Boundary-Fitted Coordinates!

Transcription

1 Computational Fluid Dnamics Computational Fluid Dnamics Computational Methods or Domains with Comple BoundariesI Grétar Trggvason Spring For most engineering problems it is necessar to deal with comple geometries consisting o arbitraril curved and oriented boundaries Computational Fluid Dnamics Outline How to deal with irregular domains Overview o various strategies Boundaritted coordinates NavierStokes equations in vorticit orm NavierStokes equations in primitive orm Grid generation or boditted coordinates Algebraic methods Dierential methods Staircasing Computational Fluid Dnamics Overview Various Strategies or Comple Geometries and to concentrate grid points in speciic regions Boundaritted coordinates Immersed boundar method no grid change Unstructured grids: triangular or tetrahedral Adaptive mesh reinement AMR Computational Fluid Dnamics Computational Fluid Dnamics Staircasing BoundarFitted Coordinates Approimate a curved boundar b a the nearest grid lines

2 Computational Fluid Dnamics BoundarFitted Coordinates Coordinate mapping: transorm the domain into a simpler usuall rectangular domain Boundaries are aligned with a constant coordinate line thus simpliing the treatment o boundar conditions The mathematical equations become more complicated Computational Fluid Dnamics BoundarFitted Coordinates Computational Fluid Dnamics BoundarFitted Coordinates Computational Fluid Dnamics BoundarFitted Coordinates First consider the D case: For the irst derivative the change o variables is straightorward using the chain rule: d d d d d d d d d d For the second derivative the derivation becomes considerabl more comple: The second derivative is given b d d d d d d d d d d d d d d d d d d d d Where we have used the epression or the ist derivative or the inal step However since the equations will be discretized in the new grid sstem it is important to end up with terms like not d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d Computational Fluid Dnamics BoundarFitted Coordinates To do so we look at the second derivative in the new sstem d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d Solving or the original derivative which we need to transorm we get: d d d d d d d d d d d d d d d d d d d d d d d d Computational Fluid Dnamics BoundarFitted Coordinates Oten we need the derivatives o the transormation itsel: B the chain rule we have d dξ dξ d For the second derivative we dierentiate the above: d d d d d d d d d d d d d d d d d d d d d d d d Giving: d d d d d d d d d d d d d d d d d d

3 Computational Fluid Dnamics Change o variables D: First Derivatives The equations will be discretized in the new grid sstem Thereore it is important to end up with terms like not BoundarFitted Coordinates Computational Fluid Dnamics Using the chain rule as we did or the D case: We want to derive epressions or in the mapped coordinate sstem BoundarFitted Coordinates Computational Fluid Dnamics Solving or the derivatives Subtracting BoundarFitted Coordinates Computational Fluid Dnamics Solving or the original derivatives ields: where is the acobian BoundarFitted Coordinates Computational Fluid Dnamics A shorthand notation: BoundarFitted Coordinates Computational Fluid Dnamics Rewriting in shorthand notation where is the acobian BoundarFitted Coordinates

4 Computational Fluid Dnamics BoundarFitted Coordinates These relations can also be written in conservative orm: Since: And similarl or the other equation Similarl Computational Fluid Dnamics BoundarFitted Coordinates D: Second Derivatives The second derivatives is ound b repeated application o the rules or the irst derivative Computational Fluid Dnamics BoundarFitted Coordinates Adding and ields an epression or the Laplacian: Computational Fluid Dnamics BoundarFitted Coordinates where Computational Fluid Dnamics BoundarFitted Coordinates q q q q q q q q q q q Computational Fluid Dnamics BoundarFitted Coordinates Epanding the derivatives ields q q q where [ q q ] [ q q ]

5 Computational Fluid Dnamics BoundarFitted Coordinates Derivation o i hence similarl Computational Fluid Dnamics BoundarFitted Coordinates Putting them together it can be shown that prove it [ q q ] q [ q q ] q Computational Fluid Dnamics BoundarFitted Coordinates We also have or an unction and g g g g g Computational Fluid Dnamics BoundarFitted Coordinates A comple domain can be mapped into a rectangular domain where all grid lines are straight The equations must however be rewritten in the new domain Thus: And more comple epressions or the higher derivatives Computational Fluid Dnamics Computational Fluid Dnamics VorticitStream Function Formulation VorticitStream Function Formulation The NavierStokes equations in vorticit orm are: t Using the transormation relations obtained earlier

6 Computational Fluid Dnamics VorticitStream Function Formulation Computational Fluid Dnamics VorticitStream Function Formulation The NavierStokes equations in vorticit orm become: t q q q q q q q q q Boundar Conditions Inlow : Inlow N : Outlow M : No slip Q Q ud vd Outlow d Lower wall Computational Fluid Dnamics VorticitStream Function Formulation Stream unction: Vorticit: noslip HOT Using that We have: [ ] Upper wall η M Computational Fluid Dnamics VorticitStream Function Formulation Stream unction: Q Vorticit: L M Q ud vd d M M [ M M ] Inlet low Computational Fluid Dnamics VorticitStream Function Formulation Considering a ulldeveloped parabolic proile u C L L L 6Q L d C C Q C 6 6Q u u L L 6Q Q L d Q L L L L Inlet low Computational Fluid Dnamics VorticitStream Function Formulation Considering a ulldeveloped parabolic proile and assume that L M 6Q u L M M Q M M Q L M 6

7 Outlow Computational Fluid Dnamics VorticitStream Function Formulation N Tpicall assuming straight streamlines n I is normal to the outlow boundar this ields I not then a proper transormation is needed or n Computational Fluid Dnamics VelocitPressure Formulation Computational Fluid Dnamics VelocitPressure Formulation Computational Fluid Dnamics VelocitPressure Formulation The NavierStokes equations in primitive orm u t uu vu p u u v t uv vv p v v and continuit equation u v Notice that we have absorbed the densit into the pressure Advection Terms uu vu [ uu uu uv uv ] [ uu uu uu uv uv uv u u v uu uv uu uv ] { } u v u { } Computational Fluid Dnamics VelocitPressure Formulation Computational Fluid Dnamics VelocitPressure Formulation Contravariant Velocit V C v u C U U u v V v u n Unit normal vector Unit tangent vector along C U u v u v Thereore U is in the direction V is in the direction Pressure Term p p p Diusion Term u u u u u u u u u u u u u

8 Computational Fluid Dnamics VelocitPressure Formulation u u u u u u u u u u q u q u q u q u q q q Computational Fluid Dnamics VelocitPressure Formulation umomentum Equation U u v V v u u t Uu Vu p p q u q u vmomentum Equation q u q u v t Uv Vv p p q v q v q v q v Computational Fluid Dnamics VelocitPressure Formulation Continuit Equation u v Using Continuit equation becomes [ u u v v ] or u v v u U V Computational Fluid Dnamics VelocitPressure Formulation The momentum equations can be rearranged to u v t t UMomentum Equation U Uu Vu Uv Vv t p q q p 4 q u q u 54 q v q v q u q u q v q v VMomentum Equation Computational Fluid Dnamics VelocitPressure Formulation V t Uu Vu Uv Vv p q q p 4 q v q v 54 where q u q u q v q v q u q u u U V v V U Computational Fluid Dnamics VelocitPressure Formulation In the plane a staggered grid sstem can be used Vv C p C Uu Same MAC grid and projection method can be used

9 Computational Fluid Dnamics Here we have eamined the elementar aspects o representing the governing equations in a mapped coordinate sstem This is a comple topics with its own language that needs to be mastered or urther studies The classic