Lecture CFD

Face reconstructionTreatment of misalignment for NVD schemes

Computation of gradients in a cell

A generalized unstructured finite volumediscretisation - Part 2

Michel VISONNEAU

LHEEA - CNRS UMR 6598Ecole Centrale de Nantes, Nantes, FRANCE

CFD in ship hydrodynamics - Kul.24-Z courseNovember - December, 2013

ECN-CNRS A generalized unstructured finite volume discretisation - Part 2



Outline of lecture 5

Centered face reconstruction

Upwinded face reconstruction

Computations of gradients within a cell




Centered reconstructionNon-centered reconstruction

Centered face reconstruction





Except terms intervening in convective and mass fluxes, quantitieslocated at the face center are rebuilt with the help of centeredapproximations based on linear interpolation between cell and facecenters.This part of the lecture is devoted to centered reconstruction of ageneric quantity Q and its normal-to-the face gradient needed fordiffusive terms.

r

l

n

Lf−

f+

R





Quantity Qf may be written from its values and its gradient in theadjacent cells L and R, by using a linear interpolation. Tworeconstructions can be considered, indicated by the subscripts + and− :

QL ' Qf −−→Lf ·−→∇ Qf QR ' Qf +

−→fR ·−→∇ Qf (1)

A decomposition of the normal-to-the-face gradient can be introducedto use the normalized vectors

−→l and −→r , for segments [Lf] and [fR] :

∀β+ −→∇ Qf ·−→n = β

+−→∇ Qf ·−→r +−→e + ·

−→∇ Qf si :−→e + ,−→n −β

+−→r(2)

∀β−−→∇ Qf ·−→n = β

−−→∇ Qf ·

−→l +−→e − ·

−→∇ Qf si :−→e − ,−→n −β

−−→l(3)





Coefficients β± are chosen to fullfill conditions −→e − ·−→n = 0 and−→e + ·−→n = 0. Let us note that vectors −→e ± vanish in case of orthogonalgrids. Reconstructions 1 yield by introducing this decomposition :

QL = Qf −h−(−→

∇ Qf ·−→n −−→∇ Qf ·−→e −

)QR = Qf + h+

(−→∇ Qf ·−→n −

−→∇ Qf ·−→e +

) (4a)

with distances h± :

h− ,−→Lf ·−→n and h+ ,

−→fR ·−→n (4b)





A second-order reconstruction can then be obtained by :

Qf =h+

hQL +

h−

hQR +

h+h−

h

−→∇ Qf · (−→e +−−→e −) (5a)

with the distance h defined by :

h , h+ + h− = (−→Lf +−→fR) ·−→n =

−→LR ·−→n (5b)

First two terms of relation (5a) yield a first order reconstruction which isused to evaluate the value of gradient at the face from values atadjacents cells :

Qf =h+

hQL +

h−

hQR +

(h−−→H +−h+

−→H −

h

)·(

h+

h

−→∇ QL +

h−

h

−→∇ QR

)(6)





with :

−→H − , h−−→e − =

(−→Lf ·−→n

)−→n −−→Lf

−→H + , h+−→e + =

(−→fR ·−→n

)−→n −−→fR (7)

It is worth emphasizing that the boxed term in (6) vanishes in case oforthogonal grids. Since it contains gradients of Q in the neighboringcells, it will be treated explicitly during the resolution.





Viscous diffusion fluxes

To rebuild at the face center the normal-to-the face gradient, one usesa similar formula :

−→∇ Q ·−→n |f =

QR−QL

h+

(−→n −

−→LR−→LR ·−→n

)·(

h+

h

−→∇ QL +

h−

h

−→∇ QR

)(8)

The boxed term, which vanishes in case of orthogonal grids, will betreated explicitly during the resolution.





Upwinded face reconstruction





Non-centered reconstructions

In order to reinforce the stability of numerical scheme and avoidunphysical oscillations, it is necessary to introduce more sophisticatedupwinded reconstructions for convective fluxes.

The order of accuracy of these reconstructions is comprised between1 and 2 and cannot be a priori specified since it depends on the usedgrid and the physical problem.





Two upwind discretisation schemes will be described:

1 Hybrid scheme2 GDS scheme3 AVLSMART scheme





Hybrid scheme

This scheme implemented in ISIS-CFD is based on a weightingbetween values linearly interpolated from neighboring cells. Contraryto classical approaches, the weighting coefficient is not uniform butdepends on the Peclet number evaluated on the face:

Pef =

.mf‖−→LR‖

2SΓQ(9)

The accuracy of this reconstruction is not necessarily uniform but islocally adapted to local flow physics.





Hybrid scheme

Moreover, the relative orientation between the face and the velocity istaken into account. The weighting factor d is computed with the help ofan exponential scheme, which ensures a smooth transition :

Qf = dLQL + dRQR + dL−→Lf ·−→∇ Q|L+dR

−→Rf ·−→∇ Q|R

d = exp(Pef )/(1 + exp(Pef )) dL , 1−dR

(10)

Once again, the boxed term is treated explicitly.





GDS scheme

The hybrid scheme does not guarantee the bounded character of thereconstruction. This property is important for multifluid flows for whichconcentration should remain between 0 and 1, for instance. The(Gamma Differencing Scheme) scheme is due to Jasak (1996).This scheme is based on an analysis using the normalized variablediagram introduced by Leonard (1988). One supposes that the valuesof the generic quantity Q are available in three points, U, C et Dlocated along the convection direction.





GDS scheme

One wants to rebuild the quantity at a face located between C and D(FIG. 7). One defines the normalized variable by :

Q̃ =Q−QU

QD−QU(11)

Q D

Q CQ f

Q U

fC DU

Flow direction

(CD)

(UD)

Variation of Q in the flow directionECN-CNRS A generalized unstructured finite volume discretisation - Part 2




GDS scheme

The GDS scheme is based on a reconstruction using these threepoints :

Q̃f = f (Q̃C) (12)

To avoid any unphysical oscillation, it is necessary that QC be boundedbetween the values Min{QU ,QD} and Max{QU ,QD}. Written in termsof normalized variables, this condition reads :

0 6 Q̃C 6 1 (13)





GDS scheme

To preserve boundedness characteristics on the face, GDS schemeshould satisfy the following constraints :

for Q̃C < 0, Q̃f = Q̃C

for 0 6 Q̃C 6 1, Q̃f is bounded by Q̃f > Q̃C and by unity

for Q̃C > 1, Q̃f = Q̃C





GDS scheme

These conditions are fullfilled in the gray area of the followingfigure 11. One observes that the first order upwind scheme (UDS)fullfills all these conditions, when the centered scheme (CDS) does notrespect these criteria in the interval Q̃C ∈ ]−∞,0[ ∪ ]1,+∞[.





GDS scheme

Monotonicity criteria in terms of normalized variables





GDS scheme

To use this strategy within an unstructured multi-dimensionalframework, some modifications should be performed. The point Ubeing unknown, the quantity QU is evaluated by projecting the gradientalong the direction

−→CD (figure 12) :

QU = QC−−→CU ·−→∇ Q|C with

−→CU ,−−→CD (14)

Definition of the fictitious point U





GDS scheme

By considering this new definition, the normalized variable at C isprovided by :

Q̃C = 1− QD−QC

2−→∇ Q|C ·

−→CD

(15)





GDS scheme

This definition being ill-posed when Q is uniform in the domain, onespecifies :

Q̃C = 0.5 si |QD−QC |6 10−6 ou |−→∇ Q|C ·

−→CD|6 10−6 (16)

The GDS scheme establishes a continuous transition between a firstorder upwind and a centered schemes in the interval 0 < Q̃C < 1. Forlower values of Q̃C , it is necessary to recover the gap between thecentered and upwind schemes. This transition is performed in theinterval [0,βm] and βm takes usually the value 1/6.





GDS scheme

{Q̃C = 0 ⇒ γ = 0 Schéma UDS

Q̃C = βm ⇒ γ = 1 Schéma CDS

The GDS scheme retains finally a linear variation for the transitioncoefficient, which leads to the following reconstruction described infigure 17 :

γ =1

βmQ̃C (17)





GDS scheme

Finally, the characteristics of this scheme are summarized in this table

including the interpolation factor fx = ‖−→fD‖‖−→CD‖

Q̃C Q̃f Qf Note

]−∞,0] Q̃C QC UDS

]0,βm[ − 12βm

Q̃2C+ (1− γ(1− fx ))QC transition

+(

1 + 12βm

)Q̃C +γ(1− fx )QD

[βm,1[ 12 + 1

2 Q̃C fxQC + (1− fx )QD CDS

[1,+∞[ Q̃C QC UDS

GDS scheme reconstruction





GDS scheme

��

��

βm

Q f

Q C1

1/2

0

1

CDS

UDS

UDS

~

~

GDS scheme in terms of normalized variables





AVLSMART scheme

Implementation in ISIS

If the base scheme for the Gamma scheme is the 2nd order CDS,

the base scheme for AVLSMART is the 3rd order QUICK scheme(Leonard 79).

��

��

βm

Q f

Q C1

1/2

0

1

CDS

UDS

UDS

~

~





AVLSMART scheme

Implementation in ISIS

Q̃C Q̃f Qf = CDQD +(1−CD)QC Note

]−∞,0] Q̃C CD = 0 UDS

]0,1/4[ 92 Q̃C CD = 5

2 (1− fx )Q̃C

1−Q̃CUDS->QUICK

[1/4,3/4[ 38 +

34 Q̃C CD = 1

4 (1− fx )3−2Q̃C

1−Q̃CQUICK

[3/4,1[ 34 +

14 Q̃C CD = 3

2 (1− fx ) QUICK->UDS

[1,+∞[ Q̃C CD = 0 UDS

The interpolation factor fx , that accounts for non-uniform grids (1D), is defined as the

ratio: fx = ‖−→fD‖‖−→CD‖

.




Misalignment with face centre

Objective

Qf = f (QU ′ ,QC′ ,QD′) to assume local 1D (aligned) configuration.

Points U’, C’ and D’ are aligned with the face centre.





Correction (1/4)

Step 1: Project C and D on axis (f,−→n ):

−→C′f = (

−→Cf .−→n )−→n = h−−→n

−→fD′ = (

−→fD.−→n )−→n = h+−→n





Correction (2/4)

Step 2: Extrapolate C’ and D’:

QC′ = QC +−−→CC′.−→∇ QC

QD′ = QD +−−→DC′.−→∇ QD





Correction (3/4)

Step 3: combine with what is known ...

QC′ = Qf−−h−−→∇ QC .

−→n , with Qf− = QC +−→Cf .−→∇ QC

QD′ = Qf+ + h+−→∇ QD.

−→n , with Qf+ = QD−−→fD.−→∇ QD

... to get the corrections

QC′ = QC +(−→

Cf −h−−→n).−→∇ QC

QD′ = QD−(−→

fD−h+−→n).−→∇ QD

No need to store the locations of C’ and D’.





Correction (4/4)

Step 4: last trick to build Q̃C′ .Instead of using the definition Q̃C′ =

QC′−QU′QD′−QU′

,

replace QU ′ with QC′−−−→C′U ′ ·

−→∇ Q|C .

Since−−→C′D′ = (h−+ h+)−→n = h−→n , and

−−→C′U ′ =−

−−→C′D′ it comes:

Q̃C′ = 1− QD′−QC′

2h−→∇ Q|C .−→n




Weighted least-square methodGauss method

Computation of a gradient in a cell





Two discretisation schemes will be described:

1 Weighted least-square method2 Gauss method





Weighted least-square method

For the sake of simplicity, this method is presented for 2D flows. Oneconsiders a control volume C0 with its center of gravity

−→X0 = (x0,y0).

One supposes that the quantity q varies linearly within this cell.Consequently, the value at each point

−→X belonging to C0 is given by :

q(−→X ) = qC0 +

−−→X0X · (

−→∇ q)C0 (18)

In 2D, this relation reads :

q(x ,y) = q(x0,y0) + (x− x0)qx + (y− y0)qy (19)






Cell C0 and its neighborhood






The evaluation of the gradient of q on cell C0 makes use of n pointswhich provide centers of neighbouring cells (Ci)i=1,...,n as illustratedby figure FIG. 2. Since there are more available points than needed,we get an over-determined linear system defined by n relations :

qi = q0 + (xi − x0)qx + (yi − y0)qy ∀i ∈ [1,n] (20)

This problem is solved in the mean-square sense by minimizing thefunctional En(qx ,qy ) :

En(qx ,qy ) =n

∑i=1

{[(q0 + (xi − x0)qx + (yi − y0)qy

)−qi

]Di

}2(21)






Quantity Di corresponds to the weight given to a point Ci over thewhole set of data. It makes possible to give an increased importanceto the points Ci which are located in the vicinity of the recontructedpoint C0. This weighting by distance is crucial to treat high-aspectratios mesh encountered in the vicinity of walls. The weight Di of a cellCi is defined as the inverse of its distance to the point C0 :

Di = (||−−→X0Xi ||)−1 ∀i ∈ [1,n]






Minimizing the functional∂En

∂qx= 0 and

∂En

∂qy= 0, leads to the gradient

of q : (qx

qy

)=

1S

(Syy −Sxy

−Sxy Sxx

)(Sxq

Syq

)(22a)






with :

Sxx =n

∑i=1

[(xi − x0)Di

]2Syy =

n

∑i=1

[(yi − y0)Di

]2

Sxy =n

∑i=1

[(xi − x0)(yi − y0)D2

i

]S = SxxSyy −Sxy

2

(23a)

and :

Sxq =n

∑i=1

[(xi − x0)(qi −q0)D2

i

]Syq =

n

∑i=1

[(yi − y0)(qi −q0)D2

i

] (23b)





Gauss method

A second more classical is to use Gauss’ theorem: For a volume Vbounded by a surface S , one gets :∫

V

−→∇ qdv =

∫S

q−→n dS (24)

Consequently, the gradient over the cell may be evaluated bydiscretising the previous relation :

−→∇ q∣∣∣C

=1

VC∑

f

qf Sf−→n f (25)

The values of the quantity on the faces are rebuilt by using 2nd orderreconstructions presented previously.


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