Partially-Averaged Navier-Stokes in PyFR Tarik Dzanic Prof. Freddie D. Witherden Department of Ocean Engineering, Texas A&M University June 19 th , 2020 1
INTRODUCTION • 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 2
FORMULATION • Two-equation closure model – unresolved kinetic energy ( k u ) and unresolved specific dissipation ( ω u ) • Unresolved and total kinetic energy/dissipation related by the parameters f k , f ω • Parameters set by the user depending on the grid • DNS at f k , f ω = 0, URANS at f k , f ω = 1 • f ω generally taken as 1/f k (f ε = 1) 3
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 4
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 P 3 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 f k parameter choices • Adaptive f k methods Top: LES (Parnaudeau et al.). Bottom: DNS (Witherden et al.) 5
CYLINDER FLOW Experiment URLES f k = 0.2 f k = 0.1 f k = 0.3 Centerline streamwise velocity Streamwise velocity contours • URLES underpredicted the size of the recirculation bubble by roughly 50% • PANS with f k = 0.1 showed excellent agreement with the DNS and experiment • PANS with f k = 0.2 and 0.3 marginally overpredicted the size of the recirculation bubble 6
CYLINDER FLOW Streamwise (top) and normal (bottom) velocity profiles at x/D = 1.06 (left), 1.54 (middle), and 2.02 (right). • URLES showed significant deviations in the predicted streamwise and normal velocity profiles • PANS with f k = 0.1-0.3 noticeably improved the predictions 7
CYLINDER FLOW Streamwise velocity variance (top) and streamwise-normal velocity covariance profiles at x/D = 1.06 (left), 1.54 (middle), and 2.02 (right). • Less variation in the second-order statistics between different f k values than first-order statistics • Optimal value between f k = 0.1 and 0.2 • Excellent agreement in the normal velocity variance profiles at all f k values (not shown) 8
ADAPTIVE PANS • Instead of tuning the f k 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 • f k calculated as the ratio of unresolved length scales to the grid scales (C PANS = 0.1) • Spatio-temporal variation in f k allowed as long as the turbulent scales are smaller 9
ADAPTIVE PANS • Information in higher-order methods can be leveraged to better predict the optimal f k • 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 f k using a numerical Kolmogorov scale with this estimate of the kinetic energy 10
ADAPTIVE PANS • Modal method more accurately predicts the low f k in the laminar separation region • Similar f k predictions by both methods toward the farfield • f k was set constant within an element for both methods • Several methods for utilizing the adaptive f k fields • On-the-fly PANS with adaptive f k • Time-averaged f k with precursor run • Time-averaged f k with frozen field • PANS simulations with time-averaged f k fields were more stable Instantaneous f k field at t = 100 – Girimaji method (top) and modal method (bottom) 11
ADAPTIVE PANS Experiment ADPANS-G Centerline streamwise velocity • Both methods were able to converge towards the f k = 0.1 ADPANS-M results • Modal method gave near identical results to the DNS • Girimaji method slightly underpredicted the size and overpredicted the strength of the recirculation region Streamwise velocity contours 12
ADAPTIVE PANS Streamwise (top) and normal (bottom) velocity profiles at x/D = 1.06 (left), 1.54 (middle), and 2.02 (right). • Modal method gave improved results over f k = 0.1 and the Girimaji method • Excellent agreement with DNS 13
ADAPTIVE PANS Streamwise velocity variance (top) and streamwise-normal velocity covariance profiles at x/D = 1.06 (left), 1.54 (middle), and 2.02 (right). • Both methods adequately predicted the second-order statistics • Better accuracy with the modal method closer to the cylinder • Better f k prediction in the laminar separation region 14
CONCLUSION • 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 f k parameters gave varying results – all were an improvement over URLES • Excellent agreement with DNS/experiment at f k = 0.1 • Improvements over lower-order PANS methods • Proposed a method for adaptively finding f k based on the modal coefficients of higher-order elements • Improvements in accuracy over current adaptive f k methods • Future work in applications to high Reynolds number flows 15
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