(Grid Generation 정렬격자 · 2017. 6. 8. · Ex) Hyperbolic marching + Elliptic smoothing 24. Structured Grid Generation Schemes • PDE-based –Mixed Scheme ... • Stretching

첨단 사이언스∙교육 허브 개발 (EDISON) 사업

격자 생성(Grid Generation_정렬격자)

이 름 : 김 병 수

소 속 : 충남대학교 항공우주공학과

Introduction to CFD

• Conservation Laws

Fundamental equations of fluid dynamics are based on the

universal laws of conservations

– Conservation of Mass

– Conservation of Momentums

– Conservation of Energy

2

Introduction to CFD

• Governing Equations

– Mass

– Momentum

– Energy

3

𝜕𝜌

𝜕𝑡+

𝜕

𝜕𝑥(𝜌𝑢) +

𝜕

𝜕𝑦(𝜌𝑣) +

𝜕

𝜕𝑧(𝜌𝑤) = 0

𝜕𝜌𝑢

𝜕𝑡+

𝜕

𝜕𝑥(𝜌𝑢2 + 𝑝 − 𝜏𝑥𝑥) +

𝜕

𝜕𝑦(𝜌𝑢𝑣 − 𝜏𝑥𝑦) +

𝜕

𝜕𝑧(𝜌𝑢𝑤 − 𝜏𝑥𝑧) = 𝜌𝑓𝑥

𝜕𝜌𝑣

𝜕𝑡+

𝜕

𝜕𝑥(𝜌𝑢𝑣 − 𝜏𝑥𝑦) +

𝜕

𝜕𝑦(𝜌𝑣2 + 𝑝 − 𝜏𝑦𝑦) +

𝜕

𝜕𝑧(𝜌𝑣𝑤 − 𝜏𝑦𝑧) = 𝜌𝑓𝑦

𝜕𝜌𝑤

𝜕𝑡+

𝜕

𝜕𝑥(𝜌𝑢𝑤 − 𝜏𝑥𝑧) +

𝜕

𝜕𝑦(𝜌𝑣𝑤 − 𝜏𝑦𝑧) +

𝜕

𝜕𝑧(𝜌𝑤2 + 𝑝 − 𝜏𝑧𝑧) = 𝜌𝑓𝑧

𝜕𝜌𝑒

𝜕𝑡+

𝜕

𝜕𝑥𝜌𝑢𝑒 + 𝑝𝑢 +

𝜕

𝜕𝑦𝜌𝑣𝑒 + 𝑝𝑣 +

𝜕

𝜕𝑧𝜌𝑤𝑒 + 𝑝𝑤 =

𝜕

𝜕𝑥𝑢𝜏𝑥𝑥 + 𝑣𝜏𝑥𝑦 + 𝑤𝜏𝑥𝑧 − 𝑞𝑥 +

𝜕

𝜕𝑦𝑢𝜏𝑦𝑥 + 𝑣𝜏𝑦𝑦 + 𝑤𝜏𝑦𝑧 − 𝑞𝑦 +

𝜕

𝜕𝑧𝑢𝜏𝑧𝑥 + 𝑣𝜏𝑧𝑦 + 𝑤𝜏𝑧𝑧 − 𝑞𝑧

Introduction to CFD

• Governing Equations

– Viscous stress tensor 𝜏𝑖𝑗 = μ𝜕𝑢𝑖

𝜕𝑥𝑗+

𝜕𝑢𝑗

𝜕𝑥𝑖−

2

3𝛿𝑖𝑗

𝜕𝑢𝑘

𝜕𝑥𝑘

– Total energy per unit mass 𝑒 = 𝑖 + 𝑉2

2+ 𝑔𝑧

– Heat transfer Ԧ𝑞 = −𝑘𝛻𝑇

– Unknowns(6) 𝜌 𝑢 𝑣 𝑤 𝑝 𝑇

– Equation of state 𝑝 = 𝑝(𝜌, 𝑇)

4

Introduction to CFD

• Governing Equations(Cont.)

– 2-D Navier-Stokes Equations in a vector form (4 scalar equations)

𝜕𝑄

𝜕𝑡+𝜕𝐸

𝜕𝑥+𝜕𝐹

𝜕𝑦=𝜕𝐸𝑣𝜕𝑥

+𝜕𝐹𝑣𝜕𝑦

– Conservation Variables

𝑄 =

𝜌𝜌𝑢𝜌𝑣𝜌𝑒

=

𝑞1𝑞2𝑞3𝑞4

– Flux vectors

𝐸 =

𝜌𝑢

𝜌𝑢2 + 𝑝𝜌𝑢𝑣

𝜌𝑒 + 𝑝 𝑢

𝐹 =

𝜌𝑣𝜌𝑢𝑣

𝜌𝑣2 + 𝑝

𝜌𝑒 + 𝑝 𝑣

𝐸𝑣=

0𝜏𝑥𝑥𝜏𝑥𝑦

𝑢𝜏𝑥𝑥 + 𝑣𝜏𝑥𝑦 − 𝑞𝑥

𝐹𝑣 =

0𝜏𝑥𝑦𝜏𝑦𝑦

𝑢𝜏𝑥𝑦 + 𝑣𝜏𝑦𝑦 − 𝑞𝑦

5

Introduction to CFD

• Solutions of Governing Equations– Non-linear PDEs(partial differential equations)

– Generally impossible to obtain analytic solutions

• Theoretical(Analytical) approach– Simplified equations with simplified physics for simple geometry

– Exact solutions for limited(specific) problems

– Asymptotic solutions for more problems (but, still limited)

– Solution is a continuous function in space (and time, if unsteady)

• Discretization methods of CFD– FDM(Finite Difference Method)

– FVM(Finite Volume Method)

– FEM(Finite Element Method)

– Solution is obtained as numbers at a finite number of discrete points

6

Discretization Methods

• FDM(Finite Difference Method)– the oldest method among CFD methods

– at each node Taylor series expansions are used

– finite-difference approximations to the derivatives of PDE

– commonly applied to structured grids

– for uniformly-spaced grid

7

𝜕𝜙

𝜕𝑥𝑖,𝑗

=𝜙𝑖+1,𝑗 − 𝜙𝑖,𝑗

∆𝑥+ 𝑂 ∆𝑥

𝜕𝜙

𝜕𝑥𝑖,𝑗

=𝜙𝑖+1,𝑗 − 𝜙𝑖−1,𝑗

2∆𝑥+ 𝑂 ∆𝑥2

𝜕𝜙

𝜕𝑥𝑖,𝑗

=−𝜙𝑖+2,𝑗 + 8𝜙𝑖+1,𝑗 − 8𝜙𝑖−1,𝑗 + 𝜙𝑖−2,𝑗

12∆𝑥+ 𝑂 ∆𝑥4

(𝐚𝐜𝐜𝐮𝐫𝐚𝐜𝐲 𝐝𝐞𝐭𝐞𝐫𝐢𝐨𝐫𝐚𝐭𝐞𝐬 𝐟𝐨𝐫 𝐧𝐨𝐧 − 𝐮𝐧𝐢𝐟𝐨𝐫𝐦 𝐠𝐫𝐢𝐝𝐬)


• FDM(Finite Difference Method)

– Euler equations in Cartesian Coordinates

𝜕𝑄

𝜕𝑡+𝜕𝐸

𝜕𝑥+𝜕𝐹

𝜕𝑦= 0

– Transformation by “Chain rule”

8

𝜕

𝜕𝑡=

𝜕

𝜕𝜏+𝜕𝜉

𝜕𝑡

𝜕

𝜕𝜉+𝜕𝜂

𝜕𝑡

𝜕

𝜕𝜂𝜕

𝜕𝑥=𝜕𝜉

𝜕𝑥

𝜕

𝜕𝜉+𝜕𝜂

𝜕𝑥

𝜕

𝜕𝜂

𝜕

𝜕𝑦=𝜕𝜉

𝜕𝑦

𝜕

𝜕𝜉+𝜕𝜂

𝜕𝑦

𝜕

𝜕𝜂

we can define the coordinate transformation

𝜉 = 𝜉 𝑥, 𝑦, 𝑡 ⇔ 𝑥 = 𝑥 𝜉, 𝜂, 𝜏𝜂 = 𝜂 𝑥, 𝑦, 𝑡 ⇔ 𝑦 = 𝑦 𝜉, 𝜂, 𝜏

𝜏 = 𝑡 𝑡 = 𝜏

𝜕𝑡𝜕𝑥𝜕𝑦

=

1 𝜉𝑡 𝜂𝑡0 𝜉𝑥 𝜂𝑥0 𝜉𝑦 𝜂𝑦

𝜕𝜏𝜕𝜉𝜕𝜂

𝒐𝒓, 𝒓𝒆𝒗𝒆𝒓𝒔𝒆𝒍𝒚

𝜕𝜏𝜕𝜉𝜕𝜂

=

1 𝑥𝜏 𝑦𝜏0 𝑥𝜉 𝑦𝜉0 𝑥𝜂 𝑦𝜂

𝜕𝑡𝜕𝑥𝜕𝑦


• FDM(Finite Difference Method)▪ The transformations are inverse of each other

1 𝜉𝑡 𝜂𝑡0 𝜉𝑥 𝜂𝑥0 𝜉𝑦 𝜂𝑦

=

1 𝑥𝜏 𝑦𝜏0 𝑥𝜉 𝑦𝜉0 𝑥𝜂 𝑦𝜂

−1

= 𝐽

𝑥𝜉𝑦𝜂 − 𝑦𝜉𝑥𝜂 −𝑥𝜏𝑦𝜂 + 𝑦𝜏𝑥𝜂 𝑥𝜏𝑦𝜉 − 𝑦𝜏𝑥𝜉0 𝑦𝜂 −𝑦𝜉0 −𝑥𝜂 𝑥𝜉

▪ Metrics of transformations : 𝜉𝑥 𝜉𝑦 𝜂𝑥 𝜂𝑦

(interpreted as the ratios of arc lengths in both space, 𝜉𝑥 =𝜕𝜉

𝜕𝑥≈

∆𝜉

∆𝑥)

▪ Jacobian of the transformations 𝐽 =𝜕 𝜉,𝜂

𝜕 𝑥,𝑦=

𝜉𝑥 𝜉𝑦𝜂𝑥 𝜂𝑦

= 𝜉𝑥𝜂𝑦-𝜉𝑦𝜂𝑥

▪ Inverse Jacobian 𝐽−1 = 𝑥𝜉𝑦𝜂 − 𝑥𝜂𝑦𝜉 (𝐽−1 ∶ 𝒄𝒆𝒍𝒍 𝒗𝒐𝒍𝒖𝒎𝒆 𝒊𝒏 𝒑𝒉𝒚𝒔𝒊𝒄𝒂𝒍 𝒅𝒐𝒎𝒂𝒊𝒏)

Euler equations in Curvilinear coordinates

9

𝜕 ෨𝑄

𝜕𝜏+

𝜕 ෨𝐸

𝜕𝜉+

𝜕 ෨𝐹

𝜕𝜂=0 with

෨𝑄 = 𝐽−1𝑄෨𝐹 = 𝐽−1(𝜉𝑡𝑄 + 𝜉𝑥𝐹 + 𝜉𝑦𝐺)෨𝐺 = 𝐽−1(𝜂𝑡𝑄 + 𝜂𝑥𝐹 + 𝜂𝑦𝐺)


• FVM(Finite Volume Method)– discretizes the integral form of the conservation equations

– computational domain is subdivided into a finite number of cells

– flow variables calculated at the centroid of each CV(Control Volume)

– interpolation is used to express variable values at the surfaces of CV

– used in most commercial codes

10

𝜕𝜙

𝜕𝑥=

1

∆𝑉න

𝑉

𝜕𝜙

𝜕𝑥dV =

1

∆𝑉න

𝑉

𝜙𝑑𝐴𝑥 ≈1

∆𝑉

𝑖=1

𝑁

𝜙𝑖𝑑𝐴𝑖𝑥

𝜕𝜙

𝜕𝑦=

1

∆𝑉න

𝑉

𝜕𝜙

𝜕𝑦dV =

1

∆𝑉න

𝑉

𝜙𝑑𝐴𝑦 ≈1

∆𝑉

𝑖=1

𝑁

𝜙𝑖𝑑𝐴𝑖𝑦

Advantages of FVM over FDM▪ it has good conservation properties▪ applicable to complicated physical domains

Grid Generation

• Grid– How the grid points are distributed affects not only the

accuracy of the flow solutions but the time it takes to obtain the flow solutions

• Grid Generation– Grid generation part of CFD analysis procedure is still a time-

consuming and labor-intensive process

– It requires experience and many man-hours

– It is usually a trial-and-error process

– Generally, it is agreed that grid generation is the bottle-neck for a routine application of CFD

11

Desirable Grid System

• A mapping which guarantees one-to-one correspondence

ensuring grid lines of the same family do not cross each other

• Smoothness of the grid point distribution with no discontinuities

• Orthogonality or near-orthogonality of the grid lines, especially to

the boundaries

• Grid point clustering in regions of interest

• In short, grid with good quality

12

Grid Type

• Structured grid

– Multi-block grid

– Patched grid

– Overset(Chimera) grid

• Unstructured grid

– Triangular grid

– Quadrilateral grid

– Polyhedral grid

• Hybrid grid

• Cartesian grid

13

Structured Grid Generation Schemes

• Algebraic scheme

• Conformal mapping

• PDE-based method

– Elliptic scheme

– Hyperbolic scheme

– Parabolic scheme

– Mixed scheme

• Variational method

14


• Algebraic scheme– Features

▪ Simplest grid generation technique

▪ Algebraic equation is used to distribute grid points

▪ Interpolation is used to generate interior points from the boundary points (boundary points should be provided)

▪ can be generated easily and takes small CPU time for calculation

▪ less smooth than grids by PDE schemes (propagation of slope discontinuities)

▪ often used as initial conditions for iterative elliptic scheme

▪ TFI(Transfinite Interpolation) is most popular

– Formulation of TFI scheme

𝑋 𝜉, 𝜂 =

𝑛=1

2

𝑎𝑛 𝜂 𝑋 𝜉, 𝜂𝑛 +

𝑚=1

2

𝑏𝑚 𝜉 𝑋 𝜉𝑚 , 𝜂 +

𝑛=1

2

𝑚=1

2

𝑎𝑛 𝜂 𝑏𝑚 𝜉 𝑋 𝜉𝑚 , 𝜂

• Linear Interpolants

𝑎1 𝜂 = 1 −𝜂−𝜂1

𝜂2−𝜂1𝑎2 𝜂 =

𝜂−𝜂1

𝜂2−𝜂1𝑏1 𝜉 = 1 −

𝜉−𝜉1

𝜉2−𝜉1𝑏2 𝜉 =

𝜉−𝜉1

𝜉2−𝜉1

15


• Conformal mapping– Features

▪ Conformal map: a function that preserves angles locally

▪ A function 𝑓: 𝑈 → 𝑉 is called conformal (or angle-preserving) at a point 𝑢0 ∈ 𝑈if it preserves oriented angles between curves through 𝑢0 with respect to their orientation

▪ Conformal map preserves both angles and the shapes of infinitesimally small figures

▪ It does not necessarily preserve their size or curvature

16


• PDE-based - Elliptic Scheme– Features

▪ entire boundary points should be specified (elliptic PDE=B.V.P.)

▪ proper for internal flows

▪ boundary slope discontinuity does not propagate into the interior

▪ slow due to its iterative solution procedure

▪ grid spacing and angle control through the control functions

▪ generates smooth grids, and most popular

– Formulation of Elliptic scheme𝛻2𝜉 = 𝑃 (𝜉𝑥𝑥 + 𝜉𝑦𝑦 = 𝑃)

𝛻2𝜂 = 𝑄 (𝜂𝑥𝑥 + 𝜂𝑦𝑦 = 𝑄)

Reversely 𝑎𝑋𝜉𝜉 + 𝑏𝑋𝜂𝜂-2c𝑋𝜉𝜂 = −𝐽−2(𝑃𝑋𝜉 + 𝑄𝑋𝜂)

Where,

a = 𝑋𝜂 ∙ 𝑋𝜂 b = 𝑋𝜉 ∙ 𝑋𝜉 c = 𝑋𝜉 ∙ 𝑋𝜂 𝐽−1 =

𝜕(𝑥,𝑦)

𝜕(𝜉,𝜂)

17


• PDE-based - Hyperbolic Scheme– Features

▪ generates by marching in the outward direction (non-iterative)

▪ boundary conditions need not be specified on all boundaries

▪ one boundary(usually along the body surface) is specified

▪ suitable for physically unbounded regions (external flows)

▪ generates orthogonal grids

▪ grid shocks can occur

– How it works▪ from selected grid points on the boundary, grid points are chosen by

marching outward with a given slope (normal to the previous grid line) and arc-length (or cell volume)

▪ solves 2 equations

∙ orthogonality (for marching direction)

∙ arc-length equation or cell-volume equation (for spacing control)

18


• PDE-based - Hyperbolic Scheme– Formulation of Hyperbolic scheme

▪ Arc-length approach : solves 2 equations given for

• Orthogonality 𝛻𝜉 ∙ 𝛻𝜂 = 0

• Arc-length specified (𝑑𝑠)2= (𝑑𝑥)2+(𝑑𝑦)2

From orthogonality relation 𝛻𝜉 ∙ 𝛻𝜂 = 𝜉𝑥𝜂𝑥 + 𝜉𝑦𝜂𝑦 = −𝐽(𝑦𝜂𝑦𝜉 + 𝑥𝜂𝑥𝜉)

Therefore, 𝑥𝜂𝑥𝜉 + 𝑦𝜂𝑦𝜉 = 0

Arc-length relation gives (𝑑𝑠)2= (𝑥𝜉𝑑𝜉 + 𝑥𝜂𝑑𝜂)2+(𝑦𝜉𝑑𝜉 + 𝑦𝜂𝑑𝜂)

2

And ∆𝑠 ≡𝑑𝑠

𝑑𝜂is specified (∆𝝃 = ∆𝜼 = 𝟏 (𝒅𝝃 = 𝒅𝜼 = 𝟏))

Then the arc-length relation becomes

(∆𝒔)𝟐= 𝑥𝜉𝟐 + 𝟐𝑥𝜂𝑥𝜉 + 𝑥𝜂

2 + 𝑦𝜉2 + 𝟐𝑦𝜂𝑦𝜉+𝑦𝜂

2=𝑥𝜉2+𝑥𝜂

2+𝑦𝜉2+𝑦𝜂

2 (← 𝑥𝜂𝑥𝜉 + 𝑦𝜂𝑦𝜉 = 0)

Introducing 𝑥𝑜, 𝑦𝑜 as values at a known position :

𝑥 = 𝑥𝑜 + ҧ𝑥 𝑦 = 𝑦𝑜 + ത𝑦

19



▪ Arc-length approach

the final equation become

𝑥𝜂𝑜 𝑦𝜂

𝑜

𝑥𝜉𝑜 𝑦𝜉

𝑜

𝑥𝑦

𝜉+


𝑜


𝑜

𝑥𝑦

𝜂=

𝑥𝜉𝑜𝑥𝜂

𝑜 + 𝑦𝜉𝑜𝑦𝜂

𝑜

1

2(∆𝑠)2+(𝑥𝜉

𝑜)2+(𝑥𝜂𝑜)2+(𝑦𝜉

𝑜)2+(𝑦𝜂𝑜)2

or simply 𝐴𝑋𝜉 + 𝐵𝑋𝜂 = Ԧ𝑓

▪ We can solve this equation once we specify the distribution of points along

the boundary, and a means for selecting Δ𝑠

▪ This method works fine on convex surfaces.

▪ However, for concave boundaries much poorer results are found

20

(a) (b)



▪ Cell-volume approach : solves 2 equations given for

• Orthogonality 𝑥𝜂𝑥𝜉 + 𝑦𝜂𝑦𝜉 = 0

• Cell-volume specified 𝑥𝜉𝑦𝜂 + 𝑥𝜉𝑦𝜂 = 𝐽−1 = 𝑉

after linearization of the equations


𝑜

𝑦𝜂𝑜 −𝑥𝜂

𝑜

𝑥𝑦

𝜉+


𝑜

−𝑦𝜉𝑜 𝑥𝜉

𝑜

𝑥𝑦

𝜂=

0𝑉 + 𝑉0

or, again simply

ሚ𝐴𝑋𝜉 + ෨𝐵𝑋𝜂 = Ԧ𝑔

21


• PDE-based - Hyperbolic Scheme– Advantages

▪ The grid system is orthogonal in two-dimensions

▪ Since a marching scheme is used for the solution of the system, computationally they are much faster compared to elliptic systems

▪ Grid line spacing may be controlled by the cell volume or arc-length functions

– Disadvantages▪ Extension to three-dimensions where complete orthogonality exists is not

possible

▪ They cannot be used for domains where the outer boundary is specified

▪ Boundary discontinuity may be propagated into the interior domain

▪ Specifying the cell-area or arc-length functions must be handled carefully. A bad selection of these functions easily leads to undesirable grid systems

22


• PDE-based - Parabolic Scheme– Features

▪ solved by marching in one direction (non-iterative)

▪ using elliptic equations(Laplace eqn, Poisson eqn) locally

▪ no grid shocks occur due to natural diffusions(2nd order derivatives)

▪ outer boundary influence can be included in the marching process

▪ difficulties in orthogonality control

▪ solves elliptic equations in a marching fashion

– Formulation of Parabolic scheme

𝜕𝑥

𝜕𝜂− 𝐴

𝜕2𝑥

𝜕𝜉2= 𝑆𝑥

𝜕𝑦

𝜕𝜂− 𝐴

𝜕2𝑦

𝜕𝜉2= 𝑆𝑦

23

𝑆𝑥 , 𝑆𝑦 : source terms

𝐴 : specific constant


• PDE-based – Mixed Scheme– Features

▪ combination of different PDE schemes

▪ take advantage of desirable features of each scheme

▪ mixing in the equation level, in the calculation level, or in the result level

▪ Ex) Hyperbolic marching + Elliptic smoothing

24


• PDE-based – Mixed Scheme– Comparison of resultant Grids

▪ C-type grid for an Airfoil

▪ Hyperbolic scheme vs. Mixed scheme

25

(a) Hyperbolic scheme (b) Mixed scheme

Surface Grid Generation

• Surface mesh generation

– Useful for surface panel method

26

Edge Point Distribution

• 1-D Point Distribution

– Grid generation as B.V.P.(Boundary Value Problem)

– Node distribution along boundary is needed

• Stretching Functions

– Exponential

– Cubic polynomial

– Hyperbolic tangent

– Hyperbolic sine

27

Grid Adaptation

• Grid Adaptation

28

(a) Before

(b) After

29

Measures of Grid

• Measures of Grid– Availability of flow solvers

– Accuracy and efficiency of flow solvers

– Turn-around time of final grids

– Ease for generation

– Block generation (structured grid)

– Automation level

– Adaptation

– Grid quality

– Surface grid generation

– Multi-body problems

– Bodies in relative motion

– CAD-CFD data interface

Documents

(Grid Generation 정렬격자 · 2017. 6. 8. · Ex) Hyperbolic marching + Elliptic smoothing 24. Structured Grid Generation Schemes • PDE-based –Mixed Scheme ... • Stretching