GRIDS OR MESHES Discretizao das Equaes As Equaes

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 v geometrias irregulares; v localização / investigação de fenômenos em locais específicos; v redução do número de volumes (rapidez de simulação, áreas mortas). • Há diversas maneiras de se personalizar a malha (grid); v técnicas como espaçamento em progressão geométrica ou segundo uma power-law; v malhas definidas por objetos (ver caso da biblioteca 290); v finegrid.

Características das Malhas utilizadas nas simulações do PHOENICS • Malhas estruturadas hexaédrica): (geometria 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 Uniforme Cartesiana Uniforme Polar Não-Uniforme Power Não-Uniforme duas regiões Não-Uniforme Fine 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 Coordinates Hexahedral 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 ativado Modelo implementado no PHOENICS que permite distinguir sólido 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 • Auto. CAD – export format: STL or DXF format files • Shapemaker • AC 3 D

Outros Recursos • interpolação para malhas mais refinadas: PINTO (ver tutorial) • refinamentos dinâmico (time-varying) – exemplo: pistão de motor; • Observações: v evitar distorções nas malhas (> 1: 3 pode ser perigoso) v sempre malhas eulerianas;

Grades BFC e Mult-Block para Geometrias Complexas Multi-Block Body Fitted Coordinates - BFC Ortogonal 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 Volumes Unstructured 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 (50 x 50) against regular grid (100 x 100) with equivalent FG cell sizes to estimate the wake length behind the cylinder and the CPU time.

Flow visualization at the wake of a cylinder at Re of 41 Experimental data: laminar flow regime

Problem Data Domain Properties: RHO 1=1 & ENUL = 0. 015 outlets Model Inlet Cylinder U=1, 025 m/s D=0. 6 m 6 m Fine Grid 10 m SIZE OBJ IN PLACE SIZE EOUT PLACE SIZE SOUT PLACE RELAX X Y Z 0 6 1 0 0 10 10 0 0 6 0 0 0 AUTO 0 1 0 ATRIBUTES U = 1, 025 P = 0 OBJ CYL X Y Z ATRIBUTES 0, 6 1 SOLID WITH FRICTION 3, 7 2, 7 0 5 3 1 FG FINENES = 2 2, 5 1, 5 0 10 0 1 NOUT P = 0 0 6 0 XMON YMON ZMON 5, 62 3 0, 5

Grid Check: 51 x 51 with FG x 2 • Use the auto-mesh and adjust the init-cell-factor to get a 51 x 51 grid. • Place the pointer downstream saddle point, choose: (x, y, z) = (5. 62, 3, 0. 5) • In Numerics box set sweeps to 3200. • It is ready to run

Wake length: numerical x experimental 51 X 51 with FG X 2 CPU time of run 30 s Rescue q 1 100 X 100 without FG CPU time of run 60 s Rescue q 1

Wake length: numerical x experimental 100 X 100 with FG X 2 CPU time of run 390 s 200 X 200 without FG CPU time of run 500 s

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 Unbounded domain Symmetric domain • 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

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

Drag & Flow visualization at different Re Re=104

Further references Go to POLIS and visit: • Documentation. . . ØHard-copy documentation > Starting with PHOENICS-VR; TR 324 > The PHOENICS-VR reference guide; TR 326

FLOW WITH ANGLED INLET/OUTLET • Air distribution inside a 2 D ware-house • The case shows a 2 D cross-section through a long warehouse. An ANGLED-IN object is used to inject air at 2 m^3/s normal to the roof, and an ANGLED-OUT object is used to create an opening on the sloping roof.

Settings • Models: – Velocity ON; Turbulence: KECHEN; Energy: OFF • Properties – Air (material 0) • Numeric: – 500 sweeps

Objects settings SET DOMAIN: X = 10 M, Y = 1 M & Z = 4 M SIZE PLACE RELAX OBJ CRATE 1 CRATE 2 ROOF 1 X Y Z 2 1 1 1, 5 0 0 2 1 1 6, 5 0 0 5 1 1 0 0 ATRIBUTES BLOCKAGE OBJ ROOF 2 IN 4 BLOCKAGE, WEDGE FALSDT, U 1 =V 1=100 & 100; KE=EP=-0, 5 OUT X Y Z ATRIBUTES 5 1 1 10 0 4 BLOCKAGE, WEDGE 1 1 0, 75 2 0 5 1 1 0, 75 7 0 4 XMON YMON ZMON 5 0, 5 2 ANGLED IN, Q=2 m 3/s & k=5% ANGLED OUT

Comments • The angle in object was set to give volumetric flow rate normal to the aperture of roof 1. • It is also possible specify a velocity. For example, try set a velocity of 2 m/s directed at 45 o to the left of the vertical. W 1 45 o U 1