BSE Public CPD Lecture – Computational Fluid Dynamics based on the Unified Coordinates on 4 December 2009 Organized by the Department of Building Services Engineering, a public CPD lecture delivered by Professor W.H. Hui on Computational Fluid Dynamics based on the Unified Coordinates was held on 4 December 2009 (Friday). Over 90 participants attended this public CPD Lecture.

Powerpoint file of the CPD lecture Professor W.H. Hui is a Professor Emeritus of the University of Waterloo of Canada and the Hong Kong University of Science and Technology. His main research interests include computational fluid dynamics, high speed aerodynamic stability theory, nonlinear water wave theory and similarity solutions of partial differential equations.

Introducing Professor W.H. Hui

Computational Fluid Dynamics (CFD) uses large scale numerical computation to solve problems of fluid flow. Traditionally, it uses either the Eulerian or the Lagrangian coordinate system. These two systems are numerically non-equivalent, but each has its advantages as well as drawbacks. In the lecture, Professor Hui explained that a unified coordinate system (UC) has recently been developed which combines the advantages of both Eulerian and Lagrangian systems, while avoiding their drawbacks. Finally, a systematic discussion on CFD using the unified coordinates was given. Many examples were also given to demonstrate the properties of the UC.

CPD public lecture by Professor Hui

08-Dec-09

1

Computational Fluid Dynamicsbased on

the Unified Coordinates

Grafton W.H. Hui (許為厚)Grafton W.H. Hui (許為厚)

Emeritus Professor of Applied MathematicsUniversity of Waterloo and

Hong Kong University of Science and Technology

Building Services Engineering Dept., HKPolyU4 December, 2009

Content

Introduction

The Unified Coordinates

1-D Flow

2 D Steady Supersonic Flow2-D Steady Supersonic Flow

2-D and 3-D Unsteady Flow

Lagrangian Gas Dynamics

Automatic Mesh Generation

Conclusions

* Eulerian coordinates (mesh) are fixed in space

* L i di t ( h) ith fl id

Two coordinate systems

for describing fluid motion have existed:

1. Introduction

* Lagrangian coordinates (mesh) move with fluid

Are they equivalent theoretically?

“Yes” for 1-D flow (D H Wagner, J. Diff. Eq., 64, 118-136, 1987)

“No” for 2- and 3-D flow (W H Hui, et al, J. Comput.Phys., 153, 596-637, 1999)

Computationally, they are not equivalent,

even for 1-D flow: each has advantages as well as drawbacks.

This motivates us toThis motivates us to

search for an “optimal” coordinate system, leading to the unified coordinates.

“Optimal” Coordinates

For compressible flow, we want a coordinate system to have the following properties:

(a) Conservation PDEs exist, as in Eulerian;

(b) Contacts are sharply resolved, as in Lagrangian;

(c) Body-fitted mesh can be generated automatically;

(d) Mesh to be orthogonal;(e) Mesh to be uniform;(f)……

The unified coordinate system satisfies these requirements

2. The Unified Corrdinates (λ,ξ,η,ζ)are given by the transformation from Eulerian (t, x, y, z)

ddt PdLdAdUddx QdMdBdVddy RdNdCdWddz

We get

So (ξ,η,ζ), and hence the computational cells, move with the

pseudo particle whose velocity is which is arbitrary

x

Q QtDt

D

),,( WVUQ

0

Dt

DQ

08-Dec-09

2

.

, ,

W

λ

R,

W

λ

N,

ξ

W

λ

C

V

λ

QV

λ

M,

V

λ

B

,U

λ

P ,

U

λ

L,

ξ

U

λ

A

Compatibility Conditions(only nine of them are independent)

Time evolution

(Geometric conservation laws)

λλξλ

. ,

, ,

, ,

RN

ξ

RC,

ξ

NC

QM

ξ

QB,

ξ

MB

PL

ξ

PA,

ξ

LA

Free divergence constraints

Special Cases: Eulerian Q = 0Lagrangian Q = q, q being fluid velocity

In the general case, we have a unified (Euler-Lagrangian) coordinate system with 3 degrees of freedom: U, V and W are arbitrary.freedom: U, V and W are arbitrary.

),,( WVUQ

Need 3 conditions to determine the mesh velocity

The coordinates (λ,ξ,η,ζ) are called

unified in the sense

(a) they unify the Lagrangian and Eulerian di tcoordinates;

(b) they unite the geometric conservation laws with the physical conservation laws.

One way to choose the mesh-moving velocity (U, V, W)

1-D thus U = u, but A is arbitrary(classical Lagrangian: U = u, A = 1/ )plus adaptive Godunov scheme

2-D plus mesh-angle preserving

3-D plus mesh-skewness preserving

0Dt

Dq

0Dt

Dq

0Dt

Dq

In this way, UC is a generalization of Lagrangian coordinates.In particular, it resolves contacts sharply without mesh tangling.

There are other ways to choose the moving velocity (examples are: ALE and aerodynamics of falling leaves).

0Dt

Dqr

Dt

3. One-Dimensional Flow

It is shown that

UC (Lagrangian coordinate +

shock-adaptive Godunov scheme)

i i t L i tis superior to Lagrangian system

which, in turn, is superior to the

Eulerian system

(W H Hui and S Kudriakov, J. Comput. Physics, 227,

2105-2117 (2008))

(Eulerian,conservation formand hyperbolic)

By transformation

duddx

ddt1

we get(Classical Lagrangian,conservation form

d h b li )0

p

u

u

1

(The seminal papers by Von Neumann(1946) and Godunov (1959)

use Lagrangian system. Equivalency established by D H Wagner (1987))

and hyperbolic)

upe

In UC, we use adaptive Godunov scheme (Hui & Lapage, JCP, 122, 291(1995)) : the shock is fitted (using the Riemann solution), hence we can use a different conservative system

0

0

1

p

u

pu

08-Dec-09

3

Ex 1. Contact Resolution in A Riemann Problem (Godunov-MUSCL)

UC (Lagrangian + Adaptive-Godunov scheme)

UC can cure all known defects of Eulerian shock-capturing methods, namely

contact smearingslow moving shockssonic-point glitch wall-overheating*start-up errors*low-density flow* strong rarefaction waves*

(*also shared in Lagrangian Computation)

So, UC is superior to Lagrangian and Eulerian system and is completely satisfactory.

4. 2-D Steady Supersonic Flow

UC reduces the 2-D steady supersonic flow problem to that of 1-D unsteady flow bymarching in the flow direction (instead of time)time).

It is robust, most accurate and efficient.

Example 2 : Steady supersonic airfoil(Space-marching method)

1.745M

0.0

Y 0

0.5

1

1.5

X

Y

0 0.5 1-1.5

-1

-0.5

0

1.0

Y 0

0.5

1

1.5

X

Y

0 0.5 1-1.5

-1

-0.5

0

08-Dec-09

4

3.0Y 0

0.5

1

1.5

X

Y

0 0.5 1-1.5

-1

-0.5

0

7.0

Y 0

0.5

1

1.5

X

Y

0 0.5 1-1.5

-1

-0.5

0

1.1

Y 0

0.5

1

1.5

X

Y

0 0.5 1-1.5

-1

-0.5

0

Flow-Generated Mesh (Close View) Y 0

0.25

0.5

X

Y

0 0.25 0.5 0.75 1-0.5

-0.25

0

chN

umbe

r

1.5

2

2.5

3

Surface Mach number distribution, 120 cellsComputing time: 1.8s (P4, 2.8GHz)

X

Mac

0 0.25 0.5 0.75 10

0.5

1

Lower SurfaceUpper SurfaceExact Solution

ach

Num

ber

1 2

1.4

1.6

1.8

2

2.2

2.4

2.6

2.8

3

X

Ma

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

1.2

Exact SolutionUpper SurfaceLower Surface

Surface Mach Number Distributions on Diamond-Shaped Airfoil.

Eulerian Computation (5th Order WENO Scheme), 100 x 200 cells.

Computing Time to 20,000 Steps: 2,393s on P4, 2.8 GHz.

08-Dec-09

5

5. 2-D & 3-D Unsteady FlowEulerian method is relatively simple, because the Euler equations of gas dynamics can be written in conservation PDE form, which is the basis for shock-capturing methods, but

(a) it smears contact interfaces badly, and

(b) it needs generating a body-fitted mesh( ) g g y

Lagrangian method resolves contact and materialinterfaces (including free surfaces) sharply, because they coincide with the coordinate surfaces, but

(a) it breaks down due to cell deformation, and(b) the gas dynamics equations could not be

written in conservation PDE form.

A typical Lagrangian mesh

Computation breaks down soon afterwards

D Serre in “Systems of Conservation Laws” (1999) states “Writing the equations of gas dynamics in Lagrangian coordinates is very complicated if ”.2D

The difficulty lies in the momentum equation

pt x

x

2

2

where

uv

v

pu

u2

0

y

G

x

F

t

E(E)

2-D Gas Dynamic Equations

In Eulerian Coordinates

pev

pv

uvG

peu

uv

puF

e

v

uE 2 , ,

p

vue1

1)(

2

1 22

Eq. (E) is: (a) in conservation form, (b) hyperbolic in t

Transforming to the unified coordinates, (E) become

0

GFE

where

0

)( ,

)( ,

uBvApYe

pAYv

pBYu

Y

GU

vLuMpXe

pLXv

pMXu

X

FA

Je

Jv

Ju

J

E

(U)

0

0

0

0

V

U

V

U

M

L

B

A

With J = AM – BL, X = (u - U)M – (v - V)L and Y = (v – V)A – (u – U)B.

Eq (U) is: (a) closed system in conservation form, (b) hyperbolic in λ, except Lagrangian when

U = u and V = v

Remarks

Consequences of having a closed system of

governing equations in conservation form are:

(a) Effects of moving mesh on the flow are fully(a) Effects of moving mesh on the flow are fully accounted for;

(b) The system can by solved as easily as the

Eulerian one.

08-Dec-09

6

UC ComputationAs we have conservation form, computations are done

like Eulerian in -- space by marching in timeλ, (eg, Godunov-MUSCL scheme plus splitting). At each time step

the mesh velocity (U, V) is given or computed.

Determination of Mesh Velocity (U, V )

(A) coordinate lines = const. shall be material lines of fluid particles, meaning .

(V – v) A - (U – u) B = 0 (a)

We place two requirements:

0Dt

D q With 0Dt

D Q

this gives

(2) As the body surface is a material line, condition (a) guarantees thatthe unified mesh is automatically a body-fitted mesh.

Observations:

(1) Contact lines, being material lines, must coincide with coordinate

lines = const and, therefore, can be resolved sharply.

(3) (x, y, t) is a level set function.

2

);(A

BB

ALA

BB

AS

P

(B) Mesh angles and orthogonality shall be preserved, yielding an ODE for U

),;(),;(

QUPU

(b)

U (η) prescribed at η= const.

222222

2

2

2

,

),;(),;(

),;(

BATMLS

uPu

Bv

AJ

LvA

uB

JT

ASQ

BAAJ

BAJT

P

Alternatively, we may require area (Jacobian)

be preserved.

6. Lagrangian Gas Dynamics

For U = u and V = v, we get

0

GFE (L)

where

v

u

uBvAp

pA

pB

G

v

u

vLuMp

pL

pM

F

M

L

B

A

Je

Jv

Ju

J

E

0

0

)(

0

,

0

0

)(

0

,

Remarks

(1) The gas dynamics equations in Lagrangian coordinates are written in conservation form for the

first time. (W. H. Hui, et al, J. Comput. Phys., 153, 1999)

(2) Lagrangian GD equation is only weakly hyperbolic: all eigenvalues are real, but there is no complete set of linearly independent eigenvectors. This is also true for 3-D case. Hence, Lagrangian GD is not equivalent to Eulerian GD.

7. Automatic Mesh GenerationExample 3. M = 0.8 past a NACA 0012 airfoil

(WH Hui & JJ Hu)

08-Dec-09

7

-CP

0

0.5

1

X0 0.25 0.5 0.75 1

-1

-0.5

Pressure at t=10.1Hafez et al. [4]

08-Dec-09

8

-Cp

0

0.5

1

X0 0.25 0.5 0.75 1

-1

-0.5

PresentHafez et al. [4]

Example 4 Two-Fluids Flow past a NACA 0012 airfoil

M = 2.2 AoA = 8 degrees

08-Dec-09

9

Example 5: Supersonic flow past a pitchingoscillating diamond-shape airfoil

( G P Zhao & W H Hui)

t sin0

3M)

Unsteady pressure distribution

Example 6

Aerodynamics ofFreely Falling Leaves

by

Changqiu Jin and Kun Xu

HKUST

falling leaves with fluttering and tumbling motion

Computed paths (10 full rotations )

-10

0

fluttering

-

-

-

X0 10 20

-40

-30

-20

tumbling

-

-

-

-

08-Dec-09

10

-0.002 0 0.002

0

2

Mesh is fixed and moved with the body;this gives mesh velocity (U, V) at all time.

-0.004 -0.002 0 0.002 0.0

4

2

Computational procedure:

(1) At time t, the body position and mesh velocity are given;

(2) Use Xu’s gas kinetic-BGK solver to find the flow and hence the aerodynamic forces on the body (Note: the effects of mesh movement on the flow are correctly and fully accounted for through theare correctly and fully accounted for through the geometric conservation laws);

(3) Use Newton’s 2nd law to compute the motion of the body under the aerodynamic and gravitational forces, giving the body position and the mesh movement at new time;

Repeat (1) – (3).

measured by experiment

Computed by Cornell

0 2 4

−1.0

−0.5

0

0.5

θ [ π ]

Fy [

m’ g

]

(a)

0 2 40

0.5

1

1.5

2

θ [ π ]

Fy [

m’ g

]

(b)

0 2 4−0.04

−0.02

0

0.02

θ [ π ]

Tor

que

[ m’ g

l ]

(c)

by Cornell group

Computed by HKUST

08-Dec-09

11

8. Conclusions and Discussions

(1) For 1-D flow, UC is superior to Eulerian and Lagrangian system and completely satisfactory. Also, with UC, 2-D steady supersonic flow is reduced to 1-D unsteady flow, hence is also completely satisfactorycompletely satisfactory.

(2) The governing equations in any moving coordinates can be written as a system of closed conservation PDE; consequently, effects of moving mesh on the flow are fully accounted for.

(3) The system of Lagrangian gas dynamics equations is written in conservation PDE form, thus providing a foundation for developing Lagrangian schemes as moving mesh schemes. (B. Despres & C. Mazeran, Arch. Rational Mech. Anal.,178, 2005)

(4) The Lagrangian system of gas dynamics equations in 2-D and 3-D are shown to be only weakly hyperbolic, in direct contrast to the Eulerian system which is fully hyperbolic; hence the two systems are not equivalent to each other.

(5) The UC possesses the advantages of Lagrangian system: contact discontinuities are resolved sharply.

(6) In using the UC, there is no need to(6) In using the UC, there is no need to generate a body-fitted mesh prior to computing flow past a body; the mesh is automatically generated by the flow.

(5) The UC is superior to other methodsA. Eulerian

is a special case of UC. It is relatively simple, because the Euler equations of gas dynamics can be written in conservation PDE form, which is the basis for shock-capturing methods, but

(a) it smears contacts badly, and(b) it needs generating a body-fitted mesh.

The UC resolves contacts sharply and generates abody-fitted mesh automatically.

B. LagrangianLagrangian method resolves contacts and materialinterfaces (including free surfaces) sharply, because they coincide with Lagarangian coordinate surfaces, but(a) it breaks down due to cell deformation, and(b) the gas dynamics equations could not be

written in conservation PDE form, rendering it difficult to capture shocks correctly (staggered p y ( ggmesh may be required).

The UC also resolves contacts sharply but without breaking down. The gas dynamics equations in UC are in conservation PDE form, making it a moving mesh in Eulerian space.

C. Arbitrary-Lagrangian-Eulerian (ALE)

The UC approach and the ALE approach share the same spirit that they both combine the best features of Lagrangian and Eulerian approach and that the coordinates move at an arbitrary velocity. However, the strategies are quite different.

In the ALE method, “the general strategy is to perform a Lagrangian step and to follow it with a remap/rezone

h h l i i h di d L istep that maps the solution in the distorted Lagrangian mesh onto the spatially fixed Eulerian mesh”. This is usually done by employing a staggered mesh, due to lack of conservation form PDE. Furthermore, the rezoning strategies are not generally prescribed; instead, “rezoning requires the intervention of the user, … , and the success of ALE depends heavily on the skill and patience of the user”.

08-Dec-09

12

In the UC approach, since the governing equations are in conservation PDE form, the computation is done in one step as if it were an Eulerian computation, but without numerical diffusion across contact discontinuities, thus avoiding the numerical diffusion arising from the g guse of staggered mesh and from remapping/rezoning. The possible computational breakdown is prevented by requiring the mesh to be orthogonal (in 2-D), or to be skewness-preserving (in 3-D).

Alternatively, the UC can be used as ALE as follows:(i) Perform a Lagangian computation, ie, take Q = q.

In doing so, we do not need a staggered meshbecause the governing equations are in

conservation form.

(ii) Before mesh tangling, remap the solution in thedistorted Lagrangian mesh onto the spatially fixedg g p yEulerian mesh, and rezone.

Repeat steps (i) and (ii).The UC version of ALE is simpler and has less numerical diffusion than the classical ALE, because the use of staggered mesh in the Lagrangian computation step is avoided.

D. Moving mesh

In methods of this type, meshes are re-distributed

statically, or moved dynamically, according to flow

properties so as to increase computational accuracy

and efficiency. To account for the effects of mesh re-

distribution or movement on the flow, it is necessary to

have one or more space conservation equations whichhave one or more space conservation equations, which

are either derived mathematically or given intuitively.

While these may be necessary conditions, they may not

be sufficient to fully and correctly account for the effects

of mesh re-distribution or movement on the flow.

In the UC, these effects are fully and correctly

accounted for through the geometric conservation

laws – the compatibility conditions – which, together

with the physical conservation laws, form a system

of closed conservation PDEs.

E. Level-set

Since , (x, y, t) is a level-set function;

there is no need for a separate level-set function.ξ0

Dt

Dq

Potential Applications

(1) Multi-fluid flow: material interfaces

(2) Fluid-solid Interactions: moving interface and small cut cells

(3) Blood flow in arteries: deforming boundary

(4) Debris flow: terrain-following coordinates and moving boundary

(5) Typhoon: bottom topography, slip surface resolution and grid uniformity

08-Dec-09

13

Snow avalanche in the Alps

“A new model of granular flows over general topography …”, by Y C Tai and C Y Kuo,

Acta Mchanica, 199: 71-96, 2008

3-D extension

Thank You

(Ref: W.H. Hui, The unified coordinate system in computational fluid dynamics,

Communications in Computational Physics,2, 577-610, 2007)

A monograph

Computational Fluid Dynamics based on the

Unified Coordinates

by W H Hui and K Xu

08-Dec-09

14

Secret of Perfection

“As I am not pressed and work more for my pleasure than from duty,………..

I make, unmake, and remake, until I am passably satisfied with my results, which happens only rarely”

Joseph-Louis Lagrange (1736-1813)

Euler equations in Cartesian coordinates of an ideal gas obeying the Gama-law

Example 3 : 2-D Steady Riemann ProblemEulerian Computation (Godunov-MUSCL)

08-Dec-09

15

UC Computation(adaptive-Godunov scheme, Lepage & Hui, J. Comput. Phys., 1995) The Unified Corrdinates (λ,ξ)

Computation cells move with velocity U

ddt AdUddx

Eulerian if U = 0

Lagrangian if U = u & A = 1/

Unified if U = u but A is arbitrary

(has more flexibility)

Example 2. Defect due to Presence of

Strong Rarefaction Waves (Tang & Liu, 2006)

Consider a Riemann Problem

Left Right

Density 1000 1Velocity 0 0Pressure 1000 1

Need use UC; classical Lagrangian is not adequatedue to large density variation

Eulerian. velocities by the MUSCL-LLF method with the minmod limiter anda third-order RK time discretization with CFL = 0.8 (tang & Liu)

Eulerian (more severe case) velocities calculated by the fifth-order WENO scheme with CFL = 0.9

08-Dec-09

16

UC Computation, 100 cellsUC Computation, 1st order, 100 cellsUC Computation, 1st order, 100 cells

UC Computation, 1st order, 100 cells Hyperbolicity of Lagrangian Gas Dynamic Equations

All eigenvalues are real.

For eigenvalue 0 (multiplicity 6), there exist 4, or 3, or 5 linearly independent eigenvectors,4, or 3, or 5 linearly independent eigenvectors,depending on how the differential constraints

are used.

ξ

MB ,

ξ

LA

In all cases, the system is weakly hyperbolic.

Lagrangian re-formulation of Eulerian hyperbolic system will

generically leads to a weak hyperbolic system.

0 xt uuu

Consider inviscid hyperbolic Burgers equation

Transformation

ddt 01

Ddtd

Aduddx

ddt 0)(

1

DtudtdxA

dor

leads to

Eigenvalues are: 0, 0. But there is only one eigenvector:

.

1

0

Therefore, the system is weakly hyperbolic

0 0 1-

0 0

A

u

A

u

Special Case: Steady Flow

0Dt

D Q

0Dt

D q

In the special case of steady flow

becomes Q * η= 0

becomes q * η= 0Dt

Hence Q = h q

For steady supersonic flow, space-marching method

is most effective.

08-Dec-09

17

Example 7. Mach contours, Inviscid flow (WH Hui, ZN Wu, B Gao)

Eulerian coordinates

M = 4

M = 4

Unified coordinates

Mach contours: Re =104, Pr = 0.72

M = 4

Eulerian 120x120

M = 4

Unified 120x120

Eulerian 240x240

M = 4

Unified 120x120

Typhoon computation

2700

km

2700

km

(a) (a) terrainterrain--following following coordinatescoordinates(b) (b) automatic grid refinementautomatic grid refinement in inner regionin inner region(c)(c) sharp resolutionsharp resolution of vortex lines of vortex lines

2700 km2700 kmBoundary layer streamlines and isotachs (m sBoundary layer streamlines and isotachs (m s--11) 10 km grid in ) 10 km grid in nested gridnested grid hurricane model (Jones, 1977)hurricane model (Jones, 1977)

Derivation of Governing Equations in

Conservation PDE Form

in Unified Coordinates

Consider conservation of mass

3,2,1)(

0

jx

u

t j

j

1,,3,2,1,0)(

00

utx

x

u

theoremdevergenceGaussbyˆ xdu

are defined similarly.d̂

where is a control volume in space, and x

2103

1032

0321

3210

ˆ

ˆ

ˆ

ˆ

dxdxdxxd

dxdxdxxd

dxdxdxxd

dxdxdxxd

gy

08-Dec-09

18

Transform Cartesian coordinates (t, x, y, z) to the unified coordinates (λ, ξ, η, ζ) via

ddt

PdLdAdUdd PdLdAdUddx

RdNdCdWddz

QdMdBdVddy

Since from the transformation, we have

dUxd ˆˆ

where

,ˆˆ

xU

2103

1032

0321

3210

,,ˆ

,,ˆ

,,ˆ

,,ˆ

xxxx

xxxx

xxxx

xxxx

we get (by Gauss divergence theorem)

Similarly for the momentum and energy equations.

we get (by Gauss divergence theorem)

3,2,1,0

This is the mass equation in conservation form, where

UuK

0 ˆˆ0

KdKxdu

in Cartesian coordinates

0)(

3,2,10)(

3,2,1,00)(

pHu

jx

p

x

uux

u

j

j

Summary

in unified coordinates

0

0

3,2,1,00

0

pUHK

pUuK

K

jj

)( UuK

0)(

0

x

p

x

peH

The physical laws are now written in

conservation PDE form in the unified

coordinates, including the Lagrangian.

0

Summary

0

0

3,2,1,00

0)(

3,2,10)(

3,2,1,00)(

0

0

pUHK

pUuK

K

x

p

x

Hu

ix

p

x

uux

u

ii

d

id

i

d

p

peH

0ˆ

0ˆ

0ˆ

0ˆˆ

0ˆˆ

0ˆ

00

dpUHK

dpUuK

dK

xdpxdHu

xdpxduu

xdu

iiii