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A HIGH-ORDER FINITE VOLUME SCHEME WITH ADAPTIVE NUMERICAL DISSIPATION FOR COMPRESSIBLE TURBULENT FLOWS X. Nogueira, L. Ram´ırez, F. Navarrina

GMNI — Group of Numerical Methods in Engineering Department of Applied Mathematics Civil Engineering School Universidade da Coru˜ na, Spain

e-mail: [email protected] web page: http://caminos.udc.es/gmni ˜ a — Group of Numerical Methods in Engineering Universidade da Corun

Outline •

Introduction

•

The ADA method

•

A Posteriori Detection

•

Numerical examples

•

Conclusions

˜ a — Group of Numerical Methods in Engineering Universidade da Corun

Outline •

Introduction

•

The ADA method

•

A Posteriori Detection

•

Numerical examples

•

Conclusions

˜ a — Group of Numerical Methods in Engineering Universidade da Corun

Introduction • Turbulent, compressible flows are more complex than incompressible ones. • Compressible turbulent fluctuations can be seen as a combination of three fundamental physical modes: vortical modes, acoustic modes and entropy modes). • Most of the SGS models have been developed for incompressible flows. • Here, we present a numerical method with Automatic Dissipation Adjustment (ADA) to act as an implicit SGS model for compressible flows. • The ADA method is combined with a modification of the MOOD approach to allow the computations of compressible flows with shocks.

˜ a — Group of Numerical Methods in Engineering Universidade da Corun

Moving Least Squares (I)

x) at a point x by using a I Reconstruction of u(x weighted LS approximation in the vicinity of x: x) ≈ u u(x ˆ(x x) =

m X

α(z) |z = x pi(x x)αi(zz ) |z =x = pT (x x)α(z)

i=1

x): base of functions with dimension m. • pT (x • α (zz ) |z =x : Parameters that minimize the error functional: Z J(α α(zz ) |z =x ) =

y ∈Ωx

 2 T W (zz − y , h) |z =x u(yy ) − p (x x)α α(zz ) |z =x dΩx

• W (zz − yy, h) |z =x : kernel (smoothing function) with compact support (Ωx ) centered in z = x. • h: smoothing length. ˜ a — Group of Numerical Methods in Engineering Universidade da Corun

Moving Least Squares (II)

I Minimization of J leads to: x)α p(yy )W (zz − y , h) u(yy )dΩx = M (x α(zz ) y ∈Ωx z =x z =x

Z

x) is the moment matrix defined as: I M (x x) = M (x p(yy )W (zz − y , h) pT (yy ) y ∈Ωx z =x Z

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Moving Least Squares (III)

I Discrete expression of the moment matrix is a m × m x) = P Ωx W (x matrix equals to M (x x)P P TΩx x)(dimension nx × nx ) are • P Ωx (dimension m × nx ), and W (x obtained by

x2) · · · p xnx x1) p (x P Ωx = p (x

W (x x) = diag {Wi (x x − xi)} i = 1, . . . , nx

I Finally, MLS approximation is written by: x)u uΩx = N T (x x)u uΩx = x) = pT (x x)M M −1(x x)P P Ωx W (x u b(x

nx X

x)uj Nj (x

j=1 ˜ a — Group of Numerical Methods in Engineering Universidade da Corun

(1)

Moving Least Squares (IV)

I Approximation can be written as: u ˆ(x x) =

nx X

x)uj Nj (x

j=1

with

x) x) = pT (x x)M M −1(x x)P N T (x P Ωx W (x

I Nj can be considered as “shape functions”. I Nj depends on the number of neighbors, the kernel and the base (ppT ). I Nj is a function of the grid. ˜ a — Group of Numerical Methods in Engineering Universidade da Corun

Moving Least Squares (V)

I Exponential Kernel. W (y, x∗, κ) =

e

−( sc )

2

−( dm c )

2

−e 2 −( dm ) c 1−e

s = |y − x∗| , dm = 2 max (|yj − x∗|) , c =

dm 2κ

˜ a — Group of Numerical Methods in Engineering Universidade da Corun

Moving Least Squares (V)

I Exponential Kernel. W (y, x∗, κ) =

e

−( sc )

2

2

−( dm c )

−e 2 −( dm ) c 1−e

s = |y − x∗| , dm = 2 max (|yj − x∗|) , c =

dm κ 2κ

˜ a — Group of Numerical Methods in Engineering Universidade da Corun

Outline •

Introduction

•

The ADA method

•

A Posteriori Detection

•

Numerical examples

•

Conclusions

˜ a — Group of Numerical Methods in Engineering Universidade da Corun

The Automatic Dissipation Adjustment (ADA) method • Recently developed as an implicit SGS model,and applied to the computation of low mach flows1. • It is based on the local energy ratio (ER) introduced by Tantikul and Domaradzki 2 in the context of the Truncated Navier-Stokes (TNS) procedure. • It uses a multiplicative coefficient () to the dissipation part of the numerical flux of the Riemann solver. • In this work, we aim to extend its range of application to all range of Mach number flows. 1

C-G Li, M. Tsubokura, An implicit turbulence model for low-Mach Roe scheme using truncated Navier-Stokes equations, Journal of Computational Physics, 345 :462-474, 2017. 2

T. Tantikul, J.A. Domaradzki,Large eddy simulations using truncated Navier-Stokes equations with the automatic filtering criterion, Journal of Turbulence, 11 :1-24, 2010.

˜ a — Group of Numerical Methods in Engineering Universidade da Corun

The Automatic Dissipation Adjustment (ADA) method • We use Roe’s numerical flux, that can be written as a central flux plus a dissipation part 4

Θi+ 1 2

1X 1 hH+ hH− ˜ k |˜ F +F )·n− α ˜ k |λ rk = (F 2 2 k=1

˜ k and r ˜k are the eigenvalues and eigenvectors of the approximated Jacobian, and αk are the wave strengths. λ

• We introduce a coefficient  to adjust the dissipation introduced by the numerical flux 4

Θi+ 1

2

1 hH+ 1 X hH− ˜ k |˜ = (F F α ˜ k |λ rk +F )·n−  2 2 k=1

• How to decide the value of ? ˜ a — Group of Numerical Methods in Engineering Universidade da Corun

The Automatic Dissipation Adjustment (ADA) method • The value of  is decided based on the ENERGY RATIO 3 X

ER =

i=1 3 X

1

(ui − ui)2 (ui − uˆi)2

i=1

• We use the following modification of ER 3 X

ER =

i=1 3 X

(ρiui − ρiui)2 (ρiui − ρiˆui)2

i=1 1

T. Tantikul, J.A. Domaradzki,Large eddy simulations using truncated Navier-Stokes equations with the automatic filtering criterion, Journal of Turbulence, 11 :1-24, 2010. ˜ a — Group of Numerical Methods in Engineering Universidade da Corun

The Automatic Dissipation Adjustment (ADA) method • In order to automatically adjust , we follow the rule proposed in 1 

ER < 0.007,  = max[( − φ), 0] ER > 0.015,  = min[( + φ), 1]

• A value of φ = 0.005 is used, to adjust the value of  continuously and gradually. • We have enlarged the range of ER according to our numerical experiments. • MLS filters have been used in the computation of the ER 1

C-G Li, M. Tsubokura, An implicit turbulence model for low-Mach Roe scheme using truncated Navier-Stokes equations, Journal of Computational Physics, 345 :462-474, 2017.

˜ a — Group of Numerical Methods in Engineering Universidade da Corun

Outline •

Introduction

•

The ADA method

•

A Posteriori Detection

•

Numerical examples

•

Conclusions

˜ a — Group of Numerical Methods in Engineering Universidade da Corun

A Posteriori Detection

I The fundamental idea behind the MOOD paradigm is to determine, a posteriori, the optimal order of the polynomial reconstruction for each particle that provides the best compromise between accuracy and stability. I Here, we adapt the MOOD approach in order to apply it to the ER-a posteriori method.

˜ a — Group of Numerical Methods in Engineering Universidade da Corun

A Posteriori Detection

I Physical Admissibility Detection (PAD) • PAD requires that the candidate solution remains physically admissible: B Positivity of the density B Positivity of the pressure

ρ?i > 0 p?i > 0

superscript ? refers to the candidate solution

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A Posteriori Detection

I Numerical Admissibility Detection (NAD) • Relaxed version of the Discrete Maximum Principle (DMP). • It checks that the solution is monotonic and new extrema are not created. min (Un (y)) − δ 6 U∗(x) 6 max (Un (y)) + δ

y∈Vi



y∈Vi



 δ = max 10−4, 10−3 · max (Un (y)) − min (Un (y)) y∈Vi

y∈Vi

• Here, it is only applied to the density. 1

M. Dumbser, O. Zanotti, R. Loub`ere and S. Diot, A posteriori subcell limiting of the discontinuous Galerkin finite element method for hyperbolic conservation laws, Journal of Computational Physics, 278 : 47-75, 2014.

˜ a — Group of Numerical Methods in Engineering Universidade da Corun

A Posteriori Detection

I Numerical Admissibility Detection (NAD) min (Un (y)) − δ 6 U∗(x) 6 max (Un (y)) + δ

y∈Vi

y∈Vi

• Vi is the set of all the cells of the stencil of cell i 1. 1

P. Tsoutsanis, Extended bounds limiter for high-order finite-volume schemes on unstructured meshes, Journal of Computational Physics, 362 :69-94, 2018.

˜ a — Group of Numerical Methods in Engineering Universidade da Corun

A Posteriori Detection

I Algorithm ADA

 FV SOLVER

storder BAD

storder BAD

Candidate solution

Uin

PAD GOOD

NAD

GOOD

Uin+1

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Outline •

Introduction

•

The ADA method

•

A Posteriori Detection

•

Numerical examples

•

Conclusions

˜ a — Group of Numerical Methods in Engineering Universidade da Corun

Decay of Compressible Isotropic Turbulence (I)

I Energy cascade

Figure taken from Davidson, “Turbulence. An introduction for scientist and engineers”, Oxford, 2006

I Effect of eddies smaller than grid spacing must be modelled ⇒ SGS models ˜ a — Group of Numerical Methods in Engineering Universidade da Corun

Decay of Compressible Isotropic Turbulence (II)

I SGS model based on Monotonically Integrated Large Eddy Simulation (MILES) I The numerical method must be designed to mimic the physics, to solve all of the energy transfer modes I The method can simulate laminar flows. I We analyze two different configurations of the decay of compressible isotropic turbulence in the nonlinear subsonic regime. I In this regime, weak shocklets develop spontaneously from the turbulent motion.

˜ a — Group of Numerical Methods in Engineering Universidade da Corun

Decay of Compressible Isotropic Turbulence (III)

I Test 1: Case 6 in Spyropoulos and Blaisdell (*) I Decay of turbulence in a periodic cube, big enough to minimize the influence of “surrounding” cubes

(*) Evaluation of the Dynamic Model for Simulations of Compressible Decaying Isotropic Turbulence. AIAA J., 34 (5),pp.990-998, 1996 ˜ a — Group of Numerical Methods in Engineering Universidade da Corun

Test 1

I The initial parameters of this case are Mt = 0.4 χ = 0.2 Reλ = 2157 2

2

(ρ0rms) / hρi = 0.032 2 2 0 (Trms ) / hT i = 0.005

I We use an initial value of the the dissipation coefficient ini = 0.15. I ini = 1 introduce excessive dissipation in the beginning of the simulation, that greatly affects the final results. χ is the fraction of energy in the dilatational part of the velocity

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Test 1

Vorticity isosurfaces and streamlines. ˜ a — Group of Numerical Methods in Engineering Universidade da Corun

Test 1

Vorticity isosurfaces and streamlines. ˜ a — Group of Numerical Methods in Engineering Universidade da Corun

Test 1

Kinetic energy decay. τ is the eddy turnover time

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Test 1

Kinetic energy decay. τ0 is the eddy turnover time. On the right, results from Visbal and Rizetta, Journal of Fluids Engineering, 124, 2002.

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Test 1

Density Fluctuations.

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Test 1

Normalized temperature fluctuations. ˜ a — Group of Numerical Methods in Engineering Universidade da Corun

Test 1

Instantaneous three-dimensional energy spectra (E(k) = ρ(u2 + v 2 + w2 )) at t/τ0 = 0.3. 323 grid. ˜ a — Group of Numerical Methods in Engineering Universidade da Corun

Test 1

Instantaneous three-dimensional energy spectra (E(k) = ρ(u2 + v 2 + w2 )) at t/τ0 = 0.3. 643 grid. ˜ a — Group of Numerical Methods in Engineering Universidade da Corun

Test 1

Numerical viscosity. ˜ a — Group of Numerical Methods in Engineering Universidade da Corun

Test 1

 values at time t/τ0 = 0.3 on a 323 grid.

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Test 1

 values at time t/τ0 = 0.3 on a 643 grid. ˜ a — Group of Numerical Methods in Engineering Universidade da Corun

Decay of Compressible Isotropic Turbulence

I Test 2: Johnsen’s case (*) I Decay of turbulence in a periodic cube, big enough to minimize the influence of “surrounding” cubes

(*) Assessment of high-resolution methods for numerical simulations of compressible turbulence with shock waves. Journal of Computational Physics 229, 1213?1237, 2010 ˜ a — Group of Numerical Methods in Engineering Universidade da Corun

Test 2

I The initial parameters of this case are Mt = 0.6 χ = 0 Reλ = 100 2

2

(ρ0rms) / hρi = 1 2 2 0 (Trms ) / hT i = 1

I We use an initial value of the the dissipation coefficient ini = 0.15. I ini = 1 introduce excessive dissipation in the beginning of the simulation, that greatly affects the final results.

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Test 2

Kinetic energy decay. τ is the eddy turnover time ˜ a — Group of Numerical Methods in Engineering Universidade da Corun

Test 2

Normalized temperature fluctuations.

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Test 2

Instantaneous three-dimensional velocity spectra (Eu (k) = u2 + v 2 + w2 ) at t/τ0 = 4. 643 grid.

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Test 2

 values at final time t/τ0 = 4. ˜ a — Group of Numerical Methods in Engineering Universidade da Corun

Test 2

Marked cells at final time t/τ0 = 4. ˜ a — Group of Numerical Methods in Engineering Universidade da Corun

Test 2

Normalized dilatation.

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Test 2

Normalized enstrophy.

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Outline •

Introduction

•

The ADA method

•

A Posteriori Detection

•

Numerical examples

•

Conclusions

˜ a — Group of Numerical Methods in Engineering Universidade da Corun

Conclusions

I A high-order finite volume scheme with adaptive numerical dissipation has been presented for the computation of compressible turbulent flows.

I It is based on the use of the ER for the adjustment of the amount of dissipation introduced by the Riemann solver.

I The stability of the numerical scheme for non-smooth flows is achieved by using the a posteriori paradigm.

I The formulation obtains accurate and very promising results in the isotropic decay of compressible turbulence.

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SHARK-FV 2018

A HIGH-ORDER FINITE VOLUME SCHEME WITH ADAPTIVE NUMERICAL DISSIPATION FOR COMPRESSIBLE TURBULENT FLOWS

Xes´ us Nogueira email: [email protected]

Thank you —

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Acknowledgments

I This work has been partially supported by: • The Ministerio de Econom´ıa y Competitividad of the Spanish Government, • Direcci´ on Xeral de I+D of the Conseller´ıa de Cultura, Educaci´ on e Ordenaci´ on Universitaria of the Xunta de Galicia, cofinanced with FEDER funds. • the Universidade da Coru˜ na (UDC), and • the Group of Numerical Methods in Engineering - GMNI

˜ a — Group of Numerical Methods in Engineering Universidade da Corun