turbulence statistics with quantified uncertainty in cold-wall supersonic channel flow
DESCRIPTION
To investigate compressibility effects in wall-bounded turbulence, a series of direct numerical simulations of compressible channel flow with isothermal (cold) walls have been conducted. All combinations of $\mbox{Re}=\left\{3000, 5000\right\}$ and $\mbox{Ma}=\left\{0.1, 0.5, 1.5, 3.0\right\}$ have been simulated where the Reynolds and Mach numbers are based on bulk velocity and sound speed at the wall temperature. Turbulence statistics with precisely quantified uncertainties computed from these simulations will be presented and are being made available in a public data base at http://turbulence.ices.utexas.edu/. The simulations were performed using a new pseudo-spectral code called Suzerain, which was designed to efficiently produce high quality data on compressible, wall-bounded turbulent flows using a semi-implicit Fourier/B-spline numerical formulation.TRANSCRIPT
Turbulence statistics with quantified uncertainty incold-wall supersonic channel flow
Rhys Ulerich and Robert D. Moser
Institute for Computational Engineering and SciencesThe University of Texas at Austin
18 November 2012
PECOS
Acknowledgment: This material is based upon work supported by the Department of Energy [NationalNuclear Security Administration] under Award Number [DE-FC52-08NA28615].
R. Ulerich & R. D. Moser (U.T. Austin) Cold-wall supersonic channel flow 18 November 2012 1 / 11
Introduction
Motivation: Complex predictions w/ quantified uncertainty
• Eddy viscosity-based models widely used in engineeringI These models well-known to be imperfect and unreliableI Higher-fidelity approaches computationally intractable,
especially for uncertainty quantification• Recent work in combining Bayesian approaches with RANS models:
Cheung et al. [2011], Oliver and Moser [2011, 2012a,b]• Combination requires rich near-wall data with uncertainty estimates
R. Ulerich & R. D. Moser (U.T. Austin) Cold-wall supersonic channel flow 18 November 2012 2 / 11
Problem definition
Compressible Navier–Stokes formulationPerfect gas with constant γ, Pr and power law viscosity
∂
∂tρ = −∇ · ρu
∂
∂tρu = −∇ · (u⊗ ρu)− 1
Ma2∇p+ 1
Re∇ · τ + f
∂
∂tρE = −∇ · ρEu−∇ · pu+
Ma2
Re∇ · τu+
1
Re Pr (γ − 1)∇ · µ∇T +Ma2f · u+ qb
p = (γ − 1)
(ρE − Ma2
2ρu2
)T = γ
p
ρ
µ = T β λ =
(α− 2
3
)µ τ = µ
(∇u+∇uT
)+ λ (∇ · u) I
R. Ulerich & R. D. Moser (U.T. Austin) Cold-wall supersonic channel flow 18 November 2012 3 / 11
Problem definition
Scenarios patterned on Coleman et al. [1995]
streamwise
z,w
x,u
spanwisew
all
norm
al y,v
bulk flow
wall
uw = 0 Tw = 1 f : (ρu)bulk = 1 qb = 0 ρbulk = 1
γ = 1.4 Pr = 0.7 α = 0 L = 4π × 2× 4π/3
Re = {3000, 5000} Ma = {0.1, 0.5, 1.5, 3.0}
R. Ulerich & R. D. Moser (U.T. Austin) Cold-wall supersonic channel flow 18 November 2012 4 / 11
Simulation results
Resolutions patterned on Coleman et al. [1995]
Fourier basis in x and z using 3/2s dealiasing: Nx = 192, Nz = 168Piecewise 7th order B-splines in y with hyperbolic tangent stretching
Identifier Re Ma β Ny tanh ∆x+ ∆y+1 ∆y+10 ∆z+ Flow throughs
c03k01 3000 0.1 2/3 128 2.25 12.5 0.22 11.8 5.0 37.1c03k05 3000 0.5 2/3 128 2.25 12.6 0.22 12.0 5.1 38.9c03k15 3000 1.5 2/3 128 2.25 14.4 0.25 13.7 5.8 39.1c03k30 3000 3.0 2/3 128 2.25 19.5 0.34 18.5 7.8 38.8c05k01 5000 0.1 2/3 144 2.50 19.4 0.26 14.1 7.8 30.6c05k05 5000 0.5 2/3 144 2.50 19.8 0.27 14.4 7.9 32.0c05k15 5000 1.5 2/3 144 2.50 22.8 0.30 16.5 9.1 48.7c05k30 5000 3.0 2/3 144 2.50 30.9 0.41 22.4 12.3 81.7
CKM95a 3000 1.5 0.7 90 17 0.1 8 10 ≥ 11.9CKM95b 4880 3.0 0.7 90 39 0.2 17 24 ≥ 11.9
CKM95{a,b} by Coleman et al. used Fourier–Legendre discretization:110× 90× 60 coefficients with 144× 119× 80 collocation points
R. Ulerich & R. D. Moser (U.T. Austin) Cold-wall supersonic channel flow 18 November 2012 5 / 11
Simulation results
Qualitative profile comparison for Re = 3000, Ma = 1.5
0
1
2
3
-1.0 -0.5 0 0.5 1.0
y
<ρ><u><T>
Simulation c03k15Simulation CKM95a
Reproduced from Coleman et al. [1995]
R. Ulerich & R. D. Moser (U.T. Austin) Cold-wall supersonic channel flow 18 November 2012 6 / 11
Simulation results
Quantitative comparison of centerline and wall meansTabulated discrepancies between c03k15 and CKM95a less than 1%, except Maτ ≤ 1.5%
Identifier Mac Maτ Rec Reτ −Bq 〈ρw〉 〈ρc〉 〈Tc〉 〈µc〉c03k01 0.116 0.006 3489 190 0.0003 1.002 0.9999 1.002 1.001c03k05 0.571 0.031 3391 193 0.0062 1.040 0.9973 1.043 1.028c03k15 1.493 0.081 2765 221 0.0491 1.365 0.9779 1.391 1.246c03k30 2.240 0.120 1764 298 0.1500 2.450 0.9274 2.665 1.922c05k01 0.115 0.006 5758 297 0.0002 1.002 0.9999 1.002 1.001c05k05 0.565 0.029 5598 303 0.0058 1.041 0.9979 1.042 1.028c05k15 1.480 0.076 4595 348 0.0462 1.366 0.9834 1.385 1.242c05k30 2.205 0.114 2973 472 0.1410 2.484 0.9486 2.600 1.891KMM87 0 0 3250 180 0 1 1 1 1CKM95a 1.502 0.082 2760 222 0.049 1.355 0.980 1.378 1.252CKM95b 2.225 0.116 2872 451 0.137 2.388 0.952 2.490 1.894
CKM95{a,b} reproduced from Coleman et al. [1995]Incompressible results from Kim et al. [1987]
R. Ulerich & R. D. Moser (U.T. Austin) Cold-wall supersonic channel flow 18 November 2012 7 / 11
Simulation results
Statistics with quantified uncertainty: c03k15
0
0.35
0.7
1.05
1.4
0 55 111 166 221
µ;
σ/µ
0%
0.25%
0.50%
0.75%
1%
0 55 111 166 221
ρ
uTν
-1
1
3
5
7
9
11
0 55 111 166 221
µ+;
σ/µ
y+
u’’u’’v’’v’’
w’’w’’u’’v’’
k
See Malaya et al. [2012] on Monday at 11:09 AM in Session H21.04:“Estimating Uncertainties in Statistics Computed from DNS”
R. Ulerich & R. D. Moser (U.T. Austin) Cold-wall supersonic channel flow 18 November 2012 8 / 11
The Suzerain framework
Suzerain: A spectral, compressible DNS frameworkDesigned for performance, extensibility, portability, and longevity
• Mixed Fourier-Galerkin /B-spline collocation method
• Low-storage, hybrid implicit/explicitRunge-Kutta time stepping[Spalart et al., 1991]
• Test-driven development of modularC99/C++03 implementation
• Self-documenting, HDF5 restartand statistics files
• Pencil decompositions forscalability [Pekurovsky, 2008]
(y, x, z) (z, y, x) (x, z, y)
SMR91 low storage RK
wave s
pace
PECOS' GRVY Toolkit, MKL/ESSL
P3DFFT (MPI+FFT)
physi
cal
spaceexplicit, nonlinear
computation (Eigen)
restart files (HDF5)
implic
it a
coust
ics
explic
it n
onlin
eari
ties
B-splines (GSL)
evaluationdifferentiation
collocation operators
implicit step solution(MKL/ESSL)
post
pro
cess
ing
& a
naly
tics
ESIO
R. Ulerich & R. D. Moser (U.T. Austin) Cold-wall supersonic channel flow 18 November 2012 9 / 11
The Suzerain framework
Suzerain: Testing and Verification26K SLOC automated regression test suite with 85% line coverage
10-15
10-12
10-9
10-6
10-3
103
104
105
106
Max
imu
m e
rro
r in
an
y c
oef
fici
ent
Degrees of freedom per scalar field
Steady solution, piecewise septic B-splines
QρQ
ρuε
ρ ρuρvρwρe
10-15
10-12
10-9
10-6
10-3
103
104
105
106
Degrees of freedom per scalar field
Unsteady solution, piecewise quartic B-splines
ρ ρuρvρwρe
Field-by-field convergence on a steady (left) and transient (right) manufactured solution problem attwo different B-spline orders. Labels Qρ and Qρu show measured relative error in the associatedfloating point manufactured forcing computations. Label ε shows machine epsilon. Ulerich et al. [2012]
R. Ulerich & R. D. Moser (U.T. Austin) Cold-wall supersonic channel flow 18 November 2012 10 / 11
Conclusions
Summary• Developed a robust, verified code for compressible, spectral DNS
• Simulated low-Re, cold-wall channels over a wide range of Ma
• Quantified uncertainties in basic statistics of interest
• Database online soon: http://turbulence.ices.utexas.edu/
Ongoing Work• Completing uncertainty estimates for derived, higher-order quantities
• Implementing features required for homogenized boundary layers:Giles [1990], Topalian et al. [2011, 2012]
• Generating database of statistics for such boundary layers
• Investigating relaminarization processes on blunt reentry vehicles
• Incorporating chemically-reacting species capabilities
R. Ulerich & R. D. Moser (U.T. Austin) Cold-wall supersonic channel flow 18 November 2012 11 / 11
P. Bradshaw. Compressible turbulent shear layers. Annual Review of Fluid Mechanics, 9(1):33–52, 1977. doi:10.1146/annurev.fl.09.010177.000341.
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John W. Eaton, David Bateman, and Søren Hauberg. GNU Octave Manual Version 3. Network Theory Limited, 2008. URL http://www.octave.org.Michael B. Giles. Nonreflecting boundary conditions for Euler equation calculations. AIAA Journal, 28(12):2050–2058, 1990. doi:
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177:133–166, 1987. doi: 10.1017/S0022112087000892.Nicholas Malaya, Todd A. Oliver, Rhys Ulerich, and Robert D. Moser. Estimating uncertainties in statistics computed from DNS. In 65th Annual
Meeting of the APS Division of Fluid Dynamics, San Diego, CA, November 2012.MASA Development Team. MASA: A library for verification using manufactured and analytical solutions.
https://red.ices.utexas.edu/projects/software/wiki/MASA, 2011.T. A. Oliver and R. D. Moser. Bayesian uncertainty quantification applied to RANS turbulence models. Journal of Physics: Conference Series, 318
(4):042032, 2011. doi: 10.1088/1742-6596/318/4/042032.Todd A. Oliver and Robert D. Moser. Accounting for uncertainty in the analysis of overlap layer mean velocity models. Physics of Fluids, 24(7):
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Fluid Dynamics, San Diego, CA, November 2012b.PECOS Development Team. ESIO: A parallel I/O library for scientific applications. https://red.ices.utexas.edu/projects/software/wiki/ESIO,
2011.Dmitry Pekurovsky. P3DFFT user guide. http://code.google.com/p/p3dfft/, 2008.Philippe R. Spalart, Robert D. Moser, and Michael M. Rogers. Spectral methods for the Navier–Stokes equations with one infinite and two periodic
directions. J. Comput. Phys., 96(2):297–324, 1991. doi: 10.1016/0021-9991(91)90238-G.Victor Topalian, Onkar Sahni, Todd Oliver, and Robert Moser. Slow growth formulation for DNS of temporally evolving boundary layers. In 64th
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