High Order Interpolation Methods and Related ...

High Order Interpolation Methods and Related URANS Schemes on Composite Grids Jean-Marie Le Gouez1, Vincent Couaillier2 and Florent Renac3 CFD and Aeroacoustics Department, ONERA, BP 72, 29 Avenue de la Division Leclerc, 92322 Chatillon France

This paper deals with high order interpolation methods and related URANS schemes under development at the CFD and Aerocaoustics Department of Onera, both in the context of Finite Volume and Discontinuous Galerkin methods, i.e. high order interpolants on respectively wide stencils and low number of degrees on freedom (collocated cell centered FV) and single cell-restrained stencil and high number of dofs (DG). Each of the interpolation problems and related spatial integration schemes are described (upwind schemes with characteristic splitting and centered schemes with scalar artificial dissipation). They are tested on a number of simple linear advection and diffusion problems, respectively on structured and unstructured grids, with a variety of boundary conditions. 2D steady flow dynamics test problems are investigated. 3D Navier-Sokes unsteady cases (Homogeneous and Isotropic Turbulence on tetrahedral grids) permit to evaluate the capacity of the reconstruction schemes to extend the non-linear response towards wave numbers close to the grid resolution, by not limiting the convection of smooth extrema. A discussion on the respective advantages and drawbacks of these methods with respect to the reference 2nd order FV URANS codes is done. This analysis permits to conduct an early estimation of the future efforts dedicated to their integration in the industrial software suite for aerodynamics problems.

I. Introduction

N

UMEROUS activities in the field of CFD method development concentrate on a new identified challenge consisting in obtaining higher order robust and cost effective numerical schemes on unstructured grid systems [1] [2], eventually hybrid, incorporating some regions of structured or partly structured meshes in the normal direction to the aerodynamic profiles [3] [4]. Besides the gain in solution efficiency expected from higher order methods, they might offer the capacity to develop advanced physical models, still needed for RANS turbulence closure, LES subgrid models and LES for acoustics, with less interference between the numerical scheme filtering effect and the physical model scales. For the more complex unsteady flow cases, with regions of higher gradients in flow, vorticity or turbulence intensity propagating within the computational domain, higher order methods would permit to limit the dynamic grid adaptivity needed by lower order methods. For higher order, developments are underway both using the DG method and MUSCL-type methods in the context of collocated cell-centered schemes for conservative variables privileged in the Finite Volume approach at Onera. The DG method was implemented for early evaluation in a prototype research code on structured multi-block grids. Concerning the developments in FV methods, the higher order upwind schemes are obtained from polynomial reconstruction on two wide stencils centered on either side of a grid interface, and upwind characteristic filtering of the information transfer associated to each wave speed. The initial lack of robustness of this process, even for lower Mach numbers flows, with smooth space variations of the conservative fields was experienced. We present here a novel approach, where the polynomial reconstruction process is formulated on a spatial support (stencil) with weighting functions that ensure a proper diagonal dominance, together with a high computing efficiency. These algorithms are presented in detail, and compared to the DG scheme implemented here, knowing that DG methods represent as of today an increasing activity in the CFD community for URANS solutions. The accuracy of the 1

Head of the CFD and Aeroacoustics Department (DSNA), [email protected], AIAA member. Head of the Research Unit on Numerical Methods for CFD, DSNA, [email protected], AIAA member. 3 Research scientist, Research Unit on Numerical Methods for CFD, [email protected]. 1 American Institute of Aeronautics and Astronautics 2

reconstructions is first evaluated on problems corresponding to the conservation of imposed analytical space profiles of test functions extending over the whole domain. Then a simple upwind convection scheme is implemented to evaluate the dissipation and dispersion associated with different orders of accuracy on reconstructing and their interaction with an explicit time integration scheme. Simple linear convection and diffusion test cases are run on 2D grids with a comparison with analytical results, both for the DG and High Order FV methods. The FV Navier-Stokes integration method is expressed and 3D and 2D flow problems are computed : Homogeneous Isotropic Turbulence on tetrahedral grids compared with previous runs on Cartesian grids, inviscid and viscous lowMach number flow around a NACA0012 airfoil. For some of these CFD cases, the same academic tests are computed with DG and FV methods.

II. Physical model Reynolds-Averaged Navier-Stokes System with Wilcox kω ω turbulence model The RANS turbulence model equations read

[ ]

r ∂ρ r + ∇ ⋅ ρV = 0 ∂t r

[ [

]

r r ∂ρV r + ∇ ⋅ ρV ⊗ V + σ + τ t = 0 ∂t r r r r ∂ρE r + ∇ ⋅ ρEV + (σ + τ t ).V − q − qt = 0 ∂t

]

r where ρ is the density, V and E are respectively the velocity and total energy. Here, σ = pI − τ is the stress tensor,

where τ is the shear stress tensor and p the static pressure. The Reynolds tensor τ t and the turbulent heat flux qt are based on the Boussinesq hypothesis : r r τ t = −2 / 3(ρk + µ t ∇. V )I + 2 µ t D , r qt = −Cpµ t / Prt ∇T , where k is the turbulent kinetic energy and µt the turbulent viscosity. Wilcox kω ω Model The equations of the kω system are given by the following relations :

[

] ]

r r r r r ∂ρk r + ∇.( ρVk ) = ∇. (µ + σ * µ t )∇k + τ t : ∇V − β * ρkω ∂t r r r r r ∂ρω r ω + ∇.( ρVω ) = ∇. (µ + σµ t )∇ω + α τ t : ∇V − βρω 2 ∂t k

[

The turbulent viscosity is evaluated by the relation : µ t = α * ρk / ω . The parameters α ,α * , β , β* ,σ ,σ * in the above equations are the closure coefficients of the model and, according to Wilcox, are functions of numerical constants and of the turbulent Reynolds number. The boundary condition implemented on the wall is expressed by setting

u2 τw and S R = 100 for smooth walls. k = 0 and ω = τ S R with uτ =

ν

ρ

These two turbulent systems of equations can be written in the following compact form :

(

)

(

r r ∂W r + ∇.( Fc (W ) − Fv W , ∇W ) = S W , ∇W ∂t

)

2 American Institute of Aeronautics and Astronautics

where FC , FV and S denote respectively the convection terms, the diffusion terms and the source terms.

III. Higher order space interpolation from volume-averaged to surface-averaged quantities In the FV context of collocated, cell centered methods, the reconstruction algorithm can in be expressed synthetically in the following manner: Given a finite span of polynomial test functions in the space coordinates, extending over a sufficiently wide stencil of cells in the vicinity of the interface, and an arbitrary space function

φ ( x, y , z ) , derive the set of linear

φ

interpolation coefficients λ relating the known volume-averaged discrete field of values c on the cells comprising the stencil C to a surface-averaged value

φg of this field on a given interface. The consistent linear interpolation

r

coefficients µ for the surface-averaged components of the gradient

r ∇φg of the field on this same interface can also

be defined in an analogous way : ns

φ g = ∑ λcφ c ,

ns

∑λ

c =1

c =1

ns r r ∇φ g = ∑ µ cφc ,

ns

c =1

r

∑µ c =1

c

c

=1

(1)

r =0

Actually two stencils are considered, from either face of the interface, and built around the left (G) and right (D) cell adjacent to the interface, by incorporating successively neighbors (sharing a common interface) of the previous cells belonging to the stencil. This recursive search can be modified by a variety of constraints : such as a true upwind (upside) search, where the c.o.g. of the cell must be on a given side of the interface by example…

G Z

C

X,Y

g d D

The interpolation algorithm will provide duplicated higher order interpolated values of the conservative variable fields associated to the 2 stencils, and hence represent slightly upside schemes. In the local reference frame (X,Y,Z), centered on the c.o.g. of the interface, with X and Y in the average plane of the interface and Z perpendicular to it, we attempt to approach a polynomial function of arbitrary order :

φa ( X , Y , Z ) = aijk X iY j Z k , where ijk

is a unique index, combining i,j,k, and varying from 1 to

nmo , number

of monomials. For accuracy reasons, the working coordinates X,Y,Z, are also non-dimensionalized by a reference cell size. The span of monomials can be complete of order n ( i + j + k ≤ n ), or of higher order in Z than in X or Y. For example :

φa ( X ,Y , Z ) = a000 + a100X + a010Y + (a001 + a101X + a011Y )Z + (a002 + a102 X + a012Y )Z 2


We seek to minimize, over the stencil volume, the square of the distance between the reconstructed polynomial field and the mean volume values in the cells, each of volume Ωc (s ) . This distance is represented by the functional :

  ψ = ∑ϖ s  Ωc( s)φc( s) − ∫ φa ( X , Y , Z )dV    s =1 Ωc ( s )  

2

ns

(2)

A number of supplementary free parameters are represented by arbitrary coefficients (weights between 0 and 1) ϖ s . They can be given values from a consistent criterion related to the topological distance from the stencil central cell, where the central cell weight has a value of 1 and the other cells of the stencil are given lower values. This has the effect of enhancing the diagonal dominance of the system obtained after balancing the fluxes, and is necessary in most non linear cases to obtain stability. The weights could also be associated to the regularity of a given field, but at the cost of having to reformulate and solve the minimization problem when the sensored fields are modified, at a given period of iteration. They can also be chosen to enhance the upside nature of the reconstruction towards a given interface. In this case the reconstruction in a given cell differs depending on each interface, and it must be checked that this does not introduce spatial modes with insufficient damping. The overall operation could lead to a case-specific pre-processing task that will be discussed later. The functional ψ

is of quadratic form with positive coefficients of the aijk . A minimum is found by expressing,

for each monomial coefficient : ns   ∂ψ ijk 2 ijk i ′j ′k ′  = ∑ϖ s  − 2 Ω c ( s )φc ( s )ℜijk c ( s ) + 2aijk ℜ c ( s ) + 2 ℜ c ( s ) ∑ ai′j ′k ′ℜ c ( s )  = 0 ∂aijk s =1  ijk ≠ i′j ′k ′ 

with

i j k ℜijk c = ∫ X Y Z dV the volume moment of order ijk of each cell. This provides a linear system that links Ωc

[ ]

the vector of coefficients A= aijk

T

Ρ A = Μ φS

[ ]

to the cell-average values of the field φS = φc(s) . T

A = Ρ Μ φS = κ g −1

φS

aijk = κ gijk ,c φc( s )

(3)

Ρ is a symmetric square matrix of order nmo and Μ a rectangular array ns ∗ nmo . The algorithm as such would be very inadequate, as the aijk would need to be recomputed at every stage of the numerical scheme, and for each field. In fact, we are not interested in the polynomial itself, but only in the evaluation of its surface integral over the interface gd.

φg =

1 S gd

∫X

∫φ

a

( X , Y , Z ) dS =

i

Y j Z k dS

∂ Ω gd

∂ Ω gd

a ijk = ν ijk a ijk

S gd

The same holds for the gradient :

r

r 1 ∇φg = S gd

∫ ∇(X

r

∫ ∇φ

( X , Y , Z ) dS =

a

∂ Ω gd

[ ]

From Eq. (3), A = aijk

T

i

Y j Z k ) dS r a ijk = η ijk a ijk

∂ Ω gd

S gd

φS =[φc(s) ]

T

being a linear combination of

, we obtain the required form (1).


φ g = ν ijk aijk = ν ijk κ g ijk ,c φ c = λ φ c

r r r r ∇ φ g = η ijk a ijk = η ijk κ g ijk ,c φ c ( s ) = µφ c

The linear interpolation coefficients

λ

r

et

µ are functions of the volume and surface moments of the cells and the

interface coordinates, and of the choice of the polynomial basis. For the array Ρ to be invertible, the stencil needs to

be balanced according to the set of monomials. The condition number of this array Ρ measures the ability to represent a given order polynomial base over the retained stencil. This can lead to different developments, the simplest ones being the extension of the stencil in the detected space direction where it is insufficient, or removing certain monomials after checking which are responsible of the singularity of Ρ . A local grid optimisation for high order can also be performed by displacing the nodes in order to decrease the objective function represented by the condition number. The linear interpolation coefficients can then be computed once in the preprocessing phase, or after each grid deformation, and stored in static arrays. The moments can be computed by analytical integration formulae for linear elements (2D triangles and planar triangular faces, tetrahedra and ruled quadrilaterals) or numerical integration. This forms the basis of the polynomial recomposition from a given stencil towards a surface. The same methodology can be used for the polynomial recomposition at arbitrary nodal points or for cell-averaged values on another staggered or overset grid, and provide projection coefficients based on cell-averaged expansions in space coordinates.

IV. Related characteristic-upwind FV scheme for the Euler and Navier-Stokes equations For the compressible Navier-Stokes equations, the conservative formulation integrated in space is written :

r r ∂(ΩW ) ni + ∑ Fi (Wg ,Wd ) − Fv (Wg ,Wd , ∇Wg , ∇Wd ) = 0 ∂t n=1 n

[

Here

]

W is the vector of conservative variables, ni the number of interfaces of the cell considered , Fi and Fv

the inviscid and viscous numerical normal fluxes integrated on an individual interface,

r r Wg ,Wd , ∇Wg , ∇Wd are

the reconstructed and face-averaged values of the conservative fields vector and the components of its gradient. The average gradient coefficients instead of

r r ∇Wg + ∇Wd 2

could also be evaluated by a unique least square operation on

the union of the two stencils. A scheme of Godunov type is used, through an intermediate characteristic combination of the vectors of conservative variables from the left and right stencils, defined by

Wgd = RgWg + RdWd with Rg and

Rd computed from the positive, resp. negative eigenvectors and eigenvalues of the Jacobian of the normal fluxes with respect to W . These eigeinvectors and eigenvalues are obtained from the Roe’s average of the 2 states, or in a nonlinear implicit extension, from the characteristics filtered

Wgd field itself. A blending of centered and partially upwind

evaluation can be chosen for each wave (acoustic, entropy) and integrated in the

Rg and Rd expression, together

with the Harten fix for near zero eigenvalues. At first order in the difference between scheme is analogous to the expression of the Roe fluxes. 5 American Institute of Aeronautics and Astronautics

Wg and Wd

, this convective

r r ∇Wg + ∇Wd  ∂(ΩW ) ni  + ∑ Fnat,i (Wgd ) − Fnat,v (Wgd , ) = 0 ∂t 2 n=1   

(4)

n

with

Fnat,i and Fnat,v

the natural inviscid and viscous fluxes. Since

Wgd and

r r ∇Wg + ∇Wd 2

are a linear

r r combination of Wg ,Wd , ∇Wg , ∇Wd , themselves linearly combined from all the W of the stencils, the linear implicit form of the time integration of the discrete scheme (4) is introduced by computing Jacobians of inviscid and viscous fluxes for

Wgd and

r r ∇Wg + ∇Wd 2

, the linearization extending by combination of partial derivatives (chain

rule) over the union of the 2 face stencils.

r r t ni  ∇Wg + ∇Wd   Ι ni  t+δt )  + ∑ Anat,i (Wgd )(Rgλg + Rd λd ) − Anat,v (Wgd )F(µg , µd ) δt (ΩcWc ) = −∑Fnat,i (Wgd ) − Fnat,v (Wgd , 2 n=1   δt n=1 n 

[

]

Anat,i , Anat,v are the jacobians of the inviscid and viscous natural fluxes integrated over the interface, and the t +δt

(

)

ΩcWc over the time step are linked linearly in all cells of the union of evolution of the conserved properties δt the left and right stencils. In difference to the Discontinuous Galerkin method, the fluxes are evaluated only once at each interface, for the average interpolated functions. To improve the order of accuracy, in regions where the tangential gradients are high, the FV interface could be subdivided into surface elements and the reconstructed duplicate values estimated by linear combination on these subsurfaces to provide contributions to the fluxes surface integral. V. Accuracy of the FV reconstruction scheme and linear convection of test functions on triangular and tetrahedral grids The accuracy of the reconstruction can be evaluated over unstructured grids in the following way. Since we want the reconstruction to be consistent, that is to represent exactly a polynomial of given order n, the interpolation scheme is applied to arbitrary polynomials. The averaged field is computed in each cell from an exact space integration over the volume of the cells, and for a check on all interfaces as the exact surface average. The cell to face reconstruction is verified at the initial state to reach machine accuracy on all internal interfaces, for the following 3rd degree polynomial, C0 and C1 continuous over the periodic interfaces, and defined in the [0,2 π ]² space, over a periodic triangular grid : 2  y − π  y − π  2  x − π  x − π  φ ( x, y, t = 0) =  − 1 +  − 1   π  π  2π  π    The linear convection of this polynomial test function can then be compared to the exact solution, given by

r

r

r

φ ( x, t ) = φ ( x − c t,0) , which is a polynomial of the same order in the space variable x at each instant, and its

cell and surface averages can be computed exactly. The transport velocity The linear convection equation

r c has components [1,0.5].

∂φ r r + ∇ • cφ = 0 is integrated in space by the polynomial reconstructions with ∂t

weights depending on the order of neighborhood

inb for each cell (number of interfaces crossed from the central

ϖ s = τ inb giving a higher weight of 1 to the central cell, and decaying weights to the others ( 0 < τ < 1 ). A value of τ less than 0.75 provides stability to the time integration, by introducing an implicit filter, cell), with the formula

at the cost of a reduction of the order of accuracy from its theoretical optimum. The stencils extend each to the 3rd neighbor, and comprise 13 to 15 cells from which to formulate the least square reconstruction for a complete basis 6 American Institute of Aeronautics and Astronautics

of 10 monomials. The time integration is explicit-implicit by a Runge-Kutta scheme with a uniform time step obtained from a CFL constraint of 1.5 from the smallest cell. The upwind reconstructed quantity is convected at each interface. Schéma convectif "Nextflow" Reconstruction polynomiale locale base complète ordre 3 Stencil 3ème voisinage

6

5

Y

4

3

2

Schéma convectif upwind ordre 4 décentrement 1/2 maille Intégration explicite RK 4 étapes dt global, CFL 1.5

1

0

0

1

2

3

4

5

6

7

8

9

10

11

12

X Solution initiale et après 20,40,60,80 convections complètes à travers le domaine (12000 dt) 6

6

17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1

Y

4

3

2

1

0

1

2

3

4

X

5

6

Phi 1 0.875 0.75 0.625 0.5 0.375 0.25 0.125 0 -0.125 -0.25 -0.375 -0.5 -0.625 -0.75 -0.875 -1

5

4

Y

5

0

Err 0.03 0.0273 0.0246 0.0219 0.0192 0.0165 0.0138 0.0111 0.0084 0.0057 0.003

Erreur à 12000 dt

3

2

1

0

0

1

2

3

4

5

6

X

Figure 1 : Isovalues of the test function after 20,40,60,80 convections over the whole domain and relative final error.

The spatial form of the final relative error can be attributed to the fact that the test function is, in its initial state, only C0 and C1 continuous across the periodic interfaces. Qualitatively, its content is of diffusive nature, and a very low dispersion error can be viewed, the global and local maxima are located at the exact positions even after 80 full convections of the test function. The same convection is done in 3D in a periodic cube, over a grid of regular tetrahedra obtained from a regular cartesian grid, by joining each edge of the cartesian grid to the two centers of adjacent cells. It provides 12*n3 tetrahedra with each 2 edges of length 1 and 4 edges of length 3 . The periodic connectivity is expressed and the 2 2 2 2       test function φ ( x, y, z, t = 0) = x − π  x − π  − 1 + y − π  y − π  − 1 + z − π  z − π  − 1 is convected at the π  π   2π  π   3π  π   velocity [1,1./2.,1./3].

Figure 2 : Tetrahedral grid 12*24**3 and isovalues of the 3D test function


Figure 3 : Max value of the cell averages as a function of time (number of domain crossings) and close view on the first period

Figure 4 : Total variation of the function over the complete set of stencils

Figure 3 presents the maximum value of the cell averages with time, for 10 full convections across the domain. It shows a very low rate of diffusion of the maximum. The close-up of the first period shows that the maximum begins to increase, this is only dependent of the grid configuration at the initial position of the maximum, and it shows that this type of smooth extremum can be convected without any smearing. The total variation of the function on all the stencils is defined as the sum of the absolute value of the difference between the central cell and all the cells belonging to its stencil. This total variation is also well tracked by the scheme, and evolves in time according to the grid arrangement only, without any noticeable decay (due to the different components of the convection velocity with time, the time real periodicity is equal to 12 times the crossing duration).

VI. Finite Volume 3D unsteady inviscid flow case : Homogeneous and Isotropic Turbulence

A case of HIT at infinite Reynolds number on a cube of span [0,2 π ]3 was computed on the same type of regular tetrahedral grid of 12*n3 cells, also with exactly matching periodic boundaries in each space direction. The time integration uses a RK3 scheme with a consistent time step based on a convective CFL number of 1.8 (a case with dual time stepping was run, but since the time step must be kept small, this approach is much more costly). The results are presented for n=40. Figure 6 represents the spectral content, at different times of the evolution from large eddies to the dissipative range of small eddies. At earlier times, the spectrum decreases more with the wave number, then the average slope is less pronounced as smaller eddies are generated. The curves tend to reach a behavior in k(-5/3) over almost the whole span of wave numbers, associated to a decrease in level in t(-2) in conformity with the theory (figures 6 and 7). The enstrophy level is multiplied by a factor 33 before the onset of the dissipation process, a value only obtained on Cartesian grids exceeding 128**3 with ENO and WENO schemes (figure 8) .


Figure 6 : Spectral content of the velocity field at different times (2, 4, 6, 8, 10) and decay of the total kinetic energy

Figure 7 : Homogeneous isotropic turbulence on 12*40**3 tetrahedra : enstrophy levels as a function of time for 2nd, 3rd and 4th degree reconstructions

Figure 8 (taken from reference 5) : HIT on Cartesian grids of 128**3 cells with directional space integration schemes

VII. Discontinuous Galerkin method for the Navier-Stokes equations in a block-structured solver The development of a discontinuous Galerkin (DG) method for turbulent flow computations presented here is performed in the context of the European project ADIGMA [6] [7] . The compressible RANS) equations coupled with a k-ω turbulence model are considered. We reformulate the equations as a first-order system in space using the mixed DG approach [8]. The boundary condition treatment is performed through a reconstruction of the solution at the physical boundary which avoids the use of a ghost cell technique and improves stability. This method allows space discretization with overlapping and non-matching multidomains as well as high order polynomial 9 American Institute of Aeronautics and Astronautics

approximations of the solution. Numerical examples for 3D inviscid and turbulent flows will be presented to demonstrate the accuracy of the method. The CANARI code [9] [10] used for this study has been developed at ONERA. It is built from the following numerical tools : multi-domain approach using structured grids with patched or overlapping interfaces, cell-centered finite-volume discretization, multigrid and implicit residual smoothing acceleration techniques and characteristic based boundary condition treatment. The code has been developed for the simulation of inviscid and viscous complex internal and external flow configurations. The FV functionalities of this code have been integrated in the elsA software environment which is actually used both by aerospace manufacturers and research laboratories. We do not recall here the numerical scheme corresponding to the second order FV scheme which is generally well known. We refer to [9] [10] for specific implementations of this FV method for more details. The DG scheme which has been used is now shortly presented. The domain Ω is discretized with hexahedral elements Ω h = ∪ iN=1 K i . The spatial discretization of the diffusive and source terms are constructed by regarding the gradient of the conservative variables as additional unknowns of the problem: G (u ) = ∇(u ) . Multiplying the RANS equations and the gradient equation by a test function and integrating by parts element by element leads to the weak formulation of the problem:

∫ Φ G dΩ − ∫ Φ uˆ h

∂K i

Ki

∫Φ Ki

(

h

⊗ ndS + ∫ ∇Φ ⊗ u h dΩ = 0 Ki

)

∂u h dΩ − ∫ Φ Fˆc − Fˆv dS − ∫ (Fc − Fv ) ⋅ ∇Φ dΩ − ∫ Φ S dΩ = 0 ∂t ∂K i Ki Ki

where now Fv = Fv (Wh , Gh (Wh )) and S v = S (Wh , Gh (Wh )) . The subscript h means that we are looking for an approximate solution of the initial problem expended in the function space of piecewise discontinuous polynomials

{

}

Vh = φ ∈ L2 (Ω h ) : φ K ∈ Pk (K i ),1 ≤ i ≤ N , where Pk (K i ) represents the space of the polynomials of degree k in i the element K i . Terms with hat symbol in the contour integrals of the weak formulation represent numerical fluxes and are chosen so as to be uniquely defined at faces of elements ∂K i and to satisfy the consistency and conservativity conditions. For the gradient construction the numerical flux

uˆ h of uh is evaluated by mean of a centered scheme :

uˆ h =

u h− + u h+ 2

+

−

where u h denotes the trace of u h on the face taken from within the interior element K i and u h denotes the trace on the same face taken from within the neighboring element. The numerical flux at a boundary is evaluated through an appropriate treatment based on a flux evaluation uˆ h = u BC which ensures that the boundary condition is verified. The boundary value

u BC is computed by imposing either physical boundary data or Riemann invariants

evaluated with a prescribed external state for incoming characteristics, and the Riemann invariants associated to

u h+ for the outgoing characteristics. The convective fluxes are discretized by using a local Lax-Friedrichs flux with artificial dissipation

 u + + u h−   ⋅ n − k 2 ρ s (u h+ − u h− ) ⋅ n Fˆc = Fc  h  2 


where ρ s

{

}

= max Vh+ + ch+ , Vh− + ch− stands for the spectral radius of the convective Jacobian, ch denotes the

k 2 = 0.25 for viscous computations and k 2 = 0.05 + for inviscid computations. The numerical flux at a boundary reads Fˆc = Fc (u BC ) ⋅ n − k 2 ρ s (u h − u BC ) ⋅ n where u BC has been defined above.

speed of sound, and the artificial viscosity parameter is set at

The viscous fluxes are replaced by a central scheme

(

)

(

 F u + , G + + Fv u h− , Gh− Fˆv =  v h h 2 

)  ⋅ n.  

The boundary treatment for the auxiliary variable is similar to that of the conservative variable:

Fˆv = Fv (u BC , GBC ) ⋅ n where GBC is imposed if there are boundary conditions on ∇u ⋅ n , otherwise we set

GBC = Gh+ . The space is discretized with a 3D structured grid with P1 and P2 approximations. The volume and contour integrals are calculated with Gauss quadrature formulae. The integrands are evaluated at the Gauss points with the polynomial approximations of the conservative variable vector and of its gradient. Finally, the time integration of the system is accomplished with an explicit three-stage Runge-Kutta technique. In order to keep the modular feature of the solver, the time integration procedure is based on a decoupling between RANS and k-ω systems of equations. At each iteration, the RANS system is integrated assuming that the coefficient µt is frozen, and then the k-ω system is solved assuming that the mean flow quantities are frozen This allows in particular the use of specific numerical flux for each system.

VIII. Evaluation of Discontinuous Galerkin methods Figure 9 presents an Euler validation of the DG method for a 3D transonic flow around the Onera M6-wing. As shown by the residual history, the computation converges to machine accuracy. The Cp-distribution, obtained with the DG method and a P1 approximation (DGP1), is in agreement with that obtained with a classical second-order Jameson FV method and demonstrates the shock capturing capability of the DG method. Since no implicit has been yet developed for DG, the computation has run with a CFL number equal to 0.5 using a 4-step Runge-Kutta timemarching method. Figure 10 presents a subsonic Euler flow performed with FV, DGP1 and DGP2 with k 2 = 0.05 , corresponding to the local Lax-Friedrichs flux which is very dissipative. We can observe that, for a given numerical flux, going from P1 to P2 strongly improves the total pressure losses at the wall.

Figure 9: Euler computation of the Onera M6-wing : (left) Iso-pressure coefficients on the upper side of the wing surface; (center) Cp distribution at y = 0.44 and comparison with experiments; (right) Logarithmic plot of the mean square residuals.


Figure 10: Subsonic Euler computation around the NACA0012: (left) Iso-mach number; (right) Comparison of total pressure losses with DGP1/DGP2-Lax Friedrichs (k2=0.05) and 2nd order finite volume.

Figure 11: Subsonic Euler computation around the NACA0012: (left) Iso-mach number; (right) Comparison of total pressure losses with DGP1/DGP2-Lax Friedrichs (k2=0.05) and 2nd order finite volume.

Figure 11 presents a transonic Euler flow performed with FV and DGP1 and k 2 = 0.05 . The structure of the shockwave is better with DGP1 than with a classical FV scheme. In particular with DGP1, this shock-wave is fitted in one cell (in x-direction) and does not present strong oscillations as for the FV result.

Figure 12: Viscous flow computation with DG-P1 method – (left) Laminar flow around the around the NACA0012airfoil; (right) Turbulent flow around the RAE2822 airfoil

Figure 12 presents viscous flows computed with DGP1. The first one corresponds to a laminar flow around the NACA0012 airfoil and the second one corresponds to a turbulent flow around the RAE2822 airfoil.


1/ Mesh convergence analysis : Mesh refinement analysis has been performed for Euler and laminar test cases, based on integrated forces : lift, drag and pitch moment coefficients. For the subsonic Euler test cases DG-P1 and DG-P2 have been used, whereas only DG-P1 has been used for the transonic Euler and the laminar test cases.

Grid CD,p CL,p CM,p 5.295e-3 2.775e-1 -2.488e-3 17×113 1.297e-3 2.816e-1 -2.528e-3 33×225 4.360e-4 2.831e-1 -2.680e-3 65×449 3.890e-4 2.831e-1 -2.525e-3 129×987 Table 1 : Drag, lift and pitching moment coefficients for the Euler subsonic flow around a NACA0012 airfoil for P1 approximations Grid CD,p CL,p CM,p 3.570 e-3 2.761 e-1 -6.880 e-4 17×113 8.900 e-4 2.795 e-1 -1.610 e-3 33×225 6.690e-4 2.741e-1 -1.040 e-3 129×987 Table 2 : Drag, lift and pitching moment coefficients for the Euler subsonic flow around a NACA0012 airfoil for P2 approximations. Grid CD,p CL,p CM,p 2.518e-2 3.361e-1 -3.457e-2 17×113 2.267e-2 3.465e-1 -3.670e-2 33×225 2.224e-2 3.476e-1 -3.825e-2 65×449 2.088e-2 3.522e-1 -3.478e-2 129×987 Table 3 : Drag, lift and pitching moment coefficients for the Euler transonic flow around the NACA0012 airfoil; P1 approximation. Grid CD,p CD,v CD 2.797e-2 2.807e-2 5.604e-2 41×161 2.518e-2 2.948e-2 5.466e-2 81×321 2.478e-2 3.126e-2 5.603e-2 161×641 Table 4 : Drag coefficients for the NACA0012 airfoil; P1 approximation. Laminar subsonic flow (M=0.5, Re=5000, alpha=2). Grid CL,p CL,v CL 3.653e-2 -1.260e-4 3.640e-2 41×161 3.806e-2 7.800e-4 3.814e-2 81×321 4.217e-2 3.070e-4 4.248e-2 161×641 Table 5 : Lift coefficients for the NACA0012 airfoil; P1 approximation. Laminar subsonic flow (M=0.5, Re=5000, alpha=2). Grid CM,p CM,v CM 1.682e-2 -3.020e-4 1.652e-2 41×161 1.639e-2 -2.370e-4 1.616e-2 81×321 1.783e-2 -1.980e-4 1.709e-2 161×641 Table 6 : Pitching moment coefficients for the NACA0012 airfoil; P1 approximation. Laminar subsonic flow (M=0.5, Re=5000, alpha=2).


IX. Comparison test cases between the two methods Convection and diffusion of a scalar function on 1D and 2D grids have been computed by the two methods and compared to analytical results. 1/ 1D convection-diffusion with Dirichlet conditions (Re=10) The exact solution of this problem

φ (1) = 1. is φ ( x ) =

φx −

1 φ xx = 0 on the domain x=[0.,1.], with φ (0) = 0. and Re

1. − e x. Re . 1. − e Re

The solution is computed for 20, 40, 80, 160, 320 cell grids. The DG solution is expressed at the center of each cell, for all orders of polynomial approximations. The L2 norm of the error at these node centers is presented in Fig; 13. For the higher order FV method, the comparison is done on the cell averages, between the computed and analytical solutions.

Figure 13 Example of solution for 16 cells over [0,1], norm of the error and residual decrease for respectively 2nd order FV(reference, blue), DG-P1 (red) and DG-P2 (green)

Figure 14 : Convergence of the various methods

The higher order FV method does not reach the theoretical order of convergence. For the cubic reconstruction, a stencil of 5 cells is used, and for the quadratic reconstruction a stencil of 3 cells for an exact reconstruction leading to a zero value for the functional (eq. 2). 14 American Institute of Aeronautics and Astronautics

The lack of accuracy is attributed to the downstream boundary condition that is of reduced order than for the computation at the central faces, for the diffusion fluxes. They are extrapolated linearly from the inner faces to the boundary face. This same procedure is used in the DG method, only the gradient extrapolation to the boundary is made from the inner degrees of freedom, located more closely to the downstream boundary. The advantages in terms of convergence and robustness of formulation of the DG method is clearly put forward. 2/ 2D convection-diffusion on a square domain (Re=10) The problem is formulated over the domain[0,1]*[0,1].

1 φ yy = 0 , convection in x direction and diffusion in y direction, with Dirichlet conditions Re on x=0 : φ ( 0, y ) = sin πy on y=0 : φ ( x,0) = 0 on y=1 : φ ( x,1) = 0 We solve φ x

−

The imposed sinus profile at inlet is damped along x by the transverse diffusion, yielding

φ ( x, y ) = e

(−

π2 Re

x)

sin πy

The DG solutions are computed on cartesian grids 20*20, 40*40, 80*80, 160*160, 320*320. The “Next Order” FV solutions are computed on the same grids, on triangular grids and on rectangular grids refined in the “walls” vicinity, with the same number of cells in y direction and a normal expansion ratio of 1.15 between successive cells. On the cartesian grids, directional stencils are used, with a polynomial reconstruction in the “Z”, i.e. normal direction only. This limits the gradient evaluation to this direction, hence the diffusive fluxes are approximated by the so-called “thin layer’ approximation. For the stretched rectangular cells, the full 2d polynomial representation is used, this is considered valid since the expansion rate between successive cells is limited.

Figure 15 : Solution on Cartesian and triangular grids

Figure 16 : Close-up on the solution near the maximum on the exit line, for 3 refined triangular grids


Figure 17 : Orders of convergence as a function of the mesh size in L2 norm

The high order FV method uses a 4th degree polynomial reconstruction, with stencils of 30 cells on the average, to compute the 15 coefficients of the least-square fitting polynomial. However, the weights decay rate is 0.5 at each new row of cells, to obtain a stable resolution. This results finally in a 3rd order convergence only. The accuracy is better than for the 1D convection-diffusion, this test case can be considered as more representative of near-wall boundary layers flows, with convection and diffusion in orthogonal directions.

X. Flow about airfoils with the higher “Next Order” FV model We present 2 types of applications using respectively triangular and structured quadrilateral grids, in 2D. The first test case is an inviscid subsonic flow about a Naca0012 at 2 degrees of incidence and a Mach number of 0.5. Three grids were generated, with respectively 8700, 26900, 49100 isotropic triangles with gmsh. The medium grid is presented on Fig. 18. For a given polynomial, more cells are added to the stencil by extending the depth of neighboring, until an excess of 50% of cells with respect to the number of monomials is reached, following the procedure suggested by Ollivier-Gooch in [2]. Ghost cells are generated and the inner stencils are allowed to gather one row of ghost cells at most in the wall tangential direction. The wall boundary conditions are prescribed in the following way : for Dirichlet conditions, the ghost cell value is modified such that the mean of the reconstructed values for the two stencils of the wall interface is equal to the prescribed value. This procedure can be made implicit by taking into account the residual in the inner cells of the stencils to compute the ghost cell residual. For normal gradients BC, the difference between the reconstructed values from the inner and outer stencil are imposed (to 0) by adjusting the ghost cell residual. Time integration is performed by a RK4 scheme with CFL 5.25, which shows the added stability associated to the wide stencil and weights. The convergence of the residuals for the fine grid is shown on Fig. 20, together with the lift and drag, for 2nd order and 4th order reconstructions.

Figure 18 medium fine grid


Figure 19 Stencils at a wall interface (left) and in the center of the grid (right) for 4th order polynomials inner stencils and quadratic reconstruction for the ghost cell stencil. Slightly offset for clarity;

Figure 20 Convergence of residuals for the 4 order reconstruction on fine grid, and of the efforts on the same grid with 2nd order and 4th order polynomials.

Figure 21 : Density at the stagnation point for the medium fine grid


Figure 22 : Density at the stagnation point for the coarse grid, and the visualization of the reconstructed space variation inside cells and interface jumps.

Figures 21 and 22 present the variation of density near the stagnation point. The variation of the conservative variables inside the cells is computed in the post-processor at inner points by means of the equation (3) that relates the coefficients of the polynomial linearly with the cell-average values of this field over the stencil. In each subelement, the visualization package then draws iso-values by linear interpolation between the nodes. The behavior of the solution at regions of high gradients is satisfactory, however downstream of the profile, as shown on the suction side of figure 21, some wiggles appear in the solution; this might indicate that the high order expression of the BC is not enough robust and the variation in orientation of the stencils near the walls, from purely centered to partly offset towards the fluid region, induces local modes that are not sufficiently dissipated. The entropy field is shown on figure 23. Although very low, it is generated away from the boundary in the region where the stencils tend to get offset towards the fluid. The improvement of this boundary conditions remains a subject of study. The lift coefficient converges towards a value of 0.257, which is a little low compared to the reference value of 0.275, while the drag coefficient is less than 1.5e-3.

Figure 23 : Entropy on the fine grid, 4th order polynomial. Another laminar test case from the ADIGMA project was computed on a structured grid, it concerns a NACA0012 at Reynolds 5000, Mach 0.5 and 2 degrees incidence. The reconstruction uses directional stencils of 3 cells for an exact quadratic polynomial fit. 18 American Institute of Aeronautics and Astronautics

The field of Mach number in the x direction is shown on figure 24. The high order treatment of the boundary is more robust with a directional stencil, and a sharp interface of the boundary layer can also be captured by the third order space scheme.

Figure 24 : Subsonic laminar flow around a NACA0012 : Mach in the x direction

XI. Conclusion A comprehensive method of conservation-driven high order polynomial reconstruction over wide stencils is developed for FV collocated cell-centered methods. A computationally efficient projection of this reconstruction on cell interfaces, nodes or overset grids is detailed. This procedure was tested on polynomial functions covering the whole unstructured composite mesh, at a fixed time, and through linear convection problems whose exact solutions are used as a reference. The explicit and implicit expressions of its application to the Navier-Stokes equations on the same grid systems was presented, and a 3D case of homogeneous and isotropic turbulence was computed, showing a high accuracy with respect to the reference schemes on structured grids. For flow about aerodynamic profiles, the main difficulty in this method resides in the ability to maintain the accuracy near walls with stencils gradually being one-sided towards the fluid as one approaches the wall. A Discontinuous Galerkin method has been developed in a block-structured code for the RANS system associated with a kω turbulence model. In the context of the European Project ADIGMA it has been evaluated on 2D and 3D test cases pointing out interesting aspects of the DG methods, as gain of accuracy and capacity to maintain high order approximation next to the computational domain boundaries. From our experience the key points for using the high order DG for complex configurations remain CPU time and memory storage. The future work consist in implementing the DG in hybrid meshes in order to be able to use local H/P refinement for accuracy optimization, and in analyzing and implementing H/P multigrid as well as implicit methods for efficiency improvement. As composite mesh and solver strategies are being developed in the CFD and Aeroacoustics Department of ONERA in a single Object Oriented software environment elsA, the objective is to combine and optimize the way of using different high order mesh/solver in order to gain in accuracy/computational cost/efficiency and robustness. Finally the reconstruction methods presented here appear as a way to treat the interface regions of different type of meshes (Cartesian, structured, unstructured) while keeping high order accuracy in these regions.

XII. Acknowledgments The work on DG methods was funded by the ADIGMA project of the European Community, within the 6th Program Framework. The authors wish to thank Professor Alain Lerat of ENSAM for his advising contribution in the preparation of this communication.


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