Advanced Computational Models Grid Adaptation Non-conformal Interfaces Moving Boundaries Deforming Boundaries…
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Grid Adaptivity
The computational grid can be refined and/or coarsened based on geometrical and numerical solution data
Useful for: Capture flow features in details Increase resolution in near-wall regions Improve grid quality
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Grid Adaptivity
The computational grid can be refined and/or coarsened based on geometrical and numerical solution data
Useful for: Capture flow features in details Increase resolution in near-wall regions Improve grid quality …
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Grid Adaptivity Example Computational Domain Flow in a complex passage
Grid Adaptivity Example Computational Domain Flow in a complex passage
A uniform triangular grid is likely to be inappropriate to capture all the feature of the flow ME469B/4/GI
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Adaptation Process
Definition of the adaptation function (based on geometrical and/or solution data) Selection of the cells to be refined or coarsened ( marking or tagging or tagging)) Selection of grid of grid refinement/coarsening scheme Adaptation Interpolation of the previous solution onto the new grid (automatic)
Adaptation Process
Definition of the adaptation function (based on geometrical and/or solution data) Selection of the cells to be refined or coarsened ( marking or tagging or tagging)) Selection of grid of grid refinement/coarsening scheme Adaptation Interpolation of the previous solution onto the new grid (automatic)
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Adaptation Functions
Geometrical Region Boundary Volume
Solution based Isovalue
Adaptation Functions
Geometrical Region Boundary Volume
Solution based Isovalue Gradient y+
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Region Adaptation This is the same option we used for “global” grid refinement to study the grid convergence of the solutions Adapt Æ Region Select a region shape
Region Adaptation This is the same option we used for “global” grid refinement to study the grid convergence of the solutions Adapt Æ Region Select a region shape
Input the geometrical definition of the region ME469B/4/GI
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Boundary Adaptation Adapt Æ Boundary Select a boundary of the computational domain
Three options: cell distance normal distance
Boundary Adaptation Adapt Æ Boundary Select a boundary of the computational domain
Three options: cell distance normal distance volume distance
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Boundary Adaptation Adaptation based on a cell’s distance from the selected boundary measured in number of cells. cells . (1 means only the cells attached to the boundary) Adaptation based on a cell’s normal distance from the selected boundary
Adaptation based on a target boundary volume (specify a target volume and a growth factor a)
Vcell = Vtarget e ad
Boundary Adaptation Adaptation based on a cell’s distance from the selected boundary measured in number of cells. cells . (1 means only the cells attached to the boundary) Adaptation based on a cell’s normal distance from the selected boundary
Adaptation based on a target boundary volume (specify a target volume and a growth factor a)
Vcell = Vtarget e ad This adaptation attempts to generate boundary-layer type grid
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Volume Adaptation Adapt Æ Volume Based on cell volume (area in 2D):
Two options: magnitude (threshold values)
Volume Adaptation Adapt Æ Volume Based on cell volume (area in 2D):
Two options: magnitude (threshold values) change (neighbor change)
Allow to compute the range in your grid
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Isovalue Adaptation Adapt Æ IsoValue All field values are available (including equation residuals, residuals, customized and gridrelated functions)
Isovalue Adaptation Adapt Æ IsoValue All field values are available (including equation residuals, residuals, customized and gridrelated functions)
Method: specify the function specify a range specify the inside/outside option ME469B/4/GI
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Gradient Adaptation As before all field values are available (including equation residuals, residuals, customized and grid-related functions)
Method: specify the function specify a range
Adapt Æ Gradient
Gradient Adaptation As before all field values are available (including equation residuals, residuals, customized and grid-related functions)
Adapt Æ Gradient
Method: specify the function specify a range Note that the option is refine/coarsen Cells above the Refine Threshold are Refined Cells below the Coarsen Threshold are Coarsened ME469B/4/GI
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y+ Adaptation Useful for turbulent flow simulations
Method: specify the wall boundaries specify the range (as before)
Adapt Æ y+
y+ Adaptation Useful for turbulent flow simulations
Adapt Æ y+
Method: specify the wall boundaries specify the range (as before)
Note y* is a friendlier version of y + y+ = r y p u t /m
y* = r Cm1/4 k p1/2 y p /m
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Mark (Tag) and display Before performing the adaptation you can mark mark the the cell selected (based on a certain adaptation function) and display them
Adapt Æ Region Æ Control Æ Display
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Mark (Tag) and display Before performing the adaptation you can mark mark the the cell selected (based on a certain adaptation function) and display them
Adapt Æ Region Æ Control Æ Display
Only the selected cells will be displayed
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Mark or Adapt? Mark allows to evaluate the effect of a refining/coarsening procedure without actually changing the grid
Marked cells are saved in registers that can be combined or manipulated before Adapting
adapt
mark
Mark or Adapt? Mark allows to evaluate the effect of a refining/coarsening procedure without actually changing the grid
Marked cells are saved in registers that can be combined or manipulated before Adapting
adapt
mark
Adapt is usually used only at the end when all The desired cells are marked
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Managing Registers Marked cells are saved in registers Adapt registers: registers: collection of cells Mask registers: registers: binary tagging to all cells in active and inactive
Operations on the registers are: Union: combine two adapt registers (or more) Union: Intersection:: combine adapt and mask registers Intersection Change Type: Type: convert a adapt in mask and viceversa Delete:: eliminate a register Delete
Adapt Æ Æ Manage
Managing Registers Marked cells are saved in registers
Adapt Æ Æ Manage
Adapt registers: registers: collection of cells Mask registers: registers: binary tagging to all cells in active and inactive
Operations on the registers are: Union: combine two adapt registers (or more) Union: Intersection:: combine adapt and mask registers Intersection Change Type: Type: convert a adapt in mask and viceversa Delete:: eliminate a register Delete Exchange : swap an adapt register (coarse Æ refine) Exchange: Invert:: swap a mask register (active Æ inactive) Invert Limit:: apply the adaptations limits to a register Limit Fill:: mark for coarsening all the cells not in the register Fill ME469B/4/GI
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Adaptation controls Set limits on the adaptation procedure (i.e. maximum number of final cells)
Select the adaptation scheme
Adapt Æ Æ Manage Æ Control
Adaptation controls Set limits on the adaptation procedure (i.e. maximum number of final cells)
Adapt Æ Æ Manage Æ Control
Select the adaptation scheme Conformal Hanging Nodes
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Hanging-Node and Conformal Adaptation
Cell to be refined
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Hanging-Node and Conformal Adaptation
Cell to be refined
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Hanging-Node Adaptation father Selected cells (father) are refined homothetically (kids) Nodes are added on the edges of the father cell and connectivity information are generated to link the kids to the father neighbors
kids
Memory penalty associated to the additional connectivity Information required and to the presence of an inactive father cell Neighboring cells are not allowed to differ more than one-level of refinement
Hanging-Node Adaptation father Selected cells (father) are refined homothetically (kids) Nodes are added on the edges of the father cell and connectivity information are generated to link the kids to the father neighbors
kids
Memory penalty associated to the additional connectivity Information required and to the presence of an inactive father cell Neighboring cells are not allowed to differ more than one-level of refinement (because of inaccuracy of inaccuracy related to large volume variations) Coarsening can be only performed on previously refined regions. Kids are deleted and the father cell becomes active
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Conformal Adaptation Selected cells are refined by splitting the longest edge It is inherently conservative because the cell connectivity is not modified No memory penalty (the old cells are deleted and the only the new are stored) Low quality meshes can be improved (refinement not homothetic) Coarsening can be applied everywhere and corresponds to a local remeshing
Conformal Adaptation Selected cells are refined by splitting the longest edge It is inherently conservative because the cell connectivity is not modified No memory penalty (the old cells are deleted and the only the new are stored) Low quality meshes can be improved (refinement not homothetic) Coarsening can be applied everywhere and corresponds to a local remeshing Can only be applied to triangular (tetrahedral) grids In is NOT as local as the hanging-node approach
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Grid Adaptivity Guidelines
Surface mesh should be fine enough to capture all the essential geometrical features of the model (especially for high curvature surfaces) The initial mesh should be fine enough to capture the overall features of the flow A reasonably converged solution must be obtained before adapting the grid Suitable flow-adaptation criteria are crucial to obtain increased resolution of
Grid Adaptivity Guidelines
Surface mesh should be fine enough to capture all the essential geometrical features of the model (especially for high curvature surfaces) The initial mesh should be fine enough to capture the overall features of the flow A reasonably converged solution must be obtained before adapting the grid Suitable flow-adaptation criteria are crucial to obtain increased resolution of selected region (i.e. velocity gradients are better than pressure gradient in incompressible flows and high values of turbulent quantities are relevant for turbulent flows)
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Boundary Grid Adaptivity Poor boundary resolution cannot be improved via grid refinement
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Boundary Grid Adaptivity Poor boundary resolution cannot be improved via grid refinement
Original geometry ME469B/4/GI
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Non-conformal grids Grid generation can be simplified in certain problems by meshing various components separately. The grids have to be coupled using a non-conformal interface
Quadrilateral grid
Non-conformal grids Grid generation can be simplified in certain problems by meshing various components separately. The grids have to be coupled using a non-conformal interface
Quadrilateral grid
Triangular grid ME469B/4/GI
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Non-conformal grid interfaces The matching conditions between the two faces have to be defined Coupled interface: interface: the interface is actually an internal face (no bc) Periodic interface: interface : the matching allows to specify pressure gradients
Define Æ Grid Interface
Non-conformal grid interfaces The matching conditions between the two faces have to be defined Coupled interface: interface: the interface is actually an internal face (no bc) Periodic interface: interface : the matching allows to specify pressure gradients
Define Æ Grid Interface
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Non-conformal grid interfaces - Example Channel flow Periodic Boundaries
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Non-conformal grid interfaces - Example Channel flow Periodic Boundaries
Non-conformal interface
Velocity contours (cell values)
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Non-conformal Interface Guidelines
Grid interface can be of any of any shape (2D and 3D) but the surfaces to be mached MUST be based on the same geometry especially for highly curved surfaces Grid resolution at the two sides of the interfaces can be different; accuracy (and fluxes conservation) degrades for highly different mesh size Non-conformal interfaces are binary are binary connectivity between two (and only
Non-conformal Interface Guidelines
Grid interface can be of any of any shape (2D and 3D) but the surfaces to be mached MUST be based on the same geometry especially for highly curved surfaces Grid resolution at the two sides of the interfaces can be different; accuracy (and fluxes conservation) degrades for highly different mesh size Non-conformal interfaces are binary are binary connectivity between two (and only two) zones.
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Moving Zones Several industrial applications involve flow through a domain which contains a moving component (propellers, turbines, etc.)
stationary
moving
Moving Zones Several industrial applications involve flow through a domain which contains a moving component (propellers, turbines, etc.)
stationary
moving
Time ME469B/4/GI
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Advanced Computational Models
• Grid Adaptivity • Non-Conformal Grid Interfaces • Moving Zones
Advanced Computational Models
• Grid Adaptivity • Non-Conformal Grid Interfaces • Moving Zones
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Modeling Moving Zones
Different approaches can be followed: 1) 2) 3) 4) 5) 6)
Sing Single le Refe Refere renc ncee Fra Frame me Mode Modell (SR (SRFM FM)) Mult Multip iple le Ref Refer eren ence ce Fra Frame me Mod Model el (MR (MRFM FM)) Mixin xing Pl Plane ane Mod Model el (MP (MPM) Slidin iding g Mes Mesh h Mod Modeel (S (SMM) Mesh De Deformation Remeshing
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Modeling Moving Zones
Different approaches can be followed: 1) 2) 3) 4) 5) 6)
Sing Single le Refe Refere renc ncee Fra Frame me Mode Modell (SR (SRFM FM)) Mult Multip iple le Ref Refer eren ence ce Fra Frame me Mod Model el (MR (MRFM FM)) Mixin xing Pl Plane ane Mod Model el (MP (MPM) Slidin iding g Mes Mesh h Mod Modeel (S (SMM) Mesh De Deformation Remeshing
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Single Reference Frame Model Simplest model available; entire computational domain is referred to a rotating reference frame (domain moves with the reference frame) The equations are rewritten in the moving reference frame and a Coriolis acceleration term appears as a source for the momentum balance Boundaries that move with the frame can assume any shape s hape BUT boundaries that are stationary MUST be surface of revolution
Single Reference Frame Model Simplest model available; entire computational domain is referred to a rotating reference frame (domain moves with the reference frame) The equations are rewritten in the moving reference frame and a Coriolis acceleration term appears as a source for the momentum balance Boundaries that move with the frame can assume any shape s hape BUT boundaries that are stationary MUST be surface of revolution
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Single Reference Frame Set-Up
Define Æ Models Æ Solver Velocity formulation Absolute: the unknowns are the field Absolute: variables in the stationary reference frame Relative: the unknowns are the field Relative: variables in the moving reference frame
Single Reference Frame Set-Up
Define Æ Models Æ Solver Velocity formulation Absolute: the unknowns are the field Absolute: variables in the stationary reference frame Relative: the unknowns are the field Relative: variables in the moving reference frame Note: the relative is only available in the Segregated solver and it is usually faster
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Single Reference Frame Boundary Conditions Define Æ Boundary Condition Æ Fluid
Define the axis of rotation
Æ
Wall
Single Reference Frame Boundary Conditions Define Æ Boundary Condition Æ Fluid
Define the axis of rotation Select moving reference frame Define the angular speed
Æ
Wall
Stationary surfaces: zero absolute rotational speed Moving surfaces: zero rotational speed relative to the adjacent zone
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Multiple Reference Frame Model
Many rotating machinery problems involve stationary components that cannot be represented as surface of revolution or move at different velocity The extension is to consider separate reference frames for each component The set-up is similar to the SRFM MRFM ignores the relative motions of subdomains and the Coriolis body-forces are local to each region (no equivalence between stationary and
Multiple Reference Frame Model
Many rotating machinery problems involve stationary components that cannot be represented as surface of revolution or move at different velocity The extension is to consider separate reference frames for each component The set-up is similar to the SRFM MRFM ignores the relative motions of subdomains and the Coriolis body-forces are local to each region (no equivalence between stationary and moving reference frame) Suitable for problems where the interaction between rotating and non-rotating components is small ME469B/4/GI
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Sliding Mesh Model Like the MRFM the domain is divided into moving and stationary components Unlike the MRFM the mesh in each subdomain moves with respect to one another and the problem is inherently unsteady The equations are solved in the stationary reference frame and the meshes are moved at each time step (no approximations to the governing equations) The sliding mesh interface is defined as a non-conformal interface
Sliding Mesh Model Like the MRFM the domain is divided into moving and stationary components Unlike the MRFM the mesh in each subdomain moves with respect to one another and the problem is inherently unsteady The equations are solved in the stationary reference frame and the meshes are moved at each time step (no approximations to the governing equations) The sliding mesh interface is defined as a non-conformal interface
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SMM Example
time
SMM Example
time
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Velocity distribution
Grid Generation
• Geometry definition (simple shapes, CAD import) • Grid generation algorithms • GAMBIT • Grid quality and improvement • Automation
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Grid Generation
• Geometry definition (simple shapes, CAD import) • Grid generation algorithms • GAMBIT • Grid quality and improvement • Automation
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Grid generation package: GAMBIT
Special Geometry Grid BC Tools
Geometry Tools
Graphics Window
Volume Tools
Visualization Tools
Text Window
The User Manual is available from the class Web Site ME469B/2/GI
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Geometrical types - Topology
• Vertex • Edge (2 or more vertices) • Face (3 or more edges) • Volume (3 or more faces)
Bottom-up approach: approach: generate low dimensional entities and build on top of them higher dimensional entities
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Vertex: • Input Coordinates…
Edge: • Line segment (connect 2 vertices) • Circular arc • Quadratic functions • NURBS: NURBS: Non-Uniform Non-Uniform Rational B-Splines (connect N vertices)
Topologically any edge is ALWAYS a connection between 2 vertices (additional vertices used to build the geometry are NOT part of the edge) ME469B/2/GI
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Face:
• Rectangular • Circular •… • Sweep (translation or rotation of an edge) • Wireframe (connecting 3 or more edges)
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Volume:
• Cuboid • Sphere • Cone • Pyramids •… • Sweep (translation or rotation of a face) • Wireframe (connecting 3 or more faces)
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Manipulate Geometry - Boolean Operations: Volume 1
• Unite • Substract • Intersect Volume 2
it generates the intersection edge
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Manipulate Geometry - Blend
smooth sharp edges ME469B/2/GI
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Create Entities - Faces
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Create Entities - Faces
Some entities generated using “primitives” have fewer lower topological entities Example: Cube volume: 6 faces, 12 edges, edges, 8 vertices Cylinder volume: 3 faces, 2 edges, 2 vertices
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Manipulate Geometry – Create Entities Create a vertex on a face Parametric representation of the face
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Manipulate Geometry – Create Entities Create two edges by splitting an edge Parametric representation of the edge
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Import Geometries • Realistic geometries are TOO complicated to be generated from “simple” shapes • Engineering design is based on CAD systems
Translation between CAD and CFD system is a major bottleneck • Gambit is based on ACIS geometrical libraries • ACIS (Andy, Charles & Ian’s System) is the most widely widely used 3D modeling software technology (http://www.spatial.com) (http://www.spatial.com) • It can also import: • STEP (ST STandard andard for E for Exchange of P of Product model data; ISO standard) standard) • IGES (Initial Graphics Exchange S pecification; ANSI standard) standard) • STL (ST STereo ereo Lithography; Rapid Prototyping Standard) • …. ME469B/2/GI
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Clean-Up a CAD Model
• Eliminate components not exposed to the flow • Eliminate duplicated entities • Eliminate small details • Water-proofing the surfaces • Rebuild geometrical connectivity between parts
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Example: Helicopter Rotor
The rotor-shaft connection is VERY complicated
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Geometrical entities
The geometry consists of 10 components 3 blades, 1 shaft support, 6 connectors ME469B/2/GI
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Example: Import IGES Model
IGES export is available from every CAD system
IGES models are a collection of “untrimmed” edges and faces
Imported geometry consists of: Imported geometry 0 volumes ~250 faces ~1100 edges ~1000 vertices ME469B/2/GI
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Example: Import STEP Model
STEP export is available from many CAD system
STEP models are a collection of parts or components
Imported geometry consists of: Imported geometry 10 volumes ~190 faces ~450 edges ~300 vertices ME469B/2/GI
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Components “exploded”
connector-1
support
blade
connector-2 For aerodynamic analysis the details of the rotor-shaft connection are not important. The geometry of the blades MUST be preserved ME469B/2/GI
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Geometry simplification
Connectors eliminated
Support generated as a “simple” cylinder with blended side
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Geometry simplification
Blade edge cleaned and sealed
Blade-support connector is a “simple” cuboid
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Clean Geometry
This example is available on the class Web site ME469B/2/GI
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Grid Generation
• Geometry definition (simple shapes, CAD import) • Grid generation algorithms • GAMBIT • Grid quality and improvement • Automation
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Grid generation techniques
• Structured grids • Ordered set of (locally orthogonal) lines • Several Techniques can be used to Map a computational domain into a physical domain: Transfinite Interpolation, Morphing, PDE Based, etc. • The grid lines are curved to fit the shape of the boundaries • Unstructured grids • Unorganized collection of polygons (polyhedron) • Three main techniques are available to generate automatically triangles triangles (tetrahedra): Delaunay triangulation, Advancing front, OCTREE • Paving for automatic generation of quads in 2D
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Structured Grids: Mapping
Side B
S i d e C
2 A e d i S
D e d i S
Side A1
Physical Domain
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j=1=NJ 2 A e d i S 1 A e d i S
Side B
j=NJ=1 k=NK C e d i S
k=1
k j
Side D
Computational Domain
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Unstructured Grids: Triangulations • Delaunay • Empty circle principle: any node must not be contained within the circumcircle (circle passing through the vertices of a triangle) on any triangle within the mesh • Automatic triangulation of random set of nodes • Nodes are inserted locally in a triangulation and triangles are redefined locally to satisfy the Delaunay criterion (available mathematical tools) + Inherent grid quality + Elegant mathematical basis - Boundary integrity
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Unstructured Grids: Triangulations • Advancing front • Triangles are built inward from the boundary surfaces • The last layer of elements constitutes the active front • An optimal location for a new nodes is generated for each segment on the front; the new node is generated by checking all existing nodes and this new optimal location • Intersection checks are required to avoid front overlap + Surface grid preserved + Specialized layers near surfaces - Computationally complex - Low quality
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Unstructured Grids: Triangulations • OCTREE • Squares containing the boundaries are recursively subdivided until desired resolution is obtained • Irregular cells (or triangulation) are generated near the surface where square intersect the boundary
+ Requires least of surface representation + Highly automated - Cannot match surface grid - Low quality near surfaces
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Unstructured Grids: Paving • Advancing front technique based on quads (instead of triangles) • Only in 2D
Triangulation ME469B/2/GI
Paving 29
Unstructured Grids: Coopering
• 2D mesh sweeping • Only for cylindrical volumes • unstructured surface mesh is generated on surface A (source face) • structured grids are generated on cylindrical surfaces C & D • mesh on surface A is sweeped in the volume to generate the full 3D m esh
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Unstructured Grids: 3D elements
ß
Standard Elements:
Hex
Tet
Pyramid
Wedge
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Hex:: Maximum Volume Covered per Edge Size Hex
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Hex:: Maximum Ratio Nodes/Elements Hex
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Hex/Wedges:: Clustering at Solid Wall with High Quality Elements Hex/Wedges
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Tets:: Automatic Meshing of Extremely Complicated Tets Complicated Regions
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Pyramids/Wedges:: Transition Between Tets & Hex Pyramids/Wedges
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Unstructured Grids: Hex or Tets? We NEED Hex-Based Meshing because: ß
Equiangular Tets are NOT Good for Thin Volumes
ß
Too Many Elements for Reasonable Resolutions (estimated >2M grid Points in conical-annular Swirler)
35K Total Elements
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Cross-Section of the Swirler
410K Total Elements
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• Structured gridding (mapping)
• Unstructured triangulation (2D/3D)
• Unstructured paving (2D)
y r t e m o e G s e x i z e s l p h m s e o C M
d e e p S
s s e n t s u b o R
l o r t n o C & y t i l a u Q
+
+
+
-
+
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+
+/-
+
-
-
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+/-
+
+
+
-
+/-
-
+
What is available in GAMBIT
• Unstructured coopering (3D)
All GAMBIT meshes are exported as unstructured collection of (mixed) elements ME469B/2/GI
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Grid generation – 1D - Edges
Straightforward Select number of points Select distribution of points Edge direction is defined from 1 st to 2nd vertex Clustering toward one side is defined accordingly
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Grid generation – 2D - Faces
Easy Select number of points • use use pred predef efin ined ed edge edge mesh meshes es • use uniform spacing Select meshing scheme • constraints on the edge meshing for mapping and paving schemes
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Grid generation – 2D - Faces
It is possible to force the cell element type at face-vertices elements forced to be triangles
Mixing element-type is one of the main advantages of unstructured mesh technology
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Grid generation – 2D - Mesh-patching options
Matching interface
Overlapping interface ME469B/2/GI
Non-conformal interface
Mixed-element interface 37
Grid generation – 3D - Volumes
Not so easy Select number of points • use use pred predef efin ined ed face face mesh meshes es • use uniform spacing Select meshing scheme • constraints on the face meshing for mapping and cooper schemes
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3D Grid generation – Advanced Cooper technique
“Creative” way of coopering: multisurface to multisurface sweep ME469B/2/GI
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Grid generation – Sizing functions Instead of the bottom-up approach (1D to 3D) grid generation Sizing functions can be specified to mesh volumes directly
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Grid generation – Clustering points Sizing functions can be used effectively to define the size of the cells BUT they cannot provide directional control (anisotropy) One option is to build (grow (grow)) elements from the boundaries and to form “viscous” layers
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Grid generation – Boundary Layers
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Example – meshing a circle Mapping
Boundary Layer Paving
Triangulation
Boundary Layer Paving
Paving
Multiblock mapping
Boundary Layer Multiblock Paving
Boundary Layer Transition Multiblock Paving
Circle defined as segments ME469B/2/GI
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Mesh linking
Edges, faces and volume meshes can be linked Define corresponding entities and ALSO reference entities Needed to “enforce” coincident grids on different entities (i.e. for periodicity bc)
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Virtual Geometry
GAMBIT operates on two different type of entities REAL: with corresponding geometrical and topological characteristics REAL: characteristics VIRTUAL:: defined only with reference to REAL entities VIRTUAL REAL entities are what we used and described so far; VIRTUAL are used to SIMPLIFY SIMPLIFY,, CLEAN UP, UP, DECOMPOSE real entities
Note that some geometry tools cannot be applied to virtual entities (boolean operation, volume blending, creation of volumes by sweeping faces, etc.)
Real entities can be transformed in virtual but NOT viceversa ME469B/2/GI
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Virtual Geometry Clean-Up Example of edge connecting operation
REAL ME469B/2/GI
VIRTUAL 46
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Grid quality Quality measures are NOT absolute but should be considered in connection with solution schemes The final accuracy of a procedure is ALWAYS a function of the grid quality
Several geometrical measures can be defined: • • •
Depe Depend ndin ing g on on the the size size of the the ele eleme ment ntss Depe Depend ndin ing g on on the the shap shapee of of the the elem elemen ents ts Depend Depending ing on on relat relative ive dimens dimension ionss of neig neighbo hborin ring g elemen elements ts
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Quality measures available in GAMBIT
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Examine meshes
• Define mesh element to examine • Define a cutting plane • Define the quality measure
Aspect ratio
Equiangular skew
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Mesh improvement • Smoothing operators are applied to redistribute nodes
Aspect ratio
Equiangular skew
Typically, improvement in 3D meshes are based on improved 2D meshes ME469B/2/GI
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Automation
• GAMBIT saves a “ journal journal”” file with the commands issues during a session • Journal file are ASCII editable files • Commands are quasi-English and easy-to-use • They are useful to trace-back sessions to find errors • They can be made general by introducing User Defined Parameters
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GAMBIT Journal file
The command: Volume create width 1 depth 1 height 1 offset 0 0 0 brick
Generates a cube of size 1 centered at 0 0 0 On the other hand the sequence $W = 2.3 $D = 1.5 $H = 4 Volume create width $W depth $D height $H offset 0 0 0 brick
Generates a cuboid of User-Specified Size centered at 0 0 0
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Example of Journal file
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/ -------------------------------------------------------/ CYCLONE GRID GENERATION / ME269B - Spring 2002 / -------------------------------------------------------/ / R1 = External radius of Cyclone / R2 = Gas Outlet Pipe (External) / R3 = Gas Outlet Pipe (Internal) / RB = Particles Outlet (Bottom) / H1 = Height of the Cylindrical Part of Cyclone / H2 = Height of the Conical Part / HE = Depth of the Outlet Channel into the Cyclone / / inletl = Length (x) of the Gas Inlet Channel / inleta = Height (z) of the Gas Inlet Channel / inletb = Span (y) of the Gas Inlet Channel / outletl = Length (z) of the outlet (gas) pipe / / cellsize = Average size of the cells / / Remark: z-axis is the Cyclone Axis / -------------------------------------------------------/ / Input Quantities / $R1 = 1.555 $R2 = 0.45 $R3 = 0.4 $RB = 0.75 $H1 = 4.5 $H2 = 5 $HE = 3.6 $inletl = 2 $inleta = 1.03 $inletb = 0.7 $outletl = 7.2 $cellsize = 0.18 / /
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Grid generation research ß
Structured grids: automatic generation of mappable subdomains NLR 2000-366 Report (PDF available from class web site)
ß
Unstructured tetrahedral grids: anisotropic Delaunay schemes Shimada et al. “High quality anisotropic tetrahedral mesh generation via ellipsoidal bubble packing” (PDF available)
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Grid generation research ß
Unstructured hex-dominant grids: OCTREE based SAMM – Computational Dynamics Ltd. Hexpress – Numeca International Inc.
ß
Unstructured purely hexahedral grids: Whisker-Weavi Whisker-Weaving ng CUBIT – Sandia National Lab. (PS report available)
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Grid generation – Links and References
• Links • Mesh generation and grid generation on the Web http://www-users.informatik http://www-u sers.informatik.rwth-aachen.de/~r .rwth-aachen.de/~roberts/meshgenerati oberts/meshgeneration.html on.html • Meshing research corner http://www.andrew.cmu.edu/use http://www. andrew.cmu.edu/user/sowen/mesh.ht r/sowen/mesh.html ml CFD: Topic mesh generation • General CFD: http://www.cfd-online.com
• References: • Handbook of grid generation. Thompson, Soni, Weatherill, CRC Press • Numerical Grid Generation: Foundation & Applications. Thompson, Warsi, Mastin. North Holland Press ME469B/2/GI
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Example #1 Geometry Modeling
Objective: generation of a swirl-generator model in GAMBIT
• •
Four Four rectan rectangul gular ar duct ductss injec injectt tange tangenti ntiall ally y in an an expan expandin ding g pipe pipe. Sele Select ct the the dim dimen ensi sion onss appr approp opri riat atel ely y
The duct offset DO is the main design variable: with fixed mass flow the swirl generated is a function of DO
DO
Use the “cyclone.jou” example as a baseline ME469B/2/GI
The shape of the pipe/duct intersection is a function of 57 the duct offset
Example #2 Grid Generation
Objective: generation of a grid for a swirl-generator model in GAMBIT • Gene Genera rate te an allall-te tetr trah ahed edra ra grid grid • Gene Genera rate te an allall-he hexa xahe hedr draa grid grid • Compar Comparee the qual quality ity of of the grid grid espec especiall ially y in the chan channel nel-pi -pipe pe inter intersec sectio tion n
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Solution methods for the Incompressible Navier-Stokes Equations
l
l l
l l
l
Discretization schemes for the Navier-Stokes equations Pressure-based approach Density-based approach Convergence acceleration Periodic Flows Unsteady Flows
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Background (from ME469A or similar)
Navier-Stokes (NS) equations Finite Volume (FV) discretization discretization Discretization of space derivatives (upwind, central, QUICK, etc.) Pressure-velocity Pressure-velocity coupling issue Pressure correction schemes (SIMPLE, SIMPLEC, PISO) Multigrid methods
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1
NS equations Conservation laws: Rate of change
+
advection
+
diffusion
= source
=0
The advection term is non-linear The mass and momentum equations are coupled (via the velocity) The pressure The pressure appears only as a source term in the momentum equation No evolution equation for the pressure There are four equations and five unknowns (r , V, p) p) ME469B/3/GI
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NS equations
Compressible flows:
The mass conservation is a transport equation for density. With an additional energy equation p equation p can be specified from a thermodynamic relation (ideal gas law)
Incompressible flows:
Density variation are not linked to the pressure. The mass conservation is a constraint on the velocity field; this equation (combined with the momentum) can be used to derive an equation for the pressure
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Pressure-based solution of the NS equation The continuity equation is combined with the momentum and the divergence-free constraint becomes an elliptic equation for the pressure
To clarify the difficulties related to the treatment of the pressure, we will define EXPLICIT and IMPLICIT schemes to solve the NS equations:
It is assumed that space derivatives in the NS are already discretized:
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Explicit scheme for NS equations
Semi-discrete form of the NS Explicit time integration The n+1 velocity field is NOT divergence free Take the divergence of the momentum Elliptic equation for the pressure
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Explicit pressure-based scheme for NS equations
Velocity field (divergence free) available at time n
Compute H Compute H n
Solve the Poisson equation for the pressure p pressure pn
Compute the new velocity field un+1
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Implicit scheme for NS equations
Semi-discrete form of the NS Implicit time integration Take the divergence of the momentum
The equations are coupled and non-linear
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Implicit scheme for NS equations Compute an intermediate velocity field (eqns are STILL non-linear)
{
Define a velocity and a pressure correction
Using the definition and combining
{
Derive an equation for u’
Enforce divergence-free condition at n+1
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Implicit pressure-based scheme for NS equations (SIMPLE) SIMPLE:: Semi-I SIMPLE emi- Implicit Method for P for Pressure-L ressure- Linked E quations
Velocity field (divergence free) available at time n Compute intermediate velocities u* Solve the Poisson equation for the pressure correction p’ correction p’ Neglecting the u*’ u*’ term term
Solve the velocity correction equation for u for u’ Neglecting the u*’ u*’ term term
Compute the new velocity un+1 and pressure p pressure pn+1 fields
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Implicit pressure-based scheme for NS equations (SIMPLEC) SIMPLE:: SIMPLE Corrected/ SIMPLE Corrected/C Consistent
Velocity field (divergence free) available at time n Compute intermediate velocities u* Solve the Poisson equation for the pressure correction p’ correction p’ u*’ (neighbor u*’ ~ Use an approximation to u*’ (neighbor values average u*’ ~ S u’ )
Solve the velocity correction equation for u for u’ Use an approximation to u*’
Compute the new velocity un+1 and pressure p pressure pn+1 fields
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Implicit pressure-based scheme for NS equations (PISO) PISO:: Pressure Implicit with S plitting O perators PISO
Velocity field (divergence free) available at time n Compute intermediate velocities u* and p’ as p’ as in SIMPLE
Solve the Poisson equation for the pressure correction p correction p(m+1)’ u*’ is u*’ is obtained from u m’
Solve the velocity correction equation for u for u(m+1)’ u*’ is u*’ is obtained from u m’
Compute the new velocity un+1 and pressure p pressure pn+1 fields ME469B/3/GI
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SIMPLE, SIMPLEC & PISO - Comments In SIMPLE under-relaxation is required due to the neglect of u*’ of u*’ un+1 = u* + a u u’
p = p n + a p p’ a p =1- a u
There is an optimal relationship
SIMPLEC and PISO do not need under-relaxation SIMPLEC/PISO SIMPLEC/PISO allow faster convergence than SIMPLE PISO is useful for irregular cells
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NS equations Conservation laws: Rate of change
+
advection
+
diffusion
= source
=0
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Segregated solver in FLUENT FV discretization for mixed elements
f W
The quantities at the cell faces can be computed using several different schemes ME469B/3/GI
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Discretization of the equations Options for the segregated solver in FLUENT Discretization scheme for convective terms 1st order upwind (UD (UD)) nd 2 order upwind (TVD (TVD)) 3rd order upwind (QUICK (QUICK ), ), only for quad and hex Pressure interpolation scheme (pressure at the cell-faces) linear (linear linear (linear between cell neighbors) second-order (similar second-order (similar to the TVD scheme for momentum) PRESTO (mimicking the staggered-variable staggered-variable arrangement) Pressure-velocity Pressure-velocity coupling SIMPLE SIMPLEC PISO ME469B/3/GI
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Solution of the equation f is one of the velocity component and the convective terms must be linearized:
This correspond to a sparse linear system for each velocity component
Fluent segregated solver uses: solver uses: Point Gauss-Seidel technique Multigrid acceleration
b b
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P
b
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Set-up of problems with FLUENT Read/Import the grid Define the flow solver option Define the fluid properties Define the discretization scheme Define the boundary condition Define initial conditions Define convergence monitors Run the simulation Analyze the results
Graphics Window
Command Menus
Text Window ME469B/3/GI
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Solver set-up Define Æ Models Æ Solver
Define Æ Controls Æ Solution
Example: text commands can be used (useful for batch execution) define/models/solver segregated define/models/steady
solve/set/discretization-scheme/mom 1 solve/set/under-relaxation/mom 0.7
…
… ME469B/3/GI
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Material properties Define Æ Materials
Quantities are ALWAYS dimensional ME469B/3/GI
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Initial and boundary conditions Solve Æ Initialize Æ Initialize
Only constant values can be specified More flexibility is allowed via patching
Define Æ Boundary Conditions
BCs will be discussed case-by-case
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Initial conditions using patching Adapt Æ Region Æ Mark
Mark a certain region of the domain (cells are stored in a register)
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Solve Æ Initialize Æ Patch
Patch desired values for each variable in the region (register) selected
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Convergence monitors Solve Æ Monitors Æ Residuals
Solve Æ Monitors Æ Surface
Convergence history of the equation residuals are stored together with the solution User-defined monitors are NOT stored by default ME469B/3/GI
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Postprocessing
Display Æ Contours
Plot Æ XY Plot
Cell-centered data are Computed This switch interpolates the results on the cell-vertices
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Detailed post-processing Define additional quantities Define plotting lines, planes and surfaces Compute integral/averaged quantities
Define Æ Custom Field Function
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Fluent GUI - Summary File: I/O Grid: Modify (translate/scale/etc.), Check Define: Models (solver type/multiphase/etc.), type/multiphase/etc.), Material (fluid properties), Boundary conditions Solve: Discretization Discretization,, Initial Condition, Condition, Convergence Monitors Adapt: Grid adaptation, adaptation, Patch marking Surface: Create zones (postprocessing/monitors) Display: Postprocessing (View/Countors/Streamlines) Plot: XY Plots, Plots, Residuals Report: Summary Summary,, Integral Typical simulation Parallel: Load Balancing, Balancing, Monitors
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Example – Driven cavity Classical test-case for incompressible flow solvers
Problem set-up
Solver Set-Up
Material Properties: r = 1kg/m3 m = 0.001kg/ms
Segregated Solver Discretization: 2nd order upwind SIMPLE
Reynolds number: H = 1m, Vslip= 1m/s Re = rVslip H/m H/m = 1,000
Vslip =1
Boundary Conditions: Slip wall (u = Vslip) on top No-slip walls the others
H
Multigrid V-Cycle
Initial Conditions: u=v=p=0 Convergence Monitors: Averaged pressure and friction on the no-slip walls
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Example – Driven cavity
The effect of the meshing scheme
Quad-Mapping 1600 cells
Tri-Paving 3600 cells
Quad-Paving 1650 cells
Edge size on the boundaries is the same ME469B/3/GI
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Example – Driven cavity
The effect of the meshing scheme – Vorticity Contours
Quad-Mapping 1600 cells
Tri-Paving 3600 cells
Quad-Paving 1650 cells
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Example – Driven cavity The effect of the meshing scheme – Convergence
Quad-Mapping 1600 cells ME469B/3/GI
Tri-Paving 3600 cells
Quad-Paving 1650 cells 30
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Example – Driven cavity The effect of the meshing scheme x-velocity component in the middle of the cavity
Quad-Mapping Tri-Paving
Quad-Paving
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Symbols corresponds to Ghia et al., 1982 31
Example – Driven cavity Grid Sensitivity – Quad Mapping Scheme
Vorticity Contours
1600 cells ME469B/3/GI
6400 cells
25600 cells 32
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Example – Driven cavity Grid Sensitivity – Quad Mapping Scheme x-velocity component in the middle of the cavity
1600 cells
6400 cells
25600 cells
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Symbols corresponds to Ghia et al., 1982 33
How to verify the accuracy?
Define a reference solution (analytical or computed on a very fine grid) Compute the solution on successively refined grids Define the error as the deviation of the current solution from the reference Compute error norms Plot norms vs. grid size (the slope of the curve gives the order of accuracy)
Problems with unstructured grids: 1) Genera Generatio tion n of a suitabl suitablee succes successio sion n of grids grids 2) Defi Defini niti tion on of of the the grid grid siz sizee
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Generation of successively refined grid 1) Modify Modify grid grid dimensio dimensions ns in GAMBIT GAMBIT and regener regenerate ate the the grid grid 2) Split Split all the cells cells in FLUEN FLUENT T Adapt Æ Region Æ Adapt
The region MUST contain the entire domain
Element shape & metric properties are preserved
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Driven Cavity - Error evaluation Reference solution computed on a 320x320 grid (~100,000 cells) Reference solution interpolated on coarse mesh to evaluate local errors
Quad-Mapping
Tri-Paving
Quad-Paving
Note that the triangular grid has more than twice as many grid cells ME469B/3/GI
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Driven Cavity – Accuracy evaluation Nominal 1st order accuracy
Nominal 2nd order accuracy
Quad and Pave meshing schemes yield very similar accuracy (close to 2nd order) Tri meshing scheme yields Slightly higher errors and lower accuracy Note that the definition of D of Dx is questionable (a change will only translate the curves not change the slope)
Quad-Mapping Tri-Paving
) m r o n 2 L ( r o r r E
(N)-1/2
Quad-Paving
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Driven Cavity – Fluent vs. other CFD codes Quad Mapping Scheme (1600 cells) x-velocity component in the middle of the cavity
FLUENT ME469B/3/GI
StarCD
NASA INS2D
Symbols corresponds to Ghia et al., 1982 38
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Techniques for the incompressible NS equations
Pressure correction schemes Fractional step methods Artificial compressibility approach Vorticity-streamfunction Vorticity-streamfunction formulation Density-based approach
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Techniques for the incompressible NS equations Fractional-step Fractional-ste p approach Similar in concept to pressure-correction schemes, but the pressure (or a related quantity) is only used in the final correction step; it is equivalent to an approximate factorization
Vorticity-streamfunction Vorticitystreamfunction approach It is effectively a change-of-variables change-of-variables;; introducing the streamfunction and the vorticity vector the continuity is automatically satisfied and the pressure disappears (if needed the solution of a Poisson-like equation is still required). It required). It is advantageous in 2D because it requires the solution of only two PDEs but the treatment of BCs is difficult. In addition in 3D the PDEs to be solved are six
Artificial compressibility approach A time-derivative (of pressure) is added to the continuity equation with the goal of transforming the incompressible NS into a hyperbolic system and then to apply schemes suitable for compressible flows. The key is the presence of a user-parameter b user-parameter b (related to the artificial speed of sound) that determines the speed of convergence to steady state ME469B/3/GI
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Density-based solvers for the NS equations The equation are written in compressible form and, for low Mach numbers, the flow is effectively incompressible incompressible
The energy equation is added to link pressure and density through the equation of state
In compact (vector) form:
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Density-based solvers for the NS equations Stiffness occurs because of the disparity between fluid velocity and speed of sound (infinite in zero-Mach limit) The equations are solved in terms of the primitive variables
Note that the continuity becomes (again) an evolution equation for the pressure
The time derivative is modified ( preconditioned preconditioned)) to force all the eigenvalues to be of the same order (similar to the artificial compressibility approach)
Several choices are available for G for G ME469B/3/GI
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FLUENT density-based solver
Explicit Scheme Multistage Runge-Kutta scheme Residual Smoothing Multigrid acceleration Implicit Scheme Euler (one-step) implicit Newton-type linearization Point Gauss-Seidel iterations Multigrid acceleration
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Example – Driven cavity Classical test-case for incompressible flow solvers
Problem set-up Material Properties: r = 1kg/m3 m = 0.001kg/ms Reynolds number: H = 1m, Vslip= 1m/s Re = rVslip H/m H/m = 1,000
Vslip =1
Solver Set-Up Coupled Solver Discretization: 2nd order upwind Implicit Multigrid V-Cycle
Boundary Conditions: Slip wall (u = Vslip) on top No-slip walls the others
H
Initial Conditions: u=v=p=0 Convergence Monitors: Averaged pressure and friction on the no-slip walls
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Example – Driven cavity
Effect of the the solver - Quad mesh (1600 cells) Segregated
Coupled
Vorticity Contours ME469B/3/GI
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Example – Driven cavity Effect of the the solver - Quad mesh (1600 cells) x-velocity component in the middle of the cavity
Segregated ME469B/3/GI
Coupled
Symbols corresponds to Ghia et al., 1982 46
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Multigrid acceleration
Basic idea: idea: the global error (low-frequency) on a fine grid appears as a local error (high-frequency) on coarse meshes. Why it is important: important: linear system solver like Gauss-Seidel are effective in removing high-frequency errors but VERY slow for global errors. Note that, on structured, grid line-relaxation (or ADI-type) schemes can be used to improve the performance of Gauss-Seidel; on unstructured grid similar concepts are extremely difficult to implement. Convergence Speed: number of iterations on the finest grid required to reach a given level of convergence is roughly independent on the number of grid nodes (multigrid (multigrid convergence) convergence)
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Two-grid scheme
1. 2. 3. 4. 5.
a smoothings are performed on the fine grid to reduce the highfrequency components of the errors (pre-smoothing, aS) the residu residual al (error) (error) is transfe transferred rred to to next coarser coarser level level (restriction, R) g iterations are performed on this grid level for the “correction” equation the prob problem lem is is transf transferr erred ed back back to the the fine fine grid grid (prolongation, P) b smoothings are performed on the fine grid to remove the highfrequency errors introduced on the coarse mesh (post-smoothing, bS)
Parameters to be defined are a, b, g
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Restriction & Prolongation Operators
Fine Level Coarse Level ME469B/3/GI
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Algebraic Multigrid
The coarse levels are generated without the use of any discretization on coarse levels; in fact no hierarchy of meshes is needed Geometric multigrid should multigrid should perform perform better than AMG because non-linearity of the problem are retained on coarse levels (correction equation) AMG is effectively a solver for linear systems and the restriction and prolongation operators might be viewed as means to modify (group or split) the coefficient matrix The same multigrid options (cycle type and a, b, g smoothing) are available
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Driven Cavity – Accuracy evaluation Nominal 1st order accuracy
Nominal 2nd order accuracy
Quad and Pave meshing schemes yield very similar accuracy (close to 2nd order) Tri meshing scheme yields Slightly higher errors and lower accuracy Note that the definition of D of Dx is questionable (a change will only translate the curves not change the slope)
Quad-Mapping Tri-Paving
) m r o n 2 L ( r o r r E
Quad-Paving
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(N)-1/2 51
Multigrid for unstructured meshes GEOMETRIC MULTIGRID ALGEBRAIC MULTIGRID
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Algebraic Multigrid Performance Convergence for the segregated solver
1600 cells
6400 cells
25600 cells
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Algebraic Multigrid Performance Convergence for the coupled solver
1600 cells ME469B/3/GI
6400 cells
25600 cells 54
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Periodic Flows Geometrical periodicity
Periodicity simply corresponds to matching conditions on the two boundaries
The velocity field is periodic BUT the pressure field is not. The pressure gradient drives the flow and is periodic. A pressure JUMP condition on the boundary must be specified
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Periodic Flows – Set-Up In the segregated solver periodicity can be imposed by fixing either the mass flow or the pressure drop In the coupled solver periodicity is enforced by fixing the pressure drop Define Æ Periodic Conditions
Segregated solver ME469B/3/GI
Define Æ Boundary Conditions
Coupled Solver 56
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Periodic Flow Example – 2D channel An analytical solution of the NavierStokes equations (Poiseuille flow) can be derived:
h
y
Solution in the form u=u(y) The pressure drop balances the viscous drag on the walls Navier-Stokes equations Velocity distribution in the channel Averaged velocity
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Periodic Flow Example – 2D channel Problem set-up
Solver Set-Up
Material Properties: r = 1kg/m3 m = 0.1kg/ms
Coupled Solver
Reynolds number: h = 2m, Vave= 1m/s Re = rVslip h/m h/m = 20
h
Periodic boundaries
Discretization: 2nd order upwind SIMPLE Multigrid V-Cycle
Boundary Conditions: Periodicity D p=0.3 No-slip top/bottom walls Initial Conditions: u = 1; v = p = 0 Exact solution: Vave = 1
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Periodic Flow Example – 2D channel x-velocity distribution in the channel
Cell-centered values showed (no interpolation)
Quad-Mapping
Tri-Paving
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Periodic Flow Example – 2D channel The error in this case CAN be computed with reference to the exact solution In this case the computed averaged velocity error is plotted
Nominal 1st order accuracy Nominal 2nd order accuracy
(N)-1/2 Quad-Mapping Tri-Paving ME469B/3/GI
This test-case is available on the class web site
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Unsteady flows The algorithms we used so far are time-marching: From an initial condition they iterate until a steady-state is reached The time-evolution of the solution is NOT accurate
Typical Implicit Time-Accurate Scheme
Unsteady transport equation 1st order time integration 2nd order time integration
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Unsteady flows – Dual time stepping At each time step we build a steady-state-like problem and we use an iterative algorithm
Outer iteration (time-step)
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Inner iteration steady-state pseudo time-step
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Unsteady flows – Set Up
Define Æ Models Æ Solver
Solve Æ Iterate Outer iteration
Inner iteration
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Unsteady Flow – Impulsive start-up of a plate Again an analytical solution of the NavierStokes equations can be derived:
y
Vwall
Solution in the form u=u(y,t) The only force acting is the viscous drag on the wall Navier-Stokes equations Velocity distribution Wall shear stress
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Unsteady Flow – Impulsive start-up of a plate Problem set-up periodic boundaries
Solver Set-Up
Material Properties: r = 1kg/m3 m = 0.1kg/ms
Segregated Solver Discretization: 2nd order upwind SIMPLE
Reynolds number: Re = r VwallL/m L/m Vwall = 5.605 twall L = mVwall/ t
H
Multigrid V-Cycle
Boundary Conditions: Slip wall (u = Vwall) on bottom No-slip wall (top) Periodicity D p=0 Initial Conditions: u=v=p=0
Vwall
Exact Solution twall = 1 @ t = 1
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H/L ~ 10
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Unsteady Flow – Impulsive start-up of a plate The error CAN be computed with reference to the exact solution In this case the computed wall shear stress is plotted
2nd order
Influence of the BCs ME469B/3/GI
Nominal 2nd order accuracy
) m r o n 2 L ( r o r r E
Time discretization 1st order
Nominal 1st order accuracy
Dt
This test-case is available on the class web site
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Overview of commercial CFD codes About 30 packages. Three major general-purpose major general-purpose products (covering ~50% of the market): FLUENT, StarCD, CFX
Grid Type
Pressure Based
Density Based
Multigrid
System Solver
Discretization
FLUENT
Unstructured Mixed
SIMPLE SIMPLEC PISO
Coupled Implicit Preconditioned
Algebraic Geometric
Gauss-Seidel
UD/TVD QUICK
StarCD
Unstructured Mixed
SIMPLE SIMPISO PISO
Conjugate -
-
Gradient
UD/TVD QUICK CD
Unstructured Mixed
SIMPLE -
Algebraic Coupled
ILU
UD/TVD QUICK CD
CFX
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Advanced Physical Models
• Heat Transfer • Buoyancy • Combustion and reaction modeling • Multiphase flows • Solidification and melting
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Heat Transfer Thermal analysis are crucial in many industrial applications
Turbulence is enhanced is internal turbine cooling passages to improve heat transfer Roughness elements (ribs) are placed in the channels
Heat Transfer Thermal analysis are crucial in many industrial applications
Turbulence is enhanced is internal turbine cooling passages to improve heat transfer Roughness elements (ribs) are placed in the channels
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Heat Transfer Modeling An energy equation must be solved together with the momentum and the continuity equations For incompressible flows the energy equation is decoupled from the others (r is NOT a function of the temperature) For laminar flows the energy equation can be solved directly; for turbulent flows after Reynolds-averaging the equation contains an unclosed correlation:
Momentum
Heat Transfer Modeling An energy equation must be solved together with the momentum and the continuity equations For incompressible flows the energy equation is decoupled from the others (r is NOT a function of the temperature) For laminar flows the energy equation can be solved directly; for turbulent flows after Reynolds-averaging the equation contains an unclosed correlation:
Momentum
Energy
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Heat Transfer Modeling
Momentum
Energy
The Prandtl number is the measure of the momentum diffusivity vs. the thermal diffusivity Pr=c p m/k
Heat Transfer Modeling
Momentum
Energy
The Prandtl number is the measure of the momentum diffusivity vs. the thermal diffusivity Pr=c p m/k Pr is order 1 for gases (typically 0.7 for air) Pr t is an additional parameter in the turbulence model (typically 0.9) ME469B/5/GI
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Set-Up for Heat Transfer Calculations Define Æ Models Æ Energy Activate the energy equation
Define Æ Materials
Set-Up for Heat Transfer Calculations Define Æ Models Æ Energy Activate the energy equation
Define Æ Materials
Specify material properties
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Wall thermal boundary conditions The options are: 1) 2)
Fixed temperature Fixe Fixed d therm thermal al flu flux x (temp (temper erat atur uree grad gradie ient nt))
wall thickness computational
BC
Wall thermal boundary conditions The options are: 1) 2)
Fixed temperature Fixe Fixed d therm thermal al flu flux x (temp (temper erat atur uree grad gradie ient nt))
wall thickness computational domain
BC
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Flow-thermal simulations For the energy equations all the numerical options (discretization, underrelaxation, etc.) are available
6
Flow-thermal simulations For the energy equations all the numerical options (discretization, underrelaxation, etc.) are available
For incompressible fluids the temperature and momentum equations are decoupled
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Periodic flows Many heat-transfer devices are characterized by geometrically periodic configurations (ribbed passages) Temperature behaves like the pressure: it varies in the streamwise direction but its variation (gradient) is periodic The energy equations can be rewritten in terms of a scaled temperature:
Periodic flows Many heat-transfer devices are characterized by geometrically periodic configurations (ribbed passages) Temperature behaves like the pressure: it varies in the streamwise direction but its variation (gradient) is periodic The energy equations can be rewritten in terms of a scaled temperature:
And the modified energy equation can be solved with periodic BC
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Example of Heat Transfer Calculatio Calculations ns Ribbed Passages
Example of Heat Transfer Calculatio Calculations ns Ribbed Passages
Wall heat transfer (temperature gradients) are strongly connected to wall friction coefficients and therefore to turbulence modeling ME469B/5/GI
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Example: Ribbed Channel Flow h h
L H
Periodic
Problem set-up
Solver Set-Up
Material Properties: r = 1kg/m3 m = 0.0001kg/ms Cp = 1000 J/Kg/ oK k = 0.142 W/m oK
Segregated Solver
Reynolds number: h = 1m, L=10m, H=L Reh = rU bh/m = 10,000 Boundary Conditions: Periodicity m=rU bH=10Kg/s
Discretization: 2nd order upwind SIMPLE Multigrid V-Cycle
Example: Ribbed Channel Flow h h
L H
Periodic boundaries
Problem set-up
Solver Set-Up
Material Properties: r = 1kg/m3 m = 0.0001kg/ms Cp = 1000 J/Kg/ oK k = 0.142 W/m oK
Segregated Solver
Reynolds number: h = 1m, L=10m, H=L Reh = rU bh/m = 10,000
Discretization: 2nd order upwind SIMPLE Multigrid V-Cycle
Boundary Conditions: Periodicity m=rU bH=10Kg/s No-slip walls Initial Conditions: u = 1; v = p = 0
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Turbulence model: k-e
Grids in Ribbed Channel Flows
Grid points are clustered at the walls and in the shear layers
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Grids in Ribbed Channel Flows
Grid points are clustered at the walls and in the shear layers
Unstructured gridding allows to separate the bottom and top BLs having different resolutions in the streamwise direction ME469B/5/GI
Heat Transfer Predictions
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Heat Transfer Predictions
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Conjugate Heat Transfer In many cases the correct prediction of the thermal field in a device requires the inclusion of conduction effects in solids Conjugate simulations are referred to coupled fluid-solid temperature calculations
Conjugate Heat Transfer In many cases the correct prediction of the thermal field in a device requires the inclusion of conduction effects in solids Conjugate simulations are referred to coupled fluid-solid temperature calculations
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Set-Up Conjugate Heat Transfer We need to specify two zones (fluid and solid) in the grid generation And then specify the material properties Define Æ Materials Æ Fluid
Define Æ Materials Æ Solid
Set-Up Conjugate Heat Transfer We need to specify two zones (fluid and solid) in the grid generation And then specify the material properties Define Æ Materials Æ Fluid
Define Æ Materials Æ Solid
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Wall thermal boundary conditions The boundary between the two zones is ALWAYS a wall and a shadow zone is created automatically by Fluent
wall thickness Fluid zone
shadow wall thickness Solid zone
Wall thermal boundary conditions The boundary between the two zones is ALWAYS a wall and a shadow zone is created automatically by Fluent
wall thickness
shadow wall thickness Solid zone
Fluid zone
coupling condition
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Conjugate Heat Transfer The temperature field is affected by the treatment of the rib walls
The rib is insulated
The rib is heated from the base
Conjugate Heat Transfer The temperature field is affected by the treatment of the rib walls
The rib is insulated
The rib is heated from the base
The rib is uniformly heated
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Effect of the Thermal Wall Bc
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Effect of the Thermal Wall Bc
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Effect of the Thermal Wall Bc
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Effect of the Thermal Wall Bc
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Comments on Thermo-fluid simulatio simulations ns
In incompressible flows the energy equation is decoupled from the momentum equations and can be solved a posteriori with the velocity field frozen Additional modeling is involved for the solution of thermal equation in the RANS context (therefore additional approximations and errors) Wall quantities (temperature and heat flux) are very sensitive to the modeling of near-wall turbulence
Comments on Thermo-fluid simulatio simulations ns
In incompressible flows the energy equation is decoupled from the momentum equations and can be solved a posteriori with the velocity field frozen Additional modeling is involved for the solution of thermal equation in the RANS context (therefore additional approximations and errors) Wall quantities (temperature and heat flux) are very sensitive to the modeling of near-wall turbulence Conjugate heat transfer (coupled fluid/solid) are often necessary to describe accurately a physical device
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User Programming
• What are User Defined Functions • Introduction to C • Set-Up C User Routines in Fluent • Implementing a Custom Turbulence Model • Programming in other CFD Commercial Codes
User Programming
• What are User Defined Functions • Introduction to C • Set-Up C User Routines in Fluent • Implementing a Custom Turbulence Model • Programming in other CFD Commercial Codes
Acknowledgement: this handout is partially based on Fluent training material ME469B/6/GI
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Introduction to UDF Programming Why programming in commercial codes? • The codes are general-purpose but cannot anticipate all needs • New (physical) models can be developed in a user-friendly environment • Large number of problems (test-cases) can be addressed with the same implementation
What is a the User Defined Function (UDF)? • C (Fluent) or FORTRAN or FORTRAN (StarCD, CFX) routines programmed by the user linked to the solver to perform certain operations: • initialization • special boundary condition (i.e. space or time dependent) • material properties • source terms • reaction rates • postprocessing and reporting • debugging • …. ME469B/6/GI
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A Simple UDF Example Specify a Parabolic Velocity Profile at the Inlet Goal: The UDF (inlet_parab) set the values of the x-velocity component at the cell faces of the inlet boundary zone
Inlet zone
1) 2) 3) 4)
Determi Determine ne the the cell-fa cell-faces ces belon belonging ging to the the inlet inlet zone zone Loop Loop on all all tho those se face facess Determi Determine ne the the coord coordinat inatee of the face centroid centroid Speci Specify fy the the x-v x-velo elocit city y comp compone onent nt
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A Simple UDF Example Parabolic Velocity Profile:
function inlet_parab Definitions Loop over all the inlet cell faces Evaluate the face centroid coordinates The function return the velocity
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A Simple UDF Example Compile/interpret Compile/interpret the UDF: Define Æ User Defined
Attach the profile to the inlet zone Define Æ Boundary Condition
Equivalent to attach the profile from a separate simulation ME469B/6/GI
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A Simple UDF Example Solve the equations Solve Æ Iterate
Final Result
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C Programming Typical C function:
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C vs. FORTRAN
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Basic Programming Rules Statements MUST end with a semicolon
Æ
;
Comments are placed anywhere in the program between Statements are grouped by curly brackets
Æ /* …… */
Æ { …… }
Variables defined within the body functions are local Variables defined outside the body functions are global and can be used by all the functions that follows Variables MUST be ALWAYS defined explicitly Integer/real/double Integer/real/double functions have to return a integer/real/double integer/real/double value C is case sensitive!
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Basic Statements Arithmetic expressions in C look like FORTRAN a = y + (i-b)/4;
Functions which return values can be used in assignment a = mycompute(y,i,b);
Functions which do not return values can be called directly mystuff(a,y,i,b);
Mathematical and various other default functions are available a = pow(b,2);
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/* pow(x,y) returns x raised to y */
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Data Types and Arrays Arrays of variables are defined using the notation int a[10]; /*define an array of ten integers */ Note the brackets are square [] not round ()!! Note that the arrays ALWAYS start from 0
Standard C types are: Integer, Real, Double, Char C allows to define additional types typedef struct list{int id, float x[3], int id_next};
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Pointers A pointer is a variable which contain the address of another variable Pointers are defined as: int *a_p; int a;
/* pointer to an integer variable */ /* an integer variable */
Set-up a pointer a = 1; a_p = &a;
/* &a return the address of variable a */
*a_p returns the content of the address pointed by a_p
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Operators Arithmetic Operators
Logical Operators
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Conditional Statements if and if-else
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Example
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Loop Procedure for loops
Example
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C Preprocessor It handles Macro substitutions #define A B #define my_f(x,y) x+y*3-5
The preprocessor replaces A with B It also handles file inclusion #include “udf.h” #include “math.h”
The files to be included MUST reside in the currect directory
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Programming in C
Of course much more than just this….
Additional information are: http://www.cs.cf.ac.uk/Dave/C/CE.html Plenty of books: “The C Programming Language”, Kernighan & Ritchie, 1988
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UDF in Commercial CFD Codes Commercial CFD codes allow the development of User Defined Functions for various applications. Anyway, the core of the software is closed. UDF must be compiled and linked to the main code Most codes provide macros and additional tools to facilitate the use of UDFs In Fluent there are two options: Interpreted The code is executed on a “line-by-line” basis at run time + does not need a separate compiler (completely automatic) - slows down the execution and uses more memory - somewhat limited in scope Compiled A library of UDF is compiled and linked to the main code Overcomes all the disadvantages reported above ME469B/6/GI
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Interpret the UDFs Define Æ User Defined Æ Interpreted
Display of code translation in assembly (and eventual compiling errors)
Default stack size might be too small for large arrays!
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Compile the UDFs
Define Æ User Defined Æ Compiled
The library MUST be precompiled in a directory tree
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Directory tree for compiled UDFs Machine dependent irix6r10 ultra hp700 …
my_library
src
Makefile
makefile
lnx86
my_udf.c
3d
2d makefile
my_udf.c
libudf.so makefile
my_udf.c
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Makefiles for UDFs In the directory /usr/local/Fluent.Inc/fluent6/src
There are two files
Makefile.udf makefile.udf
to be copied in the directory my_library to be copied in the directory my_library/src
The first one does not require modifications. In the second one two macros MUST be modified SOURCE = my_udf.c FLUENT_INC = /usr/local/Fluent.Inc
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UDFs in FLUENT Boundary Conditions Initial Conditions Adjust Function Source Terms Material Properties Execute on Demand User Defined Scalars User Defined Memory
Available MACROS
Pointers to threads Geometry Macros Cell and Face Variables Arithmetic and Trigonometric Functions
Programming Tools
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Boundary Conditions The inlet profile specification is based on the macro DEFINE_PROFILE DEFINE_PROFILE can be used for wall boundary conditions as well to impose temperature, velocity, shear stress, etc. It is possible to specify a constant value, a position-dependent or time-dependent values and to link the values on the boundary to the values in the internal cells Internal Values x x x
BC
Note that the BCs are applied to the faces of the cells (face centroids) ME469B/6/GI
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Initial Conditions The initial condition specification can be performed using the macro DEFINE_INIT Space-dependent conditions might be imposed The routine is executed once at the beginning of the solution process It is attached in the UDFs hooks
Define Æ User Defined Æ Function Hooks
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Initial Conditions Note that the difference between the DEFINE_PROFILE DEFINE_PROFILE and DEFINE_INIT is that the former performs a loop on the face-centroids belonging to a certain zone whereas the latter loops on the cell-centroids Example:
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Adjust Function Similar to the DEFINE_INIT but it is executed every iteration: DEFINE_ADJUST Can be used for postprocessing, clipping, etc. It is attached in the UDFs hooks Define Æ User Defined Æ Function Hooks
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Source Terms
To add a source term in the equations: DEFINE_SOURCE Can be used for: Continuity Momentum (component by component) Turbulent quantities Energy It is different from the previous macros because it works on a cell-by-cell basis (no need to perform loops!)
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Source Terms Define Æ Boundary Conditions Æ Fluid
Activate source terms
Attach the UDF
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Source Terms
In FLUENT the source terms are written in the form S (f) = A + B f where f is the dependent variable A is treated explicitly and B f is treated implicitly In the DEFINE_SOURCE both terms A and B have to be specified
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Material Properties
To define the properties of the materials : DEFINE_PROPERTIES Can be used for: Viscosity Density Thermal Conductivity …. As for the source terms works on a cell-by-cell basis
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Material Properties Define Æ Material Property
All the available UDFs Are shown in the menu
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Execute on Demand To perform operations instantaneously : EXECUTE_ON_DEMAND It is executed when activated by the user Can be used for: Debugging Postprocessing …. Define Æ User Defined Æ Execute on Demand
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User Defined Scalar In addition to the Continuity and Momentum Equations (and Energy) generic transport equations can be solved (up to 50!) To include scalars in the calculations 1) Define the number of scalars Define Æ User Defined Æ User Defined Scalars 2) Define the diffusion and source terms in the generic equation
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User Defined Scalar When UDS are defined in the material property panel the diffusivity of the scalars MUST be defined Define Æ Material Property
Note that the default diffusivity for the scalars is constant and equal to 1! ME469B/6/GI
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User Defined Scalar Boundary conditions for the scalars can be defined as Constant Value Constant Gradient User Defined Define Æ Boundary Conditions Æ Wall
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User Defined Scalar
The Source terms for the scalars are set using the DEFINE_SOURCE DEFINE_SOURCE macro Introduced before The scalar equations can be further customized 1) Co Conv nvec ecti tive ve te term rmss can be modified using the macro DEFINE_UDS_FLUX 2) Uns nsttea eady dy ter erm m can be modified using the macro DEFINE_UDS_UNSTEADY
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User Defined Memory
The User Defined Scalars are used to solve additional equations eventually coupled to the mean flow equations Sometimes it is useful to define temporary field variables to store and retrieve values not directly available UDM are defined:
Define Æ User Defined Æ User Defined Memory
More efficient storage compared to User Defined Scalars
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I/O in UDF User Defined Scalars and User Defined Memory are AUTOMATICALLY Stored in the Fluent Data files Additional I/O (from and to the Fluent Case and Data files) can be Accomplished using the macro DEFINE_RW_FILE
Define Æ User Defined Æ Function Hooks
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Detailed Programming The macros introduced before are interpreted/compiled and attached to the various Fluent panels The detailed programming is based on additional macros that allow to loop on cells to retrieve field variables, etc.
Loop Macros Geometry Macros Field Variable Macros Control Macros Arithmetic and Trigonometric Macros
Before looking at the Macros, the Fluent Data structure in introduced
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Data Structure It is based on a hierarchy of structures Threads (zones) are collection of cells or faces Domain is a collections of threads
Domain
Thread
Thread
Thread
Cell
Face
Face
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Threads Every zone is associated to a single ID (available in the boundary conditions menu)
zone
ID Given the ID of a thread the pointer can be retrieved as: Thread *tf = Lookup_Thread(domain, ID);
Given the thread pointer the ID of the zone can be retrieved as: ID = THREAD_ID(thread); ME469B/6/GI
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Loop Macros
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Cell-face connectivity When looping on faces the surrounding cells can be accessed using the macros: cell0_thread = F_C0_THREAD(face,thread) cell1_thread = F_C1_THREAD(face,thread) cell0_id = F_C0(face,thread) cell1_id = F_C1(face,thread)
x
cell0
cell1
face
When looping on cells adjacent faces and cells can be accessed using the macros: for(nf=0;nf
Number of faces of each cell is unknown 44
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Geometry Macros
Many more available; refer to the FLUENT UDF Manual ME469B/6/GI
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Field Variables Macros
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Field Variables Macros
Many more available; refer to the FLUENT UDF Manual ME469B/6/GI
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Control Macros
Many more available; refer to the FLUENT UDF Manual ME469B/6/GI
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An example of User Defined Programming Low-Re k-e model Development of a custom turbulence model can be accomplished using the UDFs. k-e model with damping functions formulation:
Transport Equations
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An example of User Defined Programming Eddy Viscosity P = S=
Low-Re k-e model
Turbulent Kinetic Energy Production Strain Rate Magnitude
Damping functions
y
Turbulent Reynolds number definitions
Wall boundary conditions ME469B/6/GI
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Required UDF Routines
Source Terms
DEFINE_SOURCE(k_source, thread, eqn) DEFINE_SOURCE(d_source, thread, eqn)
Diffusivity
DEFINE_PROPERTY(ke_diffusivity, domain)
Boundary Condition
DEFINE_PROFILE(wall_d_bc, domain)
Adjust Routine
DEFINE_ADJUST(turb_adjust, domain)
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Required field variables
Density
C_R(cell,thread)
Molecular viscosity
C_MU(cell,thread)
Eddy viscosity
C_MU_T(cell,thread)
Strain Rate Magnitude
Strain_rate(cell,thread)
Wall distance
C_WALL_DIST(cell,thread)
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UDF Header
#include "udf.h" /* Turbulence model constants */ #define C_MU 0.09 #define SIG_TKE 1.0 #define SIG_TDR 1.3 #define C1_D 1.44 #define C2_D 1.92 /* User-defined scalars */ enum { TKE, TDR, N_REQUIRED_UDS };
Required: includes all Fluent macros
Constant definitions (global)
Assign a number to each scalar
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Damping Functions These are defined on a cell-by-cell basis
/* Reynolds number definitions */ real Re_y(cell_t c, Thread *t) { return C_R(c,t)*sqrt(C_UDSI(c,t,TKE))*C_WALL_DIST(c,t)/C_MU_L(c,t);} real Re_t(cell_t c, Thread *t) { return C_R(c,t)*SQR(C_UDSI(c,t,TKE))/C_MU_L(c,t)/C_UDSI(c,t,TDR);} /* Damping Functions */ real f_mu(cell_t c, Thread *t) { return tanh(0.008*Re_y(c,t))*(1.+4/pow(Re_t(c,t),0.75));} real f_1(cell_t c, Thread *t) { return 1.;} real f_2(cell_t c, Thread *t) { return (1.-2/9*exp(-Re_t(c,t)*Re_t(c,t)/36))*(1.-exp(-Re_y(c,t)/12));}
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Source Term Routines The production term in the k equation is:
e is obtained from the definition of the eddy viscosity to increase the coupling between the equations and to define an implicit term
DEFINE_SOURCE(k_source, c, t, dS, eqn) { real G_k; G_k = C_MU_T(c,t)*SQR(Strainrate_Mag(c,t)); dS[eqn] = -2.*C_R(c,t)*C_R(c,t)*C_MU*f_mu(c,t)*C_UDSI(c,t,TKE)/C_MU_T(c,t); return G_k - C_R(c,t)*C_R(c,t)*C_MU*f_mu(c,t)*SQR(C_UDSI(c,t,TKE))/C_MU_T(c,t C_R(c,t)*C_R(c,t)*C_MU*f_mu(c,t)*SQR(C_UDSI(c,t,TKE))/C_MU_T(c,t); ); }
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Source Term Routines The production term in the e equation is:
It contains already both k and e. No need for manipulations! DEFINE_SOURCE(d_source, c, t, dS, eqn) { real G_k; G_k = C_MU_T(c,t)*SQR(Strainrate_Mag(c,t)); dS[eqn] = C1_D*f_1(c,t)*G_k/C_UDSI(c,t,TKE) - 2.*C2_D*f_2(c,t)*C_R(c,t)*C_UDSI(c,t,TDR)/C_UDSI(c,t,TKE); return C1_D*f_1(c,t)*G_k*C_UDSI(c,t,TDR)/C_UDSI(c,t,TKE) - C2_D*f_2(c,t)*C_R(c,t)*SQR(C_UDSI(c,t,TDR))/C_UDSI(c,t,TKE); }
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Diffusivity The diffusion terms in the scalar equations are set-up together DEFINE_DIFFUSIVITY(ke_diffusivity, c, t, eqn) { real diff; real mu = C_MU_L(c,t); real mu_t = C_R(c,t)*C_MU*f_mu(c,t)*SQR(C_UDSI(c,t,TKE))/C_UDSI(c,t,TDR); switch (eqn) { case TKE: diff = mu_t/SIG_TKE + mu; break; case TDR: diff = mu_t/SIG_TDR + mu; break; default: diff = mu_t + mu; } return diff;
But each equation can have a different value
}
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Eddy Viscosity The eddy viscosity is set in the adjust routine (called at the beginning of each Iteration) and it is used in the mean flow and in the scalar equations
DEFINE_ADJUST(turb_adjust, domain) { Thread *t; cell_t c; /* Set the turbulent viscosity */ thread_loop_c (t, domain) if (FLUID_THREAD_P(t)) { begin_c_loop(c,t) { C_MU_T(c,t) = C_R(c,t)*C_MU*f_mu(c,t)*SQR(C_UDSI(c,t,TKE))/C_UDSI(c,t,TD C_R(c,t)*C_MU*f_mu(c,t)*SQR(C_UDSI(c,t,TKE))/C_UDSI(c,t,TDR); R); } end_c_loop(c,t) } }
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Wall Boundary Conditions Only the boundary condition for e is complicated because it requires the value of the derivative of k (the square root of k) DEFINE_PROFILE(wall_d_bc, t, position) { face_t f; cell_t c0; Thread *t0 = t->t0; /* t0 is cell thread */ real xw[ND_ND], xc[ND_ND], dx[ND_ND], rootk, dy, drootkdy;
The derivative is rootk/dy begin_f_loop(f,t) { rootk is the sqrt of k in c0 = F_C0(f,t); the adjacent cell center rootk = sqrt(MAX(C_UDSI(c0,t0,TKE), 0.)); dy is the distance between F_CENTROID(xw,f,t); cell and face center C_CENTROID(xc,c0,t0); NV_VV(dx, =, xc, -, xw); dy = ND_MAG(dx[0], dx[1], dx[2]); drootkdy = rootk/dy; F_PROFILE(f,t,position) = 2.*C_MU_L(c0,t0)/C_R(c0,t0)*drootkdy*drootkdy; } end_f_loop(f,t) } ME469B/6/GI
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Set-Up the Problem
Set-Up a case using the standard k-e Define two scalars (TKE, TDR) Compile and Attach the UDFs Deactivate the Equations for k and e Initialize and solve the RANS + TKE and TDR
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Attach the UDF
Source terms
Boundary conditions
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Open UDF Library
After compiling the library It can be opened in Fluent
Output in the text window Are the functions available
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Perform the Simulation The convergence is for the mean flow + the two scalars (turbulent quantities deactivated!)
For postprocessing purposes the UDS have to be used instead of turbulent quantities ME469B/6/GI
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Additional Information
Many additional macros are available to implement different physical models combustion models particles-based models …. It is formally possible to develop additional numerical methods flux discretizations variable reconstruction and clipping …. Information are available in the FLUENT UDF manual (class web site)
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UDF in other commercial CFD Codes
CFX
CFX (v4) is a structured code; the data structure is much simpler because the field variables are accessed directly. It uses 1D arrays where the quantities are stored in succession CFX UDFs are written in FORTRAN For example:
U(I,J,K) Æ U(IJK) where IJK = (K-1)*NI*NJ+(J-1)*NI There are no macros, but examples of subroutines to perform the customization
USRBC USRSRC USRDIF …. ME469B/6/GI
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Star-CD
StarCD is an unstructured code similar to Fluent in term of data organization But similar to CFX for the organization of the UDF. StarCD has threads (as Fluent) and the UDF work normally on a cell-by-cell basis StarCD UDFs are written in FORTRAN There are no macros, but examples of subroutines to perform the customization
BCDEFW SORSCA DIFFUS ….
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Simulation of Turbulent Flows
• From the Navier-Stokes to the RANS equations • Turbulence modeling • k-e model(s) • Near-wall turbulence modeling • Examples and guidelines
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Navier-Stokes equations The Navier-Stokes equations (for an incompressible fluid) in an adimensional form contain one parameter: the Reynolds number:
Re = r Vref Lref / m it measures the relative importance of convection and diffusion and diffusion mechanisms
What happens when we increase the Reynolds number?
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Reynolds Number Effect
350K < Re
Turbulent Separation Chaotic
200 < Re < 350K
Laminar Separation/Turbulent Separation/Turbulent Wake Periodic
40 < Re < 200
Laminar Separated Periodic
5 < Re < 40
Laminar Separated Steady
Re < 5
Laminar Attached Steady
Re ME469B/3/GI
Experimental Observations
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Laminar vs. Turbulent Flow
Laminar Flow
The flow is dominated by the object shape and dimension (large scale)
Easy to compute ME469B/3/GI
Turbulent Flow
The flow is dominated by the object shape and dimension (large scale) and by the motion and evolution of small eddies (small scales) Challenging to compute 4
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Why turbulent flows are challenging?
Unsteady aperiodic motion Fluid properties exhibit quasi-random spatial variations (3D) Strong dependence from initial conditions Contain a wide range of scales (eddies)
The implication is that the turbulent simulations MUST be always three-dimensional, three-dimensional, time accurate with extremely fine grids
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Direct Numerical Simulation The objective is to solve the time-dependent NS equations resolving ALL the scale (eddies) for a sufficient time interval so that the fluid properties reach a statistical equilibrium
Grid requirement: N ~ (Re t)9/4 ~ 1x107 for Re t = 800 Time step requirement: Dt ~ (Re t)-1/2 ~ 1x10-5 for Re t = 800 ME469B/3/GI
y+ = r y p u t /m
tw/r)1/2 u t = ( t
6
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Typical DNS channel flow simulations
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Instantaneous Flow Field – Vortical Structures
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DNS of Channel Flow: Averaged Results
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Laminar and Turbulent Velocity profiles
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DNS using Commercial CFD Codes?
Channel flow @ Re t = 395 ME469B/3/GI
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Beyond DNS DNS is possible but only for low Reynolds number flows (and simple geometries) The (time and space) details provided by DNS are NOT required for design purposes Time averaged quantities are appropriate for engineering purposes Large scale resolution (not to the level of the smallest eddies) is enough for applications
Can we extract time-average and large-scale quantities at a reasonable computational cost?
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Flow over a Backstep
Istantaneous high velocity regions
low velocity regions
Time averaged
Length of the recirculation region is of engineering interest ME469B/3/GI
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Velocity & Velocity Fluctuations is a function of time with rapid fluctuations (turbulence) By re-defining it as the time average and the fluctuations
(x1)
(x2)
t
Velocity measurements in the turbulent wake of a cylinder ME469B/3/GI
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Reynolds-Averaged Navier-Stokes Equations
Define Reynolds-averaged quantities
Substitute and average:
Closure problem ME469B/3/GI
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Unsteady RANS? Every turbulent flow is unsteady BUT not all the unsteadiness is turbulence! turbulence! RANS averaging based on time average can be applied only to “statistically” steady flows. What if flow has a large scale periodicity (vortex shedding)?
We can define the Reynolds-Averaging procedure in terms of Ensemble of Ensemble Average: Average:
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Turbulence modeling Define the Reynolds stresses in terms on known (averaged) quantities 1) Boussinesq hypothesis - simple relationship between Reynolds stresses and velocity gradients through the eddy viscosity (similar to molecular viscosity) - isotropic (eddy viscosity is a scalar!) 2) Reynolds stress transport models - equations derived directly manipulating the NS equations - still contain unknown (undetermined) quantities - no assumption of isotropy - very complicated and expensive to solve 3) Non-linear 3) Non-linear Eddy viscosity models (Algebraic Reynolds stress) 4) Model directly the divergence of the Reynods Stresses ME469B/3/GI
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Eddy viscosity models Boussinesq relationship:
with:
Guidelines for defining the eddy viscosity: 1) Dime Dimens nsio iona nall arg argum umen ents ts - units are [m 2/s] - define 2 out of three scales: velocity, length, time
2)
Physical arguments - asymptotic analysis - consistency with experimental findings
3)
Numerical arguments - simple and easy to compute
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Classification of eddy viscosity models
The various models (about 200) are classified in terms of number of transport equations solved in addition to the RANS equations: 1) 2) 3) 4) 5)
zerozero-eq equat uation ion/al /algeb gebrai raicc models: models: Mixing Length, Cebeci-Smith, Baldwin-Lomax, etc oneone-eq equa uati tion on mode models ls:: Wolfstein, Baldwin-Barth, Spalart-Allmaras, k-model, etc twotwo-eq equa uati tion on mode models ls:: t, k-L, etc. k-e, k-w, k- t thre threee-eq equa uati tion on mod model els: s: k-e-A four four-e -equ quat atio ion n mode models ls:: v2-f model
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Zero-equation model
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Prandtl Mixing Length
From dimensional arguments and analogy with molecular transport Definition of L is different for each problem (boundary layes, mixing layers, etc.) Eddy viscosity is zero if the velocity gradients are zero No “history” effect; purely local L can be made “universal” using ad hoc functions of distance from the walls, pressure gradients, etc.
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One-equation model
k-model
An equation from k can be derived directly from the NS equations (using the definition)
k 1/2 is assumed to be the velocity scale it still requires a length scale L as before to define the eddy viscosity 4 out of 7 terms in the k equation require further assumptions Production is computed using the Boussinesq approximation Dissipation is modeled (using dimensionality arguments) as k 3/2/L Turbulent transport and pressure diffusion are modeled together:
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One-equation model
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k-model
The final form of the model is:
The ONLY advantage with respect to zero-equation models is the inclusion of the history effects.
Modern one-equation models abandoned the k-equation and are based on a ad-hoc Transport equation for the eddy viscosity directly. Spalart-Allmaras model:
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Two-equation model
k-f family
The main drawback of the k one-equation model is the incomplete representation of the two scales required to build the eddy viscosity; two-equation models attempt to represent both scales independently. • All models use the transport equation for the turbulent kinetic energy k • Several transport variables are used
e: turbulence dissipation rate L: turbulent length scale w: inverse of turbulent time scale w2 g y t
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k-e model The k equation is the same as before:
The e equation can be obtained from the NS equations but it contains several undetermined quantities; it is therefore derived “mimicking” the k equation
The eddy viscosity is obtained as:
There are 5 free 5 free constants
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Determining the constants?
The constants can be determined studying simple flows:
1.
Decaying Decaying homogene homogeneous ous isotrop isotropic ic turbulen turbulence ce
2.
Homo Homoge gene neou ouss she shear ar flow flow
3.
The The Loga Logari rith thmi micc Laye Layer r
4.
…
Or by comparison with experimental data
Standard k-e refers to a certain choice of the constants (Launder & Sharma 1972)
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Structure of the Turbulent Boundary Layer Universal Law (velocity profile)
At High Reynolds number the viscous dominated layer is so thin that it is very difficult to resolve it ME469B/3/GI
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Wall Function Approach (High-Re k- e) The laminar sublayer is NOT resolved First grid point is assumed to be in the logarithmic layer (y +>11) and the velocity is assumed to be described by: u+ = (1/k)ln(Ey +) A slip condition (u
≠
0) is imposed at the wall (imposed shear stress)
boundary condition is usually imposed as a zero-gradient. k boundary P k =e ) e is obtained by equilibrium condition ( P
If first grid point is too close (viscous layer) then the velocity is:
u+ = y+
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Near Wall Region Modeling
From a physical point of view: It is important because solid walls are the main source of vorticity and turbulence (local extrema of turbulent kinetic energy, large variations of turbulence dissipation, etc.) In engineering applications: Wall quantities (velocity gradients, pressure, etc.) are very important in several applications Flow separation and reattachment are strongly dependent on a correct prediction of the development of turbulence near walls
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Damping Functions Approach (Low-Re k- e) The equations are integrated to the wall WITHOUT assuming an universal law for the velocity profile and an equilibrium conditions for k for k and and e The problem is that the model predicts the wrong behavior for k for k and and e near the solid walls (from DNS and experimental observation) The equations are modified using algebraic functions to “damp” certain terms:
The damping functions are designed to correct the behavior of the eddy viscosity ME469B/3/GI
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Damping Functions We need to specify 5 constants plus 3 functions:
MANY choices are available (about 30 different formulations!). Fluent has 6 different versions Classic model is the Launder and Sharma model:
Others might be function of the distance from the wall, the pressure gradient, etc. ME469B/3/GI
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Two-Layer Approach
The computational domain is divided in two regions: viscosity affected (near wall) and fully turbulent core (outer region) Two different models are used: the complete k- e model for the outer region and a simplified model (typically a one-equation k-based model) for the near-wall The separation between between the two regions is defined in terms of a distance distance from the + wall (y ~30) The main assumption is related to the definition of e which is based on
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Near-Wall Treatments for k- e models
Approach
Physics
Grid requirement
Wall functions
-
+
+
+/-
Two layer
+/-
+/-
+
+/-
Damping functions
+/-
-
-
+/-
Numerics
Accuracy
Summary and Comparison
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Example: Turbulent Channel Flow
L h
Periodic boundaries
Problem set-up
Solver Set-Up
Material Properties: r = 1kg/m3 m = 0.0017kg/ms
Segregated Solver
Reynolds number: h = 2m, L=1m Re t = rU th/m = 590 Boundary Conditions: Periodicity D p/L=1=U t No-slip top/bottom walls Initial Conditions: u = 1; v = p = 0
Discretization: 2nd order upwind SIMPLE Multigrid V-Cycle
Grid SAME GRID USED FOR THE LAMINAR FLOW @ Re=20
Turbulence model: k-e with wall functions ME469B/3/GI
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Turbulent Channel Flow Re t = 590
Velocity and k profiles ME469B/3/GI
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Grid Convergence?
Turbulence Channel Flow
Re t = 590
Velocity and k profiles ME469B/3/GI
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From High-Re to Low-Re k-e Wall clustering y =30 y+~1
Wall boundary condition dk/dy=0 k=0
+
Grid ME469B/3/GI
Turbulent kinetic energy
Re t = 590
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Low-Re k-e model
Turbulence Channel Flow
Re t = 590
Velocity and k profiles ME469B/3/GI
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Pros & Cons of k- e model + Simple + Affordable + Reasonably accurate for wide variety of flows (without separation) + History effects - Overly diffusive - Cannot predict different flows with the same set of constants (universality) - Source terms are stiff numerically - Not accurate in the region close to no-slip walls where k and e exhibit large peaks (DNS and experimental observations) - Near wall treatment A lot of variants have been introduced to overcome these problems One (or more) constants become coefficients varying with S, distance from the walls, pressure gradient, etc.: RNG k- e, realizable k-e… ME469B/3/GI
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Alternatives/Improvements Alternatives/Improvemen ts to k- e models The k-w model was developed from the realization that most of the problems experienced by k-e-type models are due to the modeling of the e equation which is neither accurate or easy to solve (e has a local extrema close to the wall) Mathematically this is equivalent to a change of variables w~e/k
The v2-f model is based on the argument that k/e is the correct turbulent time scale in the flow (close to the wall and in the outer region) but k is not the appropriate turbulent velocity scale An additional equation for the correct velocity scale v 2 (independent from k) has to be solved. Moreover, the damping effect produced from the presence of the wall is NOT local (as assumed in the damping function approach) but must be accounted for globally using an elliptic equation ME469B/3/GI
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k-w model
Turbulence Channel Flow
Re t = 590
Velocity and k profiles ME469B/3/GI
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Reynolds Stress Models Attempt to model directly the “new” terms appearing in the RANS equations Mathematically is expensive because we have 6 additional equations:
More importantly ONLY the production term are exact, everything else MUST be modeled
RSM are extremely stiff numerically due to the coupling between the equations and suffer of the same near-wall problems of the k-e
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Models available in Fluent Define Æ Models Æ Viscous One-equation model Two-equation model k-e (3 versions + 3 wall treatments) k-w (2 versions)
Reynolds Stress model
Note that the coefficients might be adjusted!!! ME469B/3/GI
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Models in Fluent hidden behind the GUI Some turbulence models are NOT directly available in the GUI!!!
Mixing Length define/model/viscous/mixing-length y
Two-equation model k-e Low-Re (6 versions)
define/model/viscous/turbulence-expert/low-re-ke y define/model/viscous/turbulence-expert/low-re-ke-index 2
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Comparison of Models
Turbulence Channel Flow
Re t = 590
Velocity and k profiles ME469B/3/GI
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Channel Flow Summary
Wall functions are accurate if the first grid point is in the logarithmic layer Grid Convergence Study with wall functions approach FAILS Damping Functions (and Two-Layer approaches) are accurate for the velocity profiles But the turbulent kinetic energy peak is underpredicted k-w model is a viable alternative to k- e and has less sensitivity to the grid clustering SA model and v2-f model are equivalent in capturing the velocity profile v2-f model is accurate in predicting the peak of turbulence kinetic energy near the wall
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Inlet boundary conditions for turbulent quantities At inlet boundary conditions additional quantities have to be specified in turbulent flows depending on the turbulence model selected. Typically there are three options:
1)
k-e
2)
Turbulence Intensity and Turbulence Length Scale
3)
Turbulence Intensity and Turbulent Viscosity
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Example: Asymmetric Diffuser Problem set-up Material Properties: r = 1kg/m3 m = 0.0001kg/ms Reynolds number: h = 2m, Re = rUh/m = 20,000
Solver Set-Up Initial Conditions: u = 1; v = p = 0
Turbulence model: Two-equation models
Boundary Conditions: Inlet profiles available from experiments No-slip top/bottom walls
Segregated Solver Discretization: 2nd order upwind SIMPLE Multigrid V-Cycle
Outflow Inflow
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Flow in Asymmetric Diffuser
k-e Low-Re
Experiments indicated the presence of a large recirculation region v2-f
k-e models with damping function do NOT capture it
k-e Low-Re
v2-f ME469B/3/GI
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Flow in Asymmetric Diffuser
Skin friction on the bottom wall ME469B/3/GI
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Flow in Asymmetric Diffuser Streamwise Velocity
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Turbulence Kinetic Energy
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k-e with wall functions
Series of grids generated with different clustering at wall
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No separation is captured!
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k-e with wall functions In “complex” configuration it is impossible to generated a grid with the first grid point in the logarithmic layer…..
…in addition, for complicated flows with recirculation the Universal Law is inaccurate ME469B/3/GI
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Example: NLR Two Component Airfoil Problem set-up Material Properties: r = 1kg/m3 m = 3.98E-7Kg/ms Reynolds number: h = 1m, Re = rUh/m = 2,512,600
Solver Set-Up Initial Conditions:
Segregated Solver
u = cosa; v = sina, p = 0
Turbulence model: SA and k-e models
Boundary Conditions: Constant velocity at AOA=a No-slip walls
Discretization: 2nd order upwind SIMPLE Multigrid V-Cycle
a ME469B/3/GI
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Computational Grid
Due to the high Reynolds number resolution of the boundary layers requires extreme clustering
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How this grid is generated?
Multiblock Structured Grid ME469B/3/GI
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Pressure distributions
Pressure Distribution Distribution at Low and High Angle of Attack ME469B/3/GI
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Why k-e fails?
Streamwise Velocity
Turbulent kinetic energy
Spurious production of k in the stagnation regions Fix: Use of a production limiter: define/model/viscous/turbulence-expert/kato-launder-model y
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Guidelines Simulations of turbulence flows require “decisions” based on: 1) Flow Physics to characterize the flow features (turbulence, high gradients, etc.) 2) Computational requirements to evaluate the grids resolution required for a certain accuracy 3) Project Requirement to evaluate the need for sophisticated turbulence models
physics accuracy resources
Turbulence mo model
Computationa l grid
s s e c o r P D F C
Simulation ME469B/3/GI
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Guidelines
Modeling procedure: • Determine relevant Reynolds number to estimate if the flow is turbulent • Select a turbulence model option and a near-wall treatment • Estimate the physical dimension of the first grid point off the wall (y +) • Generate the grid • Perform the simulation • “Reality” check (experiments, literature, model consistency)
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Estimating y+ Definition: y+ = r y p u t /m y+ = r y p u t /m u t = Ue (cf /2)1/2 To estimate u t we can use “classic” relationships for the skin friction: • •
Flat Plate: Pipe:
c f /2 ~ 0.052 (Rex)-0.142 cf /2 ~ 0.046 (Rex)-0.2
Note that there are different laws for different Re number ranges
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“Final” Guidelines
For k-e simulations: Two-layer is preferable over wall-functions (grid dependence + accuracy) Realizable k-e or Kato&Launder limiter have to be used For k-w simulations: SST is usually better than standard Grid should be clustered at wall SA is usually a good compromise between accuracy and simplicity. It also has very good convergence properties (as compared to the two-equation models) Reynolds stress model is expensive and it require a good initial guess (typically a k-e-type simulation)
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Summary of turbulence models in Commercial Codes
Zero equation
One
Two
equation
equation
RSM
Models
Custom
FLUENT
y
y
y
y
n
y
StarCD
n
n
y
n
y
y
CFX
y
y
y
y
n
y
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