HOWARD ET AL.
Figure 9. Turbulent Reynolds number, Ret = k 2 /ν profiles at four different streamwise
stations corresponding to x = 7.8 (1), x = 25 (2), x = 35.9 (3), x = 50 (4). Lines and
symbols as for Figure 5.
In order to provide a more conclusive demonstration of this a further calcula∗
tion was carried out in which an additional space of length 50δin
was introduced
between the inlet and the suction field for the Launder and Sharma model. This
calculation gave transition at the inlet in the same way as the original result (see
Figure 2) showing that the incoming turbulence levels alone were sufficient to
cause transition for this model.
5. Analysis of Model Results
The previous section has shown that the models have considerable difficulty in
predicting the flow field of a transitional separation bubble. In this section we
investigate possible reasons for the discrepancies. All the terms of the turbulence
kinetic energy (TKE) budget (obtained from the DNS) have been compared to
identify the significance of the terms being modelled. The model performances
are also evaluated by a priori analysis.
The full equations for the balance of the TKE can be written as,
∂k
∂k
p
u
υ
= P − − (Ji,i
+ Ji,i + Ji,i
),
+ hui i
∂t
∂xi
(11)
187
TRANSITIONAL SEPARATION BUBBLE
Figure 10. Balance of TKE obtained from DNS at locations: (2) x = 25, (3) x = 35.9 and
(4) x = 50.
where the various terms are denoted by
ii
P = −hu0i u0j i ∂hu
∂xj
D 0 h 0
1 ∂ui ∂ui
= Re
+
∂xj ∂xj
∂u0j
∂xi
iE
(Dissipation)
u
Ji,i
= hu0i u0j u0j i/2
Ji,i = hp 0 u0i i
h
1
∂k
υ
Ji,i
= − Re
+
∂xi
(Turbulence transport)
p
∂k
hui i ∂x
i
(Production)
∂hu0i u0j i
∂xj
i
(Pressure transport)
(Viscous transport)
( Convection).



























.
(12)
Figure 10 shows the balance (in arbitrary units) at the locations (2), (3) and
(4). In general the distribution in the separated region resembles that of a turbulent
mixing layer (locations (2) and (3)). Here the production and turbulent transport is
balanced by convection and the dissipation terms. The distribution in the recovery
region (location (4)) is dominated by two main processes. Firstly the emergence
of a newly formed turbulent boundary layer in the near wall region and secondly
the presence of the reattaching shear layer away from the wall. At this location the
dominant terms away from the wall are production, dissipation, pressure transport
and turbulence transport whereas at the wall dissipation is balanced by the viscous
transport. The diffusion term in equation (2) is effectively modelling the convection
p
u
υ
terms Ji,i
, Ji,i , and Ji,i
from Equation (10). Knowing the exact distribution we
188
R.J.A. HOWARD ET AL.
Figure 11. Reynolds stresses at location x = 50 (4). The models have been calculated using
DNS k and .
may carry out a priori tests by comparing the form predicted by models with the
simulation data.
Figure 11 shows the plot of the Reynolds stresses obtained from the DNS and
those calculated by k–g and L–S models (in the model equation for νt we use k
and from DNS). Both the models overpredict the DNS peak by a factor of two
and the models also show a near-wall peak which is absent in the DNS. The model
peak values reach ten times as much as the DNS peak in some locations and the
profile shapes are incorrect at the end of the recovery region. The L-S model gives
peak values closer to the DNS than k–g. Figure 12 shows the comparison of the
diffusion terms at location (4). Both models predict the DNS shape reasonably
well, with k–g following the DNS data closely. However in the separated region
(not shown) and at the end of the recovery region the peak values are many times
that of the DNS peak values. Of the two models k–g is closer to DNS than the L–S
model.
In k–g and L–S models the pressure diffusion is implicitly assumed to be negligible, but referring to Figure 10 it plays a significant part in the balance of the TKE
at location (4). One model where pressure diffusion is included is by Kawamura
and Kawashima [15], which has the form (neglecting the special boundary treatment)
∂ K ∂
πK = −0.5ν
.
(13)
∂xi ∂xi
The plot of pressure diffusion from DNS and (12) is shown by Figure 13. The
model fails to follow the DNS profiles correctly. However it should be noted that
the real value of such term-by-term improvements may only come when complete
model calculations are attempted. Savill [16] found that inculsion of the pressure
189
TRANSITIONAL SEPARATION BUBBLE
p
u and DNS-2 is −(J u + J ). The
Figure 12. Diffusion at location x = 50 (4) (DNS-1 is −Ji,i
i,i
i,i
models have been calculated using DNS k and .
p
Figure 13. Pressure diffusion at location x = 50 (4) (DNS is −Ji,i ). The model (Equation (12)) has been calculated using DNS k and .
diffusion term improved attached flow transition behaviour, reinforcing the DNS
observations for the importance of that term.
It has been observed by several workers that two-equation models are capable
of predicting transitional flows in many cases without specific transition treatment
however the transition behaviour is dependent on the turbulence model stability in
responding to non-zero turbulence values. This behaviour can be forced explicitly
by ‘point’ transition and adjustment of input turbulence levels or implicitly by
sensitising individual model terms using methods such as the f t2 in the Spalart
190
R.J.A. HOWARD ET AL.
and Allmaras [17] model, the γ intermittency term of Papanicolaou and Rodi or
the Ret terms of Wilcox. Clearly the latter approach is the one that must be pursued
but the former demonstrates that it is possible for the models to capture reasonably
accurately the correct behaviour.
6. Conclusions
DNS has been used to provide a suitable turbulent inflow field and detailed data to
evaluate model performance for a transitional separation bubble problem. Without
special treatment of transition the models of Launder and Sharma [11] and k–g
(cf. [12]) were found to generate transition well before it was observed in the
DNS. This contrasts with the majority of results for the Spalart and Strelets test
case, where the models predicted transition after the separation. Suppression of the
production of turbulence kinetic energy delayed the onset of transition and allowed
the model solutions to form a separation bubble with generally similar dimensions
to the DNS and similar behaviour to the ‘naturally tripped’ model results. This
‘point’ transition treatment requires previous knowledge of where the transition
should occur and is in effect rather like adjusting the incoming turbulence levels.
An additional method derived directly from that proposed by Wilcox, 1994, based
on local turbulent Reynolds number, Ret , was also tested on the k–g model form.
This method was found to predict approximately the correct location for transition
but over predicted the turbulence levels once transition had occurred. A priori tests
demonstrated that both the eddy viscosity and the turbulence transport terms are
poorly modelled for this flow.
The ‘point’ transition results show that the two-equation models are capable
of capturing the main flow features of this problem and the sensitising approach
demonstrates a general way in which transition behaviour can be implemented
within the models.
Acknowledgments
We would like to thank Dr A.M. Savill of Cambridge University, Engineering Department and Mr A.R.B. Gould of British Aerospace for helpful discussions and
advice. This research was funded by British Aerospace (Operations) Ltd. (Sowerby
Research Centre). Time on the Cray T3D was provided by EPSRC under grants
GR/K 43902 and GR/K 43957.
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