ME 634-001
HOMEWORK #3
Spring 2016
Burgers’ equation,
1 2
1
uxx ,
(1)
u x = −px +
2
Re
is a widely-used model for studying numerical procedures intended for application to the Navier–
Stokes (N.–S.) equations. The similarities between this equation and the N.–S. equations are
obvious. Moreover, by assigning px in judicious ways it is possible to produce exact solutions to
this equation that exhibit essentially any desired mathematical properties, including turbulence-like
behaviors. In this assignment you will study a model for turbulent Poiseuille flow. It should be
observed that the mean velocity for turbulent flow does not follow the parabolic profile familiar
from the laminar case, as the figure shows.
ut +
x
1
1.0
0.8
0.6
0.4
0.2
0
0.0
0
0.2
0.4
0.6
0.8
u1
This corresponds to the empirical formula
1/n
,
u(x) = 4x(1 − x)
(2)
√
where n is often assigned the value 7. It is possible to show that n = 1/ f where f is the friction
factor of the Colebrook formula for pipe flow; and from this, the approximation
n = 2 1 + (Re − Rec )0.0933
(3)
can be obtained. Here, Rec denotes the transition Reynolds number taken to be Rec = 1750 for
the present calculations.
You will produce results for all of exact solution, DNS, RANS and LES methods for this problem,
as described below. The same Fortran 77/90 source code works for all approaches, and the solution
method is selected via flag values supplied in the input file; the file names are:
brgrs-stdnt-trblnt-16.f
inputfle.d
The Fortran code solves Eq. (1) for all methods needed for this exercise without any coding required
of the user. There will, however, be some coding needed for data post processing as described below.
Carry out the following analyses and calculations to produce the results you are expected to report
for this assignment.
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1. Exact Solution. Run the code and save output files (burgers-xctfnl.out and burgersxctavg.out) for the exact solution at the final time and the time-averaged (over a specified
time of stationarity) exact solution employing nx = 5001, nt = 100000, mthdflg = −1, nmap
= 50, and tk = 1 × 10−5 . This should be done for Re = 3 × 103 , 2 × 104 , and 1 × 105 .
These parameters are all set in the input file, and details of their meaning can be found at
the beginning of the Fortran code listing as well as in the input file, itself. You may wish
to immediately plot these results to get an intuitive feel for the form taken by the velocity
profile and how this changes with Re.
2. DNS. Repeat Prob. 1 using the DNS option of the code (mthdflg = 0), but observe that this is
not true DNS because the order of accuracy of the numerical methods employed is insufficient.
Use length-scale estimates presented in lecture to determine the grid spacing (number of grid
points) needed for a solution that is fully resolved for each value of Re, and compare this
with the numbers you will actually use, given below. Note that grid refinement will never
exactly reproduce the exact solution in this case. Nevertheless, for each value of Re, conduct
grid-function convergence tests using grids consisting of 501, 1001, and 2001 points using tk
= 2 × 10−4 , 1 × 10−4 , and 5 × 10−5 , respectively (i.e., fixed Courant number). You will need
to choose different numbers of time steps, nt, depending on Re, so that stationarity has been
achieved for a sufficiently long time for time averaging of the solution to provide meaningful
results. This may require as many as 100000, or more, time steps for higher Re cases. You can
qualitatively determine the time at which stationarity is achieved by viewing output provided
in the file time-series.out. Then you can set a value of nstat (in the input file) to permit
averaging to start only after stationarity has been reached. For each value of Re use your
most accurate (finest-grid) DNS results to construct plots showing the comparison of these to
the exact solutions for both the final time results (contained in the file burgers-dnsfnl.out)
and the time average of this (file burgers-dnsavg.out). Also, be sure to record the run time
for each calculation; you will use these data in your report for comparison with RANS and
LES run times.
3. RANS. Derive the Reynolds-averaged form of Burgers’ equation, and include this derivation
in your report. Apply the Boussinesq hypothesis to this result, and within the confines of the
mixing-length models provide a formula for eddy viscosity νT . Then repeat Prob. 2 using the
unsteady and RANS options together (mthdflg = 1, istdy = 0 in the input file) and employing
the basic mixing-length model for which ℓmix = c1 x provided in lecture. For each value of Re
perform calculations using grids of nx = 101, 201 and 401 points, and time step tk = 5× 10−4 .
Run a sufficient number of time steps, nt, and corresponding settings of nstat to guarantee
accurate time averages. Investigate the effect of changing c1 on each grid. In addition, (for
extra credit) you may wish to consider using the Van Driest damping function with possible
alteration of the constant A+
0 with code entered near line 385 in subroutine brgrs of the
Fortran code. If this proves successful, you may want to attempt adjusting the value of c1 to
produce better solutions.
4. LES. Derive the LES form of Burgers’ equation in terms of an arbitrary, unspecified filter,
and present this in your write up. Apply the Boussinesq hypothesis, in the form of the
Smagorinsky subgrid-scale (SGS) model, to this equation to obtain a formula for SGS eddy
viscosity νSGS .
Carry out LES calculations with the same Burgers’ equation code used previously, now with
mthdflg = 2. Use grids of 1001, 501 and 251 points with time step sizes tk = 1 × 10−4 ,
2
2 × 10−4 , 4 × 10−4 , respectively. Note that it may be required to alter both grid spacing and
time step size as Re increases.
a. Assume that LES calculations should be resolved down to the Taylor microscale. Use
this to determine required total arithmetic as a function of Re for these calculations. In
particular, estimate the number of grid points needed for each value of Re. (Be sure to
include this analysis in your final report.)
b. As noted above, your code employs the Smagorinsky SGS model. The Smagorinsky
constant, CS , is entered in your input data file as c2. For each value of Re, attempt to
find an optimal value of CS for a grid consisting of 1001 points. Note that because now
temporal fluctuations are contained in the resolved solution (unlike in RANS methods,
but like DNS), longer integrations (up to nt = 100000) may be needed to obtain accurate
time averages. After finding CS for the 1001-point grid, perform calculations for 501and 251-point grids. Determine whether CS is grid dependent, and discuss.
c. Construct plots, and carry out analyses, for LES analogous to those already done for
DNS and RANS.
d. (Extra Credit) You may wish to investigate implicit LES (ILES), and compare its behavior with that of usual LES and DNS. This can be done in either of two ways: i)
use mthdflg = 0 (DNS), but employ a smaller value for the filter parameter; or ii) use
mthdflg = 2 (LES) but set c2 (the Smagorinsky constant) to zero—causing νSGS to be
zero. This will also require using a smaller value of the filter parameter. Consider using
numerical experiments to find the best value for this parameter, and note whether it
changes with Re. You might also try to solve problems with higher values of Re than
can be efficiently handled with usual LES, say up to 108 , on the same relatively course
grids used for the basic LES calculations.
5. Make the following additional plots summarizing information from all (DNS, LES and RANS)
calculations: i) run times as a function of Re on the finest grid used for each of DNS, LES and
RANS—all on the same plot, ii) time-averaged (from the most highly-resolved grid for each
case) solutions, all on the same plot, for exact, DNS, LES and RANS, for each Re (separate
plots for each Re), and iii) final time solutions (again, on the most highly-resolved grid) for
exact, DNS and LES. Thoroughly discuss these results.
6. Calculate νT (or νSGS , as appropriate), and make separate (spatial) plots of these and k vs.
x. On each plot show results for all values of Re based on the finest-grid calculation and the
exact solution. Finally, compute u+ and y+ —actually x+ in our notation—(describe in detail
in your analysis section how you compute these) for all four (4) solution types: exact, DNS,
RANS and LES. Use time-averaged results in all cases, and for RANS, LES and DNS methods
do calculations based on the finest grid employed for each Re for each method. Then, again
for each value of Re, plot u+ vs. x+ for all four types of solutions on the same plot—but
separate plots for each Re. Discuss these results in the context of the corresponding plot
from boundary-layer theory.
7. To rigorously test whether RANS and LES results converge to DNS solutions, use each of
these methods with the same fine grid and number of time steps employed for DNS with
Re = 2 × 104 and 1 × 105 ; namely, use nx = 2001 spatial grid points and nt = 100000 time
steps. For each Re plot time-averaged LES and DNS results on the same graph with RANS
results. Similarly, plot the highly-resolved LES solution at the final time on the same graph
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with the corresponding DNS solution for both values of Re. Thoroughly discuss these results
in light of properties of dependent-variable decompositions and forms of small-scale models
for both RANS and LES, and with consideration of discussions associated with Prob. 5.
Your results are to be presented in the form of, at least, a rough draft—but preferrably a final
form for submission—of an archival journal paper. (It is recommended to download a style file
from any top fluids, math or physics journal, and use it.) This must include a title and author
(you) and an abstract, and the following sections: Introduction, Analysis, Results/Discussion and
Summary/Conclusions as well as a reference list formatted in a style accepted by a typical journal.
This reference list should include at least five (5) entries beyond possible citation of your texts and
lecture notes, as may or may not be appropriate. These references should present both experimental
and computational results for turbulent Poiseuille flow. It is expected that you will have read all of
the cited literature, and that you will comment on these papers in your introduction and possibly
elsewhere in the paper.
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