Turbulence statistics with quantified uncertainty in cold-wall supersonic channel flow Rhys Ulerich and Robert D. Moser Institute for Computational Engineering and Sciences The University of Texas at Austin 18 November 2012 PECOS Acknowledgment: This material is based upon work supported by the Department of Energy [National Nuclear 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
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Turbulence statistics with quantified uncertainty in cold-wall supersonic channel flow
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.
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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
• 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
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
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