Meshing and CFD Accuracy - ICM

•

Presenter Name

Presenter Title

Presentation Title

June XX, 2005

**Meshing** **and** **CFD** **Accuracy**

June 9, 2005

1

Acknowledgement

• The Major part of the materials in this lecture is

provided by

Dr. Sung-Eun Kim

with additional contributions from

Dr. Tiberiu Barbat

Dr. Peter Spicka

2

Agenda

An overview of the finite-volume method

in FLUENT

Impact of mesh quality on **CFD** solutions

Impact of cell topology on solution

accuracy

Impact of mesh resolution

Further improvement on spatial accuracy

in the FLUENT V6.2 solver

3

Finite-Volume Method in FLUENT (1)

• Co-located (cell-centered)

formulation

• Locally **and** globally

conservative

• Can h**and**le arbitrary convex

polyhedral cells

• Second-order accurate in space

**and** time

– Diffusion terms are discretized with

second-order central differencing

scheme

– Convection terms are discretized

with second-order upwind, QUICK,

or central differencing schemes

(first-order upwind **and** power-law

are still available).

4

U, V,

W,

k,

ε,

T,...

Finite-Volume Method in FLUENT (2)

Equation Discretization

Gauss divergence theorem

f

c 0

5

Finite-Volume Method in FLUENT (3)

Reconstruction of Face Value **and** Gradient

c 0

f

δr c0

Face Value (upwind )

φ

f

= φ + δr

⋅

up

( ∇φ

) up

( ∇φ )

c 0

c 0

( ∇φ )

c 1

f

c 1

Face Gradient

1

2

[ ]

( ∇φ

) = ( ∇φ

) + ( ∇φ

)

f

c

0

c

1

Therefore “linear reconstruction” is actually

hinged on the calculation of the gradients!

6

Finite-Volume Method in FLUENT (4)

Cell-Gradient Calculation

∑f

( ∇φ

) ≅ φ

f

dA

f

c0

1

V

c0

Green-Gauss’

Theorem

f

C 0

f

f

C 1

7

Case Study: Fully-Developed Laminar Flow

in a 2-D Channel (Re H = 400) (1)

• A coarse mesh (228-cell triangular mesh) was

systematically refined to verify the order of spatial

accuracy.

• L 1 -norms of the error in U-velocity were computed for

the three grids.

Coarse mesh

(228 cells)

medium mesh

(912 cells)

fine mesh

(3648 cells)

8

Case Study: Fully-Developed Laminar Flow

in a 2-D Channel (Re H = 400) (2)

• Note the grid-convergence of the solutions **and** the

reduction rate of the L 1 -norm error.

Axial velocity vs. y

L1-norm vs. cell size

9

Case Study: Taylor’s Vortex Array (1)

• Represents an periodic array of

2-D laminar vortices decaying

with time (Re = 1)

• Solved in a domain [2π x 2π]

with periodic boundaries on

systematically refined triangular

meshes

• Comparison with the analytical

solution

– L 1 -norm of the error in U-velocity

at t* = 0.32

– Evolution of total kinetic energy

10

Case Study: Taylor’s Vortex Array (2)

• Note that the error drops as O(∆x 2 ) (slope is 2 on the log-plot).

• The total kinetic energy agrees well with the exact solution for

Re = 1.

• No appreciable loss in kinetic energy for very small viscosity

(Re = 10 12 ).

11

Agenda

Introduction to the finite-volume method

in FLUENT

Impact of mesh quality on **CFD** solutions

Impact of cell topology on solution

accuracy

Impact of mesh resolution

Further improvement on spatial accuracy

in the FLUENT 6.2 solver

12

Impact of Mesh Quality (1)

• Quality measures

– Skewness

– Change in cell-size (growth rate)

– Aspect ratio

– Alignment with the flow

• Impact on convergence

– Highly skewed cells harm convergence, often leading to

solution divergence due to large source terms.

– Quite frequently, solution diverges because of just a few

problematic cells.

– Highly stretched cells make the equations stiff, delaying

convergence considerably.

13

• Impact on accuracy

Impact of Mesh Quality (2)

– Too rapid a change in cell size in regions of large gradients

degrades accuracy.

• Rule-of-thumb based on the old adage in FDM - “keep the growth

rate under 1.2”

• y + adaptation of wall-adjacent cells should be avoided.

– Gross misalignment with the flow in region of large gradients

degrades accuracy.

14

Case Study: Windshield De-icing

• Turbulent flow **and** unsteady heat

transfer prediction with melting

model

• Computed using a hybrid mesh

(tet + prisms) with a total of

112,000 cells

- “Only” 7 (seven) cells above

recommended 0.95 limit

• The steady-state solution

diverges after 73 iterations !

15

Case Study: Turbulent Flow Over a 2-D

Backward-Facing Step

• Measured by Vogel **and** Eaton

(1980) – Re H = 28,000

• Shows the impact of cell growth

rate

Triangular

30% Growth

40% Growth

16

Agenda

Introduction to finite volume method in

FLUENT

Impact of mesh quality on **CFD** solutions

Impact of cell topology on solution

accuracy

Impact of mesh resolution

Further improvement on spatial accuracy

in the FLUENT V6.2 solver

17

Impact of cell topology on accuracy (1)

• Tri **and** tet meshes have inherently larger truncation

error than quad **and** hex which are aligned with the flow

direction **and** the gradients of the transported variables

(e.g., boundary layer, jets, mixing layer, wakes)

Truncation errors for quad **and** tri cells in the boundary layer

quad

tri

δ r

∇φ

δ r

∆y

( ) ( )

2

∇φ

O δ

φ

f

= φc

+

14243

+ δ r ⋅

r

0 c0

= 0

18

( ) ( )

2

∇φ

O δ

φ

f

= φc

+

14243

+ δ r ⋅

r

0 c0

≠0

Impact of cell topology on accuracy (2)

• In thin shear layers (e.g., boundary layers, free shear

layers), tri or tet meshes are more prone to numerical

diffusion than quad **and** hex meshes that are aligned

with the flow.

U=0.1

Quad

mesh

Tri

mesh

U=1.0

Contours of axial velocity magnitude for an inviscid co-flow jet

19

Impact of cell topology on accuracy (3)

• For complex flows without dominant flow direction,

quad **and** hex meshes loose their advantage.

Contours of temperature for inviscid flow

T = 1

quad

tri

U = V = 1.0 ,

U = V = 1.0 ,

T = 1

U = V = 1.0 ,

T = 0

U = V = 1.0 ,

T = 0

20

Case Study: Recirculating Flow in a Cavity

• 3-D laminar flow driven by a moving lid (Re H = 100)

• Comparison among FLUENT results for hex (198K

cells) **and** tet (224K cells) mesh

21

Case Study: Laminar Flow Through a Pipe

with Stenosis (1)

• The stenosis has an axisymmetric constriction of

sinusoidal shape.

– The length of the constriction (L c ) is four times the pipe

diameter.

• The inlet flow is a fully-developed laminar flow.

• Two Reynolds numbers (Re D = 50, 100) were

considered.

• Three different meshes were used.

– Quad mesh

– Coarse tri mesh

– Fine tri mesh

22

Case Study: Laminar Flow Through a Pipe

with Stenosis (2)

Comparison of predicted reattachment points

x a /L C

xa/Lc

4.5

4

3.5

3

2.5

2

1.5

1

0.5

0

Re =50

Re =100

Quad

Tricoarse

Tr-fine

Exp.

23

Case Study: Turbulent Flow Over a 2-D

Backward-Facing Step

• Measured by Vogel **and** Eaton (1980)

• Re H = 28,000

• Computed using four different mesh types of comparable

resolution

Structured

Quad Pave

Tri w BL

Tri

24

Case Study: Turbulent Flow Over a 2D

Backward-Facing Step (2)

• Impact of Mesh Type

25

Impact of cell topology on accuracy (4)

• Although structured quad **and** hex

meshes are favorable from the

accuracy viewpoint, their overall

advantage over tri/tet meshes

diminishes for many industrial **and**

practical applications.

– Complex geometry **and** flow structure

• Difficulty of meshing in hex

• No dominant flow direction

– For turbulent flows, numerical diffusion is

overwhelmed by physical diffusion.

• The accuracy gap can be further

reduced by using:

– Hybrid grids

– Boundary layer (prism) mesh for walldominated

flows.

26

Case Study: Turbulent Flow in a Pipe

with 90 o bend (Re D = 43,000)

Hex mesh

15,700 cells

Tet mesh

440K cells

27

Case Study: Turbulent Flow in a Pipe

with 90 o bend (Re D = 43,000) - (cont’d)

Realizable k-ε

Axial velocity profile at θ = 60 o

hex+SOU

tet(CB)+SOU

tet (NB)+SOU tet (NB)+MUSCL

Axial velocity contour at θ = 90 o

28

Case Study: Turbulent Flow in a Pipe with

90 o bend (Re D = 43,000) (cont’d)

Hex mesh

15,700 cells

Tet mesh

440K cells

Hybrid mesh

(prism + tet)

400K cells

29

Case Study: Turbulent Flow in a Pipe with

90 o bend (Re D = 43,000) (cont’d)

hex tet prism +

tet

Axial velocity contour at θ = 90 o

Axial velocity profile at θ = 60 o

Axial velocity profile at θ = 75 o

30

Case Study: Turbulent Flow in a Pipe

with 90 o bend (Re D = 43,000) (cont’d)

• Pressure drop in the bend

0.1

0.08

0.06

∆ C

P

≡

( )

1 2

P P o ρU

− θ = 0 θ = 90 2 B

0.04

dCp

0.02

0

hex tet prism+tet

dCp 0.081 0.048 0.071

31

Case Study: Drag on a Sphere (1)

• Steady laminar flow (Re D = 250)

on the verge of vortex-shedding

• (C D ) exp = 0.70 ~ 0.72 with

frictional **and** form drag being

comparable in magnitude

• Three different meshes were

used with the near-wall mesh

resolutions kept comparable

(0.003 D)

– 970K-cell hex mesh

– 1.6M-cell tet mesh

– 1.1M-cell hybrid (prism + tet)

mesh

32

Case Study: Drag on a Sphere (2)

Cd

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

friction

pressure

total

0

hex tet hybrid exp.

33

Case Study: Windshield De-icing

Hex mesh: 283,000 cells Tet mesh: 317,000 cells Hybrid mesh (prism + tet):

54,000 cells

34

Case Study: Windshield De-icing

• Comparison in terms of the

percentage of the windshield

surface cleared of ice

– Note the time needed to melt

90% of the windshield

surface

– The prediction using hex

mesh is considered as a

reference prediction.

• The coarse hybrid mesh

gives a reasonable prediction

with much fewer cells.

• Pure tet mesh overpredicts

the melting by ~10%

35

Agenda

Introduction to finite volume method in

FLUENT

Impact of mesh quality on **CFD** solutions

Impact of cell topology on solution

accuracy

Impact of mesh resolution

Further improvement on spatial accuracy in

the FLUENT V6.2 solver

36

Impact of Mesh Resolution

General guidelines (1)

• It’s important to start with a good

surface mesh that resolves large

curvature of underlying

geometry (e.g., leading edge of

an airfoil, A-pillar, side-view

mirror, bend)

• Determine cell height **and**

growth rate such that the

resulting mesh resolves the

width of shear layers.

– At an absolute minimum, 4 - 5 cells

across the shear layer are needed.

– Remember the FDM adage for the

growth rate (“keep it under 1.2”)

37

Impact of Mesh Resolution

General guidelines (2)

• Don’t discount flow

structures away from wall

(e.g., tip-vortex, near-wake).

• Remember that underresolved

upstream flow

affects the prediction

downstream.

38

Impact of Mesh Resolution

General guidelines (3)

• When tackling a new application, it is a good practice to

first study the grid-dependency of the solutions **and**

establish a best-practice meshing strategy.

3.7E-03

Coarse

3.5E-03

Medium

C D

Fine

3.3E-03

Richardson extrapolated

3.1E-03

0.0E+00 1.0E+05 2.0E+05 3.0E+05

# of cells

39

Impact of Mesh Resolution

A Guideline for Near-Wall Resolution (1)

Wall Function

Approach

Resolving Viscous

Sublayer approach

Wall-adjacent cells in log-law

region

+

30 ~ 50 ≤ y ≤ 500

Most ideally aim at y+ = 40 ~ 100

At least 5 -6 cells inside BL

Wall-adjacent cells at y + ~ 1 or

less.

At least 10 cells within the

inner layer (log-layer, buffer &

sublayers **and** 20 cells within

BL.

40

Impact of Mesh Resolution

A Guideline for Near-Wall Resolution (2)

• Take advantage of the enhanced wall treatment (EWT).

– Predictions are less sensitive to y + than with st**and**ard wall

functions

Fully-developed turbulent flow in a pipe (Re = 110,000)

0.035

0.03

Friction factor, f

0.025

0.02

0.015

0.01

0.005

SWF

EWT

0

0 10 20 30 40 50 60

Near Wall Size, y+

41

Agenda

Introduction to finite volume method in

FLUENT

Impact of mesh (quality) on **CFD** solutions

Impact of cell topology on solution

accuracy

Impact of mesh resolution

Improvements on spatial accuracy in the

FLUENT V6.2 solver

42

The Node-Based Gradient Option

• FLUENT offers two choices

for gradient calculation since

v6.1

– Cell-based (default)

– Node-based

• The node-based gradient

scheme is aimed at improving

spatial accuracy especially

for

– Tri **and** tet meshes

– Mesh with large or abrupt

cell size changes

43

Cell-Based vs. Node-Based Gradient

Schemes

Green-Gauss’

Theorem

Cell-based gradient

Node-based gradient

c1

c0

44

The nodal values are “weighted

averages” of the surrounding cellcenter

values

Case Study: Inviscid Subsonic Flow Over

a 2-D Bump (Ma = 0.5)

• Entropy increase is an useful indicator for numerical

dissipation.

• Node-based gradient reduces the entropy increase by

more than one order-of-magnitude.

Cell-based gradient

Node-based gradient

Contours of non-dimensional entropy production

45

Spatial **Accuracy**- discretization scheme

for convection terms in V6.2

Segregated Solver (1)

• MUSCL scheme

– High-order (locally third order) convection discretization

scheme modified for the unstructured meshes

– Based on blending of central differencing (CD) **and** secondorder

upwind (SOU) **and** is adapted to unstructured meshes

– More accurate predictions of secondary flows, vortices, **and**

force/moment

• Bounded Central differencing (BCD) scheme

– An alternative scheme to CD scheme which often yields

spurious (unphysical) oscillations

– Substantially more accurate than the SOU scheme

– Available for LES **and** detached eddy simulation (DES) only

46

Spatial **Accuracy** Improvements in V6.2

Segregated Solver (2)

• Second-order reconstructed convective flux

• mainly benefits tet meshes **and** poor-quality meshes

• Resulting in better resolution of boundary layers / shear

layers

• More accurate prediction of secondary flows, swirls,

vortices, pressure drop, **and** force/moment

• Little impact on solution robustness

• Both the cell-based gradient **and** the node-based gradient

schemes benefit from using the more accurate flux

• Default for the segregated solver in 6.2

-- The coupled solvers have long adopted an equivalent

implementation for the convective flux

• Node-based pressure gradient on cell faces

– Yields more accurate prediction of form (pressure) drag

47

Bounded Central Differencing

Example: Inviscid Vortex

• Conservation of kinetic energy

in an inviscid flow

Taylor’s vortex flow – 2-D

periodic array of vortices

Evolution of total kinetic energy

48

Example: CD vs. BCD

Martinuzzi **and**

Tropea (1993)

49

Spatial **Accuracy** Improvements in V6.2

Coupled Solver

• MUSCL scheme

– Essentially the same MUSCL scheme as used in the

segregated solver

– Implemented in a slightly different context from the

segregated solver

• Low-diffusion convection flux scheme

– Second-order spatially accurate

– For coupled solver only (BCD is not unavailable for

coupled solver)

– Available for LES **and** DES only

50

Spatial **Accuracy** Improvements in V6.2

Areas of impact

• Areas where we see the most impact

– Drag, Lift, **and** Moment (external aerodynamics **and**

turbomachinery)

– Secondary Flows **and** Vortices

– Pressure drop

– Heat transfer

– Noise/Aeroacoustics

– In-Cylinder Flow Modeling

– Mixing

• The benefits are especially significant for the

tetrahedral/triangular unstructured meshes

51

Example: Heat Transfer in a Channel (1)

• Laminar flow (Re H = 700,

Pr = 0.7)

• Heat transfer at

conducing walls

predicted using

– Hex mesh (160K cells)

– Tet mesh(440K cells)

– Hybrid mesh (360K cells)

52

Example: Heat Transfer in a Channel (2)

• Tet vs. Hex --- Wall Heat Flux predictions

52.6

48

36.5

40.2

Exact Hex Tet

v6.1

53

Tet

v6.2

Example: Heat Transfer in a Channel (3)

• Hybrid vs. Hex --- Wall heat flux predictions

36.5

40.2

45

40.8

Exact Hex Hybrid

v6.1

54

Hybrid

v6.2

Example : Fully-Developed Pipe Flow

• Laminar flow, Re D = 10 3 (dp*/dx = -1.267)

• Segregated solver with full tet. mesh

• Node-based gradient without limiter

Grid size

(cells)

High-order

RC

error

Old RC

flux

error

1301

1.493

17.8%

1.8457

45.7%

10408

1.332

5.14%

1.6204

27.9%

83264

1.2772

0.813%

1.4379

13.5%

666112

1.2700

0.245%

1.3715

8.13%

55

Example: Turbulent Flow Past an Ellipsoid

• Drag predictions on an ellipsoidal body using different

meshes

– Very low profile (form) drag

– Hybrid mesh with 500K cells

56

Example: Turbulent Flow Past an Ellipsoid

(continued)

• Hybrid mesh (tet mesh +10 prism layers)

4

3.5

3

2.5

2

1.5

1

cell-based

node-based

node-based +

HORC + SOU

node-based +

HORC+MUSCL

CD Total

CDP

CDF

0.5

0

Hex

tet R1 tet R2 tet R3 tet R4

57

Example: GAW-1 Airfoil

• Turbulent flow Re C = 6.0 x 10 6 , α =12 deg.

• C D = 0.0248 from experiment (NASA TN D-7428)

• Coupled implicit solver with RNG k-ε model

• Comparson of 6.1 vs 6.2

Drag coefficient (10 x C D

) predictions

Cell-based(6.1)

Node-based, SOU

Node-based, 3 rd -

order MUSCL

2-D quad+tri

0.263 (+6.1%)

0.238 (-4.0%)

3-D hex-wedge

0.262 (+5.7%)

0.235 (-5.2%)

3-D hex-tet

0.410(+65.3%)

0.293 (+18.3%)

0.279 (+12.5%)

58

Summary

• Mesh (still) impacts convergence **and** accuracy of **CFD** solutions

significantly

• The FLUENT discretization guarantees second-order accuracy for

all types of elements, including tri- **and** tet-meshes

• Quad **and** hex cells enjoy inherently smaller truncation error when

the meshes are aligned with flow (e.g., boundary layers)

• In complex flows, however, accuracy gap between quad/hex **and**

tri/tet becomes much narrower

• The accuracy with unstructured meshes can be significantly

improved by using the hybrid **and** BL meshing capability in

Gambit/FLUENT

• Second-order convective fluxes reconstruction, MUSCL scheme,

**and** bounded central differencing in FLUENT V6.2, together with

node-based gradient scheme have been shown to further

enhance the solver’s spatial accuracy, especially for the

unstructured meshes

59