Turbulence statistics with quantified uncertainty in cold-wall supersonic channel flow

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


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


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}


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


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]


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]


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”


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


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]


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


http://turbulence.ices.utexas.edu/

P. Bradshaw. Compressible turbulent shear layers. Annual Review of Fluid Mechanics, 9(1):33–52, 1977. doi:10.1146/annurev.fl.09.010177.000341.

S. H. Cheung, T. A. Oliver, E. E. Prudencio, S. Prudhomme, and R. D. Moser. Bayesian uncertainty analysis with applications to turbulencemodeling. Reliability Engineering & System Safety, 96:1137–1149, 2011. doi: 10.1016/j.ress.2010.09.013.

G. N. Coleman, J. Kim, and R. D. Moser. A numerical study of turbulent supersonic isothermal-wall channel flow. Journal of Fluid Mechanics, 305:159–183, 1995. doi: 10.1017/S0022112095004587.

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:

10.2514/3.10521.P. G. Huang, G. N. Coleman, and P. Bradshaw. Compressible turbulent channel flows: DNS results and modelling. Journal of Fluid Mechanics, 305:

185–218, 1995. doi: 10.1017/S0022112095004599.John Kim, Parviz Moin, and Robert Moser. Turbulence statistics in fully developed channel flow at low reynolds number. Journal of Fluid Mechanics,

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):

075108+, 2012a. doi: 10.1063/1.4733455.Todd A. Oliver and Robert D. Moser. Representing turbulence model uncertainty with stochastic pdes. In 65th Annual Meeting of the APS Division of

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

Annual Meeting of the APS Division of Fluid Dynamics, November 2011.Victor Topalian, Todd A. Oliver, and Robert D. Moser. Slow growth formulation for dns of chemically reacting temporal boundary layers with forcing.

In 65th Annual Meeting of the APS Division of Fluid Dynamics, San Diego, CA, November 2012.Rhys Ulerich, Kemelli C. Estacio-Hiroms, Nicholas Malaya, and Robert D. Moser. A transient manufactured solution for the compressible

Navier-Stokes equations with a power law viscosity. In 10th World Congress on Computational Mechanics, Sao Paulo, Brazil, July 2012. URLhttp://users.ices.utexas.edu/~rhys/papers/WCCM2012-16661.pdf.


http://www.octave.org

https://red.ices.utexas.edu/projects/software/wiki/MASA

https://red.ices.utexas.edu/projects/software/wiki/ESIO

http://code.google.com/p/p3dfft/

http://users.ices.utexas.edu/~rhys/papers/WCCM2012-16661.pdf

Turbulence statistics with quantified uncertainty in cold-wall supersonic channel flow

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