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HIGHER-ORDER FV-MLS METHOD FOR THE INCOMPRESSIBLE NAVIER-STOKES EQUATIONS L. Ram´ıreza, X. Nogueiraa, S. Khelladib, J.C Chassaingc, I. Colominasa

a: Dept. of Applied Mathematics Civil Engineering School University of A Coru˜ na, Spain

b: Arts et M´ etiers Paris Tech 151 Boulevard de l’Hˆ opital 75013 Paris, France

c:Institute Jean Le Rond d’Alembert Case 162, 4 Place Jussieu 75252 Paris, France

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

Outline •

Introduction

•

The FV-MLS method

•

A high-order formulation for incompressible flows

•

High-order Fluid-Structure-Interaction techniques

•

Conclusions

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Introduction •

Introduction

•

The FV-MLS method

•

A high-order formulation for incompressible flows

•

High-order Fluid-Structure-Interaction techniques

•

Conclusions

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FV-MLS Applications

I FV-MLS Applications • • • • • • • •

All-speed Navier-Stokes Incompressible Navier-Stokes Linearized Euler Equations (acoustics) Navier-Stokes Korteweg equations Turbulence (ILES) High-order Sliding mesh applications Cavitating flows ...

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FV-MLS Applications

I FV-MLS Applications • • • • • • • •

All-speed Navier-Stokes Incompressible Navier-Stokes Linearized Euler Equations (acoustics) Navier-Stokes Korteweg equations Turbulence (ILES) High-order Sliding mesh applications Cavitating flows ...

Wednesday @11:00 by Xes´ us Nogueira

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The FV-MLS method •

Introduction

•

The FV-MLS method

•

A high-order formulation for incompressible flows

•

High-order Fluid-Structure-Interaction techniques

•

Conclusions

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The Finite Volume Method

I Let us consider a generic conservation law for the 2D domain ΩT U ∂U + ∇·F F =S ∂t

I Finite Volume discretization over ΩI : Z ΩI ΩI

UI ∂U dΩ + ∂t

Z

FH

F

FV

−F




· n dΓ = 0

ΓI

• F H → Hyperbolic-like term • F V → Elliptic-like term ˜ a — Group of Numerical Methods in Engineering Universidade da Corun

The Finite Volume Method

I Hyperbolic term: • Godunov approach U

I-1

I

j

I+1

I+2

x

• F H is the solution of a Riemann problem • Initial values → variables at both sides of the interface.

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The Finite Volume Method

I Hyperbolic term: • Godunov approach U

I-1

I

j

I+1

I+2

x

• F H is the solution of a Riemann problem • Initial values → variables at both sides of the interface. U L6=U R ˜ a — Group of Numerical Methods in Engineering Universidade da Corun

The Finite Volume Method

U

I-1

I

j

I+1

I+2

x

U L = U I + ∇U I · (x xj − x I ) xj − xI+1) U R = U I+1 + ∇U I+1 · (x

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The Finite Volume Method U

I-1

I

j

I+1

I+2

x

1 xj − xI ) + (x xj − xI )T H I (x xj − x I ) U L = U I + ∇U I · (x 2 1 ∇U I+1·(x xj − xI+1)+ (x xj − xI+1)T H I+1 (x U R = U I+1+∇ xj − xI+1) 2 ˜ a — Group of Numerical Methods in Engineering Universidade da Corun

The Finite Volume Method U

I-1

I

j

I+1

I+2

x

1 xj − xI ) + (x xj − xI )T HI (x xj − x I ) U L = U I + ∇U U I · (x 2 1 xj − xI+1)T HI+1 (x ∇U U R = U I+1+∇U U I+1 ·(x xj − xI+1)+ (x xj − xI+1) 2 ˜ a — Group of Numerical Methods in Engineering Universidade da Corun

Finite Volume Method. High-order schemes (II)

I Computation of high-order derivatives: • Easy on structured grids. • Unstructured grids⇒PROBLEM.

I We propose: • The use of Moving Least Squares (MLS) to obtain an accurate and multidimensional approximation of derivatives on unstructured grids.

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The basis: Kernel approximations (I)

I Kernel approximation is based on the properties of Dirac’s Delta distribution: Z u(x x) =

y ∈Ω

x − y )dΩ u(yy )δ(x

I Kernel approximation is defined as: x) = uh(x

Z y ∈Ω

x − y , ρ)dΩ u(yy )W (x

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The basis: Kernel approximations (II)

I In discrete form:

u ˆ(x x) =

n X

uj W (x x − xj , h)Vj

j=1

I Vj is the statistical volume of a particle j. I Compact support with r = 2h I h is the smoothing length. ˜ a — Group of Numerical Methods in Engineering Universidade da Corun

The basis: Kernel approximations (IV)

I Many functions used as kernels: splines, gaussians I An example, the cubic spline:  3 2 3 3  1 − s + s ≤1  2 4s  α  1 3 x − xj , h) = ν Wj (x x) = W (x (2 − s) 12 x − xj k kx s= h

h = k max (kx x − xj k)

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The basis: Kernel approximations (IV)

I Many functions used as kernels: splines, gaussians I An example, the cubic spline:  3 2 3 3  1 − s + s ≤1  2 4s  α  1 3 x − xj , h) = ν Wj (x x) = W (x (2 − s) 12 x − xj k kx s= h

h = k max (kx x − xj k)

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The basis: Kernel approximations (V)

I Another example: Exponential Kernel. 2 dm − c

2 s − c

( ) −e ( ) e W (x, x∗, κ) = 2 −( dm 1−e c ) ∗

s = |x − x | , dm

dm = 2 max (|xj − x |) , c = 2κ ∗

I 2D kernel⇒ product of two 1D kernels: Wj (x x, x∗, κx, κy ) = Wj (x, x∗, κx)Wj (y, y ∗, κy )

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The basis: Kernel approximations (V)

I Another example: Exponential Kernel. 2 dm − c

2 s − c

( ) −e ( ) e W (x, x∗, κ) = 2 −( dm 1−e c ) ∗

s = |x − x | , dm

dm = 2 max (|xj − x |) , c = κ 2κ ∗

I 2D kernel⇒ product of two 1D kernels: Wj (x x, x∗, κx, κy ) = Wj (x, x∗, κx)Wj (y, y ∗, κy )

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The basis: Kernel approximations (VI)

CUBIC SPLINE

EXPONENTIAL KERNEL

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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 (III)

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Moving Least Squares (IV)

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 In practice, Ω is a set of scattered points. Previous integrals are evaluated using points in Ωx as quadrature points: x)u uΩx x)P α (zz ) = M −1(x P Ωx W (x z =x

• uΩx contains nodal values of the function ux to be approximated, at nx nodes in Ωx uΩx = u(x x1) u(x x2) · · · u xnx



T

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Moving Least Squares (IV)

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 (V)

I Interpolation 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 (VI)

I A practical note about the polynomial basis p(x x) = 1

x

y

2

xy

x

y


2 T

• We define locally and scale the monomials of the basis • Better conditioning of the momentum matrix x) are evaluated at a point xI , • If MLS shape functions N (x the basis is evaluated at x−hxI • Then we can write: M −1(x xI )P P Ωx W (x xI ) = pT (00)C C (x xI ) N T (x xI ) = pT (00)M I

with

xI ) P Ωx W (x xI ) = M −1(x xI )P C (x I

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Moving Least Squares (VII)

I Computation of derivatives • First derivatives C (x x) ∂C x) ∂ppT(x x) ∂N N T(x T x) x) + p (x = C (x ∂x ∂x ∂x • Second derivatives x) ∂C C (x x) ∂ 2N T(x x) ∂ 2pT(x x) ∂ppT(x ∂ 2C (x x) x) + 2 x) = C (x + p(x 2 2 ∂x ∂x ∂x ∂x ∂x2 C (x C (x x) ∂ 2pT(x x) ∂ppT(x x) ∂C x) ∂C ∂ 2N T(x x) x) ∂ppT(x x) + = C (x + ∂x∂y ∂x∂y ∂x ∂y ∂y ∂x 2 x) ∂ C (x T x) + p (x ∂x∂y ˜ a — Group of Numerical Methods in Engineering Universidade da Corun

Moving Least Squares (VII)

• where ∂C∂x(x) is given by
x) C (x ∂W W (x ∂C x) −1 T W (x x)C x)W x) x) C (x = C (x I − p (x ∂x ∂x x) • and the second derivatives of C (x ∂ 2C (x x) ∂x2

=

 ∂C C (x x) −1 ∂W  T C (x W (x x) I − p (x x)C x) ∂x ∂x  ∂ 2 W (x x)  −1 T C (x C (x W (x I − p (x x)C x) +C x)W x) ∂x2  ∂W (x x) −1 ∂W (x x)  T −1 C (x C (x W (x W (x x) I − p (x x)C x) −C x)W x) ∂x ∂x ∂W (x x) T ∂C C (x x) C (x W −1 (x p (x x) −C x)W x) ∂x ∂x

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Moving Least Squares (VII)

x) ∂ 2C (x ∂x∂y

=

 ∂W (x x)  x) −1 ∂C C (x T C (x x) x)C x) W (x I − p (x ∂y ∂x  ∂ 2 W (x x)  T W (x C (x C (x x) x)W +C x) x)C I − p (x ∂x∂y  ∂W (x x) −1 ∂W (x x)  −1 T C (x W (x C (x −C x)W x) x) x)C x) W (x I − p (x ∂y ∂x −1

W C (x x)W −C

−1

(x x)

∂W (x x) T x) ∂C C (x x) p (x ∂x ∂y

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Moving Least Squares (VII)

I Computation of derivatives • The diffuse derivatives are obtained by neglecting all x) derivatives of C (x x) ∂ 2N T(x ∂ 2pT(x x) x) ≈ C (x 2 2 ∂x ∂x x) x) ∂ 2N T(x ∂ 2pT(x x) ≈ C (x ∂x∂y ∂x∂y ∂ nN T(x x) ∂ npT(x x) x) ≈ C (x n n ∂x ∂x

Huerta et al.,Pseudo-divergence-free Element Free Galerkin method for incompressible fluid flow,CMAME,2004 ˜ a — Group of Numerical Methods in Engineering Universidade da Corun

Moving Least Squares (VII) • However, as we already have computed the first derivatives x), it is possible to use a semi-diffuse approach of C (x without extra effort: x) C (x x) ∂C ∂ 2N T(x x) ∂ 2pT(x x) ∂ppT(x x) + 2 ≈ C (x 2 2 ∂x ∂x ∂x ∂x C (x C (x x) ∂ppT(x x) ∂C ∂ 2N T(x x) ∂ 2pT(x x) x) ∂ppT(x x) ∂C x) + ≈ C (x + ∂x∂y ∂x∂y ∂x ∂y ∂y ∂x Derivatives

L1 Error

L2 Error

Diffuse

1.631 × 10−5

4.784 × 10−5

Semi-Diffuse

1.586 × 10−5

4.710 × 10−5

Full

1.288 × 10−5

3.656 × 10−5

B It has been proved that use of diffuse or semi-diffuse derivatives does not decrease the order of accuracy. B It should be noted that the accuracy is affected

Accuracy assesment of a high-order moving least squares finite volume method for compressible flows,C&F,2013 ˜ a — Group of Numerical Methods in Engineering Universidade da Corun

The FV-MLS method (I)

I Note that due to the local scaling x−hxI pT(00) = (1

∂ppT(00) = ∂x T

∂pp (00) = ∂y

 0

0

1 h

 0

0

0

0

0

0)

 0

1 h

0

0

0

 0

0

0

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The FV-MLS method

I This scheme acknowledge the different nature of convective and diffusive terms. I We start from a high-order, continuous MLS approximation of the solution: I Convective terms discretization: • Breaks the continuous representation of the MLS approximation. • Obtains a continuous representation of the variables inside each cell.

I Diffusive terms discretization is: • Centered→ Direct interpolation at Gauss points with MLS. • Continuous. • Highly accurate. ˜ a — Group of Numerical Methods in Engineering Universidade da Corun

The FV-MLS method

I Hyperbolic term → Flux Difference Splitting U

I-1

I

j

I+1

I+2

x

1 xj − xI ) + (x xj − xI )T HI (x xj − x I ) U L = U I + ∇U U I · (x 2 1 ∇U xj − xI+1)T HI+1 (x U R = U I+1+∇U U I+1 ·(x xj − xI+1)+ (x xj − xI+1) 2 ˜ a — Group of Numerical Methods in Engineering Universidade da Corun

The FV-MLS method

I Hyperbolic-like terms: • MLS is used to compute the gradients and high-order derivatives required for the reconstruction of the variable at integration points placed at interface.

∇U I =

nx X

xI ) U j ∇N j (x

j=1

T C (x ∂N N (x x) ∂ppT(00) ∂C xI ) T xI )+pp (00) = C (x ∂x ∂x ∂x xI ) xI ) = M −1(x xI )P P Ωx W (x C (x I x=xI
C (x ∂C x) ∂W W (x x ) W −1(x x) x)C C (x x) = C (x x)W I − pT(x ∂x ∂x ˜ a — Group of Numerical Methods in Engineering Universidade da Corun

The FV-MLS method

I Hyperbolic-like terms: • MLS is used to compute the gradients and high-order derivatives required for the reconstruction of the variable at integration points placed at interface. n

x xI ) ∂ 2U I X ∂ 2Nj (x = Uj 2 2 ∂x ∂x j=1

xI ) xI )P P Ωx W (x C (x xI ) = M −1(x

2 T N x ∂ (x) ∂ 2pT(00) xI ) ≈ C (x 2 2 ∂x ∂x x=xI

I

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The FV-MLS method

I Hyperbolic-like terms: • MLS is used to compute the gradients and high-order derivatives required for the reconstruction of the variable at integration points placed at interface. n

x xI ) ∂ nU I X ∂ nNj (x = Uj n n ∂x ∂x j=1

xI ) xI )P P Ωx W (x C (x xI ) = M −1(x

n T N x ∂ (x) ∂ npT(00) ≈ C (x xI ) n n ∂x ∂x x=xI

I

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The FV-MLS method

I Elliptic-like terms: • Direct interpolation at Gauss points with MLS.

U iq =

niq X

xiq ) U j N j (x

j=1

N T (x xiq ) = pT (00)C C (x xiq )

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The FV-MLS method

I Elliptic-like terms: • Direct interpolation at Gauss points with MLS.

∇U iq =

niq X

xiq ) U j ∇N j (x

j=1

T C (x ∂N N (x x) ∂ppT(00) ∂C xiq ) T xiq )+pp (00) = C (x ∂x ∂x ∂x x=xiq

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The FV-MLS method (V)

I Vertices and/or centroids of the control cells are the “particles” to perform the MLS approximation. I We need to define stencils to “mark” the neighbor particles that define the cloud of points.

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The FV-MLS method (VI)

I How to define stencils? I There exists an optimal size nxI of points in the stencil such as Nmin < nxI Nmin

(d + order)! = d!order!

I If it is large⇒excessive dissipation I Maybe optimization??

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732

accompanied by a zero fill-in incomplete LU (ILU(0)) factorization, see [45] for more details. The obtained system using P is solved via a standard LU decomposition solver. This approximation can appear as a ‘‘crude’’ approximation of the Jacobian matrix, but using even a poor approximation can be surprisingly superior to using the Newton–Krylov method with no preconditioning [45].

733

4. Computational aspects

734

4.1. Stencils: size and distribution

735

There exists an optimal size nxI of points in the stencil such as N min 6 nxI 6 N max . The minimum number of points correspond to the dimension of the polynomial basis, it is given by

726 727 728 729 730 731

the dissipation is very small, the scheme can become unstable, this is due to the fact that waves with high wavenumber are not well solved by the numerical scheme and present high dispersion errors (see Fig. 2). This is why one should be very careful when defining the minimum diffusion criterion.

768

4.2. Boundary conditions

773

For our modeling, the boundary conditions enter in the discretized equations through a proper definition of the numerical flux. The numerical integration at the Gauss points at the boundary interfaces can be written as H(, , U+, U⁄, n), where n is the outward normal unit vector from the domain and U⁄ is the external state variable. Depending on the boundary type, the construction of U⁄ accounts for, both, the physical boundary conditions that must be enforced and the information leaving the domain. In this paper, for Navier–Stokes equations we use no-slip boundary condition to model rigid walls and far-field boundary condition to model the external boundaries of the computational domain. For LEE, a perfectly reflecting boundary condition is easily obtained by defining, at each Gauss points on the rigid wall boundaries, an external mirror fictitious state U⁄. For external boundaries, we propose a new methodology to model non-reflecting (absorbing) boundary conditions based on combining a stretching zone with MLS filtering and an upwinding technique used by Bernacki et al. [47] with DG.

774

The FV-MLS method (VII)

769 770 771 772

775

I We want stencils as compact as possible by using layers of cells around the active cell I In practice this requires a high number of points in the stencil I To overcome this inconvenient, last particles are placed such as satisfying a barycentric equilibrium 736

737 738 741

740

Nmin

ðd þ orderÞ! :¼ m ¼ d!order!

ð26Þ

742

743 744 745 746

747 749

 for d = 2: N min ¼  for d = 3: N min ¼

1 ðorder 2 1 ðorder 6

þ 1Þðorder þ 2Þ, þ 1Þðorder þ 2Þðorder þ 3Þ.

The maximum number of points in a stencil is more difficult to estimate. It can be defined as

Nmax ¼ Nmin þ nþ ;

ð27Þ

Fig. 4. MLS 5th order scheme stencil (4th order polynomial basis): Nmin = 15 points and Nmax = Nmin + 4 = 19 points (Eq. (27)). (A) Compact stencil distribution and (B) non-compact stencil distribution with respect to Nmax = 19 points.

˜ a — Group of Numerical Methods in Engineering Universidade da Corun Please cite this article in press as: S. Khelladi et al., Toward a higher order unsteady finite volume solver based on reproducing kernel methods, Comput. Methods Appl. Mech. Engrg. (2011), doi:10.1016/j.cma.2011.04.001

776 777 778 779 780 781 782 783 784 785 786 787 788 789 790 791 792

The FV-MLS method (VII)

I Boundary conditions: We impose them on the numerical fluxes

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The FV-MLS method (VII)

I However, in order to improve the reconstruction we include ghost cells in the stencil

∇U I =

nx X

xI ) U j ∇N j (x

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

The likely explanation is that by increasing the order of approximation, we obtain a more accurate solution to a wrong problem: flow around a polygon. Rarefaction waves are formed at the vertices of the polygon [2]. These are better resolved with higher p. Density plots near the top of the cylinder (with the background mesh) in Figure 1, right, demonstrate concentration of the error near vertices. Isolines take a wave-like shape instead of a smooth curve. This becomes increasingly so and affects solution in further parts of the domain as p increases.

A note on curved boundaries

Figure Mach isolines density at the (right) top of the cylinder (right) boundary with reflecting Mach isolines (left) and1:density at the(left) top and of the cylinder with reflecting conditions. p = 1, 2, boundary p = 1, 2, 3,and fromBerger, top to bottom. A wakeAccurate is formed Implementation at the rear; the of Solid Wall 3, from top to bottom. Taken conditions. from Krivodonova High-Order solution does not achieve aJCP, steady symmetric irrotational form. Boundary Conditions in Curved Geometries, 2006

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A note on curved boundaries

CELL I DG (p=2) FV-MLS MLS GHOST CELLS

Schematic representation of the differences on curved boundary discretization between FV-MLS and DG. Shaded cells represent the MLS stencil.

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A note on curved boundaries

8

J.-C. Chassaing et al. / Computers & Fluids 71 (2013) 41–53

ig. 9. Third-order accurateThird-order FV-MLS computation a 64  16 grid using boundary evaluations based on anormal straightevaluations representation (a) on or on accurate on FV-MLS computation on a 64 normal × 16 grid using boundary based a a phys b) for curved geometry. straight representation (a) or on a physical representation (b) for curved geometry.

Accuracy assesment of a high-order moving least squares finite volume method for compressible flows,C&F,2013

CELL I DG (p=2) FV-MLS

neglecting the first derivatives of C(x) (p = 1; case gives acceptable results compared to orders of accu using the full p = 1 MLS derivatives (case B in Table tained for p = 2 confirm that the use of the semi-diffu ˜ a — Group of Numerical Universidade da Corun Methods in Engineering tion not only increases the formal order of the num scheme but also decreases its error level.

Properties of the FV-MLS method

I We perform a Fourier Analysis for the 1D linear advection equation. I We obtain the dispersion-dissipation properties. I We compare MLS interpolation with Piecewise Polynomial Interpolation. I We check the order of convergence.

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Dispersion and dissipation (I)

I Dispersion error: Associated with the error in the speed of the wave propagation I Dissipation error: Associated with the error in the wave amplitude

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Dispersion and dissipation

3

1st order upwind

0

2nd order FV−MLS s =5 x

2.5

rd

3 order FV−MLS sx=5

−0.5

Exact

Im(κ ∆x)

1.5

−1

*

Re(κ*∆x)

2

−1.5

1 st

1 order upwind −2

0.5

nd

2

order FV−MLS sx=5

3rd order FV−MLS sx=5

0 0

0.5

1

1.5

κ∆x

DISPERSION

2

2.5

3

−2.5 0

0.5

1

1.5

κ∆x

DISSIPATION

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

2

2.5

3

Dispersion-Dissipation Properties. Cubic spline kernel

DISPERSION

DISSIPATION

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Dispersion-Dissipation Properties. Exponential kernel

DISPERSION

DISSIPATION

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IS MLS INTERPOLATION ACCURATE?

A COMPARISON BETWEEN Piecewise Polynomial Interpolation (PPI) AND FV-MLS

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

IS MLS INTERPOLATION ACCURATE?

I We compare interpolation with equivalent spatial resolution by using Moving Least Squares (MLS) (cubic basis) and Piecewise Polynomial Interpolation (PPI) (p = 3).

Division of a p = 3 element to obtain a FV-MLS grid with equivalent spatial resolution. ˜ a — Group of Numerical Methods in Engineering Universidade da Corun

IS MLS INTERPOLATION ACCURATE?

I We compare interpolation with equivalent spatial resolution by using Moving Least Squares (MLS) (cubic basis) and Piecewise Polynomial Interpolation (PPI) (p = 3).

Division of a p = 3 element to obtain a FV-MLS grid with equivalent spatial resolution. ˜ a — Group of Numerical Methods in Engineering Universidade da Corun

IS MLS INTERPOLATION ACCURATE?

I We compare interpolation with equivalent spatial resolution by using Moving Least Squares (MLS) (cubic basis) and Piecewise Polynomial Interpolation (PPI) (p = 3).

Division of a p = 3 element to obtain a FV-MLS grid with equivalent spatial resolution. ˜ a — Group of Numerical Methods in Engineering Universidade da Corun

Interpolation on a structured grid

I Function: u(x, y) = sin (2πx) sin (2πy) • 13 × 13 p = 3 elements on a cartesian [0, 1] × [0, 1] grid. • 39 × 39 FV-MLS elements on a cartesian [0, 1] × [0, 1] grid. • We interpolate for both grids at the same points (located at the 4 × 4 Gauss-Legendre points of each FV-MLS element). Absolute value of the error in the variable PPI

MLS

Absolute value of the error in the derivative PPI

MLS

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Interpolation on a distorted grid

I Function: u(x, y) = sin (2πx) sin (2πy)

Absolute value of the error in the variable PPI

MLS

Absolute value of the error in the derivative PPI

MLS

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Order of Convergence. Ringleb Flow (I)

I Domain: −1.15 ≤ x ≤ −0.75 , 0.15 ≤ y ≤ 0.55

Mach isolines.

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

Order of Convergence. Ringleb Flow (II) Grids

FV-MLS 15 x 15 30 x 30 60 x 60

I The order of convergence is the expected one. ˜ a — Group of Numerical Methods in Engineering Universidade da Corun

Order of Convergence. Poisson (I)

−∆u = f u = gD

in Ω on ΓD

u(x, y) = exp(α sin(Ax + By) + β cos(Cx + Dy))

Isolines of the exact solution for u. ˜ a — Group of Numerical Methods in Engineering Universidade da Corun

Order of Convergence. Poisson (II)

h/h0

DG p = 3 Error L2 u

FV-MLS Error L2 u

1

2.50 E − 04

8.34 E − 05

0.5

1.20 E − 05

5.60 E − 06

0.25

6.05 E − 07

3.75 E − 07

0.125

3.16 E − 08

2.52 E − 08

h/h0

DG p = 3 Order of Convergence u

FV-MLS Order of Convergence u

DG p = 3 Order of Convergence s

FV-MLS Order of Convergence s

1

−

−

−

−

0.5

4.38

3.86

3.83

3.54

0.25

4.31

3.99

3.69

3.52

0.125

4.26

3.89

3.60

3.46

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A first motivating example. 1D Linear advection equation • First ICASE/LaRC Workshop on Benchmark problems in CAA 2 ∂u ∂u − ln(2)( x ) 3 • We solve +a = 0 with u(x, 0) = 0.5e ∂t ∂x 0.5 Exact 2nd order FV−MLS

0.4

3rd order FV−MLS 0.3

0.2

u 0.1

0

−0.1

380

390

400

410

420

430

440

450

x

1D Linear advection equation, a = 1, t = 400, ∆x = 1, CF L = 0.6 ˜ a — Group of Numerical Methods in Engineering Universidade da Corun

A first motivating example. 1D Linear advection equation • First ICASE/LaRC Workshop on Benchmark problems in CAA 2 ∂u ∂u − ln(2)( x 3) • We solve +a = 0 with u(x, 0) = 0.5e ∂t ∂x 0.5

0.5 Exacta κx=6

t=100 0.4

u

κ =4

0.2

u

0.1

0

0

−0.1

−0.1

100

x

110

120

κx=1 κx=4

0.2

0.1

90

κx=6

0.3

x

80

Exacta

0.4

κx=1 0.3

t=400

380

390

400

410

420

x

1D Linear advection equation, a = 1, t = 400, ∆x = 1, CF L = 0.6

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A first motivating example. 1D Linear advection equation • Solution with a fourth order MacCormack scheme, ∆x = 1, CF L = 0.2

Viswanathan, Sankar, A Comparative Study of Upwind and MacCormac schemes for CAA Benchmark problems, First ICASE/LaRC Workshop on Benchmark problems in CAA, NASA Conference Publication 3300, 185-195, 1995

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

A CAA example on an unstructured grid.

I Solve the LEE for the convection of a monopolar source

Sketch of the problem.

S = e

−α[(x−xs)2+(y−ys)2]

sin wt

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A CAA example on an unstructured grid.

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

A CAA example on an unstructured grid.

Exact w ith m ass-m atrix,5th order

0.25

0.15

p

0.05

-0.05

-0.15

-0.25

-0.35 -100

-80

-60

-40

-20

0

20

40

60

80

100

x

Acoustic pressure profile across y = 0

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A high-order formulation for incompressible flows •

Introduction

•

The FV-MLS method

•

A high-order formulation for incompressible flows

•

High-order Fluid-Structure-Interaction techniques

•

Conclusions

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

A high-order formulation for incompressible flows B

Introduction

B

Formulation

B

Numerical Examples

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

A high-order formulation for incompressible flows B

Introduction

B

Formulation

B

Numerical Examples

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

Introduction

I Incompressibility assumption:

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

Introduction

I Checkerboard:

Z ΩI

∂p dΩ = ∂x

Nf NG X X

[pj n ˆ xj ]ig Wig

j=1 ig=1

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

Introduction

I Checkerboard:

I-1 w

Z ΩI

I

e I+1

∂p dΩ = (pˆ nx)e + (pˆ n x )w ∂x

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

Introduction

I Checkerboard:

I-1 w

Z ΩI

I

e I+1

∂p dΩ = (p)e − (p)w ∂x

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

Introduction

I Checkerboard:

I-1 w

(p)e =

pi+1 + pi 2

Z ΩI

I

e I+1

(p)w =

pi + pi−1 2

∂p dΩ = (p)e − (p)w ∂x

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

Introduction

I Checkerboard:

I-1 w

(p)e =

pi+1 + pi 2

Z ΩI

I

e I+1

(p)w =

pi + pi−1 2

∂p pi+1 − pi−1 dΩ = ∂x 2

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Introduction

I Checkerboard:

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Introduction

I Checkerboard:

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Introduction

I In order to solve the checkerboard: • Collocated grid arrangement → Special interpolation (MIM)

• Staggered grid arrangement → Special location of the variables

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Introduction

I Collocated grid arrangement → u, v, p located at cell centroid.

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

Introduction

I Staggered grid → u, v, p located at different locations.

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

Introduction

I Staggered grid → u control volume.

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

Introduction

I Staggered grid → v control volume.

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

Introduction

I Staggered grid arrangement: • Variables stored at different locations • No interpolations required • Drawback → Complex in unstructured and/or 3D grids

I Collocated grid arrangement: • Variables stored at cell centroid • Structured and unstructured grid • Drawback → Possibility of checkerboard

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

Introduction

I Staggered grid arrangement: • Variables stored at different locations • No interpolations required • Drawback → Complex in unstructured and/or 3D grids

I Collocated grid arrangement: • Variables stored at cell centroid • Structured and unstructured grid • Drawback → Possibility of checkerboard

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

Introduction

I Staggered grid arrangement: • Variables stored at different locations • No interpolations required • Drawback → Complex in unstructured and/or 3D grids

I Collocated grid arrangement: • • • •

Variables stored at cell centroid Structured and unstructured grids Drawback → Possibility of checkerboard Special interpolation is required

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

A high-order formulation for incompressible flows B

Introduction

B

Formulation

B

Numerical Examples

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

Incompressible Navier Stokes

I Incompressible Navier-Stokes: U 1 ∂U +U U · (∇U U ) = −∇p + (∆U U) ∂t Re ∇·U = 0

where U = (u, v)T is the velocity field, p(x, y, t) is the pressure variable and Re denotes the Reynolds number. I Resolution procedure: • A collocated Semi-Implicit Method for Pressure Linked Equations (SIMPLE). • Momentum Interpolation Method to avoid checkerboard oscillations. ˜ a — Group of Numerical Methods in Engineering Universidade da Corun

SIMPLE

Start

Set time t=t+Δt

Solve the momentum equation

Time step (n)

Solve the pressure correction equation

Inner iteration (m)

Correct velocity and pressure fields

Convergence?

t>tmax?

Exit

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SIMPLE

Start

Set time t=t+Δt

Solve the momentum equation

Time step (n)

Solve the pressure correction equation

Inner iteration (m)

Correct velocity and pressure fields

Convergence?

t>tmax?

Exit

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SIMPLE

I Momentum equation • Cell centered finite volume scheme Z ΩI

U ∂U dΩ+ ∂t

Z

Z U · (∇U U )dΩ = −

ΩI

∇pdΩ+ ΩI

1 Re

Z

 (∆U U )dΩ

ΩI

• Discretized momentum equation VI

3U U m+1,n+1 I

4U U nI

− 2∆t

+ U n−1 I

+

Nf NG h X X j=1 ig=1

Hjm,n+1U jm+1,n+1

i ig

Wig =

Nf NG h Nf NG h i i X X m,n+1 X X 1 ˆ j Wig + ˆ j Wig =− pj ·n ∇U U jm+1,n+1 · n Re j=1 ig=1 ig ig j=1 ig=1 ˜ a — Group of Numerical Methods in Engineering Universidade da Corun

SIMPLE

I Momentum equation • Cell centered finite volume scheme Z ΩI

U ∂U dΩ+ ∂t

Z

Z U · (∇U U )dΩ = −

ΩI

∇pdΩ+ ΩI

1 Re

Z

 (∆U U )dΩ

ΩI

• Discretized momentum equation VI

3U U m+1,n+1 I

4U U nI

− 2∆t

+ U n−1 I

+

Nf NG h X X j=1 ig=1

Hjm,n+1U jm+1,n+1

i ig

Wig =

Nf NG h Nf NG h i i X X m,n+1 X X 1 ˆ j Wig + ˆ j Wig =− pj ·n ∇U U jm+1,n+1 · n Re j=1 ig=1 ig ig j=1 ig=1 ˜ a — Group of Numerical Methods in Engineering Universidade da Corun

SIMPLE

I Higher-order approximations are made using MLS: • Pressure term

pj =

nx X

xj )pk Nkg (x

k=1

• Diffusive term

Uj = ∇U

nx X

Ul xj )U ∇Nlg (x

l=1

• Convective term ⇒ Deferred correction approach Uj = ( U LO = j


LO m+1,n+1 Uj UI

, Hj ≥ 0

UN

, Hj < 0

m,n+1

+ (U U HO − U LO j j ) U HO = j

nx X

Nkg (x xj )U Uk

k=1

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

SIMPLE

Start

Set time t=t+Δt

Solve the momentum equation

Time step (n)

Solve the pressure correction equation

Inner iteration (m)

Correct velocity and pressure fields

Convergence?

t>tmax?

Exit

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

SIMPLE

I Pressure correction equation Z

Z ∇ · U dΩ =

ΩI

Nf NG h i X X ˆj · n U U · nj dΓ = ˆ j Wig

ΓI

j=1 ig=1

ig

• In order to avoid checkerboard oscillations ⇒ Momentum Interpolation Method (MIM) ˆ j = U ∗j + U



 h i
VI ∇pI j − ∇pj aI j

• The MIM was proposed by Rhie and Chow in 1983.

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SIMPLE

I Pressure correction equation • Checkerboard oscillations⇒Momentum Interpolation Method ˆ j = U ∗j + U



 h i
VI ∇pI j − ∇pj aI j

B These terms are usually obtained at integration point j using linear interpolation. B We propose to use higher-order approximations using MLS     nx nx X X VI VI xj ) xj )U U ∗j = Nkg (x = Nkg (x U ∗k aI j aI k k=1

∇pI


j

=

k=1

nx X k=1

xj )∇pk Nkg (x

∇pj =

nx X

∇Nlg (x xj )pl

l=1

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SIMPLE

I Pressure correction equation • A pressure correction equation is solved in order to impose the continuity “constraint”. Nf

# Nf NG "   h  i XX XX 0 VI ˆ Wig = 0 ˆj ∇p · n Uj · n ˆ j Wig − aI j j ig j=1 ig=1 j=1 ig=1 NG

ig

0

• The pressure correction, p , is the unknown. • Approximations at integration point are obtained with MLS.

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

SIMPLE

Start

Set time t=t+Δt

Solve the momentum equation

Time step (n)

Solve the pressure correction equation

Inner iteration (m)

Correct velocity and pressure fields

Convergence?

t>tmax?

Exit

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

SIMPLE

I Correct velocity and pressure fields at cell centroids as VI  0  ∇p U =U +U =U − aI I  0 m+1,n+1 pm+1,n+1 = pm,n+1 + p 

0

• The value ∇p

 I

0

∗

m+1,n+1

∗

is approximated at cell centroid using MLS 0

∇pI =

nx X

0 g ∇Nl (x xj )pl

l=1

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

SIMPLE

Start

Set time t=t+Δt

Solve the momentum equation

Time step (n)

Solve the pressure correction equation

Inner iteration (m)

Correct velocity and pressure fields

Convergence?

t>tmax?

Exit

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

A high-order formulation for incompressible flows B

Introduction

B

Formulation

B

Numerical Examples

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

Numerical Examples

I Kovasznay Flow. u(x, y)=1 − eαx cos (2πy) v(x, y) =

α αx sin (2πy) 2π e 1 2

p(x, y) =

1−e

2αx

α=

Re 2

−

q

Re2 4

+ 4π 2




• Domain Ω = [−0.5, 0.5] × [0.5, 0.5]. Re=40

u

v

p

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

Numerical Examples

I Kovasznay Flow. −4

10

u−velocity v−velocity pressure Order 4

−5

L2 norm error

10

−6

10

−7

10

−8

10

−9

10

3

10

4

10 Number of Control Volumes

5

10

• The formal order of accuracy is recovered ˜ a — Group of Numerical Methods in Engineering Universidade da Corun

Numerical Examples

I 2D Taylor-Green Flow. u(x, y, t)=

e

cos (y) sin (x)

−2t

−e Re cos (x) sin (y)

v(x, y, t) = p(x, y, t) =

−2t Re

−4t e Re

4

(cos (2x) + cos (2y))

• Domain Ω = [0, 2π] × [0, 2π]. Re=100

u

v

p

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

Numerical Examples

I 2D Taylor-Green Flow. −2

−3

10

u−velocity v−velocity pressure Order 3

−3

−4

10

−5

10

−6

u−velocity v−velocity pressure Order 4

−4

10

L2 norm error

L2 norm error

10

−5

10

−6

10

−7

10

10

−7

10

10

−8

3

4

10

10 Number of Control Volumes

10

3

4

10

10 Number of Control Volumes

• The formal order of accuracy is recovered

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

Numerical Examples

I Cavity Flow

Re=1000 ˜ a — Group of Numerical Methods in Engineering Universidade da Corun

Numerical Examples

I Cavity Flow. 1635 cells. 0.4

0.9

0.3

0.8

0.2

Velocity component v

1

Position y

0.7 0.6 0.5 th

FV−MLS 4 order Re=100 Reference Solution (Ghia) Re=100

0.4

th

0.3

FV−MLS 4 order Re=400 Reference Solution (Ghia) Re=400

0.2

FV−MLS 4th order Re=1000 Reference Solution (Ghia) Re=1000

0.1 0 −0.5

−0.25

0

0.25

0.5

Velocity component u

0.75

1

0.1 0 −0.1 −0.2 −0.3

FV−MLS 4th order Re=100 Reference Solution (Ghia) Re=100

−0.4

FV−MLS 4th order Re=400 Reference Solution (Ghia) Re=400

−0.5

FV−MLS 4th order Re=1000 Reference Solution (Ghia) Re=1000

−0.6 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Position x

• Excellent agreement with the reference solution for different Reynolds number. The reference solution is obtained on a 128x128 structured mesh (16384 cells). ˜ a — Group of Numerical Methods in Engineering Universidade da Corun

Numerical Examples

I Laminar Flow around a cylinder • Benchmark proposed by Sch¨afer and Turek.

• Parabolic velocity profile at inlet 4Umy(H − y) u(0, y) = , v(0, y) = 0 H2

• Two test cases B Reynolds 20 B Reynolds 100 Reference Solution: Sch¨afer, M., Turek, S., Benchmark Computations of Laminar Flow Around a Cylinder, Notes on Numerical Fluid Mechanics, Volume 52 , pp. 547-566, 1996. ˜ a — Group of Numerical Methods in Engineering Universidade da Corun

Laminar Flow around a cylinder • Reynolds 20

Mesh

Order

CD

CL

La

∆p

Mesh A

2

5.5869

0.0087

0.0881

0.1149

(4968 cells)

3

5.5919

0.0108

0.0851

0.1161

Mesh B

2

5.5817

0.0113

0.0851

0.1168

(19079 cells)

3

5.5859

0.0107

0.0845

0.1174

Upper bound

−

5.5900

0.0110

0.0852

0.1176

Lower bound

−

5.5700

0.0104

0.0842

0.1172

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

Laminar Flow around a cylinder • Reynolds 20

La

Mesh

Order

CD

CL

La

∆p

Mesh A

2

5.5869

0.0087

0.0881

0.1149

(4968 cells)

3

5.5919

0.0108

0.0851

0.1161

Mesh B

2

5.5817

0.0113

0.0851

0.1168

(19079 cells)

3

5.5859

0.0107

0.0845

0.1174

Upper bound

−

5.5900

0.0110

0.0852

0.1176

Lower bound

−

5.5700

0.0104

0.0842

0.1172

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

Laminar Flow around a cylinder • Reynolds 100

Mesh

Order

CD max

CL max

St

∆p

Mesh A

2

3.2741

1.2246

0.2825

2.3548

(4968 cells)

3

3.2986

1.0451

0.2924

2.3962

Mesh B

2

3.2702

1.0662

0.2952

2.4731

(19079 cells)

3

3.2380

0.9985

0.3008

2.4858

Upper bound

−

3.2400

1.0100

0.3050

2.5000

Lower bound

−

3.2200

0.9900

0.2950

2.4600

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

High-order Fluid-Structure-Interaction techniques •

Introduction

•

The FV-MLS method

•

A high-order formulation for incompressible flows

•

High-order Fluid-Structure-Interaction techniques

•

Conclusions

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

High-order Fluid-Structure-Interaction techniques

I Incompressible flow around a cross-flow turbine.

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

High-order Fluid-Structure-Interaction techniques

I Incompressible flow around a cross-flow turbine.

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

High-order Fluid-Structure-Interaction techniques

I High-order Sliding Mesh Techniques

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

High-order Fluid-Structure-Interaction techniques

I High-order Sliding Mesh Techniques

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

High-order Fluid-Structure-Interaction techniques

I High-order Sliding Mesh Techniques

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

High-order Fluid-Structure-Interaction techniques

I High-order Sliding Mesh Techniques

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

High-order Fluid-Structure-Interaction techniques

I High-order Sliding Mesh Techniques

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

High-order Fluid-Structure-Interaction techniques

I High-order Sliding Mesh Techniques

It will be presented tomorrow ˜ a — Group of Numerical Methods in Engineering Universidade da Corun

Numerical Examples

I Incompressible flow around a cross-flow turbine.

f~ =

n

fx fy

o

I =

fN = fy cosθ − fxsinθ

~ · ~n))dΓ (p~n − ν(∇U fT = −fxcosθ − fy sinθ

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

High-order Fluid-Structure-Interaction techniques

I Fluid-Structure Interaction (FSI) • Flow driven approach → ω given by the fluid ω • • • •

n+1

(T − M ) ∆t =ω + J n

T → Torque M → Loading Moment ∆t → Time step J → Mass moment of inertia

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Further research

I Fluid-Structure Interaction (FSI)

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

Conclusions •

Introduction

•

The FV-MLS method

•

A high-order formulation for incompressible flows

•

High-order Fluid-Structure-Interaction techniques

•

Conclusions

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

Conclusions

I We have a proposed a new higher-order accurate FV

formulation for the numerical solution of incompressible fluid flows on unstructured meshes.

I We have modified the usual linear formulation of MIM to introduce higher-order approximations using MLS.

I The proposed methodology obtains excellent results. I This methodology can be easily included in existing finite volume codes which represents an additional advantage.

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

SHARK-FV 2015

HIGHER-ORDER FV-MLS METHOD FOR THE INCOMPRESSIBLE NAVIER-STOKES EQUATIONS

Luis Ram´ırez email: [email protected]

Thank you —

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

Acknowledgments

I This work has been partially supported by: • The Ministerio de Educaci´ on y Ciencia of the Spanish Government, • Direcci´ on Xeral de I+D of the Conseller´ıa de Innovaci´ on, Industria e Comercio of the Xunta de Galicia, • 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

Some FV-MLS references • • • • • • • •

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