Partially-Averaged Navier-Stokes in PyFR Tarik Dzanic Prof. Freddie - - PowerPoint PPT Presentation

Partially-Averaged Navier-Stokes in PyFR

Tarik Dzanic

• Prof. Freddie D. Witherden

Department of Ocean Engineering, Texas A&M University June 19th, 2020

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INTRODUCTION

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• Partially-averaged Navier-Stokes (PANS) is a variable resolution turbulence

closure model

• Closure model for partially averaged statistics
• Bridging method for any scale resolution
• Single framework for DNS, LES, DES, RANS, etc.
• Attempts to model the effects of the unresolved kinetic energy and dissipation
• Account for unresolved stresses with an eddy viscosity
• Can give results on par with LES at lower cost
• Higher aspect-ratios, much coarser grids away from boundaries
• Still need to be wall-resolved to predict separation

FORMULATION

• Two-equation closure model – unresolved kinetic energy (ku) and unresolved

specific dissipation (ωu)

• Unresolved and total kinetic energy/dissipation related by the parameters fk, fω
• Parameters set by the user depending on the grid
• DNS at fk, fω= 0, URANS at fk, fω = 1
• fωgenerally taken as 1/fk (fε = 1)

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IMPLEMENTATION

• Adding in turbulence transport equations to finite-element methods not as

straightforward as finite volume methods

• Physical constraints
• Low numerical diffusion
• Boundary conditions
• Some alternative approaches have to be taken to ensure stability
• Solve for log(ω) instead of ω to guarantee that it’s strictly positive
• Source/sink limiters for k
• Computational costs vary
• 2 extra transport equations to solve
• Potential time step restrictions due to eddy viscosity
• Anti-aliasing not necessary at high Reynolds numbers

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CYLINDER FLOW

• Flow around a cylinder at Re = 3900 was used as a

benchmark

• Common case for benchmarking due to the complexity of

the flow physics (laminar separation, free-shear layer, transition, turbulent wake)

• Compared to DNS (Witherden et al.) and

experimental results (Parnaudeau et al.)

• Coarse mesh with 64,000 P3 prisms (2.5m DOFs)
• Wall-resolved with large aspect ratios and growth rates
• With the same numerical setup, we compare PANS

to Navier-Stokes (URLES) simulations

• Various fkparameter choices
• Adaptive fkmethods

5 Top: LES (Parnaudeau et al.). Bottom: DNS (Witherden et al.)

CYLINDER FLOW

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• URLES underpredicted the size of the recirculation bubble by roughly 50%
• PANS with fk = 0.1 showed excellent agreement with the DNS and experiment
• PANS with fk = 0.2 and 0.3 marginally overpredicted the size of the recirculation bubble

Experiment URLES fk = 0.1 fk = 0.2 fk = 0.3

Centerline streamwise velocity Streamwise velocity contours

CYLINDER FLOW

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• URLES showed significant deviations in the predicted streamwise and normal

velocity profiles

• PANS with fk = 0.1-0.3 noticeably improved the predictions

Streamwise (top) and normal (bottom) velocity profiles at x/D = 1.06 (left), 1.54 (middle), and 2.02 (right).

CYLINDER FLOW

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• Less variation in the second-order statistics between different fk values than first-order

statistics

• Optimal value between fk = 0.1 and 0.2
• Excellent agreement in the normal velocity variance profiles at all fk values (not shown)

Streamwise velocity variance (top) and streamwise-normal velocity covariance profiles at x/D = 1.06 (left), 1.54 (middle), and 2.02 (right).

ADAPTIVE PANS

• Instead of tuning the fk constant, we want the solver to find the optimal value on

its own – Adaptive PANS

• Need to quantify how much of the turbulent kinetic energy is resolved locally
• Physical length scales vs. resolved length scales
• Girimaji & Abdol-Hamid (2005) proposed using the turbulence variables to

calculate the unresolved length scales

• fk calculated as the ratio of unresolved length scales to the grid scales (CPANS = 0.1)
• Spatio-temporal variation in fk allowed as long as the turbulent scales are smaller

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ADAPTIVE PANS

• Information in higher-order methods can be leveraged

to better predict the optimal fk

• Structured representation of the solution within elements
• Modal basis functions
• Transforming the solution within an element to a modal

basis gives the fluctuation of the solution within the element

• Integrating the non-zero modes of the velocity

magnitude gives an estimate of the resolved turbulent kinetic energy

• Calculate fk using a numerical Kolmogorov scale with this

estimate of the kinetic energy

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ADAPTIVE PANS

• Modal method more accurately predicts the low fk in

the laminar separation region

• Similar fkpredictions by both methods toward the farfield
• fk was set constant within an element for both

methods

• Several methods for utilizing the adaptive fk fields
• On-the-fly PANS with adaptive fk
• Time-averaged fk with precursor run
• Time-averaged fkwith frozen field
• PANS simulations with time-averaged fk fields were

more stable

11 Instantaneous fk field at t = 100 – Girimaji method (top) and modal method (bottom)

ADAPTIVE PANS

• Both methods were able to converge towards the fk = 0.1

results

• Modal method gave near identical results to the DNS
• Girimaji method slightly underpredicted the size and
• verpredicted the strength of the recirculation region

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Experiment ADPANS-G ADPANS-M

Centerline streamwise velocity Streamwise velocity contours

ADAPTIVE PANS

• Modal method gave improved results over fk = 0.1 and the Girimaji method
• Excellent agreement with DNS

13 Streamwise (top) and normal (bottom) velocity profiles at x/D = 1.06 (left), 1.54 (middle), and 2.02 (right).

ADAPTIVE PANS

• Both methods adequately predicted the second-order statistics
• Better accuracy with the modal method closer to the cylinder
• Better fk prediction in the laminar separation region

14 Streamwise velocity variance (top) and streamwise-normal velocity covariance profiles at x/D = 1.06 (left), 1.54 (middle), and 2.02 (right).

CONCLUSION

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• We implemented a two-equation (k-ω-SST) PANS model in PyFR
• Initial tests were done using the flow around a cylinder at Re = 3900 as a

benchmark

• Various fkparameters gave varying results – all were an improvement over URLES
• Excellent agreement with DNS/experiment at fk= 0.1
• Improvements over lower-order PANS methods
• Proposed a method for adaptively finding fk based on the modal coefficients of

higher-order elements

• Improvements in accuracy over current adaptive fk methods
• Future work in applications to high Reynolds number flows