The objectives of this study are as follows: (1) to conduct a direct numerical simulation of turbulent backward facing step flow using inflow and outflow conditions; and (2) to provide data in the form of Reynolds stress budgets for Reynolds averaged modeling. The report presents the basic statistical data and comparisons with the concurrent experiments of Jovic and Driver and budgets of turbulent kinetic energy

Le, Hung

Moin, Parviz

English

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Centerfor TurbulenceResearch
AnnualResearchBriefs199_
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- 161
/3
N94-12296
Direct numerical simulation of turbulent
flow over a backward-facing step
By H. Le AND P. Moin
1. Objectives
The objectives of this study are: (a) to conduct a direct numerical simulation
of turbulent backward facing step flow using inflow and outflow conditions and
(b) to provide data in the form of Reynolds stress budgets for Reynolds averaged
modeling. The report presents the basic statistical data and comparisons with the
concurrent experiments of Jovic and Driver (1991, 1992) and budgets of turbulent
kinetic energy.
2. Accomplishments
_.1 Method
The Navier-Stokes equations are discretized using a finite difference method on
a staggered mesh. Uniform mesh spacing is applied in the streamwise (x) and
spanwise (z) directions. In the vertical (y) direction, non-uniform mesh is employed
with mesh refinement at the wall and near the step. The fractional step method from
Le and Moin (1990) is used for time advancement. The Navier-Stokes equations
are first advanced using a second-order semi-implicit method without the pressure
terms. The pressure is calculated by solving the Poisson equation, and the velocities
are then corrected to satisfy the continuity equation.
The mean velocity profile obtained from a boundary layer simulation by Spalart
(1986) is imposed at the inlet at Reo = 667. Random velocity fluctuations u t, v',
and w' are superimposed on this profile according to a variant of the method of
Lee et al. (1992). The fluctuations are prescribed such that, at the inlet, the tur-
bulence intensities and Reynolds shear stress of Spalart's data are also duplicated.
A convective boundary condition (Pauley et al., 1988) is imposed at the exit.
The streamwise domain consists of an entry section of length 10h prior to the
step and a 20h post-expansion section, where h is the step height. The vertical
dimensions before and after the expansion are WI = 5h and W2 = 6h which give an
expansion ratio ER of 1.20. The spanwise dimension is 4h where periodic bound-
ary conditions are imposed. The simulation uses 770 × 194 x 66 grid points in the
streamwise, wall normal, and spanwise directions, respectively. The Reynolds num-
ber, based on h and the mean inlet free stream velocity U0, is Reh = 5100. The
computation uses 13 megawords of memory and requires approximately 55 CPU
seconds per time step on a single processor CRAY Y-MP at a rate of 186 mflops.
Statistical quantities are averaged over the spanwise direction and time. About
1100 CPU hours were required to obtain an adequate statistical sample. The total
computational time corresponds to approximately 4.5 flow-through times.
https://ntrs.nasa.gov/search.jsp?R=19940007824 2020-06-16T20:56:45+00:00Z
162 H. Le _ P. Moin
0 5h lOh
FIGURE 1. Instantaneous spanwise vorticity contours, wz; min = --7, max = 4.95,
increment = 1.3.
2.2 Results
Figure 1 shows contour plots of the instantaneous spanwise vorticity w_ on a typi-
cal vertical plane. The vorticity is normalized by Uo/h. A free-shear layer spreading
from the step and interacting with the lower wall near the mean reattachment lo-
cation, xR = 6h, is discernible.
The basic statistical quantities are compared to results from concurrent exper-
iments by Jovic and Driver. In 1991, they conducted a backward facing step ex-
periment at Reh = 6800 and ER = 1.09, herein referred as "JDI". However, the
results indicated that these parameters are not sufficiently close to those used in
sinndations. Thus, Jovic and Driver in 1992 conducted a second experiment with
Reh = 4950 and ER = 1.20 ("JD2"). In the following sections, JD2 results will be
used for comparison with numerical simulations.
2.2.1 Reattachment length
The instantaneous velocity fields indicate that the reattachment location oscillates
in the streamwise direction and time and also varies in the spanwise direction.
Several methods were used to determine the mean reattachment location, xn: (a)
by the location at which the mean velocity U -- 0 at the first grid point away from
the wall, (b) by the location of zero wall-shear stress (rw = 0), and (c) by the
location of the mean dividing streamline (¢ = 0). A pdf method was also used in
which the mean reattachment point is indicated by the location of 50% forward flow
fraction. The pdf method was also used experimentally by Westphal et al. (1984)
and Adams et al. (1984). The results of the first three methods are within 0.1% of
each other and about 4% from the pdf result. The calculated mean reattachment
length is 6.0h compared to 6.1h measured in JD2 experiment.
2.2.2 Pre,q,,'ure distribution
The streamwise pressure coefficient at the step-wall is compared with the JD2
results in Fig. 2. The pressure coefficient is normMized by the mean inlet free-
stream velocity. The two symbols in Fig. 2 are for the two walls of the double-sided
expansion in the JD2 experiment. The comparison shows an excellent agreement
between computation and experiment.
0.25
0.2
0.15
_D_. 0.1
0.05
0
-0.05
DNS of turbulence over a backward-facing step
I I I I I I I I
o_o_ 6 i o A a
I I l
-2.5 0 2.5 5 7.5 10 12.5 15 17.5 20
z/h
163
FIGURE 2. Step-Wall pressure coefficient as a function of x. _, computation;
o, Jovic & Driver (1992), bottom wall; ,,, Jovic & Driver (1992), top wall.
0.004
0.003
0.002
0.001
0
-0.001
-0.002
°0.003
1 I I I I
I _[t I I I I
0 2.5 5 7.5 10 12.5
I I I
I I
15 17.5 20
x/h
FIGURE 3. Step-wall coefficient of friction.
(1992).
, computation; •, Jovic & Driver
164 H. Le _ P. Moin
! I _ ! i i i
° D •
- I m_" l - "-"_" i ...... _ i
0,0 0.0 0,0 0.2 0,4 0.6 0.8 1.0
U/Uo
FIGURE 4. Mean velocity profiles. , computation; •, Jovic & Driver (1992).
_.2.3 Skin friction coe_cient
The skin friction coefficient CI is computed from the mean velocity profile and
shown in Fig. 3. Also plotted are the CI data from the JD2 experiment. The large
peak negative skin friction in the recirculation region (_ -3 × 10-3), seen in both
computation and experiment, is nearly 3 times larger than previous experimental
findings, e.g. Adams et al. (1984). Jovic and Driver (1991) also measured the skin
friction in the reverse flow region for a wide range of Reynolds numbers. Their
results show that the recirculation C1 decreases with increasing Reynolds number.
The peak negative Cy reaches a vMue of approximately -1.0 × 10 -3 as Reh ,,_ 40000
which is the Reynolds number range used in many experiments. Thus the large skin
friction in the current study is due to low Reynolds number effects.
2._._ Mean velocity profiles
Figure 4 shows the mean velocity profiles compared with JD2 experimental results
at three x-locations: x/h = 4 (reclrculation region), x/h = 6 (reattachment loca-
tion), and x/h = 10. Again, excellent agreement is obtained betwcen computation
and JD2 experiment.
The near-wall mean velocity profile in the recovery region at x/h = 19 is compared
to experimental data in Fig. 5. The shear velocity uT of the JD2 experiment is
calculated from the skin friction coefficient of Fig. 3. Above y+ = 10, both profiles
show a noticeable shift below the log-law, indicating that the turbulent boundary
layer has not fully recovered. Previous studies, however, reported a recovery of
the log-law profile even as early as x - xR = 6h, e.g. Westphal et al. (1984).
The apparent discrepancy is attributed to the method of obtaining the wall-shear
velocity uT. In these experiments, the wall-shear velocity was calculated using the
Clauser chart with the inherent assumption that the log-law of the zero-pressure
gradient turbulent boundary layer is applicablc. The result was lower values of uT.
DNS of turbulence over a backward-facing step 165
+
25
2O
15
10
5
0
I I I I I
• " i
I I I I I
101 10° 101 102 103
y+
FIGURE 5. Mean velocity profile at x/h = 19. , u + = y+;
u + = (1/.41)log(y +) + 5.0; --, computation; *, Jovic & Driver (1992).
Direct Cy measurements using a pulsed wall probe by Westphal et al. confirm that
the correct shear velocity in this flow region is indeed higher than that predicted
by the Clauser chart. In the present configuration, this discrepancy is as high as
17%, i.e., uT/u_, _ 1.17, where the subscript c denotes the value obtained using the
Clauser chart. The deviation of the velocity profile from the log-law may be due to
the effect of the strong streamwise adverse pressure gradient which is experienced
by the flow following the sudden expansion (Nagano et aI., 1991)• It may also be
due to non-equilibrium effects which are persistent after reattachment.
_.2.5 Turbulence intensities and Reynolds stress
The turbulence intensity and Reynolds shear stress profiles, normalized by the
inlet free-stream velocity U0, are shown in Figs. 6-8. They are compared with
those from JD2 experiment data at streamwise locations, x/h = 4, 6, 10. The
overall agreement is good.
_.2.6 Turbulent kinetic energy budget
The terms of the turbulent kinetic energy transport equation are evaluated from
the equation
Oq 2 2
--5/- = --2 tukUl, u ,k Z,k --2utP,t•
Ck P_ Tk Dk ek Hk
166 H. Le _ P. Moin
! i I i i i i
"0
_._ ._. _ -.---_-1o.o
• O0 •
| - • , I
0.0 0.0 0.0 0.05 O. 10 O. 15 0.20
FIGURE 6. Turbulence intensity V_, normalized to inlet mean velocity U0.
-- , computation; • , Jovic & Driver (1992).
• I • I i i
• m i I
0.0 0.0 0,0 0.05 0.10
FIGURE 7. Turbulence intensity V/-_, normalized to inlet mean velocity U0.
-- , computation; • , Jovic & Driver (1992).
DNS of turbulence over a backward-facing step 167
3 , i i
0
0.0 0.0 0.0 0.11115 0.010
FIGURE 8. Reynolds shear stress -u'v', normalized to inlet mean velocity U02.
, computation; • , Jovie & Driver (1992).
The terms on the right hand side are identified as follows:
Ck = Convection,
Pk = Production,
Tk = Turbulence Diffusion,
Dk = Viscous Diffusion,
ek = Viscous Dissipation,
IIk = Velocity Pressure-Gradient.
The budget for the turbulent kinetic energy, _ i , ,= 7(uiui), is shown in Figs. 9
and 10.
At two step heights before the separation, the energy budget (Figs. 9(a) and
10(a)) is similar to that of a turbulent boundary layer (Spalart, 1988) although
there is an enhancement of the viscous terms near the wall. The budget for the
recirculation region is shown in Fig. 9(b). This energy budget is very similar
to that of a plane mixing layer (Bradshaw and Ferriss, 1965). This budget also
agrees qualitatively with the measurement by Chandrsuda and Bradshaw (1981)
for a backward facing step flow. Both production Pk and viscous dissipation ek
have maximum values at the same point in the free-shear layer. The peak ek is
approximately 60% of the production peak. The other 40% is balanced by the
turbulence transport Tk. Far from the wall, Pk is the only major gain term. T, is
a consuming term for 0.3 < y/h < 1, but a gain term outside of this range. Above
the step (y/h > 1), Tk is in balance with Ck, the turbulence convection by the
mean flow. As one approaches the wall, the production becomes a consuming term
because of the negative gradient of the mean reverse flow (Fig. 10(b)) although its
magnitude is relatively small. Very close to the wall, the two viscous terms, Dk and
168 H. Le 0 P. Moin
.l=
"cl
o)
t_
0.01
0.005
0
-0.005
-0.01
l I I
..........
/ ill
I" I I I,
1.00 105 1.10 1.15 1.20
Z
h
,.Q
t'q
hO
0.01
0
-0.01
(b)
I I I
0 0.5 1 1.5 2
FIGURE 9. Budget terms for turbulent kinetic energy, _ 1 , ,2 = 7(uiui) ' away from
the wall, normalized by Ug/h. (a) x/h -2.0; (b) z/h = 4. --, production;
, turbulence transport; ........ , viscous diffusion; B.m , viscous dissipation;
.... , velocity pressure-gradient;-----, convection.
DNS o.f turbulence over a backward-facing step 169
0
,.Q
0.01
0.005
0
-0.005
-0.01
I I I
(c)
I
I I I
0 0.5 1 1.5
h
0.006
_'_ 0.004
;>=
0.002
o
0
_ -0.002
-0.004
I I I t
(d)
I__ " .................. --.:_-.-_:- .............
f
-i'
-0._6 I i I [
0 0.5 1 1.5 2 2.5
_L
h
FIGURE 9. (cont.) Turbulent kinetic energy budget away from the wall, normalized
by U_/h. (c) x/h = 10; (d) z/h = 18. For caption see previous page.
1'70 H. [,e PI P. Moin
"o
"0
0.3
0.2
0.1
0
-0.1
-0.2
-0.3
i 1 i i
" (a)
°".,.
.."IY::C........................................:
," s
i
[ 1 I , ,1
0 5 10 15 20 25
y+
_'1 " 1.5
,._ 1
0.5
o
o
-_ -o.s
i ' _ ' ' (b)
/ ";,...... d .'., .,.,,,................................................
• ". _ -' ........:...,_. ,.,..:_...................
, . - _ -, ....... _....-..::._ ...... _ ....... _....::--.;,,-. .............
.............":.7":2"22..........................................
a;
/"
/"
i
I
i
I I I I
0 5 10 15 20 25
y+
FIGURE 10. Budget terms for turbulent kinetic energy, q2 1 , ,2 = _(uiui), normalized
by local u_/t,. (a) x/h = -2.0; (b) x/h = 4. --, production; .... , turbulence
transport; ........ , viscous diffusion; ----- , viscous dissipation; .... , velocity
pressure-gradient;-----, convection.
DNS of turbulence over a backward-facing step 171
t_
o
,.¢j
1.5
1
0.5
0
-0.5
-1
-1.5
I I I I
/"
/
/
/
/
(c)
I 1 I i
0 5 10 15 20 25
y+
0.6
_ > 0.4
0.2
o
o
-0.2
=
..Q
-0.4%-
-0.6
1 I I I
i';":. (d)
,J
/
,/
/
/
/*
I i I I
0 5 10 15 20 25
y+
FIGURE 10. (cont.) Budget terms for turbulent kinetic energy near wall, normal-
ized by u_/v. (c) z/h = 10; (d) x/h = 18. For caption see previous page.
172 11. Le _ P. Moin
ek, grow rapidly due the high intensities in all three directions (their value at the
wall is about 40% higher than the peak production in the shear layer). The velocity
pressure-gradient term becomes significar, t only at y+ < 8 where it balances the
viscous terms plus Tk. The profiles have similar shapes through the reattachment
point.
All terms decay with x; however, the energy in the shear layer decays much faster
than at the wall (the streamwise decay of the budget terms are not apparent in Fig.
10 because they are normalized with the local u,-). By z/h = 10, the value of Dk at
the wall is about 3 times the peak production value. Near the flow exit, y/h = 18,
the turbulent kinetic energy budget resembles that of a boundary layer. However,
the effects of the free-shear layer is still apparent, e.g., Tk is still large at y/h = 1
and the peak ek is only ,._ 85% of the peak Pk.
3. Conclusions
A direct numerical simulation of turbulent flow over a backward facing step at
Reh = 5100 was successfully completed. There is good agreement in turbulence
statistics deduced from simulations and concurrent experiments of Jovic and Driver.
Of interest are two observations not previously reported for the backward-facing
step flow: (a) at the relatively low Reynolds number considered, large negative skin
friction is seen in the recirculation region; the peak [CII is about three times the
value reported at high Reynolds numbers; (b) the velocity profiles in the recovery
region fall below the universal log-law.
A large data base from this recently completed simulation has been archived. It
contains up to third-order statistics at all locations in the recirculation, reattach-
ment, and recovery zones and the budgets of all components of the Reynolds stress
tensor.
Acknowledgements
This research was funded by the National Science Foundation (NSF) and the
Center for Turbulence Research (CTR). We are grateful to Dr. J. Kim for very
insightful discussions throughout the course of this study. The authors are also spe-
cially grateful to Drs. D. Driver and S. Jovic for their collaboration and performing
the concurrent experiments.
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JovIc, S. & DRIVER, D. 1991, 1992 private communication.
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Direct numerical simulation of turbulent flow over a backward-facing step

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