BoundaryFitted CoordinatesComputational Fluid DynamicsyyfxxffyyfxxffUsing the chain rule as we did for the 1D caseWe want to derive expressions for in ID: 280958
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- Navier-Stokes equations in primitive form!¥"Grid generation for body-Þtted coordinates!!- Algebraic methods!!- Differential methods!Outline!How to deal with irregular domains!Computational Fluid Dynamics!- Staircasing!- Boundary-Þtted coordinates!- Immersed boundary method (no grid change)!- Unstructured grids: triangular or tetrahedral!- Adaptive mesh reÞnement (AMR)!Various Strategies for Complex Geometries and to concentrate grid points in speciÞc regions!Overview! Boundary-Fitted Coordinates!Computational Fluid Dynamics!!!!"""####+####=######+####=##yyfxxffyyfxxffUsing the chain rule, as we did for the 1D case:!We want to derive expressions for !in the mapped coordinate system. !yfxf!!!!/,/Boundary-Fitted Coordinates!Computational Fluid Dynamics!!"!"!"######+######=####xyyfxxxfxf!"!"!"######+######=####xyyfxxxfxfSolving for the derivatives hence! !xx=!x()x=1Jy"J#$%&'(!y")y"J#$%&'("y!*+,,-.//similarly! !yy=!y()y=1J"x#J$%&'()#x!""x#J$%&'()!x#*+,,-.//Boundary-Fitted Coordinates!Computational Fluid Dynamics!Putting them together, it can be shown that!(prove it!)! !2"=1J3q1x#y""$y#x""()$2q2x#y"#$y#x"#()+q3x#y##$y#x##()[] !2"=1J3q1y#x##$x#y##()$2q2y#x#"$x#y#"()+q3y#x""$x#y""()[]Boundary-Fitted Coordinates!Computational Fluid Dynamics! gyfx!gxfy=1Jg"f#!g#f"()We also have, for any function and!fgBoundary-Fitted Coordinates!Computational Fluid Dynamics!!!xy0=!1=!0=!1=!2=!3=!),(yx!!=),(yx!!=),(),(!"fyxf=A complex domain can be mapped into a rectangular domain where all grid lines are straight. The equations must, however, be rewritten in the new domain. yv,C=!UVC=!Unit tangent vector along!C=! x!,y!() y!,"x!()=n!Unit normal vector!!!!!vxuyxyvuU"="#=),(),(Therefore, is in the direction.!!U!Vis in the direction.!Velocity-Pressure Formulation! Velocity-Pressure Formulation!Computational Fluid Dynamics! !u!t+1J!Uu!"+!Vu!#$ % & ' ( ) =*1J+y#!p!"*y"!p!#$ % & ' ( ) +!J2""#q1u#$q2u%()+""%q3u%$q2u#()&'()*+u-Momentum Equation! !v!t+1J!Uv!"+!Vv!#$%&'()=*1J+x"!p!#*x#!p!"$%&'() +!J2""#q1v#$q2v%()+""%q3v%$q2v#()&'()*+v-Momentum Equation! U=uy!"vx!;V=vx#"uy#()Velocity-Pressure Formulation!Computational Fluid Dynamics!0=!!+!!yvxuContinuity Equation!Using!)(1),(1!""!"!!"xfxfJfyfyfJfyx#=#= 1J(u!y"#u"y!)+(v"x!#v!x")[]=0Continuity equation becomes!or! uy!"vx!()#+vx#"uy#()!=00=!!+!!"#VUVelocity-Pressure Formulation!Computational Fluid Dynamics!The momentum equations can be rearranged to!!!"#$$%&''+''(!!"#$$%&''+''+'')*)*))VvUvxVuUuytUJ =!q1"p"#!q2"p"$%&'()*++Jy$""#q1u#!q2u$()+""$q3u$!q2u#(),-./0123454!"#$%&+''(!"#$%&+''!!tvJxtuJy))U-Momentum Equation! !x"##$q1v$!q2v"()+##"q3v"!q2v$()% & ' ( ) * + , - Velocity-Pressure Formulation!Computational Fluid Dynamics! J!V!t+x"!Uu!"+!Vu!#$ % & ' ( ) *y"!Uv!"+!Vv!#$ % & ' ( ) =!q3"p"#!q2"p"$%&'()*++Jx$""$q1v$!q2v#()+""#q3v#!q2v$(),-./0123454V-Momentum Equation! !y"##"q1u"!q2u$()+##$q3u$!q2u"()% & ' ( ) * + , - where! u=1JUx!+Vx"() v=1JVy!+Uy"()Velocity-Pressure Formulation!Computational Fluid Dynamics!!"#In the plane, a staggered grid system can be used.!Same MAC grid and