GRIDS OR

MESHES

Discretização das Equações• As Equações de Transporte não são resolvidas analiticamente. Ao contrário, seu

domínio de cálculo é dividido em volumes onde se busca alcançar uma solução numérica.

• A função da malha é definir a dimensão dos volumes, das áreas das faces e das distâncias entre centros e vértices onde serão avaliados os fluxos e outros termos fonte.

• Em contornos simples (cartesianos ou cilíndricos) é relativamente fácil criar uma malha ortogonal que se adapte a este contorno

• O desafio reside em adaptar malhas em contornos complexos

Discretização das Equações

• Devido aos mais diversos motivos, a malha pode vir a ser modificada geometrias irregulares; localização / investigação de fenômenos em locais específicos; redução do número de volumes (rapidez de simulação, áreas mortas).

• Há diversas maneiras de se personalizar a malha (grid); técnicas como espaçamento em progressão geométrica ou segundo

uma power-law; malhas definidas por objetos (ver caso da biblioteca 290); finegrid.

Características das Malhas utilizadas nas simulações do PHOENICS

• Malhas estruturadas (geometria hexaédrica):

1) cartesiana / polar / cilíndrica – todas ortogonais;

2) BFC (body-fitted coordinate) – ortogonalidade variável.

Structured Mesh•Hexahedral elements orderly spaced with orthogonal or near orthogonal faces require three coordinates to locate each corner point, or 3*(I+1)*(J+1)*(K+1) values for the entire grid, which is much larger than the I+J+K values needed for rectangular grids. •In addition, other three-dimensional arrays are usually kept, such as their face areas and volumes, so these quantities don't have to be constantly recomputed. •This uses a large amount of stored memory and increases memory retrieval times. Although memory is becoming inexpensive, the amount of memory to be retrieved is becoming an important consideration in parallel computing.•The distortion of elements away from a purely rectangular shape has several consequences. For one thing, distortion may reduce numerical accuracy because numerical approximations are no longer centered (or symmetric) about the centroid of the volume element. This drawback, however, may be balanced by the increase in local grid resolution

Grades Cartesianas e Polares

UniformeCartesiana

Não-UniformePower

Não-Uniformeduas regiões

UniformePolar

Não-UniformeFine Grid Embedding

O sistema polar de coordenadas do PHOENICS• O Sistema cilíndrico polar está implementado no PHOENICS e seus termos fontes associados: centrífugo e coriolis para as equações de quantidade de movimento. • No sistema polar é necessário definir o Raio Interno, RINNER.• As demais especificações de domínio são coincidentes com aquelas do sistema cartesiano.

• A direção X do cartesiano corresponde a direção tangencial.

• A direção Y do cartesiano corresponde a direção radial.

• A direção Z do cartesiano corresponde a direção axial.

Body Fitted CoordinatesHexahedral elements with orthogonal or near orthogonal faces which adapt to the body profile. Access link to PHOENICS tutorial: BFC

Tratamento Sólido - Parede• PARSOL: partial solid – por default ativadoModelo implementado no PHOENICS que permite distinguirsólido de fluido em um mesmo volume de controle.

Malha cartesiana com volumes bloqueados pelo sólido

Malha BFC

Outros Exemplos de PARSOL com Fine-Grid

IMPORTAÇÃO & GERAÇÃO DE OBJETOS

• AutoCAD – export format: STL or DXF format files

• Shapemaker

• AC3D

Outros Recursos

• interpolação para malhas mais refinadas: PINTO (ver tutorial)

• refinamentos dinâmico (time-varying) – exemplo: pistão de motor;

• Observações:

evitar distorções nas malhas (> 1:3 pode ser perigoso)

sempre malhas eulerianas;

Grades BFC e Mult-Block para Geometrias ComplexasBody Fitted Coordinates - BFCOrtogonal ou Não Ortogonal

Multi-BlockOrtogonal ou Não Ortogonal

Grade Cartesiana com Objetos Imersos: • Iteração volume a volume tipo ‘escada’ ou;• Iteração via software com algoritmo PARSOL

Unstructured grids

Unstructured Meshing of Control VolumesUnstructured grids have the advantage of generality in that they can be made to conform to nearly any desired geometry. This generality, however, comes with a price. The grid generation process is not completely automatic and may require considerable user interaction to produce grids with acceptable degrees of local resolution while at the same time having a minimum of element distortion. Unstructured grids require more information to be stored and recovered than structured grids (e.g., the neighbor connectivity list), and changing element types and sizes can increase numerical approximation errors.

Optimization of Alumina Refinery Isolation Valves

WORKSHOP: Fine Grid Application• Fine grid increases the grid

fineness in specific regions while on other regions the domain employs a coarse grid. This strategy can reduce the computational time.

coarse grid

fine grid

• Observe on the figure that at the grids’ interface a single coarse grid cell face shares two fine grid cell faces.

• The have success on fine grid applications avoid:• placing the grid interface at regions with strong gradient,• introducing fineness greater than 3. Prefer the use of

double or tripling fineness in multiple of 2 instead

WORKSHOP: Fine Grid Application• This workshop models the laminar flow

around a circular cylinder in a free stream.

• The objective is to compare the use of FG (50x50) against regular grid (100x100) with equivalent FG cell sizes to estimate the wake length behind the cylinder and the CPU time.

Flow

visu

aliza

tion

at th

e w

ake

of a

cyl

inde

r at R

e of

41

Experimental data: laminar flow regime

Prob

lem

Dat

a

OBJ X Y Z ATRIBUTES OBJ X Y Z ATRIBUTES

SIZEIN

0 6 1U = 1,025

CYL

0,6 0,6 1 SOLID WITH FRICTIONPLACE 0 0 0 3,7 2,7 0

SIZEEOUT

0 6 1P = 0

FG

5 3 1FINENES = 2

PLACE 10 0 0 2,5 1,5 0SIZE

SOUT10 0 1

P = 0

NOUT10 0 1

P = 0PLACE 0 0 0 0 6 0

XMON YMON ZMON RELAX AUTO 5,62 3 0,5

10m

Domain Properties: RHO1=1 & ENUL = 0.015Model outlets

6m

InletU=1,025 m/s

CylinderD=0.6m

FineGrid

Grid Check: 51 x 51 with FG x 2

• In Numerics box set sweeps to 3200.• It is ready to run

• Use the auto-mesh and adjust the init-cell-factor to get a 51x51 grid.

• Place the pointer downstream saddle point, choose: (x,y,z) = (5.62, 3, 0.5)

100 X 100 without FG

Wake length: numerical x experimental

CPU time of run 60 s

CPU time of run 30 s

51 X 51 with FG X2

Rescue q1

Rescue q1

Wake length: numerical x experimental

CPU time of run 390 s

CPU time of run 500 s

200 X 200 without FG

100 X 100 with FG X2

Comments I

• The reduction on the CPU time with the use of FG is of 50% for a 50 x 50 grid. The prediction on the wake length is equivalent for a uniform grid although not coincident with the experimental value.

• A 100 x 100 grid with FG makes it hard to get a solution satisfying the residuals. The relaxation factors have to be reduced and the CPU time increases. There is still a reduction in CPU time but it is less than the 50%. Perhaps a search of optimum relaxation factor is necessary.

• The use of FG is more appropriate for problems where flow changes more quickly in a specific region while in others it remains fairly behaved.

Comments II • The use of outlets to the North and South faces of the domain is necessary to simulate the flow around a cylinder in an unbounded fluid, i.e., like the atmosphere.

• When one uses the symmetry condition, i.e., just leave these frontiers to the default condition, it is like having mirror images of the cylinder.

• The confinement of the mirror images of the cylinders increases the maximum velocity

Unbounded domain

Symmetric domain

Further Simulations

• Just get the 50 x 50 FG case and double the inlet velocity, 2,050 m/s. The Re number now should be of 82.

• Try run this case and comment your results.

Comments• When the inlet velocity changed to 2,050 the solution did not

converged. On the contrary, the monitor spot was periodic. The residuals were high after 2000 sweeps.

Comments• The X velocity field was quite distinct from the Re 41.

• The question is: this discrepancy is due to a numerical problem or the physics of the phenomena has been changed

Comments

• The physics of the phenomenon has been changed!• The flow above Re 47 is periodic.• A flow instability develops at the separated regions

shedding wall vorticity to the wake flow. • See further information on the following links– Reference 1– Reference 2

Re=104

Drag & Flow visualization at different Re

Further references

Go to POLIS and visit:• Documentation...

Hard-copy documentation

> Starting with PHOENICS-VR; TR 324

> The PHOENICS-VR reference guide; TR 326

FLOW WITH ANGLED INLET/OUTLET

• Air distribution inside a 2D ware-house• The case shows a 2D cross-section through a long warehouse.

An ANGLED-IN object is used to inject air at 2m^3/s normal to the roof, and an ANGLED-OUT object is used to create an opening on the sloping roof.

• Models:– Velocity ON; Turbulence: KECHEN; Energy: OFF

• Properties– Air (material 0)

• Numeric: – 500 sweeps

Settings

Objects settings

SET DOMAIN: X = 10M, Y = 1M & Z = 4M

OBJ X Y Z ATRIBUTES OBJ X Y Z ATRIBUTES

SIZECRATE1

2 1 1BLOCKAGE

ROOF2

5 1 1 BLOCKAGE, WEDGE

PLACE 1,5 0 0 10 0 4

SIZECRATE2

2 1 1BLOCKAGE

IN

1 1 0,75 ANGLED IN, Q=2m3/s &

k=5%PLACE 6,5 0 0 2 0 5

SIZEROOF1

5 1 1 BLOCKAGE, WEDGE

OUT

1 1 0,75ANGLED OUT

PLACE 0 0 4 7 0 4

XMON YMON ZMON RELAX FALSDT, U1 =V1=100 & 100; KE=EP=-0,5 5 0,5 2

Comments• The angle in object was set to give volumetric

flow rate normal to the aperture of roof1.• It is also possible specify a velocity. For

example, try set a velocity of 2m/s directed at 45o to the left of the vertical.

45o

U1

W1U1 2 Sin45 1.414 m/sW1 2 Cos45 1.414 m/s