The objectives for this project are as follows: (1) Modeling of the near wall turbulence: We aim to develop a second order closure for the near wall turbulence. As a first step of this project, we try to develop a kappa-epsilon model for near wall turbulence. We require the resulting model to be able to handle both near wall turbulence and turbulent flows away from the wall, computationally robust, and applicable for complex flow situations, flow with separation, for example, and (2) Modeling of the bypass transition: We aim to develop a bypass transition model which contains the effect of intermittency. Thus, the model can be used for both the transitional boundary layers and the turbulent boundary layers. We require the resulting model to give a good prediction of momentum and heat transfer within the transitional boundary and a good prediction of the effect of freestream turbulence on transitional boundary layers

Yang, Z.

English

NASA Technical Reports Server

Center for Modeling of Turbulence and Transition
Research Briefs - 1991 N93-15798
Modeling of Near Wall Turbulence
and Modeling of Bypass Transition
Z. Yang
1. Motivation and Objective
1) Modeling of the near wall turbulence. We aim to develop a second order clo-
sure for near wall turbulence. As a first step of this project, we try to develop a
k - e model for near wall turbulence. We require the resulting model to be able to
handle both near wall turbulence and turbulent flows away from the wall, compu-
tationally robust, and applicable for complex flow situations, flow with separation,
for example.
2) Modeling of the bypass transition. We aim to develop a bypass transition
model which contains the effect of intermittency. Thus, the model can be used
for both the transitional boundary layers and the turbulent boundary layers. We
require the resulting model to give a good prediction of momentum and heat transfer
within the transitional boundary and a good prediction of the effect of freestream
turbulence on transitional boundary layers.
2. Work Accomplished
In the past year, progress has been made in both topics mentioned above (i.e.,
modeling of the near wall turbulence and modeling of the bypass transition). In the
paragraphs below, these two topics will be reported separately.
2.1 Modeling of Near Wall Turbulence
Because of the wide range of scales involved in a turbulent flow, DNS (direct
numerical simulation) is limited to flows of moderate Reynolds number and simple
geometry. Turbulence modeling is the only viable approach for the calculation of
turbulent flows of engineering interest. In turbulence modeling, the k-e model is the
most widely used model in engineering calculations. The Standard k - e Model 1'2
was devised for high Reynolds number turbulent flows and is traditionally used
in conjunction with a wall function when it is applied to wall bounded turbulent
flows. However universal wall functions do not exist in complex flows and it is thus
necessary to develop a form of k - e model equations which can be integrated down
to the wall.
Jones and Launder 3 were the first to propose a low Reynolds number k - e model
for near wall turbulence, which was then followed by a number of similar k - e
models. A critical evaluation of the pre-1985 models was made by Patel et al.4. More
recently proposed models can be found in Shih 5 and Lang and Shih 6. Three major
deficiencies can be pointed out about existing k-e models. (Some of the models may
have only one or two of the three deficiencies.) First, a near wall pseudo-dissipation
rate was introduced to remove the singularity in the dissipation equation at the
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https://ntrs.nasa.gov/search.jsp?R=19930006609 2020-03-17T08:38:19+00:00Z
,7,. Yang
wall. The definition of the near wall pseudo-dissipation rate was quite arbitrary.
Second, the model constants were different from those of the Standard k - e Model,
making the near wall models less capable of handling flows containing both high
Reynolds number turbulence and near wall turbulence, which often occurs for real
flow situations. Patel et al. 4 put as the first criterion the ability of the near wall
models to predict turbulent free shear flows. Third, the variable y+ is used in the
damping function f_, of the eddy viscosity formulae. Since the definition of y+
involves u,, the friction velocity, any model containing y+ cannot be used in flows
with separation.
Effort is made to propose a new k- e model for near turbulence which is free of the
three deficiencies mentioned above. In this model, k 1/2 is chosen as the turbulent
velocity scale. The time scale is bounded from below by the Kolmogorov time scale.
The dissipation equation is reformulated using this time scale and no singularity
exists at the wall. Thus, it is no longer necessary to introduce the near wall pseudo-
dissipation rate. The model constants used are the same as in the Standard k-e
Model. Thus, the proposed model will be also suitable for flows away from the wall.
An earlier version of the model, which contains y+ in the damping function, was
proposed and reported in Yang and Shih 7. The model is now improved by using
P_ = kl/2y/v instead of y+ in the damping function. Hence, the present model can
be used for flows with separation.
In the present model, the eddy viscosity is given by
VT = c_,f_,kT (1)
where the time scale T is written as
k V)ll_
T = e + (e- " (2)
The first part is the time scale conventionally used for high Reynolds number tur-
bulent flows and the second part is the Kolmogorov time scale. Away from the wall,
the first part is much large than the second part while near the wall the second part
dominates, giving the Kolmogorov time scale as the turbulent time scale at the wall.
The time scale given is bounded from below by the Kolmogorov time scaleand is
always positive.
The damping function ,f_, is given by
.f_, = [1 - exp(-al n v - a3n_ - asR_)] 1/2 (3)"
where al = 1.5 x 10 -4, a3 = 5.0 x 10 -7, a_ = 1.0 x 10 -5°. The damping function
is chosen such that the shear stress has the correct near wall asymptotic behavior.
Away from the wall, f_, approaches one as required.
The modeled transport equations for k and e are
O"k
(4)
84
Modeling o/Near Wall Turbulence and Modeling o/Bypass Transition
+ uj .j = [(v + + (-Cl, < u uj > +vvru , kU ,j . (5)
Because T is always positive, the dissipation rate equation does not have a singu-
larity at the wall.
Turbulent channel flows and flat plate boundary layer flows at different Reynolds
numbers were calculated using the proposed model. The mean velocity, turbu-
lent kinetic energy, turbulent shear stress and turbulent dissipation rate for tur-
bulent channel flow at Rer = 395 and turbulent flat plate boundary layer flow at
Res = 1410 are shown in Fig 1 and Fig 2, respectively. DNS data for these cases
are shown for comparison. Also shown are the predictions using the Jones-Launder
model and the k - _ model proposed by Chien s. These two models are chosen
because the Jones-Launder model is the first k - e model for near wall turbulence
while Chien's model is known to perform quite well for turbulent boundary layer
flows. Overall, the proposed model is found to give a better prediction. Calcula-
tions were also made for turbulent flat plate boundary layers at larger Reynolds
numbers, turbulent boundary layers with pressure gradient (favorable pressure gra-
dient, adverse pressure gradient, and increasingly adverse pressure.) The results of
these computation and the comparisons with the available experimental data can
be found in Yang and Shih 9.
2.2 Modeling of Bypass Transition
In a quiescent environment, transition is preceeded by the amplification of Tollmien-
Schlichting waves. These waves eventually break down, giving rise to turbulent
spots, which can be viewed as the onset of transition. In an environment with high
freestream turbulence, say the flow passing over a turbine blade, turbulent spots
are formed due to the transport of turbulence from the freestream to the boundary
layer rather than the T-S wave amplification. This type of transition is called bypass
transition. Accurate prediction of bypass transitional boundary layers is very im-
portant for internal fluid mechanics because a significant proportion of the turbine
blade is in the a transitional boundary layer region. Furthermore, the performance
and the life of a turbine are directly related to the peak vales of the momentum and
heat transfer both of which occur in the transitional boundary layer.
Priddin 1° was the first to notice that the low Reynolds number two equation
models have the potential to predict transitional flows under the influence of the
freestream turbulence. This is probably due to the fact that the generation of
turbulent spots in a boundary layer is a random process and the flow is almost
fully developed turbulent within a turbulent spot. A detailed calculation procedure
was given by Rodi and Scheuerer 11, in which the Lam & Bremhorst low Reynolds
number k - e model was used. More recently, a comparative study of the perfor-
mance of existing low Reynolds number k - e models in predicting laminar-turbulent
transition was made by Fujisawa t2.
While the low Reynolds number k - e models could mimic transition, the quan-
titative predictions do not compare very well with the experimental data. This is
due to the fact that all these low Reynolds number k - e models were originally
proposed for full), developed turbulent flows and did not take into consideration the
85
Z. )rang
distinct feature of a transitional boundary layer - intermittency. Tlae intermittency
of a transitional boundary layer is measured by the intermittency factor which can
be viewed as the percentage time a transitional boundary layer is in the turbulent
state due to the passing of a turbulent spot.
We propose a model for the calculation of transitional boundary layers, which
takes the effect of intermittency into consideration. The model is based on the k - •
model for near wall turbulence we have stated above. The effect of intermittency
is introduced through the following argument: since the percentage of time that a
transitional boundary layer is turbulent is measured by the intermittency factor,
a model for transitional boundary layers could be constructed from a model for
turbulent boundary layers by multiplying all the terms due to turbulence med_anism
by a weighting factor 7, which is linearly related to the intermittency factor. Thus,
the governing equations for the flat plate transitional boundary layers axe, after
using the boundary layer approximation and the eddy viscosity assumption,
OU OV
o-7+ or 0, (6)
uov yOU_O[ ou]ox + or Oy (v + _r)_ ,
u Ok V VOk 0 [ Vr ¢Ok] iS)U _+ _ = _ (v+_)_ +_vT(_) -_•,
(r)
(s)
[ --, [ _,¢OU)2 ]I.T OU_ o• _ o• o )o•l cl_.T'-_yu--+v--= (,.,+-r ''r +'r -c,.,• +-r._,r(-8_y)_,(o)Oz Oy Oy a, OyJ
where x, y are the coordinates along and normal to the plate and U, V are the mean
velocities in the x, y directions, respectively.
In order to close the above equations, an expression for the weighting factor _t
is needed. The weighting factor is assumed to be related to both the freestream
turbulent level and the intermittency factor of the boundary layer. The intermit-
tency factor is assumed to be determined by the local state of the boundary layer.
We use H, the shape factor, to characterize the local state of the boundary layer
since both the intermittency factor and the shape factor change monotonically from
the laminar boundary layer to the turbulent boundary layer. Experimental results
by Abu-Ghannam and Shaw 13 axe used as a guide to construct this function. The
weighting factor is also assumed to change in the y direction in such a way that
outside the boundary layer, the weighting factor is one since the freestream is gov-
erned by decaying turbulence. The final expression for the weighting factor is given
in Yang and Shih 14.
The above system of parabolic equations need to be supplymented by boundary
conditions at the wall and at the freestream and by initial conditions at the starting
point of the calculation. At the wall
U=V=k=O,
86
Modeling of Near Wall Turbulence and Modeling of Bypass Transition
2 P" cokl/2 )2
,,,,= . (lO)
The boundary condition for e was obtained by applying the equation for turbulent
kinetic energy down to the wall.
At the edge of the boundary layer, the flow variables are given by the freestream
values, i.e.
U=Ue,k =ke, e-ee. (II)
The k_,e_ in the above are found from the transport equations for k and e, with
the condition that the gradient of the flow variablesin the y directionvanishes as
the free stream is approached. Thus,
dz = (12)
dee C2,ee (13)
U_-_x= T
k_0 and eeo (the values of k_ and e_ at the leading edge, for example) are needed.
ke0 is obtained from the experiment, and e_o is determined in such a way that the
resulting k¢(x) profile agrees with the experiment.
One of the issues in the calculation of transitional boundary layers through the
low Reynolds number k - e models is the prescription of the initial profiles for
the turbulence kinetic energy and its dissipation rate, the later of which could not
be found from the experiment directly. An expression for the initial profiles were
given in Rodi and Scheuerer 11. However, computations by Yang and Shih 15 which
tested the effect of the initial conditions on the transition prediction found, in
agreement with the findings of Patankar and Schmit 16, that the predicted onset of
the transition is sensitive to the initial profiles. This sensitivity of the results to the
initial conditions suggests that the only place where the initial conditions could be
specified unambiguously is at the leading edge. At the leading edge, the turbulent
kinetic energy and its dissipation rate take constant profiles, the values of which are
determined by the law for the decaying turbulence.
With the initial conditions given at the leading edge and the boundary conditions
given above, the solutions are marched downstream. Flat plate boundary layers with
free stream turbulence levels of 3% (Case T3A) and 6% (Case T3B) respectively
were calculated using the present model. These are the benchmark cases in an
ongoing project coordinated by Savil117, testing the capability of turbulence models
in predicting transitional flows. Fig. 3 shows the variation of skin friction coefficient
c/against Re_. Results from the experiment is shown for comparison. In addition,
the prediction of the Launder-Sharma model is also shown in the figure because it
was reported that among the lower Reynolds number k - e models, the Launder-
Sharma model performs best for transitional boundary layers. It is clear that the
present model gives a better prediction. Other features in the transitional boundary
layers and the calculations of the transitional boundary layers with other levels of
freestream turbulence can be found in Ref. 14.
87
Z. Yang
3. Future Plans
1) Modeling of the near wall turbulence.
The proposed k- E model is only tested for simple parabolic flows so far. Because
of the form of the model equations, the proposed model can be used in complex
flow situations, flow with separation for example. The performance of the proposed
model in those situations will be tested.
We will work on the second order closure for near wall turbulence. In particular,
we will be looking at the effect of the mean flow inhomogeneity on the pressure
strain correlation. We are hoping to represent this effect rationally, so that the
ad hoc damping functions currently being used in all the near wall second order
closures can be avoided.
2) Modeling of bypass transition.
We will apply the proposed model to transitional boundary layers with pressure
gradient and curvature. We will also extend the model to thermal boundary layers.
4. References
1 Launder, B.E. & Spalding, D.B., "The Numerical Computation of Turbulent
Flow," Computer Methods in Applied Mechanics and Engineering 3, 269-289
(1974).
2 Rodi, W., Turbulence Models and Their Application in Hydraulics, Book Pub. of
International Association for Hydraulic Research, Delft, the Netherlands, 1980.
3 Jones, W.P., and Launder, B.E., "The Calculation of Low-Reynolds Number
Phenomena with a Two-Equation Model of Turbulence," International Journal
of Heat and Mass Transfer 16, 1119-1130 (1973).
4 Patel, V.C., Rodi, W., and Scheuerer, G., "Turbulence Models for Near-Wall and
Low-Reynolds Number Flows: a Review," AIAA Journal 23, 1308-1319 (1985).
5 Shih, T.H., "An Improved k-e Model for Near-Wall Turbulence and Comparison
with Direct Numerical Simulation," NASA TM 103211 (1990).
6 Lang, N.J. and Shih, T.H., "A Critical Comparison of Two Equation Turbulence
Models," NASA TM 105237 (1991).
7 Yang, Z. and Shih, T.H., "A k - _ modeling of near wall turbulence," Proc. 4th
International Symposium on CFD, 1305-1310 (1991).
s Chien, K.Y., "Predictions of Channel and Boundary Layer Flow with a Low-
Reynolds-Number Turbulence Model," AIAA Journal 20, 33-38 (1982).
9 Yang, Z. & Shih, T.H., "A new time scale based k- c model for near wall
turbulence," submitted for publication (1992).
10 Priddin, C.H., The behaviour of the turbulent boundary layer on a curved porous
wall, PhD Thesis, Imperial College, London, 1975.
11 Rodi, W. and Scheuerer, S., "Calculation of laminar-turbulence boundary layer
transition on turbine blades," Proc. 65th AGARD-REP Symposium on Heat
Transfer and Cooling in Gas Turbine, Bergen, Norway, 18-1 to 18-12 (1985).
Modeling of Near Wall Turbulence and Modeling of Bypass Transition
12 Fujisawa, N., "Calculation of transitional boundary laye'rs with a refined low
Reynolds number version of a k - e model of turbulence," Proc. Engineering
Turbulence Modeling and Experiment, 23-32 (1990).
13 Abu-Ghannam, B.J. & Shaw, R., "Natural transition of boundary layers - the
effects of turbulence, pressure gradient, and flow history," Journal of Mechanical
Engineering Science 22,213-228 (1980).
14 Yang, Z. & Shih, T.H., "A modeling of transitional boundary layers," In prepa-
ration (1992).
15 Yang, Z. and Shih, T.H., "A k - e calculation of transitional boundary layers,"
Transition and Turbulence, Springer-Verlag, 1992.
16 Schmidt, R.C. and Patankar, S.V., "Two-equation low-Reynolds number turbu-
lence modeling of transitional boundary layer flows characteristic of gas turbine
blades," NASA CR-4145 (1988).
17 Savill, A.M., "Turbulence model predictions for transition under free-stream tur-
bulence," RAeS Transition and Boundary Layer Conference, Cambridge, England
(1991).
89
Z. Yang
30.0 I I
U +
20.0
10.0
0.0
I 0 °'°
0 DNS
- - Jones-Lounder
--- Chien
Present
I I
101"° 10 2.0
t#+
FiB ls: Mean velocity profile for channel flow st Re_ = 395.
1 0 5"0
5.0 I ' I ' I ' i '
\ o DNS
4.0 I _\ -- dones-Lounder
k_\\ ---- Chien
k+ _ \\ m. Present
3.0 .. \
2.0 " "
1.0
0,0 _ I I I . _-.. I I
0.0 100.0 200.0 _,+ 300.0 400.0
Fig Ib: Turbulent energy profde for channel flow at Re. = 395.
9O
Modeling of Near Wall Turbulence and Modeling of Bypasa Transition
1.0
' i _ I '' I
0.8
0.6
0.4
0.2
I
0.0
0.0
¢ DNS
,_ - - Jones-Launder
--._._ Chien
, I _ 1. , I _,
I00.0 200.0 y+ 300.0 400.0
Fig lc: Shear stress profile for channel flow at ReT = 395.
0.3
0.1
0.0
0.0
I I I I
¢ DNS
- - Jones-Launder
--- Chien
_ Present -
I _ l l
25.0 50.0 7,5.0 100.0
_+
Fig ld: Dissipation rate profile for channel flow at Re_ = 395.
91
Z. ]Fang
30.0
U +
20.0
10.0
0.0
I I I
¢ DNS
- - Jones-Launder
--- Chien /[- 0///
-- Present //_._v_..
/ .
J
I I !
10 °'° 101"° 10 z° y+ • 10 z°
Fig 2a: Mean velocity profile for channel flow at Reo = 1410.
1 0 4.0
5.0 'd ,
\ 0 DNS
4.0 _v-\¢\ - - Jones-Launder
k+ \0_\ ---- Chien
% %K
- _._.\ Present3.0 . _<
2.0 "_,,_,
1.0
0.0 , _,_.0I
0.0 250.0 500.0
y+
Fi 8 25: Turbulent energy profile for channel flow st Res = 1410.
750.0
92
Modeling of Near Wall Turbulence and Modeling of Byp_J Tran.,ition
I I
0 DNS
- - Jones-Launder
--- Chien
-- Present
0
I
250.0 500.0 y+
Fig 2c: Shear stress pro_Ie for channd flow at Re0 - 1410.
750.0
0.3 I I I
¢ DNS
- - Jones-Launder
--- Chien
", -- Present -
0.0 -/ i , ,
0.0 25.0 50.0 75.0 100.0
y+
Fig 2d: Dissipation rate profile for channel flow at Reo = 1410.
93
Z. Yang
0 -2.0
1 0 -2.5
Launder-Sharma
Present
I
10 5.0 Rex
Fi 8 3a: Skin friction variation for T3A.
0 G'O
I 0 -2.0
C'!
I 0 -2-5
I 0 -_'0
O EXP
--- Launder-Sharma
Present
1 0 4.0 1 0 5.0
Rex
Fi 8 3b: Skin friction variation for T3B.
0 §'0
94


Modeling of near wall turbulence and modeling of bypass transition

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